-calculate) and how to interpret the results. The video provides the formula, a discussion of the concept, and the importance of the ratio.
Return on Equity
Return on equity measures the company's ability to use its invested capital to generate income. The invested capital comes from stockholders' investments in the company's stock and its retained earnings and is leveraged to create profit. The higher the return, the better the company is doing at using its investments to yield a profit. The formula for return on equity is
„ ^ . Net Income
Return on Equity =
Average Stockholder Equity =
Average Stockholder Equity Beginning Stockholder Equity + Ending Stockholder Equity
Average stockholders' equity is found by dividing the sum of beginning and ending stockholders' equity balances found on the balance sheet. The beginning stockholders' equity balance in the current year is taken from the ending stockholders' equity balance in the prior year. Keep in mind that the net income is calculated after preferred dividends have been paid.
For Clear Lake Sporting Goods, we will use the net income figure and deduct the preferred dividends that have been paid. The return on equity for the current year is
o nJ • 90,000 + 100,000 nrnnn Average Stockholder Equity =---= 95,000
^ . $35,000 - $5,000 nnn, , JN „„w
Return on Equity =- -= 0.32 (rounded) or 32%
3)95 ,UOO
The higher the figure, the better the company is using its investments to create a profit. Industry standards can dictate what an acceptable return is.
The DuPont Method
ROE in its basic form is useful; however, there are really three components of ROE: operating efficiency (profit margin), asset usage (total asset turnover), and leverage (equity ratio). This is known as the DuPont method. It originated in 1919 when the DuPont company implemented it for internal measurement purposes.2 The DuPont method can be expressed using this formula:
ROE = Profit Margin x Total Asset Turnover x Equity Multiplier
Profit margin indicates how much profit is generated by each dollar of sales and is computed as shown:
Net Income
Profit Margin —
Net Sales
Total asset turnover indicates the number of sales dollars produced by every dollar invested in capital assets—in other words, how efficiently the company is using its capital assets to generate sales. It is computed as shown:
2 Joshua Kennon. "What Is the DuPont Method Return on Equity, or ROE, Formula?" The Balance. December 16, 2020. https://thebalance.com/the-dupont-model-return-on-equity-formula-for-beginners-357494
184 6 • Measures of Financial Health
Equity Multiplier =
Average Total Assets
Average Stockholders' Equity
The equity multiplier measures leverage. It is computed as shown:
Average Total Assets
Equity Multiplier =
Average Stockholders' Equity
Using DuPont analysis, investors can see overall performance broken down into smaller pieces, which helps them better understand what is driving ROE. We already have the computations for Clear Lake Sporting Goods' profit margin and total asset turnover:
$35,000
Profit Margin = ,/ ' ^ = 0.29 (rounded) or 29% 5 $120,000 v '
$120 000
Total Asset Turnover = ' = 0.53 times (rounded) $225,000
We can calculate the equity multiplier using the equity multiplier equation and prior calculations for Clear Lake's average total assets and average stockholder equity:
Average Total Assets
Equity Multiplier = Average Total Assets =
Average Stockholders' Equity $200,000 + $250,000
= $225,000
o ^ ■ $90,000 + $100,000 Average Stockholder Equity =-■-----= $95,000
$225 000
Equity MuMpher=,^ = 2.37
Now that we have all three elements, we can complete the DuPont analysis for Clear Lake Sporting Goods: ROE = Profit Margin X Total Asset Turnover X Equity Multiplier ROE = 29% X 0.53 X 2.37 = 0.36 or 36.4%
The DuPont Method
This video about the DuPont method (https://openstax.Org/r/video-about-DuPont-method) walks through its history, discusses its basic components, and shows how to calculate and interpret each measurement.
Performance Analysis
ROE captures the nuances of all three elements. A good sales margin and a proper asset turnover are both needed for a successful operation. Like all ratios, assessing performance is relative. It's important to look at the ratio in context of the organization, its history, and the industry. If we compare Clear Lake's ROE of 26.4% to the recreational products industry average of 12.56% for the same year, it would appear as though Clear Lake Sporting Goods is outperforming the general industry. However, recreational products can include a wide variety of businesses beyond just the outdoor gear in which Clear Lake Sporting Goods specializes. An analyst could look at other key competitors such as Cabela's or Bass Pro Shops to get even more relevant comparisons.
Clear Lake Sporting Goods is also technically a retail store, albeit a specialized one. An analyst might also consider the industry averages for general or online retail of 20.64% and 27.05%, respectively. Compared to the broader retail industry. Clear Lake Sporting Goods is still performing well, but its performance is not as
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6.6 • Profitability Ratios and the DuPont Method 185
disparate to industry average as when compared to recreational products (see Table 6.1).
Industry ROE (%)
Advertising 2.93
Air Transportation -47.03
Computer Services 13.50
Banking 8.22
Financial Services (nonbanking) 64.28
Food Processing 10.12
Renewable Energy -20.59
Hospitals/Health Care Facilities 70.64
Hotels/Gaming -30.40
Publishers -14.18
Recreational Products 12.56
Real Estate (general) 2.00
Retail: 0.00
Automotive 36.28
Building Supply 0.27
General 20.64
Grocery 30.63
Online 27.05
Rubber and Tires -26.69
Shoes 23.70
Software (systems and applications) 28.09
Transportation 21.47
Total Market Average 8.25
Table 6.1 Return on Equity by Industry in 2020 It's important to look at any ratio in context of the organization, its history, and the industry, (data source: Aswath Damodaran Online)
186 6 • Summary
a Summary
6.1 Ratios: Condensing Information into Smaller Pieces
Ratios used in financial analysis can help investors and other analysts identify trends over time, compare companies to one another, and make informed decisions about a company. There are, however, limitations to analysis, so it should be used wisely and in conjunction with other contextual information available.
6.2 Operating Efficiency Ratios
Efficiency ratios measure how well management uses the assets of the organization to earn a profit. Common efficiency ratios include accounts receivable turnover, total asset turnover, inventory turnover, and days' sales in inventory.
6.3 Liquidity Ratios
Liquidity ratios help analysts measure how well an organization can meet its short-term obligations (liabilities) as they come due. Common ratios to measure liquidity include the current ratio, the quick ratio, and the cash ratio. Each of these three ratios includes more (cash ratio) or less liquid (current ratio) current assets in its measure of liquidity.
6.4 Solvency Ratios
Solvency ratios measure how well an organization can meet its long-term obligations (liabilities) as they come due, or in more general terms, its ability to stay in business. Common solvency ratios include the debt-to-assets ratio and the debt-to-equity ratio.
6.5 Market Value Ratios
Market value ratios help analysts assess the value of publicly traded firms in the market. The most commonly used market value ratios include earnings per share (EPS), the price/earnings ratio, and book value per share.
6.6 Profitability Ratios and the DuPont Method
Profitability ratios help measure how effectively the organization earns a profit. Common profitability ratios include profit margin, return on total assets, and return on equity. The DuPont method breaks down return on equity into three smaller components for a more thorough assessment of performance: profit margin, total asset turnover, and an equity multiplier.
t Key Terms
accounts receivable turnover ratio measures how many times in a period (usually a year) a company will
collect cash from accounts receivable book value per share total book value (assets - liabilities) of a firm expressed on a per-share basis cash ratio represents the firm's cash and cash equivalents divided by current liabilities; often used by
investors and lender to asses an organization's liquidity current ratio current assets divided by current liabilities; used to determine a company's liquidity (ability to
meet short-term obligations) days' sales in inventory the number of days it takes a company to turn inventory into sales debt-to-assets ratio measures the portion of debt used by a company relative to the amount of assets debt-to-equity ratio measures the portion of debt used by a company relative to the amount of
stockholders' equity
DuPont method framework for financial analysis that breaks return on equity down into smaller elements earnings per share (EPS) measures the portion of a corporation's profit allocated to each outstanding share of common stock
efficiency ratios ratios that show how well a company uses and manages its assets
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6- CFA Institute 187
inventory turnover measures the number of times an average quantity of inventory was bought and sold during the period
liquidity ability to convert assets into cash in order to meet primarily short-term cash needs or emergencies market value ratios measures used to assess a firm's overall market price
operating cycle amount of time it takes a company to use its cash to provide a product or service and collect
payment from the customer price/earnings (P/E) ratio company's stock price divided by the company's earnings per share; indicates the
amount investors are willing to pay for one dollar of earnings profit margin represents how much of sales revenue has translated into income
quick ratio also known as the acid test ratio; ratio used to determine a firm's ability to pay short-term debts
using its most liquid assets return on equity measures the company's ability to use its invested capital to generate income return on total assets measures the company's ability to use its assets successfully to generate a profit solvency implies that a company can meet its long-term obligations and will likely stay in business in the
future
times interest earned (TIE) ratio measures the company's ability to pay interest expense on long-term debt incurred
total asset turnover measures the ability of a company to use its assets to generate revenues
m CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session (https://openstax.Org/r/cfa-study-session11). Reference with permission of CFA Institute.
Multiple Choice
Which of the following is financial statement analysis not used for?
a. identifying trends over time
b. benchmarking against other firms
c. complying with SEC (Securities and Exchange Commission) regulations
d. setting budget expectations
2. How is converting financial data to percentages helpful in financial analysis?
a. It makes the figures easier to calculate.
b. It masks actual financial data so the competition can't see it.
c. It saves time.
d. It makes comparisons to companies of varying sizes possible.
3. What is a common economic influence that has the potential to skew financial analysis figures?
a. past performance
b. inflation
c. Generally Accepted Accounting Principles
d. benchmark data
4. What is the formula for accounts receivable turnover?
a. net credit sales / average accounts receivable
b. total sales / average accounts receivable
c. net credit sales / beginning accounts receivable
d. net cash sales / ending accounts receivable
5. What is the formula for the times interest earned ratio?
188 6 • Review Questions
a. net income / interest expense
b. earnings before interest and taxes / interest expense
c. interest expense / earnings before interest and taxes
d. earnings before interest and taxes - interest expense
6. What is the formula for the calculation of earnings per share?
a. (net income + preferred dividends) / weighted average common shares outstanding
b. net income / weighted average common shares outstanding
c. (net income - preferred dividends) / weighted average common shares outstanding
d. (net income - preferred dividends) / treasury shares outstanding
7. Most analysts believe which of the following is true about earnings per share?
a. Consistent improvement in earnings per share year after year is an indication of continuous improvement in the company's earning power.
b. Consistent improvement in earnings per share year after year is an indication of continuous decline in the company's earning power.
c. Consistent improvement in earnings per share year after year is an indication of fraud within the company.
d. Consistent improvement in earnings per share year after year is an indication that the company will never suffer a year of net loss rather than net income.
8. What is the formula for profit margin?
a. net sales / net income
b. cost of goods sold / net sales
c. sales / cost of goods sold
d. net income / net sales
5» Review Questions
1. Is past performance considered a good indicator of future performance?
2. The Pony Parts Tack Shop had inventory turnover of 12.8,12.2, and 9.9 over the last three years, respectively. What can you learn about how well the Pony Parts management team is managing their inventory using this information?
3. The Pony Parts Tack Shop had total asset turnover of 1.8., 2.1, and 2.4 over the last three years, respectively. What can you learn about how well the Pony Parts management team is managing their total assets using this information?
4. Big Box Store Inc. had days' sales in inventory of 30, 32, and 34 for the last three years. Small Box Store Inc. had days' sales in inventory of 40, 38, and 36 for the last three years. What can you infer about the inventory management for the two companies based on this information? Which company is performing better?
5. Jackson's Beef Jerky Shop has a current ratio of 2.35. What does that mean? Is 2.35 a good or bad current ratio?
6. Jackson's Beef Jerky Shop has a quick ratio of 1.9. What does that mean? Is 1.9 a good or bad quick ratio?
7. Jackson's Beef Jerky Shop has a cash ratio of 1.75. What does that mean? Is 1.75 a good or bad cash ratio?
8. You have some funds that you would like to invest. Do some internet research to find two publicly traded companies in the same industry and compare their earnings per share. Would the earnings per share
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6-Problems 189
reported by each company influence your decision in selecting which company to invest in?
9. Company A has a market value per share of 19.55 and book value per share of 12.79. If you were an investor, what would you conclude about the current value of Company A stock?
10. Company B has a price/earnings ratio of 21.2. The current industry average price/earnings ratio is 20.75. What might an investor conclude about investing in Company B?
11. What are the key elements of the DuPont formula, and how do these components function to help analysts assess an organization?
0 Problems
1. Sarah's Toy Shop has total sales of $100,000, net credit sales of $70,000, beginning accounts receivable of $20,000, and ending accounts receivable of $30,000. What is Sarah's accounts receivable turnover? Assume industry average is 2.9 times. How would you interpret Sarah's turnover?
2. Fantastic Foods has total assets of $150,000, current assets of $80,000 (current assets includes $30,000 of cash, $10,000 of short term investments, $20,000 of accounts receivable, and $20,000 of inventory), total liabilities of $120,000, and current liabilities of $70,000. What is Fantastic Foods' current ratio?
3. The Big Club has total assets of $150,000, current assets of $80,000 (current assets includes $30,000 of cash, $10,000 of short-term investments, $20,000 of accounts receivable, and $20,000 of inventory), total liabilities of $120,000, and current liabilities of $70,000. What is The Big Club's quick ratio?
4. Giant Sales has total assets of $150,000, current assets of $80,000 (current assets includes $30,000 of cash, $10,000 of short-term investments, $20,000 of accounts receivable, and $20,000 of inventory), total liabilities of $120,000, and current liabilities of $70,000. What is Giant Sales' cash ratio?
5. Bonita's Bread Company has total debt of $250,000 and total assets of $150,000. What is Bonita's debt-to-assets ratio, and what can we infer about Bonita's company using the ratio?
6. Jai Company has total liabilities of $200,000 and total stockholder equity of $300,000. What is the debt-to-equity ratio for Jai Company, and what can we infer about the firm using this ratio?
7. Jamilah's Manufacturing Company has earnings before interest and taxes of $29,000 and interest expense of $4,000 for the most current period. What is Jamilah's times interest earned ratio?
8. Sarai's Sandy Beach Gear has net sales of $100,000, cost of goods sold of $60,000, and net income of $25,000. What is Sarai's profit margin?
9. Bob's Tires Inc. has sales of $100,000, net income of $50,000, beginning asset balance of $200,000, ending asset balance of $220,000, beginning stockholder equity of $160,000, and ending stockholder equity of $200,000. What is Bob's return on total assets?
10. Bob's Tires Inc. has sales of $100,000, net income of $50,000, beginning asset balance of $200,000, ending asset balance of $220,000, beginning stockholder equity of $160,000, and ending stockholder equity of $200,000. What is Bob's return on equity?
[►I Video Activity
Ratio Analysis—Limitations of Ratios
Click to view content (https://openstax.Org/r/ratio-analysis)
1. In the video, Jim walks through several key limitations in the areas of reliability, comparability, relying on only one data source, and using information based in the past. Which limitation do you feel is the most worrisome? What might you do to compensate for the limitation you identified?
190 6 • Video Activity
2. Outside of the four key areas of limitationsjim also explores a number of key elements that ratios aren't able to convey. What characteristics of a firm would you most want to know about if you were to invest that you would not be able to glean from ratios? How would you go about gathering that information if you cannot get it through financial statements and ratio analysis?
The Problem with Earnings per Share (EPS)
Click to view content (https://openstax.Org/r/problem-with-earnings)
3. What does Zach DeGregorio cite as the most beneficial characteristics of using EPS in financial analysis? And the most problematic?
4. After watching the video and listing the key benefits and problems with EPS, how do you feel about the EPS calculation? Should investors use it? Why or why not? If you were going to invest a large sum of money in a company, would you use EPS data in your decision-making process? Why or why not?
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Figure 7.1 Time has an impact on the value of money, (credit: modification of "Time is money" by Marco Verch/flickr, CC BY 2.0)
Chapter Outline
7.1 Now versus Later Concepts
7.2 Time Value of Money (TVM) Basics
7.3 Methods for Solving Time Value of Money Problems
7.4 Applications of TVM in Finance
L±J Why It Matters
One of the single most important concepts in the study of finance is the time value of money (TVM). This concept puts forward the idea that a dollar received today is worth more than, and therefore preferable to, a dollar received at some point in the future. The three primary reasons for this are that (1) money received now can be saved or invested now and earn interest or a return, resulting in more money in the future; (2) any promise of future payments of cash will always carry the risk of default; and (3) it is simple human nature for people to prefer making their purchases of goods and services in the present rather than waiting to make them at some future time.
For this reason, it is important to incentivize people to give up their present consumption patterns by offering them greater value in the future. Based on the concept of TVM, it can be said that a dollar was worth more to us, and thus carried more value, yesterday than it is to us today. It also then follows that a dollar in our possession right now carries a greater value for us than a dollar we might receive tomorrow or at some other point in the future.
The entire concept of the time value of money is particularly important because it allows savers and investors to make better-informed decisions about what to do with their money. TVM can help a person understand which option may be best based on the critical factors of overall risk, rates of interest, inflation, and return. TVM can also be used to help a person understand how much money they'll need to save in an interest-bearing account in order to reach a desired financial goal, such as saving $50,000 in 10 years in an account that earns 4% compound interest each year. TVM is the key underlying principle of such important financial analytical activities as retirement planning, corporate capital project evaluation, and even deciding on your
192 7 • Time Value of Money I: Single Payment Value
own personal investments and bank accounts.
If the main concept behind the TVM is that a specific amount of money in hand now is worth more today than that same amount of money will be worth tomorrow, you might think that a person would be better off spending their money now rather than saving it for later use. However, we know that this is not always the case. Sometimes it is simply a better idea to save your money. While inflation can have the effect of making a dollar worth less tomorrow than it is worth today, the positive effect of compound interest works in favor of savers and investors.
Now versus Later Concepts
Learning Outcomes
By the end of this section, you will be able to:
■ Explain why time has an impact on the value of money.
■ Explain the concepts of future value and present value.
■ Explain why lump sum cash flow is the basis for all other cash flows.
How and Why the Passage of Time Affects the Value of Money
The concept of the time value of money (TVM) is predicated on the fact that it is possible to earn interest income on cash that you decide to deposit in an investment or interest-bearing account. As times goes by, interest is earned on amounts you have invested [present value), which effectively means that time will add value [future value) to your savings. The longer the period of time you have your money invested, the more interest income will accrue. Also, the higher the rate of interest your account or investment is earning, again, the more your money will grow.
Understanding how to calculate values of money in the present and at different points in the future is a key component of understanding the material presented in this chapter—and of making important personal financial decisions in your future (see Figure 7.2).
Money in Deposit Account Earning Interest
Year 1 Year 2 Year 3 Year 4
Figure 7.2 The Time Value of Money It is better to be paid today, or you will lose out on the money you would have earned in interest.
The Lump Sum Payment or Receipt
The most basic type of financial transaction involves a simple, one-time amount of cash, which can be either a receipt (inflow) or a payment (outflow). Such a one-time transaction is typically referred to as a lump sum. A lump sum consists of a one-off cash flow that occurs at any single point in time, present or future. Because it is always possible to dissect more complex transactions into smaller parts, the lump sum cash flow is the basis on which all other types of cash flow are treated. According to the general principle of adding value, every type of cash flow stream can be divided into a series of lump sums. For this reason, it is critical to understand the math associated with lump sums if you wish to have a greater appreciation, and a complete understanding, of more complicated forms of cash flow that may be associated with an investment or a capital purchase by a
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7.2 • Time Value of Money (TVM) Basics 193
company.
Time Value of Money (TVM) Basics
Learning Outcomes
By the end of this section, you will be able to:
■ Define future value and provide examples.
■ Explain how future dollar amounts are calculated using a single-period scenario.
■ Describe the impact of compounding.
Because we can invest our money in interest-bearing accounts and investments, its value can grow over time as interest income accrues or returns are realized on our investments. This concept is referred to as future value (FV). In short, future value refers to how a specific amount of money today can have greater value tomorrow.
Single-Period Scenario
Let us start with the following example. Your friend is considering putting money in a bank account that will pay 4% interest per year and is particularly interested in knowing how much money they will have one year from now if they deposit $1,000 in this account. Your friend understands that you are studying finance and turns to you for help. By using the TVM principle of future value (FV), you can tell your friend that the answer is $1,040. The additional $40 that will be in the account after one year will be due to interest earned over that time. You can calculate this amount relatively easily by taking the original deposit (also referred to as the principal) of $1,000 and multiplying it by the annual interest rate of 4% for one period (in this case, one year).
Interest Earned = $1,000 X 0.04 = $40.00
By taking the interest earned amount of $40 and adding it to the original principal of $1,000, you will arrive at a total value of $1,040 in the bank account at the end of the year. So, the $1,040 one year from today is equal to $1,000 today when working with a 4% earning rate. Therefore, based on the concept of TVM, we can say that $1,040 represents the future value of $1,000 one year from today and at a 4% rate of interest. We will discuss interest rates and their importance in TVM decisions in more detail later in this chapter; for now, we can consider interest rate as a percentage of the principal amount that is earned by the original lender of funds and/or charged to the borrower of these same funds. Following are a few more examples of the single-period scenario.
If a person deposits $300 in an account that pays 5% per year, at the end of one year, they will have
FV =$300 + ($300 X 0.05) = $315
If a company has earnings of $2.50 per share and experiences a 10% increase in the following year, the earnings per share in year two are
$2.50 + $2.50 X 0.10 = $2.75 per share
If a retail store decides on a 3% price increase for the following year on an item that is currently selling for $50, the new price in the following year will be
$50 + $50 X 0.03 = $51.50
The Impact of Compounding
What would happen if your friend were willing to wait one more year to receive their lump sum payment? What would the future dollar value in their account be after a two-year period? Returning to our earlier example, assume that during the second year, your friend leaves the principal ($1,000) and the earned interest ($40) in the account, thereby reinvesting the entire account balance for another year. The quoted interest rate of 4% reflects the interest the account would earn each year, not over the entire two-year savings period. So, during the second year of savings, the $1,000 deposit and the $40 interest earned during the first year would
194 7 • Time Value of Money I: Single Payment Value
both earn 4%:
$1,000 X 0.04 + 40 X 0.04 = $41.60
The additional $1.60 is interest on the first year's interest and reflects the compounding of interest. Compound interest is the term we use to refer to interest income earned in subsequent periods that is based on interest income earned in prior periods. To put it simply, compound interest refers to interest that is earned on interest. Here, it refers to the $1.60 of interest earned in the second year on the $40.00 of interest earned in the first year. Therefore, at the end of two years, the account would have a total value of $1,081.60. This consists of the original principal of $1,000 plus the $40.00 interest income earned in year one and the $41.60 interest income earned in year two.
The amount of money your friend would have in the account at the end of two years, $1,081.60, is referred to as the future value of the original $1,000 amount deposited today in an account that will earn 4% interest every year.
Simple interest applies to year 1 while compound interest or "interest on interest" applies to year 2. This is calculated using the following method:
Year 1: 1,000 X 0.04 = 40.00 Year 2: 1,040 X 0.04 = 41.60
So, the total amount that would be in the account after two years, at 4% annual interest, would be $1,000 + $40.00 + $41.60 = $1,081.60.
To determine any future value of money in an interest-bearing account, we multiply the principal amount by 1 plus the interest rate for each year the money remains in the account. From this, we can develop the future value formula:
Future Value = Original Deposit X (1 + r) X (1 + r)
In this formula, the number of times we multiply by (1 + r) depends entirely on the number of years the money will remain in the bank account, earning interest, before it is withdrawn in a final lump sum distribution paid out from the account at the end of the chosen savings period. The 1 in the formula represents the principal amount, or the original $1,000 deposit, which will be included in the final total lump sum payment when the account is closed and all money is withdrawn at the end of the predetermined savings period.
We can write the above equation in a more condensed mathematical form using time value of money notation, as follows:
FV = Future Value PV = Present Value
r = Interest Rate
n — Number of Periods
Using these inputs, we have the following formula:
FV = PV X (1 + rf
With this equation, we can calculate the value of the savings account after any number of years. For example, suppose we are considering 3,10, and 50 years from the original deposit date at the annual 4% interest rate:
3 years: FV = $1,000 X (1.04)3 = $1,000 X 1.12486 = $1,124.86 10 years: FV = $1,000 X (1.04)10 = $1,000 X 1.48024 = $1,480.24 50 years: FV = $1,000 X (1.04)50 = $1,000 X 7.106683 = $7,106.68
How can this savings account have grown to be so large after 50 years? This question is answered by the impact of compounding interest. Every year, the interest earned in previous years will also earn interest along
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7.2 • Time Value of Money (TVM) Basics 195
with the initial deposit. This will have the effect of accelerating the growth of the total dollar value of the account.
This is the important effect of the compounding of interest: money grows in larger and larger increments the longer you leave it in an interest-bearing account. In effect, the compounding of interest over time accelerates the growth of money.
In order to determine the FV of any amount of money, it will always be necessary to know the following pieces of information: (1) the principal, initial deposit, or present value (PV); (2) the rate of interest, usually expressed on an annual basis as r, and (3) the number of time periods that the money will remain in the account (n). The interest rate is often referred to as the growth rate, or the annual percentage increase on savings or on an investment. When the rate is raised to the power of the number of periods, the formula (1 + r)n will yield a number that is commonly referred to as the future value interest factor (FVIF). As a result of this process, as n (time, or the number of periods) increases, the future value interest factor will increase. Also, as r (interest rate) increases, the FVIF will increases. For these reasons, the future value calculation is directly determined by both the interest rate being used and the total amount of time—specifically, the number of periods—being considered.
Calculating Future Values
Here's another example of calculating future values in multiple-period scenarios.
On a recent drive, you spotted your dream home, which is currently listed at $400,000. Unfortunately, you are not in a position to buy it right away and will have to wait at least another six years before you can afford it. If house values are appreciating at an annual rate of inflation of 4%, how much will a similar house cost after six years?
Solution:
In this case, PV is the current cost of the house, or $400,000; n is six years; r is the average annual inflation rate, or 4%; and we have to solve for FV.
FV = $400,000 X 1.046
= $400,000 X (1.265319) = $506,127.61
Therefore, under these circumstances, the house will cost $506,127.60 after six years. How Time Impacts Compounding
We have just seen that time will lead to the growth of our money. As long as the prevailing growth or interest rate of any account we have our money in is positive, the passage of time will have the effect of growing the value of our money. The longer the period of time, the greater the growth and the larger the future value of the money will be. This can be reinforced very clearly with the following example.
Melvin is saving money in an account at a local bank that earns 5% per year. He begins with a deposit in his account of $100 and decides to save his money for exactly one year. He will not be making any further deposits into the account during the year. Melvin will earn 5% X $100, or $5, in interest income. Adding this to the original deposit balance of $100 will give him a total of $100 + $5, or $105, in the account at the end of one year.
Melvin likes this idea and believes he may be able to keep his money in the account for a longer period of time. How much money will he have in his account, without any further deposits, at the end of years two, three, four, and five?
196 7 • Time Value of Money I: Single Payment Value
Using the future value formula, the calculation is as follows:
FV = PV X (1 + r)n Year 2: FV = $100 X (1 +0.05)2 = $110.25 Year 3: FV = $100 X (1 +0.05)3 = $115.76 Year 4: FV = $100 X (1 +0.05)4 = $121.55 Year 5: FV = $100 X (1 +0.05)5 = $127.63
How the Interest Rate Impacts Compounding
Melvin likes the idea of earning more money over time, but he also believes that what he would earn in interest may not be enough for some of the things he plans to buy in the future. His friend suggests finding an account or some form of investment with a greater interest rate than the 5% he can get at his local bank.
Melvin thinks he can leave his money in an account or investment for a total of five years. He found investments that will provide annual returns of 6%, 7%, 10%, and 12%. Using the FV = PV X (1 + r)n formula, we can complete the following calculations for him:
5%: FV = $100 X (1 + 0.05)5 = $127.63
6%: FV = $100 X (1 + 0.06)5 = $133.82
7%: FV = $100 X (1 + 0.07)5 = $140.26
10%: FV = $100 X (1+0.10)5 = $161.05
12%: FV = $100 X (1+0.12)5 = $176.23
Again, Melvin likes this information, and he states that he will try to find the highest interest rate available. This makes sense, but it's important to remember that investments are usually not guaranteed to earn you specific interest rates, or rates of return. Most investments, other than Treasury investments such as Treasury bonds, carry some form of financial risk, either small or large, and the greater the rate of return, the more likely it is that the risk associated with the investment will also be greater. This risk does not have any effect on the future calculations we have just completed, but it an important factor to bear in mind and consider well before moving ahead and putting your money in any investment or financial instrument.
Methods for Solving Time Value of Money Problems
Learning Outcomes
By the end of this section, you will be able to:
■ Explain how future dollar amounts are calculated.
■ Explain how present dollar amounts are calculated.
■ Describe how discount rates are calculated.
■ Describe how growth rates are calculated.
■ Illustrate how periods of time for specified growth are calculated.
■ Use a financial calculator and Excel to solve TVM problems.
We can determine future value by using any of four methods: (1) mathematical equations, (2) calculators with financial functions, (3) spreadsheets, and (4) FVIF tables. With the advent and wide acceptance and use of financial calculators and spreadsheet software, FVIF (and other such time value of money tables and factors) have become obsolete, and we will not discuss them in this text. Nevertheless, they are often still published in other finance textbooks and are also available on the internet to use if you so choose.
Using Timelines to Organize TVM Information
A useful tool for conceptualizing present value and future value problems is a timeline. A timeline is a visual.
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7.3 • Methods for Solving Time Value of Money Problems 197
linear representation of periods and cash flows over a set amount of time. Each timeline shows today at the left and a desired ending, or future point (maturity date), at the right.
Now, let us take an example of a future value problem that has a time frame of five years. Before we begin to solve for any answers, it would be a good approach to lay out a timeline like that shown in Table 7.1:
Year 0 (Today) 1 2 3 4 5
Table 7.1
The timeline provides a visual reference for us and puts the problem into perspective.
Now, let's say that we are interested in knowing what today's balance of $100 in our saving account, earning 5% annually, will be worth at the end of each of the next five years. Using the future value formula
FV = PV X (1 + rf
that we covered earlier, we would arrive at the following values: $105 at the end of year one, $110.25 at the end of year two, $115.76 at the end of year three, $121.55 at the end of year four, and $127.63 at the end of year five.
With the numerical information, the timeline (at a 5% interest or growth rate) would look like Table 7.2:
Year 0 1 2 3 4 5
$100.00 $105.00 $110.25 $115.76 $121.55 $127.63
Table 7.2
Using timelines to lay out TVM problems becomes more and more valuable as problems become more complex. You should get into the habit of using a timeline to set up these problems prior to using the equation, a calculator, or a spreadsheet to help minimize input errors. Now we will move on to the different methods available that will help you solve specific TVM problems. These are the financial calculator and the Excel spreadsheet.
Using a Financial Calculator to Solve TVM Problems
An extremely popular method of solving TVM problems is through the use of a financial calculator. Financial calculators such as the Texas Instruments BAH Plus™ Professional (https://openstax.org/r/baii-plus-professional) will typically have five keys that represent the critical variables used in most common TVM problems: n, i/y, pv, fv, and pmt. These represent the following:
N: Number of Periods I/Y: Interest Rate (Interest per Year) PV: Present Value of a Lump Sum V: Future Value of a Lump Sum PMT: Payment
These are the only keys on a financial calculator that are necessary to solve TVM problems involving a single payment or lump sum.
Example 1: Future Value of a Single Payment or Lump Sum
Let's start with a simple example that will provide you with most of the skills needed to perform TVM functions involving a single lump sum payment with a financial calculator.
198 7 • Time Value of Money I: Single Payment Value
Suppose that you have $1,000 and that you deposit this in a savings account earning 3% annually for a period of four years. You will naturally be interested in knowing how much money you will have in your account at the end of this four-year time period (assuming you make no other deposits and withdraw no cash).
To answer this question, you will need to work with factors of $1,000, the present value (pv); four periods or years, represented by n; and the 3% interest rate, or i/y. Make sure that the calculator register information is cleared, or you may end up with numbers from previous uses that will interfere with the solution. The register-clearing process will depend on what type of calculator you are using, but for the TI BAII Plus™ Professional calculator, clearing can be accomplished by pressing the keys 2ND and fv [clr tvm].
Once you have cleared any old data, you can enter the values in the appropriate key areas: 4 for n, 3 for i/y, and 1000 for pv. Now you have entered enough information to calculate the future value. Continue by pressing the cpt (compute) key, followed by the fv key. The answer you end up with should be displayed as 1,125.51 (see Table 7.3).
Step Description Enter Display
1 Clear calculator register ce/c 0.00
2 Enter present value (as a negative integer) 1000 + 1- pv PV = -1,000.00
3 Enter interest rate 3 i/y I/Y = 3.00
4 Enter time periods 4N N = 4.00
5 Indicate no payments or deposits 0 pmt PMT 0.00
6 Compute future value cpt fv FV = 1,125.51
Table 7.3 Calculator Steps for Finding the Future Value of a Single Payment or Lump Sum1
Important Notes for Using a Calculator and the Cash Flow Sign Convention
Please note that the PV was entered as negative $1,000 (or -$1000). This is because most financial calculators (and spreadsheets) follow something called the cash flow sign convention, which is a way for calculators and spreadsheets to keep the relative direction of the cash flow straight. Positive numbers are used to represent cash inflows, and negative numbers should always be used for cash outflows.
In this example, the $1,000 is an investment that requires a cash outflow. For this reason, -1000 is entered as the present value, as you will be essentially handing this $1,000 to a bank or to someone else to initiate the transaction. Conversely, the future value represents a cash inflow in four years' time. This is why the calculator generates a positive 1,125.51 as the end result of this calculation.
Had you entered the present value of $1,000 as a positive number, there would have been no real concern, but the ending future value answer would have been returned expressed as a negative number. This would be correct had you borrowed $1,000 today (cash inflow) and agreed to repay $1,125.51 (cash outflow) four years from now. Also, it is important that you do not change the sign of any input value by using the - (minus) key). For example, on the TI BA II Plus™ Professional, you must use the +1 - key instead of the minus key. If you enter 1000 and then hit the +1 - key, you will get a negative 1,000 amount showing in the calculator display.
An important feature of most financial calculators is that it is possible to change any of the variables in a problem without needing to reenter all of the other data. For example, suppose that we wanted to find out the future value in our bank account if we left the money from our previous example invested for 20 years instead
1 The specific financial calculator in these examples is the Texas Instruments BA II Plus™ Professional model, but you can use other financial calculators for these types of calculations.
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7.3 • Methods for Solving Time Value of Money Problems 199
of 4. Before clearing any of the data, simply enter 20 for n and then press the cpt key and then the fv key. After this is done, all other inputs will remain the same, and you will arrive at an answer of $1,806.11.
How to Determine Future Value When Other Variables Are Known
Here's an example of using a financial calculator to solve a common time value of money problem. You have $2,000 invested in a money market account that is expected to earn 4% annually. What will be the total value in the account after five years?
Solution:
Follow the recommended financial calculator steps in Table 7.4.
Step Description Enter Display
1 Clear calculator register ce/c 0.00
2 Enter present value (as a negative integer) 2000 + |-pv PV = -2,000.00
3 Enter interest rate 4 i/y I/Y = 4.00
4 Enter time periods 5N N = 5.00
5 Indicate no payments or deposits 0 pmt PMT = 0.00
6 Compute future value cpt fv FV = 2,433.31
Table 7.4 Calculator Steps for Determining Future Value
The result of this future value calculation of the invested money is $2,433.31.
Example 2: Present Value of Lump Sums
Solving for the present value (discounted value) of a lump sum is the exact opposite of solving for a future value. Once again, if we enter a negative value for the FV, then the calculated PV will be a positive amount.
Taking the reverse of what we did in our example of future value above, we can enter -1,125.51 for fv, 3 for i/y, and 4 for n. Hit the cpt and pv keys in succession, and you should arrive at a displayed answer of 1,000
An important constant within the time value of money framework is that the present value will always be less than the future value unless the interest rate is negative. It is important to keep this in mind because it can help you spot incorrect answers that may arise from errors with your input.
How to Determine Present Value When Other Variables Are Known
Here is another example of using a financial calculator to solve a common time value of money problem. You have just won a second-prize lottery jackpot that will pay a single total lump sum of $50,000 five years from now. How much value would this have in today's dollars, assuming a 5% interest rate?
Solution:
Follow the recommended financial calculator steps in Table 7.5.
200 7 • Time Value of Money I: Single Payment Value
Step Description Enter Display
1 Clear calculator register ce/c 0.00
2 Enter future value (as a negative integer) 50000 + I - fv FV = -50,000.00
3 Enter interest rate 5 I/Y I/Y = 5.00
4 Enter time periods 5N N = 5.00
5 Indicate no payments or deposits 0 pmt PMT = 0.00
6 Compute present value cpt pv PV = 39,176.31
Table 7.5 Calculator Steps for Determining Present Value
The present value of the lottery jackpot is $39,176.31.
Example 3: Calculating the Number of Periods
There will be times when you will know both the value of the money you have now and how much money you will need to have at some unknown point in the future. If you also know the interest rate your money will be earning for the foreseeable future, then you can solve for N, or the exact amount of time periods that it will take for the present value of your money to grow into the future value that you will require for your eventual use.
Now, suppose that you have $100 today and you would like to know how long it will take for you to be able to purchase a product that costs $133.82.
After making sure your calculator is clear, you will enter 5 for i/y, -100 for pv, and 133.82 for fv. Now press cpt n, and you will see that it will take 5.97 years for your money to grow to the desired amount of $133.82.
Again, an important thing to note when using a financial calculator to solve TVM problems is that you must enter your numbers according to the cash flow sign convention discussed above. If you do not make either the pv or the fv a negative number (with the other being a positive number), then you will end up getting an error message on the screen instead of the answer to the problem. The reason for this is that if both numbers you enter for the pv and fv are positive, the calculator will operate under the assumption that you are receiving a financial benefit without making any cash outlay as an initial investment. If you get such an error message in your calculations, you can simply press the ce/c key. This will clear the error, and you can reenter your data correctly by changing the sign of either pv or fv (but not both of these, of course).
THINK IT THROUGH
Determining Periods of Time
Here is an additional example of using a financial calculator to solve a common time value of money problem. You want to be able to contribute $25,000 to your child's first year of college tuition and related expenses. You currently have $15,000 in a tuition savings account that is earning 6% interest every year. How long will it take for this account grow into the targeted amount of $25,000, assuming no additional deposits or withdrawals will be made?
Solution:
Table 7.6 shows the steps you will take.
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7.3 • Methods for Solving Time Value of Money Problems 201
Step Description Enter Display
1 Clear calculator register ce/c 0.0000
2 Enter present value (as a negative integer) 15000 + |-pv PV = -15,000.0000
3 Enter interest rate 6 i/y I/Y = 6.0000
4 Enter future value 25000 fv FV = 25,000.0000
5 Indicate no payments or deposits 0 pmt PMT = 0.0000
6 Compute time periods cpt n N = 8.7667
Table 7.6 Calculator Steps for Determining Period of Time
The result of this calculation is a time period of 8.7667 years for the account to reach the targeted amount. Example 4: Solving for the Interest Rate
Solving for an interest rate is a common TVM problem that can be easily addressed with a financial calculator. Let's return to our earlier example, but in this case, we know that we have $1,000 at the present time and that we will need to have a total of $1,125.51 four years from now. Let's also say that the only way we can add to the current value of our savings is through interest income. We will not be able to make any further deposits in addition to our initial $1,000 account balance.
What interest rate should we be sure to get on our savings account in order to have a total savings account value of $1,125.51 four years from now?
Once again, clear the calculator, and then enter 4 for n, -1,000 for pv, and 1,125.51 for fv. Then, press the cpt and i/y keys and you will find that you need to earn an average 3% interest per year in order to grow your savings balance to the desired amount of $1,125.51. Again, if you end up with an error message, you probably failed to follow the sign convention relating to cash inflow and outflow that we discussed earlier. To correct this, you will need to clear the calculator and reenter the information correctly.
After you believe you are done and have arrived at a final answer, always make sure you give it a quick review. You can ask yourself questions such as "Does this make any sense?" "How does this compare to other answers I have arrived at?" or "Is this logical based on everything I know about the scenario?" Knowing how to go about such a review will require you to understand the concepts you are attempting to apply and what you are trying to make the calculator do. Further, it is critical to understand the relationships among the different inputs and variables of the problem. If you do not fully understand these relationships, you may end up with an incorrect answer. In the end, it is important to realize that any calculator is simply a tool. It will only do what you direct it to do and has no idea what your objective is or what it is that you really wish to accomplish.
Determining Interest or Growth Rate
Here is another example of using a financial calculator to solve a common time value of money problem. Let's use a similar example to the one we used when calculating periods of time to determine an interest or growth rate. You still want to help your child with their first year of college tuition and related expenses. You also still have a starting amount of $15,000, but you have not yet decided on a savings plan to use.
202 7 • Time Value of Money I: Single Payment Value
Instead, the information you now have is that your child is just under 10 years old and will begin college at age 18. For simplicity's sake, let's say that you have eight and a half years before you will need to meet your total savings target of $25,000. What rate of interest will you need to grow your saved money from $15,000 to $25,000 in this time period, again with no other deposits or withdrawals?
Solution:
Follow the steps shown in Table 7.7.
Step Description Enter Display
1 Clear calculator register ce/c 0.0000
2 Enter present value (as a negative integer) 15000 + |-pv PV = -15,000.0000
3 Enter time periods 8.5 n N = 8.5000
4 Enter future value 25000 fv FV = 25,000.0000
5 Indicate no payments or deposits 0 pmt PMT = 0.0000
6 Compute interest rate cpt i/y I/Y = 6.1940
Table 7.7 Calculator Steps for Determining Interest Rate
The result of this calculation is a necessary interest rate of 6.194%. Using Excel to Solve TVM Problems
Excel spreadsheets can be excellent tools to use when solving time value of money problems. There are dozens of financial functions available in Excel, but a student who can use a few of these functions can solve almost any TVM problem. Special functions that relate to TVM calculations are as follows:
Future Value (FV)
Present Value (PV)
Number of Periods (NPER)
Interest Rate (RATE)
Excel also includes a function called Payment (PMT) that is used in calculations involving multiple payments or deposits (annuities). These will be covered in Time Value of Money II: Equal Multiple Payments.
Future Value (FV)
The Future Value function in Excel is also referred to as FV and can be used to calculate the value of a single lump sum amount carried to any point in the future. The FV function syntax is similar to that of the other four basic time-value functions and has the following inputs (referred to as arguments), similar to the functions listed above:
Rate:Interest Rate Nper: Number of Periods Pmt: Payment PV: Present Value
Lump sum problems do not involve payments, so the value of Pmt in such calculations is 0. Another argument.
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7.3 • Methods for Solving Time Value of Money Problems 203
Type, refers to the timing of a payment and carries a default value of the end of the period, which is the most common timing (as opposed to the beginning of a period). This may be ignored in our current example, which means the default value of the end of the period will be used.
The spreadsheet in Figure 7.3 shows two examples of using the FV function in Excel to calculate the future value of $100 in five years at 5% interest.
In cell E1, the FV function references the values in cells B1 through B4 for each of the arguments. When a user begins to type a function into a spreadsheet. Excel provides helpful information in the form of on-screen tips showing the argument inputs that are required to complete the function. In our spreadsheet example, as the FV formula is being typed into cell E2, a banner showing the arguments necessary to complete the function appears directly below, hovering over cell E3.
A A B , C D E F G
1 rate 5% =FV(BlrB2rB3rB4)
z nper 5 =fv(
3 pmt 0 FV[rate, nper, pmt, [pv], [type])
4 pv -100 $127.63
5
6 =FV(.05,5,O,-100)
7
8 $127.63
Figure 7.3 Using the FV Function in Excel
Cells E1 and E2 show how the FV function appears in the spreadsheet as it is typed in with the required arguments. Cell E4 shows the calculated answer for cell E1 after hitting the enter key. Once the enter key is pressed, the hint banner hovering over cell E3 will disappear. The second example of the FV function in our example spreadsheet is in cell E6. Here, the actual numerical values are used in the FV function equation rather than cell references. The method in cell E8 is referred to as hard coding. In general, it is preferable to use the cell reference method, as this allows for copying formulas and provides the user with increased flexibility in accounting for changes to input data. This ability to accept cell references in formulas is one of the greatest strengths of Excel as a spreadsheet tool.
Download the spreadsheet file (https://openstax.Org/r/docs.google uc export) containing key Chapter 7 Excel exhibits.
Determining Future Value When Other Variables Are Known. You have $2,000 invested in a money market account that is expected to earn 4% annually. What will be the total value in the account in five years?
Present Value (PV) = ($2,000.00)
Note: Be sure to follow the sign conventions. In this case, the PV should be entered as a negative value.
Interest Rate (Rate or I/Y) = 4%
Note: In Excel, interest and growth rates must be entered as percentages, not as whole integers. So, 4 percent must be entered as 4% or 0.04—not 4, as you would enter in a financial calculator.
Number of Periods (Nper or N) = 5.00
Note: It is always assumed that if not specifically stated, the compounding period of any given interest rate is annual, or based on years.
Future Value (FV) = $2,433.31 Note: The Excel command used to calculate future value is as follows:
204 7 • Time Value of Money I: Single Payment Value
=FV(rate, nper, pmt, [pv], [type];
You may simply type the values for the arguments in the above formula. Another option is to use the Excel insert function option. If you decide on this second method, below are several screenshots of dialog boxes you will encounter and will be required to complete.
1. First, go to Formulas in the upper menu bar, and select the Insert Function option. When you do so, a dialog box will appear that looks like what you see in Figure 7.4.
Insert Function Search for a function:
X
Type a brief description of what you want to do and then click Go
Or select a category: Most Recently Used v
Go
Select a function:
RATE
ACCRINT
NPER
PV
SUM
AVERAGE
■ ™
FWate, n p er, p mt p v,ty pe)
Returns the future value of an investment based on periodic, constant payments and a constant interest rate.
Help on this function
OK
Cancel
Figure 7.4 Dialog Box to Insert FV Function
This dialog box allows you to either search for a function or select a function that has been used recently. In this example, you can search for FV by typing this in the search box and selecting Go, or you can simply choose FV from the list of most recently used functions (as shown here with the highlighted FV option).
Once you select FV and click the OK button, a new dialog box will appear for you to enter the necessary details. See Figure 7.5.
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7.3 • Methods for Solving Time Value of Money Problems 205
Function Arguments FV
X
Rate Nper Pmt
FV Type
number number number number number
Returns the future value of an investment based on periodic, constant payments and a constant interest rate.
Rate is the interest rate per period. For example, use 6%/4for quarterly payments at 6% APR.
Formula result = Help on this function
OK
Cancel
Figure 7.5 New Dialog Box for FV Function Arguments
Figure 7.6 shows the completed data input for the variables, referred to here as "function arguments." Note that cell addresses are used in this example. This allows the spreadsheet to still be useful if you decide to change any of the variables. You may also type values directly into the Function Arguments dialog box, but if you do this and you have to change any of your inputs later, you will have to reenter the new information. Using cell addresses is always a preferable method of entering the function argument data.
FV
=FV(blJb2„b4]
Á A i B i C 1
rate 4%
2 nper 5|
3 pmt 0
4 pv 2000
5
!
D
□
□
s
a
Function Argument: FV
Rate Nper Pmt
Type
b-|
0,04 5
number
2000
number
- -2433.305B05
Returns the future value of an investment based on periodic, constant payments and a constant interest rate,
Pv is the present value, orthe lump-sum amount that a series of future payments is worth now. If omitted, Pv = 0,
Formula result = -2433.305B05 Help on this function
Figure 7.6 Completed Data Entry Menu for FV Function Arguments
Additional notes:
1. The Pmt argument or variable can be ignored in this instance, or you can enter a placeholder value of
206 7 • Time Value of Money I: Single Payment Value
zero. This example shows a blank or ignored entry, but either option may be used in problems such as this where the information is not relevant.
2. The Type argument does not apply to this problem. Type refers to the timing of cash flows and is usually used in multiple payment or annuity problems to indicate whether payments or deposits are made at the beginning of periods or at the end. In single lump sum problems, this is not relevant information, and the Type argument box is left empty.
3. When you use cell addresses as function argument inputs, the numerical values within the cells are displayed off to the right. This helps you ensure that you are identifying the correct cells in your function. The final answer generated by the function is also displayed for your preliminary review.
Once you are satisfied with the result, hit the OK button, and the dialog box will disappear, with only the final numerical result appearing in the cell where you have set up the function.
The FV of this present value has been calculated as approximately $2,433.31.
Present Value (PV)
We have covered the idea that present value is the opposite of future value. As an example, in the spreadsheet shown in Figure 7.3. we calculated that the future value of $100 five years from now at a 5% interest rate would be $127.63. By reversing this process, we can safely state that $127.63 received five years from now with a 5% interest (or discount) rate would have a value of just $100 today. Thus, $100 is its present value. In Excel, the PV function is used to determine present value (see Figure 7.7).
A A B I C I D E F G
1 2 rate nper 5% =PV(BlrB2,B3,B4)
5 =pv(
3 pmt 0 PV[rate, nper, prnt, [fv], [type]]
4 fv -127.6 $100.00
5
6 =PV( .05,5,0,-100)
7
8 Si oo. oo
Figure 7.7 Using the PV Function in Excel
The formula in cell E1 uses cell references in a similar fashion to our FV example spreadsheet above. Also similar to our earlier example is the hard-coded formula for this calculation, which is shown in cell E6. In both cases, the answers we arrive at using the PV function are identical, but once again, using cell references is preferred over hard coding if possible.
Determining Present Value When Other Variables Are Known
You have just won a second-prize lottery jackpot that will pay a single total lump sum of $50,000 five years from now. You are interested in knowing how much value this would have in today's dollars, assuming a 5% interest rate.
Future Value (FV) = ($50,000.00) Interest Rate (Rate or IVY) = 5% Number of Periods (Nper or N) = 5.00 Present Value (PV) = $39,176.31
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7.3 • Methods for Solving Time Value of Money Problems 207
Notes: 1
If you wish for the present value amount to be positive, the future value you enter here should be a negative value.
In Excel, interest and growth rates must be entered as percentages, not as whole integers. So, 5 percent must be entered as 5% or 0.05—not 5, as you would enter in a financial calculator. It is always assumed that if not specifically stated, the compounding period of any given interest rate is annual, or based on years.
The Excel command used to calculate present value is as shown here:
=PV(rate, nper, pmt, [fv], [type];
Solution:
As with the FV formula covered in the first tab of this workbook, you may simply type the values for the arguments in the above formula. Another option is to again use the Insert Function option in Excel. Figure 7.8. Figure 7.9. and Figure 7.10 provide several screenshots that demonstrate the steps you'll need to follow if you decide to enter the PV function from the Insert Function menu.
1. First, go to Formulas in the upper menu bar, and select Insert Function. When you do so, the Insert Function dialog box will appear (see Figure 7.8).
Insert Function Search for a function:
X
Type a brief description of what you want to do and then click Go
Or select a category: Most Recently Used
Go
Select a function:
SUM
AVERAGE IF
HYPERLINK
COUNT
MAX
PVfrate, n per, p rut, tv, type)
Returns the present value of an investment: the total amount that a series of future payments is worth now.
Help on this function
OK
Cancel
Figure 7.8 Dialog Box to Insert PV Function
As discussed in the FV function example above, this dialog box allows you to either search for a function or select a function that has been used recently. In this example, you can search for PV by typing this into the search box and selecting Go, or you can simply choose PV from the list of the most recently used functions.
Once you have highlighted PV, click the OK button, and a new dialog box will appear for you to enter the necessary details. Similar to our FV function example, it will look like Figure 7.9.
208 7 • Time Value of Money I: Single Payment Value
Function Arguments PV
X
Rate Nper Pmt
Fv Type
number number number number number
Returns the present value of an investment: the total amount that a series of future payments is worth now.
Rate is the interest rate per period. For example, use 6%/4 for quarterly payments at 6% APR.
Formula result = Help on this function
OK
Cancel
Figure 7.9 New Dialog Box for PV Function Arguments
Figure 7.10 shows the completed data input for the function arguments. Note that once again, cell addresses are used in this example. This allows the spreadsheet to still be useful if you decide to change any of the variables. As in the FV function example, you may also type values directly in the Function Arguments dialog box, but if you do this and you have to change any of your input later, you will have to reenter the new information. Remember that using cell addresses is always a preferable method of entering the function argument data.
Function Arguments. PV
X
Rate Nper Pmt
Fv Type
b1 ± = 0.05
b2 ± = 5
b3 ± = 0
m| ± = -50000
± = number
= 39176.30332
Returns the present value of an investment: the total amount that a series of future payments is worth now,
Fv is the future value, or a cash balance you want to attain afterthe last payment is made.
Formula result = 39176.30S32 Help on this function
OK
Cancel
Figure 7.10 Completed Dialog Box for PV Function Arguments
Again, similar to our FV function example, the Function Arguments dialog box shows values off to the
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7.3 • Methods for Solving Time Value of Money Problems 209
right of the data entry area, including our final answer. The Pmt and Type boxes are again not relevant to this single lump sum example, for reasons we covered in the FV example.
Review your answer. Once you are satisfied with the result, click the OK button, and the dialog box will disappear, with only the final numerical result appearing in the cell where you have set up the function. The PV of this future value has been calculated as approximately $39,176.31.
Periods of Time
The following discussion will show you how to use Excel to determine the amount of time a given present value will need to grow into a specified future value when the interest or growth rate is known.
You want to be able to contribute $25,000 to your child's first year of college tuition and related expenses. You currently have $15,000 in a tuition savings account that is earning 6% interest every year. How long will it take for this account grow into the targeted amount of $25,000, assuming no additional deposits or withdrawals are made?
Future Value (FV) = $25,000.00 Interest Rate (Rate or I/Y) = 6% Present Value (PV) = ($15,000.00) Number of Periods (NPER) = 8.7667
Notes:
1. As with our other examples, interest and growth rates must be entered as percentages, not as whole integers. So, 6 percent must be entered as 6% or 0.06—not 6, as you would enter in a financial calculator.
2. The present value needs to be entered as a negative value in accordance with the sign convention covered earlier.
3. The Excel command used to calculate the amount of time, or number of periods, is this:
=NPER (rate, pmt, pv, [fv], [type])
As with our FV and PV examples, you may simply type the values of the arguments in the above formula, or we can again use the Insert Function option in Excel. If you do so, you will need to work with the various dialog boxes after you select Insert Function.
1. First, go to Formulas in the upper menu bar, and select the Insert Function option. When you do so, the Insert Function dialog box will appear (see Figure 7.11).
210 7 • Time Value of Money I: Single Payment Value
Insert Function ? X
Search for a function:
Type a brief description of what you want to do and then click Go
Or select a category: Financial
Select a function:
NPER
NPV
ODDFPRICE
ODDFYIELD
ODDLPRICE
ODDLYIELD
PDURATION
N P ERfrate, p rut p v.tv.ty pe) Returns the number of periods for an investment based on periodic, constant payments and a constant interest rate.
Help on this function
OK
Cancel
Figure 7.11 Dialog Box to Insert NPER Function
As discussed in our previous examples on FV and PV, this menu allows you to either search for a function or select a function that has been used recently. In this example, you can search for NPER by typing this into the search box and selecting Go, or you can simply choose NPER from the list of most recently used functions.
2. Once you have highlighted NPER, click the OK button, and a new dialog box will appear for you to enter the necessary details. As in our previous examples, it will look like Figure 7.12.
Function Arguments NPER
X
Rate Pmt Pv
Fv Type
number number number number number
Returns the number of periods for an investment based on periodic, constant payments and a constant interest rate.
Rate is the interest rate per period. For example, use 6%/4 for quarterly payments at 6% APR.
Formula result = Help on this function
OK
Cancel
Figure 7.12 New Dialog Box for NPER Function Arguments
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7.3 • Methods for Solving Time Value of Money Problems 211
Figure 7.13 shows the completed Function Arguments dialog box. Note that once again, we are using cell addresses in this example.
Function Argument: NPER
X
Rate Pmt Pv
Fv Type
hi ±
o3 ±
□4 ±
±
±
0.06 0
-15000 25000
number
S.766692911
Returns the number of periods for an investment based on periodic, constant payments and a constant interest rate.
Fv is the future value, or a cash balance you want to attain afterthe last payment is made. If omitted, zero is used.
Formularesult= B.766692911 Help on this function
OK
Cancel
Figure 7.13 Completed Dialog Box for NPER Function Arguments
As in the previous function examples, values are shown off to the right of the data input area, and our final answer of approximately 8.77 is displayed at the bottom. Also, once again, the Pmt and Type boxes are not relevant to this single lump sum example.
Review your answer, and once you are satisfied with the result, click the OK button. The dialog box will disappear, with only the final numerical result appearing in the cell where you have set up the function.
The amount of time required for the desired growth to occur is calculated as approximately 8.77 years.
Interest or Growth Rate
You can also use Excel to determine the required growth rate when the present value, future value, and total number of required periods are known.
Let's discuss a similar example to the one we used to calculate periods of time. You still want to help your child with their first year of college tuition and related expenses, and you still have a starting amount of $15,000, but you have not yet decided which savings plan to use.
Instead, the information you now have is that your child is just under 10 years old and will begin college at age 18. For simplicity's sake, let's say that you have eight and a half years until you will need to meet your total savings target of $25,000. What rate of interest will you need to grow your saved money from $15,000 to $25,000 in this time, again with no other deposits or withdrawals?
Future Value (FV) = $25,000.00 Number of Periods (Nper or N) = 8.50 Present/Value (PV) = ($15,000.00)
Note: The present value needs to be entered as a negative value.
Interest Rate (RATE)
212 7 • Time Value of Money I: Single Payment Value
Note: The Excel command used to calculate interest or growth rate is as follows:
=RATE(nper, pmt, pv, [fv], [type], [guess])
As with our other TVM function examples, you may simply type the values for the arguments into the above formula. We also again have the same alternative to use the Insert Function option in Excel. If you choose this option, you will again see the Insert Function dialog box after you click the Insert Function button.
1. First, go to Formulas in the upper menu bar, and select the Insert Function option. When you do so, the Insert Function dialog box will appear (see Figure 7.14).
Insert Function Search for a function:
X
Type a brief description of what you want to do and then click Go
Or select a category: Financial
Go
Select a function:
PRICE
PRICEDISC
PRICEMAT
PV
RATE m
RECEIVED
RRI
RATE(n per.pmtpVjtv, type, guess]
Returns the interest rate per period of a loan or an investment. For example, use 6%/4for quarterly payments at 6% APR.
Help on this function
OK
Cancel
Figure 7.14 Dialog Box to Insert RATE Function
2. This time, find and highlight RATE, and click the OK button once you have done so. The Function Arguments dialog box will look like Figure 7.15.
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7.3 • Methods for Solving Time Value of Money Problems 213
Function Argument: RATE
Nper Pmt Pv
Fv Type
? X
= number
= number
= number
= number
= number
I
Returns the interest rate per period of a loan or an investment. For example, use 6%/4for quarterly payments at 6% APR.
Nper is the total number of payment periods forthe loan or investment.
Formula result = Help on this function
OK
Cancel
Figure 7.15 New Dialog Box for RATE Function Arguments
Once we complete the input, again using cell addresses for the required argument values, we will see what is shown in Figure 7.16.
Function Arguments RATE
X
b'i
Nper |b2 Pmt Pv
Fv Type
b5
b-
I
= 3.5 = 0
= -15000 = 25000
= number v = 0.06193969
Returns the interest rate per period of a loan or an investment. For example, use 6%/4for quarterly payments at 6% APR.
Fv is the future value, or a cash balance you want to attain afterthe last payment is made. If omitted, uses Fv = 0.
Formula result = 0,06193969 Help on this function
OK
Cancel
Figure 7.16 Completed Dialog Box for RATE Function Arguments
As in our other examples, cell values are shown as numerical values off to the right, and our answer of approximately 0.0619, or 6.19%, is shown at the bottom of the dialog box.
This answer also can be checked from a logic point of view because of the similar example we worked through when calculating periods of time. Our present value and future value are the same as in that example, and our time period is now 8.5 years, which is just under the result we arrived at (8.77 years) in
214 7 • Time Value of Money I: Single Payment Value
the periods example.
So, if we are now working with a slightly shorter time frame for the savings to grow from $15,000 into $25,000, then we would expect to have a slightly greater growth rate. That is exactly how the answer turns out, as the calculated required interest rate of approximately 6.19% is just slightly greater than the growth rate of 6% used in the previous example. So, based on this, it looks like our answer here passes a simple "sanity check" review.
EH
7.4 | Applications of TVM in Finance
Learning Outcomes
By the end of this section, you will be able to:
■ Explain how the time value of money can impact your personal financial goals.
■ Explain how the time value of money is related to inflation.
■ Explain how the time value of money is related to financial risk.
■ Explain how compounding period frequency affects the time value of money.
Single-Period Scenario
Let's say you want to buy a new car next year, and the one you have your eye on should be selling for $20,000 a year from now. How much will you need to put away today at 5% interest to have $20,000 a year from now? Essentially, you are trying to determine how much $20,000 one year from now is worth today at 5% interest over the year. To find a present value, we reverse the growth concept and lower or discountthe future value back to the current period.
The interest rate that we use to determine the present value of a future cash flow is referred to as the discount rate because it is bringing the money back in time in terms of its value. The discount rate refers to the annual rate of reduction on a future value and is the inverse of the growth rate. Once we know this discount rate, we can solve for the present value (PV), the value today of tomorrow's cash flow. By changing the FV equation, we can turn
FV = PV X (1 + rf
into
1
PV = FV X
(1 + r)n
which is the present value equation. The fraction shown above is referred to as the present value interest factor (PVIF). The PVIF is simply the reciprocal of the FVIF, which makes sense because these factors are doing exactly opposite things. Therefore, the amount you need to deposit today to earn $20,000 in one year (n = 1) at 5% interest is
$20,000 X ——r = $20,000 X 0.95328 = $19,047.62 (1.05)1
The Multiple-Period Scenario
There will often be situations when you need to determine the present value of a cash flow that is scheduled to occur several years in the future (see Figure 7.17). We can again use the formula for present value to calculate a value today of future cash flows over multiple time periods.
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7.4 • Applications of TVM in Finance 215
Figure 7.17 Determining Future Cash Flow
An example of this would be if you wanted to buy a savings bond for Charlotte, the daughter of a close friend. The face value of the savings bond you have in mind is $1,000, which is the amount Charlotte would receive in 30 years (the future value). If the government is currently paying 5% per year on savings bonds, how much will it cost you today to buy this savings bond?
The $1,000 face value of the bond is the future value, and the number of years n that Charlotte must wait to get this face value is 30 years. The interest rate r is 5.0% and is the discount rate for the savings bond. Applying the present value equation, we calculate the current price of this savings bond as follows:
PV = $1,000 x -l-^r- = $1,000 X 0.231377 = $231.38
(0.05)30
So, it would cost you $231.38 to purchase this 30-year, 5%, $1,000 face-valued bond.
What we have done in the above example is reduce, or discount, the future value of the bond to arrive at a value expressed in today's dollars. Effectively, this discounting process is the exact opposite of compounding interest that we covered earlier in our discussion of future value.
An important concept to remember is that compounding is the process that takes a present valuation of money to some point in the future, while discounting takes a future value of money and equates it to present dollar value terms.
Common applications in which you might use the present value formula include determining how much money you would need to invest in an interest-bearing account today in order to finance a college education for your oldest child and how much you would need to invest today to meet your retirement plans 30 years from now.
TVM, Inflation, Compounding Interest, Investing, Opportunity Costs, and Risk
The time value of money (TVM) is a critical concept in understanding the value of money relative to the amount of time it is held, saved, or invested. The TVM concept and its specific applications are frequently used by individuals and organizations that might wish to better understand the values of financial assets and to improve investing and saving strategies, whether these are personal or within business environments.
As we have discussed, the key element behind the concept of TVM is that a given amount of money is worth more today than that same amount of money will be at any point in the future. Again, this is because money can be saved or invested in interest-bearing accounts or investments that will generate interest income over time, thus resulting in increased savings and dollar values as time passes.
Inflation
The entire concept of TVM exists largely due to the presence of inflation. Inflation is defined as a general increase in the prices of goods and services and/or a drop in the value of money and its purchasing power.
The purchasing power of the consumer dollar is a statistic tracked by the part of the US Bureau of Labor
216 7 • Time Value of Money I: Single Payment Value
Statistics and is part of the consumer price index (CPI) data that is periodically published by that government agency. In a way, purchasing power can be viewed as a mirror image or exact opposite of inflation or increases in consumer prices, as measured by the CPI. Figure 7.18 demonstrates the decline in the purchasing power of the consumer dollar over the 13-year period from 2007 to 2020.
With this in mind, we can work with the TVM formula and use it to help determine the present value of money you have in hand today, as well as how this same amount of money may be valued at any specific point in the future and at any specific rate of interest.
2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 Figure 7.18 Historical Purchasing Power Decline as a Result of Inflation, 2007-2020 (data source: Bureau of Labor Statistics)
The Relationship between TVM and Inflation
As we have seen, the future value formula can be very helpful in calculating the value of a sum of cash (or any liquid asset) at some future point in time. One of the important ideas relating to the concept of TVM is that it is preferable to spend money today instead of at some point in the future (all other things being equal) when inflation is positive. However, in very rare instances in our economy when inflation is negative, spending money later is preferable to spending it now. This is because in cases of negative inflation, the purchasing power of a dollar is actually greater in the future, as the costs of goods and services are declining as we move into the future.
Most investors would be inclined to take a payment of money today rather than wait five years to receive a payment in the same amount. This is because inflation is almost always positive, which means that general prices of goods and services tend to increase over the passage of time. This is a direct function and result of normal economic growth. The crux of the concept of TVM is directly related to maintaining the present value of financial assets or increasing the value of these financial assets at different points in the future when they may be needed to obtain goods and services. If a consumer's monetary assets grow at a greater rate than inflation over any period of time, then the consumer will realize an increase in their overall purchasing power. Conversely, if inflation exceeds savings or investment growth, then the consumer will lose purchasing power as time goes by under such conditions.
The Impact of Inflation on TVM
The difference between present and future values of money can be easily seen when considered under the effects of inflation. As we discussed above, inflation is defined as a state of continuously rising prices for goods and services within an economy. In the study of economics, the laws of supply and demand state that increasing the amount of money within an economy without increasing the amount of goods and services available will give consumers and businesses more money to spend on those goods and services. When more money is created and made available to the consuming public, the value of each unit of currency will diminish.
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7.4 • Applications of TVM in Finance 217
This will then have the effect of incentivizing consumers to spend their money now, or in the very near future, instead of saving cash for later use. Another concept in economics states that this relationship between money supply and monetary value is one of the primary reasons why the Federal Reserve might at times take steps to inject money into a stagnant, lethargic economy. Increasing the money supply will lead to increased economic activity and consumer spending, but it can also have the negative effect of increasing the costs of goods and services, furthering an increase in the rate of inflation.
Consumers who decide to save their money now and for the foreseeable future, as opposed to spending it now, are simply making the economic choice to have their cash on hand and available. So, this ends up being a decision that is made despite the risk of potential inflation and perhaps losing purchasing power. When inflationary risk is low, most people will save their money to have it available to spend later. Conversely, in times when inflationary risk is high, people are more likely to spend their money now, before its purchasing power erodes. This idea of inflationary risk is the primary reason why savers and investors who decide to save now in order to have their money available at some point in the future will insist they are paid, through interest or return on investment, for the future value of any savings or financial instrument.
Lower interest rates will usually lead to higher inflation. This is because, in a way, interest rates can be viewed as the cost of money. This allows for the idea that interest rates can be further viewed as a tax on holding on to sums of money instead of using it. If an economy is experiencing lower interest rates, this will make money less expensive to hold, thus incentivizing consumers to spend their money more frequently on the goods and services they may require. We have also seen that the more quickly inflation rates rise, the more quickly the general purchasing power of money will be eroded. Rational investors who set money aside for the future will demand higher interest rates to compensate them for such periods of inflation. However, investors who save for future consumption but leave their money uninvested or underinvested in low-interest-bearing accounts will essentially lose value from their financial assets because each of their future dollars will be worth less, carrying less purchasing power when they end up needing it for use. This relationship of saving and planning for the future is one of the most important reasons to understand the concept of the time value of money.
Nominal versus Real Interest Rates
One of the main problems of allowing inflation to determine interest rates is that current interest rates are actually nominal interest rates. Nominal rates are "stated," not adjusted for the effects of inflation. In order to determine more practical real interest rates, the original nominal rate must be adjusted using an inflation rate, such as those that are calculated and published by the Bureau of Labor Statistics within the consumer price index (CPI).
A concept referred to as the Fisher effect, named for economist Irving Fisher, describes the relationship between inflation and the nominal and real interest rates and is expressed using the following formula:
(1 + 0 = (1 + R) X (1 + h)
where / is the nominal interest rate, R is the real interest rate, and h is the expected inflation rate.
An example of the Fisher effect would be seen in the case of a bond investor who is expecting a real interest rate of return of 6% on the bond, in an economy that is experiencing an expected inflation rate of 2%. Using the above formula, we have
i = (1 + R) X (1 + h) - 1 i = (1 + 6%) X (1 + 2%) - 1 i = 8.12%
So, the nominal interest rate on the bond amounts to 8.12%, with a real interest rate of 6% within an economy that is experiencing a 2% inflation rate. This is a logical result because in a scenario of positive inflation, a real rate of return would always be expected to amount to less than the stated or nominal rate.
218 7 • Time Value of Money I: Single Payment Value
Interest and Savings
Savings are adversely affected by negative real interest rates. A person who holds money in the form of cash is actually losing future purchasing value when real interest rates are negative. A saver who decides to hold $1,000 in the form of cash for one year at a negative real interest rate of -3.65% per year will lose $1,000 X - 0.0365 or $36.50, in purchasing power by the end of that year.
Ordinarily, interest rates would rise to compensate for negative real rates, but this might not happen if the Federal Reserve takes steps to maintain low interest rates to help stimulate and stabilize the economy. When interest rates are at such low levels, investors are forced out of Treasury and money market investments due to their extremely poor returns.
It soon becomes obvious that the time value of money is a critical concept because of its tremendous and direct impact on the daily spending, saving, and investment decisions of the people in our society. It is therefore extremely important that we understand howTVM and government fiscal policy can affect our savings, investments, purchasing behavior, and our overall personal financial health.
Compounding Interest
As we discussed earlier, compound interest can be defined as interest that is being earned on interest. In cases of compounding interest, the amount of money that is being accrued on previous amounts of earned interest income will continue to grow with each compounding period. So, for example, if you have $1,000 in a savings account and it is earning interest at a 10% annual rate and is compounded every year for a period of five years, the compounding will allow for growth after one year to an amount of $1,100. This comprises the original principal of $1,000 plus $100 in interest. In year two, you would actually be earning interest on the total amount from the previous compounding period—the $1,100 amount.
So, to continue with this example, by the end of year two, you would have earned $1,210 ($1,100 plus $110 in interest). If you continue on until the end of year five, that $1,000 will have grown to approximately $1,610. Now, if we consider that the highest annual inflation rate over the last 20 years has been 3%, then in this scenario, choosing to invest your present money in an account where interest is being compounded leaves you in a much better position than you would be in if you did not invest your money at all. The concept of the time value of money puts this entire idea into context for us, leading to more informed decisions on personal saving and investing.
It is important to understand that interest does not always compound annually, as assumed in the examples we have already covered. In some cases, interest can be compounded quarterly, monthly, daily, or even continuously. The general rule to apply is that the more frequent the compounding period, the greater the future value of a savings amount, a bond, or any other financial instrument. This is, of course, assuming that all other variables are constant.
The math for this remains the same, but it is important that you be careful with your treatment and usage of rate (r) and number of periods (n) in your calculations.
For example, $1,000 invested at 6% for a year compounded annually would be worth
$1,000 X (1.06)1 =$1,060.00. But that same $1,000 invested for that same period of time—one year—and earning interest at the same annual rate but compounded monthly would grow to
$1,000 X (1.005)12 = $1,061.67, because the interest paid each month is earning interest on interest at a 6%
rate. Note that we represent r as the interest paid per period ( "T"3!inteiest = 0.005 )and n as the number r r r r \ 12 months in a year /
of periods (12 months in a year; 12 X 1 = 12) rather than the number of years, which is only one.
Continuing with our example, that same $1,000 in an account with interest compounded quarterly, or four times a year, would grow to $1,000 X (1.05)4 = $1,061.36 in one year. Note that this final amount ends up being greater than the annually compounded future value of $1,060.00 and slightly less than the monthly
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7.4 • Applications of TVM in Finance 219
compounded future value of $1.061.67, which would appear to make logical sense.
The total differences in future values among annual, monthly, and quarterly compounding in these examples are insignificant, amounting to less than $1.70 in total. However, when working with larger amounts, higher interest rates, more frequent compounding periods, and longer terms, compounding periods and frequency become far more important and can generate some exceptionally large differences in future values.
Ten million dollars at 12% growth for one year and compounded annually amounts to
$10,000 X (1.12) = $11,200,000, while 10 million dollars on the same terms but compounded quarterly will produce $10,000,000 X (1.03)4 = $11,255,088.10. Most wealthy and rational investors and savers would be very pleased to earn that additional $55,088.10 by simply having their funds in an account that features quarterly compounding.
In another example, $200 at 60% interest, compounded annually for six years, becomes $200 X (1.6)6 = $3,355.44, while this same amount compounded quarterly grows to $200 X (1.15)24 = $5,725.04.
An amount of $1 at 3%, compounded annually for 100 years, will be worth $1 X (1.03)100 = $19.22. The same dollar at the same interest rate, compounded monthly over the course of a century, will grow to $1 X (1.0025)1'200 = $397.44.
This would all seem to make sense due to the fact that in situations when compounding increases in frequency, interest income is being received during the year as opposed to at the end of the year and thus grows more rapidly to become a larger and more valuable sum of money. This is important because we know through the concept of TVM that having money now is more useful to us than having that same amount of money at some later point in time.
The Rule of 72
The rule of 72 is a simple and often very useful mathematical shortcut that can help you estimate the impact of any interest or growth rate and can be used in situations ranging from financial calculations to projections of population growth. The formula for the rule of 72 is expressed as the unknown (the required amount of time to double a value) calculated by taking the number 72 and dividing it by the known interest rate or growth rate. When using this formula, it is important to note that the rate should be expressed as a whole integer, not as a percentage. So, as a result, we have
72
Years for an Amount to Double =--—-—-—
Interest or Growth Rate
This formula can be extremely practical when working with financial estimates or projections and for understanding how compound interest can have a dramatic effect on an original amount or monetary balance.
Following are just a few examples of how the rule of 72 can help you solve problems very quickly and very easily, often enabling you to solve them "in your head," without the need for a calculator or spreadsheet.
Let's say you are interested in knowing how long it will take your savings account balance to double. If your account earns an interest rate of 9%, your money will take 72/9 or 8, years to double. However, if you are earning only 6% on this same investment, your money will take 72/6, or 12, years to double.
Now let's say you have a specific future purchasing need and you know that you will need to double your money in five years. In this case, you would be required to invest it at an interest rate of 72/5, or 14.4%. Through these sample examples, it is easy to see how relatively small changes in a growth or interest rate can have significant impact on the time required for a balance to double in size.
To further illustrate some uses of the rule of 72, let's say we have a scenario in which we know that a country's gross domestic product is growing at 4% a year. By using the rule of 72 formula, we can determine that it will
220 7 • Time Value of Money I: Single Payment Value
take the economy 72/4, or 18, years to effectively double.
Now, if the economic growth slips to 2%, the economy will double in 72/2, or 36, years. However, if the rate of growth increases to 11 %, the economy will effectively double in 72/11, or 6.55, years. By performing such calculations, it becomes obvious that reducing the time it takes to grow an economy, or increasing its rate of growth, could end up being very important to a population, given its current level of technological innovation and development.
It is also very easy to use the rule of 72 to express future costs being impacted by inflation or future savings amounts that are earning interest.
To apply another example, if the inflation rate in an economy were to increase from 2% to 3%, consumers would lose half of the purchasing power of their money. This is calculated as the value of their money doubling in 72/3, or 24, years as compared to 72/2, or 36, years—quite a substantial difference.
Now, let's say that tuition costs at a certain college are increasing at a rate of 7% per year, which happens to be greater than current inflation rates. In this case, tuition costs would end up doubling in 72/7, or about 10.3, years.
In an example related to personal finance, we can say that if you happen to have an annual percentage rate of 24% interest on your credit card and you do not make any payments to reduce your balance, the total amount you owe to the credit card company will double in only 72/24, or 3, years.
So, as we have seen, the rule of 72 can clearly demonstrate how a relatively small difference of 1 percentage point in GDP growth or inflation rates can have significant effects on any short- or long-term economic forecasting models.
It is important to understand that the rule of 72 can be applied in any scenario where we have a quantity or an amount that is in the process of growing or is expected to grow for any period of time into the future. A good nonfinancial use of the rule of 72 might be to apply it to some population projections. For example, an increase in a country's population growth rate from 2% to 3% could present a serious problem for the planning of facilities and infrastructure in that country. Instead of needing to double overall economic capacity in 72/2 or 36, years, capacity would have to be expanded in only 72/3, or 24, years. It is easy to see how dramatic an effect this would be when we consider that the entire schedule for growth or infrastructure would be reduced by 12 years due to a simple and relatively small 1% increase in population growth.
Investing and Risk
Investing is usually a sound financial strategy if you have the money to do so. When investing, however, there are certain risks you should always consider first when applying the concepts of the time value of money. For example, making the decision to take $1,000 and invest it in your favorite company, even if it is expected to provide a 5% return each year, is not a guarantee that you will earn that return—or any return at all, for that matter. Instead, as with any investment, you will be accepting the risk of losing some or even all of your money in exchange for the opportunity to beat inflation and increase your future overall wealth. Essentially, it is risk and return that are responsible for the entire idea of the time value of money.
Risk and return are the factors that will cause a rational person to believe that a dollar risked should end up earning more than that single dollar.
To summarize, the concept of the time value of money and the related TVM formulas are extremely important because they can be used in different circumstances to help investors and savers understand the value of their money today relative to its earning potential in the future. TVM is critical to understanding the effect that inflation has on your money and why saving your money early can help increase the value of your savings dollars by giving them time to grow and outpace the effects of inflation.
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7.4 • Applications of TVM in Finance 221
Opportunity Costs
The concept of opportunity cost arises from the idea that there will always be possible options that are sacrificed with every option we decide on or for every choice that we make. For example, let's consider the decision to go to college after you graduate from high school. This decision, as with just about any other, will involve evaluating opportunity costs. If you choose to go to college, this will result in your sacrificing four years of potential earnings that you could have had if you had decided to take a job instead of attending school. Also, in addition to the lost salary, you would be losing out on four years of work experience that could have had a positive impact on your resume or your future earnings prospects.
Of course, the entire idea behind furthering one's education is that you are hopeful that by choosing to go to college, you will increase the likelihood of earning a greater salary over the course of your lifetime than you would have if you had chosen to join the workforce directly out of high school. So, this ends up being a bit of a risk, but one that you have considered. The idea is that you are hoping for a more significant payoff down the road than if you had made the decision not to continue with your studies. When it comes to opportunity costs and the time value of money, it is obvious that there will always be costs associated with every forgone financial opportunity we pass on when we make a different choice. The logical individual can only hope that these choices produce a better end result than if we had made different choices and pursued any of our forgone alternatives. This also applies in situations where we may sit idly by and decide to take no action at all.
For example, if you are putting $1,000 in a savings account to save for a house, you may be giving up an opportunity to grow that money in an investment account that would earn a greater rate of return. In another example, being able to calculate the future value of your money will tell you that instead of investing, you probably should be paying down your 24% APR credit card debt that is costing you hundreds of dollars a month—hundreds of dollars more than you might earn from an investment account.
222 7 • Summary
a Summary
7.1 Now versus Later Concepts
This section discussed the underlying concepts of the time value of money (TVM). Because it is possible to earn interest income on cash that you decide to deposit in an investment or an interest-bearing account, money that you have now or receive sooner will be more valuable to you than the same amount of money received later.
7.2 Time Value of Money (TVM) Basics
Future value refers to the value that a current amount will eventually grow into at a given interest rate over a specific period of time. The single-period scenario is one way in which future amounts are calculated. Compounding, which is interest earned on interest, also affects the future value of money.
7.3 Methods for Solving Time Value of Money Problems
Calculations can be used to determine future and present dollar amounts, discount and growth rates, and periods of time required for specific growth. Time value of money problems can be solved using mathematical equations, calculators with financial functions, and spreadsheets. A useful tool for conceptualizing present value and future value problems is a timeline. A timeline is a visual, linear representation of the timing of periods and cash flows over a set amount of time.
7.4 Applications of TVM in Finance
The idea of the time value of money is often considered to be the cornerstone concept of the study of finance. TVM can help investors and savers understand the value of money today relative to its earning potential in the future. TVM is critical to understanding the effect that inflation has on money and why saving your money early can help increase the value of your savings dollars by giving them time to grow and outpace the effects of inflation. Of course, it is important to remember that there will always be possible options that are sacrificed with every option you decide on and every choice you make.
9 Key Terms
Bureau of Labor Statistics a group within the United States Department of Labor that is the primary factfinding agency for the US government in the fields of labor, statistics, and economics; serves as the principal agency of the US Federal Statistical System
compounding interest the continual addition of interest to the original principal sum of a loan or deposit, often referred to as interest on interest
compounding period the period between points in time when interest is paid or added to the principal
consumer price index (CPI) a measure that examines the weighted average of prices of a basket of consumer goods and services such as transportation, food, and medical care
discount rate the interest rate used to determine the present value of future cash inflows
Federal Reserve the central bank system of the United States
financial instrument an asset or bundle of assets, including monetary contracts between parties, that can
be bought, sold, or traded for financial gain financial risk the possibility of losing money or purchasing power on an investment, business transaction, or
venture or simply due to inflation Fisher effect an economic theory created by economist Irving Fisher that describes the relationship between
inflation and both real and nominal interest rates future value (FV) the value that a current amount will grow to at a given interest rate over a given period of
time
gross domestic product (GDP) the total value of goods produced and services provided in a country during one year
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7- CFA Institute 223
growth rate the percentage increase of a specific variable within a specific time period; synonymous with
interest rate in the context of the time value of money interest the amount of money that is paid by a borrower to a lender for the use of their money, typically
calculated from an annualized rate investment an asset or item acquired with the goal of generating financial gain through increased income
or appreciation in value liquid asset an asset that can be readily converted into cash within a short period of time money market investments low-risk financial instruments such as T-bills, federal notes, commercial paper,
certificates of deposit (CDs), repurchase agreements (repos), and bankers' acceptances, among others money supply the total dollar value of legal tender that is available to consumers within an economy at any
single point in time
opportunity cost the loss of potential gain from other alternatives when a single alternative is chosen present value (PV) the current value of a future amount, calculated by discounting the future value back at a
known discount or interest rate for a specified period of time real interest rate a rate of interest that has been adjusted to account for the effects of inflation single payment or lump sum a single payment or deposit made at one time, as opposed to a number of
smaller payments or deposits made in installments time value of money (TMV) the concept that an amount of money is worth more today than the exact same
amount of money at some point in the future Treasury investments debt obligations such as T-bills (Treasury bills), bonds, and notes issued by the US
Department of the Treasury underinvested describes an insufficient amount of investment or an investment that is earning an
insufficient rate of interest
uninvested describes cash that is being held in reserve, is not invested in an account or financial instrument, and is not earning interest or any return
a CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session (https://0penstax.0rg/r/cfainstitute link). Reference with permission of CFA Institute.
a
Multiple Choice
1. The most basic type of financial transaction involves_.
a. an amount of money that is not invested
b. a series of equal installment amounts paid or received over a period of time
c. a simple, one-time amount of cash that can be either a receipt or a payment
d. None of the above
2. If a discount (or interest) rate has a positive value, then the future value of any amount deposited in an interest-bearing account will_.
a. be less than the present value
b. be equal to the present value
c. be greater than the present value
d. decline over time
3. If the discount (or interest) rate used to calculate the present value of a future payment increases, the calculated present value will do which of the following?
a. Increase
b. Decrease
224
7 • Review Questions
c. Remain the same
d. Increase as the period of time shortens
4. The discount rate that is required to equate a future payment of $500 in three years to a present value of $400 is_.
a. 4.7%
b. 6.5%
c. 7.7%
d. 8.8%
5. If compounding periods increase in frequency and all else remains the same, the dollar values of any resulting future value calculations will_.
a. increase
b. remain the same
c. decrease
d. None of the above
Review Questions
1. All other things being the same, would you prefer a bank account that compounds interest quarterly or one that compounds interest semiannually?
2. Briefly describe the concept of future value within the context of the larger overall concept of the time value of money.
3. Which of the following two options will give you the greatest future value: (A) an initial deposit of $100 earning 20% per year, compounded annually and left to grow for 10 years, or (B) an initial deposit of $75 earning 12% per year, compounded monthly and left to grow for 15 years?
4. If a savings account pays interest on a quarterly basis and you are performing future value calculations on deposited amounts, how can you calculate the rate?
5. Briefly describe the relationship among consumer savings, purchasing power, and inflation.
Oj Problems
1. Find the future value of $100 in five years at 5% interest.
2. Find the future value of $1,800 in 3 years at 8% interest.
3. How much would you have to deposit now to have $15,000 in eight years if interest is 7%?
4. What is the present value of $5,000 that will be paid to you eight years from today at 8% interest?
5. How many years will it take a $700 balance to grow into $900 in an account earning 5%?
6. If you borrow $1,000 and pay back $1,728 in three years, what annual rate of interest are you paying?
7. How long will it take you to triple your money at 8%?
8. A company's sales were $250 million in 2019. If sales grow at 6% per year, how large will they be 10 years later, in 2029 (as expressed in millions)?
9. A US government bond in the amount of $1,000 will mature in six years, has no coupon payments, and carries an interest rate of 8%. What is the value of this bond today?
10. You spend $725.00 to purchase a $1,000 bond that will have no coupon payments and matures in 12 years.
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7 • Video Activity 225
What interest rate will you be earning on this bond?
11. At what interest rate would you be ambivalent about receiving either $50,000 10 years from now or $35,000 today?
12. Vance Corporation had earnings last year of $3.25 per share. The company has experienced 8% annual growth over the last several years, and management expects that growth rate to continue. Based on this information, after how long will earnings per share double?
13. What is the amount of total interest dollars earned on a $5,000 deposit earning 6% for 20 years?
14. You have decided that you will sell your $300,000 house when it appreciates in value to $500,000. If houses are appreciating at an average annual rate of 5% in your neighborhood, for approximately how long will you be staying in your house?
15. You just won some money in the lottery and would like to save a portion of it so that you will have $50,000 to put a down payment on a house in five years. Your bank pays a 5% rate of interest. How much money will you have to set aside from the lottery winnings?
16. Bauer Bookstore sells books before they are published. Today, they offered the book Journeys in Finance for $14.20, but the book will not be published for another two years. Upon publishing, the price of the book will be $24.00. What is the discount rate Bauer Bookstore is offering its customers for this book?
17. One of your professional goals is to one day earn a six-figure salary ($100,000). You hope to accomplish this objective within the next 30 years. Salaries grow at 3.75% per year in your field of work. What beginning salary will you need in order to reach this 30-year goal?
18. How much will $25,000 grow to in five years at a 5% annual rate that is compounded quarterly?
19. If Eisenberg Industries revenues have increased from $30 million to $90 million over a 10-year period, what has been their annual rate of growth?
20. If you are scheduled to receive $4,000 six years from today and the discount rate is 8.5%, what is the present value of this payment?
21. If you were to open a savings account that earns 3% interest and is compounded quarterly, what would be the total amount in your account after 10 years if you made an opening deposit of $9,500?
22. You are considering four possible options for your new savings account. You plan to deposit $13,000 and leave this amount in the account for 20 years with no additional deposits or withdrawals. All of these account options would earn 6% interest, but each one has a different compounding frequency, listed below. What would be the value of each account at the end of the 20-year period?
a. Annually
b. Semiannually
c. Quarterly
d. Monthly
Time Value of Money Calculations with Financial Calculator
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1. Name and describe the five primary keys on a financial calculator that are used to solve time value of money problems.
2. Following the instructions laid out in this video, practice calculating the following TVM variables using these inputs:
Video Activity
226 7 • Video Activity
Find a Future Value (FV): Calculate FVwhen
PV = -1,000, N = 5, I/Y = 4, and PMT = 0. Find a Present Value (PV): Calculate PVwhen
FV = 2,000, N = 7, I/Y = 5, and PMT = 0. Find the Number of Periods (N) when
FV = 5,000, PV = -3,000, I/Y = 3, and PMT = 0. Find the Interest Rate (I/Y) when
FV = 6,000, PV = -1,500, N = 10, and PMT = 2. Find the Payment (PMT) when
FV - 15,000, PV - -8,000, N - 12, and I/Y = 4.
Continue to practice using a financial calculator to solve various time value of money problems using different input factors until you feel comfortable with the process of using a calculator to solve TVM problems.
Time Value of Money Using Excel with 10 Examples
Click to view content (https://0penstax.0rg/r/M0ney Using Excel)
3. Where within the Excel spreadsheet and its different menus (under what menu button option) can you find the different functions that will enable you to solve for different time value of money factors?
4. Following the instructions laid out in this video, practice calculating the following TVM variables in Excel. Find a Future Value (FV): Calculate FVwhen
PV = -1,000, N = 5, I/Y = 4, and PMT = 0.
Find a Present Value (PV): Calculate PVwhen
FV = 2,000, N = 7, I/Y = 5, and PMT = 0.
Find the Number of Periods (N) when
FV = 5,000, PV = -3,000, I/Y = 3, and PMT = 0.
Find the Interest Rate (I/Y) when
FV = 6,000, PV = -1,500, N = 10, and PMT = 2.
Find the Payment (PMT) when
FV - 15,000, PV - -8,000, N - 12, and I/Y = 4.
Go back and check the answers to these questions that you ended up with under the video 77me Value of Money Calculations with Financial Calculator. You should get the exact same answers when using both methods (calculator and Excel).
Continue to practice using Excel to solve various time value of money problems using different input factors until you feel comfortable with the process of using spreadsheets to solve TVM problems.
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Figure 8.1 The value of an investment generally represents our expectations of all future cash flows from that investment, once discounted, (credit: modification of "Money" by Ervins Strauhmanis/flickr CC BY 2.0)
Chapter Outline
8.1 Perpetuities
8.2 Annuities
8.3 Loan Amortization
8.4 Stated versus Effective Rates
8.5 Equal Payments with a Financial Calculator and Excel
Why It Matters
Although this text is directed at business finance students, our daily decisions as consumers are largely based on money and finance to just as great an extent. An old adage in finance claims, "If you aren't in control of your money, your money is controlling you." Fortunately, learning to manage your money is not difficult if you're disciplined and understand some simple techniques. For example, several years ago, the author was negotiating for a three-year auto loan from a well-known regional dealer, who was offering an interest rate of 2%. When the manager left the room for a few minutes, we pulled out a financial calculator and proved in less than a minute that the actual interest rate in the payments he was proposing was nearly double the quoted and advertised rate.
In addition to understanding how the loan process works, which improves your negotiation skills when borrowing, businesses and individuals can better control their investments by understanding basic rules of finance, particularly as seen in this and the preceding chapter. Assume you pledge to invest $1,000 per year at 5% return per year and are curious about how much you will have accumulated by age 60. If you begin at age 30, you will have $69,760; if you begin at age 20, you will have $126,840. Can the extra 10 years make that much of a difference? We'll see that indeed they can, and the calculations required to prove it can take less than two minutes.
As another example, many professionals confuse income and wealth during their career growth. In their popular book The Millionaire Next Door: The Surprising Secrets of America's Wealthy, authors Thomas Stanley
228 8 • Time Value of Money II: Equal Multiple Payments
and William Danko illustrate these terms with a flowing river. A river is in constant movement, and as the flow or depth increases, this is comparable to one's income increasing through the promotions, salary increases, and bonuses one receives. Unfortunately, many individuals then increase their spending habits in response, justifying a better car, a second home, or more lavish vacations. Wealth, in contrast, is comparable to taking a bucket of water from the river and holding it aside in a tank for oneself. Financial professionals often call this "paying yourself first." Stanley and Danko list this among the "secrets" referenced in the title of their book.
Learning Outcomes
By the end of this section, you will be able to:
■ Define perpetuity.
■ Explain how perpetuities are valued.
In Time Value of Money I. we learned that the value of money changes with the passage of time. Decisionmakers consider how investments, projects, and even opportunity costs gain value as we move forward into the future. They similarly consider how value in the future can be reduced to a value in present or past periods. We saw that these value projections are called determination of future value (compounding, moving forward on a timeline) or present value (discounting, moving backward on a timeline). The easiest way to visualize this movement through time, whether forward or backward, is by use of a timeline.
Throughout the first chapter on the time value of money, we were analyzing a single amount. In this chapter, we deal with a stream of payments made periodically—in other words, payments made or received regularly over a span of time. We begin with the illustration of a perpetuity.
What Is a Perpetuity?
A perpetuity is a series of payments or receipts that continues forever, or perpetually. One of the best ways to analyze the basics of an annuity (the stream of payments to be paid or received in the future) is by starting with a perpetuity. The most common examples of perpetuities in the author's experience are college chair endowments and preferred stock.
If you gift $1,500,000 to a college to name a professor's chair for your family, you might specify that the money must be held in perpetuity and invested by the college to yield a fixed 3%. The college will take those proceeds of the investment, leaving your original $1,500,000 intact, and use the annual interest of $45,000 to fund a portion of the professor's salary.
Another common example is preferred stock. Most preferred stock issues carry a fixed and predetermined rate of dividend. If we assume that the dividend will not change in future years, then preferred dividend shareholders will receive a fixed amount of money in future years—assuming, of course, that the company's board of directors declares the dividends sufficient to fund these requirements. If we assume that the dividend is declared and paid and that it remains constant, this represents a perpetuity.
For example, Shaw Inc. has issued 100,000 shares of preferred stock with a stated value of $50 and a 4% dividend. Therefore, if they can fund and decide to declare dividends for the full amount, they will pay out $200,000, or $2.00 per preferred share. Because shares such as these are created with the intention of continuity, the owner of this preferred stock can theoretically expect this dividend income stream in perpetuity.
To place a current market value on this stream of future income, how much should an investor pay for one share of this preferred stock? The calculation is a present value. The amount the investor pays today for that one share is equal to the annual dividend (assuming it is declared and paid) divided by the rate of return. But be careful—it is not the rate on the face of the preferred stock but the required rate of return, the "market rate" that investors expect from a stock of this level of risk. We must also note an important fact affecting all
Perpetuities
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8.1 • Perpetuities 229
investment valuations: the value of an investment generally represents our expectations of all future cash flows from that investment, discounted to today's dollars.
Because a perpetuity is a stream of payments continuing indefinitely, determining the future value isn't possible. Determining the present value, however, is possible, although one might wonder how. As we learned earlier, the greater the amount of time used in a present value calculation, the smaller the amount of dollars needed at the beginning, regardless of the interest rate involved. Therefore, when we discount each payment in an infinite series, remembering that we would then add them together once we discount them to the present, the infinite payments become negligible at some point and will no longer have a significant impact on today's value. To grow to one dollar 70 years from now, even at a growth rate of 5% per year, we would only need $0.0329—not even four cents. Keeping all other facts the same, if we had 100 years to grow an investment to one dollar, we would only need 0.76 cents—not even a whole penny! There is no question that the effect of time is substantial and dramatic.
The study of perpetuities in corporate finance is a first step to understanding valuation models of certain investments, such as the dividend discount model and the constant growth model, to be addressed in other chapters. Our ability to discount future cash flows, even infinite cash flows, to a present value is a clue to the price at which a company's stock might trade. From a personal financial planning perspective, the individual investor is also better able to be certain that they are paying a fair price for holdings in their portfolio. For purposes of long-term or retirement planning, the investor must consider that a fixed and unchanging dividend, such as from preferred stock, might not adequately protect the holder from inflation in times of rising prices.
How to Value a Perpetuity
Given these facts, how do we place a value on a perpetuity? Let's keep the preferred stock example for Shaw Inc. in mind. The holder of one share will expect to receive a $2.00 dividend for every share owned. Although a perpetuity may allow for growth of that dividend, we will hold that constant now. We must know one additional fact: the required rate of return. This is our random variable, which can cause fluctuation in the price of the preferred stock. Let's assume that the required rate of return, which we'll call Rs, is 7%. This is the rate of return that the market expects in order to take on the risk of an investment such as Shaw.
Determination of the price of Shaw's preferred stock becomes quite simple because the expected annual cash flow should not change, making it a constant perpetuity. The constant perpetuity formula is
where PV is the price of the preferred stock, C is the constant dividend, and Rs is the required rate of return. By substitution,
$2.00
pv = W = $28-57
The price one should pay for a share of Shaw's preferred stock is $28.57.
Here's another constant perpetuity to try. The preferred stock of Rooney Corporation pays an annual dividend of $1.75 per share. If the required rate of return in the market for shares such as Rooney's is 5.8%, at what
$1 75
price should these preferred shares be trading? The answer is qq58, or $30.17.
Some investments might involve a growing perpetuity. In this case, some degree of change in the amount of the dividend is expected. The formula is altered slightly to include a rate of growth in the denominator, noted as G, making the growing perpetuity formula
Rs-G
230 8 • Time Value of Money II: Equal Multiple Payments
To illustrate a growing perpetuity, let's revisit Rooney Corp.'s stock, with its annual dividend of $1.75 and a required rate of return in the market of 5.8%. If the expected dividend growth rate G is 1.2%, then the value changes to 0 05g 1 0 021' or $38.04. The expectation of growth in the dividend provides incentive for the investor to pay a higher price.
THINK
KIT THR
A Growing Perpetuity
Savo Corporation. While we think of perpetuities as being static, with a constant benefit or dividend, as seen above, they might have the possibility of growth. Let's assume that our preferred stock in Savo Corporation is expected to grow at a rate of 0.2% per year. Its annual dividend per share is $4.00, and its required rate of return in the market is 3%. If this is a constant dividend stock, like most preferred stock, its price would be expected to approximate
C
PV =
or
$4.00 0.03
= $133.33
When we factor in the 0.2% annual growth in the dividend, what does the price per share become? Solution:
4
PV =
(0.03-0.002)
= $142.86.86
HINKIT THROUGH
College Endowment
In the case of an endowment for a college chair, as noted in the beginning of this section, instead of a dividend amount for a preferred stock, we would use the desired amount of the distribution that the chair would receive as part of their compensation package. Assume the college can invest this money at a fixed 3.5% annual rate. If you wish to gift the college enough funds to be held in perpetuity to produce $75,000 each year for that professor, how would you calculate this?
Solution:
We modify the constant perpetuity formula:
Gift =
Annual Distribution
In one year, $2,142,857 grows to
$2,142,857 X 1.035 = $2,217,857
The earnings (gift) of $75,000 are withdrawn to compensate the professor, and we are left with the amount originally endowed, $2,142,857:
$2,217,857 - $75,000 = $2,142,857
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8.2 • Annuities 231
Annuities
Learning Outcomes
By the end of this section, you will be able to:
■ Define annuity.
■ Distinguish between an ordinary annuity and an annuity due.
■ Calculate the present value of an ordinary annuity and an annuity due.
■ Explain how annuities may be used in lotteries and structured settlements.
■ Explain how annuities might be used in retirement planning.
Calculating the Present Value of an Annuity
An annuity is a stream of fixed periodic payments to be paid or received in the future. Present or future values of these streams of payments can be calculated by applying time value of money formulas to each of these payments. We'll begin with determining the present value.
Before exploring present value, it's helpful to analyze the behavior of a stream of payments over time. Assume that we commit to a program of investing $1,000 at the end of each year for five years, earning 7% compounded annually throughout. The high rate is locked in based partly on our commitment beginning today, even though we will invest no money until the end of the first year. Refer to the timeline shown in Table 8J..
Year 0 1 2 3 4 5
Balance Forward ($) 0.00 0.00 1,000.00 2,070.00 3,214.90 4,439.94
Interest Earned ($) 0.00 70.00 144.90 225.04 310.80
Principal Added ($) 1,000.00 1,000.00 1,000.00 1,000.00 1,000.00
New Balance ($) 1,000.00 2,070.00 3,214.90 4,439.94 5,750.74
Table 8.1
At the end of the first year, we deposit the first $1,000 in our fund. Therefore, it has not yet had an opportunity to earn us any interest. The "new balance" number beneath is the cumulative amount in our fund, which then carries to the top of the column for the next year. In year 2, that first amount will earn 7% interest, and at the end of year 2, we add our second $1,000. Our cumulative balance is therefore $2,070, which then carries up to the top of year 3 and becomes the basis of the interest calculation for that year. At the end of the fifth year, our investing arrangement ends, and we've accumulated $5,750.74, of which $5,000 represents the money we invested and the other $750.74 represents accrued interest on both our invested funds and the accumulated interest from past periods.
Notice two important aspects that might appear counterintuitive: (1) we've "wasted" the first year because we deposited no funds at the beginning of this plan, and our first $1,000 begins working for us only at the beginning of the second year; and (2) our fifth and final investment earns no interest because it's deposited at the end of the last year. We will address these two issues from a practical application point of view shortly.
Keeping this illustration in mind, we will first focus on finding the present value of an annuity. Assume that you wish to receive $25,000 each year from an existing fund for five years, beginning one year from now. This stream of annual $25,000 payments represents an annuity. Because the first payment will be received one year from now, we specifically call this an ordinary annuity. We will look at an alternative to ordinary annuities later. How much money do we need in our fund today to accomplish this stream of payments if our remaining balance will always be earning 8% annually? Although we'll gradually deplete the fund as we withdraw periodic payments of the same amount, whatever funds remain in the account will always be earning interest.
232 8 • Time Value of Money II: Equal Multiple Payments
Before we investigate a formula to calculate this amount, we can illustrate the objective: determining the present value of this future stream of payments, either manually or using Microsoft Excel. We can take each of the five payments of $25,000 and discount them to today's value using the simple present value formula:
FV
PV=-
(1 + rf
where FV is the future value, PV is the present value, r is the interest rate, and n is the number of periods.
For example, the first $25,000 is discounted by the equation as follows:
py _ $25,000 (1 + 0.08)1
PV = -^y^ = $23,148.15
Proof that $23,148.15 will grow to $25,000 in one year at 8% interest:
$23,148.15 X 1.08 = $25,000
If we use this same method for each of the five years, increasing the exponent n for each year, we see the result in Table 8.2.
Calculation Result
1 $25,000 (1.08)1 $23,148.15
2 $25,000 (1.08)2 $21,433.47
3 $25,000 -r (1.08)3 $19,846.81
4 $25,000 (1.08)4 $18,375.75
5 $25,000 -f (1.08)5 $17,014.58
TOTAL $99,818.76
Table 8.2 Present Value of Future Payments
We begin with the amount calculated in our table, $99,818.76. Before any money is withdrawn, a year's worth of interest at 8% is compounded and added to our balance. Then our first $25,000 is withdrawn, leaving us with $82,804.26. This process continues until the end of five years, when, aside from a minor rounding difference, the fund has "done its job" and is equal to zero. However, we can make this simpler. Because each payment withdrawn (or added, as we will see later) is the same, we can calculate the present value of an annuity in one step using an equation. Rather than the multiple steps above, we will use the following equation:
PVa - PYMT X --v ' J
r
where PVa is the present value of the annuity and PYMT is the amount of one payment.
In this example, PYMT is $25,000 at the end of each of five years. Note that the greater the number of periods and/or the size of the amount borrowed, the greater the chances of large rounding errors. We have used six decimal places in our calculations, though the actual time value of money factor, combining interest and time, can be much longer. Therefore, our solutions will often use ^ rather than the equal sign.
By substitution, and then following the proper order of operations:
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8.2 • Annuities 233
PVa = $25,000 x -i-(10+8°-08)5
l
PVa = $25,000 v r 1.469328) * 0.08
PVa = $25,000 v 1 - 0.680583 * 0.08
PVa = $25,000 v 0.319417 a 0.08
PVa = $25,000 x 3.9927125
PVa « $99,817. 81
In both cases, barring a rounding difference caused by decimal expansion, we come to the same result using the equation as when we calculate each of multiple years. It's important to note that rounding differences can become significant when dealing with larger multipliers, as in the financing of a multimillion-dollar machine or facility. In this text, we will ignore them.
In conclusion, five payments of $25,000, or $125,000 in total, can be funded today with $99,817.81, with the difference being obtained from interest always accumulating on the remaining balance at 8%. The running balance is obtained by calculating the year's interest on the previous balance, adding it to that balance, and subtracting the $25,000 that is withdrawn on the last day of the year. In the last (fifth) year, just enough interest will accrue to bring the balance to the $25,000 needed to complete the fifth payment.
A common use of the PVa is with large-money lotteries. Let's assume you win the North Dakota Lottery for $1.2 million, and they offer you $120,000 per year for 10 years, beginning one year from today. We will ignore taxes and other nonmathematical considerations throughout these discussions and problems. The Lottery Commission will likely contact you with an alternative: would you like to accept that stream of payments ... or would you like to accept a lump sum of $787,000 right now instead? Can you complete a money-based analysis of these alternatives? Based purely on the dollars, no, you cannot. The reason is that you can't compare future amounts to present amounts without considering the effect of time—that is, the time value of money. Therefore, we need an interest rate that we can use as a discounting factor to place these alternatives on the same playing field by expressing them in terms of today's dollars, the present value. Let's use 9%. If we discount the future stream of fixed payments (an ordinary annuity, as the payments are identical and they begin one year from now), we can then compare that result to the cash lump sum that the Lottery Commission is offering you instead.
By substitution, and following the proper order of operations:
1
PVa = $120,000 x ±-(10+0°'09)
I
l
2.3673637 ,
0.09
PVa = $120,000 x PVa =$120,000 x l~X™m PVa = $120,000 x °-5^7059892 PVa = $120,000 x 6.4176578 PVa «$770,119
All things being equal, that expected future stream of ten $120,000 payments is worth approximately $770,119 today. Now you can compare like numbers, and the $787,000 cash lump sum is worth more than the discounted future payments. That is the choice one would accept without considering such aspects as taxation, desire, need, confidence in receiving the future payments, or other variables.
234 8 • Time Value of Money II: Equal Multiple Payments
Calculating the Present Value of an Annuity Due
Earlier, we defined an ordinary annuity. A variation is the annuity due. The difference between the two is one period. That's all—just one additional period of interest. An ordinary annuity assumes that there is a one-period lag between the start of a stream of payments and the actual first payment. In contrast, an annuity due assumes that payments begin immediately, as in the lottery example above. We would assume that you would receive the first annual lottery check of $120,000 immediately, not a year from now. In summary, whether calculating future value (covered in the next section) or present value of an annuity due, the one-year lag is eliminated, and we begin immediately.
Since the difference is simply one additional period of time, we can adjust for this easily by taking the formula for an ordinary annuity and multiplying by one additional period. One more period, of course, is (1 + /). Recall from Time Value of Money I that the formula for compounding is (1 + i)N, where / is the interest rate and N is the number of periods. The superscript /vdoes not apply because it represents 1, for one additional period, and the power of 1 can be ignored. Therefore, faced with an annuity due problem, we solve as if it were an ordinary annuity, but we multiply by (1 + /) one more time.
In our original example from this section, we wished to withdraw $25,000 each year for five years from a fund that we would establish now. We determined how much that fund should be worth today if we intend to receive our first payment one year from now. Throughout this fund's life, it will earn 8% annually. This time, let's assume we'll withdraw our first payment immediately, at point zero, making this an annuity due. Because we're trying to determine how much our starting balance should be, it makes sense that we must begin with a larger number. Why? Because we're pulling our first payment out immediately, so less money will remain to start compounding to the amount we need to fund all five of our planned payments! Our rule can be stated as follows:
Whether one is calculating present value or future value, the result of an annuity due must always be larger than that of an ordinary annuity, all other facts remaining constant. Here is the stream of solutions for the example above, but please notice that we will multiply by (1 + /), one additional period, following the same order of operations:
PVa = $25,000 X
(1 + 0.08)5
0.08 1
x (1 + 0.08)
PVa = $25,000 X ^-328 ' X 1.08
PVa = $25,000 X 1 ~ %p^0583 * 1.08 PVa = $25,000 X 03Q1^17 X 1.08 PVa = $25,000 X 3.9927125 X 1.08 PVa a $107,803.24
That's how much we must start our fund with today, before we earn any interest or draw out any money. Note that it's larger than the $99,817.81 that would be required for an ordinary annuity. It must be, because we're about to diminish our compounding power with an immediate withdrawal, so we have to begin with a larger amount.
We notice several things:
1. The formula must change because the annual payment is subtracted first, prior to the calculation of annual interest.
2. We accomplish the same result, aside from an insignificant rounding difference: the fund is depleted once the last payment is withdrawn.
3. The last payment is withdrawn on the first day of the final year, not the last. Therefore, no interest is earned during the fifth and final year.
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8.2 • Annuities 235
To reinforce this, let's use the same approach for our lottery example above. Reviewing the facts, you have a choice of receiving 10 annual payments of your $1.2 million winnings, each worth $120,000, and you discount at a rate of 9%. The only difference is that this time, you can receive your first $120,000 right away; you don't have to wait a year. This is now an annuity due. We solve it just as before, except that we multiply by one additional period of interest, (1 + /):
PVa = $120,000 x ±-(10+09 09)10 x 1.09
(l__I_)
PVa = $120,000 x -2q3oJ3637 x 1-09
PVa = $120,000 x 1 " °64o94108 x 109
PVa = $120,000 x a5™2 x 1.09
PVa = $120,000 x 6.4176578 x 1.09
PVa « $839,429
Again, this result must be larger than the amount we determined when this was calculated as an ordinary annuity.
The calculations above, representing the present values of ordinary annuities and annuities due, have been presented on an annual basis. In Time Value of Money I. we saw that compounding and discounting calculations can be based on non-annual periods as well, such as quarterly or monthly compounding and discounting. This aspect, quite common in periodic payment calculations, will be explored in a later section of this chapter.
Calculating Annuities Used in Structured Settlements
In addition to lottery payouts, annuity calculations are often used in structured settlements by attorneys at law. If you win a $450,000 settlement for an insurance claim, the opposing party may ask you to accept an annuity so that they can pay you in installments rather than a lump sum of cash. What would a fair cash distribution by year mean? If you have a preferred discount rate (the percentage we all must know to calculate the time value of money) of 6% and you expect equal distributions of $45,000 over 10 years, beginning one year from now, you can use the present value of an annuity formula to compare the alternatives:
PVa = PYMT x
1 -
1
(1 + rf
By substitution:
PVa - $45,000 x
PVa = $45,000 x
PVa = $45,000 x
PVa = $45,000 x
PVa « $331,204
(1 + 0.06)10
0.06
(* 1.790848)
0.06
0.06
If the opposing attorney offered you a lump sum of cash less than that, all things equal, you would refuse it; if
236 8 • Time Value of Money II: Equal Multiple Payments
the lump sum were greater than that, you would likely accept it.
What if you negotiate the first payment to be made to you immediately, turning this ordinary annuity into an annuity due? As noted above, we simply multiply by one additional period of interest, (1 + 0.06). Repeating the last step of the solution above and then multiplying by (1 + 0.06), we determine that
PVa = $45,000 X 7.360089 X 1.06 PVa « $351,076
You would insist on that number as an absolute minimum before you would consider accepting the offered stream of payments.
To further verify that ordinary annuity can be converted into an annuity due by multiplying the solution by one additional period's worth of interest before applying the annuity factor to the payment, we can divide the difference between the two results by the value of the original annuity. When the result is expressed as a percent, it must be the same as the rate of interest used in the annuity calculations. Using our example of an annuity with five payments of $25,000 at 8%, we compare the present values of the ordinary annuity of $99,817.81 and the annuity due of $107,803.24.
$107,803.24 - $99,817.81
$99,817.81
= 0.08 = 8%
The result shows that the present value of the annuity due is 8% higher than the present value of the ordinary annuity.
Calculating the Future Value of an Annuity
In the previous section, we addressed discounting a periodic stream of payments from the future to the present. We are also interested in how to project the future value of a series of payments. In this case, an investment may be made periodically. Keeping with the definition of an annuity, if the amount of periodic investment is always the same, we may take a one-step shortcut to calculate the future value of that stream by using the formula presented below:
(1 + r)N
FVa = PYMT X --—
1
where FVa is the future value of the annuity, PYMT is a one-time payment or receipt in the series, r is the interest rate, and n is the number of periods.
As we did in our section on present values of annuities, we will begin with an ordinary annuity and then proceed to an annuity due.
Let's assume that you lock in a contract for an investment opportunity at 4% per year, but you cannot make the first investment until one year from now. This is counterintuitive for an investor, perhaps, but because it is the basis of the formula and procedures for ordinary annuities, we will accept this assumption. You plan to invest $3,000 at the end of each year. How much money will you have at the end of five years?
Let's start by placing this on a timeline like the one appearing earlier in this chapter (see Table 8.3):
Year 0 1 2 3 4 5
Balance Forward ($) 0.00 0.00 3,000.00 6,120.00 9,364.80 12,739.39
Interest Earned ($) 0.00 120.00 244.80 374.59 509.58
Principal Added ($) 3,000.00 3,000.00 3,000.00 3,000.00 3,000.00
New Balance ($) 3,000.00 6,120.00 9,364.80 12,739.39 16,248.97
Table 8.3
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8.2 • Annuities 237
As we explained earlier when describing ordinary annuities, the payment for year 1 is not invested until the last day of that year, so year 1 is wasted as a compounding opportunity. Therefore, the amount only compounds for four years rather than five. Also, our fifth payment is not made until the last day of our contract in year 5, so it has no chance to earn a compounded future value. The investor has lost on both ends. In the table above, we have made five calculations, and for a longer-term contract such as 10, 25, or 40 years, this would be tedious. Fortunately, as with present values, this ordinary annuity can be solved in one step because all payments are identical.
Repeating the formula, and then by substitution:
FVa = PYMT x (1 + r)* ~ 1 FVa = $3,000 x (1 + ~ 1
FVa = $3,000 x 121q6m ~ 1 FVa = $3,000 x 5.416325 FVa = $16,248.98
This proof emphasizes that year 1 is wasted, with no compounding because the payment is made on the last day of year 1 rather than immediately. We lose compounding through this ordinary annuity in another way: year 5's investment is made on the last day of this five-year contract and has no chance to accumulate interest. A more intuitive method would be to enter a contract for an annuity due so that our first payment can be made immediately. In this way, we don't waste the first year, and all five payments work in year 5 as well. As stated previously, this means that annuities due will yield larger results than ordinary annuities, whether one is discounting (PVa) or compounding (FVa).
Let's hold all facts constant with the previous example, except that we will invest at the beginning of each year, starting immediately upon locking in this five-year contract. We follow the same technique as in the present value section: we multiply by one additional period to convert this ordinary annuity factor into a factor for an annuity due. Whether one is solving for a future value or a present value, the result of an annuity due must always be larger than an ordinary annuity. With future value, we begin investing immediately, so the result will be larger than if we waited for a period to elapse. With present value, we begin extracting funds immediately rather than letting them work for us during the first year, so logically we would have to start with more.
Continuing our example but converting it to an annuity due, we will multiply by one additional period, (1 + /). All else remains the same:
FVa = $3,000 x (1 + ~ 1 x 1.04 FVa = $3,000 x 121^ ~ 1 x 1.04 FVa = $3,000 x 5.416325 x 1.04 FVa a $16,898.93
Let's provide one additional example of each. Assume that you have a chance to invest $15,000 per year for 10 years, earning 8% compounded annually. What amount would you have after the 10 years? If we can only make our first payment at the end of each year, our ending value will be
FVa = $15,000 x (1 + 0o0o8° " 1
FVa = $15,000 x
2.158925 - 1 0.08
FVa =$15,000 x 14.486563 FVa « $217,298
However, if we can make our first payment immediately and then make subsequent payments at the start of
238 8 • Time Value of Money II: Equal Multiple Payments
each following year, we modify the formula above by multiplying the annual payment by one additional period:
FVa = $15,000 X 14.486563 X 1.08 FVa « $234,682
Begin an Investing Program at Age 20 or 30?
In this chapter's introductory section. Why It Matters, we posed a question about pledging to invest $1,000 each year until you reach age 60. If you can earn a 5% annual rate of interest, how much will you have if you begin at age 20? What if you delay this program until age 30? The additional 10 years can make a surprisingly large difference. How can you calculate that difference?
Solution:
Perform two separate calculations comparable to the chapter examples above, using the formula for the future value of an ordinary annuity. You plan to make the first investment immediately, making this an annuity due, so you will multiply by one additional period, (1 + 0.05). Notice that the only difference between the two calculations is the exponent N, representing the number of periods.
Thirty years (starting at age 30):
FVa = $1,000 X (1 + 000o5° " 1 X 1-05 FVa =$1,000 X 66.438848 X 1.05 FVa « $69,761
Forty years (starting at age 20):
FVa = $1,000 X (1 + 0o0o54° ~ 1 x L05 FVa = $1,000 X 120.799774 x 1.05 FVa « $126,840
Waiting 10 years before committing to this program comes with a surprisingly high cost—a loss of almost 82% of the potential value.
How Annuities Are Used for Retirement Planning
On a final note, how might annuities be used for retirement planning? A person might receive a lump-sum windfall from an investment, and rather than choosing to accept the proceeds, they might decide to invest the sum (ignoring taxes) in an annuity. Their intention is to let this invested sum produce annual distributions to supplement Social Security payments. Assume the recipient just received $75,000, again ignoring tax effects. They have the chance to invest in an annuity that will provide a distribution at the end of each of the next five years, and that annuity contract provides interest at 3% annually. Their first receipt will be one year from now. This is an ordinary annuity.
We can also solve for the payment given the other variables, an important aspect of financial analysis. If the person with the $75,000 windfall wants this fund to last five years and they can earn 3%, then how much can they withdraw from this fund each year? To solve this question, we can apply the present value of an annuity formula. This time, the payment (PYMT) is the unknown, and we know that the PVa, or the present value that they have at this moment, is $75,000:
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8.3 • Loan Amortization 239
PVa = PYMT X
$75,000 = PYMT X PYMT X
1
(1 + r)"
(1 + 0.03)5
0.03
$75,000:
$75,000 = PYMT : PYMT :
i1 1.159274)
0.03
PYMT X 4.579705 $75,000 4.579705 $16,376.60
The person can withdraw this amount every year beginning one year from now, and when the final payment is withdrawn, the fund will be depleted. Interest accrues each year on the beginning balance, and then $16,376.60 is withdrawn at the end of each year.
I
LINK TO LEARNIN
Another View of Annuities
Many examples of annuities are available, with presentations as varied as the opinions as to how appropriate they are for investors, especially retirees. Math Is Fun (https://0penstax.0rg/r/Math Is Fun) is particularly interesting and potentially helpful for understanding how to apply this knowledge.
ED
8.3 | Loan Amortization
Learning Outcomes
By the end of this section, you will be able to:
■ Distinguish between different types of loans.
■ Explain how amortization works.
■ Create an amortization schedule.
■ Calculate the cost of borrowing.
Types of Loans
Funds can be loaned to businesses of any type, including corporations, partnerships, limited liability companies, and proprietorships. Bankers often refer to these lending structures as facilities, and they can be tailored to the specific needs of the borrower in a number of ways. Similarly, lenders develop loans and lines of credit for individuals. Whether for a business or an individual, the purpose of the loan, method of repayment, interest rate, specific terms, and time involved must all be tailored to the goals of the borrower and the lender. In this chapter, we will focus on fixed-rate loans, although other alternatives exist.
Typical business loans include the following:
■ Term loans generally bear a maturity date and a set rate of interest and are typically used to finance investments in assets such as equipment, buildings, and possibly other acquired firms. The length of the term loan is generally designed to match the useful life of the asset being financed, and it will usually be repaid on a monthly schedule. It's common for a term loan to be backed by collateral, such as the asset itself or other assets of the business.
■ Revolving lines of credit (revolvers), are used to finance the short-term working capital needs of a business. Revolvers will have a specific maximum but no set schedule of monthly payments. Interest accrues on the amount of cash that a company has drawn down from the facility. These credit lines may be secured by accounts receivable, inventory, other assets of the business, or sometimes simply the good
240 8 • Time Value of Money II: Equal Multiple Payments
faith and credit of the company if the firm is strong, creditworthy, and established with the lender. Revolvers must often be fully repaid and unused for a short period of time to assure the lender that the borrower is not using this facility for longer-term needs.
Personal loans also come in several types, designed for the purpose the borrower (consumer) has in mind, with assistance from the lender in determining the appropriate structure:
■ Personal lines of credit are similar to lines of credit on bank cards, with interest being charged on the outstanding balance of the credit line. These are available on the basis of personal credit scores, with data being supplied by the three best-known credit reporting firms: Experian, Equifax, and TransUnion. Individuals should check their scores with each of these companies at least once per year, which they can do for no charge. Additional requests from the same company require a small fee.
■ An unsecured personal loan is an installment loan, initially drawn for a fixed amount and repaid on a periodic schedule with interest, as we have seen in our annuity examples. Unsecured means that the loan is not secured by collateral but is instead based on the strong credit history of the borrower.
■ In contrast, a secured personal loan has an asset backing up the unpaid amount, and if the consumer defaults on the debt, the asset can be seized by the lender to satisfy their claim. A common example is an auto loan, which is secured by the car being purchased; nonpayment or default on the loan can lead to the borrower's car being repossessed.
■ A mortgage loan is another type of secured personal loan, but for a longer period, such as 20, 25, or even 30 years. The home being purchased or built is the collateral, and the home may be foreclosed upon if the borrower defaults. Full title to the home typically remains with the lender as long as an unpaid balance remains on the debt.
■ Student loans are borrowings intended to fund college or career education, and they can come from a financial institution or the federal government. Interest rates on these loans are generally low and advantageous, and repayment does not begin until after the borrower's education is complete (or if they drop below a certain level of time status, such as becoming a half-time student).
Calculating Loan Payments Using Simple Amortization
Loan amortization refers to a schedule of how and when a debt will be repaid with interest. As noted, we will focus on fixed-rate debts, such as auto loans, personal loans with installment payments, or mortgages. Before entering into a borrowing agreement, the borrower can use any of a number of tools to verify the terms being offered, such as the monthly payment on a car loan financed by the dealer. In many cases, this is accomplished by using the present value of an annuity formula:
PVa = PYMT X --v ' J
r
We've already reviewed the present value of ordinary annuities in several examples. Before we look into business or consumer loans and their repayment, we must review an area of Time Value of Money I.
We're not likely to make annual payments on a home mortgage or auto loan, as these are commonly paid on a monthly basis. Fortunately, our formulas are easily adjusted from annual to non-annual periods. You will recall that we solve for non-annual periods in the same way, with two adjustments: (1) we divide the annual interest rate by the number of periods in the year, and (2) we multiply the time periods by the number of those periods within a year. Therefore, in the case of monthly debt service, including interest and principal, we use 12 periods.
Given a three-year car loan at 6%, rather than using 6% and 3 periods in our formula, we would instead use 0.5% (6% 12) and 36 periods (3 years x 12), and then apply the present value of an annuity formula in the same way. Let's say the three-year, 6% auto loan is for $32,000. You need to know if you can squeeze the monthly payment into your budget. For our examples, we will ignore any other charges, fees, taxes, or extras that your lender might include in these payments, and we will focus only on interest and principal repayment.
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8.3 • Loan Amortization 241
You will make the first payment one month from now, making this an ordinary annuity. What is the amount of your monthly debt service? In this case, you would be solving for a different unknown: the payment amount.
By substitution into the present value of an annuity formula, adjusting for monthly payments as noted:
$32,000 = PYMT X --(10+oo5005)36
i1 ~ 1.196681 )
0.005
$32,000 = PYMT X $32,000 = PYMT x 1 "oo^645 $32,000 = PYMT x °-^555 $32,000 = PYMT X 32.871
Dividing both sides by 32.781 to isolate the payment amount (PYMT) gives us
PYMT - $32'000 riivii - 328?1
PYMT « $973.50
Solving for the payment, we find that it's approximately $973.50 per month. You consult your monthly budget and find that you can cover this monthly payment, so you conclude the deal. Ask the salesperson for the amortization table on this debt to show how your 36 payments of $973.50 will cover your interest plus repayment of the principal amount of the debt. At this point, you know how to complete your own table. Using a financial calculator or Microsoft Excel simplifies the operation above to a few keystrokes, as presented later in this chapter.
Two extracts from an amortization table are shown in Table 8.4.
Month Payment Interest Principal Remaining Balance
1 973.50 160.00 813.50 31,186.50
2 973.50 155.93 817.57 30,368.93
3 973.50 151.84 821.66 29,547.27
4 973.50 147.74 825.77 28,721.51
5 973.50 143.61 829.89 27,891.61
...continued...
33 973.50 19.23 954.27 2,891.54
34 973.50 14.46 959.04 1,932.50
35 973.50 9.66 963.84 968.66
36 973.50 4.84 968.66 0.00
Total 35,046.00 3,046.08 32,000.00
Table 8.4 Extracts from an Amortization Table ($)
This table resembles proofs we have seen of annuities, but let's focus on some details:
1. Each fixed payment contains both interest and principal repayment.
2. Because the payments are fixed and the amount of remaining debt is decreasing, the monthly interest portion is always decreasing, and the amount of principal payment therefore must be increasing.
We can conclude that the lender is making more of their revenue (interest) in the early months than in the later months. In addition, the debt is decreasing slowly in the early months and more rapidly in the later
242 8 • Time Value of Money II: Equal Multiple Payments
months. We can all agree that lenders are compensated for the risks they take earlier rather than later. Of the 36 payments of $973.50, $32,000 has been repaid as the principal borrowed. The remaining $3,046.08 is the lender's revenue, the cost of credit.
For an additional example, one that drives home the point that more interest is paid in the early months of a long-term loan, we will consider a 20-year home mortgage. Home mortgage payments are typically made monthly, and again, we will ignore additional charges by the lender, such as real estate tax and homeowner's insurance. Let's assume you buy a $200,000 home, pay $60,000 as a cash deposit, and will finance the remaining $140,000 over 20 years. The bank offers you a 3.6% annual interest rate. What will the amount of your monthly payment be for the interest and principal repayment? The bank will tell you, of course, but let's prove it for ourselves. We'll do it in exactly the same fashion as the car loan above, using the present value of an annuity formula. Remember that you are not financing the entire $200,000 purchase; you pay $60,000 in cash, so you are only financing the remaining $140,000.
We modify the periods from years to months by multiplying by 12, and we modify the annual rate to a monthly rate by dividing by 12, resulting in
n = 240 months r = 0.3%
By substitution into the present value of an annuity formula:
fi-
(1 + 0.003)240 0.003
$140,000 = PYMT X
(l__]—)
$140,000 = PYMT X ^-2g32220 ;
$140,000 = PYMT X 1 ~ooo37277 $140,000 = PYMTX °51207323 $140,000 = PYMT x 170.907667
We divide both sides by 170.907667 to isolate the payment amount (PYMT):
PYMT - $140'000 rimi 170.907667
PYMT « $819.16
Your monthly mortgage payment is $819.16. As in our auto loan example, we'll complete an amortization table of our own—though, of course, you'll remember to ask your lender for their version. Extracts from a full 240-month table are shown in Table 8.5 below. The front-end packing of interest revenue is more obvious here because of the longer time period.
Month Payment Interest Principal Remaining Balance
140,000.00
1 819.16 420.00 399.16 139,600.84
2 819.16 418.80 400.36 139,200.48
3 819.16 417.60 401.56 138,798.92
4 819.16 416.40 402.76 138,396.16
5 819.16 415.19 403.97 137,992.19
6 819.16 413.98 405.18 137,587.01
7 819.16 412.76 406.40 137,180.61
Table 8.5 Amortization Table for a Mortgage ($)
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8.4 • Stated versus Effective Rates 243
Month Payment Interest Principal Remaining Balance
8 819.16 411.54 407.62 136,772.99
9 819.16 410.32 408.84 136,364.15
...continued....
236 819.16 12.17 806.99 3,250.84
237 819.16 9.75 809.41 2,441.44
238 819.16 7.32 811.84 1,629.60
239 819.16 4.89 814.27 815.33
240 819.16 2.45 816.71 (1.38)
Total 196,598.40 56,597.02 140,001.38 (Rounding)
Table 8.5 Amortization Table for a Mortgage ($)
As with your car loan, earlier payments contain more interest than loan repayment, so the lender's revenue is at a significant peak in the early years. The length of the loan, coupled with the frequent compounding, emphasizes this. In month 10, the interest and principal amounts "pass" each other, and now the loan balance is dropping at a quicker rate. Finally, note that you will pay more than $56,000 to finance this $140,000 borrowing. If you pay off this mortgage over 240 months as planned, the interest cost represents an additional 28% of the full cost of the home!
If the borrower has the means to make an accelerated payment against this debt—for example, due to a bonus or other windfall—doing so can make a significant difference in the total cost of financing over the life of the loan. Assume that after three years (month 36), you receive a bonus of $2,000 and decide to apply the entire amount to prepay the remaining balance. Your loan agreement allows you to apply the entire amount to the remaining unpaid balance of the mortgage. While this might seem equal to just 2.5 months' worth of payment, the debt is fully paid off almost 6 months ahead of schedule, and total interest is reduced from over $56,000 to $55,000. The ability to prepay long-term debts such as this is clearly worth negotiating initially.
8.4
Stated versus Effective Rates
Learning Outcomes
By the end of this section, you will be able to:
■ Explain the difference between stated and effective rates.
■ Calculate the true cost of borrowing.
The Difference between Stated and Effective Rates
If you look at the bottom of your monthly credit card statement, you could see language such as "The interest rate on unpaid balances is 1.5% per month." You might think to yourself, "So, that's 12 months times 1.5%, or 18% per year." This is a fine example of the difference between stated and effective annual interest rates. The effective interest rate reflects compounding within a one-year period, an important distinction because we tend to focus on annual interest rates. Because compounding occurs more than once per year, the true annual rate is higher than appears. Please remember that if interest is calculated and compounded annually, the stated and effective interest rates will be the same. Keep in mind that the following principles work whether you are the debtor paying off an obligation or an investor hoping for more frequent compounding. The dynamics of the time value of money apply in either direction.
Effective Rates and Period of Compounding
Let's remain with our example of a credit card statement that indicates an interest rate of 1.5% per month on
244 8 • Time Value of Money II: Equal Multiple Payments
unpaid balances. If you use this card only once, to make a $1,000 purchase in January, and then fail to pay the bill when it comes due, the issuer will bill you $15. Now you owe them $1,015. Assume you completely ignore this bill and never pay it throughout the rest of the year. The monthly calculation of interest starts to compound on past interest assessments in addition to the $1,000 initial purchase (see Table 8.6).
Month Interest Balance
1,000.00
1 15.00 1,015.00
2 15.23 1,030.23
3 15.45 1,045.69
4 15.69 1,061.36
5 15.92 1.077.28
6 16.16 1,093.44
7 16.40 1,109.84
8 16.65 1,126.49
9 16.90 1,143.39
10 17.15 1,160.54
11 17.41 1,177.95
12 17.67 1,195.62
Table 8.6 Compounded Interest on a Credit Card Statement ($)
Because interest compounds monthly rather than annually, the effective annual rate is 19.56%, not the intuitive rate of the stated 1.5% times 12 months, or 18%. Our basic compounding formula of (1+i)An by substitution shows:
(1 + 0.015)12 = 1.19562
To isolate the effective annual rate, we then deduct 1 because our interest calculations are based on the value of $1:
(1 + 0.015)12 - 1 = 1.19562 - 1 = 0.19562 = 19.562%
Therefore, it falls to the consumer/borrower to understand the true cost of borrowing, especially when larger dollar amounts are involved. If we had been dealing with $10,000 rather than $1,000, the annual difference would be more than $156.
A Helpful Demonstration ...
From the Corporate Finance Institute (https://openstax.Org/r/Corporate Finance Institute) comes a fine visual of a similar example. Here, we see the effective annual rate that results from taking a nominal annual rate of 12%, with a benefit to an investor if they have the benefit of monthly compounding.
One example of the importance of understanding effective interest rates is an invention from the early 1990s:
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8.4 • Stated versus Effective Rates 245
the payday advance loan (PAL). The practice of offering such loans can be controversial because it can lead to very high rates of interest, perhaps even illegally high, in an act known as usury. Although some states have outlawed PALs and others place limits on them, some do not. A PAL is a short-term loan in anticipation of a person's next paycheck. A person in need of money for short-term needs will write a check on Thursday but date the check next Thursday, which is their normal payday; assume this transaction is for $200. The lender, typically operating from a storefront, will advance the $200 cash and hold the postdated check. The lender charges a fee—let's say $14—as their compensation. The following Thursday, the borrower is expected to pay off the advance, and if they do not, the lender can deposit the postdated check. If that check has insufficient funds, more fees and penalties will likely be assessed.
One primary reason that arrangements such as these are controversial is the excessively high nominal (stated) interest rate that they can represent. For a one-week loan of $200, the borrower is paying $14, or 7% of the borrowed amount. If this is annualized, with 52 seven-day periods in a year, the stated rate is 364%! While a PAL might seem to be an effective immediate solution to a cash shortfall, the mathematics behind the true cost of borrowing simply do not make sense, and a person who uses such arrangements regularly is placing themselves at a dreadful financial disadvantage.
How Tempting Is That Refund Anticipation?
Refund anticipation loans (RALs) began in 1987, and they are still available (though not from banks) and used by millions of people. Now, RALs come from private lending chains. These loans allow you to determine your April 15 personal income tax liability through a preparer and receive an advance against your expected refund. But beware: your ability to analyze the true cost of money is always critical. Like all loans, RALs bear a rate of interest. Let's assume that the firm that prepared your tax return determines that you're entitled to an $800 refund. Once they advance that amount to you, it will bear interest at a certain rate; we'll assume 0.5% per week. You might expect a tax refund in four weeks. Half a percent of $800 doesn't sound like much, but what happens when you annualize it into an effective rate, assuming your tax refund arrives exactly four weeks from when you accept the loan? Assume no compounding during those four weeks.
Solution:
A weekly rate of 0.5% on the $800 advance is $4 per week, so for four full weeks, you've paid $16 for the use of $800. Of course, that totals 2% of the amount advanced. There are 13 four-week periods in a year, so even though the interest rate appears to be small, it amounts to 26% when annualized! We assumed no compounding to keep the illustration simple, but we further assume that you are not using this advance throughout the year. If you were, then periodic compounding would drive the effective rate even higher, to just over 29.3%.
1 Michelle Singletary. "Another Reason Not to Opt for a Tax Refund Loan: It May Delay Your Next Stimulus Payment." Washington Post, February 16, 2021. https://www.washingtonpost.com/business/2021/02/16/tax-refund-loan-problems/
2 Ameliajosephson. "What Is a Refund Anticipation Loan?" SmartAsset. March 18, 2021. https://smartasset.com/taxes/what-is-a-refund-anticipation-loan
246 8 • Time Value of Money II: Equal Multiple Payments
ED
8.5 | Equal Payments with a Financial Calculator and Excel
Learning Outcomes
By the end of this section, you will be able to:
■ Use a financial calculator and Excel to solve perpetuity problems.
■ Use a financial calculator and Excel to solve annuity problems.
■ Calculate an effective rate of interest.
■ Schedule the amortization of a loan repayment.
Solving Time Value of Money Problems Using a Financial Calculator
Since the 1980s, many convenient and inexpensive tools have become available to simplify business and personal calculations, including personal computers with financial applications and handheld/desktop or online calculators with many of the functions we've studied already. This section will explore examples of both, beginning with financial calculators. While understanding and mastery of the use of time value of money equations are part of a solid foundation in the study of business and personal finance, calculators are rapid and efficient.
We'll begin with the constant perpetuity that we used to illustrate the constant perpetuity formula. A share of preferred stock of Shaw Inc., pays an annual $2.00 dividend, and the required rate of return that investors in this stock expect is 7%. The simple technique to solve this problem using the calculator is shown in Table 8.7.
Step Description Enter Display
1 Set all variables to defaults 2ND [reset] enter RST 0.00
2 Enter formula 2 - 7 % = 28.57
Table 8.7 Calculator Steps to Find the Required Rate of Return3
Earlier we solved for the present value of a 5-year ordinary annuity of $25,000 earning 8% annually. We then solved for an annuity due, all other facts remaining the same. The two solutions were $99,817.50 and $107,802.50, respectively. We enter our variables as shown in Table 8.8 to solve for an ordinary annuity:
Step Description Enter Display
1 Set all variables to defaults 2ND [reset] enter RST 0.00
2 Enter number of payments 5N N = 5.00
3 Enter interest rate per payment period 8I/Y l/Y = 8.00
4 Enter payment amount 25000 +/- pmt PMT = -25,000.00
5 Compute present value cpt pv PV = 99,817.75
Table 8.8 Calculator Steps to Solve for an Ordinary Annuity
Note that the default setting on the financial calculator is END to indicate that payment is made at the end of a period, as in our ordinary annuity. In addition, we follow the payment amount of $25,000 with the +/-keystroke—an optional step to see the final present value result as a positive value.
To perform the same calculation as an annuity due, we can perform the same procedures as above, but with two additional steps after Step 1 to change the default from payments at the end of each period to payments at the beginning of each period (see Table 8.9).
3 The specific financial calculator in these examples is the Texas Instruments BAII Plus™ Professional model, but you can use other financial calculators for these types of calculations.
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8.5 • Equal Payments with a Financial Calculator and Excel 247
Step Description Enter m Display
1 Set all variables to defaults 2nd [reset] enter RST 0.00
2 Change default to payment at end of period 2nd [bgn] 2nd [set] BGN 0.00
3 Return to calculator mode 2nd [quit] 0.00
4 Enter number of payments 5N N = 5.00
5 Enter interest rate per payment period 8 i/y I/Y = 8.00
6 Enter payment amount 25000 +/- pmt PMT = -25,000.00
7 Compute present value cpt pv PV = 107,803.17
Table 8.9 Calculator Steps to Solve for an Annuity Due
The procedures to find future values of both ordinary annuities and annuities due are comparable to the two procedures above. We begin with the ordinary annuity, with reminders that this is the default for the financial calculator and that entering the payment as a negative number produces a positive result (see Table 8.10).
Step Description Enter Display
1 Set all variables to defaults 2nd [reset] enter RST 0.00
2 Enter number of payments 5N N = 5.00
3 Enter interest rate per payment period 4 i/y I/Y = 4.00
4 Enter payment amount 3000 +/- pmt PMT= -3,000.00
5 Compute future value cpt fv FV= 16,248.97
Table 8.10 Calculator Steps to Find the Future Value of an Ordinary Annuity
Solving for an annuity due with the same details requires the keystrokes listed in Table 8.11
Step Description Enter H splay
1 Set all variables to defaults 2nd [reset] enter RST 0.00
2 Change default to payment at end of period 2nd [bgn] 2nd [set] BGN 0.00
3 Return to calculator mode 2nd [quit] 0.00
4 Enter number of payments 5n N = 5.00
5 Enter interest rate per payment period 4 i/y I/Y = 4.00
6 Enter payment amount 3000 +/- pmt PMT = -3,000.00
7 Compute future value cpt fv FV = 16,898.93
Table 8.11 Calculator Steps to Find the Future Value of an Annuity Due
Earlier in the chapter, we explored the effect of interannual compounding on the true cost of money, recalling the basic compounding formula:
We saw that when modified for monthly compounding at a stated rate of 1.5%, the actual (effective) rate of interest per year was 19.56%. One simple way to prove this is by using the calculator keystrokes listed in Table
248 8 • Time Value of Money II: Equal Multiple Payments
8.12.
Step Description Enter Display
1 Set all variables to defaults 2ND [RESET] ENTER RST 0.00
2 Set the display to four decimal places 2ND [FORMAT] 4 ENTER DEC = 4.0000
3 Return to calculator mode 2ND [QUIT] 0.0000
4 Enter (1 + the monthly interest rate) 1.015 Yx 1.0150
5 Enter the number of months 12 Yx 1.1956
Table 8.12 Calculator Steps to Prove the Actual Rate of Interest per Year
Had we assumed that the stated monthly interest rate of 1.5% could be simply multiplied by 12 months for an annual rate of 18%, we would be ignoring the effect of more frequent compounding. As indicated above, the annual interest on the money that we spent initially, accumulating at a rate of 1.5% per month, is 19.56%, not 18%:
1.1956 - $1 spent intialry = 0.1956 = 19.56%
The final example in this chapter will represent the amortization of a loan. Using a 36-month auto loan for $32,000 at 6% per year compounded monthly, we can easily find the monthly payment and the amortization of this loan on our calculator using the following procedures and keystrokes.
First, we find the monthly payment (see Table 8.13).
Step Description Enter Display
1 Set all variables to defaults 2ND [RESET] ENTER RST 0.00
2 Set payments per year to 12 2ND [P/Y] 12 ENTER P/Y = 12.00
3 Return to calculator mode 2ND [QUIT] 0.00
4 Enter number of payments with the payment multiplier 3 2ND [xP/Y] N N = 36.00
5 Enter annual interest rate 6 I/Y I/Y = 6.00
6 Enter loan amount 32000 PV PV = 32,000.00
7 Compute the monthly payment CPT PMT PMT = -973.50
Table 8.13 Calculator Steps to Find the Monthly Payment of a Loan
We've verified the amount of our monthly debt service, including both the interest and repayment of the principal, as $973.50. The next step with our calculator is to verify our amortization at any point (see Table 8.14).
Jtep Description Enter isplay
1 Set previous work as an amortization worksheet 2ND [AMORT] P1 = 1.00
2 Set beginning period to 1 1 ENTER P1 = 1.00
3 Set ending period to 12 I 12 ENTER P2 = 12.00
4 Display amortization data at the end of month 12 i BAL = 21,965.02
5 I PRN = -10,034.98
Table 8.14 Calculator Steps to Verify Amortization at the End of One Year
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8.5 • Equal Payments with a Financial Calculator and Excel 249
Description
6 i Table 8.14 Calculator Steps to Verify Amortization at the End of One Year
Display
INT= -1,647.02
Without resetting the calculator, we will try a second example, this time reviewing the second full year of amortization at the end of 24 months (see Table 8.15).
Step Description Enter Display
1 Set previous work as an amortization worksheet 2nd [amort] P1 = 1.00
2 Change beginning period to month 13 13 enter P1 = 13.00
3 Change ending period to month 24 i 24 enter P2 = 24.00
4 Display amortization data at the end of month 24 i BAL = 11,311.13
5 i PRN = -10,653.89
6 I INT = -1,028.11
Table 8.15 Calculator Steps to Verify Amortization at the End of Two Years
Solving Time Value of Money Problems Using Excel
Microsoft's popular spreadsheet program Excel is arguably one of the most common and powerful numeric and data analysis products available. Yet while mastery of Excel requires extensive study and practice, enough basics can be learned in two or three hours to provide the user with the ability to solve problems quickly and conveniently, including extensive financial capability. Most of the calculations in this chapter were prepared with Excel.
The boxes in the Excel gridwork, known individually as cells (located at the intersection of a column and a row), can contain numbers, text, and very powerful formulas (or functions) for calculations and data analytics. Cells, rows, columns, and groups of cells (ranges) are easily moved, formatted, and replicated. In the mortgage amortization table for 240 months seen in Section 8.3.2, only the formulas for month 1 were typed in. With one simple command, that row of formulas was replicated 239 more times, with each line updating itself with relevant number adjustments automatically. With some practice, a long table such as that can be constructed by even a relatively new user in less than 10 minutes.
In this section, we will illustrate how to use Excel to solve problems from earlier in the chapter, including perpetuities, ordinary annuities, effective interest rates, and loan amortization. We will omit the basic dynamics of an Excel spreadsheet because they were presented sufficiently in preceding chapters.
Revisiting the constant perpetuity from Section 8.1, in which our shares of Shaw Inc., preferred stock pay an annual fixed dividend of $2.00 and the required rate of return is 7%, we do not use an Excel function for this simple operation. The two values are entered in cells B3 and B4, respectively.
We enter a formula in cell B6 to perform the division and display the result in that cell. The actual contents of cell B6 are typed below it for your reference, in cell B8 (see Figure 8.2).
250 8 • Time Value of Money II: Equal Multiple Payments
a A E
1
2
3 Indefinite Dividend per Share ($) 2.00
4 Required Rate of Return (%) 0.07
6 Price per Share S 28.57
7 =B3/B4
Figure 8.2 Excel Spreadsheet for Valuing a Perpetuity
Download the spreadsheet file (https://0penstax.0rg/r/spreadsheet file Chapter08 finance) containing key Chapter 8 Excel exhibits.
To find the present value of an ordinary annuity, we revisit Section 8.2.1. You will draw $25,000 at the end of each year for five years from a fund earning 8% annually, and you want to know how much you need in that fund today to accomplish this. We accomplish this in Excel easily with the PV function. The format of the PV command is
=PV(rate,periods,payment,0,0)
Only the first three arguments inside the parentheses are used. We'll place them in cells and refer to those cells in our PV function. As an option, you could also type the numbers into the parentheses directly. Notice the slight rounding error because of decimal expansion. Also, the payment must be entered as a negative number for your result to be positive; this can be accomplished either by making the $25,000 in cell B5 a negative amount or by placing a minus sign in front of the B5 in the formula's arguments. In cell B3, you must enter the percent either as 0.08 or as 8% (with the percent sign). We repeated the formula syntax and the actual formula inputs in column A near the result, for your reference (see Figure 8.3).
a A B
1
2
3 Rate {%) 0.08
4 Periods 5
5 Payment (25,000)
6
7 =PV( rate, periods, payment, 0,0) S 99,817.75
8 =PV(B3,B4JB5,0,0)
9
Figure 8.3 Excel Spreadsheet Showing the Present Value of an Ordinary Annuity
We also found the present value of an annuity due. We use the same information from the ordinary annuity problem above, but you will recall that the first of five payments happens immediately at the start of year 1, not at the end. We follow the same procedures and inputs as in the previous example, but with one change to the PV function: the last argument in the parentheses will change from 0 to 1. This is a toggle switch that commands the PV function to treat this as an annuity due instead of an ordinary annuity (see Figure 8.4).
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8.5 • Equal Payments with a Financial Calculator and Excel 251
a A B
1
2
3 Rate [%) 0.08
4 Periods ~s1
5 Payment (25,000)
6
7 =PV(rate,periods,payment,0,l) $ 107,803.17
8 =PV( 83,84,65,0,1)
Figure 8.4 Excel Spreadsheet Showing the Present Value of an Annuity Due
Section 8.2 introduced us to future values. Comparable to the PV function above. Excel provides the FV function. Using the same information—$3,000 invested annually for five years, starting one year from now, at 4%—we'll solve using Excel (see Figure 8.5). The format of the command is
=FV(rate,periods,payment,0,0)
a A B i
1
2
3 Rate [%) 0.04
4 Periods 5
5 Payment (3,000)
6
7 =FV(rate,periods,payment,0,0) $ 16,248.97
8 =FV(B3,B4,B5,Q,D)
Figure 8.5 Excel Spreadsheet Showing the Future Value of an Ordinary Annuity
As with present values, using the same data but solving for an annuity due requires the fifth argument inside the parentheses to be changed from 0 to 1; all other values remain the same (see Figure 8.6).
a A B
1
2
3 Rate [%) 0.04
4 Periods 5
5 Payment (3,000)
6
7 =FV(rate,periods,payment,0,l) $ 16,898.93
8 =FV(B3,B4,B5,0,1)
Figure 8.6 Excel Spreadsheet Showing the Future Value of an Annuity Due
In Section 8.4. we explained the difference between stated and effective rates of interest to show the true cost of borrowing, in this case for a one-year period, if interest is compounded for periods within a year. The syntax for the Excel effect function to calculate this rate is
=EFFECT(rate,periods)
252 8 • Time Value of Money II: Equal Multiple Payments
where rate is the nominal rate and periods represents the number of periods within a year.
Earlier, our example showed that 1.5% compounded monthly results in not 18% per year but actually over 19.56% (see Figure 8.7).
a A B C
1
2
3 Nominal Annual Rate IS.00%
4
5
6 Period Type Periods per Year Effective Interest
7 Annual 1 18.00%
8 Quarterly 4 19.25%
9 Monthly 12 19.56%
Figure 8.7 Excel Spreadsheet Showing Effective Interest Rate
Note several things: First, the nominal interest rate is entered as a percent. Second, the actual effect function in C7 is typed as =EFFECT(rate,B7); we use the word rate because we actually assigned a name to cell B3, so Excel can use it in a function and replicate it without it changing. When cell C7 is replicated to C8 and C9, rate remains the same, but the formulas automatically adjust to use B8 and B9 for the periods.
To assign a name to a cell, keep in mind that every cell has column-row coordinates. We want cell B3 to be the anchor of our effective rate calculations. Rather than referring to cell B3, we can name it, and in this case, we use the name rate, which we can then use in formulas like any other Excel cell letter-number reference. Place the cursor in cell B3. Now, look at cell A1 on the grid: right above that cell, you see a box displaying B3, the current cursor location. If you click in that box and type "rate" (without the quotation marks), as we did, then hit the enter key, the value in that box will change to rate. Now, if you type "rate" (again, without quotation marks) into a formula. Excel knows to use the contents of cell B3.
Excel provides convenient tools for figuring out amortization. We'll revisit our 36-month auto loan for $32,000 at 6% per year, compounded monthly. A loan amortization table for a fixed interest rate debt is usually formatted as follows, with the Interest and Principal columns interchangeable:
Period Payment Interest Principal Balance
In Excel, a table is completed by using the function PMT. The individual steps follow.
1. List the information about the loan in the upper left of the worksheet, and create the column headings for the schedule of amortization. Type "B5" (without the quotation marks) in cell E9 to begin the schedule. Then enter 1 for the first month under the Payment # (or Month) column, in cell A10 (see Figure 8.8).
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8.5 • Equal Payments with a Financial Calculator and Excel 253
a A B c D E
Actual Use
2 Annual Interest Rate 6% 0.50%
3 Years 3 36
4 Payments per Year 12
5 Loan Amount $ 32,000
j_
7 Remaining
s Payment # (or Month) Payment Interest Principal Balance
$32,000.00
Figure 8.8 Step 1 of Creating an Amortization Table
2. Next, in cell B10, the payment is derived from the formula =PMT(rate,periods,pv), with PV representing the present value, or the loan amount. Because we are compounding monthly, enter C$2 and C$3 for the rate and periods, respectively. Cell B5 is used for the loan amount, but notice the optional minus sign placed in front of the entry B$5; this causes the results in the schedule to be displayed as positive numbers. The dollar sign ($) inserted in the cell references forces Excel to "freeze" those locations so that they don't attempt to update when we replicate them later; this is known in spreadsheet programs as an absolute reference (see Figure 8.9).
a A B c D E
1 Actual Use
2 Annual Interest Rate G% 0.50%
3 Years 3 36
4 Payments per Year 12
5 Loan Amount $ 32,000
6
7 Remaining
8 Payment # (or Month) Payment Interest Principal Balance
9 32,000.00
10 1 973.50
11 =PMT[C$2,C$3,-B$5)
Figure 8.9 Step 2 of Creating an Amortization Table
3. The next step is to calculate the interest. We take the remaining balance from the previous line, in this case cell E9, and multiply it by the monthly interest rate in cell C2, typing C$2 to lock in the reference. The remaining balance of the loan should always be multiplied by this monthly percentage (see Figure 8.10).
254 8 • Time Value of Money II: Equal Multiple Payments
a A B c D E
Actual Use
2 Annual Interest Rate 6% 0.50%
3 Years 3 36
4 Payments per Year 12
5 Loan Amount $ 32,000
±_
7 Remaining
a Payment # (or Month) Payment Interest Principal Balance
_±_ 32,000.00
10 973.50 160.00
11 =E9*C$2
Figure 8.10 Step 3 of Creating an Amortization Table
4. Because this is a fixed-rate loan, whatever is left from each payment after first deducting the interest represents principal, the amount by which the balance of the outstanding loan balance is reduced. Therefore, the contents of cell D10 represent B10, the total payment, minus C10, the interest portion (see Figure 8.11). No dollar signs are included because this cell reference can adjust to each row into which this formula is replicated, as will be seen in the following examples.
a A B c D E
1 Actual Use
2 Annual Interest Rate 6% 0.50%
3 Years 3 36
4 Payments per Year 12
5 Loan Amount $ 32,000
7 Remaining
8 Payment # (or Month) Payment Interest Principal Balance
32,000.00
10 973.50 160.00 B13.50
11 zBlO-CIO
Figure 8.11 Step 4 of Creating an Amortization Table
5. Because our principal portion of the last payment has reduced our outstanding balance, it is subtracted from the preceding balance in cell E9 (see Figure 8.12). The command therefore is =E9-D10.
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8.5 • Equal Payments with a Financial Calculator and Excel 255
a A B c D E
Actual Use
2 Annual Interest Rate 6% 0.50%
3 Years 3 36
4 Payments per Year 12
5 Loan Amount $ 32,000
j_
7 Remaining
s Payment # (or Month) Payment Interest Principal Balance
_±_ 32,000.00
10 973.50 160.00 813.50 31,186.50
11 =E9-D10
Figure 8.12 Step 5 of Creating an Amortization Table
Now that the first full row is defined, an amortization schedule is easily developed by Excel's replication abilities. Place the cursor on cell A10, hold down the left mouse button, and drag the cursor to cell E10. Cells A10 through E10 in row 10 should now be highlighted. Release the mouse button. Then "grab" the tiny square symbol at the bottom right of cell E10 and drag it downward as far as you need; in this case, you'll need 35 more rows because this is a 36-month loan, so it will end at row 45. We added a line for totals.
This is now a complete loan amortization schedule (see Figure 8.13). The first several periods display, followed by the last few periods, to prove that the schedule is complete (data rows for month 4 to month 22 are hidden).
a A B C D E
1 Actual Use
2 Annual Interest Rate 6% 0.50%
3 Years 3 36
4 Payments per Year 12
5 Loan Amount $ 32,000
6 7
Remaining
8 Payment # [or Month) Payment Interest Principal Balance
9 32,000.00
10 1 973.50 160.00 813.50 31,186.50
11 2 973.50 155.93 817.57 30,368.931
12 3 973.50 151.84 821.66 29,547.27
42 (continued)
43 33 973.50 19.23 954.27 2,891.54
44 34 973.50 14.46 959.04 1,932.50
45 35 973.50 9.66 963.84 968.66
46 36 973.50 4.84 968.66 0.00
47 Totals $ 35,046.07 $ 3,046.07 $ 32,000.00
Figure 8.13 Completed Amortization Schedule
This will look familiar; it's the same amortization table used as a proof in Section 8.3 (see Table 8.4). There is no rounding error because Excel uses the full decimal expansion in its calculations.
256 8 • Time Value of Money II: Equal Multiple Payments
This chapter has explored the time value of money by expanding on the concepts discussed in Time Value of Money I with additional funds being periodically added to or subtracted from our investment, either compounding or discounting them according to the situation. In all cases, the payments in the stream were identical. If they had not been identical, a separate set of operators would be required, and these will be addressed in the next chapter.
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8 • Summary 257
a Summary
8.1 Perpetuities
A perpetuity is an investment that is intended to provide an expected return indefinitely, either remaining constant or growing by an incremental amount. Preferred stock is a common example with a preestablished dividend formula. An indefinite stream of payments cannot be compounded into a future value, but it can be discounted to a present value, providing an opportunity to determine the amount an investor should be willing to pay for a share of that stock.
8.2 Annuities
An annuity is a stream of fixed periodic payments that is expected to be paid or received. Calculations of future value or present value are commonly performed on these payment streams for a wide number of reasons in business and personal financial analysis, as seen in the chapter focusing on single amounts, particularly in loan repayment. Annuities may be ordinary annuities, in which the first cash flow of a series occurs at the end of the first period, or annuities due, if the first cash flow occurs at the beginning point of the first period.
8.3 Loan Amortization
Loans are contracts between a lender and a borrower. Failure to observe the rules of that contract, such as payment of interest or repayment of the amount owed, can subject the borrower to substantial penalties as well as damage to their credit. Loan agreements bearing a fixed rate of interest have a scheduled amortization, or rate and time of repayments with interest. Several types of business and personal loans were described.
8.4 Stated versus Effective Rates
For a borrower to understand the true cost of financing, they must be familiar with interannual compounding, which can cause a stated interest rate that appears to be annual to actually be higher. The effective rate of interest was demonstrated to understand that true cost.
8.5 Equal Payments with a Financial Calculator and Excel
The use of two tools for managing and understanding the time value of money and its many applications was discussed: a professional financial calculator and the popular Microsoft Office Suite spreadsheet application Excel.
1? Key Terms
annuity a stream of regular, periodic payments to be received or paid
annuity due a stream of periodic payments in which the payment or receipt occurs at the beginning of each period
constant perpetuity a stream of periodic payments that is expected to continue indefinitely with no change
in the amount paid or received discount rate an interest rate used in time value of money calculations to determine present value; may
derive from several sources, such as stated contract rates, costs to borrow, or expected rates of return on
investments
effective interest rate the interest rate that results when compounding occurs multiple times within a year; the true cost of borrowing
growing perpetuity a stream of periodic payments that is expected to continue indefinitely with growth of
the amount paid or received in the future, usually by a fixed percentage loan amortization the scheduling of periodic repayment of a debt, typically involving regular payments or
receipts of amounts that include both interest payment and repayment of the principal of the amount owed lump sum a single cash payment made in lieu of a series of future payments, such as a lottery payout or a
258 8 -CFA Institute
legal settlement
ordinary annuity a stream of periodic payments in which the payment or receipt occurs at the end of each period
perpetuity a stream of periodic payments that is expected to continue indefinitely
preferred stock shares of ownership in a corporation that typically entitle the holder to a fixed dividend per share, if declared by the corporation, with priority over holders of that corporation's common stock
required rate of return the minimum amount of return that an investor will accept on an investment given the level of risk involved
retirement planning the process of determining one's objectives for retirement, including one's finances,
and developing strategies and tactics to achieve them structured settlements monetary legal settlements that are paid out in installments, such as an annuity,
rather than a lump sum cash amount
m CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session (https://0penstax.0rg/r/CFA Level I Study Session2). Reference with permission of CFA Institute.
a
Multiple Choice
1. The best example of a constant perpetuity would most likely be.
a. an annuity due
b. dividends from common stock
c. preferred stock
d. an ordinary annuity
2. You wish to endow a university chair of accounting for a salary of $100,000 per year to the recipient. The university will withdraw $100,000 each year for the recipient's salary. The amount of your gift will remain untouched indefinitely, in perpetuity. The university can lock in a fixed rate for your investment of 2.8% per year. In order to achieve this, what is the approximate amount of the gift you would have to make now?
a. $3,103,569
b. $3,571,429
c. $4,101,218
d. $4,227,827
3. Preferred stock in Blue Agate Inc. is issued for dividends of $3.00 per share. The dividends will increase each year at 0.178%, a growing perpetuity. The required rate of return on a stock such as this is 2.5%. At what approximate price will this preferred stock most likely sell today?
a. $117.21
b. $119.87
c. $120.00
d. $129.20
4. Julio's attorney negotiates a structured settlement after an injury, consisting of seven equal payments to Barry of $150,000 each. The first payment is due today, and the remaining payments will be received in annual amounts, with the second payment occurring one year from now. What is the approximate value of this settlement in today's dollars if Barry uses a discount rate of 5%?
a. $911,352
b. $867,960
c. $746,235
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8 • Multiple Choice 259
d. $1,050,000
5. What is the approximate present value of an ordinary annuity (beginning one year from now) of a stream of 12 annual payments of $87,000 if you use a discount rate of 6%?
a. $773,154.04
b. $747,278.92
c. $729,394.95
d. $718,974.58
6. If Maria invests $2,700 at the end of each six-month period for six years at an annual rate of 4%, what is the approximate future value of her ordinary annuity? Review Chapter 7 for the techniques of interannual compounding.
a. $17,909.10
b. $20,248.23
c. $31,755.54
d. $36,212.67
7. Assume all of the same facts as in exercise 6 above, except that Maria begins immediately and makes each of her payments at the beginning of each 6-month period instead of the end. What is the approximate future value of her annuity due at the end of the six years?
a. $17,909.10
b. $36,936.92
c. $32,707.24
d. $22,997.88
8. Rather than spending her $48,000 in casino winnings, Christy places the money in an investment that will earn her 5% per year, compounded annually. She will withdraw the money in four equal annual installments beginning one year from today. What must the approximate amount of each annual withdrawal be for this investment to be fully depleted in four years?
a. $11,136.38
b. $12,892.56
c. $12,243.47
d. $13,536.61
9. Your friend Jamal borrows $5,000 from you, agreeing to pay you back with 8% annual interest, with the first payment due to you one year from today. You ask that you be fully repaid over the next four years. However, to lower his annual payment, Jamal asks you to extend the period over five full years instead. What will be the approximate difference in his total payments to you, including interest and principal, if the debt is amortized over five years rather than four?
a. Jamal will pay $544 less.
b. Jamal will pay $544 more.
c. Jamal will pay $223 less.
d. Jamal will pay $223 more.
10. Your new El Supremo credit card arrangement indicates that you will owe interest on unpaid balances at a nominal (stated) rate of 1.2% per month. If the interest rate is compounded monthly, what is the approximate effective annual rate of interest?
a. 15.39%
b. 12.00%
c. 14.02%
260 8 • Problems
d. 14.40%
LH Problems
Use four decimal places on time value of money factors unless otherwise specified. Approximations and minor differences because of rounding are acceptable. Ignore the effect of taxes. Assume that all percentages are annual rates and that compounding occurs annually unless indicated otherwise.
1. Steve purchases preferred stock in Berklee Corporation, with each share paying a $2.50 dividend. This dividend will remain constant. If the public's required rate of return for Berklee stock is 8%, at what price should this company's stock sell?
2. Donna enters into an investment contract that will guarantee her 4% per year if she deposits $3,500 each year for the next 10 years. She must make the first deposit one year from today, the day she signs the agreement. How much will she have when she makes her last payment 10 years from now?
3. Assume the same facts as in problem 2 above, except that Donna negotiates the chance to make her first payment now and continue to pay at the beginning of each year for the 10-year period. How much will she have accumulated?
4. Bill will receive a royalty payment of $18,000 per year for the next 25 years, beginning one year from now, as a result of a book he has written. If a discount rate of 10 percent is applied, should he be willing to sell out his future rights now for $160,000? How about $162,500? $165,000?
5. Debbie won the $60 million lottery. She is to receive $1 million a year for the next 50 years beginning one year from now, plus an additional lump sum payment of $10 million after 50 years. The discount rate is 10 percent. How much cash would she need to be offered today to tempt her to take a lump-sum cash offer instead, all things equal?
6. Kim started a paper route on January 1, 2016. Every three months, she deposited $300 in her new bank account, which earned 4 percent annually but was compounded quarterly. On December 31, 2019, she placed the entire balance in her bank account in an investment that earned 5 percent annually. How much will she have on December 31, 2022?
7. You hire Thomas to work for you for five years, and you agree to put away enough money as a lump sum now to fund an annuity for him. At the end of those five years, he will retire and may begin drawing out $20,000 per year for five years, starting on the last day of each year (in this case, the end of year 6, from when this arrangement began, through year 10). How much must you invest today if your guaranteed interest rate is 3% compounded annually for all 10 years?
8. Your new boss doesn't have a pension or 401 (k) plan for your retirement, but she agrees to place aside $12,000 every year once a year for four years. She gives you the option of either starting immediately on your first day of work or starting one year from now. That makes this the difference between an ordinary annuity and an annuity due. If the plan earns 5% per year, compounded annually, what will be the difference between the two approaches after the four years / four payments?
9. Jada is borrowing $40,000 from you today. She agrees to pay you back in annual installments beginning a year from now over eight years, with interest at 3%. What would her annual payment amount be, including both interest and principal?
10. You agree to finance your new SUV with an auto loan of $38,000. This loan will be repaid over three years with monthly payments (and compounding) at a 4% annual interest rate (0.33% per month). What will your monthly loan payment be?
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8 • Video Activity 261
[►J Video Activity
Future Value of Ordinary Annuities
Click to view content (https://0penstax.0rg/r/Future Value of an Annuity)
1. What is the primary difference between this demonstration and our chapter examples, keeping the chapter "Time Value of Money I" in mind?
2. Explain the significance of Dr. van Biezen's comment at 4:32 regarding a difference when payments are made at the beginning of each pay period rather than the end.
Practical Example of Annuities
Click to view content (https://0penstax.0rg/r/What is an annuity?)
3. What is the primary difference between a fixed annuity and a variable annuity?
4. Annuities are often recommended to retirees and seniors. Why would a fixed annuity be more attractive to such a person than a variable annuity?
262 8 • Video Activity
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Figure 9.1 Capital investment in production equipment such as this robotic arm requires extensive analysis of the benefits the investment is likely to produce overtime, (credit: modification of "Webb Telescope Crew Flexes Robotic Arm at NASA" by Chris Gunn/ NASA/flickr, CC BY 2.0)
Chapter Outline
9.1 Timing of Cash Flows
9.2 Unequal Payments Using a Financial Calculator or Microsoft Excel
LziJ Why It Matters
Baseball legend Ted Williams once said, "Baseball is the only field of endeavor where a man can succeed three times out often and be considered a good performer."1 On routine or unimportant decisions, business managers might aspire to do as well as Ted Williams, making the right decisions only 30% of the time. But a professional decision maker must "hit it out of the park" when making major capital investment choices and recommendations.
As a student in a course of business studies and career development, it is highly likely that you will be a decision maker about projects that are likely to generate future cash flows but will also require a large initial expense. When you ask your manager to invest $500,000 or more in a new piece of equipment that could help your department meet or exceed its goals, you must be prepared to defend your request. Competing managers and departments will be asking for similar funding, and there simply might not be enough for everyone. This decision process requires financial analysis.
Mark Cuban of Shark Tank fame enjoys citing the series' catchphrase: "Know thy numbers." As a business professional, you must be able to assess potential profit against expenditures to be successful. In most cases, this is based on our understanding of cash flow. A major capital investment might seem initially like a gamble, but it is a gamble that can be hedged in your favor with understanding, analysis, and knowledge of your numbers.
The purpose of this chapter is to give you information and instruction on how this is done. The techniques we will discuss in this chapter will clarify decisions that must be made in the process of investing in a business. We
1 Pete Palmerand Gary Gillette, eds. The 2006 ESPN Baseball Encyclopedia. New York: Sterling Publishing, 2006, 5.
264 9 • Time Value of Money III: Unequal Multiple Payment Values
focus first on decisions we make about our own money as investors if uneven cash receipts or payments are involved.
Timing of Cash Flows
Learning Outcomes
By the end of this section, you will be able to:
■ Describe how multiple payments of unequal value are present in everyday situations.
■ Calculate the future value of a series of multiple payments of unequal value.
■ Calculate the present value of a series of multiple payments of unequal value.
Multiple Payments or Receipts of Unequal Value: The Mixed Stream
At this point, you are familiar with the time value of money of single amounts and annuities and how they must be managed and controlled for business as well as personal purposes. If a stream of payments occurs in which the amount of the payments changes at any point, the techniques for solving for annuities must be modified. Shortcuts that we have seen in earlier chapters cannot be taken. Fortunately, with tools such as financial or online calculators and Microsoft Excel, the method can be quite simple.
The ability to analyze and understand cash flow is essential. From a personal point of view, assume that you have an opportunity to invest $2,000 every year, beginning next year, to save for a down payment on the purchase of your first home seven years from now. In the third year, you also inherit $10,000 and put it all toward this goal. In the fifth year, you receive a large bonus of $3,000 and also dedicate this to your ongoing investment.
The stream of regular payments has been interrupted—which is, of course, good news for you. However, it does add a new complexity to the math involved in finding values related to time, whether compounding into the future or discounting to the present value. Analysts refer to such a series of payments as a mixed stream. If you make the first payment on the first day of next year and continue to do so on the first day of each following year, and if your investment will always be earning 7% interest, how much cash will you have accumulated—principal plus earned interest—at the end of the seven years?
This is a future value question, but because the stream of payments is mixed, we cannot use annuity formulas or approaches and the shortcuts they provide. As noted in previous chapters, when solving a problem involving the time value of money, a timeline and/or table is helpful. The cash flows described above are shown in Table 9.1 . Remember that all money is assumed to be deposited in your investment at the beginning of each year. The cumulative cash flows do not yet consider interest.
Year 0 1 2 3 4 5 6 7
Cash Invested $0.00 $2,000 $2,000 $2,000 $12,000 $2,000 $5,000 $2,000
Cumulative Cash Flows $2,000 $4,000 $6,000 $18,000 $20,000 $25,000 $27,000
Table 9.1
By the end of seven years, you have invested $27,000 of your own money before we consider interest:
■ Seven years times $2,000 each year, or $14,000
■ The extra $10,000 you received in year 3 (which is invested at the start of year 4)
■ The extra $3,000 you received in year 5 (which is invested at the start of year 6)
These funds were invested at different times, and time and interest rate will work for you on all accumulated balances as you proceed. Therefore, focus on the line in your table with the cumulative cash flows. How much cash will you have accumulated at the end of this investment program if you're earning 7% compounded
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9.1 • Timing of Cash Flows 265
annually? You could use the future value of a single amount equation, but not for an annuity. Because the amount invested changes, you must calculate the future value of each amount invested and add them together for your result.
Recall that the formula for finding the future value of a single amount is FV = PV X (1 + i)n, where FV is the future value we are trying to determine, PV is the value invested at the start of each period, / is the interest rate, and n is the number of periods remaining for compounding to take effect.
Let us repeat the table with your cash flows above. Table 9.2 includes a line to show for how many periods (years, in this case) each investment will compound at 7%.
Year 0 1 2 3 4 5 6 7
Cash Invested $0.00 $2,000 $2,000 $2,000 $12,000 $2,000 $5,000 $2,000
Cumulative Cash Flows $2,000 $4,000 $6,000 $18,000 $20,000 $25,000 $27,000
Years to Compound 7 6 5 4 3 2 1
Table 9.2
The $2,000 that you deposit at the start of year 1 will earn 7% interest for the entire seven years. When you make your second investment at the start of year 2, you will now have spent $4,000. However, the interest from your first $2,000 investment will have earned you $2,000 X 0.07 = $140, so you will begin year 2 with $4,140 rather than $4,000.
Before we complicate the problem with a schedule that ties everything together, let's focus on years 1 and 2 with the original formula for the future value of a single amount. What will your year 1 investment be worth at the end of seven years?
FVi = $2,000 X (1 + 0.07)7 « $3,211.56
You need to address the year 2 investment separately at this point because you've calculated the year 1 investment and its compounding on its own. Now you need to know what your year 2 investment will be worth in the future, but it will only compound for six years. What will it be worth?
FV2 = $2,000 X (1 + 0.07)6 « $3,001.46
You can perform the same operation on each of the remaining five invested amounts, remembering that you invest $12,000 at the start of year 4 and $5,000 at the start of year 6, as per the table. Here are the five remaining calculations:
FV3 =$2,000 X (1+0.07)5 «$2,805.10 FV4 = $12,000 X (1 + 0.07)4 « $15,729.55 FV5 = $2,000 X (1 + 0.07)3 k $2,450.09 FV6 = $5,000 X (1 + 0.07)2 * $5,724.50 FV7 =$2,000 X (1+0.07)1 «$2,140.00
Notice how the exponent representing n decreases each year to reflect the decreasing number of years that each invested amount will compound until the end of your seven-year stream. For clarity, let us insert each of these amounts in a row of Table 9.3 :
Year 0 1 2 3 4 5 6 7
Cash Invested $0.00 $2,000 $2,000 $2,000 $12,000 $2,000 $5,000 $2,000
Cumulative Cash Flows $2,000 $4,000 $6,000 $18,000 $20,000 $25,000 $27,000
Table 9.3
266 9 • Time Value of Money III: Unequal Multiple Payment Values
Year 0 1 2 3 4 5 6 7
Years to Compound 7 6 5 4 3 2 1
Compounded Value at End of Year 7 $3,211.56 $3,001.46 $2,805.10 $15,729.55 $2,450.09 $5,724.50 $2,140.00
Table 9.3
The solution to the original question—the value of your seven different investments at the end of the seven-year period—is the total of each individual investment compounded over the remaining years. Adding the compounded values in the bottom row provides the answer: $35,062.26. This includes the $27,000 that you invested plus $8,062.26 in interest earned by compounding.
It's important to note that throughout these sections on the time value of money and compounded or discounted values of mixed streams and their analysis, we are placing the valuation at the end or beginning of a period for simplicity in the examples. In reality, businesses might consider valuations happening within the period to allow for a degree of regularity in the revenue streams provided by the asset being considered. However, because this is a technique of forecasting, which is inherently uncertain, we will continue with analysis by period.
Future Value of a Mixed Stream
Assume that you can invest five annual payments of $10,000, beginning immediately, but you believe you will be able to invest additional amounts of $5,000 at the beginning of years 4 and 5. This investment is expected to earn 4% each year. What is the anticipated future value of this investment after the full five years?
Solution:
Year 0 1 2 3 4 5
Cash Invested $0.00 $10,000 $10,000 $10,000 $15,000 $15,000
Cumulative Cash Flows $10,000 $20,000 $30,000 $45,000 $60,000
Years to Compound 5 4 3 2 1
Compounded Value at End of Year 5 $12,166.53 $11,698.59 $11,248.64 $16,224.00 $15,600.00
Table 9.4
The equations to calculate each individual year's compounded value at the end of the five years are as follows:
FVi = $10,000 X (1 + 0.04)5 * $12,166.53
FV2 = $10,000 X (1 + 0.04)4 «$11,698.59
FV3 = $10,000 X (1 + 0.04)3 » $11,248.64
FV4 = $15,000 X (1 + 0.04)2 » $16,224.00
FV5 = $15,000 X (1 + 0.04)5 « $15,600.00
The sum of these individual calculations is $66,937.76, which is the total value of this stream of invested amounts plus compounded interest.
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9.1 • Timing of Cash Flows 267
Let's take the example above and review it from a different angle. Keeping in mind that we have not yet explored the use of Excel, is there another way to view our solution? The problem above takes each annual investment and compounds it into the future, then adds the results of each calculation to find the total future value of the stream of payments.
But when you break the problem down, another way to look at the problem is as a five-year annuity of $10,000 per year plus added payments in years 4 and 5. Can we solve for the future value of an annuity first and then perform two separate calculations on the additional amounts ($5,000 each in years 4 and 5)? Yes, we can.
Let's summarize:
■ Future value of a $10,000 annuity due, 4%, 5 years, plus
• Future value of a single payment of $5,000,4%, 2 years, plus
• Future value of a single payment of $5,000,4%, 1 year
This must give us the same result. The formula for the future value of an annuity due is
FVa = PYMT X (1 + f""1 X (1 + i) This problem can be solved in the three steps of the summary above. Step 1:
FVa = $10,000 X (1 + ool)5"1 X (1+0.04) FVa = $10,000 X 5.416323 X 1.04 * $56,329.76
Step 2:
FVYear 4 = $5,000 X (1 + 0.04)2 = $5,408.00
Step 3:
FVYear 5 = $5,000 X (1 + 0.04)1 = $5,200.00
Combining the results from each of the three steps gives us
56,329.76 + 5,408.00 + 5,200.00 = $66,937.76
It works. Whether you view this problem as five separate periods that can be compounded separately and then combined or as a combination of one or more annuities and/or single payment problems, we always arrive at the same solution if we are diligent about the time, the interest, and the stream of payments.
The Present Value of a Mixed Stream
Now that we've seen the calculation of a future value, consider a present value. We will begin with a personal example. You win a cash windfall through your state's lottery. You would like to take a portion of the funds and place them in a fixed investment so that you can draw $17,000 per year starting one year from now and continue to do so for the next two years. At the end of year 4, you want to withdraw $17,500, and at the end of year 5, you will withdraw the last $18,000 to close the account. When you take your last payment of $18,000, your fund will be totally depleted. You will always be earning 6% annually. How much of your cash windfall should you set aside today to accomplish this?
Let us break down the problem, remembering that we are thinking in reverse from the earlier problems that involved future values. In this case, we're bringing future values back in time to find their present values. You will recall that this process is called discounting rather than compounding.
Regardless of how we solve this, the question remains the same: How much money must we invest today (present value) to achieve this? And remember that we will always be earning 6% compounded annually on any invested balances.
268 9 • Time Value of Money III: Unequal Multiple Payment Values
We are calculating present values as we did in previous chapters, given a known future value "target," in order to determine how much money you need today to achieve that goal. Let us break this down by first reviewing the relevant equations from previous chapters.
Present value of an ordinary annuity:
Present value of a single amount:
PVa = PYMT X
PV = FV X
1-
(1 + if
(1 + 0"
where PVa is the present value of an annuity, PYMT is one payment in a consistent stream (an annuity), / is the interest rate (annual unless otherwise specified), n is the number of periods, PV is the present value of a single amount, and FV is the future value of a single amount.
You want to find out how much money you need to set aside today to accomplish your goal. You can also find out how much money you need to set aside in each period to accomplish this goal. Therefore, we can address this problem in increments. Let us look at potential solutions.
First, we will break this down into the cash flows of each year. Table 9.5 shows the timing of the future cash flows you're expecting:
Year 0 1 2 3 4 5
Expected Amount to Be Withdrawn at End of Year $0.00 $17,000 $17,000 $17,000 $17,500 $18,000
Table 9.5
One method is to take each year's cash flows, which happen at the end of the year, and discount them to today using the present value formula for a single amount:
1
PV = FV X
(l + 0"
PV! = $17,000 X---r « $16,037.74
(1 + 0.06)1
Because year 1 's withdrawal from your fund only has one year to earn interest, we discounted it for one year. The second amount is discounted for two years:
PV2 = $17,000 X---T « $15,129.94
(1 + 0.06)2
The next three years are discounted in the same way, for three, four, and five years, respectively:
PV3 = $17,000 X -1—T « $14,273.53
(1 + 0.06)3
PV4 = $17,500 X -l-—T « $13,861.64
(1 + 0.06)4
PV5 = $18,000 X -l-—r » $13,450.65
(1 + 0.06)5
Notice how we reverse our thinking on the exponent n from our approach to future value. This time, it increases each period because we discount each future amount for a longer period to arrive at the value in today's dollars.
When we add all five discounted present value amounts from above, we derive today's value of $72,753.49. Expressed more simply, if you wanted to extract the specified stream of cash flows at the end of each year
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9.1 • Timing of Cash Flows 269
($17,000 for three years, then $17,500, then $18,000), you would have to begin with $72,753.49. The thing to remember is that any amounts remaining in this fund, regardless of how you deplete it, will always be earning 6% annually. See Table 9.6.
Year 0 1 2 3 4 5
Withdrawn at End of Year $17,000.00 $17,000.00 $17,000.00 $17,500.00 $18,000.00
Interest on Balance $4,365.21 $3,607.12 $2,803.55 $1,951.76 $1,018.87
Remaining Balance $72,753.49 $60,118.70 $46,725.82 $32,529.37 $16,981.13 $0.00
Table 9.6
Let us try another approach. Because the amount of cash withdrawn in the first three years remains constant at $17,000, it can be viewed as an annuity—specifically, a three-period annuity of $17,000 and two single payments of $17,500 and $18,000. Therefore, we could also discount (bring to present value) an annuity of $17,000 for three years (the first three) and then combine it with the year 4 discounted amount and the year 5 discounted amount. We can try it using the formulas for PVa and PVused above. In Step 1, we will discount the first three years as an annuity (ordinary, as the first withdrawal is not made until one year from now); in Step 2, we will discount the year 4 single payment amount; and in Step 3, we will do the same for the year 5 single payment amount. Then we can add them together.
Step 1: Find the present value of the annuity using the PVa formula:
1
PVa = $17,000 X
1 -
(1 + 0.06)3
PVa = $17,000 X
0.06 1 - 0.839619
0.06
PVa = $17,000 X 2.673017 « $45,441.29
Step 2: Discount the year 4 amount using the formula for the present value of a single amount:
PV(Year 4) = $17,500 X + ^ « $13,861.64
Step 3: Perform the same operation as in Step 2 for the year 5 amount:
PVfYear 5) = $18,000 X ---r « $13,450.65
(1 + 0.06)5
Now that all three amounts have been discounted to today's value, we can add them:
45,441.20 + 13,861.64 + 13,450.65 = $72,753.49
Calculating the present value of cash flows is very common and critical in the analysis of capital investments in business for two compelling reasons: first, the investment is likely quite significant, and second, the risk will usually encompass a longer time frame. When the author of this chapter would purchase a large machine, it would likely take several years for that machine to justify its purchase with the revenues it would generate. This is one of the primary reasons that accountants require us to depreciate the cost of an asset over time: to assess the cost against the time it will take for that asset to produce profits and cash flow.
270 9 • Time Value of Money III: Unequal Multiple Payment Values
Present Value of a Mixed Stream
Assume that you decide to invest $450,000. All cash flows are discounted at 4%. You are told by your financial advisor to expect cash inflows from your investment of $100,000 in year 1, $125,000 in year 2, $175,000 in year 3, $90,000 in year 4, and $50,000 in year 5. Would you agree to this plan based only on the numbers? Each amount will be withdrawn at the end of every year, and interest will be compounded annually.
Solution:
Year 0 1 2 3 4 5
Expected Amount to Be Withdrawn at End of Year $0.00 $100,000 $125,000 $175,000 $90,000 $50,000
Table 9.7
Applying the formula for the present value of a single amount, we discount each amount and then add the discounted amounts. We will simplify this approach with Excel shortly, but we must understand the reasoning behind discounting uneven cash flow streams with a direct solution.
1_
PVi = $100,000 X
PV2 = $125,000 X
PV3 = $175,000 X
PV4 = $90,000 X
pv5 = $50,000 X
(1 + 0.04)1 1
(1 + 0.04)2 1
(1 + 0.04)3 1 , (1 + 0.04)4
1 , (1 + 0.04)5
i $96,153.85 * $115,569.53 i $155,574.36 $76,932.38 $41,096.36
By combining the five discounted amounts above, we get a total present value of $485,326.48. This amount represents the value today of the five expected cash inflows for as long as our remaining balance is earning 4%.
CONCEPTS IN PRACTICE
Thoughts on Cash Flow from Irina Simmons
In 2013, the author interviewed Irina Simmons, senior vice president, chief risk officer, and former treasurer of EMC Corporation. The importance and understanding of cash flow analysis is fundamental to this text, and several of her insights are highly relevant to our content and procedures here.
AA: Ms. Simmons, why is cash management so important to an existing or start-up firm, and how does it compare to the more basic and traditional focus on profitability?
Simmons: While profitability is very useful for analysis by investors to measure performance, an organization's cash flow provides superior measurement. Cash flow is easy to understand, provides a transparent way of assessing a firm's health, and is not subject to any qualifications. By focusing upon cash flow, any firm—whether it is mature or a start-up organization—can have a clear picture of its health and success.
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9.2 • Unequal Payments Using a Financial Calculator or Microsoft Excel 271
AA: In your bio, you mention liquidity management. Can you elaborate on this and why liquidity management is so important to a firm?
Simmons: Just as effective forecasting can provide superior cash management, the same holds true for liquidity management. For example, if you are able to confidently predict levels and timing of cash, then based on that forecast, you can make effective short- and long-term borrowing decisions. A disciplined approach to projecting one's cash position means that instead of investing cash in the money market to maximize day-to-day liquidity, you can look into longer-term investments that can provide a significantly higher return. This is essential to the effective matching of cash inflows and outflows for the firm.
AA: In summary, do you have any words of advice to students who might have an eye to entrepreneurial ventures?
Simmons: "Cash is king," don't forget that. Understand how cash moves through a business. It is also very important to implement and retain a cash management discipline. Never put that off until later. Many times, start-ups will say, "Well, I have all this venture money, and we can start making things happen and worry about being good cash managers later." But what I've seen is that the longer companies wait, the harder it is to break bad habits. Making cash management a priority now will serve entrepreneurs in perfect stead as their business starts to gain traction.
We closed this excellent interview with agreement that we were "kindred spirits" regarding the importance of cash flow analysis, including capital decisions such as those mentioned in this chapter. We confirmed with each other the core belief that "cash flow is the axis upon which the world of business spins."
(source: Business Finance: A Clear View, 3rd edition, by Alan S. Adams. LAD Publishing, 2015.)
Analyst Training Materials
These materials, developed to help professionals prepare for an analyst certification exam, describe the sources of return from investing in a bond (https://openstax.Org/r/return-investing-bond).
Unequal Payments Using a Financial Calculator or Microsoft Excel
Learning Outcomes
By the end of this section, you will be able to:
■ Calculate unequal payments using a financial calculator.
■ Calculate unequal payments using Microsoft Excel.
Using a Financial Calculator
A financial calculator provides utilities to simplify the analysis of uneven mixed cash streams (see Table 9.8).
Earlier, we explored the future value of a seven-year mixed stream, with $2,000 being saved each year, plus an additional $10,000 in year 4 and an additional $3,000 in year 6. All cash flows and balances earn 7% per year compounded annually, and the payments are made at the start of each year. We proved that this result totals approximately $35,062.26. We begin by clearing all memory functions and then entering each cash flow as follows:
272 9 • Time Value of Money III: Unequal Multiple Payment Values
Step Description Enter Display
1 Clear cash flow memory cf 2ND [clr work] CFO 0.00
2 Enter 0 for cash flow at Time 0 enter I CFO 0.00
3 Move to next entry I C01 0.00
4 Enter first cash flow 2000 enter C01 = 2000.00
5 Move to next entry C02 0.00
6 Enter second cash flow 2000 enter I C02 = 2000.00
7 Move to next entry 1 1 C03 0.00
8 Enter third cash flow 2000 enter C03 = 2000.00
9 Move to next entry C04 0.00
10 Enter fourth cash flow 12000 enter C04 = 12000.00
11 Move to next entry i I C05 0.00
12 Enter fifth cash flow 2000 enter C05 = 2000.00
13 Move to next entry i I C06 0.00
14 Enter sixth cash flow 5000 enter C06 = 5000.00
15 Move to next entry i I C07 0.00
16 Enter seventh cash flow 2000 enter C07 = 2000.00
17 Press npv npv I 0.00
18 Enter interest rate 7 enter 1 = 7.00
19 Press down arrow to show current NPV rate i NPV 0.00
20 Press cpt to find net present value cpt NPV 20,406.56
Table 9.8 Steps for Calculating Uneven Mixed Cash Flows2
At this point, we have found the net present value of this uneven stream of payments. You will recall, however, that we are not trying to calculate present values; we are looking for future values. The TI BAII Plus™ Professional calculator does not have a similar function for future value. This means that either we can find the future value for each payment in the stream and combine them, or we can take the net present value we just calculated and easily project it forward using the following keystrokes. Note the net present value solution in Step 20 above. We will use that and then use the simpler of the two approaches to calculate future value (see Table 9.9).
Step Description Enter Display
21 Enter npv from Step 20 20406.56 PV = 20,406.56
22 Enter number of compounding periods 8 n N= 8.00
23 Enter interest rate 7 i/y I/Y= 7.00
24 Calculate future value cptfv FV= -35,062.27
Table 9.9 Steps for Calculating Uneven Mixed Cash Flows, Continued
2 The specific financial calculator in these examples is the Texas Instruments BA II Plus™ Professional model, but you can use other financial calculators for these types of calculations.
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9.2 • Unequal Payments Using a Financial Calculator or Microsoft Excel 273
This is consistent with the solution we found earlier, with a difference of one cent due to rounding error.
We may also use the calculator to solve for the present value of a mixed cash stream. Earlier in this chapter, we asked how much money you would need today to fund the following five annual withdrawals, with each withdrawal made at the end of the year, beginning one year from now, and all remaining money earning 6% compounded annually:
Year 1 2 3 4 5
$17,000 $17,000 $17,000 $17,500 $18,000
Table 9.10
We determined these withdrawals to have a total present value of $72,753.30. Here is an approach to a solution using a financial calculator. In this example, we will store all cash flows in the calculator and perform an operation on them as a whole (see Table 9.11). Because we will use the NPV function (to be explored in more detail in a later chapter), we enter our starting point as 0 because we do not withdraw any cash until one year after we begin.
Step Description Enter Display
1 Clear cash flow memory cf 2ND [clr work] CF0 0.00
2 Enter 0 for cash flow at Time 0 enter I CF0 0.00
3 Move to next entry i C01 0.00
4 Enter first cash flow 17000 enter C01 = 17000.00
5 Move to next entry i I C02 0.00
6 Enter second cash flow 17000 enter C02 = 17000.00
7 Move to next entry i I C03 0.00
8 Enter third cash flow 17000 enter C03 = 17000.00
9 Move to next entry i I C04 0.00
10 Enter fourth cash flow 17500 enter C04 = 17500.00
11 Move to next entry i I C05 0.00
12 Enter fifth cash flow 18000 enter C05 = 18000.00
13 Press npv npv I 0.00
14 Enter interest rate 6 enter 1 = 6
15 Press down arrow to show current npv rate i NPV 0.00
16 Press cpt to find net present value cpt NPV = 72,753.49
Table 9.11 Steps for Calculating the Present Value of a Mixed Cash Stream
This result, you will remember, was calculated earlier in the chapter by the formula approach. Using Microsoft Excel
Several of the exhibits already in this chapter have been prepared with Microsoft Excel. While full mastery of Excel requires extensive study and practice, enough basics can be learned in two or three hours to provide the user with the ability to quickly and conveniently solve problems, including extensive financial applications. Potential employers and internship hosts have come to expect basic Excel knowledge, something to which you
274 9 • Time Value of Money III: Unequal Multiple Payment Values
are exposed in college.
We will demonstrate the same two problems using Excel rather than a calculator:
1. The future value of a mixed cash stream for a seven-year investment
2. The present value of a mixed cash stream of five withdrawals that you wish to make from a fund to be established today
Beginning with the future value problem, we created a simple matrix to lay out the mixed stream of future cash flows, starting on the first day of each year, with all funds earning 7% throughout. Our goal is to determine how much money you will have saved at the end of this seven-year period.
Table 9.12 repeats the data from earlier in the chaper for your convenience.
Year 0 1 2 3 4 5 6 7
Cash Invested $0.00 $2,000 $2,000 $2,000 $12,000 $2,000 $5,000 $2,000
Cumulative Cash Flows $2,000 $4,000 $6,000 $18,000 $20,000 $25,000 $27,000
Years to Compound 7 6 5 4 3 2 1
Table 9.12
Figure 9.2 is an Excel matrix that parallels Table 9.12 above.
A , B D E F H 1 1 '
1 Assumed Interest Rate 7.00?i
2 3 Year O Amount Invested at Start of Year 1 2,000.00 2 3 4 2,000.00 2,000.00 12,000.00 5 2,000.00 G 5,000.00 7 Total 2,000.00 27,000.00
4 5 Interest on Balance Cumulative Cash Flow 140.00 2,140.00 289.80 450.09 1,321.59 4,429.80 6,879.89 20,201.48 1.554.10 23,755.58 2,012.89 30,768.47 2,293,79^ 8,062.27 35,062.27 35,062.27
Figure 9.2 Compound Interest Example
Download the spreadsheet file (https://openstax.Org/r/chapter9-excel-exhibits) containing key Chapter 9 Excel exhibits.
We begin by entering the cash flow as shown in Figure 9.2. The assumed interest rate is 7%. The interest on the balance is calculated as the amount invested at the start of the year multiplied by the assumed interest rate. The cumulative cash flows of each year are calculated as follows: for year 1, the amount invested plus the interest on the balance; for years 2 through 7, the amount invested plus the interest on the balance plus the previous year's running balance. By adding up the amount invested and the interest on the balance, you should arrive at a total of $35,062.27.
We can use Excel formulas to solve time value of money problems. For example, if we wanted to find the present value of the amount invested at 7% over the seven-year time period, we could use the NPV function in Excel. The dialog box for this function (Rate, Value 1, Value2) is shown in Figure 9.3.
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9.2 • Unequal Payments Using a Financial Calculator or Microsoft Excel 275
Function Arguments NPV
Rate Valuel
Value2
X
number number number
Returns the net present value of an investment based on a discount rate and a series of future payments (negative values) and income (positive values).
Rate: is the rate of discount overthe length of one period.
Formula result = Help on this function
OK
Cancel
Figure 9.3 Dialog Box for NPV Function, Problem 1
The function argument Rate is the interest rate; Valuel, Value2, and so on are the cash flows; and "Formula result" is the answer.
We can apply the NPV function to our problem as shown in Figure 9.4.
Function Argument: NPV
Rate Valuel
Valued
X
CI
Cil3|
0.07
{2000,2000,2000,12000,2000,5000,2000}
number
= 20406.5576
Returns the net present value of an investment based on a discount rate and a series of future payments (negative values) and income (positive values).
Valuel: value1,value2,... are 1 to 254 payments and income, equally spaced in time and occurring at the end of each period.
Formula result = 520,406.56 Help on this function
OK
Cancel
Figure 9.4 Applying the NPV Function, Problem 1
Please note that the Rate cell value (C1) and the Valuel cell range (C3:I3) will vary depending on how you set up your spreadsheet.
The non-Excel version of the problem, using an assumed interest rate of 7%, produces the same result.
276 9 • Time Value of Money III: Unequal Multiple Payment Values
Year 1 2 3 4 5 6 7
Amount Invested at Start $2,000 $2,000 $2,000 $12,000 $2,000 $5,000 $2,000
NPV $20,406.56
Table 9.13
We conclude with the second problem addressed earlier in this chapter: finding the present value of an uneven stream of payments. We can use Excel's NPV function to solve this problem as well (see Figure 9.5).
Year 0 1 2 3 4 5
Expected Amount to Be Withdrawn at End of Year $0.00 $17,000 $17,000 $17,000 $17,500 $18,000
Table 9.14
Function Argument: NPV
Rate Valuel
Value2
X
number number number
Returns the net present value of an investment based on a discount rate and a series of future payments (negative values) and income (positive values).
Rate: is the rate of discount overthe length of one period.
Formula result = Help on this function
OK
Cancel
Figure 9.5 Dialog Box for NPV Function, Problem 2
Again, Rate is the interest rate; Valuel, Value 2, and so on are the cash flows; and "Formula result" is the answer.
Let us apply the NPV function to our problem, as shown in Figure 9.6 and Figure 9.7.
A A B c a E F G
1 Year 0 1 2 3 4 5
2 Desired Withdrawals Each Year 17,000.00 17,000.00 17,000.00 17,500.00 18,000.00
3 Interest Rate 6.00%
4 NPV 72,753.49
Figure 9.6 Applying the NPV Function, Problem 2: Excel Data
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9.2 • Unequal Payments Using a Financial Calculator or Microsoft Excel 277
Function Arguments NFV
Rate Valuel
ValueZ
X
E3
C2:G2|
0.06
{17000,17000,17000,17500,1BOO0}
number
= 72753.43936
Returns the net present value of an investment based on a discount rate and a series of future payments (negative values) and income (positive values).
Valuel: value1,value2,... are 1 to Z54 payments and income, equally spaced in time and occurring at the end of each period.
Formula result = 572,753.49 Help on this function
Figure 9.7 Applying the NPV Function, Problem 2: Function Argument
Please note that the Rate cell value (B3) and the Valuel cell range (C2:G2) will vary depending on how you set up your spreadsheet.
The non-Excel version of the problem produces the same result: an NPV of $72,753.49.
Year 0 1 2 3 4 5
Desired Withdrawals Each Year $17,000 $17,000 $17,000 $17,500 $18,000
Interest Rate 0.06
NPV $72,753.49
Table 9.15
OK
Cancel
278 9 • Summary
a Summary
9.1 Timing of Cash Flows
To understand the true value and strength of cash, it is necessary to consider its timing. This is relevant for investments for the future and for analyses of the value of projects that require investment today to produce expected flows of cash later. These future cash flows could involve inflows or outflows of cash in unequal amounts. This section analyzed the determination of present and future value of these uneven or mixed cash flows.
9.2 Unequal Payments Using a Financial Calculator or Microsoft Excel
This section discussed the use of two tools for managing and understanding the time value of money and its many applications when the flows of cash are unequal: the TI BAII Plus™ Professional financial calculator and the spreadsheet application Excel.
¥ Key Terms
capital investment a major expenditure that requires a large up-front investment and is expected to
generate substantial cash inflow in return cash flow the amount of cash actually flowing into and out of a business, as opposed to income (which is
based on accounting rules, accruals, and reports) future value the value of an asset or holding at a point in the future based on expectations of that asset's
growth at a certain rate of return mixed stream a set of cash flows over a period of time that can vary in amount from one period to the next present value the value of an asset in today's dollars based on future growth expectations at an assumed
interest rate
m CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session (https://openstax.Org/r/cfa-level1-study-session). Reference with permission of CFA Institute.
Multiple Choice
General instructions: Approximations and minor differences because of rounding are acceptable. Ignore the effect of taxes. Please assume that all percentages are annual rates and compounding occurs annually unless otherwise indicated.
1. For which of the following situations would you NOT be able to use the one-step shortcut provided by an annuity calculation?
a. Planning an amortization table for a three-year car loan at 2%
b. Investing $2,000 per year for 10 years at 5%
c. Calculating the monthly payment on a 20-year home mortgage of $150,000 at 3%
d. Finding the present value of a fund that will pay $1,000 the first year, $2,000 in the second year, and $3,500 in the third year
2. If you invest $3,000 today, $5,000 one year from now, and $3,000 two years from now, approximately how much money will you have at the end of the third year if you invest in a fund paying 7%?
a. $11,770.24
b. $12,609.63
c. $13,187.79
d. $14,274.50
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9 • Review Questions 279
You intend to invest four annual payments, starting immediately: $10,000 now, another $10,000 at the end of year 1, $12,000 at the end of year 2, and $15,000 at the end of year 3. How much will you have at the end of year4 if the fund is always earning 6% compounded annually?
a. $53,918.13
b. $54,417.52
c. $57,685.30
d. $57,894.77
You can invest a windfall of $40,000 today at 4% compounded annually. You don't believe you can make another investment one year from now, but you believe that you can save enough that two years from now, you can deposit another $10,000 in the same investment, which will still be earning 4% compounded annually. How much will you have accumulated in this investment three years from now?
a. $50,400.00
b. $54,875.32
c. $55,394.56
d. $57,123.33
Donna can invest today in a fund that will guarantee her a 5% return compounded annually for five years. She can invest $1,200 today, $1,400 one year from now, $1,800 two years from now, and $2,100 three years from now. If she can make these investments as scheduled, how much will she have accumulated five years from now?
a. $6,878.97
b. $7,632.23
c. $8,142.52
d. $8,799.49
Jane wants to create a fund today that will provide her a 4% guaranteed compounded annual rate. She wants to withdraw $15,000 one year from now and $27,000 two years from now, after which the fund will be depleted. How much must she invest today to achieve this goal?
a. $42,400.00
b. $41,912.43
c. $40,775.89
d. $39,386.10
You wish to draw from a fund you're creating today with a 3% guaranteed compounded annual rate. You hope to withdraw $40,000 one year from now, $35,000 two years from now, and $25,000 three years from now, at which point the fund will be depleted. How much must you invest today to achieve this?
a. $94,704.35
b. $95,346.98
c. $95,255.12
d. $95,945.78
Q Review Questions
1. Refer to Question 3 above. Solve using the financial calculator, documenting your steps/keystrokes.
2. Refer to Question 7 above. Solve using the financial calculator, documenting your steps/keystrokes.
3. You agree to repay a loan over five years with the following stream of cash payments: $1,000; $1,100; $1,250; $1,280; and $1,300. If you wish to discount these payments to their present value today using 4%, why can you not use one annuity calculation, as seen in previous chapters?
280 9 • Problems
Q Problems
Your aunt promises to gift you $1,500 now, $1,700 one year from now, $1,900 two years from now, and $2,500 three years from now. You will deposit all four amounts in an account that bears 3% interest compounded annually. How much will be in the account at the end of the fourth year?
You believe that you can set aside $1,200 each year for the next four years, starting immediately, in order to buy a small fishing boat for your retirement. Your friend Luis promises that he'll pay you back $4,900 that he owes you three years from now, so you will add that to the payment you make at the start of year 4. Then, at the start of year 5, you will increase your payment to $1,400; in year 6, to $1,500; and in year 7, to $1,600. Every payment will be deposited in a fund bearing 4% interest compounded annually. How much will you have set aside for your boat at the end of the seventh year?
Assume you win a lottery that will pay you $10,000 immediately, plus $20,000 one year from now, $30,000 two years from now, $40,000 three years from now, and $75,000 four years from now. You're curious about how much the lottery commission might offer you as an immediate lump sum instead. What is the minimum amount you should consider accepting today instead? Use a 5% annual discount rate. Please remember that we are ignoring taxation and other considerations and basing this only on the math itself.
Assume you win a lottery, and you are offered the following stream of payments by the lottery commission: $25,000 today, $32,000 one year from now, another $32,000 two years from now, and a final payment of $55,000 three years from now. You accept the offer. If you invest all of these proceeds at 6% compounded annually and extract nothing from the investment, how much will you have at the end of the fourth year?
You are offered a business partnership that guarantees you cash returns of $150,000 one year from now, nothing at the end of year 2, and $350,000 at the end of year 3. After year 3, the partnership will be dissolved, and there will be no more expected returns on your investment. If you analyze this plan expecting 7% compounded annually, what is this potential deal worth to you today?
Tony owns a small business that he is attempting to sell. A potential buyer offers him $500,000 today, plus $1,500,000 two years from now and a balance of $1,700,000 three years from now. Tony always analyzes cash flows using a rate of 4% compounded annually. To compare this to other offers, which would be paid in cash immediately, he wants to know the present value of these future cash flows. Show the calculation and solution that you would present to Tony to use in his decision.
Continuing Problem 6 above, assume that as Tony receives each payment from the buyer, he can immediately invest it in a fund that returns a guaranteed 3.5% compounded annually. If he accepts the terms of this offer, how much will he have accumulated in this investment four years from now?
You hire Susan for a three-year term. She has no retirement plan, so you agree to invest money immediately to allow her a stream of three payments beginning one year after her employment term ends. Draw a timeline! The money you invest for the full six years of this arrangement (three years of employment and three years of withdrawals) always earns 4% compounded annually. When Susan receives her third and last payment, the fund will be depleted and equal zero. The three payments she will receive are as follows: End of year 4: $25,000 End of year 5: $30,000 End of year 6: $37,000
Therefore, your goal is to have enough money in this account at the end of Susan's three-year employment term to assure her of receiving these payments.
How much money must you invest today to accomplish this strategy?
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9 • Video Activity 281
9. Abby owns a business that projects two years of strong cash flow, followed by a two-year hiatus while she pursues a graduate degree. She then plans to resume her business and is confident about the following two years of cash flow. At the end of each year, she will invest her profits in a fund that returns 3% compounded annually. Each investment will be made at the end of the year. Her expected investments are as follows:
Year 1: $50,000 Year 2: $78,000 Year 3: 0 Year 4: 0 Year 5: $84,000 Year 6: $89,000
If Abby meets her expectations, how much money will she have in this fund at the end of year 7?
Calculate the Present Value for Multiple Cash Flows
Click to view content (https://openstax.Org/r/calculate-the-present-value)
1. Provide a practical example from your own personal financial management of why an understanding of the present value of future cash flows would be important.
2. You expect to receive $10,800 one year from now, $17,400 two years from now, nothing at the end of the third year, and $24,000 four years from now. If you discount all of your cash flows at 7%, then with annual compounding, what are these future cash amounts worth to you in total today?
Future Value of Uneven Cash Flows
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3. Using Excel, determine the future value of this series of expected unequal receipts five years from now if each payment is received at the end of each year, beginning one year from now, and the interest rate is 6% compounded annually.
End of year 1: $3,800 End of year 2: $4,400 End of year 3: $5,100 End of year 4: $5,800
Video Activity
4. When using Excel's built-in Future Value function, why does Dr. Konners enter dollar amounts as negative numbers?
282 9 • Video Activity
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II.I..I.I..I-II-I-ML ■ ■■ üf fl
JANE S DOE PO BOX
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Figure 10.1 Bonds are very useful investments to add to a portfolio, (credit: modification of "US Treasury Checks - 3D Illustration" by DonkeyHotey/flickr, CC BY 2.0)
10.1 Characteristics of Bonds
10.2 Bond Valuation
10.3 Using the Yield Curve
10.4 Risks of Interest Rates and Default
10.5 Using Spreadsheets to Solve Bond Problems
When an investor purchases a bond, that person is, for all intents and purposes, making a loan to the bond issuer. Bonds issues are used to raise funds and can be issued by corporations, governments, or even subagencies of governments (including local municipalities).
As with any type of loan, the borrowing party is expected to offer something to the lender in exchange for their time and trouble. In this case, the bond-issuing entity will agree not only to repay the original face value of the loan on a specific date (the maturity of the bond) but also to pay the lender interest—or, in bond terminology, coupon payments.
Coupon payments are designed to make a bond purchase more acceptable for investors by helping compensate them for the time value of money. Because investors are parting with money that they have right now in order to make the initial bond purchase but will not see repayment of principal until the maturity date of the bond, they will experience the negative impact of time value over the bond term. When a bond issuer offers periodic coupon payments, this helps offset the negative effect of the delayed receipt of the principal amount for the investor. Also, because coupon payments will be coming to the investor throughout the term of the bond, essentially in installment payments (an annuity), the time value of money plays a critical role in bond transactions and in calculating bond valuation.
Bonds, along with stocks and mutual funds, are considered to be one of the most basic financial instruments available to any investor. It is quite common for investors to round out their portfolios by purchasing bonds.
Chapter Outline
J
Why It Matters
284 10 • Bonds and Bond Valuation
adding a degree of safety and diversity to their investment mix.
Characteristics of Bonds
10.1
Learning Outcomes
By the end of this section, you will be able to:
■ List and define the basic characteristics of bonds.
■ List and describe the various types of bonds available.
■ Explain how a bond price is inversely related to its return (yield).
Bonds as Investments
One way to look at bond investments is to consider the fact that any investor who purchases a bond is essentially buying a future cash flow stream that the bond issuer (or borrower) promises to make as per agreement.
Because bonds provide a set amount of cash inflow to their owners, they are often called fixed-income securities. Thus, future cash flows from the bond are clearly stated per agreement and fixed when the bond sale is completed.
Bonds are a basic form of investment that typically include a straightforward financial agreement between issuer and purchaser. Nevertheless, the terminology surrounding bonds is unique and rather extensive. Much of the specialized vocabulary surrounding bonds is designed to convey the concept that a bond is similar to other financial instruments in that it is an investment that can be bought and sold. Much of this unique terminology will be covered later in this chapter, but we can set out some of the basics here with an example.
Let's say that you buy a $1,000 bond that was issued by Apple Inc. at 5% interest, paid annually, for 20 years. Here, you are the lender, and Apple Inc. is the borrower.
LINK TO LEARNING
Informational Basics on Bonds
This video (https://openstax.Org/r/the-basics-of-bonds) from MoneyWeek editor Tim Bennett provides information about the basics of bonds—what they are, how they work, why companies and governments issue them, and why investors might buy them.
Basic Terminology
We need to know the following basic bond terms and pricing in order to apply the necessary time value of money equation to value this Apple, Inc. bond issue:
■ Par value: A bond will always clearly state its par value, also called face amount or face value. This is equal to the principal amount that the issuer will repay at the end of the bond term or maturity date. In our example, the par value of the bond is $1,000.
■ Coupon rate: This is the interest rate that is used to calculate periodic interest, or coupon payments, on the bond. It is important to note that coupon rates are always expressed in annual terms, even if coupon payments are scheduled for different periods of time. The most common periods for coupon payments other than annual are semiannual and quarterly. Coupon rates will typically remain unchanged for the entire life of the bond. In our example, the coupon rate is the 5% interest rate.
■ Coupon payment: This refers to the regular interest payment on the bond. The coupon or periodic interest payment is determined by multiplying the par value of the bond by the coupon rate. It is important to note that no adjustment needs to be made to the coupon rate if the bond pays interest
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10.1 • Characteristics of Bonds 285
annually. However, if a bond pays interest on a semiannual or quarterly basis, the coupon rate will have to
be divided by 2 or 4, respectively, to convert the stated annual rate to the correct periodic rate. In our
example, coupons are paid annually, so the periodic or annual interest that is paid is equal to
$1,000 X 0.05 = $50. You may notice that because these payments are the same amount and made at
regular intervals, they constitute an annuity stream (refer to Time Value of Money II: Equal Multiple
Payments.
Maturity date: The maturity date is the expiration date of the bond, or the point in time when the term of a bond comes to an end. On the maturity date, the issuer will make the final interest or coupon payment on the bond and will also pay off its principal, or face value. In our example, the maturity date is at the end of the 20-year period.
Yield to maturity (YTM): The YTM is essentially the discount rate used to bring the future cash flows of a bond into present value terms. It also equals the return that the investor will receive if the bond is held to maturity. The YTM helps quantify the overall investment value of a bond. We will explore how to compute this rate later in the chapter.
Table 10.1 displays a selected listing of bonds available for purchase or sale. First, let's review the columns so you can learn how to read this table.
Issuer Bond Type Current Price % Callable? Coupon Rate % Maturity Date Yield% Rating
3M Co. Corp 105.120 Yes 2.25 9/19/2026 1.240 A+
Alteryx Corp 98.818 No 1.00 8/1/2026 1.206 None
Anheuser-Busch Cos. LLC Corp 125.319 Yes 5.95 1/25/2033 3.340 BBB+
City of Chicago Muni 103.164 Yes 5.00 1/1/2033 1.030 BBB+
Coca-Cola Co. Corp 95.206 Yes 1.00 3/15/2028 1.731 A+
DuPont De Nemours Inc. Corp 100.815 Yes 2.17 5/1/2023 1.775 BBB+
Exxon Mobil Corp. Corp 107.325 Yes 3.18 3/15/2024 0.483 AA-
Ford Motor Co. Corp 114.880 No 7.13 11/15/2025 3.623 BB+
Nordstrom Inc. Corp 112.905 No 6.95 3/15/2028 4.758 BB+
Tennessee Energy Acquisition Corp. Muni 102.168 No 5.25 9/1/2021 0.451 BBB+
Table 10.1 Bond Information, March 2021 (source: FINRA-Markets.Morningstar.com)
Column 1: Issuer. The first column shows the company, city, or state issuing the bond. This bond listing includes two municipal issuers (City of Chicago and Tennessee Energy) as well as several corporate issuers.
Column 2: Bond Type. This describes the issue of the bond and indicates whether it is a corporation or a municipality.
Column 3: Current Price. The third column shows the price as a percent of par value. It is the price someone is willing to pay for the bond in today's market. We quote the price in relation to $100. For example, the Nordstrom bond is selling for 112.905% of its par value, or $112,905 per $100.00 of par value. If this bond has a $1,000.00 par value, it will sell for $1,129.05 ($1,000 X 1.12905). Note: Throughout this chapter, we use $1,000 as the par value of a bond because it is the most common par value for corporate bonds.
Column 4: Callable? This column states whether or not the bond has a call feature (if it can be retired or ended before its normal maturity date).
Column 5: Coupon Rate. The fifth column states the coupon rate, or annual interest rate, of each bond. Column 6: Maturity Date. This column shows the maturity date of the issue—the date on which the
286 10 • Bonds and Bond Valuation
corporation will pay the final interest installment and repay the principal.
■ Column 7: Yield. The seventh column indicates each bond's yield to maturity—the yield or investment return that you would receive if you purchased the bond today at the price listed in column 3 and held the bond to maturity. We will use the YTM as the discount rate in the bond pricing formula.
■ Column 8: Rating. The final column gives the bond rating, a grade that indicates credit quality. As we progress through this chapter, we will examine prices, coupon rates, yields, and bond ratings in more detail
Types of Bonds
There are three primary categories of bonds, though the specifics of these different types of bond can vary depending on their issuer, length until maturity, interest rate, and risk.
Government Bonds
The safest category of bonds are short-term US Treasury bills (T-bills). These investments are considered safe because they have the full backing of the US government and the likelihood of default (nonpayment) is remote. However, T-bills also pay the least interest due to their safety and the economics of risk and return, which state that investors must be compensated for the assumption of risk. As risk increases, so should return on investment. Treasury notes are a form of government security that have maturities ranging from one to 10 years, while Treasury bonds are long-term investments that have maturities of 10 to 30 years from their date of issue.
Savings bonds are debt securities that investors purchase to pay for certain government programs. Essentially, the purchase of a US savings bond involves the buyer loaning money to the government with a guaranteed promise that they will earn back the face value of the bond plus a certain amount of interest in the future. Savings bonds are backed by the US government, meaning that there is virtually no possibility of the buyer losing their investment. For this reason, the return on savings bonds is relatively low compared to other forms of bonds and investments.
Government Bonds
Review this introductory video (https://openstax.Org/r/treasury-bonds-prices) about government bonds.
Municipal bonds ("munis") are issued by cities, states, and localities or their agencies. Munis typically will return a little more than Treasury bills while being just a bit riskier.
Corporate Bonds
Corporate bonds are issued by companies. They carry more risk than government bonds because corporations can't raise taxes to pay for their bond issues. The risk and return of a corporate bond will depend on how creditworthy the company is. The highest-paying and highest-risk corporate bonds are often referred to as non-investment grade or, more commonly, junk bonds.
Corporate bonds that do not make regular coupon payments to their owners are referred to as zero-coupon bonds. These bonds are issued at a deep discount from their par values and will repay the full par value at their maturity date. The difference between what the investor spends on them in original purchase price and the par value paid at maturity will represent the investor's total dollar value return.
Convertible bonds are similar to other types of corporate bonds but have a feature that allows for their conversion into a predetermined number of common stock shares. The conversion from the bond to stock can be done at certain times during the bond's life, usually at the discretion of the bondholder.
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10.1 • Characteristics of Bonds 287
The Global Bond Market versus the Global Stock Market
Bonds have long been a trusted investment vehicle for many investors. Though the global fixed-income debt market remains considerably larger than the global stock market, this is not an entirely fair comparison. Bond markets include sovereign bonds, or bonds that are issues by governments, while stock markets do not. Some experts believe that a more relevant comparison is between the value of corporate bond markets only (excluding sovereign bonds) and total stock market value.
The chart in Figure 10.2 provides global market value information by category so that we may make our own conclusions about these markets.
140 n
Stock Market Corporate Sovereign Total Bonds Bonds Bonds
Figure 10.2 Global Bonds versus Stocks: Total Market Values (STrillions), November 2020 (data source: Nasdaq)
While the total value of bond markets continues to exceed that of stocks, the prevailing trend over the past several years has been that stock markets are gaining in terms of total market size. The primary reason for this is that stocks have traditionally outperformed bonds in terms of return on investment over extended periods of time and are likely to continue to do so. This makes them more attractive to investors, despite the higher risk associated with stock.
The Two Sides of a Bond Investment
There are essentially two sides to a bond investment, meaning the bondholder will receive two types of cash inflow from the bond investment over its term. These are the payment of the par value at maturity (often referred to as payment of the face value of the bond at term end), and the periodic coupon payments (also called interest income) from the bond. These coupon payments are contractually determined and clearly indicated in the bond issue documentation received by the bondholder upon purchase.
As a result of these two types of inflow, bond valuation requires two different time value of money techniques—specifically, present value calculations—to be computed separately and then added together.
The Relationship between Bond Prices and Interest Rates
Bond price and interest rate have an inverse relationship. When interest rates fall, bond prices rise, and vice versa (see Figure 10.3). If interest rates increase, the value of bonds sold at lower interest rates will decline. Similarly, if interest rates decline, the value of fixed-rate bonds will increase. An exception to this general rule is floating-rate bonds (often referred to as "floaters," floating-rate notes, or FRNs).
288 10 • Bonds and Bond Valuation
As interest rates change, the value of existing bonds goes up or down.
Figure 10.3 The Seesaw Effect of Bond Prices and Interest Rates
A floating-rate note is a form of debt instrument that is similar to a standard fixed-rate bond but has a variable interest rate. Rates for floating-rate bonds are typically tied to a benchmark interest rate that exists in the economy. Common benchmark rates include the US Treasury note rate, the Federal Reserve funds rate (federal funds rate), the London Interbank Offered Rate (LIBOR), and the prime rate.
So, investors who decide to purchase normal (fixed rate) bonds may not be thrilled to hear that the economy is signaling inflation and that interest rates are forecasted to rise. These bond investors are aware that when interest rates rise, their bond investments will lose value. This is not the case with floating- or variable-rate bonds.
Bonds with very low coupon rates are referred to as deep discount bonds. Of course, the bond that has the greatest discount is the zero-coupon bond, with a coupon rate of zero. The smaller the coupon rate, the greater the change in price when interest rates move.
INK TO LEARNING
Relationship between Bond Prices and Interest Rates
Review this video (https://openstax.Org/r/relationship-bond-prices) explaining the relationship between bond prices and interest rates within financial and capital markets.
10.2
Bond Valuation
Learning Outcomes
By the end of this section, you will be able to:
■ Determine the value (price) of a bond.
■ Understand the characteristics of and differences between discount and premium bonds.
■ Draw a timeline indicating bond cash flows.
■ Differentiate between fixed-rate and variable-rate bonds.
■ Determine bond yields.
Pricing a Bond in Steps
Why do we want to learn how to price a bond? The answer goes to the heart of finance: the valuation of assets. We need to ascertain what a given bond is worth to a willing buyer and a willing seller. What is its value to these interested parties? Remember that a bond is a financial asset that a company sells to raise money from willing investors. Whether you are the company selling the bond or the investor buying the bond, you want to make sure that you are selling or buying at the best available price.
Let's begin our pricing examples with the 3M Company corporate bond listed in Table 10.1 above. The table
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10.2 • Bond Valuation 289
information tells us that 3M issued a series of corporate bonds that promise to pay coupons annually on September 19 and to pay back the principal, or face value, on the maturity date of September 19, 2026. While this is not specified in the table, let's say these are 15-year corporate bonds. In that case, we know that they were issued on September 20, 2011.
The 3M bonds have an annual coupon rate of 2.25%, which indicates that the annual interest payment on the bond will be the face value (assumed to be $1,000.00 multiplied by 2.25%), or $22.50. The appropriate discount rate to apply to these future payments is the yield to bond maturity, 1.24%.
Note that the 3M bond is selling at a premium (above par or face value) due to the fact that its coupon rate is greater than the YTM percentage. This means that the bond earns more value in interest than it loses due to discounting its cash flows to allow for the time value of money principle.
Finally, the table tells us some of the bond's features. For example. Standard & Poor's, an international rating agency, rates 3M Co. as A+ (high credit quality). Additionally, the bonds are designated as callable, meaning that 3M has the option of redeeming them before their maturity on September 19, 2026.
We can price a bond using the same methods from earlier chapters: an equation, a calculator, and a spreadsheet. Let's start with the equation method (see Figure 10.4).
Find the present values of the lump sum principal and the annuity stream of coupons.
^^fl Add the present value of the lump sum principal and the present
I value of coupons.
Figure 10.4 Steps in Pricing a Bond
The first step is to identify the amounts and the timing of the two types of future cash flows to be received on the bond. Any bond that pays interest or coupon payments (coupon bonds) will have two sources of future cash flow to its bondholder/investor: the periodic coupon payments, which are a form of annuity, and the final lump sum payment of the face value amount at maturity.
As discussed above, the principal or face value is paid in a one-time lump sum payment at bond maturity. In our example with 3M Co., this is the $1,000 par value of the bond that will be paid on the maturity date of September 19, 2026. Step 1 is to lay out the timing and amount of the future cash flows. The first future cash flow we need to determine is the annual interest payment. Here, it is the coupon rate of 2.25% times the par value of the bond. As mentioned above, we will use $1,000 as the par value of this bond, so the annual coupon or interest payment will equal $22.50:
Par Value X Annual Coupon Rate = Annual Coupon Payment
$1,000 X 2.25% = $22.50
The next future cash flow that we need to determine is the payment of the par value or principal—in this case, the $1,000 par value of the bond—at the maturity date of September 19, 2026. We can set out the future cash flows for the bond as shown in Table 10.2:
290 10 • Bonds and Bond Valuation
Year (Period) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Coupons $22.50 $22.50 $22.50 $22.50 $22.50 $22.50 $22.50 $22.50 $22.50 $22.50 $22.50 $22.50 $22.50 $22.50 $22.50
Principal $1,000
Table 10.2
Note that annual coupon payments are made each year on September 19, and the first annual coupon payment date is September 19, 2012. The annual payments continue for 15 years, with the last payment being made on September 19, 2026. At this point, we can apply previously learned concepts: the coupon payments constitute an annuity stream, or payments of the same amount at regular intervals.
The principal of $1,000 is also paid out at maturity. Here, we recognize another key concept: the final amount is a lump sum payment. So, we now have the promised set of future cash flows for the 3M Co. bond.
In Step 2, we will need to decide on a discount rate to use on these future bond cash payments. For now, we will jump to the answer and simply use the YTM of 1.24% from the bond data in Table 10.1. Later in the chapter, we will develop the concepts behind how an appropriate discount rate is determined.
For Step 3, we now apply two equations to the set of future cash flows from the bond. This will then provide us with the present values of these cash flows, or the expected present-day value of the bond. Because we know that the coupon payments constitute an annuity stream, we can use the equation for the present value of an annuity (discussed in Time Value of Money II: Equal Multiple Payments. To value the one-time par value payment, we use the equation for the present value of a lump sum payment. So, by combining these, we will have the present value of the coupon payment stream, or
Coupon Payment Amount X
So, for our example above, this becomes
1
1
(1 + r)»
= PVC,
oupon
$22.50 X
l -
l
(1 + 0.0124)15
0.0124 $22.50 X 13.61099
— PVcoupon « $306.25
Next, we need to determine the present value of the payment of the par or face value of the bond at maturity. This is calculated as follows:
Par Value X
Inserting our values into this formula gives us
$1,000 x
1
(1 + r)
1
n ~ PVprincipal
(1+0.0124)15 ~ PVPrin«Pal $1,000 x 0.831224 « $831.22 Adding the present values of the two payment streams gives us
$306.25+ $831.22 = $1,137.47
Our bond price is $1,137.47. This bond price represents the value of the financial asset to both a willing buyer and a willing seller.
In this example, the willing seller is 3M Company. The willing buyer is an investor who is demanding a 1.24% yield on the investment. As per Table 10.1 above, the 3M bond sold for $1,051.20 in March 2021. However, we
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10.2 • Bond Valuation 291
display the price as a percentage of the par value, so we have the displayed price as
$''0l20 = 1.05120or 105.120% $1,000
Because we round the percent of par, we do not see the cents digit in the quoted price. Pricing a Bond Using a Financial Calculator
A financial calculator can also be used to solve common types of bond valuations. For example, what would be the current price (value) of a 4% coupon bond, paid semiannually, with a face value of $1,000 and a remaining term to maturity of 15 years, assuming a required YTM rate of 5%? The steps to solve this problem are shown in Table 10.3 below.
Step Description Enter Display
1 Clear calculator register ce/c 0.0000
2 Enter future or par value as a negative amount 1000 +|-FVFV= -1,000.0000 Enter interest rate
3 (5%^^=2.5%semiannualrate) " I/Y I/Y= 2.5000
4 Enter periods Enter periods (15 years X 2 = 30 semiannual periods) 30 N N = 30.0000
Enter coupon payment $1'°°^x4% = $20
5 2 20 +1 - pmt PMT = -20.0000 as a negative amount
6 Compute present value or price cptpv PV= 895.3485 Table 10.3 Calculator Steps for Bond Valuations1
The current price is $895.35. Time Value Connection
As we have briefly discussed, bond valuation is determined by time value of money techniques, most notably present value calculations. This makes logical sense when one considers that an investment in a bond involves a series of future cash inflows, or payments from the bond issuer to the bondholder over the term of the bond's maturity.
To determine the value of a bond today, the two-step time value of money calculation we discussed earlier must be used, and the present value of a series of coupon payments (or an annuity) must be determined. This present value amount will then be added to the present value of a single lump sum payment (the principal or face value) that will come to the bondholder at the end of the bond's term (maturity).
Fixed Income
Because standard fixed-rate bonds have their coupon payments and maturity amounts locked in, they are often referred to as fixed-income investments. This is because their values are relatively straightforward to calculate. Bonds are generally viewed as stable investments that offer income and a lower amount of volatility compared to stocks.
1 The specific financial calculator in these examples is the Texas Instruments BA II Plus™ Professional model, but you can use other financial calculators for these types of calculations.
292 10 • Bonds and Bond Valuation
While yields provided by corporate and government bonds such as US T-bills and municipal bonds are currently low because the Federal Reserve System (the Fed) has kept interest rates low for several years, investors may still consider adding bonds to their portfolios.2 This is especially true as investors enter their retirement years and seek to generate income while avoiding the volatility of the stock market. Such investors can add a mix of individual bonds, mutual funds, or exchange-traded funds to their portfolios, thus generating potential return while keeping risks at a minimum. Fixed-income investments such as intermediate- or longer-term bond funds are still providing good yields despite the low-interest-rate state of the economy.
It is important to note, however, that even though bonds are generally thought of as safer investments, they still are subject to a number of risks. Because income from most bonds is fixed, such instruments can have their values eroded by external factors such as interest rates and inflation. We will discuss some of these risks after the next section.
Period 1 2 3 4 5 6 7 8 9 10
End Date 6/30/20 12/31/20 6/30/21 12/31/21 6/30/22 12/31/22 6/30/23 12/31/23 6/30/24 12/31/24
Coupon $4,500 $4,500 $4,500 $4,500 $4,500 $4,500 $4,500 $4,500 $4,500 $4,500
Principal $100,000
Table 10.4
Table 10.4 shows the cash inflow of a five-year, 9%, $100,000 corporate bond dated January 1, 2020. The bond will have coupon (interest) payment dates of June 30 and December 31 for each of the following five years. Because the bond was issued on January 1, 2020, the year 2020 is the first full year of the bond, followed by the years 2021, 2022, 2023, and 2024, with the bond maturing in December of the latter year.
Cash inflows will be (1) the coupon or interest payments of 9% x ^"".OOO = $4,500, paid to the bondholder every six months, and (2) the one-time principal or face-value payment of $100,000 upon maturity on December 31, 2024.
Yields and Coupon Rates
The two interest rates that we associate with a bond are often confusing to students when they first begin to work with bonds. The coupon rate is the interest rate printed on the bond; this is only used to determine the interest or coupon payments. The yield to maturity (YTM) is an interest rate that is used to discount the bond's future cash flow. The YTM is derived from the marketplace and is based on the riskiness of future cash flows.
As we have seen when pricing bonds, a bond's YTM is the rate of return that the bondholder will receive at the current price if the investor holds the bond to maturity.
Yield to Maturity
As noted above, the market sets this discount rate, or the yield to maturity. The YTM reflects the going rate in the bond market for this type of bond and the bond issuer's perceived ability to make the future payments. Hence, we base the yield on a mutually agreeable price between seller and buyer. The bond market determines the YTM and the available supply of competing financial assets. By competing against other available financial assets, the YTM reflects the risk-free rate and inflation, plus such premiums as maturity and default specific to the issued bond.
The YTM is the expected return rate on the bond held to maturity. How do we determine the bond's YTM? We can use our same three trusty methods: equations, a financial calculator, and Microsoft Excel (as shown at the end of the chapter).
2 Adam Hayes. "What Do Constantly Low Bond Yields Mean forthe Stock Market?" Investopedia.June 15, 2021. https://www.investopedia.com/ask/answers/061715/how-can-bond-yield-influence-stock-market.asp
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10.2 • Bond Valuation 293
Determining Bond Yield Using an Equation
The solution, when solving for discount rates, requires us to revisit the bond pricing formula, which is
1-
l
(1 + rf
Bond Price = Par Value X ,„ ^ + Coupon Payment Amount X
(l+r)n r
Of course, with one equation, we can solve for only one unknown, and here the variable of concern is r, which is the YTM. Unfortunately, it is difficult to isolate r on the left-hand side of the equation. Therefore, we need to use a calculator or spreadsheet to solve for the bond's YTM.
Let's take another bond, the Coca-Cola bond, from Table 10.1 above and again back up our time to March 2021. If the Coca-Cola bond has just been issued in March 2021, then it would be a seven-year, semiannual bond with a coupon rate of 1.0% and an original price of $952.06 at the time of issue (Table 10.5).
Bond Characteristic
Price (% of par) Coupon rate
Maturity date March 15, 2028
Standard 8< Poor's Rating A+ Coupon payment frequency Semiannual First coupon date September 1, 2021
Type Corporate
Callable? Yes
Table 10.5 Overview of Coca-Cola Bond (as of March 2021) (credit: FINRA-Markets.Morningstar)
Determining Bond Yield Using a Calculator
For the Coca-Cola bond above, what was the bond's YTM at its issue date? This is not an easy problem to solve with a mathematical formula. It is far more practical, not to mention easier, to use a financial calculator or an Excel spreadsheet to solve for bond prices, yields, and maturity periods.
We will cover Excel applications later, but we can jump into some calculator examples right now. So, to calculate the yield on the Coca-Cola bond, we'll start by entering the values we have for this bond into a calculator. The values we know are as follows:
$1,000 par value 14 payments (n = 7 years X 2 payments per year)
($1 000 x 1% \ —-—--= $5.00 J
If the bond's selling price was $952.06 at issue, we have all the information we need to determine the bond's YTM at issue. Table 10.6 shows the steps for using a calculator to come to an answer.
Description
1 Clear calculator register ce/c
2 Enter present value or price as a negative amount 952.06
Display
0.0000
PV PV = -952.0600
Table 10.6 Calculator Steps for Bond's Yield to Maturity at Issue
294 10 • Bonds and Bond Valuation
Description
3 Enter future or par value 1000 fv
4 Enter periods 7 years x 2 = 14 semiannual periods 14 n
5 Enter coupon payment ($1'°°°x 1% = $5.00 ) 5 pmt
6 Compute interest rate cpt i/y
Display
FV= 1,000.0000 N = 14.0000
PMT= 5.0000
I/Y= 0.8651
Table 10.6 Calculator Steps for Bond's Yield to Maturity at Issue
The calculated I/Y (interest rate or YTM) of 0.8651 is a semiannual figure because the periods and coupon payments we entered for the calculation are semiannual values. To covert the semiannual value into an annual rate, we will need multiply the calculated I/Y by 2. This gives us an amount of 1.73%.
So, the YTM of the Coca-Cola bond at issue date was 1.73%. It is important to know that unless otherwise indicated, bond yields are expressed in annual percentage terms.
We have just demonstrated how a calculator can be used to determine the YTM or interest rate of a bond. Let's look at a few more examples that cover the most common types of bond problems. These are determining a YTM, calculating a bond's current price (or value), and determining a bond's maturity period.
First, let's work through another example of calculating a YTM, but this time with a bond that has annual interest payments instead of semiannual coupons.
Let's say you are considering buying a bond, but you want to calculate the YTM to determine if it will meet your overall return requirements. Some facts you have on the bond are that it has a $1,000 face value and that it matures in 12 years. Assume that the current price of the bond is $675 and it pays coupons annually at 3.5%. See Table 10.7 for the steps to calculate the YTM.
Step Description Enter Display
1 Clear calculator register ce/c 0.0000
2 Enter present value or price as a negative amount 675 +1 - : pv PV = -675.0000
3 Enter future or par value 1000 fv FV = 1,000.0000
4 Enter periods (12 years) 12 n N = 12.0000
5 Enter coupon payment ($1,000 x 3.5% = $35.00) 35 pmt PMT = 35.0000
6 Compute interest rate CPT i/y I/Y = 7.7589
Table 10.7 Calculator Steps for Computing Yield to Maturity
By following the steps in the table above, you will arrive at a YTM of 7.76%.
Using a calculator is fast and accurate for finding bond yields. Thus, if you know the bond's current price and all of the future cash flows, you can find the YTM, or the return rate that the bond buyer is receiving on the funds loaned to the bond issuer. As mentioned. Excel spreadsheets are as easy and accurate as a financial calculator for determining bond rates, and we will cover these later in the chapter.
Determining Bond Price or Value Using a Calculator
Let's say a friend recommends a 20-year bond that has a face value of $1,000 and a 6% annual coupon rate. If similar bonds are yielding 4% annually, what would be a fair price for this bond today? Table 10.8 shows the steps to make this determination.
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10.2 • Bond Valuation 295
Step Description Display
1 Clear calculator register CE/C 0.0000
2 Enter future or par value as a negative amount 1000 + I - FV FV = -1,000.0000
3 Enter interest rate (4% annual rate) 4 I/Y I/Y = 4.0000
4 Enter periods (20 years) 20 N N = 20.0000
5 Enter coupon payment ($1,000 X 6% = $60) a negative amount 35 60 + |- PMT PMT -60.0000
6 Compute present value or price CPT PV PV = 1,271.8065
Table 10.8 Calculator Steps for Computing Present Value
So, the bond should be priced today at $1,271.81. Determining Bond Maturity Using a Calculator
Imagine you are considering investing in a bond that is selling for $820, has a face value of $1,000, and has an annual coupon rate of 3%. If the YTM is 10%, how long would it take for the bond to mature? See Table 10.9 for the steps to calculate the time to maturity.
Step Description Display
1 Clear calculator register CE/C 0.0000
2 Enter present value or price as a negative amount 820 +1 - PVPV= -820.0000
3 Enter interest rate (10% annual rate) 10 I/Y I/Y= 10.0000
4 Enter future or par value 1000 FV FV= 1,000.0000
5 Enter coupon payment ($1,000 X 3% = $30.00) 30 PMT PMT= 30.0000
6 Compute periods until maturity CPT N N= 3.1188
Table 10.9 Calculator Steps for Computing Time to Maturity
So, the bond's time to maturity would be 3.12 years.
Using a Calculator
If a $1,000 face value bond is selling for $595, has 20 years until it matures, and has a YTM of 6.5%, what are the coupon rate and the periodic coupon payment of the bond? Follow the steps in Table 10.10.
Step Description Enter Display
1 Clear calculator register CE/C 0.0000
2 Enter present value or price as a negative amount 595 + I - PV PV = -595.0000
3 Enter periods 20 N N = 20.0000
4 Enter future or par value 1000 FV FV = 1,000.0000
296 10 • Bonds and Bond Valuation
Step Description
5 Enter interest rate or YTM 6.5 i/Y I/Y= 6.5000
6 Compute coupon payment CPT PMT PMT = 28.2437
Table 10.10 Calculator Steps for Computing Coupon Payment
Solution:
The annual coupon payment amount is $28.24. This means the coupon rate on the bond is yjj2^ — 2.824%. The Coupon Rate
The coupon rate is the rate that we use to determine the amount of a bond's coupon payments. The issuer states the rate as an annual rate, even though payments may be made more frequently. Thus, for semiannual bonds, the most common type of corporate and government bond, the coupon payment is the par value of the bond multiplied by the annual coupon rate and then divided by the number of payments per year, 2.
We have already seen the coupon rate. The first bond we reviewed, the 3M Co. bond, was an annual coupon bond with a coupon rate of 2.25%. Using a par value of $1,000, we determined that the annual coupon payments would be $1,000 X 0.0225 = $22.50.
For the Coca-Cola bond, we note from Table 10.5 that it has a coupon rate of 1% and is paid semiannually.
$1 000 x 1%
Using a par value of $1,000, we can determine that the coupon payments would be ——^-= $5.00.
The Relationship of Yield to Maturity and Coupon Rate to Bond Prices
The value or price of any bond has a direct relationship with the YTM and the coupon rate.
■ When the coupon rate of a bond exceeds the YTM, the bond sells at a premium compared to its par value. That is, market demand will push the price of the bond to an amount greater that than its face or par value. We call this kind of bond a premium bond.
■ When the coupon rate is less than the YTM, the bond sells at a discounted amount, or less than its par value. We refer to such a bond as a discount bond.
■ When the coupon rate and YTM are identical, a bond will sell at its par value. Bonds that experience this scenario in the market are referred to as par value bonds.
The interest or coupon payments of a bond are determined by its coupon rate and are calculated by multiplying the face value of the bond by this coupon rate.
The inverse relationship of interest rates and bond prices is an important concept for investors to know. Because interest rates fluctuate and can change significantly over time, it is important to understand how these changes will impact bond values.
10.3
Using the Yield Curve
Learning Outcomes
By the end of this section, you will be able to:
■ Use the yield curve to show the term structure of interest rates.
■ Describe and define changes in the yield curve shape.
■ Explain the importance of the yield curve shape.
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10.3 • Using the Yield Curve 297
Term Structure of Interest Rates
The expected yields of various bonds across different maturity periods are referred to as the term structure of interest rates. This is because they represent interest rates for different periods of time, maturities, or terms.
When interest rate yields are plotted against their respective maturity periods and these plotted points are connected, the resulting line is called a yield curve. Essentially, the yield curve is a result of this plotting process and becomes a graphical representation of the term structure of interest rates. A yield curve will always be constructed by showing the value of yields (rates) on they-axis and maturities or time periods on the x-axis (see Figure 10.5).
To create a useful graph of the yield curve, interest rate yields should be computed for all government bonds at all remaining times to maturity. For example, the yields on all government bonds with a single year remaining until maturity should be calculated. This value is then plotted on the y-axis against the one-year term on the x-axis. Similarly, yields on government bonds with two years remaining until maturity are calculated and plotted on the y-axis against two years on the x-axis, and so on, until a point of critical mass of information is reached and the resulting graph displays useful information.
The yield curve for government bonds is also known as the risk-free yield curve because these securities are thought of as safe investments that are not expected to fail or default and will in all likelihood repay or otherwise meet all financial obligations made through the bond issuance.
Cash 1y 3y 5y 10y 30y Term Rate
Figure 10.5 A Normal Yield Curve: Long-Term Rates Are Higher Than Short-Term Rates
A normal yield curve slopes upward, with yield increasing as the term increases. This is because yields on fixed-income investments such as bonds will rise as maturity periods increase and produce greater levels of risk.
Corporate issuers of bonds will usually offer bond issues at higher yields that the government, which is understandable because they are potentially riskier for investors. Government securities are guaranteed by governments and have little to no chance of default or nonpayment. This is not the case for corporate bonds, where there is always a chance of default, though the likelihood of this occurring will vary by individual company or issuer as well as by bond type and term. We will discuss bond default and default risk next.
The Yield Curve
Review this video (https://openstax.Org/r/introduction-to-the-yield-curve) that introduces the concept of the yield curve.
298 10 • Bonds and Bond Valuation
Different Shapes of the Yield Curve
There are two important elements to any yield curve that will define its shape: its level and its slope. The level of a yield curve directly relates to the yield rates depicted on the y-axis of the graph (see Figure 10.6). The slope of the yield curve indicates the difference between yields on short-term and longer-term investments. The difference in yields is primarily due to investors' expectations of the direction of interest rates in the economy and how the federal funds rate (referred to as cash rate in many countries) is uncertain and may differ significantly over time. As an example, yields on three-year bonds incorporate the expectations of investors on how bank rates might move over the next three years, combined with the uncertainty of those rates over the three-year period.
Higher Cash Rate
Lower Cash Rate
Cash Rate
1y 3y 5y 10y 30y Term
Figure 10.6 Changes in Yield Curve Depending on Cash Rate
As we briefly discussed above, a positive or normal yield curve is indicative of the investment community's requirements for higher rates of return as financial consideration for assuming the risk of entering into fixed-income investments, such as the purchase of bond issues. Typically, as a bond term increases, so will the potential interest rate risk to the bondholder. Therefore, bonds with longer terms will usually carry higher coupon rates to make returns greater for investors. Additionally, economists have come to believe that a steep positive yield curve is a sign that investors anticipate relatively high inflation in the future and thus higher interest rates accompanied by higher investment yields over shorter (inflationary) periods of time.
Normal yield curves are generally observed during periods of economic expansion, when growth and inflation are increasing. In any expansionary economy, there is a greater likelihood that future interest rates will be higher than current rates. This tends to occur because investors will anticipate the Fed or the central bank raising its short-term rates in response to higher inflation rates within the economy.
CONCEPTS IN PRACTICE
How C0VID-19 Impacted the Yield Curve
Figure 10.7 shows the relatively normal-shaped yield curve effective in February 2021.
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10.3 • Using the Yield Curve 299
Maturity Figure 10.7 Yield Curve in February 2021
A yield curve with an inverted (downward-sloping) shape is considered unusual and will occur when long-term rates are lower than short-term rates. This causes the yield curve to assume an inverted shape with a negative slope. An inverted yield curve has historically been observed as a prelude to a general decline in economic activity and interest rate levels. In some countries, such as the United States, an inverted yield curve has been associated with upcoming recession and economic contraction.
This may occur because central banks, such as the Federal Reserve in the United States, will often attempt to stimulate a stagnant economy by reducing interest rates. Essentially, the potential actions of the central bank to improve the economy have the effect of lowering overall money rates with the economy, which is exactly what investors anticipated would happen and why the yield curve was inverted to begin with.
The yield curve was considered normal with an upward slope in August 2018, as shown in Figure 10.8. but the curve inverted in March 2019 as yields on short-term bonds exceeded those of longer-term bonds, resulting in concerns surrounding impending recession and other economic problems. This inverted shape to the yield curve continued into 2020, as evidenced in Figure 10.9.
3.2%
ust13, 2019
1 Mo 3 Mo 6 Mo 1Yr 2Yr 3Yr 5Yr 7Yr 10Yr 20Yr 30Yr
Figure 10.8 The Economy Shifts to an Inverted Yield Curve (data source: US Department of the Treasury, Resource Center, Daily Treasury Yield Curve Rates)
Yield curves constructed on different days in early 2020 appeared similar to the examples below. Again, these are obviously not normal yield structures. As a specific example, note on the February 21, 2020, curve
300 10 • Bonds and Bond Valuation
that rates on five-year securities are lower than those of one-year and even three-month securities. 2.00% -|
1.75%
1.50%
1.25%
Figure 10.9 Elements of Inversion in Recent Yield Curves (data source: US Department of the Treasury, Resource Center, Daily Treasury Yield Curve Rates)
This inverted yield curve signaled the beginning of a recessionary period in the United States, which was compounded by the COVID-19 pandemic and the closing of many restaurants and businesses.
In March and April 2020, the US economy experienced a significant decline. Most economic indicators dropped so badly that the National Bureau of Economic Research's Business Cycle Dating Committee, the US agency that officially declares recessions, was required to intervene.
The recession declaration process by the committee is completed over the course of four months, but in this instance, it only took a total of 15 weeks for the committee to make its declaration. This remains the fastest declaration by the committee on record since the founding of the National Bureau of Economic Research (NBER) in 1978 4
In July 2021, the committee declared that the economy had reached a trough in April 2020, marking the end of the recession of the early 2020s and making it the shortest US recession on record as well as the most quickly identified one.5
(sources: www.nytimes.com/2019/11/08/business/yield-curve-recession-indicator.html; www.nber.org/ news/business-cycle-dating-committee-announcement-june-8-2020; fredblog.stlouisfed.org/2020/11/are-we-still-in-a-recession/)
A flat shape for the yield curve occurs when there is not a great deal of difference between short-term and long-term yields (see Figure 10.10). A flat curve is usually not long lasting and is often observed when the curve is transitioning between a normal and an inverted shape, or vice versa.
A flat yield curve has also been observed as a result of low interest rate levels or some types of unconventional monetary policy.
3 National Bureau of Economic Research. "Business Cycle Dating Committee Announcement June 8, 2020." NBER News.
4 Jeffrey Frankel. "The US Is Officially in Recession Thanks to the Corona Virus Crisis." The Belfer Center for Science and International Affairs. Harvard Kennedy School, June 16, 2020. https://www.belfercenter.org/publication/us-officially-recession-thanks-corona-virus-crisis
5 National Bureau of Economic Research. "Business Cycle Dating Committee Announcement July 19, 2021." NBER News. July 19, 2021. https://www.nber.org/news/business-cycle-dating-committee-announcement-july-19-2021
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10.4 • Risks of Interest Rates and Default 301
2
Normal
Flat
Inverted
Cash Rate
1y 3y 5y 10y 30y Term
Figure 10.10 Graph Depicting Normal, Flat, and Inverted Yield Curves
Why Is the Yield Curve Important?
Market technicians, brokers, and investment analysts will study the yield curve in great detail by keeping track of its many changes and movements. This is because of the overall importance of the yield curve as an economic indicator and how it can be representative of the ideas, attitudes, and bond market expectations of individuals as well as large institutional investors that exert significant influence on investment markets and the economy as a whole.
10.4
Risks of Interest Rates and Default
Learning Outcomes
By the end of this section, you will be able to:
■ Define interest rate, default, and other common forms of bond risk.
■ Calculate the primary indicator of interest rate risk.
■ Determine factors impacting default risk.
■ Understand bond laddering as an investment strategy.
■ List major rating agencies and their indications of default.
■ Define and calculate the yield to maturity (YTM) on a bond.
Bond Risks
As we touched on earlier, bonds are fixed-income investments, and because of this, they are subject to a number of risks that could have negative effects on their market value. The most common and best-known risks are interest rate risk and default risk, but other risks exist that should be understood. Among these risks are the following:
■ Credit risk. If investors believe that a bond issuer is unlikely to meet its payment commitments, they may demand a higher yield to purchase the bond issue in the first place. Due to the relative stability of governments compared to corporations, government bonds are considered to have low credit risk.
■ Liquidity risk. If investors believe that a bond may be difficult to sell, it will likely have a higher yield. This has the effect of compensating the bondholder for the lack of liquidity (the ability to cash out of the bond). Government bonds usually have the lowest yields of all investments available and are typically among the most liquid in any country where they are traded. Government securities will only face significant liquidity risk in times of great economic distress.
■ Duration risk. Duration risk is the risk associated with the sensitivity of a bond's price to a single 1 % change in interest rates. A bond's duration is expressed in numerical measurements. The higher the duration number, the more sensitive a bond investment will be to changes in interest rates.
302 10 • Bonds and Bond Valuation
■ Call risk and reinvestment risk. Call risk is the risk of bonds being redeemed or called by the issuing firm before their maturity dates. Corporations may elect to call a bond issue (provided the bond issue has a call feature) when interest rates drop and companies are in a position to save a great deal of money by issuing new bonds with lower coupon rates. To investors, this is a risk in and of itself, but call risk also has the effect of potentially causing reinvestment risk. Reinvestment risk is defined as the risk to investors when they find themselves facing unfavorable alternatives for investing the proceeds from their called bonds in new, lower-paying investments. This can potentially lead to substantial financial loss for the original bond investors.
■ Term risk. Investors will generally demand higher returns for lending funds at fixed interest rates. This is because doing so exposes them to the risks presented by rising interest rates and the negative impact of these higher rates on their bond holdings. In a scenario of rising interest rates, investors will find that their return from lending money through a bond purchase just once, at a fixed interest rate, will be lower than the return they might have realized from making several different investments for much shorter periods of time. Term risk is usually measured by a special indicator referred to as the term premium.
As mentioned above, however, the most common forms of bond risk are interest rate risk and default risk.
Interest Rate Risk
As we have discussed, when interest rates rise, bond values will fall. This is the general concept behind interest rate risk. Any investor in fixed-income securities (such as bonds) will have to contend with interest rate risk at one time or another. Interest rate risk is also referred to as market risk and usually increases the longer an investor maintains a bond investment.
Default Risk
Any time a bond is purchased, the investor is taking a risk that the bond issuer may be late in making scheduled payments on a bond issue—or, in the worst case, may not be able to make payments at all. This is the underlying idea behind the concept of default risk.
Because US Treasury securities have the full backing of the government, they are generally considered free of default risk. However, most corporate bonds will face some possibility of default. Obviously, some bonds and their issuing companies are riskier in this respect than others.
To assist potential bond investors in understanding some of these risks, bond ratings are regularly published by a number of organizations to express their assessment of the risk quality of various bond issues. We will discuss these bond ratings and the companies that issue them next.
Bond Ratings and Rating Providers
It is important for investors to know the risks they are assuming when investing in bonds. Many investors will take advantage of information provided by bond rating services to assess the likelihood of borrowers (bond issuers) defaulting on the financial obligations of their bond issues.
To help investors evaluate the default risks of bonds, rating agencies (bond rating services) were established to evaluate bonds and other fixed-income investments, taking into consideration and then analyzing any information that has been published or otherwise made available to the investing public. These services then apply a rating system that has been developed for measuring the quality of bonds and assign individual grades to each bond and its issuing company.
The three largest and best-known bond rating providers are Fitch Ratings, Moody's Investors Service, and Standard & Poor's (S&P) Global Ratings. The rating system used by these services identifies the very highest-quality bonds (the least likely to default) as triple-A (AAA or Aaa), followed next in quality level by double-A bonds (AA or Aa), and so on. Any bond that is rated BBB (S&P, Fitch) / Baa (Moody's) or higher is referred to as investment grade and is considered strong and stable by the investment community (see Table 10.11).
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10.4 • Risks of Interest Rates and Default 303
It is important to note that investment-grade bonds are among the most popular due to the fact that many commercial banks, as well as several pension funds, are only allowed to trade bonds that are investment grade.
Any bond that is below investment grade, or rated lower than BBB (S&P, Fitch) or Baa (Moody's), is referred to as a high-yield bond or a junk bond. Junk bonds have had mixed levels of success for companies wishing to issue them to raise capital. In the early 1990s, the market for junk bonds collapsed, due in part to a political movement involving influential people who had been dominating corporate debt markets. This movement, combined with illegal insider trading activities conducted by investments banks, ultimately resulted in the bankruptcy of former financial giant Drexel Burnham Lambert.
s&P / Fitch Moody's Grade Meaning
AAA Aaa Investment Risk almost zero
AA Aa Investment Low risk
A A Investment Risky if economy declines
BBB Baa Investment Some risk; more if economy declines
BB Ba Speculative Risky
B B Speculative Risky; expected to get worse
ccc Caa Speculative Probably bankruptcy
cc Ca Speculative Probably bankruptcy
c C Speculative In bankruptcy or default
D Speculative In bankruptcy or default
Table 10.11 Bond Ratings (sources: S&P Global Ratings; Moody's)
The market for junk bonds enjoyed a brief resurgence in popularity when the economy improved later in the 1990s. However, in 2001, the junk bond market shrank once again, resulting in 11% of US junk bond issues defaulting.
In general, it is important to understand that bond ratings are only judgments on corporations' future ability to repay debt obligation and their growth prospects. There is no fixed methodology or basis for calculating a bond rating. However, some financial analysts can get a strong indication of how a bond will be rated by examining certain financial ratios of the issuing firm, such as company debt ratio, earnings-to-interest ratio, and their return on assets.
CONCEPTS IN PRACTICE
The Collapse of Enron: Bond and Credit Ratings and How They Change
The bond rating system is not infallible. A perfect example of this is when energy giant Enron failed in 2001. After the collapse, investors correctly pointed out that a mere two months before this occurred, the company's bonds were rated as investment grade and considered relatively safe, having little risk for investors.
Background on Enron
From its formation in 1985 through the late 1990s, Enron grew to become an energy mega-conglomerate.
6 Lawrence Delevingne. "The Drexel Collapse, 25 Years Later." CNBC. February 13, 2015. https://www.cnbc.com/2015/02/13/the-drexel-collpase-25-years-later.html
304 10 • Bonds and Bond Valuation
expanding into areas such as trading of futures contracts, paper products, electricity, water, pipelines, and broadband services. Reported revenues grew at an exceptional pace, Enron's stock price continued to rise, and business was proceeding exceptionally well.
However, things were not as they appeared on the surface. Enron's financial statements were often very confusing to shareholders and analysts. Additionally, Enron's unscrupulous business practices included revenue misstatements and other questionable accounting practices to indicate favorable financial performance. On top of this, some of Enron's speculative business ventures proved to be disastrous, resulting in substantial financial losses.
Initial allegations against Enron also focused on the role of their public accountants, Arthur Andersen. Andersen was one of the Big Five accounting firms in the United States at that time and had served as Enron's auditing firm for over 16 years. According to court documents, Enron and Arthur Andersen had improperly categorized hundreds of millions of dollars of transactions as increases to the company's shareholder equity. It was also later discovered that Andersen failed to follow generally accepted accounting principles (GAAP) when considering Enron's dealings with related partnerships. As a result, Enron was able to conceal some of its losses from the investing public. After investigation by the United States Justice Department, the firm was indicted on obstruction of justice charges in March 2002. The combination of all of these irregularities and issues resulted in the December 2, 2001, bankruptcy of the corporation.' It was later determined that the majority of these unethical issues had been perpetuated with the indirect knowledge or even, in some cases, by the direct actions of the board of directors or senior operational management of the company.
Specifics and Enron's Bond Ratings
On October 27, 2001, the company began buying back all its commercial paper, valued at around $3.3 billion, in an effort to calm investor fears about Enron's supply of cash. On November 8, Enron announced that restatements to its financial statements for the years 1997-2000 were necessary to correct several accounting violations. However, by November 28, 2001, credit rating agencies had reduced Enron's bond rating to junk status.8
Other Examples of Significant Bond Ratings Downgrades
Some companies that have recently experienced downgrades (or potential downgrades) to their credit and bond ratings include Delta Airlines, Ford, Occidental Petroleum, Carnival Cruises, and T-Mobil. Some of these businesses, such as Delta and Carnival, are suffering the effects of the COVID-19 pandemic, but the hope is that they don't experience the same disastrous fate as Enron.
(sources: www.britannica.com/event/Enron-scandal; www.journalofaccountancy.com/issues/2002/apr/ theriseandfallofenron.html; corporatefinanceinstitute.com/resources/knowledge/other/enron-scandal/; www.wsj.com/articles/corporate-bond-downgrades-grow-as-coronavirus-spreads-11585849497)
Concepts of Bond Returns
Bond investors earn profits through two different means: collecting interest income and generating capital gains. These are important concepts for any investor who considers putting their money in fixed-income securities such as bonds.
7 Douglas O. Linder. "Enron (Lay & Skilling) Trial (2006)." Famous Trials. Accessed November 24, 2021. https://famous-trials.com/ enron
8 Paul M. Healy and Krishna G. Palepu. "The Fall of Enron. "Journal of Economic Perspectives 17, no. 2 (Spring 2003): 3-26. https://doi.org/10.1257/089533003765888403
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10.4 • Risks of Interest Rates and Default 305
Collecting Interest Income
As we have covered, when investors buy bonds, they are lending money to bond issuers. The coupon rate of a bond is determined by the issuer and is generally tied to the overall level of interest rates in the economy at the time of issue as well as the maturity period of the bond and the credit rating of the issuer. The established coupon rate then governs how much periodic interest is paid to bondholders. For example, if an investor purchases a 5%, $1,000 bond with a 20-year maturity and annual coupon payments, that investor will receive 20 coupon payments equal to $1,000 X 5% or 20 X $50 for a total of $1,000.
Depending on interest and inflation rates over the 20-year period, this could be a very favorable situation resulting in significant realized return for the investor. However, if interest rates and inflation over the investment period are at high levels, the investment is not nearly as attractive.
Generating Capital Gains
Many bonds are not held until their maturity dates. Should an investor require funds before maturity, they have the option to sell them through a broker in the secondary market. When this situation occurs, the investor may earn a capital gain or experience a capital loss, depending on whether the bond ends up being sold at a premium (above face value) or at a discount (below face value).
For example, if an investor bought a corporate bond yielding 7% and then the economy changed so that comparable bonds yielded 10%, the investor would have to lower their price on the original 7% bond until it also yielded the 10% market rate. Potential investors would not be very likely to buy the bond if they could simply buy a newly issued bond from an alternate issuer and receive a higher coupon rate.
It is equally possible that prevailing bond rates could fall and an investor could end up selling their bond at a higher price, thus earning a capital gain.
Bond Laddering as an Investment Strategy
There are several successful strategies for successful bond investments, but perhaps one of the most common yet ingenious of these strategies is called bond laddering. Bond ladders help investors achieve diversity in their portfolios and reduce risk while helping maintain regular cash inflows in the form of coupon payments or interest. In a bond ladder, an investor will divide their total investment dollars among various bonds that mature at regular intervals, thereby balancing risk and return. An example of a bond ladder would be to purchase 10 different bonds that have maturities of one year, two years, three years, and so on, all the way through to 10 years.
When the first bond matures, the investor will purchase a new bond that matures in 10 years to take its place in the ladder and continue the overall laddering strategy.
This strategy has several benefits. First, the shorter-term bonds in the ladder provide stability because they are less sensitive to risk than longer-term bonds. The longer-term bonds within the ladder will generally provide higher returns but with higher risk due to such factors as rising interest rates. So, by investing in bonds with different maturities and creating a bond ladder, investors can realize superior financial returns to what they would earn by only investing in short-term bonds. Also, the general level of risk from a bond ladder is reduced by the shorter-term component of the investment mix, making the bond ladder less risky than an investment that only included long-term bonds.
It is easy to see why bond laddering has become such a highly adopted bond investment strategy with investors ranging from novice to the most well-seasoned and experienced.
Interest Rate Movements and Bond Prices
We now know that when investors buy bonds, either directly or through mutual funds, they are lending money to bond-issuing firms or governments. In turn, issuers promise to pay back the principal (par or face value) when the loan is due at the bond's maturity date.
306 10 • Bonds and Bond Valuation
Issuers also promise to pay bondholders periodic interest or coupon payments to compensate them for the use of their money over the term of the bond. The rate at which issuers pay investors, or the bond's stated coupon rate, is typically fixed at the time of issuance.
We have also covered the concept that bond values have an inverse relationship with interest rates. As interest rates rise, bond prices fall, and when interest rates fall, bond values increase. Movement of interest rates can have a dramatic effect on a bond's value and presents the typical bondholder with a number of different financial risks that we have described in detail.
Also in this chapter, we have discussed how bond values can be estimated through the use of several different factors. Prevailing interest rates are among the most critical of these, but also important are factors such as maturity periods, the taxability of bond interest, the credit standing of bond issuers, and the likelihood of bond call, or issuers paying off their debt early.
When considering purchasing bonds or any such fixed-income investment, investors should remain aware that interest rates are always in a state of flux and can change at any time. The movements of bond values and bond yields will be significantly affected by these changes and can be favorable or unfavorable for any investor.
10.5
Using Spreadsheets to Solve Bond Problems
Learning Outcomes
By the end of this section, you will be able to:
■ Demonstrate bond valuations using Excel.
■ Demonstrate bond yield calculations using Excel.
Calculating the Price (Present Value) of a Bond
The following examples illustrate how Microsoft Excel can be used to calculate common bond problems. Please be sure to refer to the chapters on the time value of money for examples of using spreadsheets to solve present value problems, as these same concepts are also used in solving bond problems.
You can use the following steps in Excel to determine the price or present value of a coupon bond. Suppose that a bond has a par or face value of $1,000, pays coupons semiannually at a 4% annual rate, and matures in 15 years. We can assume a YTM rate of 5%.
1. First, select Formulas from the Excel upper menu bar, and from the dialog box, select PV (see Figure 10.11).
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10.5 • Using Spreadsheets to Solve Bond Problems 307
Insert Function ? X
Search for a function:
Type a brief description of what you want to do and then click Go
Or select a category: Financial
Select a function:
PRICEMAT_ _ A
PV
RATE
RECEIVED
RRI
SLN
STOCKHISTORY
P Vfrate, n p er, p mtfyty pe)
Returns the present value of an investment: the total amount that a series of future payments is worth now.
Help on this function
OK
Cancel
Figure 10.11 Using Excel to Enter a PV (Present Value) Function
2. When the PV function is selected, another dialog box will appear (see Figure 10.12). It is here that the function variables, or arguments, will be entered. It is preferable to use cell addresses to refer to these arguments so that the spreadsheet can be easily used again if inputs/arguments change.
Function Argument: PV
X
Rate Nper Pmt
Fv Type
number number number number number
Returns the present value of an investment: the total amount that a series of future payments is worth now.
Rate is the interest rate per period, For example, use 6%/4for quarterly payments at 6% APR.
Formula result = Help on this function
OK
Cancel
Figure 10.12 Function Arguments Dialog Box
3. Enter the function inputs or arguments (see Figure 10.13). We refer to the cell addresses as per our example spreadsheet.
308 10 • Bonds and Bond Valuation
Bond Face Value
Annual Interest Rate (Yield to Maturity) Semiannual Interest Rate Periods Years
Semiannual Periods Coupon Rate (Annual) Coupon Rate (Semiannual) Semiannual Coupon Payment
Calculate Price (or PV)
2.50% 15
30 4.00% 2.00% 20.00
=PV(FĚrF8,FllrF4) I
Function Argument:
Rale Pmt
= 20 - 1000
= -395.343537
Returns the present value of an investment: the total amount that a series of future payments is
Fv is the future value, or a cash balance you want to attain payment is made,
Formula result = Help on this function
worth now. afterthe last
Figure 10.13 Completed Data Entry Menu
Note that the result, the price or present value, will appear in the bottom left section of the Function Arguments box once the arguments are entered. It will appear as a negative value because of the sign convention and because the bond face value in cell F4 was entered as a positive value.
Calculating the Yield to Maturity (Interest Rate) of a Bond
Use the following steps in Excel to determine the YTM (interest rate) of a bond. Assume that you want to find the YTM of a $1,000, 3.5% bond with annual coupon payments that is selling for $675.00 and will mature in 12 years.
1. First, select Formulas from the Excel upper menu bar, and from the dialog box, select Rate (see Figure 10.14).
Insert Function Search for a function:
X
Type a brief description of what you want to do and then click Go
Or select a category: Financial
Go
Select a function:
PRICE
PRICEDISC
PRICEMAT
PV
RATE
RECEIVED
RRI
RATEfn per.prmtpVjtv, type, guess:]
Returns the interest rate per period of a loan or an investment. For example, use 6%/4for quarterly payments at 6% APR.
Help on this function
OK
Cancel
Figure 10.14 Using Excel to Enter a Rate Function
2. After the dialog box appears, enter the variables or arguments. As with our earlier example, we will use the preferred method of identifying the arguments with cell addresses (see Figure 10.15).
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10.5 • Using Spreadsheets to Solve Bond Problems 309
-5—I
Current Bond Price/Value or PV (entered as a negative)
Band -ace Value or FV
Coupon Rate
Coupon Payment (1,000 * 3.5% = 35.00) YTM or Rate
(675.DO) 1,000. DO
3.50% 35.00
= RATE(H6rH8,H4rH5) 1
Function Arguments
rate
= 12 - 35
i
= 0.Q775SB5O5 ■nt. For example. use !r:: - for quarterly paymer
Formula result = Ů.C"5!£í:5 Hele on this function
Figure 10.15 Completed Data Entry Menu
3. Again, after all arguments are entered through their correct cell references, the answer will appear in the lower left corner of the box. Once satisfied with the result, you can hit Enter to insert this final calculated value in your spreadsheet. This has been set up in this sheet in cell H10.
Calculating a Coca-Cola Bond to Maturity
Earlier, we covered how a financial calculator could be used to determine the YTM of our Coca-Cola bond example. If we wanted to use an Excel spreadsheet to perform this calculation instead of a calculator, we would set up our spreadsheet as shown in the steps below. The current bond price, entered as a negative, is ($952.06). The bond face value of FV is $1,000; the time period is 7 years X 2, or 14 semiannual periods; the coupon rate is or 0.05%; and the coupon payment is $5.00.
1. First, select Formulas from the Excel upper menu bar, and from the dialog box, select Rate (see Figure 10.16).
Insert Function Search for a function:
X
Type a brief description of what you want to do and then click Go
Or select a category: Financial
Go
Select a function:
PRICE
PRICEDISC
PRICEMAT
PV
RATE
RECEIVED
RRI
RATEfn p er, p mt pv, retype, g uessO
Returns the interest rate per period of a loan or an investment. For example, use 6%/4for quarterly payments at 6% APR,
Help on this function
OK
Cancel
Figure 10.16 Using Excel to Enter a Rate Function
310 10 • Bonds and Bond Valuation
2. After the dialog box appears, enter the variables or arguments. As with our earlier examples, we will use the preferred method of identifying arguments with cell addresses (see Figure 10.17).
A B C D E F H 1 J K L M N 0 P Q R
1 2 Data: Function Argument: ľ X
3 RATE
4 Current Bond Price/Value or PV (entered as a negative) (952.06) Nper H~ |+| = 14 *
5 Bond Face Value o r FV 1,000.00 Pint Hfl | + | = 5
6 Periods (7 years x 2-14 semiannual periods) 14 ľ Pv H4 I + I = -952.06
7 Coupon rate (1% / 2 = 0.05%) 0.500% Fv H5j | ♦ | = 1000
S Coupon Payment (1,000 x 3.5% = 35.00) 5.00 Type |4| - number _*J
10 YTM or Rate = RATE(H6,H8,H4rH5) 1 Returns the interest rate per per at ffifc APR, - O.O0SSS0597 od of a loan oran investment. For example, use 6%.''4for quarterly payments
'2 Fv is the future value, or a cash balance you want to attain after the last
'3 payment Is made. If omitted, uses Fv = 0,
'4
'5 Formula result = O.GOB55G59'7
' Č
Help on this function OK 1 Cancel
'7
Figure 10.17 Completed Data Entry Menu
3. Again, after all arguments are entered through their correct cell references, the answer will appear in the lower left corner of the box. Once satisfied with the result, you can hit Enter to insert this final calculated value into your spreadsheet. This has been set up in this sheet in cell H10.
As noted above, remember that this is a semiannual rate because it was calculated using semiannual coupon payments and periods. To express it as an annual YTM rate, you must multiply it by 2.
Calculating the Maturity Period (Term) of a Bond
You can use the following steps in Excel to determine the maturity period or term of a bond. Assume that you are considering investing in a bond that is selling for $820.00, has a face value of $1,000, and has an annual coupon rate of 3%. If the YTM is 10%, how long will it be until the bond matures?
1. First, select Formulas from the Excel upper menu bar, and from the dialog box, select Nper (see Figure 10.18).
Insert Function ? X
Search for a function:
Type a brief description of what you want to do and then click Go
Or select a category: Financial v
Select a function:
NPER MMMMNMMNMMNMMNMB *
NPV
ODDFPRICE
ODDFYIELD
ODDLPRICE
ODDLYIELD
PDURATION
N P ERfrate, p rut p Vjtv.ty pe)
Returns the number of periods for an investment based on periodic, constant payments and a constant interest rate.
Help on this function
OK
Cancel
Figure 10.18 Using Excel to Calculate Bond Time to Maturity
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10.5 • Using Spreadsheets to Solve Bond Problems 311
When the dialog box appears, enter function arguments (see Figure 10.19). Once again, we will use the preferred method of using cell addresses as reference points.
Data:
Current Bond Price/Value or PV (entered as a negative)
YTM or Interest Rate Face Value or FV Coupon Rate
Coupon Payment
Calculate Periods or Time to Maturity (Nper) |=NPER(H5,H8,H4,H6) |
(820.00) 10.00% 1,000 3.00% 30.00
Function Arguments NPER
Rate
PiTit
Hi 1 * 1
H3 1 ± 1
H4 1 ~ 1
H5J 1 * 1
1 - 1
= 0.1 = 30
- 3.11s7s0429
:r or periods for an investment based on periodic, constant payments and a constant
rant to attain afterthe last
Formula result - 3.1187&M29 Help on this function
Figure 10.19 Completed Data Entry Menu
3. When arguments have all been entered, the answer will appear in the lower left of the Function Arguments box, as per the above. We arrive at a final answer of 3.12 years until this bond matures.
Calculating Coupon Rate and Interest (Coupon) Payments
Here is how you would determine the coupon or interest rate and coupon payment using Excel. Assume a $1,000 face value bond is selling for $595, has 20 years until it matures, and has a YTM of 6.5%. What are the coupon rate and the periodic coupon payment amount of the bond?
1. First, select Formulas from the Excel upper menu bar, and from the dialog box, select PMT (see Figure 10.20).
Insert Function Search for a function:
X
Type a brief description of what you want to do and then click Go
Or select a category: Financial
Go
Select a function:
PDURATIQN
PPMT
PRICE
PRICEDISC
PRICEMAT
PV
P MTfrate, n per, pVafv, type)
Calculates the paymentfor a loan based on constant payments and a constant interest rate.
Help on this function
OK
Cancel
Figure 10.20 Using Excel to Enter a PMT or Payment Function
2. When the dialog box appears, enter function arguments. Once again, we will use the preferred method of using cell addresses as reference points (see Figure 10.21).
312 10 • Bonds and Bond Valuation
-H
Data:
Current Bond Price/Value or PV (entered as a negative)
Periods
Face Value or FV YTIA or Interest Rate
Calculate Payment (Coupon Payment)
1,000.00 6.50%
= Plv1TiH7,H5,H4,H5i
Function Arguments
Rate
th* pa, m*n11oi ■
|H7 ± = 0.0&5
I Hi ± = 20
|H4 = -595
H ± - 1000
- - number
n after the last
Formula result - $23.24 Help on this function
Figure 10.21 Completed Data Entry Menu
3. When arguments have all been entered, the answer will appear in the lower left of the Function Arguments box, as per the above. We arrive at a final answer of $28.24 as the coupon payment.
The coupon rate can be calculated by taking this coupon payment amount and dividing it by the face value:
$28.24
$1,000
- 2.824%
So, the coupon rate is 2.824%.
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10 •Summary 313
a Summary
10.1 Characteristics of Bonds
Bonds are typically a basic form of investment that entails a straightforward financial agreement between issuer and purchaser. There are three primary categories of bonds: government bonds, corporate bonds, and convertible bonds. These different types of bond vary depending on their issuer, length until maturity, interest rate, and risk.
10.2 Bond Valuation
It is important to ascertain what a given bond is worth to a willing buyer and a willing seller. We can price a bond using an equation, a calculator, or a spreadsheet. The essential steps are (1) identify the amount and timing of the future cash flow; (2) determine the discount rate; (3) find the present values of the lump sum principal and the annuity stream of coupons; and (4) add the present value of the lump sum principal and the present value of the coupons.
10.3 Using the Yield Curve
When interest rate yields are plotted against their respective maturity periods and these plotted points are connected, the resulting line is called the yield curve. The yield curve is a graphical representation of the term structure of interest rates. A yield curve always shows the value of yields (rates) on they-axis and maturities or time periods on the x-axis.
10.4 Risks of Interest Rates and Default
Because bonds are fixed-income investments, they are subject to a number of risks that could have negative effects on their market value. The most common and best-known risks are interest rate risk and default risk, but there are some others risks that should be understood, such as credit risk, liquidity risk, duration risk, call risk, investment risk, and term risk. To assist potential bond investors in understanding some of these risks, bond ratings have been developed and are regularly published by a number of organizations to express their assessment of the risk quality of various bond issues.
10.5 Using Spreadsheets to Solve Bond Problems
Microsoft Excel can be used to solve common bond problems. It can be used to calculate the value of a coupon bond, the yield to maturity (interest rate) of a bond, the maturity period of a bond, and the coupon rate and interest (coupon) payments of a bond.
¥ Key Terms
bond call a feature of certain bonds or other fixed-income instruments that allows the issuer to repurchase
and retire these instruments before maturity bond price the present, discounted value of the future cash stream generated by a bond; the sum of the
present values of all likely coupon payments and the present value of the par value at maturity bond ratings grades assigned to bonds by rating services that indicate their overall credit quality Business Cycle Dating Committee a subdivision of the National Bureau of Economic Research (NBER), the
US government agency that maintains a chronology of US business cycles call risk the risk that a bond issuer will redeem a callable bond prior to maturity capital gains the increase in a capital asset's value that is realized when the asset is sold cash rate the interest rate that a central bank, such as the Reserve Bank of Australia or the US Federal
Reserve System, will charge commercial banks for loans; also known as the bank rate or the base interest
rate
convertible bonds fixed-income corporate debt securities that yield interest payments but can be converted into a predetermined number of common stock or equity shares
314 10-Key Terms
coupon payment the periodic dollar value of interest that is paid to a bondholder by the bond issuer coupon rate the amount of annual interest paid by the bond issuer; is multiplied by the face value of a bond
to determine annual interest or coupon payment amounts credit risk the risk taken by a bond investor that the bond issuer will default by failing to pay interest and
repay the principal on schedule deep discount bonds bonds that sell at significantly lower values than their par values default when an issuer fails to make scheduled interest or principal payments on its bonds default risk the risk taken by investors that payments will be delayed or will not occur discount bond a bond currently trading for less than its par value in the secondary market; offers a coupon
rate that is lower than prevailing interest rates duration a measure of how much bond prices are likely to change if and when interest rates move duration risk the risk associated with the sensitivity of a bond's price to a 1 % change in interest rates Federal Reserve funds rate (federal funds rate) the target interest rate, set by the Federal Reserve, at
which commercial banks borrow and lend their excess reserves to each other Federal Reserve System (the Fed) the central banking system of the United States, responsible for
administering fiscal policy for the country fixed-income securities investments that provide a return in the form of fixed, periodic interest payments
and the eventual return of principal at maturity; the most common forms are bonds floating-rate bonds bonds with variable interest rates that allow investors to benefit from rising interest
rates
interest income annual interest amounts paid, or coupon payments made, on a bond between its issue date
and the date of maturity interest rate risk the risk of investment losses that result from changes in interest rates investment grade describes a municipal or corporate bond with a rating that indicates it presents a low risk
of default
junk bonds bonds that have been given a low credit rating, below investment grade; riskier than other
bonds due to a greater chance that the issuer will default or experience a credit event liquidity risk risk that stems from the lack of marketability of an investment, meaning that it cannot be
bought or sold quickly enough to prevent or minimize a loss London Interbank Offered Rate (LIBOR) a benchmark interest rate at which major global banks lend to one
another in the international interbank market for short-term loans maturity date the date on which a bondholder ceases to receive interest payments on a bond investment
and instead is repaid its par, or face, value municipal bonds ("munis") debt securities issued by state and local governments; can be thought of as
loans that investors make to local governments to fund infrastructure par value also called the face amount or face value; the value written on the front of the bond, which is the
amount of money that bond issuers promise to be paid at maturity premium bond a bond that is trading above its par value in the secondary market; offers a coupon rate that
is higher than the current prevailing interest rates being offered prime rate the interest rate that banks charge creditworthy corporate customers; among the most widely
used benchmarks for setting home equity lines of credit and credit card rates, based on the federal funds
rate set by the Federal Reserve rating agencies (bond rating services) independent service agencies, such as Fitch, Moody's, or Standard &
Poor's, that perform the isolated function of credit risk evaluation realized return the actual return that an investor earns over a given time period through the buying and
selling of a security
reinvestment risk the risk that an investor will be unable to reinvest cash flows received from an investment (e.g., coupon payments or interest) at a rate comparable to their current rate of return
savings bonds debt securities purchased by investors, as a personal investments or as gifts, that the US government issues to pay for certain public or government programs
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10- CFA Institute 315
term risk the risk of potentially earning lower returns on longer-term bond holdings compared to those
potentially available when making several shorter-term investments over the same period of time US Treasury bills (T-bills) short-term US government debt obligations backed by the Treasury Department
with a maturity of one year or less US Treasury note rate the interest rate that the US government pays to borrow money for different lengths
of time; notes are issued in terms of two, three, five, seven, and 10 years yield curve a line that plots yields (interest rates) of bonds having equal credit quality but differing maturity
dates; gives an idea of future interest rate changes and economic activity yield to maturity (YTM) the total return anticipated on a bond if the investment is held until maturity zero-coupon bonds bonds that are issued at a deep discount from face value and offer no interest or
coupon payments
m CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session (https://openstax.Org/r/study-session-14). Reference with permission of CFA Institute.
a
Multiple Choice
1. When solving bond problems relating to a bond that pays interest on a quarterly basis, the_before
being applied.
a. quoted annual yield to maturity should be multiplied by 4
b. quoted number of years until maturity should be divided by 4
c. quoted annual coupon payments should be divided by 4
d. stated face value should be divided by 4
2. Which of the following is NOT an adjustment that must be made when interest is paid semiannually instead of annually?
a. Dividing the annual coupon payment by 2
b. Dividing the annual interest rate by 2
c. Dividing the total number of years by 2
d. Dividing the annual yield to maturity by 2
3. Which of the following is NOT considered a factor that influences a bondholder's required rate of return?
a. Financial risk
b. Other investments by the bondholder
c. Risk premium
d. Business risk
4. How might an investment in a bond fund be affected by a decline in interest rates?
a. The fund investment would not be affected.
b. The fund investment would likely decrease in value.
c. The fund investment would likely increase in value.
d. Coupon payments from bonds in the fund would decline.
5. Interest rates and bond prices_.
a. are unrelated
b. have an inverse relationship
c. have a direct relationship
d. are both economic factors set by central banks
316 10- Multiple Choice
6. The coupon rate of a bond is typically_.
a. fixed at the time of bond issuance
b. subject to change based on the federal funds rate
c. zero in the case of zero-coupon bonds
d. Both A and C
7. A zero-coupon bond is a bond that_.
a. has no value
b. has no periodic coupon payments
c. has been rated below investment grade
d. Both A and C
8. A bond that has a coupon rate less than prevailing interest rates will_.
a. sell at par value
b. sell at a discount
c. sell at a premium
d. be overpriced
9. Determining bond prices often involves using which two TVM (time value of money) equations?
a. The future value of a lump sum and the present value of a lump sum
b. The present value of an annuity and the future value of a lump sum
c. The future value of an annuity and the present value of a lump sum
d. None of the above
10. A normal yield curve will_.
a. slope downward as it moves along its x-axis (term).
b. slope upward as it moves along its x-axis (yield).
c. fluctuate depending on the federal funds rate
d. slope upward as it moves along its x-axis (term).
11. An inverted yield curve is an indication that_.
a. long-term yields and interest rates are higher than short-term rates
b. the economy is in the process of a significant recovery
c. short-term yields and interest rates are higher than long-term rates
d. the yields to maturity on all bonds are less than market interest rates
12. Bond laddering is_.
a. a risky bond investment strategy that may yield tremendous returns
b. a strategy in which bonds with several different maturity periods are added to a portfolio
c. a strategy that involves replacing equity investments with bonds in a portfolio
d. a bond strategy that sacrifices diversity for potential capital gains
13. A call feature_.
a. is desirable to an investor
b. may cause additional risk for the bond issuer
c. may cause additional risk for an investor
d. Both A and B
14. The duration of a bond is_.
a. a measurement of the bond's overall risk
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10 • Review Questions 317
b. synonymous with the bond's term
c. a measurement of how long an investor holds the bond
d. a measurement of the bond's sensitivity to interest rate changes
5. Review Questions
1. What is a junk bond?
2. Briefly describe interest income within the context of bond investments.
3. Briefly describe the two types of cash flow that a bondholder will receive from the issuer.
4. What is an inverted yield curve, and what is its significance?
5. Describe reinvestment risk for a bondholder.
D Problems
1. A $1,000 Expo Corp. bond has a coupon rate of 5%, pays interest semiannually, and matures in six years. If the yield to maturity is 7%, what is the bond's value today?
2. A $1,000 Omega Corp. bond has an 8% coupon rate that is paid semiannually. The bond matures in three years. If the current price of the bond is $1,125, what is the yield to maturity?
3. You are considering buying a bond that is currently priced at $830, has a face value of $1,000, and matures in seven years. If interest is paid semiannually and the bond has a yield to maturity of 6%, what is the bond's annual coupon rate?
4. A $1,000 Noah Corp. bond has a coupon rate of 5% with semiannual payments, matures in 10 years, and has a yield to maturity of 6.5%. What is the bond's current price?
5. Chronowerx Inc. has issued a bond that has a face value of $1,000, a 3% coupon rate (with semiannual interest), and a maturity date four years from now. If the bond's current price is $895, what is its yield to maturity?
6. You are considering adding a $1,000, 25-year bond to your portfolio. It has a coupon rate of 8%, which is paid annually, and your required return is 10%. What is the current price of the investment?
7. A Cameron Corp. bond has a $1,000 par value, a 5 percent coupon rate paid semiannually, and nine years until maturity. If similar investments yield 6%, what is the current value of Cameron Corp. bonds?
8. McLaren Motors just issued a series of $1,000.00 bonds with a 10-year maturity and an 8% coupon rate, paid quarterly. If you purchase a McLaren bond at a price of $920.00, what is your required rate of return?
9. Three years ago. Petty Partners Inc. issued 15-year, $1,000 bonds that are currently priced at $911.37. If the prevailing rate of return on similar investments is 5%, what is the coupon rate on Petty Partners bonds, and what is the annual interest payment?
10. A $1,000 Riker Corp. bond has a 20-year maturity and a 6% coupon rate, with interest paid annually. If similar bonds from Riker Corp. are yielding 4%, what is the current market value of the Riker issue?
1. A $1,000 bond that matures in eight years, has quarterly coupon payments of $25, and is currently priced at $962.00 will have a yield to maturity of_.
318 10-Video Activity
[►J Video Activity
Using TI BA11+ to Price a Bond
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1. If you are provided with annual information in a bond problem, which bond variables must be adjusted to accommodate semiannual compounding periods, and how must they be adjusted? Review the video to follow along as this information is provided.
2. Following the instructions laid out in the video, practice using a calculator to find the price of a bond under two scenarios. The first scenario should be when compounding periods are annual, or once a year, and the second scenario should be when compounding periods are semiannual, or once every six months.
Start out this exercise with the bond factor input variables that are used in the video: par value of $1,000, annual coupon rate of 4%, time to maturity of 10 years, and yield to maturity of 8%. Calculate the current value or price of the bond under both of the different compounding period scenarios.
Once you have arrived at the same results demonstrated in the video, practice calculating prices using different bond factor variables for inputs, changing time or years to maturity, coupon rate (and coupon payments), and yield to maturity until you are completely confident using a financial calculator to find the price or value of any bond.
Bond Pricing, Valuation, Formulas, and Functions in Excel
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Review the examples included in this video, and practice setting up spreadsheets that solve for each of the five primary bond variables using the values in the videos (maturity 10 years, coupon rate 10%, coupon payment $100, yield to maturity 8%, and par value $1,000). Parallel the spreadsheets that are set up in the video, ensuring that you arrive at the same results for each bond variable amount.
3. Name and describe the five primary bond variables that can be solved using Excel spreadsheets.
4. What are the Excel functions that allow you to perform these calculations, and where in the standard Excel spreadsheet are they located?
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Figure 11.1 In addition to bonds, corporations will often issue common stock as a means of raising necessary capital to finance future operations, (credit: modification of "New York Stock Exchange Huge US Flag Photo i018" by Grant Wickes/flickr, CC BY 2.0)
Chapter Outline
11.1 Multiple Approaches to Stock Valuation
11.2 Dividend Discount Models (DDMs)
11.3 Discounted Cash Flow (DCF) Model
11.4 Preferred Stock
11.5 Efficient Markets
Why It Matters
Similar to bonds, shares of common stock entitle investor owners to a portion of a company's future earnings and cash flows. However, stocks differ significantly from bonds in how they are issued and managed by companies, the methodology used to calculate their values in public markets, and how they can generate income and eventual value for individual investors.
With common stock, there is no specific promise of how much cash investors will receive or when they will receive it. This differs from bond investments, which are valued entirely on the basis of their guaranteed timing of future cash flows to bondholders.
This means that with stocks, there are no maturity dates, face values, or coupon payment guarantees. It also means that stocks do not promise any specified cash flows in the form of coupons or a face value payment at some point in the future. Instead, stocks (only some, not all) may pay dividends. These dividends are declared after shares of stock have been issued by a company and then purchased by the investing public. Following a dividend declaration, the designated per-share amounts are paid to shareholders of record on a specified date, also determined by a company's board.
Because stock investments carry no guarantee of payments to investors, they are far riskier than bonds and other forms of fixed-income investments.
While there are many reasons for an investor to choose to purchase common stock, three of the most
320 11 • Stocks and Stock Valuation
common reasons are
■ to use stocks as instruments or repositories for maintaining value;
■ to accumulate wealth over the term of the stock investment; and
■ to earn income through capital gains and dividend payments.
As with any financial instrument, common stock purchases offer advantages and disadvantages to investors. Important advantages include the following:
■ Returns through dividends and price appreciation of shares can be substantial.
■ Stocks are a liquid form of investment and can be bought or sold within secondary markets relatively easily.
■ Information about companies, markets, and important trends are widely published and readily available to the investing public.
These advantages are significant and lead many individuals to move into stock investments. Yet it is important to realize that stock has some significant disadvantages, which can include the following:
■ General risk levels are greater than with bonds or other fixed-income investments.
■ Timing the buy-and-sell transactions of stock can be tricky and may lead to losses or not taking full advantage of share price opportunities.
■ Dividends (provided that the stock does indeed pay them, as not all do) are uncertain and subject to change based on decisions of company management.
We will discuss these topics in this chapter and cover many of the details regarding why corporations issue common stock and why investors purchase that stock.
11.1
Multiple Approaches to Stock Valuation
Learning Outcomes
By the end of this section, you will be able to:
■ Define and calculate a P/E (price-to-earnings) ratio given company data.
■ Determine relative under- or overvaluation indicated by a P/E (price-to-earnings) ratio.
■ Define and calculate a P/B (price-to-book) ratio given company data.
■ Determine relative under- or overvaluation indicated by a P/B (price-to-book) ratio.
■ Define and detail alternative valuation multipliers, including P/S (price-to-sales) ratio, P/CF (price-to-cashflow) ratio, and dividend yield.
The Price-to Earnings (P/E) Ratio
Experienced investors use a number of different methods to evaluate information on companies and their common stock before deciding on any potential purchase. One of the most popular techniques used by investors and analysts is to study a company's financial statements in order to uncover basic fundamental information on the company. This involves calculating a number of financial ratios that help identify trends, bringing elements of operational performance to light and allowing for clearer analysis and evaluation.
A well-proven analytical approach for investors to use in evaluating common stock is to review the overall market value of the company that issues a stock. One of the most consistently used calculations in this analysis, which has important applications in company and common stock evaluation, is the price-to-earnings (P/E) ratio.
The P/E ratio is computed using the following formula:
Price per Share
P/E Ratio =
Earnings per Share
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11.1 • Multiple Approaches to Stock Valuation 321
The P/E ratio is extremely useful to analysts in that it shows the expectations of the market. Essentially, the P/E ratio is representative of the price an investor must pay for every unit of current (or future) corporate earnings.
Bottom-line earnings are a critical factor in valuing common stock. Investors will always want to know how profitable a company is now as well as how profitable it will be in the future. When a company's bottom line remains relatively flat over a period of time, leaving earnings per share (EPS) relatively unchanged, the P/E ratio can be interpreted as the payback period for the original amount paid for each share of common stock.
For example, the common stock price of Cameo Corp. is currently at $24.00 a share, and its EPS for the year is $4.00. Cameo's P/E ratio is calculated as
P/B - $24-00 _ fx
IVUCameo- $4 0Q -°
This ratio would typically be expressed in the form 6 X. Essentially, this means that investors are willing to pay up to six dollars for every one dollar of earnings. It can also be stated that Cameo stock is currently trading at a multiple of six.
The P/E ratio is typically expressed in two primary ways. The first is as a metric listed by most finance websites and often carries the notation P/E (ttm). This refers to the Wall Street acronym for "trailing 12 months" and signals the company's operating performance over the past 12 months.
Another form of the P/E ratio is known as the forward (or leading) P/E. This uses future earnings projections rather than actual trailing amounts. The leading P/E, sometimes called the estimated price to earnings, is useful for comparing current earnings to future earnings and helps provide a clearer picture of what earnings may look like, assuming there are no major changes in the company's operations or accounting treatments.
Referring back to our calculation for Cameo Corp. above, because the current EPS was used in the calculation, this ratio would be classified as a trailing P/E ratio. If we had used an estimated or projected EPS as the denominator in the calculation, it would then be considered a leading P/E ratio.
Analyzing a company's P/E ratio alone or within a vacuum will actually tell an analyst very little. It is only when a company's P/E is compared to historical P/E ratios or the P/E ratios of other companies in the same industry that it becomes a useful tool for analysis. One of the most important benefits of using comparative P/E ratios is that they can standardize stocks with different prices and various earnings levels.
Generally speaking, it is very difficult to make any conclusions about a stand-alone stock value, such as whether a stock that has a ratio of 8 X is a good buy at its current price or if a stock with a P/E ratio of 35 X is too expensive, without performing any relevant comparisons or further analysis.
Analysts have many different ways to interpret P/E ratio data. One of the most common interpretations is that firms with high P/E ratios should be growth companies. Also, a high P/E ratio could mean that a stock's price is high relative to earnings and possibly overvalued. This could signal a possible undesired downward adjustment in market price in the future.
We can extrapolate from the argument above to put forward the idea that stocks with low P/E ratios should be stabler, more mature organizations. A low P/E might indicate that the current stock price is low relative to earnings and that there may be an opportunity to take advantage of upward price movements and potential investment gains through stock price appreciation.
While this information is often very useful for evaluating stocks and making investment decisions, caution must always be used, as a current stock price may simply be out of line with the company's earning potential, which would mean that price adjustments are likely to occur in the short term. This is why experienced analysts and investors will use multiple evaluation techniques when conducting stock analysis and evaluation and not rely solely on insights provided by a single set of facts or one form of statistical measurement.
322 11 • Stocks and Stock Valuation
The Price-to-Book (P/B) Ratio
Another financial ratio commonly used by investors and analysts is the price-to-book (P/B) ratio, also called the market-to-book (M/B) ratio. This is a financial metric used to evaluate a company's current market value relative to its book value.
The market value, or market capitalization, of a company is defined as the current price of all its outstanding shares of common stock. This is essentially equal to the total value of the company as perceived by the market. For all intents and purposes, the book value is representative of the residual of a company after it has liquidated all assets and paid off all of its liabilities.
Book value can be determined by performing some financial analysis on a company's balance sheet. Essentially, analysts will use the P/B ratio to compare a business's available net assets relative to the current sales price of its stock. The price-to-book-value ratio formula is
„__. Market Price per Share
P/B Ratio = „ , „ ,——--
Book Value per Share
One of the primary uses of the P/B ratio is to understand market perceptions of a particular stock's value. It is often the metric of choice for evaluating financial services firms such as real estate firms, insurance companies, and investment trusts. The P/B ratio has a notable shortcoming, however, in that it does not evaluate companies that have a high level of intangible assets such as patents, trademarks, and copyrights.
Ultimately, this ratio will tell an analyst exactly how much potential investors are willing to pay for each dollar of asset value. The PB(M/B) ratio is computed by dividing the current closing price of the stock by the company's current book value per share, which is calculated by either of the following two equations:
P/R Bctin - Market Capitalization P/B Ratio- NetBookValue
P/B Ratio = Price per Share
Net Book Value per Share
Net book value is equal to net assets, or the total assets minus the total liabilities of the company.
Analysts often consider a low P/B ratio (less than 1) to indicate that a stock is undervalued and a higher ratio (greater than 1) to mean that a stock is overvalued. A low ratio may be an indication that something is wrong with the company or that an investor may be paying too much for any residual value should the company be liquidated.
However, many market experts will argue the exact opposite of the above interpretations. Because of these discrepancies in interpretation and overall variance of opinion, the use of alternate stock valuation metrics, either in addition to or in place of the P/B ratio, is always worth exploring.
In conclusion, the P/B ratio can help a company understand if its net assets are comparable to the market price of its stock. However, as with the P/E ratio, it is always a good idea to compare P/B ratios of companies within the same industry and use them in conjunction with other metrics and analytical methodologies.
1
LINK TO LEARNING
Price-to-Earnings Ratio and Price-to-Book Ratio
Two videos cover the price-to-earnings (P/E) ratio (https://openstax.Org/r/price-to-earnings) and the price-to-book (P/B) ratio (https://0penstax.0rg/r/price-t0-b00k ratio). These are excellent introductory videos that will provide you with helpful information on what these ratios represent and how they can be used in stock valuation.
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11.1 • Multiple Approaches to Stock Valuation 323
Alternative Multipliers
There are two main types of valuation metrics multiples used to value common stock. These are equity multiples and enterprise value (EV) multiples. Additionally, there are two primary methods by which to perform analysis using these multiples. These methods are comparable company analysis (comps) and precedent transaction analysis (precedents).
Experienced financial analysts advocate the use of multiples in valuation analysis for a number of reasons, the most important being that they help generate realistic and sound judgments of enterprise values (total company values), they are relatively easy to use and interpret, and they can provide helpful information on a company's overall financial condition when used appropriately.
However, it should be noted that simplicity may have some important disadvantages. When such complex information is reduced to a single equation or final value, it can easily be misunderstood, and the influence of important factors may be masked or lost in the evaluation process.
What's more, the calculation of multiples represents a snapshot in time for a firm and cannot easily show how a company grows or progresses. Thus, these calculations are only applicable to short-term analysis, not to long-term scenarios.
Equity Multiples
Equity multiples are especially useful for investment decisions when an investor aspires to minority positions in companies. Below are some common equity multiples used in valuation analyses.
The price-to-earnings (P/E) ratio, which we discussed earlier, is probably the most common equity multiple used in stock valuation because it is relatively simple to calculate and all necessary data are easily accessible by analysts and investors. The market-to-book (M/B), or price-to-book (P/B), ratio is also useful if assets primarily drive a company's earnings. Again, it is computed as the proportion of share price to book value per share.
Dividend yield is another form of equity multiple and is primarily used when conducting comparisons between cash returns and investment types. Dividend yield is computed as the proportion of dividend per share to share price. The price-to-sales (P/S) ratio is an additional metric used for firms that are experiencing financial losses. The P/S ratio is often used for quick estimates and is computed as the proportion of share price to sales (or revenue) per share.
Another useful metric is the price-to-cash-flow (P/CF) ratio. The P/CF ratio is used to compare a company's market value to its operating cash flow (or the company's stock price per share to its operating cash flow per share). This measurement is suitable only in certain cases, such as when a company has substantial noncash expenses (e.g., depreciation or amortization). In some situations, companies may have positive cash flows but still show a bottom-line loss due to large noncash expenses. The P/CF ratio is helpful for arriving at a less distorted view of such a company's value.
While these various metrics are important, a financial analyst must always consider that companies often operate under their own unique sets of circumstances that ultimately will influence many of these equity multiples.
Enterprise Value (EV) Multiples
The following are some common EV multiples used in valuation analyses: Gordon growth model is as follows
Enterprise Value
Revenue Enterprise Value EBIT
324 11 • Stocks and Stock Valuation
Enterprise Value
EBITDA Enterprise Value
EBITDAR
EV multiples take an increasingly important role when value decisions surround recent mergers and acquisitions. Enterprise value (EV) is a measurement of the total value of a company. Companies often believe that EV offers a more accurate representation of a firm's total value than a basic market capitalization method. Generally, EV is perceived to offer an aggregate value of the firm as an enterprise, which is a more comprehensive measurement (see Figure 11.2).
Figure 11.2 Enterprise Value: The Complete Picture
Following are two of the most common enterprise values metrics used in valuing companies and their common stock:
EV/Revenue (EV/R). Also called EV/Sales, EV/R is a valuation metric used to understand a company's total valuation compared to its annual sales levels. EV/R can help provide an analyst with an idea of exactly how much investors pay for every dollar of a company's sales revenue.
EV/R is considered a relatively crude metric but can be useful when analyzing companies that have different methods of revenue recognition. P/E ratios, for example, can be significantly affected by changes in the accounting policies of companies being evaluated. This is another reason why multiple metrics should be used in valuations.
Additionally, EV/R is a useful measure for companies that are consuming cash or experiencing financial losses. Such companies may be start-ups or emerging technology firms that have not fully matured and are still in a growth stage of development.
EV/EBIT. A firm's earnings before interest and taxes (EBIT) is an indicator of its profitability before the effects of interest or taxes. EBIT is also referred to as operating earnings, operating profit, and profit before interest and tax.
■ EV/EBITDA. EV/EBITDA is a ratio that compares a company's enterprise value (EV) to its earnings before interest, taxes, depreciation, and amortization (EBITDA). EBITDA is often used by analysts as a substitute for cash flow and can be applied to capital analysis using tools such as net present value and internal rate of return. It is relatively easy to calculate, as all information required to compete the calculation is
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11.2- Dividend Discount Models (DDMs) 325
available from any publicly traded company's financial statements. Because of this, the EV/EBITDA ratio is a commonly used metric to compare the relative values of different businesses. ■ EV/EBITDAR. Another form of valuation based on enterprise value is EV/EBITDAR. This metric divides enterprise value by earnings before interest, tax, depreciation, amortization, and rental costs (EBITDAR). This multiple is used in businesses that have substantial rental and lease expenses, such as hotel chains and airlines. Capital investment can differ significantly for these firms, and when assets are leased, these companies tend to have artificially lower debt and operating income compared to firms that actually own their assets.
EV/Capital Employed. The EV-to-capital employed ratio is a measure of enterprise value compared to the level of capital used by a business. For example, a business with a large capital basis is bound to carry a large enterprise value simply due to its large capital holdings.
Final Thoughts on Valuation Ratios and Multiples
There are many equity and enterprise value multiples used in company valuation, but the discussions above cover those that are most commonly used. In any case, gaining a thorough understanding of each multiple and its related concepts can help analysts make better use of these metrics in their stock analysis and valuation efforts. Also, as discussed, it is important that analysts and technicians use multiple ratios and alternate measures for any evaluation of a company and its common stock. Not doing so will limit the ultimate interpretation of the results, can lead to incorrect conclusions, and may cause fundamental mistakes in overall investment strategy.
LINK TO LEARNIN
Graph of Historical P/E for S&P 500
Look at the 90-year historical average P/E ratio of the S&P 500 (https://0penstax.0rg/r/
90-year historical average P/E). When viewing this information on historical P/E ratios, think about the
following:
■ Note that the general trend for historical P/E ratios has been one of growth.
■ Note that in May 2009, the P/E ratio reached a staggering 123.73 X, the highest ratio in US history. This was primarily due to depressed earnings during the Great Recession.
11.2
Dividend Discount Models (DDMs)
Learning Outcomes
By the end of this section, you will be able to:
Identify and use DDMs (dividend discount models). Define the constant growth DDM.
List the assumptions and limitations of the Gordon growth model. Understand and be able to use the various forms of DDM. Explain the advantages and limitations of DDMs.
The dividend discount model (DDM) is a method used to value a stock based on the concept that its worth is the present value of all of its future dividends. Using the stock's price, a required rate of return, and the value of the next year's dividend, investors can determine a stock's value based on the total present value of future dividends.
This means that if an investor is buying a stock primarily based on its dividend, the DDM can be a useful tool to determine exactly how much of the stock's price is supported by future dividends. However, it is important to
326 11 • Stocks and Stock Valuation
understand that the DDM is not without flaws and that using it requires assumptions to be made that, in the end, may not prove to be true.
The Gordon Growth Model
The most common DDM is the Gordon growth model, which uses the dividend for the next year (D-i), the required return (r), and the estimated future dividend growth rate (g) to arrive at a final price or value of the stock. The formula for the Gordon growth model is as follows:
Stock Value =-—
r ~ g
This calculation values the stock entirely on expected future dividends. You can then compare the calculated price to the actual market price in order to determine whether purchasing the stock at market will meet your requirements.
Dividend Discount Model
Watch this short video on the dividend discount model (https://0penstax.0rg/r/ short video on the dividend) and how it is used it in stock valuation and analysis.
The Gordon growth model equation is presented and then applied to a sample problem to demonstrate how the DDM yields an estimated share price for the stock of any company.
Now that we have been introduced to the basic idea behind the dividend discount model, we can move on to cover other forms of DDM.
Zero Growth Dividend Discount Model
The zero growth DDM assumes that all future dividends of a stock will be fixed at essentially the same dollar value forever, or at least for as long as an individual investor holds the shares of stock. In such a case, the stock's intrinsic value is determined by dividing the annual dividend amount by the required rate of return:
Annual Dividends
Stock Value =
Required Rate of Return
When examined closely, it can be seen that this is the exact same formula that is used to calculate the present value of a perpetuity, which is
„ , Dividend
Present Value = —:-
Discount Rate
For the purpose of using this formula in stock valuation, we can express this as
PV=^ r
where PV is equal to the price or value of the stock, D represents the dividend payment, and r represents the required rate of return.
This makes perfect sense because a stock that pays the exact same dividend amount forever is no different from a perpetuity—a continuous, never-ending annuity—and for this reason, the same formula can be used to price preferred stock. The only factor that might alter the value of a stock based on the zero-growth model would be a change in the required rate of return due to fluctuations in perceived risk levels.
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11.2- Dividend Discount Models (DDMs) 327
Example:
What is the intrinsic value of a stock that pays $2.00 in dividends every year if the required rate of return on similar investments in the market is 6%?
Solution:
We can apply the zero growth DDM formula to get
«2 00
Stock Value = -—- = $33.33 0.6
While this model is relatively easy to understand and to calculate, it has one significant flaw: it is highly unlikely that a firm's stock would pay the exact same dollar amount in dividends forever, or even for an extended period of time. As companies change and grow, dividend policies will change, and it naturally follows that the payout of dividends will also change. This is why it is important to become familiar with other DDMs that may be more practical in their use.
Constant Growth Dividend Discount Model
As indicated by its name, the constant growth DDM assumes that a stock's dividend payments will grow at a fixed annual percentage that will remain the same throughout the period of time they are held by an investor. While the constant growth DDM may be more realistic than the zero growth DDM in allowing for dividend growth, it assumes that dividends grow by the same specific percentage each year. This is also an unrealistic assumption that can present problems when attempting to evaluate companies such as Amazon, Facebook, Google, or other organizations that do not pay dividends. Constant growth models are most often used to value mature companies whose dividend payments have steadily increased over a significant period of time. When applied, the constant growth DDM will generate the present value of an infinite stream of dividends that are growing at a constant rate.
The constant growth DDM formula is
Stock Value = ™±A = ^L.
r - g r - g
where D0 is the value of the dividend received this year, D-| is the value of the dividend to be received next year, g is the growth rate of the dividend, and r is the required rate of return.
As can be seen above, after simplification, the constant growth DDM formula becomes the Gordon growth model formula and works in the same way. Let's look at some examples.
Constant Growth DDM: Example 1
If a stock is paying a dividend of $5.00 this year and the dividend has been steadily growing at 4% annually, what is the intrinsic value of the stock, assuming an investor's required rate of return of 8%?
Solution:
Apply the constant growth DDM formula:
Stock Value
Simplify to the Gordon growth model:
Dl = A)(l
Dp (1 + g) _ $5.00(1 + 0.04) r-g ~ 0.08-0.04
+ g) = $5.00 X 1.04 = $5.20
328 11 • Stocks and Stock Valuation
Stock Value = —— = * " , = $130.00 r-g 0.08 -0.04
HINK IT THROUG
Constant Growth DDM: Example 2
If a stock is selling at $250 with a current dividend of $10, what would be the dividend growth rate of this stock, assuming a required rate of return of 12%?
Solution:
Apply the constant growth DDM formula:
$250 =
$10 X (1 + g) 0.12 - g
Simplify and continue the calculation:
30 - 250g = 10 + lOg 0 = 10 + 260g 20 = 260g 7.69% = g
So, the growth rate is 7.69%.
LINK TO LEARNIN
Dividend Discount Model: A Complete Animated Guide
In this video for the Investing for Beginners course and podcast, Andrew Sather introduces the DDM. (https://0penstax.0rg/r/Andrew Sather introduces the DDM) demonstrating both the constant growth DDM (Gordon growth model) and the two-stage DDM.
Variable or Nonconstant Growth Dividend Discount Model
Many experienced analysts prefer to use the variable (nonconstant) growth DDM because it is a much closer approximation of businesses' actual dividend payment policies, making it much closer to reality than other forms of DDM. The variable growth model is based on the real-life assumption that a company and its stock value will progress through different stages of growth.
The variable growth model is estimated by extending the constant growth model to include a separate calculation for each growth period. Determine present values for each of these periods, and then add them all together to arrive at the intrinsic value of the stock. The variable growth model is more involved than other DDM methods, but it is not overly complex and will often provide a more realistic and accurate picture of a stock's true value.
As an example of the variable growth model, let's say that Maddox Inc. paid $2.00 per share in common stock dividends last year. The company's policy is to increase its dividends at a rate of 5% for four years, and then the growth rate will change to 3% per year from the fifth year forward. What is the present value of the stock if the required rate of return is 8%? The calculation is shown in Table 11.1.
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11.2- Dividend Discount Models (DDMs) 329
... Growth % Dividend ($) Value after Year 4 PV Discount Factor at 8% Present Value of Dividend ($)
0 5% 2.00
1 5% 2.10 1.0800 1.9444
2 5% 2.21 1.1664 1.8904
3 5% 2.32 1.2597 1.8379
4 5% 2.43 1.3605 1.7869
5 3% 2.50 50.07886 1.4693 35.7870
Total: $43.2466
Table 11.1 Value of Stock with 8% Required Rate of Return
Note:
ValueafterYear4 = 2A3t}?PS ^i°3 = 50.078858 0.08 - 0.03
The value of Maddox stock in this example would be $43.25 per share. Two-Stage Dividend Discount Model
The two-stage DDM is a methodology used to value a dividend-paying stock and is based on the assumption of two primary stages of dividend growth: an initial period of higher growth and a subsequent period of lower, more stable growth.
The two-stage DDM is often used with mature companies that have an established track record of making residual cash dividend payments while experiencing moderate rates of growth. Many analysts like to use the two-stage model because it is reasonably grounded in reality. For example, it is probably a more reasonable assumption that a firm that had an initial growth rate of 10% might see its growth drop to a more modest level of, say, 5% as the company becomes more established and mature, rather than assuming that the firm will maintain the initial growth rate of 10%. Experts tend to agree that firms that have higher payout ratios of dividends may be well suited to the two-stage DDM.
As we have seen, the assumptions of the two-stage model are as follows:
■ The first period analyzed will be one of high initial growth.
■ This stage of higher growth will eventually transition into a period of more mature, stable, and sustainable growth at a lower rate than the initial high-growth period.
■ The dividend payout ratio will be based on company performance and the expected growth rate of its operations.
Let's use an example. Lore Ltd. estimates that its dividend growth will be 13% per year for the next five years. It will then settle to a sustainable, constant, and continuing rate of 5%. Let's say that the current year's dividend is $14 and the required rate of return (or discount rate) is 12%. What is the current value of Lore Ltd. stock?
Step 1:
First, we will need to calculate the dividends for each year until the second, stable growth rate phase is reached. Based on the current dividend value of $14 and the anticipated growth rate of 13%, the values of dividends (D-|, D2, Q3, D4, D5) can be determined for each year of the first phase. Because the stable growth rate is achieved in the second phase, after five years have passed, if we assume that the current year is 2021, we can lay out the profile for this stock's dividends through the year 2026, as per Figure 11.3.
330 11 • Stocks and Stock Valuation
1
2
3
4
5
6
Step 1
Current 2021
Dividend at 13% Growth
14.00
Growth Rate of Dividends
N/A
2022 15.82
13%
2023 17.88
13%
2024 20.20
2025 22.83
13%
13%
2026 25.79
13%
Figure 11.3 Profile of Stock Dividend through 2026
Step 2:
Next, we apply the DDM to determine the terminal value, or the value of the stock at the end of the five-year high-growth phase and the beginning of the second, lower growth-phase.
We can apply the DDM formula at any point in time, but in this example, we are working with a stock that has constant growth in dividends for five years and then decreases to a lower growth rate in its secondary phase. Because of this timing and dividend structure, we calculate the value of the stock five years from now, or the terminal value. Again, this is calculated at the end of the high-growth phase, in 2026. By applying the constant growth DDM formula, we arrive at the following:
DN(l + g) DN+l
Stock Valuejv =
r ~ g
r - g
The terminal value can be calculated by applying the DDM formula in Excel, as seen in Figure 11.4 and Figure 11.5. The terminal value, or the value at the end of 2026, is $386.91.
A B C D E F G H
9 Step 2A
10 Current
11 2D21 2022 2023 2024 2025 2026 2027
12
13 Dividend at 13% Growth 14.00 15.82 17.88 20.20 22.83 25.79 27.0B
14
15 Growth Rate of Dividends N/A 13% 13% 13% 13% 13% 5%
16 Terminal Value =H13/(B17-H15)
17 Required Rate of Return 0.12
Figure 11.4 Terminal Value at the End of 2026 (Showing Formula)
A B C D E F G H
19 Step 2B
20 Current
21 2021 2022 2023 2024 2025 2026 2027
22
23 Dividend at 13% Growth 14.00 15.82 17.88 20.20 22.83 25.79 27.08
24
25 Growth Rate of Dividends N/A 13% 13% 13% 13% 13% 5%
26 Terminal Value 386.91
21 Required Rate of Return 0.12
Figure 11.5 Terminal Value at the End of 2026
Step 3:
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11.2- Dividend Discount Models (DDMs) 331
Next, we find the PV of all paid dividends that occur during the high-growth period of 2022-2026. This is shown in Figure 11.6. Our required rate of return (discount rate) is 12%.
A E C D E F G H
Step 3 Current
2021 2022 2023 2024 2025 2026 2027
Dividend at 13% Growth 14.00 15.82 17.88 20.20 22.83 25.79 27.08
Growth Rate of Dividends N/A 13% 13% 13% 13% 13% 5%
Terminal Value 386.91
Required Rate of Return 0.12
Present Value of Dividends N/A 14.13 14.25 14.33 14.51 14.64
Figure 11.6 Present Value of All Paid Dividends, 2022-2026
Step 4:
Next, we calculate the PV of the single lump-sum terminal value:
Future Value (FV) = 386.91 Interest Rate (I/Y) = 12 Number of Periods (N) = 5 Payment (PMT) = 0
Compute Present Value (CPTPV) = 219.54
Remember that due to the sign convention, either the FV must be entered as a negative value or, if entered as a positive value, the resulting PV will be negative. This example shows the former.
Step 5:
Our next step is to find the current fair (intrinsic) value of the stock, which comprises the PV of all future dividends plus the PV of the terminal value. This is represented in the following formula, with all factors shown in Figure 11.7:
Fair Value = PV Projected Dividends + PV Terminal Value
A B C D E F G H I
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
Step 5
Dividend at 13% Growth
Growth Rate of Dividends Terminal Value
Current
2021 2022 2023 2024 2025 2026 2027 14.00 15.82 17.88 20.20 22.83 25.79 27.08
Required Rate of Return
N/A 0.12
13% 13%
13% 13% 13% 386.91
1%
Present Value of Dividends N/A 14.13 14.25 14.33 14.51 14.64
Present Value of Terminal Value 219.54
Fair Value
71.90 219.54
291.44
Figure 11.7 Fair Value of the Stock
332 11 • Stocks and Stock Valuation
So, we end up with a total current fair value of Lore Ltd. stock of $291.44 (due to Excel's rounding), although the sum can also be calculated as shown below:
Fair Value = 14.13 + 14.25 + 14.38 + 14.51 + 14.64 + 219.54 = 291.45
Determining Stock Value
Take a few minutes to review this video, which covers methods used to determine stock value (https://openstax.Org/r/covers methods used to determine stock value) when dividend growth is nonconstant.
Advantages and Limitations of DDMs
Some of the primary advantages of DDMs are their basis in the sound logic of present value concepts, their consistency, and the implication that companies that pay dividends tend to be mature and stable entities. Also, because the model is essentially a mathematical formula, there is little room for misinterpretation or subjectivity. As a result of these advantages, DDMs are a very popular form of stock evaluation that most analysts show faith in.
Because dividends are paid in cash, companies may keep making their dividend payments even when doing so is not in their best long-term interests. They may not want to manipulate dividend payments, as this can directly lead to stock price volatility. Rather, they may manipulate dividend payments in the interest of buoying up their stock price.
To further illustrate limitations of DDMs, let's examine the Concepts in Practice case.
CONCEPTS IN PRACTICE
Limitations of DDMs
A major limitation of the dividend discount model is that it cannot be used to value companies that do not pay dividends. This is becoming a growing trend, particularly for young high-tech companies. Warren Buffett, CEO of Berkshire Hathaway, has stated that companies are usually better off if they take their excess funds and reinvest them into infrastructure, evolving technologies, and other profitable ventures. The payment of dividends to shareholders is "almost a last resort for corporate management,"1 says Buffett, and cash balances should be invested in "projects to become more efficient, expand territorially, extend and improve product lines or... otherwise widen the economic moat separating the company from its competitors." Berkshire follows this practice of reinvesting cash rather than paying dividends, as do tech companies such as Amazon, Google, and Biogen.3 So, rather than receiving cash dividends, stockholders of these companies are rewarded by seeing stock price appreciation in their investments and ultimately large capital gains when they finally decide to sell their shares.
The sensitivity of assumptions is also a drawback of using DDMs. The fair price of a stock can be highly sensitive to growth rates and the required rates of return demanded by investors. A single percentage point change in either of these two factors can have a dramatic impact on a company's stock, potentially changing it by as much as 10 to 20%.
1 Dan Caplinger. "Why Don't These Winning Stocks Pay Dividends?" The Motley Fool. Updated October 3, 2018. https://www.fool.com/investing/general/2015/03/01/why-dont-these-winning-stocks-pay-dividends.aspx
2 The Motley Fool. "Why Warren Buffet's Berkshire Hathaway Won't Pay a Dividend in 2015." Nasdaq. November 16, 2014. https://www.nasdaq.com/articles/why-warren-buffetts-berkshire-hathaway-wont-pay-dividend-2015-2014-11-16.
3 Caplinger, "Why Don't These Winning Stocks Pay Dividends?" The Motley Fool.
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11.2- Dividend Discount Models (DDMs) 333
Finally, the results obtained using DDMs may not be related to the results of a company's operations or its profitability. Dividend payments should theoretically be tied to a company's profitability, but in some instances, companies will make misguided efforts to maintain a stable dividend payout even through the use of increased borrowing and debt, which is not beneficial to an organization's long-term financial health.
(sources: www.wallstreetmojo.com/dividend-discount-model/; pages.stern.nyu.edu/~adamodar/pdfiles/ valn2ed/ch13d.pdf; www.managementstudyguide.com/disadvantages-of-dividend-discount-model.htm)
Stock Valuation with Changing Growth Rates and Time Horizons
Before we move on from our discussion of dividend discount models, let's work through some more examples of how the DDM can be used with a number of different scenarios, changing growth rates, and time horizons.
As we have seen, the value or price of a financial asset is equal to the present value of the expected future cash flows received while maintaining ownership of the asset. In the case of stock, investors receive cash flows in the form of dividends from the company, plus a final payout when they decide to relinquish their ownership rights or sell the stock.
Let's look at a simple illustration of the price of a single share of common stock when we know the future dividends and final selling price.
Steve wants to purchase shares of Old Peak Construction Company and hold these common shares for five years. The company will pay $5.00 annual cash dividends per share for the next five years.
At the end of the five years, Steve will sell the stock. He believes that he will be able to sell the stock for $25.00 per share. If Steve wants to earn 10% on this investment, what price should he pay today for this stock?
The current price of the stock is the discounted cash flow that Steve will receive over the next five years while holding the stock. If we let the final price represent a lump-sum future value and treat the dividend payments as an annuity stream over the next five years, we can apply the time value of money concepts we covered in earlier chapters.
Method 1: Using an Equation
Problem:
Solution:
1 -
1
Price = Future Price X
+ Dividend Stream x
(l + ry
(1 + r)n
r
1 -
1
= $25.00 X -1—r + $5.00 X
(1 + 0.10)5
(1 + 0.10)5
0.10
= $25.00 X 0.6209 + $5.00 X 3.7908 = $15.52 + $18.95 = $34.47
Method 2: Using a Financial Calculator
We can also use a calculator or spreadsheet to find the price of the stock (see Table 11.2).
334 11 • Stocks and Stock Valuation
Step Description Enter Display
1 Clear calculator register CE/C 0.00
2 Enter number of periods (5) 5N N = 5.00
3 Enter rate of return or interest rate (10% 10 I/Y I/Y = 10.00
4 Enter eventual sales price ($25) 25 FV FV = 25.00
5 Enter dividend amount ($5) 5 PMT PMT = 5.00
6 Compute present value CPT PV PV = - 34.47
Table 11.2 Calculator Steps for Finding the Price of the Stock4
The stock price is calculated as $34.47.
Note that the value given is expressed as a negative value due to the sign convention used by financial calculators. We know the actual stock value is not negative, so we can just ignore the minus sign.
In cases such as the above, we find the present value of a dividend stream and the present value of the lumpsum future price. So, if we know the dividend stream, the future price of the stock, the future selling date of the stock, and the required return, it is possible to price stocks in the same manner that we price bonds.
Method 3: Using Excel
Figure 11.8 shows a spreadsheet setup in Excel to reach a solution to this problem.
A B C D E F
1
2 Nper (N) or Number of Periods 5
3 Rate of Return or Interest Rate 10%
4 Future Value or Stock Sale Price $ 25.00
5 Payment (Dividend Amount) $ 5.00
6 $ (34.48)
Figure 11.8 Excel Solution for Finding the Price of the Stock
Due to the sign convention in Excel, we can ignore the parentheses around the solution, which indicate a negative value. Therefore, the price is $34.48. The Excel command used in cell F6 to calculate present value is as follows:
=PV(rate,nper,pmt,[fv],[type])
Finding Stock Price with Constant Dividends Example 1:
Four Seasons Resorts pays a $0.25 dividend every quarter and will maintain this policy forever. What price should you pay for one share of common stock if you want an annual return of 10% on your investment?
Solution:
You can restate your annual required rate of 10% as a quarterly rate of 2.5% (^p-)- Apply the quarterly dividend amount and the quarterly rate of return to determine the price:
4 The specific financial calculator in these examples is the Texas Instruments BA II Plus™ Professional model, but you can use other financial calculators for these types of calculations.
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11.2- Dividend Discount Models (DDMs) 335
Price —
Dividend
Price = |m = $10.00
Even though we anticipate that companies will be in business "forever," we are not going to own a company's stock forever. Therefore, the dividend stream to which we would have legal claim is only for that period of the company's life during which we own the stock. We need to modify the dividend model to account for a finite period when we will sell the stock at some future time. This modification brings us from an infinite to a finite dividend pricing model, which we will use to price a finite amount of dividends and the future selling price of the stock. We will maintain a constant dividend assumption. Let's assume we will hold a share in a company that pays a $1 dividend for 20 years and then sell the stock.
Method 1: Using an Equation
The dividend pricing model under a finite horizon is a concept we have seen earlier. It is a simple present value annuity stream application:
Value of Future Dividends for Specific Periods = Dividend X PVIFA (Present Value Interest Factor of an Annuity)
l_
Dividend Stream X
l
(1 + r)»
= $1.00 X
1 -
(1 + 0.10)20
0.10
= $1.00 X 8.5136 = $8.51
We now need to determine the selling price that we would get in 20 years if we were to sell the stock to someone else at that time. What would a willing buyer give us for the stock 20 years from now? This price is difficult to estimate, so for the sake of this exercise, we will assume that the price in 20 years will be $30. So, what is the present value of the price in 20 years with a 10% discount rate? Again, this is just a simple application of the PV formula we covered earlier in the text:
Pnc^O _ $30
PV =
(1 + r}
,20
(1.10)
,20
= $4.46
We can now price the stock as if it were a bond with a dividend stream of 20 years, a sales price in 20 years, and a required return of 10%:
■ The dividend stream is analogous to the coupon payments.
■ The sales price is analogous to the bond's principal.
■ The 20-year investment horizon is analogous to the bond's maturity date.
■ The required return is analogous to the bond's yield.
Carrying on with the PV calculations, we have
Price = $30 X
+
$1.00 x 1 -
(1 + 0.10)20
(1 + 0.10)20 = $4.46 + $8.51 = $12.97
0.10
Method 2: Using a Financial Calculator
We can also use a calculator or spreadsheet to find the price of the stock using constant dividends (see Table 11.3).
Enter Display
1 Clear calculator register
CE/C
0.00
Table 11.3 Calculator Steps for Finding the Price of Stock Using Constant Dividends
336 11 • Stocks and Stock Valuation
Step Description Enter Display
2 Enter number of periods (20) 20 N N = 20.00
3 Enter rate of return or interest rate (10%] 10 I/Y I/Y= 10.00
4 Enter eventual sales price (30) 30 FV FV = 30.00
5 Enter dividend amount ($1) 1 PMT PMT= 1.00
6 Compute present value CPT PV PV= -12.97
Table 11.3 Calculator Steps for Finding the Price of Stock Using Constant Dividends
The stock price resulting from the calculation is $12.97. Method 3: Using Excel
This same problem can be solved using Excel with a setup similar to that shown in Figure 11.9.
a b C d e f
e
9 Nper (N) or Number of Periods 20
10 Rate of Return or Interest Rate 10%
11 Payment (Dividend Amount) $ 1.00
12 Future Value or Stock Sale Price $ 30.00
13 $ (12.97)
Figure 11.9 Excel Solution for Finding the Price of Stock Using Constant Dividends
Once again, we can ignore the negative indicator that is generated by the Excel sign convention because we know that the stock will not have a negative value 20 years from now. Therefore, the price is $12.97. The Excel command used in cell F13 to calculate present value is as follows:
=PV(rate,nper,pmt,[fv],[type]:
Example 2:
Let's look at an example and estimate current stock price given a 10.44% constant growth rate of dividends forever and a desired return on the stock of 13.5%. We will assume that the current stock owner has just received the most recent dividend, D0, and the new buyer will receive all future cash dividends, beginning with D-|. This part of the setup of the model is important because the price reflects all future dividends, starting with D-|, discounted back to today. (Price0 refers to the price at time zero, or today.) The first dividend the buyer would receive is one full period away. Using the discounted cash flow approach, we have
Dpxg + g)1 , Dpxg + g)2 , Joxg + g)3 , A>x(i + g)4
rnceo =-;- + -^- + -^- +
(1 + r)1
(1 + rY
(1 + ff
where g is the annual growth rate of the dividends and r is the required rate of return on the stock. We can simplify the equation above into the following:
D0 x (1 + g) Pnceo = ur_g-
Dl = D0 X (1+g) Pnce0 = y^i
As we discussed above, this classic model of constant dividend growth, known as the Gordon growth model, is
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11.2- Dividend Discount Models (DDMs) 337
a fundamental method of stock pricing. The Gordon growth model determines a stock's value based on a future stream of dividends that grows at a constant rate. Again, we assume that this constantly growing dividend stream will pay forever. To see how the constant growth model works, let's use our example from above once again as a test case. The most recent dividend (Do) is $1.76, the growth rate (g) is 10.44%, and the required rate of return (r) is 13.5%, so applying our PV equation, we have
p • _ $1.76 X (1 + 0.1044) _ $1.943774 _ *^ so rnceo - 0.135 - 0.1044 ~~ 0.0306 ~~
Our estimated price for this example is $63.52. Notice that the formula requires the return rate rto be greater than the growth rate g of the dividend stream. If g were greater than r, we would be dividing by a negative number and producing a negative price, which would be meaningless.
Let's pick another company and see if we can apply the dividend growth model and price the company's stock with a different dividend history. In addition, our earlier example will provide a shortcut method to estimate g, although you could still calculate each year's percentage change and then average the changes over the 10 years.
Estimating a Stock Price from a Past Dividend Pattern Problem:
Phased Solutions Inc. has paid the following dividends per share from 2011 to 2020:
2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
$0.070 $0.080 $0.925 $1.095 $1.275 $1.455 $1.590 $1.795 $1.930 $2.110
Table 11.4
If you plan to hold this stock for 10 years, believe Phased Solutions will continue this dividend pattern forever, and you want to earn 17% on your investment, what would you be willing to pay per share of Phased Solutions stock as of January 1, 2021 ?
Solution:
First, we need to estimate the annual growth rate of this dividend stream. We can use a shortcut to determine the average growth rate by using the first and last dividends in the stream and the time value of money equation. We want to find the average growth rate given an initial dividend (present value) of $0.70, the most recent dividend (future value) of $2.11, and the number of years (n) between the two dividends, or the number of dividend changes, which is 9. So, we calculate the average growth rate as follows:
= (S&)* " 1 = 3.0142857*-1 = 0.1304, or 13.04%
Table 11.5 shows the step-by-step process of using a financial calculator to solve for the growth rate.
Step Description Enter Display
1 Clear calculator register ce/c 0.0000
2 Enter number of periods (9) 9n n= 9.0000
Enter present value or initial dividend ($0.70) as a neqative . _,
3 K 0.7 +1 - PV PV = -0.7000 value
Table 11.5 Calculator Steps for Solving the Growth Rate
338 11 • Stocks and Stock Valuation
Step Description Enter Display
4 Enter future value or the most recent dividend ($2.11) 2.11 FV FV= 2.1100
5 Enter a zero value for payment as a placeholder, as this factor is not used here 0 PMT PMT 0.0000
6 Compute annual growth rate CPT I/Y I/Y= 13.0427
Table 11.5 Calculator Steps for Solving the Growth Rate
The calculated growth rate is 13.04%.
We can also use Excel to set up a spreadsheet similar to the one in Figure 11.10 that will calculate this growth rate.
D
15
21
Present Value or Current Dividend Amount (enter as a negative value) S (0.70)
Nper (N) or Number of Periods 9
Payment or PMT (unused here, so enter zero as a placeholder)
Future Value (dividend amount in nine years) 13.04% $ 2.11
13.04-27%
Growth Rate
Figure 11.10 Excel Solution for Growth Rate
The Excel command used in cell G22 to calculate the growth rate is as follows:
=RATE(nper,pmt,pv,[fv],[type],[guess])
We now have two methods to estimate g, the growth rate of the dividends. The first method, calculating the change in dividend each year and then averaging these changes, is the arithmetic approach. The second method, using the first and last dividends only, is the geometric approach. The arithmetic approach is equivalent to a simple interest approach, and the geometric approach is equivalent to a compound interest approach.
To apply our PV formula above, we had to assume that the company would pay dividends forever and that we would hold on to our stock forever. If we assume that we will sell the stock at some point in the future, however, can we use this formula to estimate the value of a stock held for a finite period of time? The answer is a qualified yes. We can adjust this model for a finite horizon to estimate the present value of the dividend stream that we will receive while holding the stock. We will still have a problem estimating the stock's selling price at the end of this finite dividend stream, and we will address this issue shortly. For the finite growing dividend stream, we adjust the infinite stream in our earlier equation to the following:
/1 4.
1
1 + £ Price0 = Do X-- X
r - g
- (mi
where n is the number of future dividends.
This equation may look very complicated, but just focus on the far right part of the model. This part calculates the percentage of the finite dividend stream that you will receive if you sell the stock at the end of the nth year. Say you will sell Johnson & Johnson after 10 years. What percentage of the $60.23 (the finite dividend stream) will you get? Begin with the following:
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11.2- Dividend Discount Models (DDMs) 339
10 Years: Percent = 1 - ()W
= 1 - 0.96615410 = 1-0.7087 = 0.2913
Now, multiply the result by the price for your portion of the infinite stream:
Price = $60.23 X 0.2913 = $17.55
The next step is to discount the selling price of Johnson &Johnson in 10 years at 17% and then add the two values to get the stock's price. So, how do we estimate the stock's price at the end of 10 years? If we elect to sell the stock after 10 years and the company will continue to pay dividends at the same growth rate, what would a buyer be willing to pay? How could we estimate the selling price (value) of the stock at that time?
We need to estimate the dividend in 10 years and assume a growth rate and the required return of the new owner at that point in time. Let's assume that the new owner also wants a 17% return and that the dividend growth rate will remain at 13.04%. We calculate the dividend in 10 years by taking the current growth rate plus one raised to the tenth power times the current dividend:
Dio = $2.11 X (1.1304)10 =$2.11 X 3.4066 =$7.1879
We then use the dividend growth model with infinite horizon to determine the price in 10 years as follows:
Your price for the stock today—given that you will receive the growing dividend stream for 10 years and sell for $258.10 in 10 years, and also given that you want a 17% return over the 10 years—is as shown below:
/ 1 + 0.1304 \10' v 1 + 0.1700 /
= $42.68 + $17.55 = $60.23
Why did you get the same price of $60.23 for your stock with both the infinite growth model and the finite model? The reason is that the required rate of return of the stock remained at 17% (your rate) and the growth rate of the dividends remained at 13.04%. The infinite growth model gives the same price as the finite model with a future selling price as long as the required return and the growth rate are the same for all future sales of the stock.
Although this point may be subtle, what we have just shown is that a stock's price is the present value of its future dividend stream. When you sell the stock, the buyer purchases the remaining dividend stream. If that individual should sell the stock in the future, the new owner would buy the remaining dividends. That will always be the case; a stock's buyer is always buying the future dividend stream.
LINK TO LEARNIN
Determining Stock Value Using Different Scenarios
This video explains methods for determining stock value (https://0penstax.0rg/r/ explains methods for determining stock value) using scenarios of constant dividends and scenarios of constant dividend growth.
pri-. _ $205.18 , $2.11 x (1 +0.1304)
- (ll70Q)io (0.1700 - 0.1304) *
Nonconstant Growth Dividends
One final issue to address in this section is how we price a stock when dividends are neither constant nor growing at a constant rate. This can make things a bit more complicated. When a future pattern is not an annuity or the modified annuity stream of constant growth, there is no shortcut. You have to estimate every
340 11 • Stocks and Stock Valuation
future dividend and then discount each individual dividend back to the present. All is not lost, however. Sometimes you can see patterns in the dividends. For example, a firm might shift into a dividend stream pattern that will allow you to use one of the dividend models to take a shortcut for pricing the stock. Let's look at an example.
JM and Company is a small start-up firm that will institute a dividend payment—a $0.25 dividend—for the first time at the end of this year. The company expects rapid growth over the next four years and will increase its dividend to $0.50, then to $1.50, and then to $3.00 before settling into a constant growth dividend pattern with dividends growing at 5% every year (see Table 11.6). If you believe thatJM and Company will deliver this dividend pattern and you desire a 13% return on your investment, what price should you pay for this stock?
TO T1 T2 T3 T4 T5
$0.25 $0.50 $1.50 $1.275 $3.00 $3.00 X 1.05
Table 11.6
Solution: To price this stock, we will need to discount the first four dividends at 13% and then discount the constant growth portion of the dividends, the first payment of which will be received at the end of year 5. Let's calculate the first four dividends:
py _ $0.25 + $0.50 + $1.50 + $3.00 (1.13)1 (1.13)2 (1.13)3 (1.13)4
= $0.22 + $0.39 + $1.04 + $1.84 = $3.49
We now turn to the constant growth dividend pattern, where we can use our infinite horizon constant growth model as follows:
_ $3.00 X (1 + 0.05) $3.15 ^ " 0.13 - 0.05 " 0.08 -$39375
This figure is the price of the constant growth portion at the end of the fourth period, so we still need to discount it back to the present at the 13% required rate of return:
Price = = $24.15
(1.13)4
So, the price of this stock with a nonconstant dividend pattern is
Price = $3.49 + $24.15 = $27.64
Some Final Thoughts on Dividend Discount Models
The dividend used to calculate a price is the expected future payout and expected future dividend growth. This means the DDM is most useful when valuing companies that have long, consistent dividend records.
If the DDM formula is applied to a company with a limited dividend history, or in an industry exposed to significant risks that could affect a company's ability to maintain its payout, the resulting derived value may not be entirely accurate.
In most cases, dividend models, whether constant growth or constant dividend, appeal to a fundamental concept of asset pricing: the future cash flow to which the owner is entitled while holding the asset and the required rate of return for that cash flow determine the value of a financial asset. However, problems can arise when using these models because the timing and amounts of future cash flows may be difficult to predict.
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11.3 • Discounted Cash Flow (DCF) Model 341
11.3
Discounted Cash Flow (DCF) Model
Learning Outcomes
By the end of this section, you will be able to:
■ Explain how the DCF model differs from DDMs.
■ Apply the DCF model.
■ Explain the advantages and disadvantages of the DCF model.
When investors buy stock, they do so in order to receive cash inflows at different points in time in the future. These inflows come in the form of cash dividends (provided the stock does indeed pay dividends, because not all do) and also in the form of the final cash inflow that will occur when the investor decides to sell the stock.
The investor hopes that the final sale price of the stock will be higher than the purchase price, resulting in a capital gain. The hope for capital gains is even stronger in the case of stocks that do not pay dividends. When securities have been held for at least one year, the seller is eligible for long-term capital gains tax rates, which are lower than short-term rates for most investors. This makes non-dividend-paying stocks even more attractive, provided that they do indeed appreciate in value over the investor's holding period. Meanwhile, short-term gains, or gains made on securities held for less than one year, are taxed at ordinary income tax rates, which are usually higher and offer no particular advantage to an investor in terms of reducing their taxes.
Understanding How the DCF Model Differs from DDMs
The valuation of an asset is typically based on the present value of future cash flows that are generated by the asset. It is no different with common stock, which brings us to another form of stock valuation: the discounted cash flow (DCF) model. The DCF model is usually used to evaluate firms that are relatively young and do not pay dividends to their shareholders. Examples of such companies include Facebook, Amazon, Google, Biogen, and Monster Beverage. The DCF model differs from the dividend discount models we covered earlier, as DDM methodologies are almost entirely based on a stock's periodic dividends.
The DCF model is an absolute valuation model, meaning that it does not involve comparisons with other firms within any specific industry but instead uses objective data to evaluate a company on a stand-alone basis. The DCF model focuses on a company's cash flows, determining the present value of the entire organization and then working this down to the share-value level based on total shares outstanding of the subject organization. This highly regarded methodology is the evaluation tool of choice for experienced financial analysts when evaluating companies and their common stock. Many analysts prefer DCF methods of valuation because these are based on a company's cash flows, which are far less easily manipulated through accounting treatments than revenues or bottom-line earnings.
The DCF model formula in its mathematical form is presented below:
Stock Value = _^L_ + + ... +
(1 + r)1 ' (1 + rf (1 + r)"'1
where CF-| is the estimated cash flow in year 1, CF2 is the estimated cash flow in year 2, and so on; TCF is the terminal cash flow, or expected cash flow from the ending asset sale; r is the discount rate or required rate of return; g is the anticipated growth rate of the cash flow; and n is the number of years covered in the model.
Applying the DCF Model
We can apply the DCF model to an example to demonstrate this methodology and how the formula works. Calculate the value of Mayweather Inc. and its common stock based on the next six years of cash flow results. Assume that the discount rate (required rate of return) is 8%, Mayweather's growth rate is 3%, and the terminal value (TCF) will be two and one-half times the discounted value of the cash flow in year 6.
342 11 • Stocks and Stock Valuation
Mayweather has a cash flow of $2.0 million in year 1, so its discounted cash flow after one year (CF-|) is $1,851,851.85. We arrive at this amount by applying the discount rate of 8% for a one-year period to determine the present value.
In subsequent years, Mayweather's cash flow will be increasing by 3%. These future cash flows also must be discounted back to present values at an 8% rate, so the discounted cash flow amounts over the next six years will be as follows:
Year 1: $1,851,852 Year 2: $1,766,118 Year 3: $1,684,353 Year 4: $1,606,374 Year 5: $1,532,005 Year 6: $1,461,079
Our earlier assumption that the terminal value will be 2.5 times the value in the sixth year gives us a total terminal cash flow (TCF) of $1,461,079 X 2.5, or $3,652,697. Now, if we take all these future discounted cash flows and add them together, we arrive at a grand total of $13,554,477. So, based on our DCF model analysis, the total value of Mayweather Inc. is just over $13.5 million.
At this point, we have the estimated value of the entire company, but we need to work this down to the level of per-share value of common stock.
Let's say that Mayweather is currently trading at $12 per share, and it has 1,000,000 common shares outstanding. This tells us that the market capitalization of the company is $12 X 1,000,000, or $12 million, and that a $12 share price may be considered relatively low. The reason for this is that based on our DCF model analysis, investors would theoretically be willing to pay $13,554,477 divided by 1,000,000 shares, or $13.55 per share, for Mayweather. The overall conclusion would be that at $12.00 per share, Mayweather common stock would be a good buy at the present time. Figure 11.11 shows the Excel spreadsheet approach for arriving at the total value of Mayweather.
A A B c D E
1
2 Discount Rate = 8.00%
3
4 Year Cash Flow Present Value
5
6 CF 1 2,000,000 1,851,852 =-PV($C$2,B6,0,C6)
7 CF 2 2,060,000 1,766,118
8 CF 3 2,121,800 1,684,353
CF 4 2,185,454 1,606,374
10 CF 5 2,251,018 1,532,005
11 CF 6 2,318,548 1,461,079
12
13 Totals $12,936,820 9,901,780
14 TCF 3,652,697
15 Grand Total $13,554,477
Figure 11.11 Excel Solution for Total Value
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11.4 • Preferred Stock 343
Cell E6 displays the present value formula that is active in cell D6. Advantages and Limitations of the DCF Model
Due to several corporate accounting scandals in recent years, many analysts have given increasing credence to the use of cash flow as a metric for determining accurate corporate valuations. However, it should be noted that cash flow is not always the best means of measuring financial health. A company can always sell a large portion of its assets to generate a positive cash flow, even if it is operating at a loss or experiencing other financial difficulties. Additionally, investors prefer to see companies reinvesting their cash back into their businesses rather than sitting on excessive balances of idle cash.
Similar to other models, the discounted cash flow model is only as good as the information entered. As the common expression goes, "garbage in, garbage out." This can often be the case if reasonably accurate cash flow estimates are not available or if an unrealistic discount rate or required rate of return is used in the calculations. It is always best to use several different methods when valuing companies and their common stock.
INK TO LEARNING
Discounted Cash Flow
Please view this short video on discounted cash flow (https://openstax.Org/r/short video on discounted) and how this method can be applied in stock valuation.
11.4
Preferred Stock
Learning Outcomes
By the end of this section, you will be able to:
■ Define preferred stock.
■ Calculate the intrinsic value of preferred stock.
■ Understand the difference between common stock and preferred stock. Features of Preferred Stock
Preferred stock is a unique form of equity sold by some firms that offers preferential claims in ownership. Preferred stock will often feature a dividend that a company is obligated to pay out before it makes any dividend payments to common stockholders. In cases of bankruptcy and liquidation of the issuing company, preferred stockholders have priority claim to assets before common stockholders. Additionally, preferred stockholders are usually entitled to a set (or constant) dividend every period.
Preferred stock carries a stated par value, but unlike bonds, they have no maturity date, and consequently, there is no final payment of the par value. The only time a company would pay this par value to the shareholder would be if the company ceased operations or retired the preferred stock. Many preferred stock issues are cumulative in nature, meaning that if a company skips or is otherwise unable to pay a cash dividend, it becomes a liability to the company and must eventually be paid out to preferred shareholders at some point in the future. Other preferred stocks may be noncumulative, in which case if the company skips dividends, they are forever lost to the shareholder.
The term preferred comes from preferred shareholders receiving all past (if cumulative) and present dividends before common shareholders receive any cash dividends. In other words, preferred shareholders' dividend claims are given preferential treatment over those of common shareholders. Preferred stock is usually a form of permanent funding, but there are circumstances or covenants that could alter the payoff stream.
344 11 • Stocks and Stock Valuation
For example, a company may convert preferred stock into common stock at a preset point in the future. It is not uncommon for companies to issue preferred stock that has a conversion feature. Such conversion features give preferred shareholders the right to convert to common shares after a predetermined period.
A review of the characteristics of preferred stock will lead to the conclusion that the constant growth dividend model is an excellent approach for valuing such stock. Because shares of preferred stock provide a constant cash dividend based on original par value and the stated dividend rate, these may be considered a form of perpetuity.
It is this constant, preferred dividend stream that makes preferred stock seem more like bonds or another form of debt than like stock. In addition, the constant dividend stream leads nicely to the pricing of preferred stock with the four dividend models we presented earlier in this chapter.
Determining the Intrinsic Value of Preferred Stock
We can apply a version of the present value of a perpetuity formula to value preferred stock, as in the following example. Oh-Well Heath Services Inc. has issued preferred stock that has a par value of $1,000 and pays an annual dividend rate of 5%. If the market considers the risk of Oh-Well to warrant a 10% discount rate, what would be a fair market price for Oh-Well preferred stock?
First, we find the dividend value of Oh-Well:
Dividend Dollar Value = Par Value X Annual Dividend Rate = $1,000 X 0.05 = $50
We then use the constant dividend model with infinite horizon because we have g equal to zero and n equal to infinity:
Stock Value = Dividend Discount Rate
_ $50 ~~ .10
= $500
We can also rearrange the formula to determine the required return on this stock, given its annual dividend and current price.
HINK IT THR
Calculating the Return on Preferred Stock
Data Forge Inc. has just issued preferred stock (cumulative) with a par value of $100.00 and an annual dividend rate of 7%. The preferred stock is currently selling for $35.00 per share. What is the yield or return on this preferred stock?
Solution:
The first step is to determine the annual dividend by multiplying the dividend rate by the par value:
$100 x 0.07 = $7.00
Now, using this $7.00 annual dividend, the $35.00 current price, and the equation above, we calculate the rate of return as follows:
$7.00
r = = 0.20 or 20%
$35.00
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11.5 • Efficient Markets 345
We have introduced the concept of return here, which should be thought of as both the anticipated return for the preferred stockholder and the company's cost of borrowing money for this particular type of capital.
Differences between Preferred and Common Stock
As we have discussed, preferred stock has important differences from common stock that apply to issuing firms and to investors. Some of the most important of these differences are listed in Table 11.7.
lommon Stock
Paid only after preferred stockholders are paid
Dividends Variable and may increase or decrease
High potential but tied to company performance
Paid out last, after all creditors and preferred stockholders are paid
Growth
Liquidation
Voting Rights
Arrears Certainty
Yes
No accrual of missed dividends
Dividends potentially not paid if company earns no profits
Preferred Stock
Highest priority, paid first
Predetermined rates, so constant dividend amounts
Given preference in terms of payments, similar to bonds
No
If cumulative, unpaid dividends become liability that must be paid out eventually
Dividends paid even when company experiences financial losses
Table 11.7 Differences between Common Stock and Preferred Stock
11.5
Efficient Markets
Learning Outcomes
By the end of this section, you will be able to:
■ Understand what is meant by the term efficient markets.
• Understand the term operational efficiency when referring to markets.
■ Understand the term informational efficiency when referring to markets.
■ Distinguish between strong, semi-strong, and weak levels of efficiency in markets.
Efficient Markets
For the public, the real concern when buying and selling of stock through the stock market is the question, "How do I know if I'm getting the best available price for my transaction?" We might ask an even broader question: Do these markets provide the best prices and the quickest possible execution of a trade? In other words, we want to know whether markets are efficient. By efficient markets, we mean markets in which costs are minimal and prices are current and fair to all traders. To answer our questions, we will look at two forms of efficiency: operational efficiency and informational efficiency.
Operational Efficiency
Operational efficiency concerns the speed and accuracy of processing a buy or sell order at the best available price. Through the years, the competitive nature of the market has promoted operational efficiency.
In the past, the NYSE (New York Stock Exchange) used a designated-order turnaround computer system known as SuperDOT to manage orders. SuperDOT was designed to match buyers and sellers and execute trades with confirmation to both parties in a matter of seconds, giving both buyers and sellers the best
346 11 • Stocks and Stock Valuation
available prices. SuperDOT was replaced by a system known as the Super Display Book (SDBK) in 2009 and subsequently replaced by the Universal Trading Platform in 2012.
NASDAQ used a process referred to as the small-order execution system (SOES) to process orders. The practice for registered dealers had been for SOES to publicly display all limit orders (orders awaiting execution at specified price), the best dealer quotes, and the best customer limit order sizes. The SOES system has now been largely phased out with the emergence of all-electronic trading that increased transaction speed at ever higher trading volumes.
Public access to the best available prices promotes operational efficiency. This speed in matching buyers and sellers at the best available price is strong evidence that the stock markets are operationally efficient.
Informational Efficiency
A second measure of efficiency is informational efficiency, or how quickly a source reflects comprehensive information in the available trading prices. A price is efficient if the market has used all available information to set it, which implies that stocks always trade at their fair value (see Figure 11.12). If an investor does not receive the most current information, the prices are "stale"; therefore, they are at a trading disadvantage.
All Public Information
Information about Past Stock Prices
Partially Strong
Weak
Figure 11.12 Forms of Market Efficiency A market is efficient if it provides all information that is available.
Forms of Market Efficiency
Financial economists have devised three forms of market efficiency from an information perspective: weak form, semi-strong form, and strong form. These three forms constitute the efficient market hypothesis. Believers in these three forms of efficient markets maintain, in varying degrees, that it is pointless to search for undervalued stocks, sell stocks at inflated prices, or predict market trends.
In weak form efficient markets, current prices reflect the stock's price history and trading volume. It is useless to chart historical stock prices to predict future stock prices such that you can identify mispriced stocks and routinely outperform the market. In other words, technical analysis cannot beat the market. The market itself is the best technical analyst out there.
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11 • Summary 347
a Summary
11.1 Multiple Approaches to Stock Valuation
This section introduced common stock and some of the models and calculation methods used by investors and financial analysts to determine the prices or values of common shares. The most evaluative ratios that can be computed from a company's financial statements include the price-to-earnings (P/E), price-to book (P/B), price-to-sales (P/S), and price-to cash-flow (P/CF) ratios.
11.2 Dividend Discount Models (DDMs)
The dividend discount model, or DDM, is a method used to value a stock based on the concept that its worth is the present value of all of its future dividends. The most common DDM is the Gordon growth model, which values stock entirely on expected future dividends. Other techniques include the zero growth DDM, which depends on fixed dividends; the constant growth DDM, which assumes that dividends will grow at a constant rate; and the variable growth or nonconstant growth DDM, which is based on the assumption that stock value will progress through different stages of growth. There is also the two-stage DDM, which is based on the assumption of two stages of dividend growth: an initial period of higher growth and a subsequent period of lower, more stable growth.
11.3 Discounted Cash Flow (DCF) Model
Investors buy stock to receive cash inflows at different points in the future. These inflows may come in the form of dividends or a final cash inflow. If the investor chooses to wait for a final cash flow, the hope is that capital gains will be even stronger. The DCF model is usually used to evaluate firms that are relatively young and do not pay dividends to their shareholders. The DCF model focuses on a company's cash flows, determining the present value of an entire organization using objective data and then working this down to the share-value level based on total shares outstanding of the subject organization.
11.4 Preferred Stock
Preferred stock is a unique form of equity sold by some firms that offers preferential claims in ownership. Preferred stock carries a stated par value, but unlike bonds, there is no maturity date, and consequently, there is no final payment of the par value. The term preferred comes from preferred shareholders receiving all past (if cumulative) and present dividends before common shareholders receive any cash dividends.
11.5 Efficient Markets
Efficient markets are markets in which costs are minimal and prices are current and fair to all traders. There are two forms of efficiency: operational efficiency and informational efficiency. Operational efficiency concerns the speed and accuracy of processing a buy or sell order at the best available price. Informational efficiency concerns how quickly a source reflects comprehensive information in the available trading prices. Financial economists have devised three forms of efficient markets from an information perspective: weak form, semi-strong form, and strong form.
¥ Key Terms
amortization the process of spreading business costs in accounting; similar to depreciation, but differs in
that it generally refers to intangible assets such as patents or copyrights capital employed also known as funds employed; the total amount of capital used for the acquisition of
profits by a firm or on a project common stock a security that represents partial ownership of a corporation
comparable company analysis (comps) a method for valuating a company using the metrics of other
businesses of similar size in the same industry depreciation the process of spreading business costs in accounting; similar to amortization, but differs in
348 11 • Key Terms
that it generally refers to fixed, tangible assets such as buildings, machinery, furniture, fixtures, and equipment
discounted cash flow (DCF) a method for estimating the value of an investment based on the present value
of its expected future cash flows dividend a sum of money paid regularly (typically quarterly) by a company to its shareholders out of its
profits or reserves
dividend discount model (DDM) a quantitative method for predicting the price of a company's stock based
on the theory that its present-day price is worth the sum of all of its future dividend payments when
discounted back to their present value dividend yield a financial ratio (dividend/price) that shows how much a company pays out in dividends each
year relative to its stock price, expressed as a percentage earnings per share (EPS) the ratio of a company's profits to the outstanding shares of its common stock;
serves as an indicator of a company's profitability EBIT short for earnings before interest and taxes; an indicator of a firm's profitability before the effects of
interest or taxes; also referred to as operating earnings, operating profit, and profit before interest and tax efficient markets markets in which costs are minimal and prices are current, fair, and reflective of all
available relevant information enterprise value (EV) a company's total value; often used as a more comprehensive alternative to equity
market capitalization
enterprise value (EV) multiples also known as company value multiples; ratios used to determine the
overall value of a company and, by extension, the value of its common stock equity multiples metrics that calculate the expected or achieved total return on an initial investment Gordon growth model a methodology used to determine the intrinsic value of a stock based on a future
series of dividends that grow at a constant rate growth rate the rate at which the dollar amount of dividends paid on a specific stock holding increases intangible assets assets that are not physical in nature, such as goodwill, brand recognition, and intellectual
property (i.e., patents, trademarks, and copyrights) intrinsic value the value of a firm's stock based entirely on internal factors, such as products, management,
and the strength of company brands in the marketplace market capitalization the value of a company traded on the stock market, calculated by multiplying the
total number of shares by the current share price NASDAQ (National Association of Securities Dealers Automated Quotations) short for National
Association of Securities Dealers Automated Quotations; an American stock exchange based in New York
City, ranked second behind the New York Stock Exchange in terms of total market capitalization of shares
traded
net book value also called net asset value (NAV); the total assets of a company minus its total liabilities NYSE (New York Stock Exchange) the world's largest stock exchange by market capitalization of its listed
companies, based in New York City; often referred to as the "Big Board" perpetuity in terms of investments, an annuity with no end date
precedent transaction analysis (precedents) a method for valuating a company in which the price paid for
similar companies in the past is considered an indicator of the company's current value preferred stock stock that entitles the holder to a fixed dividend, the payment of which takes priority over
payment of common stock dividends price-to-book (P/B) ratio also called the market-to-book (M/B) ratio; a metric used to compare a company's
market capitalization, or market value, to its book value price-to-cash-flow (P/CF) ratio a stock valuation indicator or multiple that measures the value of a stock's
price relative to its operating cash flow per share price-to-earnings (P/E) ratio a ratio that indicates the dollar amount an investor can expect to invest in a
company in order to receive one dollar of that company's earnings price-to-sales (P/S) ratio a ratio that indicates how much investors are willing to pay for a company's stock
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11 • CFA Institute 349
per dollar of that company's sales required return the minimum return an investor expects to achieve by investing in a project secondary markets markets where investors buy and sell securities they already own stock value also called intrinsic value; the fundamental, objective value of a share of stock
iQ CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session (https://0penstax.0rg/r/CFA Level I Study Session). Reference with permission of CFA Institute.
a
Multiple Choice
The price-to-earnings (P/E) ratio is_.
a. used to calculate a company's book value
b. a multiplier used in enterprise valuation
c. only applied when valuing preferred stock
d. a metric for showing the expectations of the market
2. The price-to-book (P/B) ratio is also called the_ratio.
a. market-to-book
b. enterprise-to-book
c. asset-to-price
d. liability-to-book
3. The two primary types of stock valuation multiples are_.
a. enterprise value multiples and comparable company value multiples
b. equity multiples and enterprise value multiples
c. asset value multiples and liability value multiples
d. None of the above
4. Dividend yield is a form of equity multiple that is primarily used when_.
a. the subject company is operating at a loss
b. conducting comparisons between companies in different industries
c. conducting comparisons between cash returns and investment types
d. analyzing mature companies that have been paying dividends for several years
5. Dividend yield is computed as the proportion of dividend per share to_.
a. share price
b. earnings per share
c. market value per share
d. book value per share
6. Which of the following statements about enterprise value metrics is NOT true?
a. EV to revenue can be used to assess companies with negative cash flows or firms that are currently experiencing financial losses.
b. ROCE is a profitability ratio that measures the return on the equity in a company in comparison to its financing over a short period of time.
c. EBIT allows investors to assess the core operations of the business without worrying about the costs of the capital structure.
d. EBITDAR is a metric that is used in businesses that have substantial rental and least expenses.
350 11 • Review Questions
7. If an investor's required rate of return increases and all other characteristics of a stock remain the same, the value of the stock will_.
a. remain the same
b. increase
c. decrease
d. None of the above
8. A major limitation of the dividend discount model (DDM) is that_.
a. it is extremely complicated to use
b. it cannot be used with companies that do not pay dividends
c. it must always be used in conjunction with another metric
d. None of the above
9. In which of the following ways does preferred stock differ from common stock?
a. Preferred stock carries voting rights for its ownership.
b. Preferred stock must be purchased through a broker dealer.
c. Preferred stock may have a cumulative dividend feature.
d. In the event of corporate liquidation, preferred stockholders are paid last.
10. The term efficient markets refers to the idea that_.
a. publicly traded companies always file their financial reports on time
b. investors are able to identify underpriced stocks for purchase
c. stocks trade with minimal costs and prices are current and fair to all traders
d. financial information on companies and their stock is only available to efficient traders
Q Review Questions
1. Briefly discuss one of the primary benefits of using comparative P/E ratios.
2. Name an important characteristic of companies for which the price-to-book (P/B) ratio does not work well.
3. Briefly describe the main type of scenario in which the two-stage DDM approach might be used to value a firm and its stock.
4. Briefly describe a major shortcoming of the zero growth DDM model.
5. Briefly describe the required inputs for the discounted cash flow (DCF) model.
6. Briefly describe preferred stock and some of its ownership advantages compared to common stock.
7. Briefly explain what is meant by the terms cumulative and noncumulative as they relate to preferred stocks.
8. What were SuperDOT and SOES, and what were they designed to do?
9. What are operational efficiency and informational efficiency, and how do they differ in terms of trading markets?
10. What is meant by informational efficiency, and how does it affect the price of a stock?
Problems
1. Today, Sysco Enterprises paid dividends on its common stock of $1.25 per share. If dividends per share are expected to increase to $3.50 per share six years from now, what is the percentage dividend growth rate?
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11-Video Activity 351
2. Let's say you want to purchase shares of Fontaine Ltd. and then hold this stock for six years. The company has a stated dividend policy of $2.00 annually per share for the next six years, at the end of which time you will sell the stock. You expect to be able to sell the stock for $35.00 at that time. If you want to earn an 8% return on this investment, what price should you pay today for this stock?
3. Damian Painting Systems has established a dividend policy of $3.00 per share per year. If the company plans to be in business forever, what is the value of this stock if an investor wants a 10% return?
4. Wilk Productions wants its shareholders to earn a 12% return on their investment in the company. At what value would Wilk stock be priced if the company paid $2.75 per share in constant annual dividends
5. Dax Industrial Systems has stock currently priced at $50.00 per share. If investors are earning a 7% return on Dax Industrial, what is the company's annual dividend payment per share?
6. If a stock is selling at $400 with a current dividend of $40 and a potential investor's required rate of return is 15%, what would be the anticipated dividend growth rate?
7. Mind Max Inc. has a dividend policy that increases annual dividends by 3% each year. If last year's dividend was $2.00, the company intends to stay in business for 50 years, and an investor wants a 9% return, what would be the price of Mind Max stock?
8. Odon Corp. paid dividends today in the amount of $1.50 per share. If Odon will pay dividends of $5.00 10 years from today, what is the annual dividend growth rate over this 10-year period?
9. The Kirkson Distributors common stock is currently selling at $52.00 per share, pays dividends annually at $2.50 per share, and has an annual dividend growth rate of 2%. What is the required return?
10. If a preferred share of stock pays dividends of $2.50 per year and the required rate of return for the stock is 6%, what is its intrinsic value?
Efficient Markets
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1. In the efficient market hypothesis (EMH), describe what is meant by the terms weak form efficiency, semi-strong form efficiency, and strong form efficiency. How do these forms of market efficiency differ from each other, and what are their characteristics?
2. What is meant by the term random walk, and how does this concept relate to the EMH? What Is Preferred Stock?
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3. Discuss the relative risks of the following financial instruments and how they compare to each other: bonds, common stocks, and preferred stocks. How and why will these three investment types typically carry different levels of risk to an investor?
4. Discuss some of the important differences between preferred stocks and common stocks.
forever?
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Figure 12.1 Stock markets are affected by many factors, such as interest rates, inflation, and politics, (credit: modification of "That was supposed to be going up, wasn't it? by Rafael Matsunaga/flickr, CC BY 2.0)
Chapter Outline
12.1 Overview of US Financial Markets
12.2 Historical Picture of Inflation
12.3 Historical Picture of Returns to Bonds
12.4 Historical Picture of Returns to Stocks
LzfJ Why It Matters
Author Mark Twain spoke for many when he wrote, "October—this is one of the peculiarly dangerous months to speculate in stocks. The others are July, January, September, April, November, May, March, June, December, August, and February." Twain's comment, though humorous, reflects the serious risks associated with investing in the stock market. So why should we study the history of the US financial markets? Financial experts regularly remind us that past performance is no guarantee of future results. However, past performance can provide targets or benchmarks around which to build expectations. We can learn about the past to prepare for future possibilities, or we can suffer what Winston Churchill warned, "Those that fail to learn from history are doomed to repeat it."
Carlos Slim Helu, a Mexican businessman and the richest person in the world from 2010 to 2013, once said, "With a good perspective on history, we can have a better understanding of the past and present, and thus a clear vision of the future."1 In this chapter, we examine current and historical performance in money, bond, and stock markets. Studying past market risk and return also allows investors to understand what is reasonable and what is not.
1 Dan Western. "45 Carlos Slim Helu Quotes About Wealth and Success." Wealthy Gorilla, https://wealthygorilla.com/carlos-slim-helu-quotes/
354 12 • Historical Performance of US Markets
12.1
Overview of US Financial Markets
Learning Outcomes
By the end of this section, you will be able to:
■ Identify the various aspects of the US money markets.
■ Characterize government and corporate bond markets.
■ Detail the structure and operations of US equity markets.
Money Markets
The money market is a multitrillion-dollar market. Features of money market securities include being short-term (with maturities of less than one year) and very low risk (rarely failing to make their required payments). Further, money market securities are also liquid, which means that they trade easily without losing value.
Financial institutions, corporations, and governments that have short-term borrowing and/or lending needs issue securities in the money market. Most of the transactions are quite large, with typical amounts of $10,000, $100,000, $1 million, or more. Money market securities are available in smaller amounts if you choose to invest in money market mutual funds (MMMFs) or certain types of exchange-traded funds (ETFs).
Treasury bills (T-bills) are short-term debt instruments issued by the federal government. T-bills are auctioned weekly by the Treasury Department through the trading window of the Federal Reserve Bank of New York, with maturities of 4, 8,13, or 26 weeks. The Treasury also auctions 52-week T-bills once every four weeks. The federal government uses T-bills to meet short-term liquidity needs. T-bills have very short maturities and a broad secondary market and are default-risk free. T-bills are also exempt from state and local income taxes. As a result, they carry some of the lowest effective interest rates on publicly traded debt securities.
The volume of T-bills auctioned depends upon government borrowing needs. Much of the money raised at weekly T-bill auctions goes to repay the money borrowed 4, 8,13, 26, or 52 weeks earlier. The gross amount of new T-bills issued in December 2020 was $1,591.1 billion, and the amount of T-bills retired in the same month was $1,570.6 billion, resulting in net new borrowing of "only" $20.5 billion.
In addition to the regular auction of new T-bills, there is also an active secondary market where investors can trade used or previously issued T-bills. Since 2001, the average daily trading volume for T-bills has exceeded $75 billion.
Commercial paper (CP) is a short-term, unsecured debt security issued by corporations and financial institutions to meet short-term financing needs such as inventory and receivables. For example, credit card companies use commercial paper to finance credit card payments. Commercial paper is a short-term debt instrument, with a typical maturity of 30 days and up to 270 days. The short maturity reduces US Securities and Exchange Commission (SEC) oversight. The lesser oversight and the unsecured nature of CP means that only highly rated firms are able to issue the uninsured paper.
Commercial paper typically carries a minimum face value of $100,000 and sells at a discount, with the face value as the repayment amount. Corporations and financial institutions, not the government, issue CP; thus, returns are taxable. Further, unlike T-bills, there is not a robust secondary market for CP. Most purchasers are large, such as mutual fund investment companies, and they tend to hold commercial paper until maturity. The default rate on commercial paper is typically low, but default rates did increase into the double-digit range during the financial crisis of 2008.
Negotiable certificates of deposit (NCDs) are very large CDs issued by financial institutions. They are redeemable only at maturity, but they can and often do trade prior to maturity in a broad secondary market. NCDs, or jumbo CDs, are so called because they sell in increments of $100,000 or more. However, typical amounts are $1 million, with a maturity of two weeks to six months.
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12.1 • Overview of US Financial Markets 355
NCDs differ in some important ways from the typical CD you may be familiar with from your local bank or credit union. The typical CD has a maturity date, interest rate, and face amount and has FDIC insurance. However, if an investor wishes to cash out prior to maturity, they will incur a substantial penalty from the issuer (bank or credit union). An NCD also has a maturity date and amount, but it is much larger than a regular CD and appeals to institutional investors. The principal is not insured. When the investor wishes to cash out early, there is a robust secondary market for trading the NCD. The issuing institution can offer higher rates on NCDs compared to CDs because it knows it will have use of the purchase amount for the entire maturity of the NCD and because the reserve requirements on NCDs by the Federal Reserve is lower than for other types of deposits.
Investment companies such as Vanguard and Fidelity, among many others, sell shares in money market mutual funds (MMMFs). The investment company purchases money market instruments, such as T-bills, CP, or NCDs; pools them; and then sells shares of ownership to investors (see Table 12.1). Generally, MMMFs invest only in taxable securities, such as commercial paper and negotiable certificates of deposit, or only in tax-exempt government securities, such as T-bills. Investors can then choose which type of short-term liquid securities they would like to hold, taxable or nontaxable. MMMFs provide smaller firms and investors the opportunity to participate in the money market by facilitating smaller individual investment amounts.
Financial Instrument Typical Maturity Minimum Amount Issuer
Treasury bills 4-52 weeks $10,000 Federal government
Commercial paper 270 days $100,000 Businesses
Negotiable CDs 6 months $100,000 Financial institutions
MMMFs Investment companies
Federal funds 1 day $1 million Financial institutions
Table 12.1 Money Market Instrument Characteristics (Selected)
The market for federal funds is notable because the Federal Reserve (Fed) targets the equilibrium interest rate on federal funds as one of its most important monetary policy tools. The federal funds market traditionally consists of the overnight borrowing and lending of immediately available funds among depository financial institutions, notably domestic commercial banks. Financial institutions such as banks are required to keep a fraction of their deposits on reserve with the Fed. When banks find they are short of reserves and immediately need cash to meet reserve requirements, they can borrow directly from the Fed through the so-called "discount window" or purchase excess reserves from other banks in the federal funds market. Often, the maturity of a federal funds contract is as short as a single day or overnight. The participants in the market negotiate the federal funds interest rate. However, the Federal Reserve effectively sets the target interest rate range in the federal funds market by controlling the supply of funds available for use in the market.
Since the financial crisis of 2008, the activities and functioning of the federal funds market has changed. The federal funds rate is still the rate targeted by the Fed for monetary policy, but the participants have evolved for several reasons. The market now includes foreign banks and non-depository financial institutions, such as the Federal Home Loan Banks. These institutions do not need to meet Fed reserve requirements and are not required to keep reserves with the Fed. In addition, the Fed now pays interest to commercial banks for reserves held at the Federal Reserve banks. Paying interest on reserves reduces the incentive for domestic commercial banks to enter the federal funds market since they can already earn interest on their excess reserves.
356 12 • Historical Performance of US Markets
Daily trading volume in the federal funds market from 2016 through 2020 ranged from a high of $115 billion in March of 2018 to a low of only $34 billion on December 31, 2020.2 The volume of federal funds activity is lower in periods of slower economic growth because banks have fewer good opportunities to issue loans and are less likely to be short of required reserves.
Bond Markets
Bond markets are financial markets that make payments to investors for a specific period of time. Investors decide how much to pay for a bond depending on how much they expect inflation to affect the value of the fixed payment. There are several types of bonds: government bonds, corporate bonds, and municipal bonds.
Government Bond Markets
In the section on money markets, we discussed T-bills, and we now discuss longer-term government securities in the form of Treasury notes and Treasury bonds.
We learned in Bonds and Bond Valuation that the federal government issues Treasury notes and bonds to raise money for current spending and to repay past borrowing. The size of the Treasury market is quite large, as the US federal government over the years has accumulated a total indebtedness of over $28 trillion dollars. The debt has grown so large we even have a real-time debt calculator online at https://www.usdebtclock.org/ (https://openstax.Org/r/usdebtclock).
Treasury notes (T-notes) are US government debt instruments with maturities of 2, 3, 5, 7, or 10 years. The Treasury auctions notes on a regular basis, and investors may purchase new notes from TreasuryDirect.gov (https://openstax.0rg/r/treasurydirect.gov) in the same way they would a T-bill. T-notes differ from T-bills in that they are longer term and pay semiannual coupon interest payments, as well as the par or face value of the note at maturity. T-bills, as you will recall, sell at a discount and pay the face value at maturity with no explicit interest payments. Upon issue of a note, the size, number, and timing of note payments is fixed. However, prices do change in the secondary market as interest rates change. Like T-bills, T-notes are generally exempt from state and local taxes.
Economists and investors keep a close eye on the 10-year T-note for several reasons. Mortgage lenders use it as a basis for setting and adjusting mortgage interest rates. In general, the rate on the 10-year T-note is a reliable market indicator of investor confidence.
There is an active secondary market for Treasury notes. From 2001 to 2020, the daily trading volume for Treasury notes has averaged $395 billion, or roughly five times the daily trading volume of T-bills. Treasury notes are the largest single type of government debt instrument, with over $11 trillion outstanding. As you can see from Figure 12.2. the Treasury dramatically increased borrowing by issuing notes following the 2008 financial crisis.
Brokers, dealers, and investment companies provide secondary market opportunities for individual and institutional investors. Exchange-traded funds (ETFs) are popular investment vehicles for many types of government T-bill, T-note, and T-bond portfolios. An ETF is a basket of securities that can trade like stocks on a stock exchange. For example, IEI is an iShares ETF managed by BlackRock that invests in Treasury securities with three to seven years to maturity. When investors buy this ETF, they purchase a small bundle of Treasury notes that they can buy or sell, just as if they owned an individual share of stock. ETFs are a convenient way for investors to own broad portfolios of securities while still being able to trade the whole group in a single transaction if they choose.
2 Federal Reserve Bank of New York. "Effective Federal Funds Rate." Federal Reserve Bank of New York. https://www.newyorkfed.org/markets/reference-rates/effr
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12.1 • Overview of US Financial Markets 357
CY> & <& C?> <•£ \N O Review Questions
1. Define the competitive and noncompetitive bid process for US Treasury bills.
2. How does a negotiable certificate of deposit (NCD) differ from the typical certificate of deposit you may see advertised by your local bank?
3. If you are an investor concerned about unexpected inflation in the coming years which of the following investments offers the greatest protection against inflation, and why: T-notes, T-bonds, or TIPS?
4. Debentures are more common than mortgage bonds issued by corporations. Why do you think debentures are more popular with investors? Be sure to define each bond contract in your discussion.
5. Market capitalization is a common way to rank firm size. Search the internet to identify and define at least two other ways to rank firms based on size. Identify at least one reason you prefer market capitalization as the method of choice to rank firm size.
6. Compare and contrast an SEO, IPO, and SPAC. If Ford Motor Company wished to raise new equity capital, which of these vehicles would they use?
7. Compared to a "best efforts" form of underwriting, how does "firm commitment" underwriting transfer risk from the issuing firm to the underwriter?
8. How would a decrease in inflation affect the interest rate on an adjustable-rate debenture?
9. If inflation unexpectedly rises by 3%, would a corporation that had recently borrowed money by issuing fixed-rate bonds to pay for a new investment benefit or lose?
10. If wages on average rise at least as fast as inflation, why do people worry about how inflation affects incomes?
11. Identify at least one item that you use regularly whose price has changed significantly.
12. What has been the average annual rate of inflation between 1985 and 2020? What is the long-run average annual rate of inflation over the last century?
13. Between 1985 and 2020, what year had the lowest realized annual rate of inflation in the United States? Why do you think inflation was so low in this particular year?
14. Go to https://www.usinflationcalculator.com/ (https://openstax.Org/r/usinflationcalculator). How much money would it take today to purchase what one dollar would have bought in 1950, in 1975, and in your birth year?
15. Are US Treasury bonds truly risk free?
16. At the end of 2020 and the beginning of 2021, coupon rates on long-term T-notes and T-bonds were near historic lows. Further, the federal government was running a historically large budget deficit in an effort to stimulate an economy battered by COVID-19 and to support millions of unemployed workers. Some investment advisers warned that this could be a particularly bad time to invest in government bonds or bonds in general. Why?
17. Which group of securities earned a higher average annual return from 2000 to 2020, T-bonds or Baa bonds? Why do think this was so?
18. Which earned a higher average annual return, a portfolio of T-bonds from 1980 to 2000 or from 2000 to 2020? Why do think this was so?
19. Why is standard deviation of returns a reasonable measure of risk for a portfolio of equity securities?
20. Many popular-press articles claim that growth investing is "clearly better" than value investing or that
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12-Video Activity 379
value investing is "dead." How would you respond to proponents of growth investing after observing Figure 12.15?
21. Over the last 120 years, few countries have achieved the realized rate of returns enjoyed by US equity markets. Does this mean investors should ignore international investments and focus only on domestic markets in an effort to maximize returns?
How Private Companies Are Bypassing the IPO Process
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1. Can you identify three apparent advantages and three disadvantages for investors in a SPAC (special purpose acquisition company) versus a traditional IPO process?
2. The video concludes with a question about whether SPACs are a current fad doomed to fade away or a new and growing method of publicly financing firms. What do you think? Search for information related to SPACs and proposals for SPAC regulation, and report your conclusion.
A Secret Meeting and the Birth of the Federal Reserve
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3. How can an institution like the United State Federal Reserve System prevent bank runs?
4. After the passage of the Federal Reserve Act in 1913 (https://0penstax.0rg/r/federal-reserve-act). the United States has suffered through three great global financial crises: the Great Depression of the 1930s, the Great Recession of 2007-2009 (https://openstax.Org/r/the-worlds-most-devastating). and the COVID-19 pandemic of 2020-2021 (https://openstax.Org/r/tracking-the-covid-19-recessions-effects). Research one of the latter two crises, and identify and discuss some of the tools used by the Fed to lessen the length and economic severity of the economic hardships. In what ways has this research project supported or changed your opinion about the United States having the Federal Reserve System?
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Figure 13.1 Graphical displays are used extensively in the finance field, (credit: modification of "Analysing target market" by Marco Verch/flickrCCBY2.0)
Chapter Outline
13.1 Measures of Center
13.2 Measures of Spread
13.3 Measures of Position
13.4 Statistical Distributions
13.5 Probability Distributions
13.6 Data Visualization and Graphical Displays
13.7 The R Statistical Analysis Tool
Why It Matters
Statistical analysis is used extensively in finance, with applications ranging from consumer concerns such as credit scores, retirement planning, and insurance to business concerns such as assessing stock market volatility and predicting inflation rates. As a consumer, you will make many financial decisions throughout your life, and many of these decisions will be guided by statistical analysis. For example, what is the probability that interest rates will rise over the next year, and how will that affect your decision on whether to refinance a mortgage? In your retirement planning, how should the investment mix be allocated among stocks and bonds to minimize volatility and ensure a high probability for a secure retirement? When running a business, how can statistical quality control methods be used to maintain high quality levels and minimize waste? Should a business make use of consumer focus groups or customer surveys to obtain business intelligence data to improve service levels? These questions and more can benefit from the use and application of statistical methods.
Running a business and tracking its finances is a complex process. From day-to-day activities such as managing inventory levels to longer-range activities such as developing new products or expanding a customer base, statistical methods are a key to business success. For finance considerations, a business must manage risk versus return and optimize investments to ensure shareholder value. Business managers employ a wide range of statistical processes and tools to accomplish these goals. Increasingly, companies are also
382 13 • Statistical Analysis in Finance
interested in data analytics to optimize the value gleaned from business- and consumer-related data, and statistical analysis forms the core of such analytics.
13.1
Measures of Center
By the end of this section, you will be able to:
■ Calculate various measures of the average of a data set, such as mean, median, mode, and geometric mean.
■ Recognize when a certain measure of center is more appropriate to use, such as weighted mean.
■ Distinguish among arithmetic mean, geometric mean, and weighted mean.
Arithmetic Mean
The average of a data set is a way of describing location. The most widely used measures of the center of a data set are the mean (average), median, and mode. The arithmetic mean is the most common measure of the average. We will discuss the geometric mean later.
Note that the words mean and average are often used interchangeably. The substitution of one word for the other is common practice. The technical term is arithmetic mean, and average technically refers only to a center location. Formally, the arithmetic mean is called the first moment of the distribution by mathematicians. However, in practice among non-statisticians, average is commonly accepted as a synonym for arithmetic mean.
To calculate the arithmetic mean value of 50 stock portfolios, add the 50 portfolio dollar values together and divide the sum by 50. To calculate the arithmetic mean for a set of numbers, add the numbers together and then divide by the number of data values.
In statistical analysis, you will encounter two types of data sets: sample data and population data. Population data represents all the outcomes or measurements that are of interest. Sample data represents outcomes or measurements collected from a subset, or part, of the population of interest.
The notation x is used to indicate the sample mean, where the arithmetic mean is calculated based on data taken from a sample. The notation ^x is used to denote the sum of the data values, and n is used to indicate the number of data values in the sample, also known as the sample size.
The sample mean can be calculated using the following formula:
n
Finance professionals often rely on averages of Treasury bill auction amounts to determine their value. Table 13.1 lists the Treasury bill auction amounts for a sample of auctions from December 2020.
Maturity Amount (SBillions)
4-week T-bills $32.9 8-week T-bills 38.4
13-week T-bills 63.1
26-week T-bills 59.6
52-week T-bills 39.7
Total $233.7
Table 13.1 United States Treasury Bill Auctions, December 22 and 24, 2020
(source: Treasury Direct)
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13.1 • Measures of Center 383
To calculate the arithmetic mean of the amount paid for Treasury bills at auction, in billions of dollars, we use the following formula:
,= Il = ^l =$46.74 n 5
Median
To determine the median of a data set, order the data from smallest to largest, and then find the middle value in the ordered data set. For example, to find the median value of 50 portfolios, find the number that splits the data into two equal parts. The portfolio values owned by 25 people will be below the median, and 25 people will have portfolio values above the median. The median is generally a better measure of the average when there are extreme values or outliers in the data set.
An outlier or extreme value is a data value that is significantly different from the other data values in a data set. The median is preferred when outliers are present because the median is not affected by the numerical values of the outliers.
The ordered data set from Table 13.1 appears as follows:
32.9, 38.4, 39.7, 59.6, 63.1 The middle value in this ordered data set is the third data value, which is 39.7. Thus, the median is $39.7 billion.
You can quickly find the location of the median by using the expression " * 1. The variable n represents the total number of data values in the sample. If n is an odd number, the median is the middle value of the data values when ordered from smallest to largest. If n is an even number, the median is equal to the two middle values of the ordered data values added together and divided by 2. In the example from Table 13.1. there are five data values, so n = 5. To identify the position of the median, calculate " ^ 1, which is 5 ^ 1 • or 3- This indicates that the median is located in the third data position, which corresponds to the value 39.7.
As mentioned earlier, when outliers are present in a data set, the mean can be nonrepresentative of the center of the data set, and the median will provide a better measure of center. The following Think It Through example illustrates this point.
Finding the Measure of Center
Suppose that in a small village of 50 people, one person earns a salary of $5 million per year, and the other 49 individuals each earn $30,000. Which is the better measure of center: the mean or the median?
Solution:
The mean, in dollars, would be arrived at mathematically as follows:
_ 5,000,000 + 49(30,000)
x =-—-= 129,400
However, the median would be $30,000. There are 49 people who earn $30,000 and one person who earns $5,000,000.
The median is a better measure of the "average" than the mean because 49 of the values are $30,000 and one is $5,000,000. The $5,000,000 is an outlier. The $30,000 gives us a better sense of the middle of the data set.
384 13 • Statistical Analysis in Finance
Mode
Another measure of center is the mode. The mode is the most frequent value. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A data set with two modes is called bimodal. For example, assume that the weekly closing stock price for a technology stock, in dollars, is recorded for 20 consecutive weeks as follows:
50, 53, 59, 59, 63, 63, 72, 72, 72, 72, 72, 76, 78, 81, 83, 84, 84, 84, 90, 93
To find the mode, determine the most frequent score, which is 72. It occurs five times. Thus, the mode of this data set is 72. It is helpful to know that the most common closing price of this particular stock over the past 20 weeks has been $72.00.
Geometric Mean
The arithmetic mean, median, and mode are all measures of the center of a data set, or the average. They are all, in their own way, trying to measure the common point within the data—that which is "normal." In the case of the arithmetic mean, this is accomplished by finding the value from which all points are equal linear distances. We can imagine that all the data values are combined through addition and then distributed back to each data point in equal amounts.
The geometric mean redistributes not the sum of the values but their product. It is calculated by multiplying all the individual values and then redistributing them in equal portions such that the total product remains the same. This can be seen from the formula for the geometric mean, x (pronounced x-tilde):
x = ■ x2 — x„
The geometric mean is relevant in economics and finance for dealing with growth—of markets, in investments, and so on. For an example of a finance application, assume we would like to know the equivalent percentage growth rate over a five-year period, given the yearly growth rates for the investment.
For a five-year period, the annual rate of return for a certificate of deposit (CD) investment is as follows: 3.21%, 2.79%, 1.88%, 1.42%, 1.17%. Find the single percentage growth rate that is equivalent to these five annual consecutive rates of return. The geometric mean of these five rates of return will provide the solution. To calculate the geometric mean for these values (which must all be positive), first multiply1 the rates of return together—after adding 1 to the decimal equivalent of each interest rate—and then take the nth root of the product. We are interested in calculating the equivalent overall rate of return for the yearly rates of return, which can be expressed as 1.0321,1.0279,1.0188,1.0142, and 1.0117:
5c = yX\ ■x2-x„ = ^/1.0321 • 1.0279 • 1.0188 • 1.0142 • 1.0117 = 1.0209
Based on the geometric mean, the equivalent annual rate of return for this time period is 2.09%.
Arithmetic versus Geometric Means
In this video on arithmetic versus geometric means (https://openstax.Org/r/arithmetic-versus-geometric) the returns of the S&P 500 are tracked using an arithmetic mean versus a geometric mean, and the difference between these two measurements is discussed.
Weighted Mean
A weighted mean is a measure of the center, or average, of a data set where each data value is assigned a corresponding weight. A common financial application of a weighted mean is in determining the average price
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13.2 • Measures of Spread 385
per share for a certain stock when the stock has been purchased at different points in time and at different share prices.
To calculate a weighted mean, create a table with the data values in one column and the weights in a second column. Then create a third column in which each data value is multiplied by each weight on a row-by-row basis. Then, the weighted mean is calculated as the sum of the results from the third column divided by the sum of the weights.
THINK IT THROUG
Calculating the Weighted Mean
Assume your portfolio contains 1,000 shares of XYZ Corporation, purchased on three different dates, as shown in Table 13.2. Calculate the weighted mean of the purchase price for the 1,000 shares.
Date Purchased Purchase Price ($) Number of Shares Purchased Price ($) Times Number of Shares
January 17 78 200 15,600
February 10 122 300 36,600
March 23 131 500 65,500
Total NA 1,000 117,700
Table 13.2 1,000 Shares of XYZ Corporation
Solution:
In this example, the purchase price is weighted by the number of shares. The sum of the third column is $117,700, and sum of the weights is 1,000. The weighted mean is calculated as $117,700 divided by 1,000, which is $117.70.
Thus, the average cost per share for the 1,000 shares of XYZ Corporation is $117.70.
13.2
Measures of Spread
By the end of this section, you will be able to:
■ Define and calculate standard deviation for a data set.
■ Define and calculate variance for a data set.
■ Explain the relationship between standard deviation and variance.
Standard Deviation
An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated close to the mean; in other data sets, the data values are more widely spread out. For example, an investor might examine the yearly returns for Stock A, which are 1 %, 2%, -1 %, 0%, and 3%, and compare them to the yearly returns for Stock B, which are -9%, 2%, 15%, -5%, and 0%.
Notice that Stock B exhibits more volatility in yearly returns than Stock A. The investor may want to quantify this variation in order to make the best investment decisions for a particular investment objective.
The most common measure of variation, or spread, is standard deviation. The standard deviation of a data set is a measure of how far the data values are from their mean. A standard deviation
■ provides a numerical measure of the overall amount of variation in a data set; and
386 13 • Statistical Analysis in Finance
■ can be used to determine whether a particular data value is close to or far from the mean.
The standard deviation provides a measure of the overall variation in a data set. The standard deviation is always positive or zero. It is small when the data values are all concentrated close to the mean, exhibiting little variation or spread. It is larger when the data values are more spread out from the mean, exhibiting more variation.
Suppose that we are studying the variability of two different stocks. Stock A and Stock B. The average stock price for both stocks is $5. For Stock A, the standard deviation of the stock price is 2, whereas the standard deviation for Stock B is 4. Because Stock B has a higher standard deviation, we know that there is more variation in the stock price for Stock B than in the price for Stock A.
There are two different formulas for calculating standard deviation. Which formula to use depends on whether the data represents a sample or a population. The notation s is used to represent the sample standard deviation, and the notation a is used to represent the population standard deviation. In the formulas shown below, x is the sample mean, fi is the population mean, n is the sample size, and N is the population size.
Formula for the sample standard deviation: Formula for the population standard deviation:
V n
Variance
Variance also provides a measure of the spread of data values. The variance of a data set measures the extent to which each data value differs from the mean. The more the individual data values differ from the mean, the larger the variance. Both the standard deviation and the variance provide similar information.
In a finance application, variance can be used to determine the volatility of an investment and therefore to help guide financial decisions. For example, a more cautious investor might opt for investments with low volatility.
Similar to standard deviation, the formula used to calculate variance also depends on whether the data is collected from a sample or a population. The notation s1 is used to represent the sample variance, and the notation o2 is used to represent the population variance.
Formula for the sample variance:
S ~ n-1
Formula for the population variance:
2 £(*-/*)2 ^ =-n-
This is the method to calculate standard deviation and variance for a sample:
1. First, find the mean x of the data set by adding the data values and dividing the sum by the number of data values.
2. Set up a table with three columns, and in the first column, list the data values in the data set.
3. For each row, subtract the mean from the data value (x — x), and enter the difference in the second
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13.2 • Measures of Spread 387
column. Note that the values in this column may be positive or negative. The sum of the values in this column will be zero.
4. In the third column, for each row, square the value in the second column. So this third column will contain the quantity (Data Value - Mean)2 for each row. We can write this quantity as (x — x)2. Note that the values in this third column will always be positive because they represent a squared quantity.
5. Add up all the values in the third column. This sum can be written as ^(x — x)2.
6. Divide this sum by the quantity [n - 1), where n is the number of data points. We can write this as
I(*-*)2 n - 1
7. This result is called the sample variance, denoted by s2. Thus, the formula for the sample variance is
2 = I(x-x)2 n - 1
8. Now take the square root of the sample variance. This value is the sample standard deviation, called s.
Thus, the formula for the sample standard deviation is s =
9. Round-off rule: The sample variance and sample standard deviation are typically rounded to one more decimal place than the data values themselves.
THINK IT THROUG
Finding Standard Deviation and Variance
A brokerage firm advertises a new financial analyst position and receives 210 applications. The ages of a sample of 10 applicants for the position are as follows:
40, 36, 44, 51, 54, 55, 39, 47, 44, 50
The brokerage firm is interested in determining the standard deviation and variance for this sample of 10 ages.
Solution:
Find the sample variance and sample standard deviation by creating a table with three columns (see Table 13.3).
1. The data set is 40, 36, 44, 51, 54, 55, 39, 47, 44, 50.
2. This data set has 10 data values. Thus, n = 10.
3. The mean is calculated as x — 46.
4. Column 1 will contain the data values themselves.
5. Column 2 will contain x — x.
6. Column 3 will contain (x — x)2.
Column 1 X Column 2 (X - X) Column 3 (x - x)2
40 40 - 46 = - 6 (-6)2 = 36
36 36 - 46 = - 10 (-10)2 = 100
44 44 - 46 = - 2 (-2)2 = 4
51 51 - 46 = 5 (5)2 = 25
Table 13.3 Calculations for Standard Deviation for Age Example
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Column 1 X Column 2 (x — x) Column 3 (x - x)2
54 54 - 46 = 8 (8)2 = 64
55 55 - 46 = 9 (9)2 = 81
39 39 - 46 = - 7 (-7)2 = 49
47 47 - 46 = 1 (i)2 = 1
44 44 - 46 = - 2 (-2)2 = 4
50 50 - 46 = 4 (4)2 = 16
Sum = 0 Sum = 380
Table 13.3 Calculations for Standard Deviation for Age Example
7. To calculate the sample variance, use the sample variance formula:
2 = Z(* ~ *)2 = 380 = 380
n - 1 10-1 9 ~ '
8. To calculate the sample standard deviation, use the sample standard deviation formula:
As the above example illustrates, calculating the variance and standard deviation is a tedious process. A financial calculator can calculate statistical measurements such as mean and standard deviation quickly and efficiently.
There are two steps needed to perform statistical calculations on the calculator:
1. Enter the data in the calculator using the [data] function, which is located above the 7 key.
2. Access the statistical results provided by the calculator using the [stat] function, which is located above the 8 key.
Follow the steps in Table 13.4 to calculate mean and standard deviation using the financial calculator. The ages data set from the Think It Through example above is used again here: 40, 36,44, 51, 54, 55, 39,47,44, 50.
Enter Display
2ND [data] X01 0.00
2ND [clr work] X01 0.00
Description
1 Enter [data] entry mode
2 Clear any previous data
3 Enter first data value of 40
4 Move to next data entry
5 Move to next data entry
6 Enter second data value of 36
40 enter
X01 :
Y01 :
X02
40.00 1.00 0.0
36 enter
X02 = 36.00
Table 13.4 Calculator Steps for Mean and Standard Deviation
2 The specific financial calculator in these examples is the Texas Instruments BA II Plus™ Professional model, but you can use other financial calculators for these types of calculations.
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13.2 • Measures of Spread 389
Step Description Enter Display
7 Move to next data entry i Y02 = 1.00
8 Move to next data entry i X03 0.00
9 Enter third data value of 44 44 enter X03 = 44.00
10 Move to next data entry 1 Y03 = 1.00
11 Continue to enter remaining data values
12 Enter [stat] mode 2nd[stat] LIN
13 Move to first statistical result i n = 10.00
14 Move to next statistical result I x = 46.00
15 Move to next statistical result I Sx = 6.50
Table 13.4 Calculator Steps for Mean and Standard Deviation2
From the statistical results, the mean is shown as 46, and the sample standard deviation is shown as 6.50.
Excel provides a similar analysis using the built-in functions =AVERAGE (for the mean) and =STDEV.S (for the sample standard deviation). To calculate these statistical results in Excel, enter the data values in a column. Let's assume the data values are placed in cells A2 through A11. In any cell, type the Excel command =AVERAGE(A2:A11) and press enter. Excel will calculate the arithmetic mean in this cell. Then, in any other cell, type the Excel command =STDEV.S(A2:A11) and press enter. Excel will calculate the sample standard deviation in this cell. Figure 13.2 shows the mean and standard deviation for the 10 ages.
A A B C D E
1 Ages
2 40
3 36 Excel command to calculate sample mean:
4 44 =AVERAGE(A2:A11)
5 51
6 54 Result: 46
7 55
S 39 Excel command to calculate sample standard deviation:
9 47 =STDEV.S(A2:A11)
10 44
11 50 Result: 6.50
Figure 13.2 Mean and Standard Deviation in Excel.
Relationship between Standard Deviation and Variance
In the formulas shown above for variance and standard deviation, notice that the variance is the square of the standard deviation, and the standard deviation is the square root of the variance.
Once you have calculated one of these values, you can directly calculate the other value. For example, if you know the standard deviation of a data set is 12.5, you can calculate the variance by squaring this standard deviation. The variance is then 12.52, which is 156.25.
In the same way, if you know the value of the variance, you can determine the standard deviation by calculating the square root of the variance. For example, if the variance of a data set is known to be 31.36, then the standard deviation can be calculated as the square root of 31.36, which is 5.6.
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One disadvantage of using the variance is that the variance is measured in square units, which are different from the units in the data set. For example, if the data set consists of ages measured in years, then the variance would be measured in years squared, which can be confusing. The standard deviation is measured in the same units as the original data set, and thus the standard deviation is used more commonly than the variance to measure the spread of a data set.
13.3
Measures of Position
By the end of this section, you will be able to:
■ Define and calculate z-scores for a measurement.
■ Define and calculate quartiles and percentiles for a data set.
■ Use quartiles as a method to detect outliers in a data set.
z-Scores
A z-score, also called a z-value, is a measure of the position of an entry in a data set. It represents the number of standard deviations by which a data value differs from the mean. For example, suppose that in a certain year, the rates of return for various technology-focused mutual funds are examined, and the mean return is 7.8% with a standard deviation of 2.3%. A certain mutual fund publishes its rate of return as 12.4%. Based on this rate of return of 12.4%, we can calculate the relative standing of this mutual fund compared to the other technology-focused mutual funds. The corresponding z-score of a measurement considers the given measurement in relation to the mean and standard deviation for the entire population.
The formula for a z-score calculation is as follows:
x — n
z =-
a
where x is the measurement, fi is the mean, and a is the standard deviation.
THINK IT THROUG
Interpreting a z-Score
A certain technology-based mutual fund reports a rate of return of 12.4% for a certain year, while the mean rate of return for comparable funds is 7.8% and the standard deviation is 2.3%. Calculate and interpret the z-score for this particular mutual fund.
Solution:
In this example, the measurement x = 12.4, // = 7.8, and a = 2.3. Using the z-score formula, the calculation is performed as follows:
_ x - n _ Y2A - 7.8 _ 4.6
Z~ a ~ 23 ~ 23 ~
The resulting z-score indicates the number of standard deviations by which a particular measurement is above or below the mean. In this example, the rate of return for this particular mutual fund is 2 standard deviations above the mean, indicating that this mutual fund generated a significantly better rate of return than all other technology-based mutual funds for the same time period.
Quartiles and Percentiles
If a person takes an IQ test, their resulting score might be reported as in the 87th percentile. This percentile indicates the person's relative performance compared to others taking the IQ test. A person scoring in the 87th percentile has an IQ score higher than 87% of all others taking the test. This is the same as saying that the
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13.3 • Measures of Position 391
person is in the top 13% of all people taking the IQ test.
Common measures of location are quartiles and percentiles. Quartiles are special percentiles. The first quartile, Q1f is the same as the 25th percentile, and the third quartile, Q3, is the same as the 75th percentile. The median, M, is called both the second quartile and the 50th percentile.
To calculate quartiles and percentiles, the data must be ordered from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. If you score in the 90th percentile of an exam, that does not necessarily mean that you receive 90% on the test. It means that 90% of the test scores are the same as or less than your score and the remaining 10% of the scores are the same as or greater than your score.
Percentiles are useful for comparing values. In a finance example, a mutual fund might report that the performance for the fund over the past year was in the 80th percentile of all mutual funds in the peer group. This indicates that the fund performed better than 80% of all other funds in the peer group. This also indicates that 20% of the funds performed better than this particular fund.
Quartiles are values that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first find the median, or second quartile. The first quartile, Q1f is the middle value, or median, of the lower half of the data, and the third quartile, Q3, is the middle value of the upper half of the data. As an example, consider the following ordered data set, which represents the rates of return for a group of technology-based mutual funds in a certain year:
5.4, 6.0, 6.3, 6.8, 7.1, 7.2, 7.4, 7.5, 7.9, 8.2, 8.7
The median, or second quartile, is the middle value in this data set, which is 7.2. Notice that 50% of the data values are below the median, and 50% of the data values are above the median. The lower half of the data values are 5.4, 6.0, 6.3, 6.8, 7.1 Notice that these are the data values below the median. The upper half of the data values are 7.4, 7.5, 7.9, 8.2, 8.7, which are the data values above the median.)
To find the first quartile, Qi, locate the middle value of the lower half of the data. The middle value of the lower half of the data set is 6.3. Notice that one-fourth, or 25%, of the data values are below this first quartile, and 75% of the data values are above this first quartile.
To find the third quartile, Q3, locate the middle value of the upper half of the data. The middle value of the upper half of the data set is 7.9. Notice that one-fourth, or 25%, of the data values are above this third quartile, and 75% of the data values are below this third quartile.
The interquartile range (IQR) is a number that indicates the spread of the middle half, or the middle 50%, of the data. It is the difference between the third quartile, Q3, and the first quartile, Q-\.
IQR = Qs - Qi
In the above example, the IQR can be calculated as
IQR = Q3- Qi = 7.9 - 6.3 = 1.6
Outlier Detection
Quartiles and the IQR can be used to flag possible outliers in a data set. For example, if most employees at a company earn about $50,000 and the CEO of the company earns $2.5 million, then we consider the CEO's salary to be an outlier data value because is significantly different from all the other salaries in the data set. An outlier data value can also be a value much lower than the other data values, so if one employee only makes $15,000, then this employee's low salary might also be considered an outlier.
To detect outliers, use the quartiles and the IQR to calculate a lower and an upper bound for outliers. Then any data values below the lower bound or above the upper bound will be flagged as outliers. These data values should be further investigated to determine the nature of the outlier condition.
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To calculate the lower and upper bounds for outliers, use the following formulas:
Lower Bound for Outliers = Qi - (1.5 ■ IQR) Upper Bound for Outliers = Q3 + (1.5 • IQR)
Calculating the IQR
Calculate the IQR for the following 13 portfolio values, and determine if any of the portfolio values are potential outliers. Data values are in dollars.
389,950; 230,500; 158,000; 479,000; 639,000; 114,950; 5,500,000; 387,000; 659,000; 529,000; 575,000; 488,800; 1,095,000
Solution:
Order the data from smallest to largest.
114,950; 158,000; 230,500; 387,000; 389,950; 479,000; 488,800; 529,000; 575,000; 639,000; 659,000; 1,095,000; 5,500,000
M = 488,800 Qi = 230,500 + 387,000 =3Q8750
Q3 = 639,000 + 659,000 = 649 0(X)
IQR = 649,000 - 308,750 = 340,250 (1.5)(IQR) = (1.5)(340,250) = 510,375 Lower Bound = Qx - (1.5)(/Q«) = 308,750 - 510,375 = -201,625 Upper Bound = Q3 + (1.5)(IQR) = 649,000 + 510,375 = 1,159,375
No portfolio value price is less than -201,625. However, 5,500,000 is more than 1,159,375. Therefore, the portfolio value of 5,500,000 is a potential outlier. This is important because the presence of outliers could potentially indicate data errors or some other anomalies in the data set that should be investigated.
13.4
Statistical Distributions
By the end of this section, you will be able to:
■ Construct and interpret a frequency distribution.
■ Apply and evaluate probabilities using the normal distribution.
■ Apply and evaluate probabilities using the exponential distribution.
Frequency Distributions
A frequency distribution provides a method to organize and summarize a data set. For example, we might be interested in the spread, center, and shape of the data set's distribution. When a data set has many data values, it can be difficult to see patterns and come to conclusions about important characteristics of the data. A frequency distribution allows us to organize and tabulate the data in a summarized way and also to create graphs to help facilitate an interpretation of the data set.
To create a basic frequency distribution, set up a table with three columns. The first column will show the intervals for the data, and the second column will show the frequency of the data values, or the count of how
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13.4 • Statistical Distributions 393
many data values fall within each interval. A third column can be added to include the relative frequency for each row, which is calculated by taking the frequency for that row and dividing it by the sum of all the frequencies in the table.
Graphing Demand and Supply
A financial consultant at a brokerage firm records the portfolio values for 20 clients, as shown in Table 13.5. where the portfolio values are shown in thousands of dollars.
278 318 422 577 618
735 798 864 903 944
1,052 1,099 1,132 1,180 1,279
1,365 1,471 1,572 1,787 1,905
Table 13.5 Portfolio Values for 20 Clients at a Brokerage Firm (SOOOs)
Create a frequency distribution table using the following intervals for the portfolio values:
0-299 300-599 600-899 900-1,199 1,200-1,499 1,500-1,799 1,800-2,099
Solution:
Create a table where the intervals for portfolio value are listed in the first column. For this example, it was decided to create a frequency distribution table with seven rows and a class width set to 300. The class width is the distance from the start of one interval to the start of the next interval in the subsequent row. For example, the interval for the second row starts at 300, the interval for the third row starts at 600, and so on.
In the second column, record the frequency, or the number of data values that fall within the interval, for each row. For example, for the first row, count the number of data values that fall between 0 and 299. Because there is only one data value (278) that falls in this interval, the corresponding frequency is 1. For the second row, there are 3 data values that fall between 300 and 599 (318,422, and 577). Thus, the frequency for the second row is 3.
For the third column, called relative frequency, take the frequency for each row and divide it by the sum of the frequencies, which is 20. For example, in the first row, the relative frequency will be 1 divided by 20, which is 0.05. The relative frequency for the second row will be 3 divided by 20, which is 0.15. The resulting frequency distribution table is shown in Table 13.6.
394 13 • Statistical Analysis in Finance
Portfolio Value Interval ($000s) Frequency Relative Frequency
0-299 1 0.05
300-599 3 0.15
600-899 4 0.20
900-1,199 6 0.30
1,200-1,499 3 0.15
1,500-1,799 2 0.10
1,800-2,099 1 0.05
Table 13.6 Frequency Distribution of Portfolio Values for 20 Clients at a Brokerage Firm (SOOOs)
The frequency table indicates that most customers have portfolio values between $300,000 and $599,000, as this row in the table shows the highest frequency. Very few customers have portfolios with a value below $299,000 or above $1,800,000, as these frequencies in these rows are very low. Because the highest frequency corresponds to the row in the middle of the table and the frequencies decrease with each interval below and above this middle interval, the frequency table indicates that this distribution is a bell-shaped distribution.
The following is a summary of how to create a frequency distribution table (for integer data). Note that the number of classes in a frequency table is the same as the number of rows in the table.
1. Calculate the class width using the formula Class Width = „ Mf* ~ ff"
3 Number of Classes
° Note: For integer data, round the class width up to the next whole number.
2. Create a table with a number of rows equal to the number of classes. Create columns for Lower Class Limit, Upper Class Limit, Frequency, and Relative Frequency.
3. Set the lower class limit for the first row equal to the minimum value from the data set, or some other appropriate value.
4. Calculate the lower class limit for the second row by adding the class width to the lower class limit from the first row. Add the class width to each new lower class limit to calculate the lower class limit for each subsequent row.
5. The upper class limit for each row is 1 less than the lower class limit of the subsequent row. You can also add the class width to each upper class limit to determine the upper class limit for the subsequent row.
6. Record the frequency for each row by counting how many data values fall between the lower class limit and the upper class limit for that row.
7. Calculate the relative frequency for each row by taking the frequency for that row and dividing by the total number of data values.
Normal Distribution
The normal probability density function, a continuous distribution, is the most important of all the distributions. The normal distribution is applicable when the frequency of data values decreases with each class above and below the mean. The normal distribution can be applied to many examples from the finance industry, including average returns for mutual funds over a certain time period, portfolio values, and others. The normal distribution has two parameters, or numerical descriptive measures: the mean, n, and the standard deviation, a. The variable x represents the quantity being measured whose data values have a
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13.4 • Statistical Distributions 395
normal distribution.
Figure 13.3 Graph of the Normal Distribution
The curve in Figure 13.3 is symmetric about a vertical line drawn through the mean, fi. The mean is the same as the median, which is the same as the mode, because the graph is symmetric about //. As the notation indicates, the normal distribution depends only on the mean and the standard deviation. Because the area under the curve must equal 1, a change in the standard deviation, a, causes a change in the shape of the normal curve; the curve becomes fatter and wider or skinnier and taller depending on a. A change in n causes the graph to shift to the left or right. This means there are an infinite number of normal probability distributions.
To determine probabilities associated with the normal distribution, we find specific areas under the normal curve, and this is further discussed in Apply the Normal Distribution in Financial Contexts. For example, suppose that at a financial consulting company, the mean employee salary is $60,000 with a standard deviation of $7,500. A normal curve can be drawn to represent this scenario, in which the mean of $60,000 would be plotted on the horizontal axis, corresponding to the peak of the curve. Then, to find the probability that an employee earns more than $75,000, you would calculate the area under the normal curve to the right of the data value $75,000.
Excel uses the following command to find the area under the normal curve to the left of a specified value:
=N0RM.DIST(XVALUE, MEAN, STANDARD_DEV, TRUE)
For example, at the financial consulting company mentioned above, the mean employee salary is $60,000 with a standard deviation of $7,500. To find the probability that a random employee's salary is less than $55,000 using Excel, this is the command you would use:
=N0RM.DIST(55000, 60000, 7500, TRUE)
Result: 0.25249
Thus, there is a probability of about 25% that a random employee has a salary less than $55,000. Exponential Distribution
The exponential distribution is often concerned with the amount of time until some specific event occurs. For example, a finance professional might want to model the time to default on payments for company debt holders.
An exponential distribution is one in which there are fewer large values and more small values. For example, marketing studies have shown that the amount of money customers spend in a store follows an exponential distribution. There are more people who spend small amounts of money and fewer people who spend large amounts of money.
Exponential distributions are commonly used in calculations of product reliability, or the length of time a product lasts. The random variable for the exponential distribution is continuous and often measures a
396 13 • Statistical Analysis in Finance
passage of time, although it can be used in other applications. Typical questions may be. What is the probability that some event will occur between x-\ hours and x2 hours? or What is the probability that the event will take more than x-\ hours to perform? In these examples, the random variable x equals either the time between events or the passage of time to complete an action (e.g., wait on a customer). The probability density function is given by
1 -±x fix) = -e fx
H
where n is the historical average of the values of the random variable (e.g., the historical average waiting time). This probability density function has a mean and standard deviation of i.
To determine probabilities associated with the exponential distribution, we find specific areas under the exponential distribution curve. The following formula can be used to calculate the area under the exponential curve to the left of a certain value:
-±x F(x) = 1-e I*
Calculating Probability
At a financial company, the mean time between incoming phone calls is 45 seconds, and the time between phone calls follows an exponential distribution, where the time is measured in minutes. Calculate the probability of having 2 minutes or less between phone calls.
Solution:
To calculate the probability, find the area under the curve to the left of 1 minute. The mean time is given as 45 seconds, which is the same as 0.75 minutes. The probability can then be calculated as follows:
Probability = 1 - e 0.75(l) = 0.736
13.5
Probability Distributions
By the end of this section, you will be able to:
■ Calculate portfolio weights in an investment.
■ Calculate and interpret the expected values.
■ Apply the normal distribution to characterize average and standard deviation in financial contexts. Calculate Portfolio Weights
In many financial analyses, the weightings by asset category in a portfolio are a key index used to assess if the portfolio is meeting allocation metrics. For example, an investor approaching retirement age may wish to shift assets in a portfolio to more conservative and lower-volatility investments. Weightings can be calculated in several different ways—for example, based on individual stocks in a portfolio or on various sectors in a portfolio. Weightings can also be calculated based on number of shares or the value of shares of a stock.
To calculate a weighting in a portfolio based on value, take the value of the particular investment and divide it by the total value of the overall portfolio. As an example, consider an individual's retirement account for which the desired portfolio weighting is determined to be 40% stocks, 50% bonds, and 10% cash equivalents. Table 13.7 shows the current assets in the individual's portfolio, broken out according to stocks, bonds, and cash equivalents.
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13.5 • Probability Distributions 397
Asset Value ($)
Stock A 134,000
Stock B 172,000
Bond C 38,000
Bond D 102,000
Bond E 96,000
Cash in CDs 35,700
Cash in savings 22,500
Total Value 600,200
Table 13.7 Portfolio Assets in
Stocks, Bonds, and Cash Equivalents
To determine the weighting in this portfolio for stocks, bonds, and cash, take the total value for each category and divide it by the total value of the entire portfolio. These results are summarized in Table 13.8. Notice that the portfolio weightings shown in the table do not match the target, or desired, allocation weightings of 40% stocks, 50% bonds, and 10% cash equivalents.
Asset Category Category Value ($) Portfolio Weighting
Stocks 306,000 306,000 n S1 600,200 - U-:U
Bonds 236,000 236,000 n o0 600,200 ~~ UJy
Cash 58,200 58,200 0 10 600,200
Total Value 600,200
Table 13.8 Portfolio Weightings for Stocks, Bonds, and Cash Equivalents
Portfolio rebalancing is a process whereby the investor buys or sells assets to achieve the desired portfolio weightings. In this example, the investor could sell approximately 10% of the stock assets and purchase bonds with the proceeds to align the asset categories to the desired portfolio weightings.
Calculate and Interpret Expected Values
A probability distribution is a mathematical function that assigns probabilities to various outcomes. For example, we can assign a probability to the outcome of a certain stock increasing in value or decreasing in value. One application of a probability distribution function is determining expected value.
In many financial situations, we are interested in determining the expected value of the return on a particular investment or the expected return on a portfolio of multiple investments. To calculate expected returns, we formulate a probability distribution and then use the following formula to calculate expected value:
Expected Value = Pi ■ R\+P2- 2?2 + P3 • 2*3 + ... + P„ ■ Rn
where P1f P2, P3, ■■■ Pn are the probabilities of the various returns and /?1f R2, R3, ■■■ Rn are the various rates of return.
In essence, expected value is a weighted mean where the probabilities form the weights. Typically, these values for Pn and Rn are derived from historical data. As an example, consider a probability distribution for
398 13 • Statistical Analysis in Finance
potential returns for United Airlines common stock. Assume that from historical data gathered over a certain time period, there is a 15% probability of generating a 12% return on investment for this stock, a 35% probability of generating a 5% return, a 25% probability of generating a 2% return, a 14% probability of generating a 5% loss, and an 11 % probability of resulting in a 10% loss. This data can be organized into a probability distribution table as seen in Table 13.9.
Using the expected value formula, the expected return of United Airlines stock over an extended period of time follows:
Expected Value = Pi • Ri + P2 ■ R2 + P3 ■ R3 + ... + Pn ■ Rn ExpectedValue = 0.15(0.12) + 0.35(0.05) + 0.25(0.02) + 0.14(-0.05) + O.ll(-O.lO) = 0.0225
Based on the probability distribution, the expected value of the rate of return for United Airlines common stock over an extended period of time is 2.25%.
Historical Return (%) Associated Probability (%)
12 15
5 35
2 25
-5 14
-10 11
Table 13.9 Probability Distribution for Historical Returns on United Airlines Stock
We can extend this analysis to evaluate the expected return for an investment portfolio consisting of various asset categories, such as stocks, bonds, and cash equivalents, where the probabilities are associated with the weighting of each category relative to the total value of the portfolio. Using historical return data for each of the asset categories, the expected return of the overall portfolio can be calculated using the expected value formula.
Assume an investor has assets in stocks, bonds, and cash equivalents as shown in Table 13.10.
Asset Category Value ($) Portfolio Weighting Historical Return (%)
Stocks Bonds Cash Total Value
Table 13.10 Portfolio Weightings and Historical Returns for Various Asset Categories
ExpectedValue = Pi • Ri + P2 • R2 + P3 • R3 + ... + P„ ■ R„ ExpectedValue = 0.51(0.130) + 0.39(0.040) + 0.10(0.025) = 0.0844
Based on the probability distribution, the expected value of the rate of return for this portfolio over an extended period of time is 8.44%.
Apply the Normal Distribution in Financial Contexts
The normal, or bell-shaped, distribution can be utilized in many applications, including financial contexts.
306,000 236,000 58,200 $600,200
306,000 _ n 600,200 _ 236,000
600,200
= 0.39
58,200 0 1Q 600,200 u,lu
13.0
4.0
2.5
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13.5 • Probability Distributions 399
Remember that the normal distribution has two parameters: the mean, which is the center of the distribution, and the standard deviation, which measures the spread of the distribution. Here are several examples of applications of the normal distribution:
IQ scores follow a normal distribution, with a mean IQ score of 100 and a standard deviation of 15. Salaries at a certain company follow a normal distribution, with a mean salary of $52,000 and a standard deviation of $4,800.
Grade point averages (GPAs) at a certain college follow a normal distribution, with a mean GPA of 3.27 and a standard deviation of 0.24.
The average annual gain of the Dow Jones Industrial Average (DJIA) over a 40-year time period follows a normal distribution, with a mean gain of 485 points and a standard deviation of 1,065 points. The average annual return on the S8 - AN Portfolio Values Figure 13.6 Histogram of Portfolio Values
Graphing Bivariate Data
Bivariate data refers to paired data, where each value of one variable is paired with a value of a second variable. An example of paired data would be if data were collected on employees' years of experience and their corresponding salaries. Typically, it is of interest to investigate possible associations or correlations between the two variables under analysis.
Time Series Graphs
Suppose that we want to track the consumer price index (CPI) over the past 10 years. One feature of the data that we may want to consider is the element of time. Because each year is paired with the CPI value for that year, we do not have to think of the data as being random. We can instead use the years given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing CPI value as the decade progresses is called a time series graph.
To construct a time series graph, we must look at both pieces of our paired data set. We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a point in time and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.
Example: The following data set shows the annual CPI for 10 years. We need to construct a time series graph for the (rounded) annual CPI data (see Table 13.12). The time series graph is shown in Figure 13.7.
Year CPI
2012 226.65
2013 230.28
2014 233.91
Table 13.12 Data for Time Series Graph of Annual CPI, 2012-2021 (source: US Bureau of Labor Statistics)
404 13 • Statistical Analysis in Finance
Year CPI
2015 233 70
2016 236 91
2017 242 84
2018 247 87
2019 251 71
2020 257 97
2021 261 58
Table 13.12 Data for Time Series Graph of Annual CPI, 2012-2021 (source: US Bureau of Labor Statistics)
Q.
U
2012 2013 2014 2015 2016 2017 2018 2019 2020 2021
Year
Figure 13.7 Time Series Graph of Annual CPI, 2012-2021
Scatter Plots
A scatter plot, or scatter diagram, is a graphical display intended to show the relationship between two variables. The setup of the scatter plot is that one variable is plotted on the horizontal axis and the other variable is plotted on the vertical axis. Then each pair of data values is considered as an (x, y) point, and the various points are plotted on the diagram. A visual inspection of the plot is then made to detect any patterns or trends. Additional statistical analysis can be conducted to determine if there is a correlation or other statistically significant relationship between the two variables.
Assume we are interested in tracking the closing price of Nike stock over the one-year time period from April 2020 to March 2021. We would also like to know if there is a correlation or relationship between the price of Nike stock and the value of the S8 , which indicates that R is ready to accept commands.
Type 'demoO' for some demos, 'helpO' for on-line help, or ' help. start() ' for an HTML browser interface to help. Type 'qO' to quit R.
R is a command-driven language, meaning that the user enters commands at the prompt, which R then executes one at a time. R can also execute a program containing multiple commands. There are ways to add a graphic user interface (GUI) to R. An example of a GUI tool for R is RStudio (https://openstax.0rg/r/RStudio).
The R command line can be used to perform any numeric calculation, similar to a handheld calculator. For
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13.7 • The R Statistical Analysis Tool 407
example, to evaluate the expression 10 + 3-7, enter the following expression at the command line prompt and hit return:
> 10+3*7 [1] 31
Most calculations in R are handled via functions. For statistical analysis, there are many preestablished functions in R to calculate mean, median, standard deviation, quartiles, and so on. Variables can be named and assigned values using the assignment operator <-. For example, the following R commands assign the value of 20 to the variable named x and assign the value of 30 to the variable named y.
> x <- 20
> y <- 30
These variable names can be used in any calculation, such as multiplying xby yto produce the result 600:
> x*y
[1] 600
The typical method for using functions in statistical applications is to first create a vector of data values. There are several ways to create vectors in R. For example, the cfunction is often used to combine values into a vector. The following R command will generate a vector called salaries that contains the data values 40,000, 50,000, 75,000, and 92,000:
> salaries <- c(40000, 50000, 75000, 92000)
This vector salaries can then be used in statistical functions such as mean, median, min, max, and so on, as shown:
> mean(salaries) [1] 64250
> median(salaries) [1] 62500
> min(salaries) [1] 40000
> max(salaries)
[1] 92000
408 13 • Statistical Analysis in Finance
Another option for generating a vector in R is to use the seq function, which will automatically generate a sequence of numbers. For example, we can generate a sequence of numbers from 1 to 5, incremented by 0.5, and call this vector examplel, as follows:
> examplel <- seq(l, 5, by=0.5)
If we then type the name of the vector and hit enter, R will provide a listing of numeric values for that vector name.
> salaries
[1] 40000 50000 75000 92000
> examplel
[1] 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Often, we are interested in generating a quick statistical summary of a data set in the form of its mean, median, quartiles, min, and max. The R command called summary provides these results.
> summary(salaries)
Min. 1st Qu. Median Mean 3rd Qu. Max. 40000 47500 62500 64250 79250 92000
For measures of spread, R includes a command for standard deviation, called sd, and a command for variance, called var. The standard deviation and variance are calculated with the assumption that the data set was collected from a sample.
> sd(salaries) [1] 23641.42
> var(salaries) [1] 558916667
To calculate a weighted mean in R, create two vectors, one of which contains the data values and the other of which contains the associated weights. Then enter the R command weighted.mean(values, weights).
The following is an example of a weighted mean calculation in R:
Assume your portfolio contains 1,000 shares of XYZ Corporation, purchased on three different dates, as shown in Table 13.14. Calculate the weighted mean of the purchase price, where the weights are based on the number of shares in the portfolio.
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13.7 • The R Statistical Analysis Tool 409
Date Purchased Purchase Price ($) Number of Shares Purchased
January 17 78 200
February 10 122 300
March 23 131 500
Total 1,000
Table 13.14 Portfolio of XYZ Shares
Here is how you would create two vectors in R: the price vector will contain the purchase price, and the shares vector will contain the number of shares. Then execute the R command weighted.mean(price, shares), as follows:
> price <- c(78, 122, 131)
> shares <- c(200, 300, 500)
> weighted.mean(price, shares) [1] 117.7
A list of common R statistical commands appears in Table 13.15.
R Command Result
mean() Calculates the arithmetic mean
median() Calculates the median
min() Calculates the minimum value
max() Calculates the maximum value
weighted.mear () Calculates the weighted mean
sum() Calculates the sum of values
summary() Calculates the mean, median, quartiles, min, and max
sd() Calculates the sample standard deviation
var() Calculates the sample variance
IQR() Calculates the interquartile range
barplot() Plots a bar chart of non-numeric data
boxplot() Plots a boxplot of numeric data
hist() Plots a histogram of numeric data
plot() Plots various graphs, including a scatter plot
freq() Creates a frequency distribution table
Table 13.15 List of Common R Statistical Commands
Graphing in R
There are many statistical applications in R, and many graphical representations are possible, such as bar graphs, histograms, time series plots, scatter plots, and others. The basic command to create a plot in R is the
410 13 • Statistical Analysis in Finance
plot command, plot(x, y), where x is a vector containing the x-values of the data set and y is a vector containing they-values of the data set.
The general format of the command is as follows:
>plot(x, y, main="text for title of graph", xlab="text for x-axis label", ylab="text for y-axis label")
For example, we are interested in creating a scatter plot to examine the correlation between the value of the S&P 500 and Nike stock prices. Assume we have the data shown in Table 13.13. collected over a one-year time period.
Note that data can be read into R from a text file or Excel file or from the clipboard by using various R commands. Assume the values of the S&P 500 have been loaded into the vector SP500 and the values of Nike stock prices have been loaded into the vector Nike. Then, to generate the scatter plot, we can use the following R command:
>plot(SP500, Nike, main="Scatter Plot of Nike Stock Price vs. S&P 500", xlab="S&P 500", ylab="Nike Stock Price")
As a result of these commands, R provides the scatter plot shown in Figure 13.10. This is the same data that was used to generate the scatter plot in Figure 13.8 in Excel.
150
140 -
m u •r
°- 120
u o
5) no
* 100
90
80
o o
° 0 D
o
o
oo o
o
2,800 3,000 3,200 3,400 3,600 3,800 4,000 S&P 500
Figure 13.10 Scatter Plot Generated by R for Nike Stock Price versus S&P 500
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13 •Summary 411
1
Summary
13.1 Measures of Center
Several measurements are used to provide the average of a data set, including mean, median, and mode. The terms mean and average are often used interchangeably. To calculate the mean for a set of numbers, add the numbers together and then divide the sum by the number of data values. The geometric mean redistributes not the sum of the values but the product by multiplying all of the individual values and then redistributing them in equal portions such that the total product remains the same. To calculate the median for a set of numbers, order the data from smallest to largest and identify the middle data value in the ordered data set.
13.2 Measures of Spread
The standard deviation and variance are measures of the spread of a data set. The standard deviation is small when the data values are all concentrated close to the mean, exhibiting little variation or spread. The standard deviation is larger when the data values are more spread out from the mean, exhibiting more variation. The formula used to calculate the standard deviation depends on whether the data represents a sample or a population, as the formulas for the sample standard deviation and the population standard deviation are slightly different.
13.3 Measures of Position
Several measures are used to indicate the position of a data value in a data set. One measure of position is the z-score for a particular measurement. The z-score indicates how many standard deviations a particular measurement is above or below the mean. Other measures of position include quartiles and percentiles. Quartiles are special percentiles. The first quartile, Qi, is the same as the 25th percentile, and the third quartile, Qi, is the same as the 75th percentile. The median, M, is called both the second quartile and the 50th percentile. To calculate quartiles and percentiles, the data must be ordered from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths.
13.4 Statistical Distributions
A frequency distribution provides a method of organizing and summarizing a data set and allows us to organize and tabulate the data in a summarized way. Once a frequency distribution is generated, it can be used to create graphs to help facilitate an interpretation of the data set. The normal distribution has two parameters, or numerical descriptive measures: the mean, //, and the standard deviation, a. The exponential distribution is often concerned with the amount of time until some specific event occurs.
13.5 Probability Distributions
A probability distribution is a mathematical function that assigns probabilities to various outcomes. In many financial situations, we are interested in determining the expected value of the return on a particular investment or the expected return on a portfolio of multiple investments. When analyzing distributions that follow a normal distribution, probabilities can be calculated by finding the area under the graph of the normal curve.
13.6 Data Visualization and Graphical Displays
Data visualization refers to the use of graphical displays to summarize a data set to help to interpret patterns and trends in the data. Univariate data refers to observations recorded for a single characteristic or attribute, such as salaries or blood pressure measurements. When graphing univariate data, we can choose from among several types of graphs. The type of graph to be used for a certain data set will depend on the nature of the data and the purpose of the graph. Examples of graphs for univariate data include line graphs, bar graphs, and histograms. Bivariate data refers to paired data where each value of one variable is paired with a value of a second variable. Examples of graphs for bivariate data include time series graphs and scatter plots.
412 13-Key Terms
13.7 The R Statistical Analysis Tool
R is an open-source statistical analysis tool that is widely used in the finance industry. It provides an integrated suite of functions for data analysis, graphing, and statistical programming. R is increasingly being used as a data analysis and statistical tool as it is an open-source language, and additional features are constantly being added by the user community. This tool can be used on many different computing platforms and can be downloaded at The R Project for Statistical Computing (https://openstax.Org/r/The-R-Project).
1? Key Terms
arithmetic mean a measure of center of a data set, calculated by adding up the data values and dividing the
sum by the number of data values bar graph a chart that presents categorical data in a summarized form based on frequency or relative
frequency
bivariate data paired data in which each value of one variable is paired with a value of a second variable data visualization the use of graphical displays, such as bar charts, histograms, and scatter plots, to help
interpret patterns and trends in a data set empirical rule a rule that provides the percentages of data values falling within one, two, and three standard
deviations from the mean for a bell-shaped (normal) distribution expected value a weighted average of the values of a variable where the weights are the associated
probabilities
exponential distribution a continuous probability distribution that is useful for calculating probabilities
within the time between events frequency distribution a method of organizing and summarizing a data set that provides the frequency
with which each value in the data set occurs geometric mean a measure of center of a data set, calculated by multiplying the data values and then
raising the product to the exponent i, where n is the number of data values histogram a graphical display of continuous data showing class intervals on the horizontal axis and
frequency or relative frequency on the vertical axis interquartile range (IQR) a number that indicates the spread of the middle half, or middle 50%, of the data;
the difference between the third quartile (Q_) and the first quartile (Q-|) median the middle value in an ordered data set mode the most frequently occurring data value in a data set
normal distribution a bell-shaped distribution curve that is used to model many measurements, including
IQ scores, salaries, heights, weights, blood pressures, etc. outliers data values that are significantly different from the other data values in a data set percentiles numbers that divide an ordered data set into hundredths; often used to indicate position of a
data value in a data set population data data representing all the outcomes or measurements that are of interest portfolio a collection of financial investments, such as stocks, bonds, mutual funds, certificates of deposit,
etc.
probability distribution a mathematical function that assigns probabilities to various outcomes quartiles numbers that divide an ordered data set into quarters; the second quartile is the same as the median
sample data data representing outcomes or measurements collected from a subset or part of a population scatter plot (or scatter diagram) a graphical display that shows the relationship between a dependent
variable and an independent variable standard deviation a measure of the spread of a data set that indicates how far a typical data value is from
the mean
time series graph a graphical display used to show measurement data plotted versus time, where time is
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13- CFA Institute 41
displayed on the horizontal axis variance the measure of the spread of data values calculated as the square of the standard deviation weighted mean a measure of center of a data set in which each data value has a corresponding weighting x-axis the horizontal axis in a rectangular coordinate system y-axis the vertical axis in a rectangular coordinate system
z-score (or z-value) a measure of the position of a data value in the data set, calculated by subtracting the mean from the data value and then dividing the difference by the standard deviation
CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session (https://openstax.Org/r/cfa-level1-study-session). Reference with permission of CFA Institute.
Multiple Choice
1. A data set of salaries contains an outlier salary. The best measure of center to use for this data set is the
a. mean
b. median
c. mode
d. standard deviation
2. A portfolio includes shares of United Airlines stock that were purchased at different times and different prices. Which measure is best to determine the average cost of the shares of the stock?
a. mean
b. median
c. weighted mean
d. standard deviation
3. Standard deviation is a measure of the _
a. center of a data set
b. position of a data value in a data set
c. area under a normal curve
d. spread of a distribution
4. How are standard deviation and variance related?
a. The two measures are equal to one another.
b. Variance is the square root of the standard deviation.
c. Standard deviation is the square root of the variance.
d. The two squared measures are equal to one another.
5. Which of the following is the best definition of a z-score?
a. the distance of a data value from the mean
b. the number of standard deviations that a data value is from the mean
c. the distance of a data value from the mean divided by the sample size
d. the number of quartiles that a data value is from the mean
6. The results of a standardized test indicate that you are in the 85th percentile. What is the best interpretation of this result?
a. You scored in the top 85% of all students taking the test.
414 13 • Review Questions
b. You scored in the top 15% of all students taking the test.
c. Your score on the test is 85 when measured on a scale from 0 to 100.
d. You scored in the bottom 15% of all students taking the test.
7. The interquartile range is_
a. the middle 50% of a data set
b. the upper 50% of a data set
c. the lower 50% of a data set
d. equal to the median
8. In a frequency distribution table, the sum of the relative frequencies must be equal to
a. the sample size
b. 1, or 100%
c. zero
d. the standard deviation of the distribution
9. A change in the standard deviation of a normal distribution will result in_
a. a change in the location of the peak of the curve
b. a change in the area under the curve
c. a change in the shape of the curve
d. a change that shifts the graph to the left or the right
10. When calculating an expected value,_.
a. the result should always be 1
b. the result should always be a positive value
c. the result should always be a negative value
d. the result can be a positive or negative value
11. The area under a normal curve between a z-score of-2 and a z-score of+2 is
a. 0.68
b. 0.95
c. 0.997
d. dependent on the mean and standard deviation
12. A scatter plot is a visualization for_
a. univariate data only
b. bivariate data only
c. either univariate or bivariate data
d. test scores
13. Which of the following is NOT a benefit of using the R statistical analysis tool?
a. Additional features are constantly being added by the user community.
b. It can be used on many computer platforms, including Mac, Windows, and Linux.
c. It is free to download.
d. Users pay an annual subscription fee.
5> Review Questions
Explain the considerations that determine whether the mean or the median is the best measure of central tendency for a data set.
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13-Problems 415
2. Explain the difference between a mean and a weighted mean.
3. Explain why the standard deviation of a data set cannot be a negative value.
4. Explain what a negative z-score, a positive z-score, and a z-score of zero imply.
5. Explain how quartiles can be used to detect outliers in a data set.
0 Problems
1. You purchased 1,000 shares of a stock for $12 per share. Then, two months later, you purchased an additional 500 shares of the same stock at $9 per share. Calculate the weighted mean of the purchase price for the total of 1,500 shares.
2. You score a 60 on a biology test. The mean test grade is 70, and the standard deviation is 5. Calculate and interpret your corresponding z-score.
3. You score a 60 on a biology test. The mean test grade is 70, and the standard deviation is 5. What percentile does your grade correspond to?
4. A fast food restaurant has measured service time for customers waiting in line, and the service time follows an exponential distribution with a mean waiting time of 1.9 minutes. The restaurant has a guarantee that if customers wait in line for more than 5 minutes, their meal is free. What is the probability that a customer will receive a free meal?
5. The total value of your portfolio consists of approximately 65% stock assets, 25% bonds, and 10% cash equivalents. Historical returns have shown that stocks provide a return of 12%, bonds provide a return of 3.5%, and cash savings provide a return of 1.5%. What is the expected value of the return on this portfolio?
6. The distribution of the average annual return of the S&P 500 over a 50-year time period follows a normal distribution with a mean rate of return of 10.5% and a standard deviation of 14.3%. What is the probability that an average annual return will fall between -3.8% and 24.8%?
7. Write a short R program to find the expected return for the data set in Table 13.16.
Historical Return on United Airlines Stock Associated Probability
12% 15%
5% 35%
2% 25%
-5% 14%
-10% 11%
Table 13.16
| ►! Video Activity
Normal Distribution Stock Return Calculations
Click to view content (https://openstax.Org/r/stock-return)
1. Assume the return on stocks follows a normal distribution. Is it more likely that a stock will return between -1 and +1 standard deviations from the mean or between -2 and +2 standard deviations from the mean? Why?
416 13-Video Activity
2. Would an investor be likely to prefer a stock that has a smaller standard deviation for annual stock returns or one with a larger standard deviation for annual stock returns? Why?
Portfolio Weights
Click to view content (https://openstax.Org/r/port-weights)
3. What are the reasons for calculating portfolio weights? What useful information does this provide to the investor?
4. What are the advantages and disadvantages of the equal weighting approach and the market cap weighting approach for portfolio allocation strategy?
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^^^^^^^^^^
Figure 14.1 Regression analysis is used in financial decision-making, (credit: modification of "Stock exchange" by Jack Sem/flickr, CC BY 2.0)
Chapter Outline
14.1 Correlation Analysis
14.2 Linear Regression Analysis
14.3 Best-Fit Linear Model
14.4 Regression Applications in Finance
14.5 Predictions and Prediction Intervals
14.6 Use of R Statistical Analysis Tool for Regression Analysis
LzlJ Why It Matters
Correlation and regression analysis are used extensively in finance applications. Correlation analysis allows the determination of a statistical relationship between two numeric quantities. Regression analysis can be used to predict one quantity based on a second quantity, assuming there is a significant correlation between the two quantities. For example, in finance, we use regression analysis to calculate the beta coefficient of a stock, which represents the volatility of the stock versus overall market volatility, with volatility being a measure of risk.
A business may want to establish a correlation between the amount the company spent on advertising versus its recorded sales. If a strong enough correlation is established, then the business manager can predict sales based on the amount spent on advertising for a given time period.
Finance professionals often use correlation analysis to predict future trends and mitigate risk in a stock portfolio. For example, if two investments are strongly correlated, an investor might not want to have both investments in a certain portfolio since the two investments would tend to move in the same directions during up markets or down markets. To diversify a portfolio, an investor might seek investments that are not strongly correlated with one another.
Regression analysis can be used to establish a mathematical equation that relates a dependent variable (such as sales) to an independent variable (such as advertising expenditure). In this discussion, the focus will be on
418 14 • Regression Analysis in Finance
analyzing the relationship between one dependent variable and one independent variable, where the relationship can be modeled using a linear equation. This type of analysis is called linear regression.
14.1
Correlation Analysis
Learning Outcomes
By the end of this section, you will be able to:
■ Calculate a correlation coefficient.
■ Interpret a correlation coefficient.
■ Test for the significance of a correlation coefficient.
Calculate a Correlation Coefficient
In correlation analysis, we study the relationship between bivariate data, which is data collected on two variables where the data values are paired with one another.
Correlation is the measure of association between two numeric variables. For example, we may be interested to know if there is a correlation between bond prices and interest rates or between the age of a car and the value of the car. To investigate the correlation between two numeric quantities, the first step is to create a scatter plot that will graph the (x, y) ordered pairs. The independent, or explanatory, quantity is labeled as the x-variable, and the dependent, or response, quantity is labeled as the y-variable.
For example, we may be interested to know if the price of Nike stock is correlated with the value of the S&P 500 (Standard & Poor's 500 stock market index). To investigate this, monthly data can be collected for Nike stock prices and value of the S&P 500 for a period of time, and a scatter plot can be created and examined. A scatter plot, or scatter diagram, is a graphical display intended to show the relationship between two variables. The setup of the scatter plot is that one variable is plotted on the horizontal axis and the other variable is plotted on the vertical axis. Each pair of data values is considered as an (x, y) point, and the various points are plotted on the diagram. A visual inspection of the plot is then made to detect any patterns or trends on the scatter diagram. Table 14.1 shows the relationship between the Nike stock price and its S&P value over a one-year time period.
To assess linear correlation, the graphical trend of the data points is examined on the scatter plot to determine if a straight-line pattern exists. If a linear pattern exists, the correlation may indicate either a positive or a negative correlation. A positive correlation indicates that as the independent variable increases, the dependent variable tends to increase as well, or, as the independent variable decreases, the dependent variable tends to decrease (the two quantities move in the same direction). A negative correlation indicates that as the independent variable increases, the dependent variable decreases, or, as the independent variable decreases, the dependent variable increases (the two quantities move in opposite directions). If there is no relationship or association between the two quantities, where one quantity changing does not affect the other quantity, we conclude that there is no correlation between the two variables.
Nike :ock Price
4/1/2020 2,912.43 87.18 5/1/2020 3,044.31 98.58 6/1/2020 3,100.29 98.05
Table 14.1 Nike Stock Price ($) and Value of S&P 500 over a One-Year Time Period (source: Yahoo! Finance)
Date S&P 500
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14.1 • Correlation Analysis 419
Date S&P 500 Nike Stock Price
7/1/2020 3,271.12 97.61
8/1/2020 3,500.31 111.89
9/1/2020 3,363.00 125.54
10/1/2020 3,269.96 120.08
11/1/2020 3,621.63 134.70
12/1/2020 3,756.07 141.47
1/1/2021 3,714.24 133.59
2/1/2021 3,811.15 134.78
3/1/2021 3,943.34 140.45
3/12/2021 3,943.34 140.45
Table 14.1 Nike Stock Price ($) and Value of S&P 500 over a One-Year Time Period (source: Yahoo! Finance)
From the scatter plot in the Nike stock versus S&P 500 example (see Figure 14.2). we note that the trend reflects a positive correlation in that as the value of the S&P 500 increases, the price of Nike stock tends to increase as well.
160 T-1-1-1-1-1-1
« 140
Q.
J* U
o
120 -
= 100
80
• it
2,900 3,100 3,300 3,500 3,700 3,900 4,100
S&P 500
Figure 14.2 Scatter Plot of Nike Stock Price ($) and Value of S&P 500 (data source: Yahoo! Finance)
When inspecting a scatter plot, it may be difficult to assess a correlation based on a visual inspection of the graph alone. A more precise assessment of the correlation between the two quantities can be obtained by calculating the numeric correlation coefficient (referred to using the symbol r).
The correlation coefficient, which was developed by statistician Karl Pearson in the early 1900s, is a measure of the strength and direction of the correlation between the independent variable x and the dependent variable/.
The formula for r is shown below; however, technology, such as Excel or the statistical analysis program R, is typically used to calculate the correlation coefficient.
w£xy-(2»(£:y)
420 14 • Regression Analysis in Finance
where n refers to the number of data pairs and the symbol ^ x indicates to sum the x-values.
Table 14.2 provides a step-by-step procedure on how to calculate the correlation coefficient r.
Step Representation in Symbols
1. Calculate the sum of the x-values. 2>
2. Calculate the sum of they-values. 2>
3. Multiply each x-value by the corresponding y-value and calculate the sum of these xy products.
4. Square each x-value and then calculate the sum of these squared values.
5. Square each y-value and then calculate the sum of these squared values. Xy2
6. Determine the value of n, which is the number of data pairs. n
7. Use these results to then substitute into the formula for the »2>y-(2»(E y)
correlation coefficient. v —
Table 14.2 Steps for Calculating the Correlation Coefficient
Note that since r is calculated using sample data, r is considered a sample statistic used to measure the strength of the correlation for the two population variables. Sample data indicates data based on a subset of the entire population.
Given the complexity of this calculation. Excel or other software is typically used to calculate the correlation coefficient.
The Excel command to calculate the correlation coefficient uses the following format:
=CORREL(A1:A10, B1:B10)
where A1 :A10 are the cells containing the x-values and B1 :B10 are the cells containing the y-values.
Download the spreadsheet file (https://0penstax.0rg/r/excel exhibits) containing key Chapter 14 Excel exhibits.
Interpret a Correlation Coefficient
Once the value of r is calculated, this measurement provides two indicators for the correlation:
a. the strength of the correlation based on the value of r
b. the direction of the correlation based on the sign of r
The value of r gives us this information:
■ The value of r is always between —1 and +1: —1 < r < 1.
■ The size of the correlation r indicates the strength of the linear relationship between the two variables. Values of r close to —1 or to +1 indicate a stronger linear relationship.
■ If r = 0, there is no linear relationship between the two variables (no linear correlation).
■ If r = 1, there is perfect positive correlation.
■ If r — —1, there is perfect negative correlation. In both of these cases, all the original data points lie on a straight line.
The sign of r gives us this information:
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14.1 • Correlation Analysis 421
A positive value of r means that when x increases, y tends to increase, and when x decreases, y tends to decrease [positive correlation).
A negative value of r means that when x increases, y tends to decrease, and when x decreases, y tends to increase [negative correlation).
LINK TO LEARNIN
Correlation in Finance Applications
This video on correlation concepts (https://0penstax.0rg/r/vide0 correlation) discusses them with a specific focus on finance applications.
The Excel command used to find the value of the correlation coefficient for the Nike stock versus S&P 500 example (refer back to Table 14.1) is
=C0RREL(B2:B14,C2:C14) In this example, the value of r is calculated by Excel to be r = 0.928.
Since this is a positive value close to 1, we conclude that the relationship between Nike stock and the value of the S&P 500 over this time period represents a strong, positive correlation.
The correlation coefficient r can also be determined using the statistical capability on the financial calculator:
■ Step 1 is to enter the data in the calculator (using the [data] function that is located above the 7 key).
■ Step 2 is to access the statistical results provided by the calculator (using the [stat] function that is located above the 8 key) and scroll to the correlation coefficient results.
Follow the steps in Table 14.3 for calculating the correlation data for the data set of Nike stock price and value of the S&P 500 shown previously.
Step Description Enter Display
1 Enter [data] entry mode 2nd [data] X01 0.00
2 Clear any previous data 2nd [clr work] X01 0.00
3 Enter first x-value of 2912.43 2912.43 enter X01 = 2,912.43
4 Move to next data entry i Y01 = 1.00
5 Enter first y-value of 87.18 87.18 enter Y01 = 87.18
6 Move to next data entry i X02 0.00
7 Enter second x-value of 3044.31 3044.31 enter X02 = 3,044.31
8 Move to next data entry i Y02 = 1.00
9 Enter second y-value of 98.58 98.58 enter Y02 = 98.58
10 Move to next data entry i X03 0.00
11 Continue to enter remaining data values
12 Enter [stat] mode 2nd [stat]
Table 14.3 Calculator Steps for Finding the Relationship between Nike Stock Price and Value of S&P 5001
1 The specific financial calculator in these examples is the Texas Instruments BA II Plus ™ Professional model, but you can use other financial calculators for these types of calculations.
422 14 • Regression Analysis in Finance
Step Description Enter Display
13 Press [set] until LIN appears 2ND [set] LIN
14 Move to 1st statistical result i n= 13.00
15 Move to next statistical result I x = 3,480.86
16 Continue to scroll down until the value of r is displayed i r = 0.93
Table 14.3 Calculator Steps for Finding the Relationship between Nike Stock Price and Value of S&P 5001
From the statistical results shown on the calculator display, the correlation coefficient r is 0.93, which indicates that the relationship between Nike stock and the value of the S8 -^=, then this implies that the correlation between the two variables demonstrates that a linear V"
relationship exists and is statistically significant at approximately the 0.05 level of significance. As the formula indicates, there is an inverse relationship between the sample size and the required correlation for significance of a linear relationship. With only 10 observations, the required correlation for significance is 0.6325; for 30 observations, the required correlation for significance decreases to 0.3651; and at 100 observations, the required level is only 0.2000.
NOTE:
If r is significant and the scatter plot shows a linear trend, the line can be used to predict the value of y for values of xthat are within the domain of observed x-values.
If r is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
If r is significant and the scatter plot shows a linear trend, the line may not be appropriate or reliable for prediction outside the domain of observed x-values in the data.
THINK
KIT THRO
Determining If a Correlation Is Significant
Suppose that the chief financial officer (CFO) of a corporation is investigating the correlation between stock prices and unemployment rate over a period of 10 years and finds the correlation coefficient to be -0.68. There are 10 (x, y) data points in the data set. Should the CFO conclude that the correlation is significant for the relationship between stock prices and unemployment rate based on a level of significance of 0.05?
Solution:
When using a level of significance of 0.05, if | r \ > -^=, then this implies that the correlation between the
V"
two variables demonstrates a linear relationship that is statistically significant at approximately the 0.05
level of significance. In this example, we check if I —0.681 is greater than or equal to -7=- where n = 10.
V"
Since ^_ is approximately 0.632, this indicates that the absolute value of r of —0.68 is greater than —7^, and thus the correlation is deemed as significant.
Correlations may be helpful in visualizing the data, but they are not appropriately used to explain a relationship between two variables. Perhaps no single statistic is more misused than the correlation coefficient. Citing correlations between health conditions and everything from place of residence to eye color have the effect of implying a cause-and-effect relationship. This simply cannot be accomplished with a correlation coefficient. The correlation coefficient is, of course, innocent of this misinterpretation. It is the duty of analysts to use a statistic that is designed to test for cause-and-effect relationships and to report only those results, if they are intending to make such a claim. The problem is that passing this more rigorous test is difficult, therefore lazy and/or unscrupulous researchers fall back on correlations when they cannot make their case legitimately.
424 14 • Regression Analysis in Finance
14.2
Linear Regression Analysis
Learning Outcomes
By the end of this section, you will be able to:
■ Analyze a regression using the method of least squares and residuals.
■ Test the assumptions for linear regression.
Method of Least Squares and Residuals
Once the correlation coefficient has been calculated and a determination has been made that the correlation is significant, typically a regression model is then developed. In this discussion we will focus on linear regression, where a straight line is used to model the relationship between the two variables. Once a straight-line model is developed, this model can then be used to predict the value of the dependent variable for a specific value of the independent variable.
Recall from algebra that the equation of a straight line is given by
y = mx + b
where m is the slope of the line and b is they-intercept of the line.
The slope measures the steepness of the line, and the y-intercept is that point on the y-axis where the graph crosses, or intercepts, the y-axis.
In linear regression analysis, the equation of the straight line is written in a slightly different way using the model
y = a + bx
In this format, b is the slope of the line, and a is the y-intercept. The notation y is called y-hat and is used to indicate a predicted value of the dependent variable y for a certain value of the independent variable x.
If a line extends uphill from left to right, the slope is a positive value, and if the line extends downhill from left to right, the slope is a negative value. Refer to Figure 14.3.
(a) (b) (c)
Figure 14.3 Three Possible Graphs of y = a + bx (a) If b > 0, the line slopes upward to the right, (b) If b = 0, the line is horizontal, (c) If b < 0, the line slopes downward to the right.
When generating the equation of a line in algebra using y — mx + b, two (x, y) points were required to generate the equation. However, in regression analysis, all (x, y) points in the data set will be utilized to develop the linear regression model.
The first step in any regression analysis is to create the scatter plot. Then proceed to calculate the correlation coefficient r, and check this value for significance. If we think that the points show a linear relationship, we would like to draw a line on the scatter plot. This line can be calculated through a process called linear regression. However, we only calculate a regression line if one of the variables helps to explain or predict the other variable. If xisthe independent variable andythe dependent variable, then we can use a regression line to predict y for a given value of x
As an example of a regression equation, assume that a correlation exists between the monthly amount spent
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14.2 • Linear Regression Analysis 425
on advertising and the monthly revenue for a Fortune 500 company. After collecting (x,y) data for a certain time period, the company determines the regression equation is of the form
y = 9,376.7 + 61.8*
where x represents the monthly amount spent on advertising (in thousands of dollars) and y represents the monthly revenues for the company (in thousands of dollars).
A scatter plot of the (x, y) data is shown in Figure 14.4.
25,000 -i-1
20,000 -
tc 10,000
5,000 -
0
—i—
20
—i—
40
—i—
60
—i—
80
100
Advertising Spend
120
140
160
Figure 14.4 Scatter Plot of Revenue versus Advertising for a Fortune 500 Company (SOOOs)
The Fortune 500 company would like to predict the monthly revenue if its executives decide to spend $150,000 in advertising next month. To determine the estimate of monthly revenue, let x = 150 in the regression equation and calculate a corresponding value for y:
y = 9,376.7 + 61.8* y = 9,376.7 + 61.8(150) y = 18,646.7
This predicted value of/indicates that the anticipated revenue would be $18,646,700, given the advertising spend of $150,000.
Notice that from past data, there may have been a month where the company actually did spend $150,000 on advertising, and thus the company may have an actual result for the monthly revenue. This actual, or observed, amount can be compared to the prediction from the linear regression model to calculate a residual.
A residual is the difference between an observed y-value and the predicted y-value obtained from the linear regression equation. As an example, assume that in a previous month, the actual monthly revenue for an advertising spend of $150,000 was $19,200,000, and thus y - 19,200. The residual for this data point can be calculated as follows:
Residual = (observed y-value)-(predicted y-value) Residual = y — y
Residual = 19,200- 18,646.7 = 553.3
Notice that residuals can be positive, negative, or zero. If the observed y-value exactly matches the predicted y-value, then the residual will be zero. If the observed y-value is greater than the predicted y-value, then the residual will be a positive value. If the observed y-value is less than the predicted y-value, then the residual will be a negative value.
When formulating the linear regression line of best fit to the points on the scatter plot, the mathematical
426 14 • Regression Analysis in Finance
analysis generates a linear equation where the sum of the squared residuals is minimized. This analysis is referred to as the method of least squares. The result is that the analysis generates a linear equation that is the "best fit" to the points on the scatter plot, in the sense that the line minimizes the differences between the predicted values and observed values for y.
THINK IT THROUG
Calculating a Residual
Suppose that the chief financial officer of a corporation has created a linear model for the relationship between the company stock and interest rates. When interest rates are at 5%, the company stock has a value of $94. Using the linear model, when interest rates are at 5%, the model predicts the value of the company stock to be $99. Calculate the residual for this data point.
Solution:
A residual is the difference between an observed y-value and the predicted y-value obtained from the linear regression equation
Residual = (observed y-value)-(predicted y-value) Residual = y — y Residual = 94 - 99 = -5
The goal in the regression analysis is to determine the coefficients a and b in the following regression equation:
y = a + bx
Once the (x, y) has been collected, the slope [b) and y-intercept [a) can be calculated using the following formulas:
b _ n Z xy - (£ x) (Z y) where n refers to the number of data pairs and £ x indicates sum of the x-values.
Notice that the formula for the y-intercept requires the use of the slope result [b), and thus the slope should be calculated first and the y-intercept should be calculated second.
When making predictions for y, it is always important to plot a scatter diagram first. If the scatter plot indicates that there is a linear relationship between the variables, then it is reasonable to use a best-fit line to make predictions for y, given x within the domain of x-values in the sample data, but not necessarily for x-values outside that domain.
Note: Computer spreadsheets, statistical software, and many calculators can quickly calculate the best-fit line and create the graphs. The calculations tend to be tedious if done by hand.
Assumptions for Linear Regression
Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between x and y in the sample data provides strong enough evidence that we can conclude that there is a linear relationship
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14.3 • Best-Fit Linear Model 427
between xand y in the population.
The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population (Figure 14.5). Examining the scatter plot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.
These are the assumptions underlying the test of significance:
1. There is a linear relationship in the population that models the average value of yfor varying values of x. In other words, the expected value of yfor each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
2. The y-values for any particular x-value are normally distributed about the line. This implies that there are morey-values scattered closer to the line than are scattered farther away. Assumption (1) implies that these normal distributions are centered on the line: the means of these normal distributions of y-values lie on the line.
3. The standard deviations of the population y-values about the line are equal for each value of x. In other words, each of these normal distributions of y-values has the same shape and spread about the line.
4. The residual errors are mutually independent (no pattern).
5. The data are produced from a well-designed, random sample or randomized experiment.
Figure 14.5 Best-Fit Line The y-values for each x-value are normally distributed about the line with the same standard deviation. For each x-value, the mean of the y-values lies on the regression line. Morey-values lie near the line than are scattered further away from the line.
14.3
Best-Fit Linear Model
Learning Outcomes
By the end of this section, you will be able to:
■ Calculate the slope and y-intercept for a linear regression model using technology.
■ Interpret and apply the slope and y-intercepts.
Calculate the Slope and y-Intercept for a Linear Regression Model Using Technology
Once a correlation has been deemed as significant, a best-fit linear regression model is developed. The goal in the regression analysis is to determine the coefficients a and b in the following regression equation:
y = a + bx
The slope (b) and y-intercept (a) can be calculated using the following formulas:
428 14 • Regression Analysis in Finance
b_ » Z *y ~ (Z *) (Z y) «Z*2-(Z*)2
n n
These formulas can be quite cumbersome, especially for a significant number of data pairs, and thus technology is often used (such as Excel, a calculator, R statistical software, etc.).
Using Excel: To calculate the slope and y-intercept of the linear model, start by entering the (x, y) data in two columns in Excel. Then the Excel commands =SLOPE and INTERCEPT can be used to calculate the slope and intercept, respectively.
The following data set will be used as an example: the monthly amount spent on advertising and the monthly revenue for a Fortune 500 company for 12 months (data is shown in Table 14.4).
Month Advertising Expenditure Revenue
Jan 49 12,210
Feb 145 17,590
Mar 57 13,215
Apr 153 19,200
May 92 14,600
Jun 83 14,100
Jul 117 17,100
Aug 142 18,400
Sep 69 14,100
Oct 106 15,500
Nov 109 16,300
Dec 121 17,020
Table 14.4 Revenue versus Advertising for Fortune 500 Company (SOOOs)
To calculate the slope of the regression model, use the Excel command
=SL0PE(y-data range, x-data range)
It's important to note that this Excel command expects that the y-data range is entered first and the x-data range is entered second. Since revenue depends on amount spent on advertising, revenue is considered the y-variable and amount spent on advertising is considered the x-variable. Notice the y-data is contained in cells C2 through C13 and the x-data is contained in cells B2 through B13. Thus the Excel command for slope would be entered as
=SL0PE(C2:C13, B2:B13) In the same way, the Excel command to calculate the y-intercept of the regression model is
=INTERCEPT(y-data range, x-data range)
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14.3 • Best-Fit Linear Model 429
For the data set shown in the above table, the Excel command would be
=INTERCEPT(C2:C13, B2:B13) The results are shown in Figure 14.6. where
slope b = 61.8 intercept a = 9,376.7
a AB C
1 Month Advertising Expenditure Revenue
2 Jan 49 12,210
3 Feb 145 17,590
4 Mar 57 13,215
5 Apr 153 19,200
6 May 92 14,600
7 Jun 83 14,100
8 Jul 117 17,100
9 A-Jg 142 18,400
10 Sep 69 14,100
11 Oct 106 15,500
12 Nov 109 16,300
13 Dec 121 17,020
14
15 =SLOPE(C2:C13,B2:B13) =INTERCEPT(C2:C13,B2:B13)
16 61.8 9,376.7
Figure 14.6 Revenue versus Advertising for Fortune 500 Company (SOOOs) Showing Slope and y-Intercept Calculation in Excel
Based on this, the regression equation can be written as
y = a + bx
y = 9,376.7 + 61.8x
where x represents the amount spent on advertising (in thousands of dollars) and y represents the amount of revenue (in thousands of dollars).
Using a Financial Calculator
The financial calculator provides the slope and y-intercept for the linear regression model once the (x, y) data is inputted into the calculator.
Follow the steps in Table 14.5 for calculating the slope and y-intercept for the data set of amounts spent on advertising and revenue shown previously.
Description
1 Enter [data] entry mode
2 Clear any previous data
3 Enter first x-value of 49
4 Move to next data entry
Enter
2ND [data] X01 2ND [CLR WORK] X01 49 ENTER X01 I Y01
Display
0.00 0.00 49.00 1.00
Table 14.5 Calculator Steps for the Slope and y-Intercept
430 14 • Regression Analysis in Finance
Step Description Enter m splay
5 Enter first y-value of 12210 12210 enter Y01 = 12,210.00
6 Move to next data entry I X02 0.00
7 Enter second x-value of 145 145 enter X02 = 145.00
8 Move to next data entry I Y02 = 1.00
9 Enter second y-value of 17590 17590ENTER Y02 = 17,590.00
10 Move to next data entry i X03 0.00
11 Continue to enter remaining data values
12 Enter [stat] mode 2ND [stat]
13 Press [set] until LIN appears 2ND [set] LIN
14 Move to 1st statistical result 1 n = 12.00
15 Move to next statistical result I x = 103.58
16 Continue to scroll down until the value of a is displayed i a = 9,376.70
17 Continue to scroll down until the value of b is displayed i b = 61.80
Table 14.5 Calculator Steps for the Slope and y-Intercept
From the statistical results shown on the calculator display, the slope b is 61.8 and the y-intercept a is 9,367.7.
Based on this, the regression equation can be written as
y = a + bx
y = 9,376.7 + 61.8x
Interpret and Apply the Slope and y-Intercept
The slope of the line, b, describes how changes in the variables are related. It is important to interpret the slope of the line in the context of the situation represented by the data. You should be able to write a sentence interpreting the slope in plain English.
Interpretation of the Slope
The slope of the best-fit line tells us how the dependent variable (y) changes for every one unit increase in the independent (x) variable, on average.
In the previous example, the linear regression model for the monthly amount spent on advertising and the monthly revenue for a Fortune 500 company for 12 months was generated as follows:
y — a + bx
y = 9,376.7 +61.8x
Since the slope was determined to be 61.8, the company can interpret this to mean that for every $1,000 dollars spent on advertising, on average, this will result in an increase in revenues of $61,800.
The intercept of the regression equation is the corresponding y-value when.
Interpretation of the Intercept
The intercept of the best-fit line tells us the expected mean value of y in the case where the x-variable is equal to zero.
However, in many scenarios it may not make sense to have the x-variable equal zero, and in these cases, the Access for free at openstax.org
14.4 • Regression Applications in Finance 431
intercept does not have any meaning in the context of the problem. In other examples, the x-value of zero is outside the range of the x-data that was collected. In this case, we should not assign any interpretation to the y-intercept.
In the previous example, the range of data collected for the x-variable was from $49 to $153 spent per month on advertising. Since this interval does not include an x-value of zero, we would not provide any interpretation for the intercept.
14.4
Regression Applications in Finance
Learning Outcomes
By the end of this section, you will be able to:
■ Calculate the regression model for a single independent variable as applied to financial forecasting.
■ Extract measures of slope and intercept from regression analysis in financial applications.
Regression Model for a Single Independent Variable
Regression analysis is used extensively in finance-related applications. Many typical applications involve determining if there is a correlation between various stock market indices such as the S&P 500, the Dow Jones Industrial Average (DJIA), and the Russell 2000 index.
As an example, suppose we would like to determine if there is a correlation between the Russell 2000 index and the DJIA. Does the value of the Russell 2000 index depend on the value of the DJIA? Is it possible to predict the value of the Russell 2000 index for a certain value of the DJIA? We can explore these questions using regression analysis.
Table 14.6 shows a summary of monthly closing prices of the DJIA and the Russell 2000 for a 12-month time period. We consider the DJIA to be the independent variable and the Russell 2000 index to be the dependent variable.
Monthly Close DJIA Russell 2000
1-Apr-21 34,200.67 2,262.67
1-Mar-21 32,981.55 2,220.52
1-Feb-21 30,932.37 2,201.05
1-Jan-21 29,982.62 2,073.64
1-Dec-20 30,606.48 1,974.86
1-Nov-20 29,638.64 1,819.82
1-Oct-20 26,501.60 1,538.48
1-Sep-20 27,781.70 1,507.69
1-Aug-20 28,430.05 1,561.88
1-Jul-20 26,428.32 1,480.43
1-Jun-20 25,812.88 1,441.37
1-May-20 25,383.11 1,394.04
Table 14.6 Monthly Closing Prices of the DJIA and the Russell 2000 for a 12-Month Time Period (source: Yahoo! Finance)
The first step is to create a scatter plot to determine if the data points appear to follow a linear pattern. The scatter plot is shown in Figure 14.7. The scatter plot clearly shows a linear pattern; the next step is to calculate
432 14 • Regression Analysis in Finance
the correlation coefficient and determine if the correlation is significant.
■ Using the Excel command =CORREL, the correlation coefficient is calculated to be 0.947. This value of the correlation coefficient is significant using the test for significance referenced earlier in Correlation Analysis.
■ Using the Excel commands =SLOPE and INTERCEPT, the value of the slope and y-intercept are calculated as 0.11 and —1,496.34, respectively, when rounded to two decimal places.
The Excel output is shown below:
=C0RREL(C3:C14,B3:B14): 0.947 =SL0PE(C3:C14,B3:B14): 0.113 =INTERCEPT(C3:C14,B3:B14): -1,496.340
2,600
x
01
-a c >—I
o o o
CM
3
2,400 -
2,200
2,000
1,800
1,600
1,400
1,200 -
1,000
24,000
26,000
28,000
30,000 DJ IA
32,000
34,000
36,000
Figure 14.7 Scatter Plot for Monthly Closing Prices of the DJIA versus the Russell 2000 for a 12-Month Time Period (data source: Yahoo! Finance)
Based on these results, the corresponding linear regression model is
y — a + bx
y = -1,496.34+ 0.11x
Assume the DJIA has reached a value of 32,000. Predict the corresponding value of the Russell 2000 index. To determine this, substitute the value of the independent variable, x = 32,000 (this is the given value of the DJIA), and calculate the corresponding value for the dependent variable, which is the predicted value for the Russell 2000 index:
y = -1,496.34 + 0.11(32,000) y = 2,023.66
Thus the predicted value for the Russell 2000 index is approximately 2,024 when the DJIA reached a value of 32,000.
Measures of Slope and Intercept from Regression Analysis
An important application of regression analysis is to determine the systematic risk for a particular stock, which is referred to as beta. A stock's beta is a measure of the volatility of the stock compared to a benchmark such as the S&P 500 index. If a stock has more volatility compared to the benchmark, then the stock will have a beta greater than 1.0. If a stock has less volatility compared to the benchmark, then the stock will have a beta less
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14.4 • Regression Applications in Finance 433
than 1.0.
Beta can be determined as the slope of the regression line when the stock returns are plotted versus the returns for the benchmark, such as the S&P 500. As an example, consider the calculation for beta of Nike stock based on monthly returns of Nike stock versus monthly returns for the S&P 500 over the time period from May 2020 to March 2021. The monthly return data is shown in Table 14.7.
S&P Nike Nike
Date S&P 500 Monthly Stock Monthly
Return (%) Price ($) Return(%
4/1/2020 2,912.43 N/A 87.18 N/A
5/1/2020 3,044.31 0.05 98.58 0.13
6/1/2020 3,100.29 0.02 98.05 -0.01
7/1/2020 3,271.12 0.06 97.61 0.00
8/1/2020 3,500.31 0.07 111.89 0.15
9/1/2020 3,363.00 -0.04 125.54 0.12
10/1/2020 3,269.96 -0.03 120.08 -0.04
11/1/2020 3,621.63 0.11 134.70 0.12
12/1/2020 3,756.07 0.04 141.47 0.05
1/1/2021 3,714.24 -0.01 133.59 -0.06
2/1/2021 3,811.15 0.03 134.78 0.01
3/1/2021 3,943.34 0.03 140.45 0.04
3/12/2021 3,943.34 0.00 140.45 0.00
Table 14.7 Monthly Returns of Nike Stock versus Monthly
Returns for the S&P 500 (source: Yahoo! Finance)
The scatter plot that graphs S8 salaries <- c(40000, 50000, 75000, 92000) To calculate the correlation coefficient r, we use the R command called cor.
As an example, consider the data set in Table 14.8. which tracks the return on the S8 SP500 <- c(8,1,0,2,-3,7,4)
> CocaCola <- c(6,0,-2,1,-1,8,2)
The R command called cor returns the correlation coefficient for the x-data vector and y-data vector:
> cor(SP500, CocaCola)
Generate Linear Regression Models Using the R Statistical Tool
To create a linear model in R, assuming the correlation is significant, the command Im (for linear model) will provide the slope and y-intercept for the linear regression equation.
The format of the R command is
lm(dependent_variable_vector - independent_variable_vector)
Notice the use of the tilde symbol as the separator between the dependent variable vector and the independent variable vector.
We use the returns on Coca-Cola stock as the dependent variable and the returns on the S8 ImtCocaCola - SP500) Call:
lm(formula = CocaCola - SP500) Coefficients: (Intercept) SP500 -0.3453 0.8641
The R output provides the value of the y-intercept as —0.3453 and the value of the slope as 0.8641. Based on this, the linear model would be
y = a + bx
y = -0.3453 + 0.8641x
where x represents the value of S&P 500 return and y represents the value of Coca-Cola stock return.
The results can also be saved as a formula and called "model" using the following R command. To obtain more detailed results for the linear regression, the summary command can be used, as follows:
> model <- ImtCocaCola - SP500)
> summary(model) Call:
lm(formula = CocaCola - SP500) Residuals: 1 2 3 4 5 6 7
-0.5672 -0.5188 -1.6547 -0.3828 1.9375 2.2969 -1.1109 Coefficients:
Estimate Std. Error t value Pr(>|t|) (Intercept) -0.3453 0.7836 -0.441 0.67783 SP500 0.8641 0.1734 4.984 0.00416 **
Signif. codes: 0 0.001 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.658 on 5 degrees of freedom
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14.6 • Use of R Statistical Analysis Tool for Regression Analysis 441
Multiple R-squared: 0.8325, Adjusted R-squared: 0.7989 F-statistic: 24.84 on 1 and 5 DF, p-value: 0.004161
In this output, the y-intercept and slope is given, as well as the residuals for each x-value. The output includes additional statistical details regarding the regression analysis.
Predicted values and prediction intervals can also be generated within R.
First, we can create a structure in R called a data frame to hold the values of the independent variable for which we want to generate a prediction. For example, we would like to generate the predicted return for Coca-Cola stock, given that the return for the S8 a <- data.frame(SP500=6)
> predict(model, a) 1
4.839062
The output from the predict command indicates that the predicted return for Coca-Cola stock will be 4.8% when the return for the S8 predict(model,a, interval="predict") fit Iwr upr
1 4.839062 0.05417466 9.62395 Thus the 95% prediction interval for Coca-Cola return is (0.05%, 9.62%) when the return for the S8 Review Questions
1. A correlation coefficient is calculated as —0.92. Provide an interpretation for this correlation coefficient.
2. Explain what a residual is and how this relates to the best-fit regression model.
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14 • Problems 445
3. Explain how to interpret the slope of the best-fit line.
4. Explain how to generate a prediction using a linear regression model.
5. Will the sign of the correlation coefficient always be the same as the sign of the slope of the best-fit linear regression model?
a Problems
1. A Fortune 500 company is tracking revenues versus cash flow for recent years, and the data is shown in the table below. Consider cash flow to be the dependent variable. Create a scatter plot of the data set, comment on the correlation between these two variables, and comment on the correlation for this data (all dollar amounts are in thousands).
Revenues ($000s) Cash Flow ($000s)
237 82
241 86
229 77
284 94
307 93
Table 14.9
2. A Fortune 500 company is tracking revenues versus cash flow for recent years, and the data is shown in the table below. Consider cash flow to be the dependent variable. Calculate the correlation coefficient for this data (all dollar amounts are in thousands).
Revenues ($000s) Cash Flow ($000s)
237 82
241 86
229 77
284 94
307 93
Table 14.10
3. A chief financial officer calculates the correlation coefficient for bond prices versus interest rate as -0.71. The data set contained nine (x, y) data points. Determine if the correlation is significant or not significant at the 0.05 level of significance.
4. A Fortune 500 company is tracking revenues versus cash flow for recent years, and the data is shown in the table below. Consider cash flow to be the dependent variable. Determine the best-fit linear regression equation for this data set (all dollar amounts are in thousands).
Revenues ($000s) Cash Flow ($000s)
237 82
241 86
Table 14.11
446 14 • Problems
229 77
284 94
307 93
Table 14.11
5. A Fortune 500 company is tracking revenues versus cash flow for recent years, and the data is shown in the table below. Consider cash flow to be the dependent variable. Assume the correlation is significant. Predict the cash flow for company revenues of $250,000 (all dollar amounts are in thousands).
Revenues ($000s) Cash Flow ($000s)
237 82
241 86
229 77
284 94
307 93
Table 14.12
6. A Fortune 500 company is tracking revenues versus cash flow for recent years, and the data is shown in the table below. Consider cash flow to be the dependent variable. Assume the correlation is significant. Predict the cash flow for company revenues of $750,000 (all dollar amounts are in thousands).
Revenues ($000s) Cash Flow ($000s)
237 82
241 86
229 77
284 94
307 93
Table 14.13
7. A Fortune 500 company is tracking revenues versus cash flow for recent years, and the data is shown in the table below. Consider cash flow to be the dependent variable. Calculate the residual for the revenue value of $284,000 (all dollar amounts are in thousands):
Revenues ($000s) Cash Flow ($000s)
237 82
241 86
229 77
284 94
307 93
Table 14.14
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14-Video Activity 447
[►J Video Activity
Simple Linear Regression
Click to view content (https://0penstax.0rg/r/linear regression)
1. Based on the scatter plot shown, will the correlation coefficient be a positive value or negative value? Would you estimate that the correlation is significant for the relationship between radio ads and revenue?
2. For the linear regression model for ads versus revenue, the slope is shown as 78.075. How would this slope be interpreted (that is, provide a verbal description for the meaning of the slope of 78.075)?
How to Calculate Correlation for Stocks, Bonds, and Funds
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3. Based on the presentation shown in the video, is the FTSE 100 index correlated with the value of sterling? Or are the two measures uncorrelated? What data leads to your conclusion?
4. Based on the presentation in the video, is there a correlation between stock funds and bond funds? Why is this information important to an investor trying to design a portfolio?
448 14 • Video Activity
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Figure 15.1 Investing can often provide great returns, but it can also be a risk, (credit: modification of work "E-ticker" by klip game/ Wikimedia Commons, Public Domain)
Chapter Outline
15.1 Risk and Return to an Individual Asset
15.2 Risk and Return to Multiple Assets
.3 The Capital Asset Pricing Model (CAPM)
15.4 Applications in Performance Measurement
15.5 Using Excel to Make Investment Decisions
Why It Matters
Having finished her college degree and embarked on her career, Maria is now contemplating her financial future. She is considering how she might invest some of her hard-earned money. As a short-term goal, she wants to build an emergency fund so that she could cover her expenses for six months if she became ill or injured and had to take time off of work. She would also like to save money for a down payment on a home and to purchase new furniture. Although she is not yet 30 years old, Maria also knows that it is prudent to begin saving for retirement.
What should she do with her savings? Maria has some friends who have told her how successful they have been investing in stocks. Bart bragged about doubling his money in just over a year when he purchased Facebook stock, and Tiffany quickly tripled her money when she purchased shares in Netflix. But Maria also knows that her uncle lost a significant amount of money when his Boeing stock dropped from over $300 per share to under $150 within a couple of months at the beginning of 2020. Just how risky would it be to invest in stocks? What type of return might Maria expect? Are there strategies she could follow that would allow her to avoid her uncle's fate?
450 15 • How to Think about Investing
15.1
Risk and Return to an Individual Asset
Learning Outcomes
By the end of this section, you will be able to:
■ Compute the realized return from an individual investment.
■ Compute the average return and volatility of returns from historical data.
■ Describe firm-specific risk.
Measuring Historical Returns
Risk and return are often referred to as the two Rs of finance. Investors are interested in both risk and return because understanding one without the other is really meaningless. In terms of investment, the concept of return is fairly straightforward; return is the benefit, or profit, the investor expects from an expenditure. It is the reward for investing—the reason an investment is made in the first place. However, no investment is a sure thing. The return may not be what the investor was expecting. This uncertainty about what the return will be is referred to as risk.
We begin by looking at how to measure both risk and return when considering an individual asset, such as one stock. If your grandparents bought 100 shares of Apple, Inc. stock for you when you were born, you are interested in knowing how well that investment has done. You may even want to compare how that investment has fared to how an investment in a different stock, perhaps Disney, would have done. You are interested in measuring the historical return.
Individual Investment Realized Return
The realized return of an investment is the total return that occurs over a particular time period. Suppose that you purchased a share of Target (TGT) at the beginning of January 2020 for $128.74. At the end of the year, you sold the stock for $176.53, which was $47.79 more than you paid for it. This increase in value is known as a capital gain. As the owner of the stock, you also received $2.68 in dividends during 2020. The total dollar return from your investment is calculated as
Total Dollar Return = Dividend Income + Capital Gain = 2.68 + $47.79 = $50.47
It is common to express investment returns in percentage terms rather than dollar terms. This allows you to answer the question "How much do I receive for each dollar invested?" so that you can compare investments of different sizes. The total percent return from your investment is
Total Percent Return = Dividend Yield + Capital Gain Yield
= ^gj^j + = 0.0208 + 0.3712 = 0.3920 = 39.20%
The dividend yield is calculated by dividing the dividends you received by the initial stock price. This calculation says that for each dollar invested in TGT in 2020, you received $0.0208 in dividends. The capital gain yield is the change in the stock price divided by the initial stock price. This calculation says that for each dollar invested in TGT in 2020, you received $0.3712 in capital gains. Your total percent return of 39.20% means that you made $0,392 for every dollar invested when your gains from both dividends and stock price appreciation are totaled together.
Calculating Return
You purchased 10 shares of 3M (MMM) stock in January 2020 for $175 per share, received dividends of $5.91
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15.1 • Risk and Return to an Individual Asset 451
per share, and sold the stock at the end of the year for $169.72 per share. Calculate your total dollar return, your dividend yield, your capital gain yield, and your total percent yield.
Solution:
Because you purchased 10 shares, you received $5.91 X 10 = $59.10 in dividend income. You spent $175.00 X 10 = $1,750.00 to purchase the stock, and you sold it for $169.72 X 10 = $1,697.20. Your total dollar return is
Total Dollar Return = Dividend Income + Capital Gain
= $59.10 + ($1,697.20 - $1,750.00) = $6.30
Your dividend yield is 0Q = 0.0338, or 3.38%, and your capital gain yield is
$169 $175 0075"°° = -°-0302> °r 3 02%-Your total Percent return is 3.38% + (-3.02%) = 0.36%.
Notice that you sold MMM for a price lower than what you paid for it at the beginning of the year. Your capital gain is negative, or what is often referred to as a capital loss. Although the price fell, you still had a positive total dollar return because of the dividend income.
Of course, investors seldom purchase a stock and then sell it exactly one year later. Assume that you purchased shares of Facebook (FB) on June 1, 2020, for $228.50 per share and sold the shares three months later for $261.90. You received no dividends. In this case, your holding period percentage return is calculated as
$261lVafn28-50 =0-1462 = 14.62% $228.50
This 14.62% is your return for a three-month holding period. To compare them to other investment opportunities, you need to express returns on a per-year, or annualized, basis. The holding period returned is converted to an effective annual rate (EAR) using the formula
EAR = (1 + Holding Period Percentage Return)"1 - 1
where m is the number of holding periods in a year.
There are four three-month periods in a year. So, the EAR for this investment is
EAR = (1 + 0.1462)4 - 1 = 0.7260 = 72.60%
What happens if you own a stock for more than one year? Your holding period return would have occurred over a period longer than a year, but the process to calculate the EAR is the same. Suppose you purchased shares of FB in May 2015, when it was selling for $79.30 per share. You held the stock until May 2020, when you sold it for $224.59. Your holding period percentage return would be $224^9~ j*7930 = 183.22%. You more
than tripled your money, but it took you five years to do so. Your EAR, which will be smaller than this five-year holding period return rate, is calculated as
I
EAR = (1 + 1.8322)5 - 1 = 23.15%
Average Annual Returns
Suppose that you purchased shares of Delta Airlines (DAL) at the beginning of 2011 for $11.19 and held the stock for 10 years before selling it for $40.21. You made $40.21 - $11.19 = $29.02 on your investment over a 10-year period. This is a 259.34% holding period return. The EAR for this investment is
J_
EAR = (1 + 2.5934) 10 - 1 = 13.65%
452 15 • How to Think about Investing
To calculate the EAR using the above formula, the holding period return must first be calculated. The holding period return represents the percentage return earned over the entire time the investment is held. Then the holding period return is converted to an annual percentage rate using the formula.
You can also use the basic time value of money formula to calculate the EAR on an investment. In time value of money language, the initial price paid for the investment, $11.19, is the present value. The price the stock is sold for, $40.21, is the future value. It takes 10 years for the $11.19 to grow to $40.21. Using the time value of money will result in a calculation of
PV x (1 + 0" = FV 1.19 x (1 + 010 =40.21
1 + / = 3.59340-10 / = 13.65%
The EAR formula and the time value of money both result in a 13.65% annual return. Mathematically, the two formulas are the same; one is simply an algebraic rearrangement of the other.
If you earned 13.65% each year, compounded for 10 years, you would have converted your $11.19 per share investment to $40.21 per share. Of course, DAL stock did not increase by exactly 13.65% each year. The returns for DAL for each year are shown in Table 15.1. Some years, the return was much higher than 13.65%. In 2013, the return was almost 133%! Other years, the return was much lower than 13.65%; in fact, in the return was negative in four of the years.
Year Return Value of Investment ($)
Initial investment of 11.19
2011 -0.3579 7.19
2012 0.4672 10.54
2013 1.3261 24.52
2014 0.8053 44.27
2015 0.0405 46.06
2016 -0.0135 45.44
2017 0.1623 52.81
2018 -0.0866 48.24
2019 0.2038 58.07
2020 -0.3077 40.20
Table 15.1 Yearly Returns for DAL, 2011-2020: Value of Initial Investment at Each Year End
Although an investment in DAL of $11.19 at the beginning of 2011 grew to $40.20 by the end of 2020, this growth was not consistent each year. The amount that the stock was worth at the end of each year is also shown in Table 15.1. During 2011, the return for DAL was -35.79%, resulting in the value of the investment falling to $11.19 x [1 + (-0.3579)] = $7.19. The following year, 2012, the return for DAL was 46.72%. Therefore, the value of the investment was $7.19 x 1 + (1 + 0.4672) = $10.54 at the end of 2012. This process continues each year that the stock is held.
The compounded annual return derived from the EAR and time value of money formulas is also known as a geometric average return. A geometric average return is calculated using the formula
J_
Geometric Average Return = [(1 + Ri) x (1 + R2) x ... x (1+«jv)] N - 1
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15.1 • Risk and Return to an Individual Asset 453
where RN is the return for each year in the time period for which the average is calculated.
The calculation of the geometric average return for DAL is shown in the right column of Table 15.2. (The slight difference in the geometric average return of 13.64% from the 13.65% derived from the EAR and time value of money calculations is due to rounding errors.)
Year Return 1+ Return
2011 -0.3579 0.6421
2012 0.4672 1.4672
2013 1.3261 2.3261
2014 0.8053 1.8053
2015 0.0405 1.0405
2016 -0.0135 0.9865
2017 0.1623 1.1623
2018 -0.0866 0.9134
2019 0.2038 1.2038
2020 -0.3077 0.6923
Arithmetic Avg 0.2240 3.5928 Product of (1 + Return)
Std Dev 0.5190 1.1364 Product raised to 1/N
0.1364 Geometric Average
Table 15.2 Yearly Returns for DAL, 2011-2020, with Calculation of the Arithmetic Mean, Standard Deviation, and Geometric Mean
Looking at Table 15.2. you will notice that the geometric average return differs from the mean return. Adding each of the annual returns and dividing the sum by 10 results in a 22.4% average annual return. This 22.4% is called the arithmetic average return.
The geometric average return will be smaller than the arithmetic average return (unless the returns for all years are identical). This is due to the basic arithmetic of compounding. Think of a very simple example in which you invest $100 for two years. If you have a positive return of 50% the first year and a negative 50% return the second year, you will have an arithmetic average return of 05 + (~0,5°) — 0.0%, but you will have a geometric average return of [(1 + 0.5) X (1 - 0.5)]0.5 - 1 = - 13.4%. With a 50% positive return the first year, you ended the year with $150. The second year, you lost 50% of that balance and were left with only $75.
Another important fact when studying average returns is that the order in which you earn the returns is not important. Consider what would have occurred if the returns in the two years were reversed, so that you faced a loss of 50% in the first year of your investment and a gain of 50% in the second year of your investment. With a -50% return in the first year, you would have ended that year with only $50. Then, if that $50 earned a positive 50% return the second year, you would have a $75 balance at the end of the two-year period. A negative return of 50% followed by a positive return of 50% still results in an arithmetic average return of 0% and a geometric average return of [(1 - 0.5) X (1 + 0.5)]0.5 - 1 = -13.4%.
454 15 • How to Think about Investing
Calculating Arithmetic and Geometric Average Return
The annual returns for CVS Health Corp. (CVS) for the 10-year period of 2011-2020 are shown in Table 15.3.
Year Returns
2011 18.94%
2012 20.28%
2013 50.38%
2014 37.12%
2015 2.90%
2016 -17.83%
2017 -5.75%
2018 -7.04%
2019 17.26%
2020 -5.14%
Table 15.3 CVS Annual Returns, 2011-2020 (source: Yahoo! Finance)
What was the arithmetic average return during the decade? What was the geometric average return during the decade?
Solution:
See Table 15.4 for the arithmetic mean and geometric mean calculations.
Arithmetic Average Calculation Geometric Average Calculation
Year Return 1 + Return
2011 0.1894 1.1894
2012 0.2028 1.2028
2013 0.5038 1.5038
2014 0.3712 1.3712
2015 0.0290 1.0290
2016 -0.1783 0.8217
2017 -0.0575 0.9425
2018 -0.0704 0.9296
2019 0.1726 1.1726
2020 -0.0514 0.9486
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15.1 • Risk and Return to an Individual Asset
Arithmetic Average Calculation Geometric Average Calculation
Sum of Returns 1.1112 2.4309 Product of (1 + Return)
Sum Divided by 10 0.1111 1.0929 Raise Product to 1/10
Arithmetic Average 11.11% 0.0929 Subtract 1
9.29% Geometric Average
Table 15.4 Yearly Returns for CVS, 2011-2020
The arithmetic average return for CVS was 11.11 %, and the geometric average return was 9.29%.
Both the arithmetic average return and the geometric average return are "correct" calculations. They simply answer different questions. The geometric average tells you what you actually earned per year on average, compounded annually. It is useful for calculating how much a particular investment grows over a period of time. The arithmetic average tells you what you earned in a typical year. When we are looking at the historical description of the distribution of returns and want to predict what to expect in a particular year, the arithmetic average is the relevant calculation.
Measuring Risk
Although the arithmetic average return for Delta Airlines (DAL) for 2011-2020 was 22.4%, there is not a year in which the return was exactly 22.4%. In fact, in some years, the return was much higher than the average, such as in 2013, when it was 132.61 %. In other years, the return was negative, such as 2011, when it was -35.79%. Looking at the yearly returns in Table 15.2. the return for DAL varies widely from year to year. In finance, this volatility of returns is considered risk.
Volatility of Returns
The most commonly used measure of volatility of returns in finance is the standard deviation of the returns. The standard deviation of returns for DAL for the sample period 2011-2020 is 51.9%. Remember that if the normal distribution (a bell-shaped curve) describes returns, then 68% (or about two-thirds) of the time, the return in a particular year will be within one standard deviation above and one standard deviation below the arithmetic average return. Given DAL's average return of 22.4%, the actual yearly return will be somewhere between -29.5% and 74.29% in two out of three years. A very high return of greater than 74.29% would occur 16% of the time; a very large loss of more than 29.5% would also occur 16% of the time.
As you can see, there is a wide range of what can be considered a "typical" year for DAL. Although we can calculate an average return, the return in any particular year is likely to vary from that average. The larger the standard deviation, the greater this range of returns is. Thus, a larger standard deviation indicates a greater volatility of returns and, hence, more risk.
Calculating the Standard Deviation of Returns
You calculated the arithmetic average return for CVS to be 11.11% for the 10-year period of 2011-2020. Calculate the standard deviation of returns for CVS for the same period (see Table 15.5). What does this tell you about what an investor in CVS experienced in a typical year during that decade?
Solution:
456 15 • How to Think about Investing
Year Return Return - Mean Squared Difference
2011 0.1894 0.0783 0.0061
2012 0.2028 0.0917 0.0084
2013 0.5038 0.3927 0.1542
2014 0.3712 0.2601 0.0676
2015 0.0290 -0.0821 0.0067
2016 -0.1783 -0.2894 0.0838
2017 -0.0575 -0.1686 0.0284
2018 -0.0704 -0.1815 0.0329
2019 0.1726 0.0615 0.0038
2020 -0.0514 -0.1625 0.0264
Mean 0.1111 0.4185 Sum of Squared Differences
0.0465 Sum Divided by N - 1
0.2156 Square Root
Table 15.5 Standard Deviation of CVS Returns
The standard deviation of returns for CVS during the sample period of 2011-2020 was 21.56%. With an arithmetic average return of 11.11%, the return would lie between -10.45% and 32.67% in about two out of three years. Even though the average return is 11.11 %, a return in a particular year might be much higher or much lower than that average. In fact, a loss of more than 10.45% would be expected about once every six years. Also, about once every six years, a return greater than 32.67% would be expected.
Firm-Specific Risk
Investors purchase a share of stock hoping that the stock will increase in value and they will receive a positive return. You can see, however, that even with well-established companies such as ExxonMobil and CVS, returns are highly volatile. Investors can never perfectly predict what the return on a stock will be, or even if it will be positive.
The yearly returns for four companies—Delta Airlines (DAL), Southwest Airlines (LUV), ExxonMobil (XOM), and CVS Health Corp. (CVS)—are shown in Table 15.6. Each of these stocks had years in which the performance was much better or much worse than the arithmetic average. In fact, none of the stocks appear to have a typical return that occurs year after year.
Two-Stock Portfolio
Year DAL LUV XOM CVS
2011 -35.79% -33.93% 18.67% 18.94%
2012 46.72% 20.03% 4.70% 20.28%
2013 132.61% 85.38% 20.12% 50.38%
2014 80.53% 126.47% -6.06% 37.12%
2015 4.05% 2.43% -12.79% 2.90%
Table 15.6 Yearly Returns for DAL, LUV, XOM, and CVS (source: Yahoo! Finance)
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15.1 • Risk and Return to an Individual Asset 457
Two-Stock Portfolio
Year DAL LUV XOM CVS
2016 -1.35% 16.72% 19.88% -17.83%
2017 16.23% 32.41% -3.81% -5.75%
2018 -8.66% -28.28% -15.09% -7.04%
2019 20.38% 17.69% 7.23% 17.26%
2020 -30.77% -13.04% -36.21% -5.14%
Average 22.40% 22.59% -0.34% 11.11%
Std Dev 51.90% 49.84% 18.18% 21.56%
Table 15.6 Yearly Returns for DAL, LUV, XOM, and CVS (source: Yahoo! Finance)
Figure 15.2 contains a graph of the returns for each of these four stocks by year. In this graph, it is easy to see that DAL and LUV both have more volatility, or returns that vary more from year to year, than do XOM or CVS. This higher volatility leads to DAL and LUV having higher standard deviations of returns than XOM or CVS.
140% -i
-40%
2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
■ DAL ■ LUV ■ XOM ■ CVS
Figure 15.2 Yearly Returns for DAL, LUV, XOM, and CVS (data source: Yahoo! Finance)
Standard deviation is considered a measure of the risk of owning a stock. The larger the standard deviation of a stock's annual returns, the further from the average that stock's return is likely to be in any given year. In other words, the return for the stock is highly unpredictable. Although the return for CVS varies from year to year, it is not subject to the wide swings of the returns for DAL or LUV.
Why are stock returns so volatile? The value of the stock of a company changes as the expectations of the future revenues and expenses of the company change. These expectations may change due to a number of events and new information. Good news about a company will tend to result in an increase in the stock price. For example, DAL announcing that it will be opening new routes and flying to cities it has not previously serviced suggests that DAL will have more customers and more revenue in future years. Or if CVS announces that it has negotiated lower rent for many of its locations, investors will expect the expenses of the company to fall, leading to more profits. Those types of announcements will often be associated with a higher stock price. Conversely, if the pilots and flight attendants for DAL negotiate higher salaries, the expenses for DAL will increase, putting downward pressure on profits and the stock price.
458 15 • How to Think about Investing
Peloton and Risk
An example of how news can impact the price of a stock occurred on May 5, 2021, when Peloton recalled all of its Tread+ and Tread products after the tragic death of a child and 70 injuries associated with use of its products.1 The previous day, Peloton stock traded for $96.70 per share. The stock price dropped approximately 15% when the recall was announced. The closing price for a share of Peloton on May 5 was $82.62.2 You can read the company's statement about this recall (https://0penstax.0rg/r/ the companys recall) online.
15.2
Risk and Return to Multiple Assets
Learning Outcomes
By the end of this section, you will be able to:
■ Explain the benefits of diversification.
■ Describe the relationship between risk and return for large portfolios.
■ Compare firm-specific and systematic risk.
■ Discuss how portfolio size impacts risk.
Diversification
So far, we have looked at the return and the volatility of an individual stock. Most investors, however, own shares of stock in multiple companies. This collection of stocks is known as a portfolio. Let's explore why it is wise for investors to hold a portfolio of stocks rather than to pick just one favorite stock to own.
We saw that investors who owned DAL experienced an average annual return of 20.87% but also a large standard deviation of 51.16%. Investors who used all their funds to purchase DAL stock did exceptionally well during 2012-2014. But in 2020, those investors lost almost one-third of their money as COVID-19 caused a sharp reduction in air travel worldwide. To protect against these extreme outcomes, investors practice what is called diversification, or owning a variety of stocks in their portfolios.
Suppose, for example, you have saved $50,000 that you want to invest. If you purchased $50,000 of DAL stock, you would not be diversified. Your return would depend solely on the return on DAL stock. If, instead, you used $5,000 to purchase DAL stock and used the remaining $45,000 to purchase nine other stocks, you would be diversifying. Your return would depend not only on DAL's return but also on the returns of the other nine stocks in your portfolio. Investors practice diversification to manage risk.
It is akin to the saying "Don't put all of your eggs in one basket." If you place all of your eggs in one basket and that basket breaks, all of your eggs will fall and crack. If you spread your eggs out across a number of baskets, it is unlikely that all of the baskets will break and all of your eggs will crack. One basket may break, and you will lose the eggs in that basket, but you will still have your other eggs. The same idea holds true for investing. If you own stock in a company that does poorly, perhaps even goes out of business, you will lose the money you placed in that particular investment. However, with a diversified portfolio, you do not lose all your money because your money is spread out across a number of different companies.
1 Trefis Team and Great Speculations. "Is Peloton's Tread+ Recall an Opportunity to Buy the Stock?" Forbes, May 7, 2021. https://www.forbes.com/sites/greatspeculations/2021/05/07/is-pelotons-tread-recall-an-opportunity-to-buy-the-stock/
2 Tomi Kilgore. "Peloton Stock Sinks to 8-Month Low after 125,000 Treadmills Recalled for 'Risk of Injury or Death.'" MarketWatch. May 6, 2021. https://www.marketwatch.com/story/peloton-stock-sinks-toward-9-month-low-after-125-000-treadmills-recalled-for-risk-of-injury-or-death-11620233715
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15.2 • Risk and Return to Multiple Assets 459
Diversification
Fidelity Investments Inc. is a multinational financial services firm and one of the largest asset managers in the world. In this educational video for investors (https://0penstax.0rg/r/educati0nal investors). Fidelity provides an explanation of what diversification is and how it impacts an investor's portfolio.
Table 15.7 shows the returns of investors who placed 50% of their money in DAL and the remaining 50% in LUV, XOM, or CVS. Notice that the standard deviation of returns is lower for the two-stock portfolios than for DAL as an individual investment.
Two-Stock Portfolio
Year DAL DAL and LUV DAL and XOM DAL and CVS
2011 -35.79% -34.86% -8.56% -8.43%
2012 46.72% 33.38% 25.71% 33.50%
2013 132.61% 109.00% 76.36% 91.50%
2014 80.53% 103.50% 37.24% 58.83%
2015 4.05% 3.24% -4.37% 3.47%
2016 -1.35% 7.69% 9.27% -9.59%
2017 16.23% 24.32% 6.21% 5.24%
2018 -8.66% -18.47% -11.88% -7.85%
2019 20.38% 19.03% 13.81% 18.82%
2020 -30.77% -21.90% -33.49% -17.95%
Average 22.40% 22.49% 11.03% 16.75%
Std Dev 51.90% 49.11% 30.43% 35.10%
Table 15.7 Yearly Returns for DAL Versus a Two-Stock Portfolio Containing DAL and LUV, XOM, or CVS (data source: Yahoo! Finance)
As investors diversify their portfolios, the volatility of one particular stock becomes less important. XOM has good years with above-average returns and bad years with below-average (and even negative) returns, just like DAL. But the years in which those above-average and below-average returns occur are not always the same for the two companies. In 2014, for example, the return for DAL was greater than 80%, while the return for XOM was negative. On the other hand, in 2011, when DAL had a return of-35.15%, XOM had a positive return. When more than one stock is held, the gains in one stock can offset the losses in another stock, washing away some of the volatility.
When an investor holds only one stock, that one stock's volatility contributes 100% to the portfolio's volatility. When two stocks are held, the volatility of each stock contributes to the volatility of the portfolio. However, the volatility of the portfolio is not simply the average of the volatility of each stock held independently. How correlated the two stocks are, or how much they move together, will impact the volatility of the portfolio.
You will recall from our study of correlation in Regression Analysis in Finance that a correlation coefficient describes how two variables move relative to each other. A correlation coefficient of 1 means that there is a perfect, positive correlation between the two variables, while a correlation coefficient of -1 means that the two variables move exactly opposite of each other. Stocks that are in the same industry will tend to be more
460 15 • How to Think about Investing
strongly correlated than stocks that are in much different industries. During the 2011-2020 time period, the correlation coefficient for DAL and LUV was 0.87, the correlation coefficient for DAL and XOM was 0.35, and the correlation coefficient for DAL and CVS was 0.79. Combining stocks that are not perfectly positively correlated in a portfolio decreases risk.
Notice that investors who owned DAL and LUV from 2011 to 2020 would have had a lower portfolio standard deviation, but not much lower, than investors who just owned DAL. Because the correlation coefficient is less than one, the standard deviation fell. However, because the two stocks are in the same industry and exposed to many of the same economic issues, the correlation coefficient is relatively high, and combining those two stocks provides only a small decrease in risk.
This is because, as airlines, DAL and LUV face many of the same market conditions. In years when the economy is strong, the weather is good, fuel prices are low, and people are traveling a lot, both companies will do well. When something such as bad weather conditions reduces the amount of air travel for several weeks, both companies are harmed. By holding LUV in addition to DAL, investors can reduce exposure to risk that is specific to DAL (perhaps a problem that DAL has with its reservation system), but they do not reduce exposure to the risk associated with the airline industry (perhaps rising jet fuel prices). DAL and LUV tend to experience positive returns in the same years and negative returns in the same years.
On the other hand, investors who added XOM to their portfolio saw a significantly lower standard deviation than those who held just DAL. In years when jet fuel prices rise, harming the profits of both DAL and LUV, XOM is likely to see high profits. Diversifying a portfolio across firms that are less correlated will reduce the standard deviation of the portfolio more.
LINK TO LEARNINI
How to Build a Diversified Portfolio
TV personality, former hedge fund manager, and author Jim Cramer encourages investors to build a diversified portfolio, having no more than 20% of a portfolio in one sector.3 Watch this CNBC video (https://0penstax.0rg/r/CNBC video) to learn more about how he suggests investors can build a diversified portfolio by purchasing five to 10 stocks.
Portfolio Size and Risk
As you add more stocks to a portfolio, the volatility, or standard deviation, of the portfolio decreases. The volatility of individual assets becomes less and less important. As we discussed earlier, the risk that is associated with events related to a particular company is called firm-specific risk, or unsystematic, risk. Examples of unsystematic risk would include a company facing a product liability lawsuit, a company inventing a new product, or accounting irregularities being detected. Holding a portfolio of stocks means that if one company you have invested in goes out of business because of poor management, you do not lose all your savings because some of your money is invested in other companies. Portfolio diversification protects you from being significantly impacted by unsystematic risk.
However, there is a level below which the portfolio risk does not drop, no matter how diversified the portfolio becomes. The risk that never goes away is known as systematic risk. Systematic risk is the risk of holding the market portfolio.
We have talked about reasons why a firm's returns might be volatile; for example, the firm discovering a new technology or having a product liability lawsuit brought against it will impact that firm specifically. There are also events that broadly impact the stock market. Changes in the Federal Reserve Bank's monetary policy and
3 Abigail Stevenson. "Jim Cramer Shares His #1 Rule for Investing." Make It. CNBC, March 15, 2016. https://www.cnbc.com/2016/03/ 03/cramer-f orget-sectors-a-better-way-to-diversify.htm I
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15.3 • The Capital Asset Pricing Model (CAPM) 461
interest rates impact all companies. Geopolitical events, major storms, and pandemics can also impact the entire market. Investors in stocks cannot avoid this type of risk. This unavoidable risk is the systematic risk that investors in stocks have. This systematic risk cannot be eliminated through diversification.
In addition, as per research conducted by Meir Statman,4 the standard deviation of a portfolio drops quickly as the number of stocks in the portfolio increases from one to two or three (see Figure 2 illustration in this subsequent article by Statman (https://0penstax.0rg/r/subsequent Statman) for context). Increasing the size of the portfolio decreases the standard deviation, and thus the risk, of the portfolio. However, as the portfolio increases in size, the amount of risk reduced by adding one more stock to the portfolio will decrease. How many stocks does an investor need for a portfolio to be well-diversified? There is not an exact number that all financial managers agree on. A portfolio of 15 highly correlated stocks will offer less benefits of diversification than a portfolio of 10 stocks with lower correlation coefficients. A portfolio that consists of American Airlines, Spirit Airlines, United Airlines, Southwest Airlines, Delta Airlines, and Jet Blue, along with a few other stocks, is not very diversified because of the heavy concentration in the airline industry. The term diversified portfolio is a relative concept, but the average investor can create a reasonably diversified portfolio with approximately a dozen stocks.
15.3
The Capital Asset Pricing Model (CAPM)
Learning Outcomes
By the end of this section, you will be able to:
■ Define risk premium.
■ Explain the concept of beta.
■ Compute the required return of a security using the CAPM. Risk-Free Rate
The capital asset pricing model (CAPM) is a financial theory based on the idea that investors who are willing to hold stocks that have higher systematic risk should be rewarded more for taking on this market risk. The CAPM focuses on systematic risk, rather than a stock's individual risk, because firm-specific risk can be eliminated through diversification.
Suppose that your grandparents have given you a gift of $20,000. After you graduate from college, you plan to work for a few years and then apply to law school. You want to use the $20,000 your grandparents gave you to pay for part of your law school tuition. It will be several years before you are ready to spend the money, and you want to keep the money safe. At the same time, you would like to invest the money and have it grow until you are ready to start law school.
Although you would like to earn a return on the money so that you have more than $20,000 by the time you start law school, your primary objective is to keep the money safe. You are looking for a risk-free investment. Lending money to the US government is considered the lowest-risk investment that you can make. You can purchase a US Treasury security. The chances of the US government not paying its debts is close to zero. Although, in theory, no investment is 100% risk-free, investing in US government securities is generally considered a risk-free investment because the risk is so miniscule.
4 Meir Statman. "How Many Stocks Make a Diversified Portfolio?" Journal of Financial and Quantitative Analysis 22, no. 3 (September 1987): 353-363. https://doi.org/10.2307/2330969
462 15 • How to Think about Investing
The rate that you can earn by purchasing US Treasury securities is a proxy for the risk-free rate. It is used as an investing benchmark. The average rate of return for the three-month US Treasury security from 1928 to 2020 is 3.36%.5 You can see that you will not become immensely wealthy by investing in US Treasury bills. Another characteristic of US Treasury securities, however, is that their volatility tends to be much lower than that of stocks. In fact, the standard deviation of returns for the US Treasury bills is 3.0%. Unlike the returns for stocks, the return on US Treasury bills has never been negative. The lowest annual return was 0.03%, which occurred in 2014.6
US Treasury Securities
Visit the website of the US Department of the Treasury (https://openstax.Org/r/home.treasury) to learn more about US Treasury securities. You will find current interest rates for both short-term securities (US Treasury bills) and long-term securities (US Treasury bonds).
Risk Premium
You know that if you use your $20,000 to invest in stock rather than in US Treasury bills, the outcome of the investment will be uncertain. Your investments may do well, but there is also a risk of losing money. You will only be willing to take on this risk if you are rewarded for doing so. In other words, you will only be willing to take the risk of investing in stocks if you think that doing so will make you more than you would make investing in US Treasury securities.
From 1928 to 2020, the average return for the S8
Figure 15.7 Creating a Scatter Plot in Excel (data source: Yahoo! Finance)
When the trendline is inserted, a formatting box will appear on the right of your screen (see Figure 15.8). If it is not already selected, choose the Linear trendline option. Scroll down and select the "Display Equation on chart" option. You will see the equation y = \.\Allx + 0.0186 appear on the screen. This is the equation for the best-fit line that shows how AMZN moves when the market moves. The slope of this line, 1.1477, is the beta for AMZN. This tells you that for every 10% move the overall market makes, AMZN tends to move 11.477%. Because AMZN tends to move a little more than the broader market, it has a little more systematic risk than the average stock in the market.
P Comments
iif hü
Add Chart Quick Element - Layout -
ft ffl dl I H
Switch Row/ Select i i 1 i 11 Change Move
Column Data Chart Type Chart
Data Type Location
Date SPY AMZN
1/1/2018 0.06176 0.24064
2/1/2018 -0.03636 0.04243
3/1/2018 -0.03129 -0.04305
3/1/2020 -0.12999 O.Q3502Y 4/1/2020 0.13361 0.2689
5/1/2020 0.04765 -0.01278
6/1/2020 0.01328 0.12957
0.06355 0.14711
0.0698 0.09046
■0.04128 -0.087581
-0.02103 -0.03575T
0.10878 0.04344
0.0-3265 0.02806
7/1/2020 8/1/2020 9/1/2020 10/1/2020 11/1/2020 12/1/2020
v-1.117 K+ 0.0186
Format Trendli
Tren-dlineOptions
<5> Q ill
Lo g a rith in Polynom i č
Líc- i n g Avera g e
■•■ Autom atii Custom
Figure 15.8 Inserting a Trendline to Determine Beta (data source: Yahoo! Finance)
472 15-Summary
a Summary
15.1 Risk and Return to an Individual Asset
Investors are interested in both the return they can expect to receive when making an investment and the risk associated with that investment. In finance, risk is considered the volatility of the return from time period to time period. Historical returns are measured by the arithmetic average, and the risk is measured by the standard deviation of returns.
15.2 Risk and Return to Multiple Assets
As investors hold multiple assets in a portfolio, they are able to eliminate firm-specific risk. However, systematic or market risk remains, even if an investor holds the market portfolio. The return to a portfolio is measured by the arithmetic average, and the risk is measured by the standard deviation of the returns of the portfolio. The risk of the portfolio will be lower than the weighted average of the risk of the individual securities because the returns of the securities are not perfectly correlated. A low or negative return for one stock in a period can be offset by a high return for another stock in the same period.
15.3 The Capital Asset Pricing Model (CAPM)
The capital asset pricing model (CAPM) relates the expected return of an asset to the systematic risk of that asset. Investors will be rewarded for taking on systematic risk. They will not be rewarded for taking on firm-specific risk, however, because that risk can be diversified away.
15.4 Applications in Performance Measurement
Because investors are not simply interested in returns but are also interested in risk, the success of a portfolio cannot be measured simply by considering the portfolio's return. In order to compare investment portfolios, risk and return must both be taken into consideration. The Sharpe ratio and the Treynor ratio are two measures that provide a reward-to-risk measure of a portfolio. Jensen's alpha provides a measure of the abnormal return of a portfolio, considering the portfolio's risk level.
15.5 Using Excel to Make Investment Decisions
Using Excel to manipulate publicly available stock data makes calculating the average return of a stock and the standard deviation of returns easy. The average return for a portfolio and the standard deviation of the portfolio returns can also be calculated easily. By comparing the returns of a stock with the returns of the overall market using Excel charting tools, the beta for a stock, which measures systematic risk, can be determined.
Key Terms
arithmetic average return the sum of an asset's annual returns over a number of years divided by the
number of years beta a measure of how a stock moves relative to the market
capital asset pricing model (CAPM) the expected return of a security, equal to the risk-free rate plus a
premium for the amount of risk taken capital gain yield the difference between the price a stock is sold for and the price that was originally paid
for it divided by the price originally paid diversification holding a variety of assets in a portfolio
dividend yield the total dividends received by the owner of a share of stock divided by the price originally paid for the stock
effective annual rate (EAR) returns expressed on an annualized or yearly basis; allows for the comparison of various investments
firm-specific risk the risk that an event may impact the expected revenue or costs of a firm, thereby
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15- CFA Institute 473
impacting the returns to investors; also known as diversifiable risk geometric average return the compound annual return derived from the effective annual rate and time value of money formulas
holding period percentage return the gain received from holding a stock, calculated by adding the amount received when the stock is sold to any dividends earned while holding the stock, subtracting the price originally paid for the stock, then dividing the difference by the price originally paid
Jensen's alpha a measure of portfolio performance, calculated as the raw portfolio return minus the expected portfolio return predicted by the CAPM
market risk premium the reward for taking on the average amount of market risk
portfolio a collection of owned stocks
realized return the total return of an investment that occurs over a particular time period risk premium the extra return earned by taking on risk
risk-free rate the reward for lending money when there is no risk of not receiving the principal and interest as promised
Sharpe ratio a reward-to-risk measure of portfolio performance, calculated by subtracting the risk-free rate from the average portfolio return and then dividing by the standard deviation of the portfolio
systematic risk risk that impacts the entire market and cannot be diversified away; also known as market risk
Treynor ratio a reward-to-risk measure of portfolio performance, calculated by subtracting the risk-free rate from the average portfolio return and then dividing by the beta of the portfolio
iQ CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session (https://0penstax.0rg/r/cfa-inst-study-sessi0n). Reference with permission of CFA Institute.
a
Multiple Choice
The total dollar return equals_.
a. the EPS of a stock
b. capital gains income plus dividend income
c. the price paid for a share of stock minus the selling price of the stock
d. the price paid for a share of stock divided by the selling price of the stock
2. The dividend yield is calculated by_.
a. dividing the price of the stock by the EPS
b. subtracting any capital loss from the capital gain
c. dividing the annual dividend by the initial stock price
d. dividing the annual dividend by the net income for the year
3. Which of the following is the best example of firm-specific risk?
a. A global pandemic causes major disruptions in the economy.
b. The Federal Reserve increases the money supply dramatically, leading to massive inflation.
c. AAA Pharmaceuticals withdraws a medication as it studies whether strokes five people suffered after taking the medication were related to the medication.
d. As an arctic blast descends on North America, most of the United States is blanketed in snow or ice.
4.
Investors diversify their portfolio in order to
a. reduce risk
b. increase risk
474 15 • Review Questions
c. increase return
d. increase the standard deviation
5. Which of the following would be the best example of systematic risk?
a. An error in the company's computer system miscalculates the amount of inventory that Monique's Boutique is holding.
b. Bluejay Air has a reduction in new reservations following a crash of one of its jets.
c. The spokesperson for Serena's Sports Shoes is involved in an ethical scandal.
d. Interest rates rise after the Federal Reserve announces it will slow down the rate of growth of the money supply.
6. As the number of stocks in a portfolio increases,_
a. firm-specific risk increases
b. systematic risk becomes zero
c. systematic risk decreases and returns increase
d. firm-specific risk is reduced but systematic risk remains
7. Beta is a measure of_
a. systematic risk
b. firm-specific risk
c. a firm's profitability
d. a stock's dividend yield
8. Which of the following would be the best estimate of the risk-free rate?
a. The rate of inflation
b. The average return on the S&P 500
c. The average return on Amazon's stock
d. The average return on US Treasury bills
9. The Sharpe ratio can be considered a measure of_.
a. the reward of an investment in relation to the risk
b. the systematic risk of a stock
c. the total return of a stock investment
d. the historical return of an individual security
10. A positive Jensen's alpha indicates that a portfolio has_.
a. a negative beta
b. an abnormal return
c. more total risk than the average portfolio
d. more systematic risk than the average portfolio
11. If an equally weighted portfolio contains 10 stocks, then.
a. the stocks in the portfolio will each have a weight of 0.10
b. the return of the portfolio must be multiplied by 10 to get the annualized return
c. the standard deviation of the portfolio will be one-tenth the standard deviation of one of the stocks
d. the standard deviation of the portfolio will be 10 times the standard deviation of one of the stocks
% Review Questions
1. What is the difference between firm-specific risk and unsystematic risk?
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15-Problems 475
2. Explain why diversification reduces unsystematic risk but not systematic risk.
3. Explain what happens to the standard deviation of returns of a portfolio as the number of stocks in the portfolio increases.
4. Enrique owns five stocks: Alaska Airlines, American Airlines, Delta Airlines, Southwest Airlines, and Ford. Radha also owns five stocks: Apple, McDonald's, Tesla, Facebook, and Disney. Does Enrique or Radha have a more diversified portfolio?
5. You are considering purchasing shares in a company that has a beta of 0.8. Explain what this beta means.
6. Explain how the Sharpe ratio and the Treynor ratio can be considered reward-to-risk measures.
0 Problems
1. You purchase 100 shares of COST (Costco) for $280 per share. Three months later, you sell the stock for $290 per share. You receive a dividend of $0.57 a share. What is your total dollar return?
2. You purchase 100 shares of COST for $280 per share. Three months later, you sell the stock for $290 per share. You receive a dividend of $0.57 a share. What are your dividend yield, capital gain yield, and total percentage return?
3. You purchase 100 shares of COST for $280 per share. Three months later, you sell the stock for $290 per share. You receive a dividend of $0.57 a share. What is the EAR of your investment?
4. You invest in a stock for four years. The returns for the four years are 20%, -10%, 15%, and -5%. Calculate the arithmetic average return and the geometric average return.
5. You are considering purchasing shares in a company that has a beta of 0.9. The average return for the S8
Maximum Value of Firm ~V ^-----i
PV of Interest Tax Shields Costs of Financial Distress
Value of Unlevered Firm Optimal Debt Level
Debt Level
Figure 17.4 Maximum Value of a Levered Firm
As the firm increases debt and increases the value of the tax benefit of debt, it also increases the probability of facing financial distress. The magnitude of the costs of financial distress increases as the debt level of the company rises. To some degree, these costs offset the benefit of the interest tax shield.
The optimal debt level occurs at the point at which the value of the firm is maximized. A company will use this optimal debt level to determine what the weight of debt should be in its target capital structure. The optimal capital structure is the target. Recall that the market values of a company's debt and equity are used to determine the costs of capital and the weights in the capital structure. Because market values change daily due to economic conditions, slight variations will occur in the calculations from one day to the next. It is neither practical nor desirable for a firm to recalculate its optimal capital structure each day.
Also, a company will not want to make adjustments for minor differences between its actual capital structure and its optimal capital structure. For example, if a company has determined that its optimal capital structure is 22.5% debt and 77.5% equity but finds that its current capital structure is 23.1 % debt and 76.9% equity, it is close to its target. Reducing debt and increasing equity would require transaction costs that might be quite significant.
Table 17.7 shows the average WACC for some common industries. The calculations are based on corporate information at the end of December 2020. A risk-free rate of 3% and a market-risk premium of 5% are assumed in the calculations. You can see that the capital structure used by firms varies widely by industry. Companies in the online retail industry are financed almost entirely through equity capital; on average, less than 7% of the capital comes from debt for those companies. On the other hand, companies in the rubber and tires industry tend to use a heavy amount of debt in their capital structure. With a debt weight of 63.62%, almost two-thirds of the capital for these companies comes from debt.
Industry Name Equity Weight Debt Weight Beta Cost of Equity Tax Rate After-Tax Cost of Debt WACC
Retail (online) 93.33% 6.67% 1.16 8.82% 2.93% 2.19% 8.38%
Computers/peripherals 91.45% 8.55% 1.18 8.92% 3.71% 1.88% 8.32%
Household products 87.07% 12.93% 0.73 6.65% 5.06% 2.19% 6.07%
Drugs (pharmaceutical) 84.62% 15.38% 0.91 7.54% 1.88% 2.19% 6.72%
Table 17.7 Capital Structure, Cost of Debt, Cost of Equity, and WACC for Selected Industries (data source: Aswath Damodaran Online)
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17.6 • Alternative Sources of Funds 523
Industry Name Equity Weight Debt Weight Beta Cost of Equity Tax Rate After-Tax Cost of Debt WACC
Retail (general) 82.41% 17.59% 0.90 7.49% 12.48% 1.88% 6.51%
Beverages (soft) 82.24% 17.76% 0.79 6.96% 3.32% 2.19% 6.11%
Tobacco 76.74% 23.26% 0.72 6.61% 8.69% 1.88% 5.51%
Homebuilding 75.34% 24.66% 1.46 10.29% 15.91% 2.19% 8.30%
Food processing 75.18% 24.82% 0.64 6.18% 8.56% 2.19% 5.19%
Restaurants/dining 74.79% 25.21% 1.34 9.72% 3.19% 2.19% 7.82%
Apparel 71.74% 28.26% 1.10 8.49% 4.75% 2.19% 6.71%
Farming/agriculture 68.94% 31.06% 0.87 7.37% 6.45% 1.88% 5.67%
Packaging & containers 64.47% 35.53% 0.92 7.61% 15.67% 1.88% 5.57%
Food wholesalers 64.10% 35.90% 1.03 8.17% 0.52% 2.19% 6.02%
Hotels/gaming 63.60% 36.40% 1.56 10.82% 2.02% 2.19% 7.68%
Telecom, services 54.60% 45.40% 0.66 6.30% 3.93% 1.40% 4.07%
Retail (grocery and food) 51.46% 48.54% 0.24 4.21% 13.52% 2.19% 3.23%
Air transport 38.26% 61.74% 1.61 11.04% 6.00% 2.19% 5.58%
Rubber & tires 36.38% 63.62% 1.09 8.47% 5.30% 1.88% 4.28%
Table 17.7 Capital Structure, Cost of Debt, Cost of Equity, and WACC for Selected Industries (data source: Aswath Damodaran Online)
Industries that have high betas, such as hotels/gaming and air transport, have high equity costs of capital. More recession-proof industries, such as food processing and household products, have low betas and low equity costs of capital. The WACC for each industry ends up being influenced by the weights of equity and debt the company chooses, the riskiness of the industry, and the tax rates faced by companies in the industry.
17.6
Alternative Sources of Funds
By the end of this section, you will be able to:
■ Calculate the required return to preferred shareholders.
■ Calculate the WACC of a firm that issues preferred shares.
■ Discuss how issuing new equity impacts the cost of equity capital.
■ Explain the functionality of convertible debt.
A company can finance its assets in two ways: through debt financing and through equity financing. Thus far, we have treated these sources as two broad categories, each with a single cost of capital. In reality, a company may have different types of debt or equity, each with its own cost of capital. The same principle would apply: the WACC of the firm would be calculated using the weights of each of these types multiplied by the cost of that particular type of debt or equity capital.
Preferred Shares
Although our calculations of WACC thus far have assumed that companies finance their assets only through debt and common equity, we saw at the beginning of the chapter that the basic WACC formula is
WACC = D% X rd(l - T) + P% X rpfd + E% X re In addition to common stock, a company can raise equity capital by issuing preferred stock. Owners of
524 17 • How Firms Raise Capital
preferred stock are promised a fixed dividend, which must be paid before any dividends can be paid to common stockholders.
In the order of claimants, preferred shareholders stand in line between bondholders and common shareholders. Bondholders are paid interest before preferred shareholders are paid annual dividends. Preferred shareholders are paid annual dividends before common shareholders are paid dividends. Should the company face bankruptcy, the same priority of claimants is followed in settling claims—first bondholders, then preferred stockholders, with common stockholders standing at the end of the line.
Preferred stock shares some characteristics with debt financing. It has a promised cash flow to its holders. Unlike common equity, the dividend on preferred stock is fixed and known. Also, there are consequences if those preferred dividends are not paid. Common shareholders cannot receive any dividends until preferred dividends are paid, and in some cases, preferred shareholders receive voting rights until they are paid the dividends that are due. However, preferred shareholders cannot force the company into bankruptcy as debt holders can. For tax and legal purposes, preferred stock is treated as equity.
The cost of the preferred equity capital is calculated using the formula
rpfd -
Divp{i
Ppfd
Suppose that Greene Building Company has issued preferred stock that pays a dividend of $2.00 each year. If this preferred stock is selling for $21.80 per share, then the company's cost of preferred stock is
$2.00
^ = $200 = 9'17%
KIT THROUGH
Calculating Common Equity Financing
Greene Building Company uses 40% debt, 15% preferred stock, and 45% common stock in its capital structure. The yield to maturity on the company's bonds is 7.2%. The cost of preferred equity is 9.17%. In the most recent year, Greene paid a dividend of $3.15 to its common shareholders. This dividend has been growing at a rate of 3.0% per year, which is expected to continue in the future. The company's common stock is trading for $32.25 per share. Greene pays a corporate tax rate of 21 %. Estimate the WACC for Greene.
Solution:
The cost of Greene's common equity financing must be calculated. This can be done using the constant dividend growth model:
Dm ($3.15x1.03) nM _____ ___
re = + g =---~ + 0.03 = 0.1006 + 0.03 = 13.06
P0 $32.25
The weights and the costs for each component of capital are placed in the WACC formula: WACC =D% X rd(l -T) + P% X rpfd + E% X re
= 40% X 0.0702(1-0.21) + 15% X 0.0917 + 45% X 0.1306 = 2.218% + 1.3755% + 5.8770% = 9.4705%
The WACC for Greene Building Company is estimated to be 9.47%. Note that debt financing is the cheapest cost of capital for Greene. The reason for this is twofold. First, because debt holders face the least amount of risk because they are paid first in the order of claimants, they require a lower return. Second, because interest payments are tax-deductible, the interest tax shield lowers the effective cost of debt to the
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17.6 • Alternative Sources of Funds 525
company. Preferred shareholders will require a higher rate of return than debt holders, 9.17%, because they are later in the order of claimants. Common shareholders are the residual claimants, standing at the end of the line to receive payment. After all other claimants are paid, any remaining money belongs to the shareholders. If this residual amount is small, the common shareholders receive a small payment. If there is nothing left after all other claimants have been paid, common shareholders receive nothing. Thus, common shareholders have the greatest amount of risk and require the highest rate of return.
Also, note that the weights for debt, preferred stock, and common stock in the capital structure sum to 100%. The company must finance 100% of its assets.
Issuing New Common Stock
An existing firm can acquire equity capital to expand its assets in two ways: the retention of earnings or the sale of new shares of stock. Thus far in the chapter, the cost of equity capital calculations have assumed that the earnings were being retained for equity capital financing.
The net income that is left after all expenses are paid is the residual income that belongs to the shareholders. Instead of receiving a fixed payment for letting the firm use their capital (like bondholders who receive fixed interest payments), the reward to shareholders for letting the company use their capital varies from year to year. In a good year, net income and the reward to shareholders is high. In a poor year, net income is low or perhaps even negative.
The net income can either be paid immediately and directly to shareholders in the form of dividends or be retained within the company to fund growth. Shareholders are willing to allow the company to retain these earnings because they expect that the money will be used to fund profitable projects, leading to an even larger reward for shareholders in future years.
Although managers do not need to actively solicit the funds that are retained to fund the business, managers cannot view these funds as costless. The shareholders will require a return on those funds to entice them to allow the company to delay paying the dollars to them immediately in terms of a dividend.
Suppose a company has $1 million in net income one year. If it pays $250,000 in dividends and retains $750,000, then it can finance $750,000 more in assets. If the company has a capital structure of 25% debt and 75% equity and wants to maintain that capital structure, it must increase its debt by $250,000 to balance the increase in equity. Thus, the company would be increasing its total financing by $1 million. Of that financing, 25% would be debt financing, and 75% would be equity financing.
To increase its assets by more than $1 million, the company would need to decide to either change its capital structure or issue new stock. Consider the firm represented by the market-value balance sheet in Figure 17.5. The firm has $900 million in assets. These assets are financed by $225 million in debt capital and $675 million in equity capital, resulting in a capital structure of 25% debt and 75% equity.
Assets Liabilities and Equity
Market Value of Debt
= $225,000,000
Market Value of Assets
= $900,000,000 Market Value of Equity
= $675,000,000
Figure 17.5 Market-Value Balance Sheet for a Company with $900 Million in Assets and a Capital Structure of 25% Debt and 75% Equity
The retained earnings of $750,000 cause the equity on the balance sheet to increase to $675.75 million. The
526 17 • How Firms Raise Capital
company could sell $250,000 in bonds, increasing its debt to $225.25 million. Figure 17.6 shows the impact on the balance sheet. The company has increased its financing by $1,000,000 and can expand assets by $1,000,000. The capital structure remains 25% debt and 75% equity.
Assets Liabilities and Equity
$225,000,000 + $250,000 in
New Debt = $225,250,000
$901,000,000 $675,000,000 + $750,000 in
Retained Earnings
= $675,750,000
Figure 17.6 Balance Sheet with $1 Million Growth Financed through Retained Earnings and New Debt
What if the economy is in an expansionary period and this company thinks it has the opportunity to grow at a rate of 5%? The company knows that it will need more assets to be able to grow. If it needs 5% more assets, its assets will need to increase to $945 million. To increase the left-hand side of its balance sheet, the company will also need to increase the right-hand side of the balance sheet.
Where does the company get the $45 million in capital? With $750,000 in retained earnings, the company can increase its equity to $675.75 million, but if the remainder of $44.25 million was financed through debt, the company's capital structure would change. Its weight of debt would increase to
$225,000,000 + $44,250,000 mo/io oq/ioc -$945j000000- - 0.2849-28.49/c.
If the company has determined that its optimal capital structure is 25% debt and 75% equity, financing the majority of the growth through debt would cause it to stray from these levels. Funding the growth while keeping the capital structure the same would require the firm to issue new shares. Figure 17.7 shows how the firm would need to finance $45 million in growth while maintaining its desirable capital structure. The firm would need to increase equity capital to $708.75 million; retained earnings could provide $750,000, but $33 million of new equity would need to be sold.
Assets Liabilities and Equity
$225,000,000 + $11,250,000 in New Debt = $236,250,000
$675,000,000 + $750,000 in
Retained Earnings + $33,000,000 in New Equity $708,750,000
Figure 17.7 Balance Sheet with $45,000,000 in Financing Coming from Debt, Retained Earnings, and New Stock
Investors who are providing common equity financing require a return to entice them to let the company use their money. If this company has paid $0.50 per share in dividends to shareholders and this dividend is expected to increase by 3% each year, we can use the constant dividend growth model to estimate how much common shareholders require. If the stock is trading for $8.00 per share, the cost of common equity financing is estimated as
Div\ $0,515 re = —i- + g = + 0.03 = 0.0644 + 0.03 = 0.0944 = 9.44%
Pq $8
If, however, the firm must issue more equity, its cost of equity for those additional shares will be higher than 9.44%. Even if shareholders are willing to pay $8.00 per share for the stock, the firm will incur flotation costs; this means the firm will not receive the entire $8.00 to use to finance new assets and generate a profit for shareholders. Flotation costs include the costs of filing with the Securities and Exchange Commission (SEC) as
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17.6 • Alternative Sources of Funds 527
well as the fees paid to investment bankers to place the new shares.
When new equity must be issued to finance the company, the flotation costs must be subtracted from the price of the stock to determine the net proceeds the firm will receive. The cost of this new equity capital is calculated as
_ Divx r& ~ new ~ P0-F 8
where F represents the flotation costs of the new stock issue. If, in this example, the flotation cost is $0.25 per share, then the cost of raising new equity capital is
_ Dwi + g = °5^5 + 0.03 = 0.0665 + 0.03 = 0.09 = 9.65%
* P0-F ° $8-0.25
Issuing new common equity is the most expensive form of raising capital. Equity capital is already expensive because the common shareholders are the residual claimants who will only be paid if all other claimants are paid. Because of this risk, they require a higher rate of return than providers of capital who have precedence in the order of claimants. Flotation costs must be added to this equity cost when new shares are issued to grow the company.
Cost of Issuing New Equity
You are a financial manager for American Motor Works (AMW). The target capital structure for the company is 30% debt and 70% equity. You know that your company's after-tax cost of debt is 4.6%. Your company paid a dividend of $3 per share last year, and it has a policy of increasing its dividend at a rate of 1.5% each year. AMW stock is currently trading for $27.50 per share. You estimate that the company's retained earnings will be $10 million this year. If the company needs to issue new shares of stock, flotation costs are expected to be $0.75 per share.
Given this information, you are tasked with calculating the company's WACC. You need to provide an estimate of WACC if retained earnings are used and an estimate if new equity must be issued.
Solution:
First, calculate the cost of equity capital for AMW using the constant dividend growth model:
re = EiEL+g = $3™(™l5) + 0.015 = 0.1107 + 0.015 = 0.1257 = 12.57% Pq $27.50
Using 12.57% as the equity cost of capital and 4.6% as the after-tax cost of debt, the WACC is calculated as
WACC = D% x rd(l —T) + E% X re = 0.30(0.0460) + 0.70(0.1257) = 0.0138 + 0.0880 = 0.1018 = 10.18%
The WACC for AMW when it is using retained earnings for equity financing is 10.18%. If the company has $10 million in retained earnings this year, its equity will increase by $10 million. Given its target capital structure of 30% debt and 70% equity, AMW will be able to increase its overall financing by $100TO°00 = $14>285>714 by usin9 its retained earnings and issuing new debt of $4,285,714.
If AMW wants to expand its assets by more than $14,285,714 during the next year, it will need to issue new stock or increase the weight of debt in its capital structure. The company will incur flotation costs of $0.65 per share to issue new stock. The cost of new equity capital will be
528 17 • How Firms Raise Capital
DiVi
+ g =
$3.00(1.015)
+ 0.015 = 0.1138 + 0.015 = 0.1288 = 12.88%
$27.50 - $0.75
With this more expensive newly issued equity capital, AMW's WACC will become
WACC =D% X rd(l -T) + E% X re = 0.30(0.0460) + 0.70(0.1288) = 0.0138 + 0.0902 = 0.1040 = 10.40%
If AMW wants to take on a large project that requires investment in more than $14,285,714 worth of assets, such as building a new production facility, the company will need to issue more equity and will face a higher WACC than when using retained earnings as its equity financing.
Convertible Debt
Some companies issue convertible bonds. These corporate bonds have a provision that gives the bondholder the option of converting each bond held into a fixed number of shares of common stock. The number of common shares the bondholder would receive for each bond is known as the conversion ratio.
Suppose that you own a convertible bond issued by Sheridan Sodas with a face value of $1,000 and a
conversion ratio of 20 shares that matures today. If you convert the bond today, you will receive 20 shares of
Sheridan common stock. If you do not convert, you will receive $1,000. If you convert, you are basically paying
$1 000
$1,000 for 20 shares of Sheridan stock. The conversion price is = $50. If Sheridan is trading for more
than $50 per share, you would want to convert. If Sheridan is trading for less than $50 per share, you would not want to convert; you would prefer the $1,000. In other words, you will choose to convert whenever the stock price exceeds the conversion price at maturity.
A convertible bond gives the holder an option; the bondholder is able to choose between the face value cash or receiving shares of stock. Options always have a positive value to holders. It is always preferable to be able to choose $1,000 or shares of stock than to simply be given $1,000. There is a possibility that the shares of stock will be more valuable, and there is no way the choice can put you in a worse position.
Because holders of convertible bonds have the valuable option of conversion that holders of nonconvertible bonds do not have, convertible debt can be offered with a lower interest rate. It might seem as if the firm could lower its weighted average cost of capital by issuing convertible debt rather than nonconvertible debt. However, this is not the case. Remember that holders of convertible bonds choose whether they would prefer to convert the bond and become a stockholder or receive the face value of the bond at maturity.
If a bond has a face value of $1,000, the convertible bond holders will consider whether the stock they can convert to is worth more than $1,000. Only when the price of the stock has increased enough that the value of the stock received is more than $1,000 will the bondholders convert. However, this means that instead of paying $1,000, the firm is paying the bondholder in stock worth more than $1,000. In essence, the firm (and the current shareholders) would be selling an equity position in the company for less than the market price of that equity position. The lower interest rate compensates for the possibility that conversion will occur.
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17 •Summary 529
a Summary
17.1 The Concept of Capital Structure
Capital structure refers to how a company finances its assets. The two main sources of capital are debt financing and equity financing. A cost of capital exists because investors want a return equivalent to what they would receive on an investment with an equivalent risk to persuade them to let the company use their funds. The market values of debt and equity are used to calculate the weights of the components of the capital structure.
17.2 The Costs of Debt and Equity Capital
The yield to maturity (YTM) on a company's outstanding bonds represents the return that debt holders are requiring to lend money to the company. Because interest expenses are tax-deductible, the cost of debt to the company is less than the YTM. The cost of equity capital is not directly observed, so financial managers must estimate this cost. Two common methods for estimating the cost of equity capital are the constant dividend growth model and the capital asset pricing model (CAPM).
17.3 Calculating the Weighted Average Cost of Capital
Calculate the weighted average cost of capital (WACC) using the formula
WACC = D% X rd(l - T) + P% X rpfd + E% X re
Remember that the WACC is an estimate; different methods of estimating the cost of equity capital can lead to different estimations of WACC.
17.4 Capital Structure Choices
An unlevered firm uses no debt in its capital structure. A levered firm uses both debt and equity in its capital structure. In perfect financial markets, the value of the firm will be the same regardless of the firm's decision to use leverage. With the tax deductibility of interest expenses, however, the value of the firm can increase through the use of debt. As the level of debt increases, the value of the interest tax shield increases.
17.5 Optimal Capital Structure
A company wants to choose a capital structure that maximizes its value. Although increasing the level of financial leverage, or debt, in the capital structure increases the value of the interest tax shield, it also increases the probability of financial distress. As the weight of debt in the capital structure increases, the return that providers of both debt and equity capital require to entice them to provide money to the firm increases because their risk increases. Trade-off theory suggests that the value of a company that uses debt equals the value of the unlevered firm plus the value of the interest tax shield minus financial distress costs.
17.6 Alternative Sources of Funds
Preferred stock is a type of equity capital; the owners of preferred stock receive preferential treatment over common stockholders in the order of claimants. A fixed dividend is paid to preferred shareholders and must be paid before common shareholders receive dividends. Equity capital can be raised through either retaining earnings or selling new shares of stock. Significant flotation costs are associated with issuing new shares of stock, making it the most expensive source of financing. Convertible debt allows the debt holders to convert their debt into a fixed number of common shares instead of receiving the face value of the stock at maturity.
Key Terms
after-tax cost of debt the net cost of interest on a company's debt after taxes; the firm's effective cost of debt
capital a company's sources of financing
530 17 -CFA Institute
capital structure the percentages of a company's assets that are financed by debt capital, preferred stock
capital, and common stock capital conversion price the face value of a convertible bond divided by its conversion ratio conversion ratio the number of shares of common stock receivable for each convertible bond that is
converted
convertible bonds bonds that can be converted into a fixed number of shares of common stock upon maturity
financial distress when a firm has trouble meeting debt obligations financial leverage the debt used in a company's capital structure flotation costs costs involved in the issuing and placing of new securities
interest tax shield the reduction in taxes paid because interest payments on debt are a tax-deductible
expense; calculated as the corporate tax rate multiplied by interest payments levered equity equity in a firm that has debt outstanding
net debt a company's total debt minus any cash or risk-free assets the company holds
preferred stock equity capital that has a fixed dividend; preferred shareholders fall in between debt holders
and common stockholders in the order of claimants trade-off theory a theory stating that the total value of a levered company is the value of the firm without
leverage plus the value of the interest tax shield less financial distress costs unlevered equity equity in a firm that has no debt outstanding
weighted average cost of capital (WACC) the average of a firm's debt and equity costs of capital, weighted by the fractions of the firm's value that correspond to debt and equity
CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session (https://0penstax.0rg/r/CFA-Level-I-Study). Reference with permission of CFA Institute.
Multiple Choice
1. Sandage Auto Parts has debt outstanding with a market value of $2 million. The company's common stock has a book value of $3 million and a market value of $8 million. What weight is equity in Sandage's capital structure?
a. 11%
b. 20%
c. 60%
d. 80%
2. The capital structure of a company refers to_.
a. whether the company purchases assets or liabilities with its equity
b. the proportion of debt and equity the company uses in financing is assets
c. the ability of the company to use its assets to generate equity for the owners
d. whether the company uses short-term assets or long-term assets to create its product
3. Which of the following should be used when calculating the weights for a company's capital structure?
a. Book values
b. Current market values
c. Historic accounting values
d. Par and face values
4. Two methods for estimating a company's cost of common stock capital are
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17-Multiple Choice 531
a. the historic method and the current method
b. the weighted valuation model and the beta model
c. the constant dividend growth model and the CAPM
d. the balance sheet method and the face value method
5. Which of the following would be the most reasonable approach to calculating the cost of debt for a company?
a. Using the coupon rate on the company's existing bonds
b. Using the interest amount reported on the income statement
c. Using the yield to maturity on the company's existing bonds
d. Multiplying the amount of debt on the company's balance sheet by the risk-free rate
6. Net debt equals_.
a. Debt/Equity
b. Debt x (1 - Tax Rate)
c. total debt minus the cash and risk-free assets the company owns
d. the yield to maturity of a company's bonds divided by the tax rate
7. Unlevered equity refers to_.
a. the equity in a firm with no debt
b. a firm's equity minus the firm's debt
c. the equity in a firm in the absence of taxation and transaction costs
d. the portion of a firm's capital structure that is financed by its owners
8. In perfect capital markets,_.
a. a company's WACC does not change as it changes its capital structure
b. a company can lower its WACC by using more debt in its capital structure
c. a company can lower its WACC by using more equity in its capital structure
d. a company's cost of debt capital is exactly equal to its cost of equity capital when the company uses 50% debt and 50% equity in its capital structure
9. The interest tax shield occurs because_.
a. interest payments are a tax-deductible expense
b. interest payments are made from after-tax income
c. investors require a lower rate of return the higher the company's tax rate
d. investors require a lower rate of return the more debt the company incurs
10. As a company increases the weight of debt in its capital structure,_.
a. its cost of debt capital falls
b. the weight of equity capital also increases
c. the value of the interest tax shield decreases
d. its possibility of financial distress increases
11. A company is said to be in financial distress if_.
a. it is not fully exploiting the interest tax shield
b. it needs to raise capital to finance a new project
c. it has difficulty meeting its debt obligations
d. its cost of equity capital exceeds its cost of debt capital
12. Issuing new stock
532 17 • Review Questions
a. costs the same as retaining earnings
b. will not impact a company's WACC
c. is the most expensive source of capital because of flotation costs
d. is the cheapest source of capital because dividends do not have to be paid each year
13. If a bond with a face value of $1,000 has a conversion ratio of 10 shares, the conversion price is.
a. $0.01
b. $10
c. $100
d. $1,000
5. Review Questions
1. Why does a company's capital have a cost?
2. Why is the rate that debt holders require to entice them to lend money to a company different from the company's effective cost of debt capital?
3. Assume that the corporate tax rate is 21 %. Congress is discussing increasing the corporate tax rate to 32%. How might this change the capital structures that companies choose?
4. Describe the order of claimants and how it impacts the returns that various providers of capital require to entice them to provide funding to a company.
5. Explain what is meant by trade-off theory.
Problems
1. SodaFizz has debt outstanding that has a market value of $3 million. The company's stock has a book value of $2 million and a market value of $6 million. What are the weights in SodaFizz's capital structure?
2. The yield to maturity on SodaFizz's debt is 7.2%. If the company's marginal tax rate is 21 %, what is SodaFizz's effective cost of debt?
3. SodaFizz paid a dividend of $2 per share last year; its dividend has been growing at a rate of 2% per year, and that growth rate is expected to continue into the future. The stock of SodaFizz is currently trading at $19.50 per share. According to the constant dividend growth model, what is the cost of equity capital for SodaFizz?
4. SodaFizz has a beta of 1.1. If the risk-free rate is 3% and the market risk premium is 11%, what is the cost of equity capital for SodaFizz according to the capital asset pricing model?
5. Given the answers to Problems 1, 2, and 3, what is SodaFizz's WACC when the constant dividend growth model is used to calculate its equity cost of capital?
6. Given the answers to Problems 1, 2, and 4, what is SodaFizz's WACC when the CAPM is used to calculate SodaFizz's equity cost of capital?
7. Shirley Manufacturing paid $1 million in interest payments last year. The company is in the 21% tax bracket and has $15 million in debt outstanding. How much was the company's interest tax shield last year?
8. King Medical Supplies has issued preferred stock that pays a yearly dividend of $4 per share. This preferred stock is trading at a price of $47 per share. What is King's cost of preferred stock capital?
9. McPherson Pharmaceutical has common stock that is trading for $75 per share. The company paid a
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17-Video Activity 533
dividend of $5.25 last year. This dividend is expected to increase at a rate of 3% per year. What is the cost of equity capital for McPherson? If McPherson issues new shares with a flotation cost of $2 per share, what is the company's cost of new equity?
Calculating the Weighted Average Cost of Capital
Click to view content (https://0penstax.0rg/r/WACC)
1. What is the formula for calculating WACC? What do each of the components of this formula represent?
2. In the video, the tax rate for Brick and Mortar Co. was 30%. What would your calculation of the company's WACC be if there was a change in the tax code and the tax rate for Brick and Mortar Co. fell to 15%? Why does the tax rate impact a firm's WACC? Do you think the managers of Brick and Mortar Co. should consider making any changes to its capital structure if the tax rate falls to 15%? Why or why not?
Capital Structure for Real Estate Companies
Click to view video content (https://0penstax.0rg/r/0ptimal-capital)
3. Why doesn't one optimal capital structure exist for commercial real estate businesses?
4. How do you think a family that runs a multigenerational commercial real estate business will think about risk compared to a young entrepreneur who is beginning to build a commercial real estate business? How do you think the capital structures of these two entities are likely to compare? How would those capital structures likely be linked to the risk profiles of the two companies?
Video Activity
534 17-Video Activity
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Financial Forecasting
Figure 18.1 Forecasts are an important financial tool, (credit: modification of "Red Post-It Label, Calculator and Ballpen" by photosteve101/flickr, CC BY 2.0)
Chapter Outline
18.1 The Importance of Forecasting
18.2 Forecasting Sales
18.3 Pro Forma Financials
18.4 Generating the Complete Forecast
18.5 Forecasting Cash Flow and Assessing the Value of Growth
18.6 Using Excel to Create the Long-Term Forecast
J
Why It Matters
Though no one in business has a crystal ball, managers must often do all they can to predict the future as accurately as possible. This is called forecasting. Accounting and finance professionals use past performance along with what they know about the business, its competitors, the economy, and the company's plans for the future to assemble detailed financial forecasts. Forecasts are useful to many individuals for different reasons. A budget, a type of static forecast, helps accountants and managers see how their plans for the coming year can be achieved. It outlines sales targets and how much can be spent on cost of goods sold and expenses to achieve the company's bottom-line (net income) targets. Investors use financial forecasts to help guide their decisions to buy, sell or hold stocks or to estimate future potential income through dividends. Perhaps most importantly, for our purposes in finance, forecasts are used to help predict and manage cash flows.
A business can have all the profit in the world at the end of the year, but if it doesn't raise enough cash (liquidity) to pay the bills and pay its employees halfway through the year, it could still go bankrupt despite being profitable. Forecasting sales and expenses helps assemble a cash forecast—when sales will be collected and when expenses will be paid—so that financial managers can look forward far enough to have enough time to react accordingly and secure short- or long-term financing to meet gaps in cash flow.
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18.1
The Importance of Forecasting
Learning Outcomes
By the end of this section, you will be able to:
■ Discuss how to use financial statements in forecasting firm financials.
■ Explain why balance sheet items are important in forecasting a firm's financial result.
■ Explain why income statement items are important in forecasting a firm's financial result.
In this section, we will briefly review some of the basic elements of financial statements and how we can analyze historical statements to help assemble financial forecasts. Financial forecasting is important to short-and long-term firm success. It helps a firm plan for the resources it will need, ensuring it will have enough cash on hand at the right time to cover daily operations and capital expenditures. It helps the firm communicate its future potential and manage its shareholders' expectations. It also helps management assess future risk and set plans in place to mitigate that risk.
Financial forecasting involves using historical data, analysis tools, and other information we can gather to make an educated guess about the future financial performance of the firm. Historical figures provide a reasonable starting point. We use tools such as ratios, common size, and trend analysis to fine-tune our forecast. And finally, we assess what we know about the firm, its competitors, the economy, and anything else that might impact performance and further fine-tune our forecast from there.
It's important to take a moment to consider the role of ethics in forecasting. Ethics is a huge issue in the world of accounting and finance in general, and forecasting is no different. There can be tremendous pressure on management to perform, to deliver certain levels of profit, and to meet shareholder expectations.
Forecasting, as you will learn throughout this chapter, is not an exact science. There is a great deal of subjectivity that can come into play when forecasting sales and expenses. Ethical behavior is crucial in this area. Those who create forecasts must have a firm understanding of where their data comes from, how reliable it is, and whether or not their assumptions and projections are reasonably justified.
Financial Statement Foundations
In Financial Statements, you were introduced to a firm called Clear Lake Sporting Goods. You learned about the four key financial statements: the income statement, balance sheet, statement of stockholders' equity, and statement of cash flows. Each one provides a different view of the firm's financial health and performance.
Clear Lake Sporting Goods is a small merchandising company (a company that buys finished goods and sells them to consumers) that sells hunting and fishing gear. It uses financial statements to understand its profitability and current financial position, to manage cash flow, and to communicate its finances to outside parties such as investors, governing bodies, and lenders. We will use Clear Lake's company information and historical financial statements in this chapter as we explore its forecasting process. It's important to note that in this chapter, we are focusing on just one firm and the one method its managers have chosen to forecast financial performance. There are a variety of types of firms in actual application, and they may choose to forecast their financial performance differently. We are demonstrating just one approach here.
The balance sheet shows all the firm's assets, liabilities, and equity at one point in time. It also supports the accounting equation in a very clear and transparent way. We find one section of the balance sheet contains all current and noncurrent assets that must total the other section of the balance sheet: total liabilities and equity. In Figure 18.2. we see that Clear Lake Sporting Goods has total assets of $250,000 in the current year, which balances with its total liabilities and equity of $250,000.
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18.1 • The Importance of Forecasting 537
Clear Lake St )ortinq Goods
Comparative Year-End Balance Sheets
Prior Year Current Year
Assets
Current Assets:
Cash $ 90,000 $110,000
Accounts Receivable 20,000 30,000
Inventory 35,000 40,000
Short-Term Investments 15,000 20,000
Total Current Assets 160,000 200,000
Noncurrent Assets:
Equipment 40,000 50,000
Total Assets $200,000 $250,000
Liabilities
Current Liabilities:
Accounts Payable 60,000 75,000
Unearned Revenue 10,000 25,000
Total Current Liabilities 70,000 100,000
Noncurrent Liabilities:
Notes Payable 40,000 50,000
Total Liabilities $110,000 $150,000
Stockholder Equity
Common Stock 75,000 80,000
Ending Retained Earnings 15,000 20,000
Total Stockholder Equity 90,000 100,000
Total Liabilities and $200,000 $250,000
Stockholder Equity
Figure 18.2 Balance Sheet
The income statement reflects the performance of the firm over a period of time. It includes net sales, cost of goods sold, operating expenses, and net income. In Figure 18.3. we see that Clear Lake had $120,000 in net sales, $60,000 in cost of goods sold, and $35,000 in net income in the current year.
Comparative Year-End Income Statements
Prior Year Current Year
Gross Sales $105,000 $126,000
Sales Returns & Allowances 5,000 6,000
Net Sales 100,000 120,000
Cost of Goods Sold 50,000 60,000
Gross Profit 50,000 60,000
Rent Expense 5,000 5,500
Depreciation Expense 2,500 3,600
Salaries Expense 3,000 5,400
Utility Expense 1,500 2,500
Operating Income 38,000 43,000
Interest Expense 3,000 2,000
Income Tax Expense 5,000 6,000
Net Income $ 30,000 $ 35,000
Figure 18.3 Full Income Statement
Finally, the statement of cash flows is used to reconcile net income to cash balances. The statement begins with net income, then reflects adjustments to balance sheet accounts and noncash expenses. The statement of
538 18 • Financial Forecasting
cash flows is broken down into three key categories: operating, investing, and financing. This allows users to clearly see what elements of the business are generating or using cash. In Figure 18.4. we see that Clear Lake had cash flow from operating activities of $53,600, cash used for investing activities of ($18,600), and cash used for financing activities of ($15,000).
Statement of Cash Flows Indirect Method
Cash Flow from Operating Activities
Net Income $ 35,000
Adjustment to Reconcile Net Income to Net Cash Flow from Operating Activities:
Depreciation $3,600
Accounts Receivable Increase (10,000)
Inventory Increase (5,000)
Accounts Payable Increase 15,000
Unearned Revenue Increase 15,000 18,600
Net Cash Flow from Operating Activities 53,600
Cash Flow from Investing Activities
Purchase of Short-Term Investments (5,000)
Cost of New Plant Assets (13,600)
Net Cash Flow: Investing Activities (18,600)
Cash Flow from Financing Activities
Additional Notes Payable 10,000
Issuance of Common Stock 5,000
Payment of Dividends (30,000)
Net Cash Flow from Financing Activities (15,000)
Total Cash Flow Increase/Decrease 20,000
Beginning Cash Balance 90,000
Ending Cash Balance $110,000
Figure 18.4 Statement of Cash Flows
Another key concept to remember about the financial statements is that the statement of cash flows is necessary to truly understand how the firm is using and generating cash. A common misconception is that if a firm reports net income on its income statement, then it must have plenty of cash, and if it reports a loss, it must be short on cash. Although this can be true, it's not necessarily the case. Historically speaking, we need the statement of cash flows to get the full picture of how cash was used or generated in the past. Looking to the future, we need a cash flow forecast to plan for possible gaps in cash flow and, potentially, how to make the best use of any cash surplus. Throughout this chapter, we will see how to use historical financial statements to help develop the future cash forecast.
It's also important to remember that the four financial statements are tied together. Net income from the income statement feeds into retained earnings, which live on the balance sheet. Equity balances on the balance sheet feed information to the statement of stockholders' equity. And information from both the income statement (net income and noncash expenses) and the balance sheet (changes in working capital accounts) all feed into the statement of cash flows. These relationships will be helpful to understand when using historical statements and preparing forecasts.
Balance Sheet Analysis
Fully understanding the items that are on the balance sheet and how they relate to one another and to other financial statements will help you create a financial forecast. In Financial Statements, you learned that on the classified balance sheet, both assets and liabilities are broken down into current and noncurrent categories. You also know that the balance sheet must live up to its name—it must balance. This means that total assets (what the company owns) must equal total liabilities and equity (what the company owes).
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18.1 • The Importance of Forecasting 539
You continued your financial statement development in Measures of Financial Health, where you saw how to use elements of the balance sheet to assess financial health. Ratios based on balance sheet accounts can be useful for understanding relationships between balance sheet items—how they related in the past and then, in forecasting, how those relationships might change or remain the same in the future. Examples of balance sheet ratios include the current ratio, quick ratio, cash ratio, debt-to-assets ratio, and debt-to-equity ratio.
In Financial Statements, you also explored common-size analysis. To prepare a common-size analysis of the balance sheet, every item on the statement must be expressed as a percentage of total assets. Seeing each item as a percentage—that is, seeing its relationship to total assets—is also helpful for assessing historical statements and how those percentages or relationships can be used to predict future balances in the forecast. For example, in Figure 18.5. you can see that Clear Lake's current assets represented 80% of its total assets in both the current and prior years.
Comparative Year-End Balance Sheets
Prior Year Prior Year% Current Year Current Year %
Assets
Current Assets:
Cash $ 90,000 45% $110,000 44%
Accounts Receivable 20,000 10% 30,000 12%
Inventory 35,000 18% 40,000 16%
Short-Term Investments 15,000 8% 20,000 8%
Total Current Assets 160,000 80% 200,000 80%
Noncurrent Assets:
Equipment 40,000 20% 50,000 20%
Total Assets $200,000 100% $250,000 100%
Liabilities
Current Liabilities:
Accounts Payable 60,000 30% 75,000 30%
Unearned Revenue 10,000 5% 25,000 10%
Total Current Liabilities 70,000 35% 100,000 40%
Noncurrent Liabilities:
Notes Payable 40,000 20% 50,000 20%
Total Liabilities $110,000 55% $150,000 60%
Stockholder Equity
Common Stock 75,000 38% 80,000 32%
Ending Retained Earnings 15,000 8% 20,000 8%
Total Stockholder Equity 90,000 45% 100,000 40%
Total Liabilities and $200,000 100% $250,000 100%
Stockholder Equity
Figure 18.5 Common-Size Balance Sheet
Income Statement Analysis
Like balance sheet analysis, income statement analysis is also quite helpful in preparing for the forecasting process. In Financial Statements, you learned that the income statement is commonly broken down into a few sections. Cost of goods sold is deducted from net sales to arrive at gross margin. Gross margin refers to the profits earned solely on the sale of the product itself, without consideration for the expenses incurred to run the business. Next, operating expenses are deducted to reflect operating income. Operating income reflects the profits of the core business function. Finally, other items, such as interest expense, tax expense, and other gains and losses, are deducted to arrive at net income, a.k.a. the bottom line. Each segment of the income statement is helpful for assessing past performance and estimating future expenses for a forecast.
You continued your financial statement development in Measures of Financial Health, where you saw how to
540 18 • Financial Forecasting
use elements of the income statement to assess historical financial performance. Ratios based on the income statement can be useful for understanding relationships between net sales and expenses—how they related in the past and then, in forecasting, how those relationships might change or remain the same. Examples of income statement ratios include gross margin, operating margin, and profit margin. Common ratios that incorporate items from both the balance sheet and the income statement include return on assets (ROA), return on equity (ROE), inventory turnover, accounts receivable turnover, and accounts payable turnover.
Performance Trends
Review the most recent annual report for Big 5 Sporting Goods (https://0penstax.0rg/r/ annual report for Big 5 Sporting-Goods). Review the company's sales and gross margins for the current and past two years. How is their performance? Are their sales trending up or down? Why might the contribution margin have increased or decreased?
In Financial Statements, you also explored common-size analysis. To prepare a common-size analysis of the income statement, every item on the statement must be expressed as a percentage of net assets. Seeing each item as a percentage, in terms of its relationship to total sales, is also helpful for assessing historical statements and how those percentages or relationships can be used to predict future balances in the forecast. For example, in Figure 18.6. you can see that Clear Lake's cost of goods sold represented 50% of its net sales in both the current and prior years.
Comparative Year-End Income Statements
Prior Year Prior Year%* Current Year Current Year %*
Gross Sales $105,000 $126,000
Sales Returns & Allowances 5,000 6,000
Net Sales 100,000 100% 120,000 100%
Cost of Goods Sold 50,000 50% 60,000 50%
Gross Profit 50,000 50% 60,000 50%
Rent Expense 5,000 5% 5,500 5%
Depreciation Expense 2,500 3% 3,600 3%
Salaries Expense 3,000 3% 5,400 5%
Utility Expense 1,500 2% 2,500 2%
Operating Income 38,000 38% 43,000 36%
Interest Expense 3,000 3% 2,000 2%
Income Tax Expense 5,000 5% 6,000 5%
Net Income $ 30,000 30% $ 35,000 29%
*Some figures are rounded to the nearest whole percent, which may alter the total percentage to plus or minus 100%.
Figure 18.6 Common-Size Income Statement
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18.2 • Forecasting Sales 541
18.2
Forecasting Sales
Learning Outcomes
By the end of this section, you will be able to:
Explain how sales are the main driver for a financial forecast. Determine a past time period to formulate the basis for a financial forecast.
Explain the advantages and disadvantages of using past data to forecast future financial performance. Calculate past sales growth averages.
Justify adjusting relationships when forecasting future financial performance.
In this section of the chapter, you will begin to explore the first step of creating a forecast: forecasting sales. We will discuss common time frames for sales forecasts and why we use historical data in our forecasts (but only with caution), and we will work through the process of forecasting future sales. We will be using the percent-of-sales method to forecast some expenses for Clear Lake Sporting Goods, the example used throughout the chapter. This method relies on sales data, further highlighting why accuracy in forecasting sales is crucial.
Sales as the Driver
A significant portion of a business's costs are driven by how much it sells. Thus, the sales forecast is the necessary first step in preparing a financial forecast. Common costs driven by sales include direct product costs, direct labor costs, and other key variable costs (i.e., costs that vary proportionately to sales), such as sales commissions.
Looking to the Past
Forecasting sales is not always an easy task, as no one knows the future. We can, however, use the information we do have to forecast future sales with the greatest accuracy possible. Most firms start by looking at the past. A firm may look at past sales from a variety of prior periods. It's common to look at the past 12 months to estimate the coming 12 months. Looking at 12 consecutive months helps identify seasonality of sales trends, what time of year sales tend to drop off and when they increase, possible sales spikes that might reoccur, and any other trends that tend to appear over a 12-month period. In Figure 18.7. we see Clear Lake's sales by month for the past 12 months.
Past data is often used in conjunction with probabilities and weighted average calculations derived from probabilities. Though used in several areas of forecasting, this approach is particularly common in drafting the sales forecast. Using multiple scenarios and the probability of each scenario occurring is a common approach to estimating future sales.
Clear Lake Sporting Goods
Monthly Sales
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Current Year
Gross Sales 9,000 11,000 9,000 10,000 14,000 19,000 18,000 12,000 3,000 6,000 5,000 5,000 $126,000
Figure 18.7 Historical Sales Data
We can see at first glance that sales remain fairly steady from January to April. Sales then goes up significantly in April and May, seem to peak in June, taper off a bit in July, then decline steeply from August to the end of the year, with the lowest sales being in November and December. Though not exact, it's easy to quickly see that sales follow a seasonal pattern. We will focus on just one year of data here to keep things simple. However, it's important to note that when a firm has a seasonal sales pattern, it normally uses more than one year of data to detect and evaluate the pattern. It's not uncommon for firms to have a seasonal sales pattern that fluctuates based on an external factor such as weather patterns, patterns in business or demand, or other
542 18 • Financial Forecasting
factors such as holidays. Common examples might include farm-based businesses that function on a weather pattern for harvesting and selling crops or a toy company that fluctuates around gift-giving holidays.
This knowledge is helpful when assembling a first pass at the next year's sales forecast. Using common-size and horizontal (trend) analyses on sales is also helpful, as shown in Figure 18.8. We can see the exact percentages that sales went up or down each month:
■ In January, the company had sales of $9,000, which was 7.1% ($9.000/$ 126,000) of the total annual sales.
■ In June, the company had $19,000 sales, which was 15.1 ($19,000/$126,000) of the total annual sales and 211% ($19,000/$9,000) of January sales.
Clear Lake Sporting Goods
Monthly Sales as Percentages
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Current Year
Gross Sales 9,000 11,000 9,000 10,000 14,000 19,000 18,000 12,000 3,000 6,000 5,000 5,000 $126,000
% ofAnnua 7.1% 8.7% 7.1% 7.9% 11.1% 15.1% 14.3% 9.5% 6.3% 4.8% 4.0% 4.0% 100.0%
Sales
% Change 100% 122% 100% 111% 156% 211% 200% 133% 89% 67% 56% 56%
from January
Baseline
Figure 18.8 Historical Sales Data as Percentages
Once a baseline in the 12-month period is assessed, it can also be helpful to look for trends in other ways. For example, the past several years might be assessed to see if there is a trend in total growth or decline for those years on a summary basis or by period. Clear Lake Sporting Goods had sales in the current year of $126,000, in the prior year of $105,000, and two years ago of $89,000. This reflects a 20% increase and an 18% increase, respectively. It might be reasonable to expect a roughly 18 to 20% increase in total sales in the future with only this information in mind. Keep in mind that we will learn about many other factors to consider in the forecast, so the 18 to 20% increase is a good general guideline to consider along with other factors.
Sales Forecast for Big 5 Sporting Goods
Review the 2020 annual report for Big 5 Sporting Goods (https://0penstax.0rg/r/
2020 annual report for Big 5 Sporting Goods). Locate the consolidated statements of operations on page F-7. Using the company's net sales figures for the current and prior years, what percentage might you recommend for their sales forecast for the next year?
Solution:
Big 5's sales for 2020 and 2019 were $1,041,212,000 and $996,495,000, respectively. This represents a 4.5% growth from the prior year. Forecasted sales, based solely on this information, might be appropriate at 4.5% growth for the next year, keeping in mind that many other factors should also be considered along with this information. Historical sales are only one set of data to use as a starting point.
Looking at Figure 18.9. assume that Clear Lake Sporting Goods decides to take its first pass at a forecast using the more conservative estimate of 18% total sales growth. The company could consider last year's sales of $126,000 and increase them by 18% to arrive at total forecasted sales for next year of
$148,680 ($126,000 X 118%). Next, to get the monthly sales, the company could use the same percent of the total for each month that it did for the previous year. For example, sales in January of last year were 7.1 % of the full year's sales. To find the forecast for the next year, the company would take the forecasted sales of $148,680 for the year and multiply that by 7.1% to get $10,620 for January. The process is repeated for each
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18.2 • Forecasting Sales 543
month to get the full year.
Clear Lake Soortina Goods
Forecasted Monthly Income Statements
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Next Year
Gross Sales 10,620 12,980 10,620 11,800 16,520 22,420 21,240 14,160 9,440 7,080 5,900 5,900 $148,680
% of Annual 7.1% 8.7% 7.1% 7.9% 11.1% 15.1% 14.3% 9.5% 6.3% 4.8% 4.0% 4.0% 100.0%
Sales
% Change 100% 122% 100% 111% 156% 211% 200% 133% 89% 67% 56% 56%
from January
Baseline
Figure 18.9 Forecasted Sales Data
Keep in mind that this is only a starting point. These estimates will be reviewed, assessed, and updated as more information and other factors are taken into consideration.
It can also be helpful to look at a shorter period, perhaps just the last few months, on a more detailed basis (by department, by customer, etc.) to see if there are any possible new trends beginning to develop that might be an indicator of performance in the coming year. For example. Clear Lake Sporting Goods might look at detailed sales records for October, November, and December and see that it had an old product line that was discontinued in early October, which contributed to a 2% reduction in monthly sales. This reduction in monthly sales will likely continue into the new year until the new line the company has signed on begins arriving in stores. Thus, the management team feels they should reduce their first quarter monthly estimates by 2%, as reflected in Figure 18.10. lanuary is now $10,408 ($10,620 X 98%), for example.
Forecasted Monthly Income Statements
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Total
Gross Sales 10,408 12,720 10,408 11,800 16,520 22,420 21,240 14,160 9,440 7,080 5,900 5,900 $147,996
Figure 18.10 Adjusted Forecasted Sales Data
Changes for the Future
It's important to note that the past is not always a reliable predictor of the future. Circumstances can often change to make the future quite different from the past. The business itself may change, the economy can change, the customer base may undergo a shift in demographics or a change in buying habits, new competition may emerge, and so on. So while past performance is helpful, it is only one step in the process of forecasting sales.
Big 5 Sporting Goods MD&A Report
Review the most recent annual report for Big 5 Sporting Goods (https://0penstax.0rg/r/ annual report for Big 5 Sporting Goods). Review the management's discussion and analysis (MD&A) report (Item 7). What information does the report share about the firm, the economy, and other factors that might be useful for forecasting sales growth for next year?
Most firms first look to the past to target some form of baseline estimate for the coming year; then, managers begin making adjustments based on what they know about the future. Assume that Clear Lake Sporting Goods will be adding a new brand to its collection of fishing supplies in March. The manufacturer plans to begin running its commercials in late February, which managers anticipate will increase Clear Lake's monthly sales
544 18 • Financial Forecasting
by about $500 in March, $1,000 in April, $1,400 in May, and $2,000 per month in Junejuly, and August. We see the monthly adjustments to Clear Lake's latest sales forecast in Figure 18.11. March, for example, is now $10,908 ($10,408 prior estimate plus $500 increase from new brand).
Forecasted Monthly Income Statements
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Total
GrossSales 10,408 12,720 10,908 12,800 18,020 24,420 23,240 16,160 9,440 7,080 5,900 5,900 $156,996
Figure 18.11 Forecasted Sales Data with New Brand
What we have discussed here are only some brief examples of the myriad factors that might impact a sales budget for the coming year. It's critical that all members of the team take the time and effort to research their customers and the factors that impact their business in order to effectively assess the impact of these factors on future sales. Though only two adjustments were made here, it's likely that a large firm would have to consider many, many factors that would ultimately impact monthly sales figures before arriving at a conclusion.
18.3
Pro Forma Financials
Learning Outcomes
By the end of this section, you will be able to:
■ Define pro forma in the context of a financial forecast.
■ Describe the factors that impact the length of a financial forecast.
■ Explain the risks associated with a financial forecast.
In this section of the chapter, we will move beyond the sales forecast and look at the general nature, length, and timeline of forecasts and the risks associated with using them. We'll look at why we use them, how long they generally are, what the key variables in a forecast are, and how we pair those variables with common-size analysis to develop the forecast.
Purpose of a Forecast
As mentioned earlier in the chapter, forecasts serve different purposes depending on who is using them. Our focus here, however, is the world of finance. In this realm, the key purpose of pro forma (future-looking) financial statements is to manage a firm's cash flow and assess the overall value that the firm is generating through future sales growth. Growing just for the sake of growing doesn't always yield favorable income for the firm. A larger top-line sales figure that results in lower net income doesn't make sense in the grand scheme of things. The same is true of profitable sales that don't generate enough cash flows at the right time. The firm may make a profit, but if it doesn't manage the timing of its cash flows, it could be forced to shut down if it can't cover the costs of payroll or keep the lights on. Forecasting helps assess both cash flow and the profitability of future growth. Managers can forecast cash flow using data from forecasted financial statements; this allows them to identify potential gaps in cash and plan ahead in order to either alter collection and payment policies or obtain funding to cover the gap in the timing of cash flows.
I
LINK TO LEARNIN
Pro Forma Financial Statements
Review the video Business Plan and Pro-Forma Financial Statements (https://0penstax.0rg/r/ Business Plan and Pro-Forma Financial Statements) to learn about the basics of pro forma financial statements and why they are helpful.
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18.3 • Pro Forma Financials 545
Length of a Forecast
Forecasts can generally be for any length of time. The length generally depends on the user's needs. A one-year forecast, broken down by month, is quite typical. A firm will often go through a formal budgeting process near the end of its calendar or fiscal year to project financial plans and goals for the coming year. Once that is done, a rolling financial forecast is then done monthly to adjust as time moves on, more information becomes available, and circumstances change.
To be useful, the future forecast for financial planning purposes is almost always calculated as monthly increments rather than one total figure for the next 12 months. Breaking the data down by month allows finance managers to more clearly see fluctuations in cash flows in and out, identify potential gaps in cash flow, and plan ahead for their cash needs.
Forecasts can also be done for several years into the future. In fact, they commonly are. However, once the firm is looking out beyond 12 months, it gets difficult to forecast items with a great degree of accuracy. Often, forecasts beyond a year will be completed only to quarterly or even annual figures rather than monthly. Forecasts that far into the future are often strategic in nature, made more to communicate future plans for the firm than for more detailed decision-making and cash flow planning.
Common-Size Financials
As we saw earlier in the chapter, common-size analysis involves using historical financial statements as a basis for future forecasts. Financial statements provide a great starting point for analysis, as we can see the relationships between sales and costs on the income statement and the relationships between total assets and line items on the balance sheet.
For example, in Figure 18.6. we saw that for the past two years, cost of goods sold has been 50% of sales. Thus, in the first draft of a forecast for Clear Lake, it's likely that managers would estimate cost of goods sold at 50% of their forecasted sales. We can begin to see why forecasting sales first is crucial and why doing so as accurately as possible is also important.
Select Variables to Use
A simple way to begin a full financial statement forecast might be to simply use the common-size statements and forecast every item using historical percentages. It's a logical way to begin a very rough draft of the forecast. However, several variables should be taken into consideration. First, managers must address the cost of an account and determine if it's a variable or fixed item. Variable costs tend to vary directly and proportionally with production or sales volume. Common examples include direct labor and direct materials. Fixed costs, on the other hand, do not change when production or sales volume increases or decreases within the relevant range. Granted, if production were to increase or decrease by a large amount, fixed costs would indeed change. However, in normal month-to-month changes, fixed costs often remain the same. Common examples of fixed costs include rent and managerial salaries.
So, if we were to approach our common-size income statement, for example, we would likely use the percentage of sales as a starting point to forecast variable items such as cost of goods sold. However, fixed costs may not be accurately forecast as a percentage of sales because they won't actually change with sales. Thus, we would likely look at the history of the dollar values of fixed costs in order to forecast them.
CONCEPTS IN PRACTICE
COVID-19 Makes Forecasting Difficult for Big 5 Sporting Goods
Big 5 Sporting Goods announced record earnings in the third quarter of 2020, attributing its huge success that quarter to the impact of people's reactions to the COVID-19 pandemic. With so many people in
546 18 • Financial Forecasting
quarantine still wanting to make healthy lifestyle choices, sporting goods stores were making record sales. Record-breaking sales, however, are not certain in the future. The impacts of the pandemic are extremely difficult to predict, making it a challenge for Big 5 Sporting Goods and other companies to assemble pro forma financial statements.
(sources: https://www.globenewswire.eom/news-release/2020/10/27/2115470/0/en/Big-5-Sporting-GoodS" Corporation-Announces-Record-Fiscal-2020-Third-Quarter-Results.html; https://finance.yahoo.com/news/ investors-want-big-5-sporting-054658521.html; https://www.cpapracticeadvisor.com/accounting-audit/ news/21206691 /four-ways-covid19-will-impact-2021 -financial-forecasting-and-planning)
Determine Potential Changes in Variables
So far, we have focused on using historical common-size statements to create a draft (not a final version) of the forecast. This is because the past isn't always a perfect indicator of the future, and our finances don't always follow a linear pattern. We use the past as a good starting point; then, we must assess what else we know to fine-tune and make adjustments to the forecast.
Many items impact the forecast, and they will vary from one organization to another. The key is to do research, gather data, and look around at the market, the economy, the competition, and any other factors that have the potential to impact the future sales, costs, and financial health of the company. Though certainly not an exhaustive list, here are a few examples of items that may impact Clear Lake Sporting Goods.
■ It has an old product line that was discontinued in early October, contributing to a 2% reduction in monthly sales that will likely continue into the new year until a new line begins arriving in stores.
■ It will be adding a new brand to its collection of fishing supplies in March. The manufacturer plans to begin running commercials in late February. Managers anticipate that this will increase Clear Lake's monthly sales by about $500 in March, $1,000 in April, $1,400 in May, and $2,000 per month in June, July, and August.
■ The company has just finished updating its employee compensation package. It goes into effect in January of the new year and will result in an overall 4% increase in the cost of labor.
■ The landlord indicated that rent will increase by $50 per month starting July 1.
■ Some fixed assets will be fully depreciated by the end of March. Thus, depreciation expense will go down by $25 per month beginning in April.
■ There are rumors of new regulations that will impact the costs of importing some of the more difficult-to-obtain hunting supplies. Managers aren't entirely sure of the full impact of the new legislation at this time, but they anticipate that it could increase cost of goods sold for the affected product line when the new legislation goes into effect in the last quarter. Their best estimate is that it could increase the overall cost of goods sold by up to 2%.
We will use all of this data later in the chapter when we are ready to compile a complete forecast for Clear Lake.
18.4
Generating the Complete Forecast
Learning Outcomes
By the end of this section, you will be able to:
■ Generate a forecasted income statement that incorporates pertinent sales, functional, and policy variables.
■ Generate a forecasted balance sheet.
■ Connect the balance sheet and income statement forecasts with appropriate feedback linkages. In this section of the chapter, we will tie together what we have learned so far about forecasting sales.
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18.4 • Generating the Complete Forecast 547
common-size analysis, and using what we know about the company and its environment to create a full set of pro forma (forward-looking or forecasted) financial statements.
Forecast the Income Statement
To arrive at a fully forecasted income statement, we use historical income statements, common-size income statements, and any additional information we have about future sales and costs, such as the effects of the economy and competition. As we saw earlier in the chapter, we begin with forecasted sales because they are the basis for many of the forecasted costs.
Let's begin with the sales forecast for Clear Lake Sporting Goods that we saw earlier in the chapter, in Figure 18.9. and use it along with the prior year income statement by month shown in Figure 18.12. We will consider other data we have about the business to begin creating a full income statement (see Figure 18.13).
Prior Year Monthly Income Statements
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Current Year Current Year %
Gross Sales 9,000 11,000 9,000 10,000 14,000 19,000 18,000 12,000 8,000 6,000 5,000 5,000 $126,000
% of Annual Sales 7.1% 8.7% 7.1% 7.9% 11.1% 15.1% 14.3% 9.5% 6.3% 4.8% 4.0% 4.0% 100.0%
% Change from 100% 122% 100% 111 % 156% 211% 200% 133% 89% 67% 56% 56%
January Baseline
Sales Returns & 300 400 400 600 800 1,000 800 600 400 300 200 200 6,000
Allowances
Net Sales 8,700 10,600 8,600 9,400 13,200 18,000 17,200 11,400 7,600 5,700 4,800 4,800 120,000 100%
Cost of Goods Sold 4,785 5,830 4,730 4,700 5,940 8,100 7,740 5,700 3,800 2915 2,880 2,880 60,000 50%
Gross Margin 3,915 4,770 3,870 4,700 7,260 9,900 9,460 5,700 3,800 2,785 1,920 1,920 60,000 50%
Rent Expense 458 458 458 458 458 458 458 458 458 458 458 458 5,500 5%
Depredation Expense 300 300 300 300 300 300 300 300 300 300 300 300 3,600 3%
Salaries Expense 450 450 450 450 450 450 450 450 450 450 450 450 5,400 5%
Utility Expense 179 218 179 198 278 377 357 238 159 119 99 99 2,500 2%
Operating Income 2,528 3,343 2,483 3,293 5,774 8,315 7,895 4,254 2,433 1,458 612 612 43,000 36%
Interest Expense 167 167 167 167 167 167 167 167 167 167 167 167 2,000 2%
Income Tax Expense 353 467 346 460 806 1,160 1,102 594 339 203 85 85 6,000 5%
Net Income 2,009 2,710 1,970 2,667 4,802 6,988 6,626 3,493 1,927 1,088 360 360 $ 35,000 29%
Figure 18.12 Prior Year Monthly Income Statement by Month
The first two key points regarding product lines have already been built into the sales forecast. Notice that the cost of goods sold was 50% in the prior year. However, based on possible future legislation, to be conservative, we should increase the cost of goods sold by 2% in the last quarter of the year. Thus, we will forecast cost of goods sold at 50% of sales in the first nine months and increase it to 52% in the last three months of the year.
Rent is a fixed cost that historically amounts to $458 per month. However, we know that the landlord is increasing rent by $50 starting on July 1. Thus, we will forecast rent at the same fixed cost of $458 per month for the first six months and increase it to $508 per month for the second half of the year.
Depreciation, also a fixed cost, was historically $300 per month. However, we know that depreciation expense will go down by $25 beginning in April. Thus, we forecast depreciation at $300 for the first three months and at $275 for the last 9 months.
Salaries expense has historically been $450 per month. However, we know that the company is implementing a new compensation program on January 1 that will increase salaries expense by 4% ($18). Thus, we will forecast salaries for the whole year at $468.
Utilities expense seems to vary somewhat by sales from month to month, as shops are open longer hours during their busy season. However, the total utilities expense is not expected to change for the coming year. Thus, the forecast for utilities expense remains at $2,500, broken down by month as a percentage of sales.
Interest expense is a fixed cost and isn't anticipated to change. Thus, the same $167 interest expense per month is forecast for the coming year.
548 18 • Financial Forecasting
Finally, income tax expense is forecasted as a percentage of operating income because tax liability is incurred as a direct result of operating income. Figure 18.13 shows the next 12 months' forecast for Clear Lake Sporting Goods using all of this data.
Forecasted Monthly Income Statements
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Total
Gross Sales $10,408 $12,720 $10,908 $12,800 $18,020 S24.420 $23,240 $16,160 $9,440 $7,080 $5,900 $5,900 $156,996
% of Annual Sales 7.1% 8.7% 7.1% 7.9% 11.1% 15.1% 14.3% 9.5% 6.3% 4.8% 4.0% 4.0% 100%
% Change from 100% 122% 105% 123% 173% 235% 223% 155% 91% 68% 57% 57%
January Baseline
Sales Returns & 300 400 400 600 800 1,000 800 600 400 300 200 200 6,000
Allowances
Net Sales 10,108 12,320 10,508 12,200 17,220 23,420 22,440 15,560 9,040 6,780 5,700 5,700 150,996
Cost of Goods Sold 5,054 6,160 5,254 6,100 8,610 11,710 11,220 7,780 4,520 3,526 2,964 2,964 75,861
Gross Margin 5,054 6,160 5,254 6,100 8,610 11,710 11,220 7,780 4,520 3,254 2,736 2,736 75,134
Rent Expense 458 458 458 458 458 458 508 508 508 508 508 508 5,798
Depreciation Expense 300 300 300 275 275 275 275 275 275 275 275 275 3,375
Salaries Expense 468 468 468 468 468 468 468 468 468 468 468 468 5,616
Utility Expense 175 214 183 215 303 411 391 272 159 119 99 99 2,640
Operating Income 3,652 4,720 3,844 4,683 7,106 10,098 9,578 6,257 3,110 1,884 1,386 1,386 57,705
Interest Expense 167 167 167 167 167 167 167 167 167 167 167 167 2,000
Income Tax Expense 510 659 536 654 991 1,409 1,336 873 434 263 193 193 8,052
Net Income $ 2,976 $ 3,895 $ 3,141 $ 3,863 $ 5,948 $ 8,522 $ 8,075 $ 5,218 $2,510 $1,455 $1,026 $1,026 f 47,653
Figure 18.13 Forecasted Income Statement
Forecast the Balance Sheet
Now that we have a reasonable income statement forecast, we can move on to the balance sheet. The balance sheet, however, is entirely different from the income statement. It requires a bit more research and additional assumptions. Just like the income statement, it's often a work in progress. A first draft is a good starting point, but adjustments must be made once it is created, and all the interrelationships between the statements, cash flow in particular, are taken into consideration.
The balance sheet is a bit more difficult to forecast because the statement reflects balances at just a given point in time. Account balances change daily, so forecasting just one snapshot in time for each month can be a challenge. A good starting point is to assess general company financial policies or rules of thumb. For example, assume that Clear Lake pays most of its vendors on net 30-day terms. A good way to forecast accounts payable on the balance sheet might be to add up the cost of goods sold from the forecasted income statement for the prior month. For example, in Figure 18.14. we see that Clear Lake has forecasted its accounts payable for March as the cost of goods sold in March from its forecasted income statement.
For accounts receivable. Clear Lake generally receives payment from customers within net 90-day terms. Thus, it uses the sum of the current and prior two months' forecasted sales to estimate its accounts receivable balance.
Inventory will vary throughout the year. For the first six months, the company tries to build inventory for four months of sales. Once the busy season hits, inventory goes down to three months' worth of future sales, then finally drops to only two months of sales in December. Thus, managers use their sales forecast by month to estimate their inventory ending balance each month.
The equipment balance is forecasted by reducing the prior month's balance by the forecasted depreciation expense on the forecasted income statement.
Unearned revenue is historically around 50% of the current month's sales. Thus, Clear Lake estimates its unearned revenue balance each month by taking the current month's net sales from the forecasted income statement and multiplying it by 50%.
Short-term investments, notes payable, and common stock are not anticipated to change, so the current balance is forecasted to remain the same for the next 12 months.
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18.4 • Generating the Complete Forecast 549
To forecast the ending balance for retained earnings for each month, managers add the monthly net income from the forecasted balance sheet to the prior balance and subtract a quarterly $10,000 dividend.
Once all of these accounts are completed, the balance sheet is out of balance. Given that all of these events are somewhat related but are not tied together dollar for dollar, it's not surprising when the forecasted balance sheet is finished and does not balance. To complete the first draft (see Figure 18.14). the cash account is used as a variable and plugged in to make the balance sheet balance. Notice that by the end of the year, the company has $59,905 in cash. However, look at what happens midyear—the cash account falls to only $8,782. In the next section, we will generate a cash flow forecast, which will allow Clear Lake to update its balance sheet forecast once it estimates what it will do to cover the cash flow gaps.
Forecasted Monthly Balance Sheets
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Assets
Current Assets:
Cash $42,581 $38,259 $19,083 $11,150 $12,345 $8,782 $16,175 $20,842 $21,733 $40,099 $60,834 $59,905
Accounts Receivable 25,408 32,028 37,736 45,136 52,248 63,348 75,280 78,640 70,460 53,820 37,080 27,220
Inventory 22,568 26,124 31,674 37,640 39,320 35,230 23,520 15,826 11,010 9,454 5,928 8,018
Short-Term Investments 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000
Total Current Assets 110,557 116,411 108,493 113,925 123,913 127,360 134,975 135,308 123,202 123,373 123,842 115,142
Noncurrent Assets:
Equipment 50,000 49,700 49,400 49,100 48,825 48,550 48,275 48,000 47,725 47,450 47,175 46,900
Total Assets 160,557 166,111 157,893 163,025 172,738 175,910 183,250 183,308 170,927 170,823 171,017 162,042
Liabilities
Current Liabilities:
Accounts Payable 5,054 6,160 5,254 6,100 8,610 11,710 11,220 7,780 4,520 3,526 2,964 2,964
Unearned Revenue 2,527 3,080 2,627 3,050 4,305 5,855 5,610 3,890 2,260 1,695 1,425 1,425
Total Current Liabilities 7,581 9,240 7,881 9,150 12,915 17,565 16,830 11,670 6,780 5,221 4,389 4,389
Noncurrent Liabilities:
Notes Payable 50,000 50,000 50,000 50,000 50,000 50,000 50,000 50,000 50,000 50,000 50,000 50,000
Total Liabilities 57,581 59,240 57,881 59.150 62,915 67.565 66,830 61,670 56,780 55,221 54,389 54,389
Stockholder Equity
Common Stock 80,000 80,000 80,000 80,000 80,000 80,000 80,000 80,000 80,000 80,000 80,000 80,000
Ending Retained Earnings 22,976 26,871 20,012 23,875 29,823 28,345 36,420 41,638 34,147 35,602 36,628 27,653
Total Stockholder Equity 102,976 106,871 100,012 103,875 109,823 108,345 116,420 121,638 114,147 115,602 116,628 107,653
Total Liabilities and $160,557 $166,111 $157,893 $163,025 $172,738 $175,910 $183,250 $183,308 $170,927 $170,823 $171,017 $162,042
Stockholder Equity
Figure 18.14 Forecasted Balance Sheet Draft
Linkages between the Forecasted Balance Sheet and the Income Statement
Notice that in the discussion in the prior section on the balance sheet forecast, a lot of the information in the forecasted income statement was used to generate the forecasted balance sheet. The balance sheet accounts generally depend on activity reported in the income statement. For example, for many firms, the balance in their accounts receivable account is tied to their sales. Looking at historical balances in the accounts receivable account and how those relate to historical sales will help determine how to use the forecasted future sales to estimate the future balance of accounts receivable.
The same is true of accounts payable. Looking at past balances, past expenses (normally cost of goods sold), and the firm's payment terms for its vendors allows managers to use forecasted cost of goods sold or other expenses to estimate the balance in the accounts payable account.
We learned in Financial Statements that net income flows into retained earnings. Thus, the net income from the forecasted income statement can be used to help estimate the ending balance in retained earnings. If the firm intends to issue any dividends in the coming year, managers should also estimate that reduction in their forecast.
It's also common to find other general policies or procedures that help drive performance and aid in forecasting balances. For example, if the company has a goal of maintaining a certain level of inventory or a minimum balance in its cash account, that information can be used to guide the estimate for those accounts.
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18.5
Forecasting Cash Flow and Assessing the Value of Growth
Learning Outcomes
By the end of this section, you will be able to:
■ Generate a cash flow forecast.
■ Assess a cash flow forecast to determine future cash funding needs.
■ Use pro forma financial statements and cash flow forecasts to assess the value of growth to the firm.
In this section of the chapter, we will use the forecasted income statement, forecasted balance sheet, and other information we know about the firm's policies and goals for the coming year to generate and assess a cash flow forecast.
Create a Cash Flow Forecast
A cash flow forecast isn't overly complex, yet it is not easy to assemble because it requires making many assumptions about the future. A cash forecast begins with the beginning cash balance, adds anticipated cash inflows, and deducts anticipated cash outflows. This identifies cash surpluses and shortages.
For Clear Lake Sporting Goods, for example, we see in Figure 18.15 that the company begins with cash of $42,581,000 in January of the new year. Next, it lists the cash inflows, or cash received from customers. Given the assumption that customers pay in 90-day terms, the cash flow is filled in by plugging in the sales forecast for the three prior months. For example, the cash flow from customers of $10,508 for June is the same as the net sales forecast for March (see Figure 18.13).
ake Sporting Goods
Cash Forecast
Jan Feb Mar Apr May Jun Jul Aug Sep
Beginning Cash $42,581 $44,394 $43,039 $35,038 $35,782 $40,861 $35,030 $35,133 $35,766 $36,658 $53,444 $64,383
Balance
Cash Inflows:
Accounts Receivable 5,700 4,800 4,800 10,108 12,320 10,508 12,200 17,220 23,420 22,440 15,560 9,040 Collected
Total Cash Inflows: $ 5,700 $ 4,800 $ 4,800 $10,108 $12,320 $10,508 $12,200 $17,220 $23,420 $22,440 $15,560 $9,040
Figure 18.15 Forecasted Cash Inflows
Next, Clear Lake identifies cash outflows, which include accounts payable, salaries, rent, utilities, dividends, and interest payments. Accounts payable are normally paid within 30 days, so the forecast for cost of goods sold for the prior month is used as an estimate of amount paid for payables. For example, in Figure 18.16. we see that the accounts payable settled in June of $8,610 is the cost of goods sold for May from the forecasted income statement.
Salaries are paid monthly and thus represent the same recurring monthly cash outflow, as does rent. Utilities, like accounts payable, are assumed to be paid within 30 days. Thus, the cash outflow for utilities is the utilities expense for the prior month from the forecasted income statement.
Management intends to pay a quarterly dividend of $10,000. Thus, in Figure 18.16. we see $10,000 cash outflows forecasted for March, June, September, and December. Interest on the long-term liability is paid quarterly. Thus, the $500 cash outflows in March, June, September, and December are simply the monthly interest expense of $167 from the income statement, summed for each quarter.
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18.5 • Forecasting Cash Flow and Assessing the Value of Growth 551
Cash Forecast
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Beginning Cash $42,581 $44,394 $43,039 $35,038 $35,782 $40,861 $35,030 $35,133 $35,766 $36,658 $53,444 $64,383
Balance
Cash Inflows:
Accounts Receivable 5,700 4,800 4,800 10,108 12,320 10,508 12,200 17,220 23,420 22,440 15,560 9,040
Collected
Total Cash Inflows: 5,700 4,800 4,800 10,108 12,320 10,508 12,200 17,220 23,420 22,440 15,560 9,040
Cash Outflows:
Accounts Payable Paid 2,880 5,054 6,160 5,254 6,100 8,610 11,710 11,220 7,780 4,520 3,526 2,964
Salaries 450 468 468 468 468 468 468 468 468 468 468 468
Rent 458 458 458 458 458 458 508 508 508 508 508 508
Utilities 99 175 214 183 215 303 411 391 272 159 119 99
Dividends 0 0 10,000 0 0 10,000 0 0 10,000 0 C 10,000
Interest Payments 0 0 500 0 0 500 0 0 500 0 C 500
Total Cash Outflows: $ 3,888 $ 6,155 $17,800 $ 6,364 $ 7,242 $20,339 $13,097 $12,587 $19,528 $ 5,655 $ 4,621 $14,539
Figure 18.16 Forecasted Cash Inflows and Outflows
Using a Cash Forecast to Determine Additional Funds Needed
Finally, at the end of the cash flow forecast, cash outflows are subtracted from the cash inflows. This identifies whether a cash surplus (extra) or cash deficit (not enough) exists for each month. For example, in Figure 18.17. we see that in March, Clear Lake is forecasting $4,800 of cash inflows and $17,800 of total cash outflows, which results in a cash deficit of $13,000.
Clear Lake has a general policy to not let its cash balance fall below $35,000. Thus, managers need to assess their monthly balances and potential deficits and identify months when financing is necessary. For example, the deficit of $13,000 in March is enough to push the cash balance lower than $35,000. Thus, it's estimated that the company will need $5,000 in short-term financing in March. It has an estimated surplus in April, so $3,000 of the borrowing is returned.
Clear Lake Sporting Goods
Cash Forecast
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Beginning Cash $42,581 $44,394 $43,039 $35,038 $35,782 $40,861 $35,030 $35,133 $35,766 $36,658 $53,444 $64,383
Balance
Cash Inflows:
Accounts Receivable 5,700 4,800 4,800 10,108 12,320 10,508 12,200 17,220 23,420 22,440 15,560 9,040
Collected
Total Cash Inflows: 5,700 4,800 4,800 10,108 12,320 10,508 12,200 17,220 23,420 22,440 15,560 9,040
Cash Outflows:
Accounts Payable Paid 2,880 5,054 6,160 5,254 6,100 8,610 11,710 11,220 7,780 4,520 3,526 2,964
Salaries 450 468 468 468 468 468 468 468 468 468 468 468
Rent 458 458 458 458 458 458 508 508 508 508 508 508
Utilities 99 175 214 183 215 303 411 391 272 159 119 99
Dividends 0 0 10,000 0 0 10,000 0 0 10,000 0 C 10,000
Interest Payments 0 0 500 0 0 500 0 0 500 0 C 500
Total Cash Outflows: 3,888 6,155 17,800 6,364 7,242 20,339 13,097 12,587 19,528 5,655 4,621 14,539
Cash Surplus or Deficit 1,812 (1,355) (13,000) 3,744 5,079 (9,832) (897) 4,633 3,892 16,785 10,939 (5,499)
Short-Term Financing 0 0 5,000 (3,000) 0 4,000 1,000 (4,000) (3,000) 0 0 0
Borrowed (Repaid)
Ending Cash Balance $44,394 $43,039 $35,038 $35,782 $40,861 $35,030 $35,133 $35,766 $36,658 $53,444 $64,383 $58,884
Figure 18.17 Forecasted Cash Surplus or Deficit
Assessing the Value of Growth
It's a fairly common assumption that most, if not all, businesses want to grow. While it certainly can be good as a firm to grow in size, growth just for the sake of growth isn't necessarily a good goal. A firm can grow in size based on customers, employees, locations, or simply sales. However, that doesn't mean that the growth will increase profits. Growth may increase profits, but this is not a safe assumption. Scaling up operations takes careful planning, which includes monitoring the profitability of the sales and, of course, the cash flow it
552 18 • Financial Forecasting
would require. Growing a business can require more inventory, more locations, more equipment, and more manpower, all of which cost money. Even if the forecasted growth is profitable, it may pose problems from a cash flow perspective. It's important that the firm review not only its forecasted income statement and balance sheet but also its cash forecast, as this can reveal some serious gaps in funding depending on the extent, timing, and nature of the planned growth.
For example, assume that Clear Lake Sporting Goods intends to run a large-scale ad campaign to boost sales in its busy season. Historically, the store relied primarily on its prime location for high volumes of retail foot traffic. Managers felt, however, that given the increase in competition, they could boost sales significantly by running the ad campaign in the first quarter. The campaign would cost $30,000. Forecasts already reflect a cash deficit at the end of the first quarter of $13,000, so the additional $30,000 ad campaign, which would require payment up front, would create a much larger need for funding. It's also important that managers look at the increased cost of doing business along with the increased cost in advertising to ensure that the move would be profitable. Fortunately, Excel or other forecasting software can be used to create a forecast with formulas that tie together, making scenario analysis such as this a much easier process.
Scenarios in Forecasting
Forecasting is almost never a linear process. In other words, we don't do one forecast and call it good. The first draft is completed using historical data, and then changes are made a bit at a time as all potential variables are assessed for their impact on the forecast. It's quite common to then use the work-in-progress forecast to complete scenario analysis. This is particularly true when the forecast is completed in Excel or other budgeting or forecasting software. Elements of the forecast can be changed to see what the overall impact would be to the firm. Assuming the forecast is set up using formulas in Excel or other software, a change to one figure or one variable would then "ripple" through the forecast to reflect the overall impact.
Often, a firm may complete an initial forecast (scenario) under the assumption that the economy is in a "normal state." The firm can then alter the initial forecast for different scenarios, such as the economy in a recession or the economy in a state of expansion. This helps the firm understand different possible future states and highlights how changes in the economy such as inflation may cause revenue and expenses to increase.
Assume that Clear Lake's initial forecast is created under the assumption that the economy will remain average. Management also wants to know the worst-case scenario. What will their financial results look like if the economy were in a recession, for example? If management assumes their sales would drop to only 60% of the prior year sales in a recessionary economy, they could alter the formula in Excel driving their sales and variable costs, resulting in a new pro forma income statement. In Figure 18.18, we can see that net income would drop to $16,391 under this assumption, compared to the net income of $47,653 forecasted under average economy assumptions in Figure 18.13.
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18.5 • Forecasting Cash Flow and Assessing the Value of Growth 553
Forecasted Monthly Income Statements: Worst-Case Scenario
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Total
Gross Sales $5,292 $6,468 $5,792 $7,000 $9,900 $13,400 $12,800 $9,200 $4,800 $3,600 $3,000 $3,000 $84,252
% of Annual Sales 7.1% 8./% 7.1% 7.9% 11.1% 15.1% 14.3% 9.5% 6.3% 4.8% 4.0% 4.0% 100%
% Change from 100% 122% 109% 132% 187% 253% 242% 174% 91% 68% 57% 57%
January Baseline
Sales Returns & 300 400 400 600 800 1,000 800 600 400 300 200 200 6,000
Allowances
Net Sales 4,992 6,068 5,392 6,400 9,100 12,400 12,000 8,600 4,400 3,300 2,800 2,800 78,252
Cost of Goods Sold 2,496 3,034 2,696 3,200 4,550 6,200 6,000 4,300 2,200 1,716 1,456 1,456 39,304
Gross Margin 2,496 3,034 2,696 3,200 4,550 6,200 6,000 4,300 2,200 1,584 1,344 1,344 38,948
Rent Expense 458 458 458 458 458 458 508 508 508 508 508 508 5,798
Depreciation Expense 300 300 300 275 275 275 275 275 275 275 275 275 3,375
Salaries Expense 468 468 468 468 468 468 468 468 468 468 468 468 5,616
Utility Expense 175 214 192 231 327 443 423 304 159 119 99 99 2,786
Operating Income 1,095 1,594 1,278 1,767 3,021 4,556 4,326 2,745 790 214 (6) (6) 21,373
Interest Expense 167 167 167 167 167 167 167 167 167 167 167 167 2,000
Income Tax Expense 153 222 178 24/ 422 63 b 604 383 110 30 (1) (1) 2,982
Net Income $ 775 $1,205 t 933 $1,354 $2,433 $ 3,753 $ 3,555 $2,195 $ 513 $ 17 $(172) $(172) $16,391
Figure 18.18 Forecasted Cash Surplus or Deficit
Though creating a full forecast in Excel can be a bit complex, it is a powerful tool that is useful for analysis. Elements can be used to vary just about anything, from something small such as a 1% increase in the cost of a product to a company-wide increase in salaries, the introduction of an entire new product line, or the purchase of a new production machine, among other possibilities.
For example, assume that Clear Lake has completed a first pass at its forecast and is reviewing the forecasted profit for the next 12 months. Managers feel the profit is currently low, as they always want to target a certain percentage. They might tinker with variables in the forecast file to see the impact on profits of potential changes they are considering. They may reduce the new salaries package by a percentage point to see if it gets them closer to their goal. They may adjust cost of goods sold by a certain percentage if they feel they can negotiate with vendors to work down their costs. They may adjust rent and see if they can find a better retail location to either reduce costs or increase sales due to increased foot traffic in a new location. They may save an entirely new version of the forecast and change it drastically to see what investing in opening a second retail location would do.
As you can see, the list of possibilities is endless. Though the main goal of financial managers may be cash planning, the power of a well-developed forecast is tremendous. It can help assess potential growth, new opportunities, and even small changes in the business as well.
Sensitivity Analysis in Forecasting
Sensitivity analysis will often look at the change in just one variable rather than the entire scenario. It examines how sensitive a particular output (commonly net income) will be to a change in a particular underlying input (sales or costs, for example). What if sales are 10% more or less than forecasted? What if the prices the firm can charge its customers are 10% more or less? What if the cost of goods sold increases by 10%? The purpose is to see which variables are crucial to "get right." It isn't worth spending a lot of research dollars to make sure you are accurately predicting a variable if that variable won't notably change the outcome. However, a slight change in other variables may have significant impact.
Using pro forma financial statements created in Excel allows management to quickly generate new pro forma financials and see the impact that each possible variable might have on the overall financial results.
554 18 • Financial Forecasting
18.6
Using Excel to Create the Long-Term Forecast
Learning Outcomes
By the end of this section, you will be able to:
■ Generate a financial statement forecast using spreadsheet tools.
■ Connect the balance sheet and income statement using appropriate formula referencing.
■ Use spreadsheet functions to generate appropriate iterations that balance financial forecasts.
Throughout this chapter, we have seen forecasted financial statements for Clear Lake Sporting Goods along with its forecasted cash flow. These statements could all have been generated by hand, of course, but that wouldn't be an effective use of time. As mentioned in prior sections, several different types of software can be quite effective in making the forecasting process faster and more flexible. In this section, we will review just one common option, Microsoft Excel.
Download the spreadsheet file (https://0penstax.0rg/r/spreadsheet file) containing key Chapter 18 Excel exhibits.
Using the "Sheet"
Creating a budget in Excel can be very simple or extremely complex, depending on the size and complexity of the business and the number of formulas and dependencies that are written into the Excel workbook.
Creating the forecast in Excel follows the same steps and flow we just explored in this chapter but with the power of a software program to do the math for you. We begin with the sales forecast, which uses several key formulas in Excel.
1. First, sales are projected to be 18% higher than the prior year. Thus, a total projection for the year is calculated using a simple link and multiplication function tied to last year's total sales. In Figure 18.19. you can see the formula in cell 04 is "='Figure 18.12'!N4*1.18". This formula simply does the math to increase the prior year's sales by 18%.
2. Next, the sales are distributed by month. In Figure 18.19. we see in cell B5 that the forecasted income statement sheet is linked to the percent of annual sales from the Prior Year Income Statement (Figure 18.12) sheet. Then, in cell B4, January sales are estimated with a formula that multiplies the total forecasted sales in 04 by the percent of annual sales for January of the prior year. Notice that the formula then multiplies that product by 0.98. This is because Clear Lake discontinued a product line in the last quarter of the prior year, and management feels that this will reduce sales in the first quarter of the new year by roughly 2%.
A B 0
1 Clear Lake Sporting Goods
2 Forecasted Monthly Income Statements
3 Jan Next Year
4 Gross Sales =($O4*B5)*0.98 = Figure 18.12'!N4*1.18
5 % of Annual Sales =rFigure 18.12r!FJ5 =04/$04
Figure 18.19 Forecasted Sales Formulas in Excel
3. As Clear Lake continues to fill out its forecasted income statement, the next formula we see is a simple sum formula to calculate net sales in B8 (see Figure 18.20). It's a simple formula that subtracts sales returns and allowances in B7 from gross sales in B4. Similar formulas are also found in B10 for gross margin and B18 for net income.
4. In cell B9, we see a multiplication formula that multiplies sales from B4 by 0.5, or 50%. This is because
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18.6 • Using Excel to Create the Long-Term Forecast 555
management feels that cost of goods sold will remain the same as last year, in most quarters at least, and last year's percentage was 50%.
Rent, depreciation, and salaries are all simply typed in, as they are fixed expenses that remain the same as last year.
The utilities calculation, found in cell B14, is somewhat similar to the sales calculation. The total utilities expense from 014 is multiplied by the current month's sales in B4 divided by the total annual sales in 04. This spreads out the utility cost by month based on the percentage of annual sales.
A B
3
4
5
6
7
e
9
10
11
12
13
14
15
16
17
18
Gross Sales
% of Annual Sales
Jan
=($O4*B5)*0.98
=rFigure 18.12r!B5
% Change from January Baseline
Sales Returns & Allowances Net Sales
Cost of Goods Sold Gross Margin Rent Expense
=B4/$B4 300
=B4-B7
= B8*0.5 =B8-B9
Depreciation Expense
Salaries Expense Utility Expense Operating Income
Interest Expense
Income Tax Expense Net Income
458 300 468
=$Q14*(B4/$Q4) = B10-SUM(B11:B14) 1=2000/12 =$Oj.7*(B15/$Oj.5)
=B15-B16-B17
Figure 18.20 Forecasted Income Statement Formulas
Clear Lake's forecasted balance sheet ties very closely to both the forecasted income statement and the prior year's income statement. In Figure 18.21. we see in C7 an addition formula using the sum of the current month and three months of prior sales as an estimate of the ending accounts receivable balance. The formula for inventory is similar but forward looking. In C8, inventory is estimated by adding the cost of goods sold for the current month and next three months from the forecasted income statement.
Total current assets in C10 is calculated with a SUM formula that adds together the values in all the selected cells. Amounts such as short-term investments and common stock that are not anticipated to change are simply typed as a number in the cell. Much like in the income statement, subtotals are found in C13 for total assets, C17 for current liabilities, C24 for total equity, and C25 for total liabilities and equity. Retained earnings in C23 pulls the ending retained earnings balance from the end of last year (hidden in column B) and adds the
556 18 • Financial Forecasting
net income for January in the forecasted income statement to get the current month's ending balance.
Assets
Noncurrent assets:
Jan
Cash
Accounts Receivable Inventory
Short -Term Investments
42581.4534883721
='Figure lS.^'lBS+'Figure 18.12r!M8+rFigure 18.12'iLS+'Figure 18.12r!K8 ='Figure 18.13'!B9+'Figure 18.13r!C9+'Figure 18.13r!D9+rFigure 18.13'!E9 20000
10
'2
13
14
15
16
17
18
19 2C
r
22
23
Total Current Assets
Noncurrent Assets: Equipment
Total Assets
Current Liabilities: Accounts Payable Unearned Revenue
Total Current Liabilities
Noncurrent Liabilities:
Notes Payable
=SUM(C6:C9)
50000
=C10+C12
'Figure 18.13'!B9
=0.25*'Figure 18.13'!B8 =C15+C16
50000
Total Liabilities Stockholder Equity
Common Stock
=C17+C19
End ng Reta ned Earnings
80000
= B23+'Figure 18.13'!B18
2^
25
Total Stockholder's Equity
Total Liabilities and Stockholder Equity
=C22+C23
=C24+C20
Figure 18.21 Forecasted Balance Sheet Formulas
Much like the balance sheet, the cash forecast also relies heavily on data from the forecasted income statement as well as the forecasted balance sheet. To begin the year, in Figure 18.22. we see that the formula in B4 pulls the cash balance from the forecasted balance sheet. In B6, the formula pulls the sales for the three months prior from the previous year's income statement. This is because it's assumed that cash is collected from customers 90 days after the sale. The same approach is used for accounts payable, rent, salaries, and utilities. The formulas pull the expenses from a prior month depending on the assumed timing for payment. Utilities, for example, are assumed to be paid within 30 days, so the cash outflow in February is assumed to be the utilities expense for January from the forecasted income statement. Note that interest payments are assumed to be zero injanuary and February, but in March, the formula in D14 sums the interest expenses on the forecasted income statement for January, February, and March. This is because interest is paid quarterly.
Finally, note the formula in C4. The beginning cash balance for a given month is the same as the ending cash balance from the prior month; thus, the figure in B18 is linked to C4 to start the new month.
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18.6 • Using Excel to Create the Long-Term Forecast 557
10
12
13
14
15
16
17
18
Jan Feb Mar
Beginning Cash Balance ='Figure 18.21'!C6 =B18 =C18
Cash Inflows:
Accounts Receivable Collected ='Figure 18.12'IKS =rFigure 18.12r!L8 ='Figure 18.12'!M8
Total Cash Inflows: =SUM(B6:B6) =SUM(C6:C6) =SUM(D6:DG)
Cash Outflows:
Accounts Payable paid ='Figure 18.12'!M9 =rFigure 18.13'!B9 ='Figure 18.13'!C9
Salaries ='Figure 18.12'!M13 =rFigurel8.13'!B13 ='Figure 18.13'!C13
Rent ='Figure 18.13'lBll =rFigurel8.13'!Cll ='Figuire 18.13'!D11
Utilities ='Figure 1&.12'!M14 =rFigurel8.13'!B14 ='Figuire 18.13'!C14
Dividends 0 0 10000
Interest Payments 0 0 =SUM(rFigure 18.13r!B
Total Cash Outflows: =SUM(B9:B14) =SUM(C9:C14) =SUM(D9:D14)
Cash surplus or deficit =B7-B15 =C7-C15 =D7-D15
Short term financing borrowed (repaid) 0 0 5000
Ending Cash Balance =B4+B16+B17 =C4+C16+C17 =D4+D16Hn7
Figure 18.22 Cash Forecast Formulas
Using Excel Functions to Balance
Once we get a draft of the forecasts outlined, then the tinkering starts. Additional information can be used to adjust the formulas, as we saw with the 2% reduction in January sales for the forecasted income statement. Because we have linked most (though not all) of our expenses, subtotals, and statements together using formulas, management can also use the forecast workbook to perform scenario and sensitivity analyses, essentially asking "what if?" and looking at the results. When completed, however, before finalizing the forecast, it's important that the financial statements are in balance (particularly the balance sheet, just as the name implies).
Notice that throughout, we used formulas to calculate subtotals to ensure they are correct and change as needed. We also linked figures, such as the ending and beginning cash balances, to ensure they are in balance. Perhaps the easiest but most important thing to do is to ensure that the balance sheet balances. We can do this with a simple formula that compares total assets to total liabilities and equity. We can see in Figure 18.23 that subtracting one from the other in cell C27 should result in $0. If there is a difference, the formula will highlight it, forcing us to investigate and correct the sheet so that it balances.
558 18 • Financial Forecasting
A C
10 Total Current Assets =SUM(C6:C9)
11 Noncurrent Assets:
12 Equipment 50000
13 Total Assets =C10+C12
14 Current Liabilities:
15 Accounts Payable ^FigurelB.lS'lBS
16 Unearned Revenue =0.25*rFigure 18.13r!B8
17 Total Current Liabilities =C15+C16
18 Noncurrent Liabilities:
19 Notes Payable 50000
20 Total Liabilities =C17+C19
21 Stockholder Equity
22 Common Stock 80000
23 End ng Reta ned Earnings = B23+rFigure 18.13r!B18
24 Total Stockholder's Equity =C22+C23
25 Total Liabilities and Stockholder Equity =C24+C20
26
27 Balance =C13-C25
Figure 18.23 Forecasted Balancing Formula
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18 •Summary 559
a Summary
18.1 The Importance of Forecasting
Forecasting financial statements is important to different users for different reasons. In finance, it's most important for assessing the value of future growth plans and planning for future cash flow needs.
18.2 Forecasting Sales
The sales forecast is the foundation on which much of the rest of the forecast is built. Thus, the sales forecast is completed first. Historical sales data and any other information on the firm, its products, the economy, its customers, and its competitors are all used to create the most accurate sales forecast possible.
18.3 Pro Forma Financials
Pro forma financial statements are forward looking in nature. They use the sales forecast, historical data, financial statement analyses, relationships between accounts and statements, and any other information known about the firm, the environment, and the future to create the most accurate financial statement forecast possible.
18.4 Generating the Complete Forecast
Interrelationships among historical data, the forecasted income statement, and the forecasted balance sheet are all used to estimate each line item in the financial statements.
18.5 Forecasting Cash Flow and Assessing the Value of Growth
Once the income statement and balance sheet forecasts are complete, data from those statements, information on company policies, and account relationships are used to generate a cash forecast. The cash forecast is important for identifying any gaps in cash flow so that financial managers can plan for cash needs. It's also important to review not only the cash forecast but all forecasted financial statements to assess the overall impact and value of proposed firm growth.
18.6 Using Excel to Create the Long-Term Forecast
Excel can be a powerful tool for creating financial forecasts. Formulas that complete mathematical functions and tie accounts and financial statements together are used to create the statements, ensure that they balance, and facilitate scenario and sensitivity analyses.
¥ Key Terms
balance sheet a financial statement that reflects a firm's asset, liability, and equity account balances at a given point in time
cash deficit an excess of cash outflows over cash inflows for a given period
cash forecast a financial statement that estimates a firm's future cash inflows and outflows
cash surplus an excess of cash inflows over cash outflows for a given period
common-size describes a financial statement in which each element is expressed as a percentage of a base amount
financing activities cash business transactions reported on the statement of cash flows that reflect the use of financed funds
forecast an estimate of future performance based on historical performance and other contextual information
income statement a financial statement that measures a firm's financial performance over a given period of time
investing activities cash business transactions reported on the statement of cash flows that reflect the acquisition or disposal of long-term assets
560 18- Multiple Choice
operating activities cash business transactions reported on the statement of cash flows that relate to
ongoing day-to-day operations pro forma in the context of financial statements, forward-looking
scenario analysis analysis of how various situations and circumstances would impact the financial forecast sensitivity analysis analysis of the sensitivity of an output variable to a change in an input variable statement of cash flows a financial statement that lists a firm's cash inflows and outflows over a given period of time
statement of stockholders' equity a financial statement that reports the difference between the beginning and ending balances of each of the stockholders' equity accounts during a given period
a
Multiple Choice
1. Which type of financial statement analysis is most commonly used to create a baseline estimate for a financial forecast?
a. Trend analysis
b. Common-size analysis
c. Ratio analysis
d. Liquidity analysis
2. What key element of the income statement is used to estimate several other key income statement lines?
a. Cost of goods sold
b. Gross margin
c. Sales
d. Fixed costs
Jamal wants to forecast sales for the first quarter of next year. His first assumption is that sales will likely grow by 3% in the coming year. Ifjamal's monthly sales were $10,000, $9,000, and $11,000 in the first quarter of this year, what should his sales forecast be for the first quarter of next year?
a. $30,000
b. $30,900
c. $33,000
d. $33,500
4. In the context of a firm's financial statements, what does pro forma mean?
a. Forward looking
b. Historical
c. Board approved
d. Audited
5. What is the most common length of a forecast if the goal is to forecast cash and assess possible short-term growth?
a. 3 months
b. 12 months
c. 3 years
d. 5 years
6. When completing a first pass at a forecasted income statement, which type of costs are assumed to be tied directly to sales?
a. Fixed costs
b. Period costs
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18 • Review Questions 561
c. Variable costs
d. Sunk costs
7. In the cash forecast, if cash inflows exceed cash outflows, what does this create?
a. A cash surplus
b. A cash deficit
c. A long-term liability
d. An undeclared dividend
8. Amelia wants to use a formula in Excel to estimate her utilities expense for each month. She normally pays her utilities within 30 days. What formula or link might she use in Excel to estimate her cash outflow for utilities?
a. Sum the past three months' cost of goods sold from the forecasted income statement
b. Link to the prior month's accounts payable from the forecasted balance sheet
c. Link to the prior month's utilities expense from the forecasted balance sheet
d. Link to the prior month's ending cash balance from the cash flow forecast
9. Amelia wants to use a formula in Excel to estimate her sales for each month. She believes her sales for the next year will be about 7% higher than this year's. She also has a big new ad campaign running late this year that she thinks will add another $5,000 to January sales. Which of the following is an appropriate Excel formula for Amelia's January sales?
a. =(lastyearsalesA2*1.07)+5000
b. =(lastyearsales+5000*1.07)
c. =lastyearsalesA2+5000*.07
d. =lastyearsalesA2*5000*1.07
5. Review Questions
1. Javier's firm has created a forecasted income statement that shows the firm with a net profit of $25,000 for the coming year. What can we assume about Javier's cash flows?
2. Lulu's firm's sales grew by 9%, 11%, and 10% over the past three years, respectively. Lulu wants to take her first pass at forecasting sales for next year. What percent sales growth would you recommend she use, and why?
3. Aria wants to create a set of pro forma financial statements. Her goal is to plan for future cash flows and operations as well as help envision her long-term strategy. What time frames should Aria consider for her operations and cash flows versus her long-term strategy?
4. What information might you use to calculate the ending balance for retained earnings on a forecasted balance sheet?
5. Damon estimates his beginning cash balance forjune to be $10,000, with cash inflows of $4,000 and cash outflows of $6,000 for the month. What is Damon's forecasted ending balance forjune?
6. Tanneh wants to use an Excel formula to help her estimate sales for January in her forecasted income statement. She already has her sales estimate for the full year. Assuming she wants to use the past year's income statement percentages to forecast next year's sales, how would she calculate estimated sales for January?
562 18-Problems
QjJ Problems
1. ABC Company has the following data for its monthly sales. Complete the % of Annual Sales row.
Monthly Income Statements
Jan
Feb
Mar Apr May Jun
Jul Aug Sep
Oct
Nov
Dec
Total
Gross Sales $40,000 $42,000 $47,000 $56,000 $60,000 $71,000 $53,000 $53,000 $46,000 $37,000 $39,000 $31,000 $575,000
% of Annual Sales
2. Using the same data as in Problem 1, assume that ABC Company expects a 10% increase in sales in the coming year (10% more than the $575,000 it had in the past year). Prepare its sales forecast, assuming the company breaks its sales down by month using the same percentages as the actual sales from the past year, which you calculated in the first problem.
Monthly Income Statements
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Total
Gross Sales
% of Annual Sales
ABC Company anticipates its sales being a bit lower than normal in January and February of the coming year due to major road construction on the street where it is located, which will draw away foot traffic from the store. The company anticipates that this will reduce its sales in these two months by 5%. Use the information from Problems 1-2 to update the sales forecast.
ABC Company's cost of goods sold last year was 60%. It anticipates that this will be the same in the coming year. Its sales returns and allowances are small, normally 1% of sales. Use the information from Problems 1-3 to estimate the company's sales returns and allowances, net sales, and cost of goods sold and calculate its gross margin.
Monthly Income Statements
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Total
Gross Sales $41,800 $43,890 $51,700 $61,600 $66,000 $78,100 $58,300 $58,300 $50,600 $40,700 $42,900 $34,100 $627,990
% of Annual Sales 7.0% 7.3% 8.2% 9.7% 10.4% 12.3% 9.2% 9.2% 8.0% 6.4% 6.8% 5.4% 100.0%
Sales Returns & Allowances
Net Sales
Cost of Goods Sold
Gross Margin
Use the partial income statement generated in Problem 4 along with the following additional information to complete ABC Company's forecasted income statement in Excel.
a. Rent expense is $1,000 per month. However, the landlord has indicated that rent will go up to $1,250 in the fourth quarter.
b. Depreciation expense is $2,250 per month and does not change throughout the year.
c. Salaries expense is $1,500 per month and is expected to go up by 10% in the second half of the year, when a new compensation plan will be implemented.
d. Utilities expense is $5,000 for the entire year and should be allocated to each month based on that month's percentage of annual sales. Interest expense is $500 per month.
Income tax is 25% of operating income less interest expense.
e.
f.
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18-Video Activity 563
Monthly Income Statements
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Total
Gross Sales $41,800 $43,890 $51,/00 $61,600 $66,000 S/8,100 $58,300 $58,300 $50,600 $40, mo $42,900 $34,100 $627,990
% of Annual Sales 7.0% 7.3% 8.2% 9.7% 1 0.4% 12.3% 9.2% 9.2% 8.0% 6.4% 6.8% 5.4% 100.0%
Sales Returns & Allowances $ 418 $ 439 $ 517 $ 616 $ 660 $ 781 $ 583 $ 583 $ 506 $ 407 $ 429 $ 341 $ 6,280
Net Sales 41,382 43,451 51,183 60,984 65,340 77,319 57,717 57,717 50,094 40,293 42,471 33,759 621,710
Cost of Goods Sold 24,829 26,071 30,710 36,590 39,204 46,391 34,630 34,630 30,056 24,176 25,483 20,255 373,026
Gross Margin 16,553 17,380 20,473 24,394 26,136 30,928 23,087 23,087 20,038 16,117 16,988 13,504 248,684
Rent Expense
Depreciation Expense
Salaries Expense
Utility Expense
Operating Income
Interest Expense
Income Tax Expense
Net Income
[►J Video Activity
What Is a Pro Forma?
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1. What is a pro forma financial statement? What are some scenarios in which you might find a pro forma financial statement helpful?
2. Why might someone compile a pro forma financial statement that is intentionally inaccurate? What factors contribute to the accuracy of a pro forma?
Cash Flow Forecasting Explained: How to Complete a Cash Flow Forecast Example
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3. Assume you are the financial manager for a large electronics retailer. What benefits could you gain from preparing a cash forecast?
4. Assume you are the financial manager for a large electronics retailer. You are going to prepare a cash forecast. What key cash inflows and outflows do you anticipate will be in your forecast?
564 18-Video Activity
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The Importance of Trade Credit and Working Capital in Planning
Figure 19.1 Working capital describes the resources that are needed to meet the daily, weekly, and monthly operating cash flow needs, (credit: modification of "Sealey Power Products Warehouse" by Mark Hunter/flickr, CC BY 2.0)
Chapter Outline
19.1 What Is Working Capital?
19.2 What Is Trade Credit?
19.3 Cash Management
19.4 Receivables Management
19.5 Inventory Management
19.6 Using Excel to Create the Short-Term Plan
LzlJ Why It Matters
During the COVID-19 pandemic, many families and small businesses realized the importance of financial resiliency. In personal finance, financial resiliency is the ability to overcome financial difficulties such as sudden job loss or significant unexpected expenses—to spring back quickly.
To help promote resiliency, personal financial planners advise clients to maintain liquid assets equal to three to six months of living expenses, keep debt levels low, manage the household budget, keep insurance in force (health, property, and life), establish a solid credit history, and make wise use of credit cards and home equity lines of credit.
In business finance, financial resiliency is not important only during pandemics but is important through the ups and downs of seasonal cycles and economic downturns. Managing cash, accounts receivable, and inventory while making optimal use of trade credit (accounts payable) makes for a business that meets its operating needs and pays its debts when due.
Working capital management is also critical during good times. Even though profits might be rising, a business with growing demand for its products and services still needs to have working capital management tools to pay its bills. Growth in sales and profits do not immediately mean sufficient cash flow, so planning ahead with tools such as a cash budget is key.
566 19 • The Importance of Trade Credit and Working Capital in Planning
19.1
What Is Working Capital?
By the end of this section, you will be able to:
■ Define working capital.
■ Calculate a firm's operating cycle and cash cycle.
■ Compute inventory days, accounts receivable days, and accounts payable days.
The concept of business capital is often associated with the cash and assets (such as land and equipment) that the owners contributed to the business. Early political economists like Adam Smith and Karl Marx identified this concept of capital, along with labor and entrepreneurship, to be the factors of production.
That general idea of capital is important and critical to a company's productive capacity. This chapter is about a specific type of capital— working capital—that is just as important as long-term capital. Working capital describes the resources that are needed to meet the daily, weekly, and monthly operating cash flow needs. Employees are paid out of working capital as well as cash from operations, the fulfillment of merchandise orders is possible because of working capital, and the liquidity of a company hinges upon how well management plans and controls working capital.
Understanding working capital begins with the concept of current assets—those resources of a business that are cash, near cash, or expected to be turned into cash within a year through the normal operations of the business. Current assets are necessary for the everyday operation of the firm, and they are synonymous with term gross working capital.
Cash is needed to pay the bills and meet the payroll. Excess cash is invested in cash alternatives such as marketable securities, creating liquidity that can be tapped when operating cash flow needs exceed the amount of cash on hand (checking account balances). Investment in inventory is necessary to meet the demand for products (sales), and if the firm extends credit to its customers so that a sale can be made, the balance sheet will also show accounts receivable—a very common current asset that derives its value from the probability that customers will pay their bills.
Working capital is often spoken about in two versions: gross working capital and net working capital. As was previously stated, gross working capital is equivalent to current assets, particularly those that are cash, cashlike, or will be converted to cash within a short period of time (i.e., in less than one year).
Net working capital (NWC) is a more refined concept of working capital. It is best understood by examining its formula:
Current Assets - Current Liabilities = Net Working Capital Goal of Working Capital Management
The goal of working capital management is to maintain adequate working capital to
■ meet the operational needs of the company;
■ satisfy obligations (current liabilities) as they come due; and
■ maintain an optimal level of current assets such as cash (provides no return), accounts receivable, and inventory.
Working capital management encompasses all decisions involving a company's current assets and current liabilities. One very important aspect of working capital management is to provide enough cash to satisfy both maturing short-term obligations and operational expenditures—keeping the company sufficiently liquid.
In summary, working capital management helps a company run smoothly and mitigates the risk of illiquidity. Well-run companies make effective use of current liabilities to finance an optimal level of current assets and maintain sufficient cash balances to meet short-term operating goals and to satisfy short-term obligations. Working capital management is accomplished through
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19.1 • What Is Working Capital? 567
■ cash management;
■ credit and receivables management;
■ inventory management; and
■ accounts payable management.
Components of Working Capital Management
In contrast to net working capital, gross working capital is synonymous with current assets, particularly those current assets that are either cash or cash equivalents or that will be converted to cash within a short period of time (i.e., in less than one year).
Below is a list of the components of gross working capital.
■ Cash and cash equivalents
■ Marketable securities
■ Accounts receivable
■ Inventory
Here is an example. On December 31, a company has the following balances and gross working capital:
Cash and Equivalents $ 100 000
Marketable Securities 250 000
Accounts Receivable 330 000
Inventory 425 000
Gross Working Capital $1,105 000
Think of the $1,105,000 of gross working capital as a source of funds for the most pressing obligations (i.e., current liabilities) of the company. Gross working capital is available to pay the bills. However, some of the current assets would need to be converted to cash first. Accounts receivable need to be collected, and inventory would need to be sold before it too can become cash. What if the company had $600,000 of current liabilities? That amount of current obligations could not be paid out of cash until the marketable securities were sold and a significant portion of accounts receivable were collected.
The second, more refined and useful concept of working capital is net working capital:
NWC = Current Assets - Current Liabilities
For example, if a company has $1,000,000 of current assets and $750,000 of current liabilities, its net working capital would be $250,000 ($1,000,000 less $750,000).
NWC provides a better picture because it takes into account the liability "coverage" provided by the current assets. As the above example shows, the current assets would "cover" the current liabilities with an excess of $250,000. Think of it this way: if the current assets could be converted to cash, they could be used to meet the current obligations with another $250,000 of cash leftover.
Current liabilities include
■ accounts payable;
■ dividends payable;
■ notes payable (due within a year);
■ current portion of deferred revenue;
■ current maturities of long-term debt;
■ interest payable;
■ income taxes payable; and
■ accrued expenses such as compensation owed to employees.
Net working capital possibilities can be thought of as a spectrum from negative working capital to positive, as
568 19 • The Importance of Trade Credit and Working Capital in Planning
explained in Table 19.1.
Negative Networking Capital
Current liabilities are greater than current assets.
Could indicate a liquidity problem. There is difficulty satisfying current obligations.
Zero Net Working Capital Positive Net Working Capital
Current assets equal current liabilities.
Indicates that current assets could cover current obligations. However, there is no positive margin (safety cushion) or "liquid reserve" to satisfy unexpected cash needs.
Current assets are greater than current liabilities.
Indicates that the company can meet its current obligations. However, excessively high net working capital could mean too little cash and therefore an opportunity cost (forgoing rates of return on alternative investment).
Table 19.1 Spectrum of Net Working Capital
Measures of Financial Health provides information on a variety of financial ratios to help users of financial statements understand the strengths and weakness of companies' financial statements. Three of the financial ratios covered in that chapter are brought back into this chapter's discussion to demonstrate how financial managers examine working capital and liquidity. Liquidity is the ease with which an asset can be converted into cash. Those ratios are the current ratio, the quick ratio, and the cash ratio. A higher ratio indicates a greater level of liquidity.
The formulas for the three liquidity ratios are:
Current Ratio Quick Ratio (Acid Test) Cash Ratio
_ Current Assets
Current Liabilities _ Current Assets - Inventory
— Current Liabilities
_ (Cash + Marketable Securities)
— Current Liabilities
Notice how the current ratio includes the two elements of net working capital—current assets and current liabilities. It makes for a quick comparison of relative size or proportion.
Current Ratio
A company has $2,000,000 of current assets, while its current liabilities are $1,000,000. What is the current ratio, and what does it mean?
Solution:
The current ratio would be 2 = $2,000,000/$1,000,000, which is a 2:1 proportion of current asset value to the amount of the current liabilities. This means that if all the current assets could be converted to cash, then the current liabilities could be satisfied two times.
There are two drawbacks to the current ratio: (1) it is a working capital analytic as of a point in time but is not indicative of future liquidity or future cash flows and (2) as an indicator of liquidity, it can be deceptive if a significant proportion of the current assets are inventory, supplies, or prepaid expenses. Inventory is not very liquid as it can take an extended time period to convert to cash, and assets such as supplies and prepaid expenses never become cash and therefore are not a source of funds to pay bills.
The quick ratio is considered a more conservative indication of liquidity since it does not include a firm's
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19.1 • What Is Working Capital? 569
inventory: (Current Assets - Inventory)/Current Liabilities. THINK IT THROUG
Quick Ratio
A company's current assets total $2,000,000, but $500,000 of that is inventory and the current liabilities total $1,000,000. What is the quick ratio, and what does it mean?
Solution:
The quick ratio would be 1.5 = $1,500,500/$1,000,000 and would indicate a smaller cushion of net working capital.
Cash Ratio
The cash ratio is even more conservative in that it presents a picture of liquidity by excluding all current assets except cash and marketable securities.
A company's total current assets are $2,000,000, but only $1,100,000 of the current assets consist of cash and marketable securities. Assuming $1,000,000 of current liabilities, what would be the cash ratio and what does it mean?
Solution:
The cash ratio would be 1.1 = $1,100,000/$1,000,000, and the amount of cash is enough to pay the current bills by $100,000.
Working capital ratios, like any financial ratio, are most valuable when examined in light of trends and in comparison to industry/peer averages. For example, a deteriorating current ratio over several quarters (a decline in the company's current ratio) could indicate a reduced ability to pay bills.
Working capital ratios are also compared to industry averages, which are available in databases produced by such financial publishers as Dun & Bradstreet, Dow Jones Company, and the Risk Management Association (RMA). These information services are available via subscriptions and through many libraries. For example, if a company's current ratio is 0.9 while the industry average is 2.0, then the company is less liquid than the average company in its industry and strategies, and techniques need to be considered to change things and to better compete with peer groups. Industry averages can be aspirational, motivating management to set liquidity goals and best practices for working capital management.
It is common to think about working capital with a simple assumption: current assets are being "financed" by current liabilities. However, such an assumption may be an oversimplification. Some level of current assets is often necessary to meet longer-term obligations, and in that way, you could think of some amount of current assets as a permanent based of working capital that may need to be financed with longer-term sources of capital.
Think of a company with seasonal business. During busy times, more working capital will be needed than during certain other portions of the year, such as less busy times. But there will always be some level—a permanent base—of working capital needed. Think of it this way: the total working capital of many companies will ebb and flow depending on many variables such as the operating cycle, production needs, and the growth of revenue. Therefore, working capital can be thought of as having a permanent base that is always needed
570 19 • The Importance of Trade Credit and Working Capital in Planning
and a total working capital amount that increases when activity levels (i.e., production and sales volume) are higher (see Figure 19.2).
The Cash Cycle
The cash cycle, also called the cash conversion cycle, is the time period between when a business begins production and acquires resources from its suppliers (for example, acquisition of materials and other forms of inventory) and when it receives cash from its customers. This is offset by the time it takes to pay suppliers (called the payables deferral period).
More Working Capital Needed for High Activity
Total Working Capital
Time
| Temporary Working Capital
| Permanent Working Capital Figure 19.2 Working Capital Needs Can Vary: Temporary and Permanent Working Capital
The cash cycle is measured in days, and it is best understood by examining its formula:
Cash Cycle = Inventory Conversion Period +
Receivables Collection Period - Payables Deferral Period
The inventory conversion period is also called the days of inventory. It is the time (days) it takes to convert inventory to sales and is calculated by following these steps:
1. First, calculate the Inventory Turnover Ratio using this formula:
_, _, . Cost of Goods Sold
Inventory Turnover Ratio = —---
Average Inventory
The Average Inventory is arrived at as follows:
(Beginning Inventory + Ending Inventory) Average Inventory =---
2. Then, use the Inventory Turnover Ratio to calculate the Inventory Conversion Period:
365
Inventory Conversion Period =--—-
Inventory Turnover
The receivables collection period, also called the days sales outstanding (DSO) or the average collection period, is the number of days it typically takes to collect cash from a credit sale. It is calculated by following these steps:
1. First, calculate the Accounts Receivable Turnover using this formula:
„ . ,, m Credit Sales
Accounts Receivable Turnover =
Average Accounts Receivable
The Average Accounts Receivable is arrived at as follows:
„ . ,, (Beginning Accounts Receivable + Ending Accounts Receivable) Average Accounts Receivable =---
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19.1 • What Is Working Capital? 571
2. Then, use the Accounts Receivable Turnover to calculate the Receivables Collection Period:
Receivables Collection Period =---r—n-
Accounts Receivable Turnover
The payables deferral period, also known as days in payables, is the average number of days its takes for a company to pay its suppliers. It is calculated by following these steps:
1. First, calculate the Accounts Payable Turnover using this formula:
„ ,, m Cost of Goods Sold
Accounts Payable Turnover =
Average Accounts Payable
The Average Accounts Payable is arrived at as follows:
,, (Beginning Accounts Payable + Ending Accounts Payable) Average Accounts Payable =---
Then, use the Accounts Payable Turnover to calculate the Payables Deferral Period:
365
Payables Deferral Period =--—-
Accounts Payable Turnover
Periods of the Cash Cycle
Scenario 1: King Sized Products (KSP) Inc. has annual credit sales of $40,000,000. The average inventory is $3,000,000, and the company has average accounts receivable of $6,000,000 and average accounts payable of $2,800,000. The cost of goods sold for KSP Inc. is $30,000,000. The cash cycle for the company is 57.2 days. Calculate the inventory conversion, receivables collection, and payable deferral periods.
Solution:
Inventory conversion period:
■ Inventory Turnover Ratio = ^SooSS = 10
■ Inventory Conversion Period = ^ = 36.5 days
Receivables collection period:
■ Accounts Receivable Turnover = ^'f^'f^ = 6.67 times
■ Receivables Collection Period = = 54.7 days
Payables deferral period:
■ Accounts Payable Turnover = gfff^ff = 10.71 times
■ Payables Deferral Period = y|p^- = 34 days
Cash Conversion Cycle = 36.5 days + 54.7 days - 34 days = 57.2 days The solution (the entire cash conversion cycle) is also illustrated in a chart. Figure 19.3.
572 19 • The Importance of Trade Credit and Working Capital in Planning
Inventory
Conversion
Cycle
Receivable Conversion Cycle
Payable Deferral Cycle
Cash
Conversion Cycle
Acquire the Inventory
Pay the Sell the Trade Credit Inventory
Collect the Receivable
36.5 Days
34 Days
54.7 Days
57.2 Days Figure 19.3 Cash Conversion Cycle
Shortening the inventory conversion period and the receivables collection period or lengthening the payables deferral period shortens the cash conversion cycle. Financial managers monitor and analyze each component of the cash conversion cycle. Ideally, a company's management should minimize the number of days it takes to convert inventory to cash while maximizing the amount of time it takes to pay suppliers.
Quickly converting inventory to sales speeds up cash inflows and shortens the cash cycle, but it also could help reduce inventory losses as a result of obsolescence. Inventory becomes obsolete because of a variety of factors including time—inventory that has not been sold for a long period of time and is not expected to be sold in the future has to be written down or written off according to accounting rules. Write-offs of inventory can result in significant losses for a company. In the food business, inventory conversion periods take on great importance because of spoilage of perishable goods; in retailing, seasonal items lose value the longer they stay on the shelves.
Various inventory management techniques are used to shorten production time in manufacturing, and in retailing, strategies are used to reduce the amount of time a product sits on the shelf or is stored in the warehouse. Production techniques such as just-in-time inventory systems and marketing and pricing strategies can have an impact on the number of days in the inventory conversion cycle.
For the receivables collection period, a relatively long receivables collection period means that the company is having trouble collecting cash from its customers and so whatever can be done to speed up collections while still offering competitive credit terms should be pursued by financial managers. For example, companies that converted paper invoicing to e-invoicing most likely reduce the average collection period by some number of days, as it makes sense that if a bill is transmitted electronically, lag time is cut (no delays because of "snail mail") and collections (payments back to the company from customers) may happen sooner. Other credit management techniques, some of which are explained in subsequent sections, can help minimize and control the receivables collection period.
The payables deferral period is the one element that probably cannot be optimized without violating credit terms. Certainly, cash balances can be conserved by delaying payments to vendors for as long as possible; however, payments on trade credit need to be made on time or the company's relationship with the supplier can suffer. In a worst-case scenario, the company's credit rating could also deteriorate.
A credit rating, also called a credit score, is a measure produced by an independent agency indicating the
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19.1 • What Is Working Capital? 573
likelihood that a company will meet its financial obligations as they come due; it is an indication of the company's ability to pay its creditors. Three business credit rating services are Equifax Small Business, Experian Business, and Dun & Bradstreet.
JK IT THROUGH
The Cash Conversion Cycle
Considering the previous Think It Through (Scenario 1), what if you could reduce inventory levels, hold lower accounts receivable balances, and rely more heavily on accounts payable while maintaining the same sales level?
Here's Scenario 2. Because of better inventory management, credit and collections management, and negotiation of longer payment periods with vendors. King Sized Products (KSP) Inc. needs less investment in inventory and accounts receivable and is able to utilize a greater amount of trade credit financing.
Annual credit sales are $40,000,000, average inventory is $2,800,000, average accounts receivable are $5,500,000, average accounts payable are $3,300,000, and cost of goods sold is $30,000,000. What is the cash conversion cycle?
Solution:
Inventory conversion period:
■ Inventory Turnover Ratio = $$32°^0ff0° = 10.71 times
■ Inventory Conversion Period = j^fff = 34.07 days
■ Inventory Conversion Period = -^p^ = 34.07 days
Receivables collection period:
■ Accounts Receivable Turnover = $55^|^ = 7.27 times
■ Receivables Collection Period = ip^j = 50.21 days
Payables deferral period:
irnrtvpr — _
$3,300,000
Accounts Payable Turnover = $,30'°°°'°y = 9.09 times
■ Payables Deferral Period = |f§ = 40.15 days
Cash Conversion Cycle = 34.07 days + 50.21 days - 40.15 days = 44.13 days
Notice that the investment in inventory and accounts receivable is less and the average accounts payable is more with no change in credit sales and cost of goods sold—you would certainly anticipate a reduction in the cash conversion cycle. The improvement would be about 13 days (from 57.2 in Scenario 1 to 44.1 days in Scenario 2). Figure 19.4 shows a bar chart comparison of the two scenarios.
Scenario 1 Scenario 2
Inventory Conversion Period 36.50 34.07
Receivables Collection Period 54.75 50.21
Payables Deferral Period 34.07 40.15
Cash Conversion Cycle 57.18 44.10
Table 19.2
574 19 • The Importance of Trade Credit and Working Capital in Planning
Comparison of Cash Conversion Cycles
Scenario 1 Scenario 2
Inventory Receivables Payables Cash Conversion Collection Deferral Conversion
Period Period Period Cycle
Figure 19.4 Scenario 1 and 2 Comparison: Shortening the Cash Conversion Cycle
LINK TO LEARNIN
A Harvard Business School blog post. How Amazon Survived the Dot-Com Bubble (https://0penstax.0rg/r/ how-amazon-survived). discusses how Amazon managed its cash conversion cycle to the point where it was receiving payment for the things it sold before Amazon had to pay for them. In that way, Amazon had a negative cash conversion cycle (which is really a huge positive for a company trying to manage positive cash flow!).
Working Capital Needs by Industry
When comparing working capital needs by industry, you can see some variation. For example, some companies in the grocery business can have very low cash conversion cycles, while construction companies can have very high cash conversion cycles. And some companies, like those in the restaurant business, can have very low numbers and even have negative cash conversion cycles.
Working capital can also differ from one industry to another. An often cited general rule is that a current ratio of 2 is considered optimal. However, general rules of thumb must be treated with caution. A better benchmarking approach is to compare a firm's ratios—current ratio and quick ratios—to the average of the industry in which the subject company operates.
Take, for example, a home construction company. Such as firm has a long operating cycle because of the production process (building homes), and the "storage of finished goods" can result in very high current ratios—such as 11 or 12 times current liabilities—whereas a retailer like Walmart or Target would have much lower current ratios.
In recent years, Walmart Stores Inc. (NYSE: WMT) has had a current ratio of around 0.9 and has been able to manage its working capital needs by efficient management of its supply chain, quick turnover of inventory, and a very small investment in accounts receivables.1 Big retailers like Walmart are effective at negotiating favorable payment terms with their vendors. The ability to generate consistent positive cash flow from operations allows a retailer like Walmart to operate with relatively low amounts of working capital.
The credit policies of a company also affect working capital. A company with a liberal credit policy will require a greater amount of working capital, as collection periods of accounts receivable are longer and therefore tie up
1 Walmart Inc. "2020 Annual Report." 2020. https://corporate.walmart.com/media-library/document/2020-walmart-annual-report/_proxyDocument?id=00000171-a3ea-dfc0-af71-b3fea8490000
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19.2-What Is Trade Credit? 575
more dollars in receivables.
Almost all businesses will have times when additional working capital is needed to pay bills, meet the payroll (salaries and wages), and plan for accrued expenses. The wait for the cash to flow into the company's treasury from the collection of receivables and cash sales can be longer during tough times.
During the COVID-19 pandemic, the US government made paycheck protection program (PPP) loans available to help alleviate working capital problems for small and large business when the economy slowed because of shutdowns and social distancing. And although 60 percent of the PPP loan proceeds were to go to cover payroll-related costs, 40 percent could be used to bolster working capital to meet rent, utilities costs, and some interest expense while companies were "treading water"—waiting for positive cash flow to pick up under a recovery.2
It isn't just during downturns that working capital is strained. Growing companies, even if they are extremely profitable, need additional working capital as they ramp up operations by acquiring raw materials, component parts, supplies, or other forms of inventory; hiring temporary or additional employees; and taking on new projects. Whenever additional resources are needed, working capital is also needed.
Some of the current assets and expenditures needed in a growing company may need to be financed from sources that are not spontaneous financing—trade credit (accounts payable). Such forms of external financing such as lines of credit, short-term bank loans, inventory-based loans (also called floor planning), and the factoring of accounts receivables might have to be relied upon.
19.2
What Is Trade Credit?
By the end of this section, you will be able to:
■ Compute the cost of trade credit.
■ Define cash discount.
■ Define discount period.
■ Define credit period.
Trade credit, also known as accounts payable, is a critical part of a business's working capital management strategy. Trade credit is granted by vendors to creditworthy companies when those companies purchase materials, inventory, and services.
A company's purchasing system is usually integrated with other functions such production planning and sales forecasting. Purchasing managers search for and evaluate vendors, negotiate order quantities, and prepare purchase orders. In carrying out the purchasing process, credit terms are granted by the company's vendors and purchases of inventory and services can be made on trade credit accounts—allowing the purchaser time to pay. The purchaser carries an accounts payable balance until the account is paid.
Trade credit is referred to as spontaneous financing, as it occurs spontaneously with the gearing up of operations and the additional investment in current assets. Think of it this way: If sales are increasing, so too is production. Increased sales mean more current assets (accounts receivable and inventory), and increased sales mean increases in accounts payable (financing happening spontaneously with increased sales and inventory purchases). Compared to other financing arrangements, such as lines of credit and bank loans, trade credit is convenient, simple, and easy to use.
Once a company is approved for trade credit, there is no paperwork or contracts to sign, as is the case with various forms of bank financing. Invoices specify the credit terms, and there is usually no interest expense associated with trade credit. Accounts payable is a type of obligation that is interest-free and is distinguished from debt obligations, such as notes payable, that require the creditor to pay back principal and interest.
2 US Small Business Administration. "PPP Loan Forgiveness." n.d. https://www.sba.gov/funding-programs/loans/covid-19-relief-options/paycheck-protection-program/ppp-loan-forgiveness
576 19 • The Importance of Trade Credit and Working Capital in Planning
How Trade Credit Works
Trade credit is common in B2B (business to business) transactions and is analogous to consumer spending using a credit card. With a credit card, a consumer opens an account with a credit limit. Most trade credit is offered to a company with an open account that has a credit limit up to which the company can purchase goods or services without having to pay the cash up front. As long as the payments are made in accordance with the terms of the agreement (also called credit terms), no interest or additional fees are charged on the credit balance except possibly for a fee for late payment.
Initially, the vendor's credit department approves both a trade credit limit and credit payment terms (i.e., number of days after the invoice date that payment is due). Timely payments on accounts payable (trade credit) helps create a credit history for the purchasing firm.
Trade Credit Terms
Trade credit arrangements often carry credit terms that offer an incentive, called a discount, for a company (the buyer) to pay its bill within a relatively short period of time. Net terms, also referred to as the full credit period, are the number of days that a business (purchaser) has before they must pay their invoice. A common net term is Net 30, with payment due in full within 30 days of the invoice.
Many vendors also offer cash discounts to customers that pay their bill early. A company's invoice that specifies payment terms of "2/10 n/30" (stated as: "two ten net 30") would allow a 2 percent discount if the buyer's account balance is paid within 10 days of the invoice date; otherwise, the net amount owed would be due in 30 days. The "10 days" in the example is the discount period—the number of days the buyer has to take advantage of the cash discount for an early payment, also known as quick payment.
For example, Jackson's Premium Jams Inc. received a $10,500 invoice for the purchase of jelly jars. The invoice has payment terms of 2/10 n/30. Jackson's pays the bill within 10 days of the invoice date. Jackson's payment would be $10,290 = ($10,500 X (100% - 2%)). The effect of taking a discount because of a quick payment is a lowering of the cost of inventory in the case of purchases of materials (for a manufacturer), merchandise (for a retailer or wholesaler), and operating expenses (for any company that "buys" services using trade credit). In Cost of Trade Credit, there is an example that shows the high annualize opportunity cost (36.73 percent) of not taking advantage of cash discounts.
CONCEPTS IN PRACTICE
Trade Credit of International Trade
When international trade occurs, two important documents are commonly required: a letter of credit and a bill of lading. A letter of credit is issued by a financial institution on behalf of the foreign buyer (importer). The bill of lading is a legal document that gives proof of a contract between a transportation company and the buyer and is one important piece of documentation that allows the buyer to draw on the letter of credit. A bill of lading serves as a document of title and proof of receipt of goods by the shipper.
The letter of credit secures a promise of payment to the seller (exporter) provided that the terms of the sale are met. For an international trade transaction, the letter of credit is the main mechanism that establishes a liability for the buyer. Instead of a trade payable, the buyer uses a line of credit from a bank.
Cost of Trade Credit
Trade credit is often referred to as a no-cost type of financing. Unlike with other credit arrangements (e.g., bank loans, lines of credit, and commercial paper), there is usually no interest expense associated with trade credit, and as long as your account does not become delinquent, there are no special fees. Some accounts payable arrangements specify an interest penalty or a late fee when the account goes delinquent, but as long
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19.3 • Cash Management 577
as payments are made on time, trade credit is thought of as a low-cost source of working capital.
However, there is one possible cost associated with trade credit for companies that don't take advantage of cash discounts when offered by sellers. Using accounts payable to purchase goods and services can involve an opportunity cost—a cost of the forgone opportunity of making a quick payment and benefiting from a cash discount. A business that does not take advantage of a cash discount for early payment of trade credit will pay more for goods and services than a business that routinely takes advantage of discounts.
The annual percentage rate of forgoing quick payment discounts can be estimated with the following formula:
AT^„ „ . _ . , „ ^. 360 Discount APR of Forgoing Quick Payment Discounts = — -—- . x
Full Credit Period - Discount Period 100 - Discount %
Example: Novelty Accessories Inc. (NAI) purchases products from a vendor that offers credit payment terms of 2/10, net 30. The annual cost to NAI of not taking advantage of the discount for quick payment is 36.73 percent.
360 2% 360 2%
APR of Quick Payment Discounts = —-— x ———— = —— X ——- = 36.73%
v y 30 -10 100% - 2% 20 nocf
19.3
Cash Management
By the end of this section, you will be able to:
■ Explain why firms hold cash.
■ List instruments available to a financial manager for investing cash balances.
Cash management means efficiently collecting cash from customers and managing cash outflows. To manage cash, the cash budget—a forward-looking document—is an important planning tool. To understand cash management, you must first understand what is meant by cash holdings and the motivations (reasons) for holding cash. A cash budget example is covered in Using Excel to Create the Short-Term Plan.
Cash Holdings
The cash holdings of a company are more than the currency and coins in the cash registers or the treasury vault. Cash includes currency and coins, but usually those amounts are insignificant compared to the cash holdings of checks to be deposited in the company's bank account and the balances in the company's checking accounts.
Motivations for Holding Cash
The initial answer to the question of why companies hold cash is pretty obvious: because cash is how we pay the bills—it is the medium of exchange. The transactional motive of holding cash means that checks and electronic funds transfers are necessary to meet the payroll (pay the employees), pay the vendors, satisfy creditors (principal and interest payments on loans), and reward stockholders with dividend payments. Cash for transaction is one reason to hold cash, but there is another reason—one that stems from uncertainty and the precautions you might take to be ready for the unexpected.
Just as you keep cash balances in your checking and savings accounts and even a few dollars in your wallet or purse for unexpected expenditures, cash balances are also necessary for a business to provide for unexpected events. Emergencies might require a company to write a check for repairs, for an unexpected breakdown of equipment, or for hiring temporary workers. This motive of holding cash is called the precautionary motive.
Some companies maintain a certain amount of cash instead of investing it in marketable securities or in upgrades or expansion of operations. This is called the speculative motive. Companies that want to quickly take advantage of unexpected opportunities want to be quick to purchase assets or to acquire a business, and a certain amount of cash or quick access to cash is necessary to jump on an opportunity.
578 19 • The Importance of Trade Credit and Working Capital in Planning
Sometimes cash balances may be required by a bank with which a company conducts significant business. These balances are called compensating balances and are typically a minimum amount to be maintained in the company's checking account.
For example. Jack's Outback Restaurant Group borrowed $500,000 from First National Bank and Trust. As part of the loan agreement. First National Bank required Jack's to keep at least $50,000 in its company checking account as a way of compensating the bank for other corporate services it provides to Jack's Outback Restaurant Group.
Cash Alternatives
Cash that a company has that is in excess of projected financial needs is often invested in short-term investments, also known as cash equivalents (cash alternatives). The reason for this is that cash does not earn a rate of return; therefore, too much idle cash can affect the profitability of a business.
Table 19.3 shows a list of typical investment vehicles used by corporations to earn interest on excess cash. Financial managers search for opportunities that are safe and highly liquid and that will provide a positive rate of return. Cash alternatives, because of their short-term maturities, have low interest rate risk (the risk that an investment's value will decrease because of changes in market interest rates). In that way, prudent investment of excess cash follows the risk/return trade-off; in order to achieve safe returns, the returns will be lower than the possible returns achieved with risky investments. Cash alternative investments are not committed to the stock market.
Security Description
US Treasury bills Obligations of the US government with maturities of 3 and 6 months
Federal agency securities
Certificates of deposit
Commercial paper
Obligations of federal government agencies such as the Federal Home Loan Bank and the Federal National Mortgage Association
Issued by banks, a type of savings deposit that pays interest
Short-term promissory notes issued by large corporations with maturities ranging from a few days to a maximum of 270 days
Table 19.3 Typical Cash Equivalents
Figure 19.5 shows a note within the 2021 Annual Report (Form 10-K) of Target Corporation. The note discloses the amount of Target's cash and cash equivalent balances of $8,511,000,000 for January 30, 2021, and $2,577,000,000 for February 1, 2020.
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19.4 • Receivables Management 579
8. Cash and Cash Equivalents
Cash equivalents include highly liquid investments with an original maturity of three months or less from the time of purchase. Cash equivalents also include amounts due from third-party financial institutions for credit and debit card transactions. These receivables typically settle in five days or less.
Cash and Cash Equivalents
(millions)
January 30, 2021
February 1, 2020
Cash $ 307 $ 326
Short-term investments 7,644 1,810 Receivables from third-party financial institutions for credit
and debit card transactions 560 441
Cash and cash equivalents (a)
$8,511
$2,577
(a) We have access to these funds without any significant restrictions, taxes or penalties.
Figure 19.5 Note from Target Corporation 2021 10-K Filing (source: US Securities and Exchange Commission/EDGAR)
In that note, which is a supplement to the company's balance sheet, receivables from third-party financial institutions is also considered a cash equivalent. That is because purchases by Target's customers who use their credit cards (e.g., VISA or MasterCard) create very short-term receivables—amounts that Target is waiting to collect but are very close to a cash sale. So instead of being reported as accounts receivable—a line item on the Target balance sheet that is separate from cash and cash equivalents—these amounts receivable from third-party financial institutions are considered part of the cash and cash equivalents and are a very liquid asst. For example, the amount of $560,000,000 for January 30, 2021, is considered a cash equivalent since the settlement of these accounts will happen in a day or two with cash deposited in Target's bank accounts. When a retailer sells product and accepts a credit card such as VISA, MasterCard, or American Express, the cash collection happens very soon after the credit card sale—typically within 24 to 72 hours.3
Companies also invest excess funds in marketable securities. These are debt and equity investments such as corporate and government bonds, preferred stock, and common stock of other entities that can be readily sold on a stock or bond exchange. Ford Motor Company has this definition of marketable securities in its 2019 Annual Report (Form 10-K):
"Investments in securities with a maturity date greater than three months at the date of purchase and other securities for which there is more than an insignificant risk of change in value due to interest rate, quoted price, or penalty on withdrawal are classified as Marketable securities."
19.4
Receivables Management
By the end of this section, you will be able to:
■ Discuss how decisions on extending credit are made.
■ Explain how to monitor accounts receivables.
For any business that sells goods or services on credit, effective accounts receivable management is critical for cash flow and profitability planning and for the long-term viability of the company. Receivables management begins before the sale is made when a number of factors must be considered.
Can the customer be approved for a credit sale?
3 Creditcardprocessing.com. "How Long Does it Take for a Merchant to Receive Funds?" n.d. https://www.creditcardprocessing.com/resource/article/long-take-merchant-receive-funds/#:~:text=The%20time%20that%20it%20takes,days%20to%20process%20the%20payment
4 Ford Motor Company. "2019 Annual Report." n.d. https://s23.q4cdn.com/725981074/files/doc_downloads/Ford-2019-Printed-Annual-Report.pdf
580 19 • The Importance of Trade Credit and Working Capital in Planning
■ If the credit is approved, what will be the credit terms (i.e., how long do we give customers to pay their bills)?
■ Will there be a cash discount for quick payment?
■ How much credit should be extended to each customer (credit limit)?
Accounts receivable is not about accepting credit cards. Credit card sales are not technically accounts receivable. When a credit card is accepted, it means that the credit card company (e.g., VISA, MasterCard, or American Express) will guarantee the payment. The cash will be deposited in the merchant's bank account in a very short period of time.
When a business makes a sale on account, management (e.g., a credit manager or analyst) does its best to distinguish between customers who have a high likelihood of paying and customers who have a low likelihood. Customers with low credit risk are approved; the decision is based on an effective analysis of creditworthiness.
Creditworthiness is judged by looking at a number of factors including an evaluation of the customer's financial statements, financial ratios, and credit reports (credit scores) based on a customer's payment history on credits owed to other firms. If a company has a prior relationship with a customer seeking trade credit, the customer's payment history with the firm is also carefully evaluated before additional credit is granted.
LINK TO LEARNIN
Corporate Finance Institute
Credit managers use various tools and techniques to evaluate creditworthiness of customers. The Corporate Finance Institute's website states that "the '5 Cs of Credit' is a common phrase used to describe the five major factors [character, capacity, collateral, capital, and conditions] used to determine a potential borrower's creditworthiness." It goes on to say that "a credit report provides a comprehensive account of the borrower's total debt, current balances, credit limits, and history of defaults and bankruptcies, if any."5 More on the 5 Cs of credit can be found on the Corporate Finance Institute's website (https://openstax.org/ r/corporate-finance).
Determining the Credit Policy
A company's credit policy encompasses rules of credit granting and procedures for the collections of accounts. It's how a company will process credit applications, utilize credit scoring and credit bureaus, analyze financial statements, make credit limit decisions, and conduct collection efforts when accounts become delinquent (still outstanding after their due date).
Establishing Credit Terms
Trade credit terms were discussed earlier. Recall that part of the terms and conditions of a sale are the credit terms—elements of a sales agreement (contract) that indicate when payment is due, possible discounts (for quick payments), and any late fee charges.
If open credit is for a sales transaction, an agreement is made as to the length of time for which credit is to be granted (payment period) and a discount for early payment. Although companies are free to establish credit terms as they see fit, most companies look to the practice of the particular industry in which they operate. The credit terms offered by the competition are a factor. Net terms usually range between 30 days and 90 days, depending on the industry. Discounts for early payments also differ and are typically from 1 to 3 percent.
Establishing credit terms offered can be thought of as a decision process similar to setting a price for products and services. Just as a price is the result of a market forces, so too are credit terms. If credit terms are not
5 Corporate Finance Institute. "What Are the 5 Cs of Credit?" n.d. https://corporatefinanceinstitute.com/resources/knowledge/ credit/5-cs-of-credit/
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19.4 • Receivables Management 581
competitive within the industry, sales can suffer. Typically, companies follow standard industry credit terms. If most companies in an industry offer a discount for early payments, then most companies will follow suit and also offer an equal discount.
Once credit terms are established, they can be changed based on both marketing strategies and financial management goals. For example, discounts for early payments can be more generous, or the full credit period can be extended to stimulate additional sales. Both discount periods and full credit periods can be tightened to try to speed up collections. The establishment of and changes to credit terms are usually made in consultation with the sales and financial management departments.
Monitoring Accounts Receivables
Financial managers monitor accounts receivables using some basic tools. One of those tools is the accounts receivable aging schedule (report). To prepare the aging schedule, a classifying of customer account balances is performed with age as the sorting attribute.
An account receivable begins its life as a credit sale. The age of a receivable is the number of days that have transpired since the credit sale was made (the date of the invoice). For example, if a credit sale was made on June 1 and is still unpaid on July 15, that receivable is 45 days old. Aging of accounts is thought to be a useful tool because of the idea that the longer the time owed, the greater the possibility that individual accounts receivable will prove to be uncollectible.
An aging schedule is a report that organizes the outstanding (unpaid) receivable balances into age categories. The receivables are grouped by the length of time they have been outstanding, and an uncollectible percentage is assigned to each category. The length of uncollectible time increases the percentage assigned. For example, a category might consist of accounts receivable that are 0-30 days past due and is assigned an uncollectible percentage of 6 percent. Another category might be 31-60 days past due and is assigned an uncollectible percentage of 15 percent. All categories of estimated uncollectible amounts are summed to get a total estimated uncollectible balance.
The aging of accounts is useful to the credit and collection managers, both from a global view—estimating how much of the accounts receivable asset might be bad debts—and on a micro basis—being able to drill down to see which specific customers are slow paying or delinquent so as to implement collection tactics.
Accountants and auditors also find the aging of accounts to determine a reasonable amount to be reported as bad debt expense and to establish a sufficient balance in the allowance for doubtful accounts. Bad debt expense is the cost of doing business because some customers will not pay the amounts they owe (accounts receivable), while the allowance for doubtful accounts is a contra-asset (it will be deducted from accounts receivable on the balance sheet) that contains management's best guess (management's estimate) as to how much of its accounts receivable will never be collected.
In Figure 19.6. Foodinia Inc.'s accounts receivable aging report shows that the total receivables balance is $189,000. The company splits its accounts into four age categories: not due, 30 to 60 days past due, 61 to 90 days past due, and more than 90 days past due. Of the $189,000 owed to Foodinia by its customers, $75,500 ($189,000 less $113,500) of invoices have been outstanding (not paid yet) beyond their due dates.
582 19 • The Importance of Trade Credit and Working Capital in Planning
30 to 60 61 to 90 More than
Days Past Days Past 90 Days Total
Customer Name Not Due Due Due Past Due Receivables
Arithmio $ 15,000 $ 15,000
Bakely Inc. $16,000 $ 5,000 $4,000 25,000
Bakeville 30,000 30,000
Boxfully 27,000 27,000
City Shop 5,000 5,000
Deskmeter 5,000 5,000
Flavorica 6,000 6,000
Merchain 7,000 7,000
Mshops 8,500 8,500
Next Dessert 12,000 12,000
Pichets 5,000 5,000
Shopahoo 19,000 19,000
SportsBuzz Ltd 2,000 1,500 3,500
Wearsy USA 21,000 21,000
Totals $113,500 $49,500 $17,000 $9,000 $189,000
Figure 19.6 Foodinia Inc. Aging of Accounts Receivable Schedule
In addition to preparing aging schedules, financial managers also use financial ratios to monitor receivables. The accounts receivable turnover ratio determines how many times (i.e., how often) accounts receivable are collected during an operating period and converted to cash. A higher number of times indicates that receivables are collected quickly. In contrast, a lower accounts receivable turnover indicates that receivables are collected at a slower rate, taking more days to collect from a customer.
Another receivables ratio is the number of days' sales in receivables ratio, also called the receivables collection period—the expected days it will take to convert accounts receivable into cash. A comparison of a company's receivables collection period to the credit terms granted to customers can alert management to collection problems. Both the accounts receivable turnover ratio and receivables collection period are covered, including the formulas for calculating the ratios, in the previous section of this chapter.
Accounts Receivables and Notes Receivable
An accounts receivable is an informal arrangement between a seller (a company) and customer. Accounts receivable are usually paid within a month or two. Accounts receivable don't require any complex paperwork, are evidenced by an invoice, and do not involve interest payments. In contrast, a note receivable is a more formal arrangement that is evidence by a legal contract called a promissory note specifying the payment amount and date and interest.
The length of a note receivable can be for any time period including a term longer than the typical account receivable. Some notes receivable have a term greater than a year. The assets of a bank include many notes receivable (a loan made by a bank is an asset for the bank).
A note receivable can be used in exchange for products and services or in exchange for cash (usually in the case of a financial lender). Sometimes a company might request that a slow-paying customer sign a note promissory note to further secure the receivable, charge interest, or add some type of collateral to the arrangement, in which case the receivable would be called a secured promissory note. Several characteristics of notes receivable further define the contract elements and scope of use (see Table 19.4).
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19.5 • Inventory Management 583
Accounts Receivable Notes Receivable
■ An informal agreement between customer and ■ A legal contract with established payment terms company ■ Receivable beyond one year and outside of a
■ Receivable in less than one year or within a company's operating cycle company's operating cycle ■ Includes interest
■ Does not include interest ■ Could stipulate collateral
Table 19.4 Key Feature Comparison of Accounts Receivable and Notes Receivable
19.5
Inventory Management
By the end of this section, you will be able to:
■ Outline the costs of holding inventory.
■ Outline the benefits of holding inventory.
Financial managers must consider the impact of inventory management on working capital. Earlier in the chapter, the concept of the inventory conversion cycle was covered. The number of days that goods are held by a business is one of the focal points of inventory management.
Managers look to minimize inventory balances and raise inventory turnover ratios while trying to balance the needs of operations and sales. Purchasing personnel need to order enough inventory to "feed" production or to stock the shelves. The sales force wants to meet or surpass their sales budgets, and the operations people need inventory for the factories, warehouses, and e-commerce sites.
The days in inventory ratio measures the average number of days between acquiring inventory (i.e., purchasing merchandise) and its sale. This ratio is a metric to be watched and monitored by inventory managers and, if possible, minimized. A high days in inventory ratio could mean "aging" inventory. Old inventory could mean obsolesce or, in the case of perishable goods, spoilage. In either case, old inventory means losses.
Imagine a company selling high-tech products such as consumer electronics. A high days in inventory ratio could mean that technologically obsolete products will be sold at a discount. There are similar issues with older inventory in the fashion industry. Last year's styles are not as appealing to the fashion-conscious consumer and are usually sold at significant discounts. In the accounting world, lower of cost or market value is a test of inventory value to determine if inventory needs to be "written down," meaning that the company takes an expense for inventory that has lost significant value. Lower of cost or market is required by Generally Accepted Accounting Principles (GAAP) to state inventory valuations at realistic and conservative values.
Inventory is a very significant working capital component for many companies, such as manufacturers, wholesalers, and retailers. For those companies, inventory management involves management of the entire supply chain: sourcing, storing, and selling inventory. At its very basic level, inventory management means having the right amount of stock at the right place and at the right time while also minimizing the cost of inventory. This concept is explained in the next section.
Inventory Cost
Controlling inventory costs minimizes working capital needs and, ultimately, the cost of goods sold. Inventory management impacts profitability; minimizing cost of goods sold means maximizing gross profit (Gross Profit = Net Sales Less Cost of Goods Sold).
There are four components to inventory cost:
■ Purchasing costs: the invoice amount (after discounts) for inventory; the initial investment in inventory
584 19 • The Importance of Trade Credit and Working Capital in Planning
■ Carrying costs: all costs of having inventory in stock, which includes storage costs (i.e., the cost of the space to store the inventory, such as a warehouse), insurance, inventory obsolescence and spoilage, and even the opportunity cost of the investment in inventory
■ Ordering costs: the costs of placing an order with a vendor; the cost of a purchase and managing the payment process
■ Stockout costs: an opportunity cost incurred when a customer order cannot be filled and the customer goes elsewhere for the product; lost revenue
Minimizing total inventory costs is a combination of many strategies, the scope and complexity of which are beyond the scope of this text. Concepts such as just-in-time (JIT) inventory practices and economic order quantity (EOQ) are tools used by inventory managers, both of which help keep a company lean (minimizing inventory) while making sure the inventory resources are in place in time to complete the sale.
Benefit of Holding Inventory
Brick-and-mortar stores need goods in stock so that the customer can see and touch the product and be able to acquire it when they need it. Customers are disappointed if they cannot see and touch the item or if they find out upon arrival at the store that it is out of stock.
Customers of all kinds don't want to wait for the delivery of a purchase. We have become accustomed to Amazon orders being delivered to the door the next day. Product fulfillment and availability is important. Inventory must be in stock, or sales will be lost.
In manufacturing, the inventory of materials and component parts must be in place at the start of the value chain (the conversion process), and finished goods need to be ready to meet scheduled shipments. Holding sufficient inventory meets customer demand, whether it is products on the shelves or in the warehouse that are ready to move through the supply chain and into the hands of the customer.
19.6
Using Excel to Create the Short-Term Plan
By the end of this section, you will be able to:
■ Create a one-year budget.
■ Create a cash budget.
A cash budget is a tool of cash management and therefore assists financial managers in the planning and control of a critical asset. The cash budget, like any other budget, looks to the future. It projects the cash flows into and out of the company. The budgeting process of a company is a really an integrated process—it links a series of budgets together so that company objectives can be achieved. For example, in a manufacturing company, a series of budgets such as those for sales, production, purchases, materials, overhead, selling and administrative costs, and planned capital expenditures would need to be prepared before cash needs (cash budget) can be predicted.
Just as you might budget your earnings (salary, business income, investment income, etc.) to see if you will be able to cover your expected living expenses and planned savings amounts, to be successful and to increase the odds that sufficient cash will be available in the months ahead, financial managers prepare cash budgets to
■ meet payrolls;
■ allocate dollars for contingencies and emergencies;
■ analyze if planned collections and disbursements policies and procedures result in adequate cash balances; and
■ plan for borrowings on lines of credit and short-term loans that might be needed to balance the cash budget.
A cash budget is a model that often goes through several iterations before managers can approve it as the
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19.6 • Using Excel to Create the Short-Term Plan 585
plan going forward. Changes in any of the "upstream" budgets—budgets that are prepared before the cash budget, such as the sales, purchases, and production budgets—may need to be revised because of changing assumptions. New economic forecasts and even cost-cutting measures will require a revision of the cash budget.
Although a budget might be prepared for each month of a future 12-month period, such as the upcoming fiscal year, a rolling budget is often used. A rolling budget changes often as the planning period (e.g., a fiscal year) plays out. When one month ends, another month is added to the end (the next column) of the budget. For example, if in your budget January is the first month of the planning period, once January is over, next January's cash budget column would be added—right after December's column (at the far right of the budget).
Sample One-Year (Annual) Operating Budget
Preparing an annual operating budget can be a complex task. In essence, a company budget is a series of budgets, many of which are interrelated.
The sales budget is prepared first and has an impact on many other budgets. Take the example of a production budget of a manufacturer. The sales budget impacts what needs to be produced (production budget), and the production budget influences planned purchases of material (purchases budget), overhead resources (overhead budget), and the amount of labor costs for the year ahead (direct labor budget.)
For a merchant (such as a wholesaler or retailer), the annual budget would be less complex than that of a manufacturing firm but would still require an inventory purchases budget and an operating expense budget (such as selling and administrative expenses). For a service firm, a purchase budget for inventory would not be necessary, but an operating budget would be. All businesses need a cash budget, which is the topic of the next section of this chapter.
The example operating budget presented here is of a merchandising company. Budgets are prepared following a process that begins with a sales (or revenue) forecast. The sales forecast is normally based on information obtained from both internal and external sources and predicts the amount of units to be sold in the planning period—usually one year into the future.
A company's management, in consultation with its marketing and sales executives, would prepare a sales budget by making assumptions about the number of units that are expected to be sold and the prices that will be charged. From the sales budget, projections are made as to cash receipts each month, and therefore assumptions have to be made as to how much of each month's sales will be cash sales and how much cash will flow into the company from the collection credit sales (including cash flow in from the prior month's sales). Figure 19.7 provides an example of a sales budget and projected accounts receivable collections and cash sales for the months of January through December. Keep in mind that projected monthly sales amounts are not equal to cash collected from sales. Because of sales on credit, some cash from sales lags credit sales as collections can extend beyond the month of sale. Credit terms such terms such as net 30 (net amount owed to be paid in 30 days) have to be considered when developing a forecasted cash collection pattern.
586 19 • The Importance of Trade Credit and Working Capital in Planning
A A c E F
10 January February $ 500,000 460,300 March $ 600,000 513,000 April $ 540,000 568,000 May $ 486,000 518,700 June $ 534,600 491,130
11 Total Sales $ 460,000 400,000
12 Accounts Receivable Collections/Cash Sales
13
14 July August 667,000 $ 587,880 September 653,000 $ 645,810 October 640,000 $ 637,390 November 576,000 $ 611,850 December 460,000 $ 538,680
15 Total Sales 580,000
16 Accounts Receivable Collections/Cash Sales $ 532,828
Figure 19.7 Example of a Sales and Collections Budget
Download the spreadsheet file (https://openstax.Org/r/spreadsheet-file1) containing key Chapter 19 Excel exhibits.
Sales budgets "drive" the preparation of other budgets. If sales are expected to increase, purchases of inventory and some operating expenses would also increase. To meet the demand for goods and services (as defined in the sales budget), a purchases (inventory) budget would be prepared. In this example (Figure 19.8). the purchases budget shows projected purchases of inventory (merchandise) and the projected payments (also called disbursements) for each month.
Cash outflows as a result of purchases often do not equal the projected purchase amount. That is because payments for purchases are usually on credit (accounts payable), and so purchases for one month typically get spread out over a period of time that encompasses the current month and the month (or months) thereafter. To keep this example simple, the assumption is that the purchases are paid for in the following month (an average days payable outstanding of 30 days). However, in other cases, payment patterns may be based on other payment periods such as 45, 60, or even 90 days, depending on the trade credit terms.
A E F G
1 January February March April May June
2 Purchases $ 225,000 $ 270,000 S 243,000 S 219,000 S 241,000 $ 260,000
3 Payments 190,000 225,000 270,000 243,000 219,000 241,000
4
5 July August September October November December
e Purchases 300,000 293,000 288,000 259,000 207,000 200,000
7 Payments $ 260,000 $ 300,000 S 293,000 S 288,000 s 259,000 $ 207,000
Figure 19.8 Purchases Budget
An operating expense budget is prepared next and is basically a prediction of the selling and administrative expenditures of the company. Notice in Figure 19.9 that in the operating expense budget, cost of goods sold (an expense) is not included, nor are noncash expenses such as depreciation. The cash outlays related to goods sold, at least in a merchandising operation, are accounted for in the purchases budget (payments for purchases of inventory.)
With the sales, purchases, and operating expense budgets prepared, the cash budget can be prepared. Some of the "inputs" to the cash budget are from the sales (collections of cash), purchases (payments), and the operating expense budget (cash expenditures for selling and administrative expenses). A sample cash budget and a discussion of its preparation follows in the next section of this chapter.
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19.6 • Using Excel to Create the Short-Term Plan 587
Sample Cash Budget
A cash budget is the last budget to be prepared and is often part of the financial budget (cash budget, budgeted income statement, and budgeted balance sheet). The purpose of the cash budget is to estimate cash flows, to help ensure sufficient cash balances are maintained during the planning period, and to plan for external financing during periods of cash deficits.
When a budget is prepared in Excel, cash budget analysts can play "what if" with different scenarios to see when cash surpluses and deficits are expected. Cash surpluses means that funds can be invested in marketable securities to earn a rate of return, while cash deficits mean that financing, such as a line of credit, will be necessary (assuming forecasts are accurate).
Although the example shown in Figure 19.10 is a monthly cash budget, a cash budget could be prepared using any useful time elements: weekly, monthly, or quarterly.
One common practice is to use a rolling cash budget. A rolling cash budget is continually updated to add a new budget period, such as a month's amount of cash flow activity, as the most recent budgeted month expires. For example, assume that a 12-month cash budget is prepared for a period covering January 20X1 to December 20X1. Once the month of January 20X1 has concluded, a 12-month planning period continues by add January 20X2 to the last column of the budget. The rolling cash monthly budget is an extension of the initial cash budget model, adding one month and thereby always extending cash flow projections one year into the future.
A B E F G
1 January February March April May June
2 Wages and Salaries $200,000 $200,000 $200,000 $200,000 $200,000 $200,000
3 Payroll Taxes 20,000 20,000 20,000 20,000 20,000 20,000
4 Advertising and Marketing 50,000 50,000 50,000 50,000 50,000 50,000
5 Repairs and Maintenance 12,000 12,000 12,000 12,000 12,000 12,000
6 Rent 3,000 3,000 3,000 3,000 3,000 3,000
7 Shipping 5,520 6,000 7,200 6,480 5,832 6,415
S Utilities 1,500 1,500 1,500 1,500 1,500 1,500
9 Insurance 2,050 2,050 2,050 2,050 2,050 2,050
10 Supplies 400 400 400 400 400 400
11 Taxes 7,000 7,000 7,000 7,000 7,000 7,000
12 Total S3 01.470 S301.950 5303.150 S302.430 S301.7S2 S302.365
13
14 July August September October November December
15 Wages and Salaries $200,000 $200,000 $200,000 $200,000 $200,000 $200,000
16 Payroll Taxes 20,000 20,000 20,000 20,000 20,000 20,000
17 Advertising and Marketing 50,000 60,000 60,000 55,000 50,000 50,000
18 Repairs and Maintenance 12,000 12,000 12,000 12,000 12,000 12,000
19 Rent 3,000 3,000 3,000 3,000 3,000 3,000
20 Shipping 6,960 8,004 7,836 7,680 6,912 5,520
21 Utilities 1,500 1,500 1,500 1,500 1,500 1,500
22 Insurance 2,050 2,050 2,050 2,050 2,050 2,050
23 Supplies 400 400 400 400 400 400
24 Taxes 7,000 7,000 7,000 7,000 7,000 7,000
25 Total S302.910 S313.954 S313.7S6 S30S.630 S3 02.862 S3 01.470
Figure 19.9 Operating Expenses Budget
588 19 • The Importance of Trade Credit and Working Capital in Planning
A A B E F G
1 January February March April May June
2 Beginning Cash Balance $350,000 $253,530 $191,330 $131,730 $154,300 $152,213
3 Cash Collections 400,000 460,300 513,000 563,000 518,700 491,130
4 Cash Disbursements 491,470 526,950 573,150 545,430 520,782 543,365
5 Net Cash Flow (91,470) (66,650) (60,150) 22,570 (2,032) (52,235)
6 Preliminary Ending Cash Balance 258,530 isi,aso 131,730 154,300 152,218 99,983
7 Less: Minimum Cash Balance 50,000 50,000 50,000 50,000 50,000 50,000
S Cash Surplus (Deficiency) $208,530 $141, SSO $81,730 $104,300 $102,218 $49,983
9 Ending Cash Balance $258,530 $191,880 $131,730 $154,300 $152,218 $»,983
10
11 July August September October November December
12 Beginning Cash Balance $99,983 $69,901 $50,000 $89,024 $129,784 $179,772
13 Cash Collections 532,828 587,880 645,810 637,390 611,850 538,680
14 Cash Disbursements 562,910 613,954 606,786 596,630 551,862 508,470
15 Net Cash Flow (30,082) (26,074) 39,024 40,760 49,988 30,210
16 Preliminary Ending Cash Balance 69,901 43,327 39,024 129,734 179,772 209,982
17 Less: Minimum Cash Balance 50,000 50,000 50,000 50,000 50,000 50,000
18 Cash Surplus (Deficiency) $19,901 ($6,173) $39,024 $79,734 $129,772 $159,982
19 Ending Cash Balance $69,901 $50,000 $89,024 $129,734 $179,772 $209,982
Figure 19.10 Sample Cash Budget
Using Figure 19.9 as an example. Table 19.5 shows the formulas that form the skeleton of a monthly cash budget.
This is the amount of cash the company expects to have on the first day of the month. For example, in Figure 19.10. cell B2 is the amount of cash on Jan. 1 to start the year (the planning period). The remaining beginning cash balances for the months February through December are the ending cash balances of the previous month. For example, February's beginning cash balance (C2) is referenced from cell B9 (ending cash balance for January).
These are the projected cash inflows from collections from customers (accounts receivable), cash sales, and any other significant cash inflows, such as dividends and interest on investments or sale of fixed assets. For example, the Cash Collections shown in the Sample Cash Budget (Figure 19.10) are referenced from the Sales and Collections Budget (Figure 19.7). January's Cash Collections (cell B3) in the Sample Cash Budget are from cell B12 of the Sales and Collection Budget.
Beginning Cash Balance
Cash
Collections
Cash
Disbursements
Net Cash Flow
Preliminary Ending Cash Balance
Less:
Minimum Cash Balance
Cash disbursements are the projected cash outflows, such as those for operating expenses and payment of payables. For example. Cash Disbursements in the Sample Cash Budget for January (Figure 19.10) are the sum of January's payments for purchases in the Purchases Budget (Figure 19.8. cell B3) and the January operating expenses (Operating Expenses Budget. Figure 19.9. cell B12).
The formula for net cash flow is For example, in Figure 19.10. the January Net Cash Flow is calculated in cell B5.
Beginning Cash Balance + or - Net Cash Flow. This is the projected cash balance before taking into account the target cash balance to be maintained (minimum cash balance). In the Sample Cash Budget (Figure 19.10). the preliminary ending cash balance formula for January is =B2+B5 (B2 is the Beginning Cash Balance and B5 is the Net Cash Flow for the month).
This is a target cash balance that management sets; it is the minimum amount of cash that should be maintained by the company (in Figure 19.10. cells B2:G7 and B17:G17).
Table 19.5 Excel Formulas for Monthly Cash Budget
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19.6 • Using Excel to Create the Short-Term Plan 589
A cash surplus means that cash can be invested in marketable securities. A cash deficiency means that some type of financing, such as a line of credit or bank loan, will be needed to provide enough cash for operations and to maintain a minimum cash balance. This number is
Cash Surplus found by subtracting the minimum cash balance from the preliminary ending cash balance.
(Deficiency) For example, the cash surplus for January in Figure 19.10 is calculated with this formula:
=B6-B7. Notice that all months in the Sample Cash Budget show a surplus except for August's forecast of a deficit, which may require drawing on a line of credit to provide enough cash to meet obligations in August.
Table 19.5 Excel Formulas for Monthly Cash Budget
590 19-Summary
ID Summary
19.1 What Is Working Capital?
Working capital is not only necessary to run a business; it is a resource that will expand and contract with business cycles and must be carefully managed and monitored. The daily, weekly, and monthly needs of business operations are met by cash. Financial managers understand the significance of net working capital (current assets - current liabilities) and various liquidity ratios as they attempt to ensure that bills can be paid. The cash conversion cycle and the cash budget provide additional working capital management tools.
19.2 What Is Trade Credit?
Trade credit is very prevalent in the business world, especially in B2B (business-to-business) transactions. Many business exchanges (sales) could not take place without trade credit and the credit terms that are offered. Like any component of working capital, trade credit must be planned and managed. The creditor (the company granting the credit) does so based on an analysis of creditworthiness and must monitor payments and manage slow-paying accounts. The debtor (accounts payable) needs to make payments on time to keep a clean credit history and to take advantage of discounts.
19.3 Cash Management
Cash management is simply making sure you have enough cash to meet expected obligations and for contingencies (unexpected or emergency cash needs). Excess cash should be invested low-risk and highly liquid marketable securities. The cash budget is a critical tool of cash management.
19.4 Receivables Management
Accounts receivables are monitored by management with tools such as the ratios accounts receivable turnover and average collection period and the aging of receivables. The credit managers' mantra rings true: "The older the receivable, the greater the likelihood that the account will not be collected."
19.5 Inventory Management
Inventory, usually the least liquid of the current assets, presents its own set of management challenges. Finding the optimal level of inventory is probably more of an art than a science. JIT helps to reduce the investment in inventory and lower the costs of storage, but stockout costs can be very damaging to profitability.
19.6 Using Excel to Create the Short-Term Plan
Short-term plans of a business are funded with cash, with cash budgets being a critical tool of planning. The cash budget takes into account a target amount of cash, factoring in all the motives for holding cash. A cash budget looks ahead—predicting cash inflows and outflows, allocating for minimum cash balances to be maintained, and helping management determine short-term financing needed. Although it is the last budget prepared, the preparation of the cash budget is an important financial planning exercise of companies small and large.
? Key Terms
accounts receivable aging schedule a report that shows amounts owed by customers by the age of the
account, as measured by the number of days since the sale allowance for doubtful accounts an account that contains the estimated amount of accounts receivable
that will not be collected
bad debt expense an expense that a business incurs as a result of uncollectible accounts receivables bankruptcies federal court procedures that protect distressed businesses from creditor collection efforts while allowing the debtor firm to liquidate its assets or devise a reorganization plan
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19-Key Terms 591
benchmarking the process of performance analysis that involves comparing financial condition and
operating results against a standard, called a benchmark bill of lading a document that is a detailed list of a goods that have been shipped; a receipt given by the
carrier (shipping company) to the seller as evidence that the goods have been shipped to the buyer carrying costs all costs associated with having inventory in stock including storage costs, insurance,
inventory obsolescence, and spoilage cash budget a report that shows an estimation of cash inflows, outflows, and cash balances over a specific
period of time, such as monthly, quarterly, or annually cash cycle or cash conversion cycle the time period (measured in days) between when a business begins
production and acquires resources from its suppliers (for example, acquisition of materials and other forms
of inventory) and when it receives cash from its customers; offset by the time it takes to pay suppliers
(called the payables deferral period) cash discount discount granted to a customer who has purchased goods or services on account (credit) and
pays the invoice within a certain number of days as specified by credit terms compensating balance minimum balance of cash that a business must deposit and maintain in a bank
account to obtain a loan
contra-asset an account with a balance that is used to offset (reduce) its related asset on the balance sheet (for example, allowance for doubtful accounts reduces the value of accounts receivable reported on the balance sheet)
credit period the number of days that a business purchaser has before they must pay their invoice credit rating a type of score that indicates a business's creditworthiness
credit terms the terms that are part of a sales credit agreement that indicate when payment is due, possible
discounts, and any fees that will be charged for a late payment current assets assets that are cash or cash equivalents or are expected to be converted to cash in a short
period of time and will be consumed, used, or expire through business operations within one year or the
business's operating cycle, whichever is shorter discount period the number of days the buyer has to take advantage of the cash discount for an early
payment
factoring the process of selling accounts receivables to a financial institution or, in some cases, using the
accounts receivables as security for a loan from a financial institution floor planning a type of inventory financing whereby a financial institution provides a loan so that the
company can acquire inventory with proceeds from the sale of inventory used to pay down the loan; a
common method of financing inventory for automobile dealers and sellers of other big-ticket (high-priced)
items
gross working capital synonymous with the current assets of a company, those assets that include cash and
other assets that can be converted into cash within a period of 12 months just-in-time inventory inventory management method in which a company maintains as little inventory on
hand as possible while still being able to satisfy the demands of its customers letter of credit a letter issued by a bank that is evidence of a guarantee for payments made to a specified
entity (such as a supplier) under specified conditions; common in international trade transactions liquidity ability to convert assets into cash in order to meet primarily short-term cash needs or emergencies marketable securities investments that can be converted to cash quickly; short-term liquid securities that
can be bought or sold on a public exchange (market) and tend to mature in a year or less net terms also referred to as the full credit period; the number of days that a business purchaser has before
they must pay their invoice
net working capital the difference between current assets and current liabilities (Current Assets - Current
Liabilities = Net Working Capital) operating cycle the time it takes a company to acquire inventory, sell inventory, and collect the cash from
the sale of said goods; synonymous with cash cycle opportunity cost the cost of a forgone opportunity
592 19 • Multiple Choice
ordering costs costs associated with placing an order with a vendor or supplier
precautionary motive a reason to hold cash balances for unexpected expenditures such as repairs, costs associated with unexpected breakdown of equipment, and hiring temporary workers to meet unexpected production demands
quick payment a payment made on an account payable during a period of time that falls within the discount period
ratios numerical values taken from financial statements that are used in formulas to examine financial relationships and create metrics of performance, strengths, weaknesses; help analysts gain insight and meaning
speculative motive a reason for holding an amount of cash—to be able to take advantage of investment opportunities
stockout costs an opportunity cost (lost revenue) incurred when a customer order cannot be filled because
the item is out of stock and the customer goes elsewhere for the product supply chain the network of participants and activities between a company and its suppliers and the
company and its customers; exists to distribute a product or to provide a service to the final buyer trade credit credit granted to a business, also called accounts payable; allows a business to buy goods and
services on account and pay the cash at some point in the future transactional motive holding an amount of cash to meet operational expenditures such as payroll,
payments to vendors, and loan payments working capital the resources that are needed to meet the daily, weekly, and monthly operating cash flow
needs
Multiple Choice
1. The term working capital is synonymous with.
a. accounts payable
b. current assets
c. equity
d. current liabilities
2. The formula for net working capital is _
a. Current Assets - Current Liabilities
b. Fixed Assets - Current Assets
c. Assets - Liabilities
d. Current Assets - Liabilities
3. When sales are made on credit, which current assets typically increase at the time of the sale?
a. cash
b. notes receivable
c. accounts receivable
d. marketable securities
4. Which of the following is NOT a goal of working capital management?
a. meet the operational needs of the company
b. satisfy obligations (current liabilities) as they come due
c. maintain an optimal level of current assets
d. maximize the investment in current assets
5. Accelerated Growth Inc. has the following account balances at year-end.
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19 • Multiple Choice 593
Property, Plant, and Equipment $500,000
Accounts Payable 6,000
Intangible Assets 5,000
Cash 10,000
Marketable Securities (maturing in 6 months or less) 30,000
Retained Earnings 25,000
Interest Payable 2,000
Notes Payable (due in 24 months) 10,000
Accounts Receivable 20,000
Notes Payable (due in 6 months) 20,000
Common Stock 45,000
Dividends Payable 3,000
Inventory $ 50,000
What is the cash ratio?
a. 1.29
b. 1.43
c. 1.71
d. .088
6. Which of the following is true of these credit terms: 3/15, n/30?
a. 15 percent discount if the payable is paid within 3 days of the invoice date
b. 3 percent discount if the payable is paid in the period between 15 days and 30 days after the invoice date
c. 3 percent discount if the payable is paid within 15 days of the invoice date
d. 30 percent discount if cash is paid on the sale date and 15% discount if paid 3 days after the invoice date
7. When reviewing its budgets, including the cash budget, management of Transcend Inc. have considered best-case and worst-case scenarios. As they completed their analysis, it was decided because of the possibility of unexpected repairs and unanticipated higher labor costs to add another $30,000 to the amount of the target cash balance to maintain throughout the year. The reason for this action would be which of these motives for holding cash?
a. transaction motive
b. opportunity cost mitigation motive
c. precautionary motive
d. speculative motive
8. A large retailer has more than $100 million of cash and cash equivalents on its balance sheet. Which of the following would not be part of the cash equivalents?
a. cash in banks (checking account balances)
b. US Treasury bond maturing in two years
c. receivables from a bank that processes credit card payments
d. commercial paper
9. An account receivable is created when_.
a. a customer pays its bill
b. a company accepts a credit card, such as VISA or MasterCard
c. a company sells to a customer on an open account
d. a company sells to a customer only on a cash basis
594 19 • Review Questions
10. Jackson's Moonshine LLC has a receivables collection period of 47 days. Which the following would be reasonable conclusions?
a. Jackson's Moonshine LLC is most likely experiencing serious liquidity issues.
b. Jackson's Moonshine LLC is most overinvested in marketable securities.
c. If the industry average is 31 days, Jackson's management should attempt strategies that will lower their receivables collection period.
d. If the industry average is 53 days, Jackson's management should attempt strategies that will raise their receivables collection period.
11. Two Way Power Ltd. (2WP) stocks an inventory item, BB3, that is projected to be in great demand over the next 12 months. In discussing its sales forecasts with its suppliers, a reasonable estimate shows that 2WP could lose about $30,000 of sales in month 3 due to inventory financing difficulties. Which, if any, of the following inventory costs would be affected by this development?
a. purchase cost
b. carrying costs
c. ordering costs
d. stockout costs
e. none of these costs because loss of sales is not an inventory cost
12. If a company has significant inventory in each element of the value chain, it most likely is descriptive of
a. the cost of goods sold of a retailer
b. the inventory balances held by wholesalers and service firms
c. the materials, work in process, and finished goods of a manufacturer
d. the inventory on the shelves of an e-commerce retailer
5» Review Questions
1. Intelligent Cookies Inc. (ICI) sold $30,000,000 of product in a year that had a cost of goods sold of
$10,000,000. The average inventory carried by ICI was $500,000. On average, it takes 35 days for ICI's customers, such as grocery stores and restaurants, to pay on their accounts. ICI buys ingredients, including flour, spices, and eggs, from its vendors on credit, and ICI takes about 40 days to pay its suppliers. How many days is ICI's cash conversion cycle?
2. Shown below are account balances for Electra Engines Inc., a manufacturer. The accounts are shown in a random order. What is the amount of net working capital?
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19 • Review Questions 595
NetWorkinq Capital $
Property, Plant, and Equipment $500,000
Accounts Payable 6,000
Intangible Assets 5,000
Cash 10,000
Marketable Securities (maturing in 6 months or less) 30,000
Retained Earnings 25,000
Interest Payable 2,000
Notes Payable (due in 24 months) 10,000
Accounts Receivable 20,000
Notes Payable (due in 6 months) 20,000
Common Stock 45,000
Dividends Payable 3,000
Inventory $ 50,000
3. Shown below are account balances for Electra Engines Inc., a manufacturer. The accounts are shown in a random order. What is the current ratio and the quick ratio?
Property, Plant, and Equipment $500,000
Accounts Payable 9,000
Intangible Assets 5,000
Cash 12,000
Marketable Securities (maturing in 6 months or less) 25,000
Retained Earnings 25,000
Interest Payable 2,500
Notes Payable (due in 24 months) 10,000
Accounts Receivable 25,000
Notes Payable (due in 6 months) 20,000
Common Stock 45,000
Dividends Payable 3,000
Inventory 55,000
4. Imagine that these are the cash collection cycles for some well-run companies:
Cash Conversion
Company Cycle
Grocery Chain 6.20
Building/Construction Company 162.69
Restaurant Chain (4.06)
Department Store 63.60
What types of conclusions can you reach when you see this kind of variability?
Imagine that those cash conversion cycles are based on this information:
Accounts Accounts
Inventory Receivable Payable
Company Type Turnover Turnover Turnover
Grocery Chain 14.60 91.00 16.00
Building/Construction Company 2.00 121.67 16.00
Restaurant Chain 150.00 100.00 36.00
Department Store 2.78 91.00 3.00
596 19 • Review Questions
What would be your analysis of the cash conversion cycles based on the above information (inventory turnover, accounts receivable turnover, and accounts payable turnover)? Use the worksheet below to summarize your conclusions.
Worksheet
- Company Cash Conversion Cycle Analysis
Grocery Chain 6.20
Building/Construction Company 162.69
Restaurant Chain (4.06)
Department Store 63.60
5. What is the estimated annual percentage rate (APR) of not taking advantage of the early payment discount based on these terms: 4/15, n/45?
6. If you were a credit manager reviewing a potential customer's request for a $20,000 line of credit, what would you analyze? Generally, how would the 5Cs of Credit guide your analysis and help lead you to a prudent decision to accept or reject the request?
7. Aspire Excellent Inc. is a book publisher. On March 1, Aspire sells $25,000 of books to Get Your Books Inc. (YBI), a large bookstore chain. The sale is made on account with terms net 60. Aspire's customers usually take the full 60 days to pay their invoices. The books cost Aspire $10,000 to manufacture. Below, summarize the effect on the accounts on March 1 from the standpoint of the seller. Aspire Excellent Inc., and the buyer, YBI.
Change (Amount and Direction of Change: + or -)
Aspire's Accounts in the Account
Accounts Receivable
Inventory
Sales
Change (Amount and Direction of Change: + or -)
YBI's Accounts in the Account
Accounts Payable
Inventory
8. The financial manager of New England Blissful Dairies, a distributor of milk, cream, and ice cream
products, has finished the 12-month operating budget. For the month of June, the following projections were made:
June 1 Cash Balance $90,000 Cash Receipts $300,000 Cash Disbursement $350,000
Taking into account an amount of cash that the firm likes to maintain as a target (minimum cash balance) of $75,000, prepare the cash budget for June using the format below. Assume that, if necessary, the company will draw upon a preestablished line of credit with their bank to be able to maintain the target cash balance.
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19-Video Activity 597
June
Beginning Cash Balance
Cash Collections
Cash Disbursements
Net Cash Flow
Preliminary Ending Cash Balance
Less: Minimum Cash Balance
Cash Surplus (Deficiency)
Ending Cash Balance
Will the company need short-term financing?
9. The sales for Re-Works Inc., a company that fabricates iron fencing from recycled metals, are all on account. For the first three months of the year, Re-Works management expects the following sales:
Sales
January $120,000
February $130,000
March $140,000
Based on past collection patterns, management expects the following:
Month of Sale 10% Month after Sale 50% Remainder - Second Month after Sale 40%
Also, based on past experience, management forecasts that 5 percent of accounts receivable will be uncollectible and will eventually be written off.
What are the expected cash receipts for March?
10. With the same sales forecasts as in question 9, Re-Works Inc. management would like to implement some changes to credit policy and credit terms that they believe would change the collection pattern going forward and would lower the uncollectible accounts prediction to 3 percent.
What would be the expected cash receipts for March?
Month of Sale 15%
Month after Sale 55%
Remainder 30%
Bad Debt Expense 3%
| ►! Video Activity
How Companies Report Cash Flow
Click to view content (https://openstax.Org/r/how-companies-report-cash-flow)
1. Why isn't the net income reported on a corporate balance sheet a good estimate of the increase in cash that occurred during the year?
2. What is the difference between a corporate cash budget and a projected statement of cash flows?
598 19-Video Activity
Trade Credit and Interest Rates on Short-Term Borrowing
Click to view content (https://0penstax.0rg/r/c0st-0f-trade-credit)
3. Explain this statement: Accounts payable and accounts receivables are essentially financial opposites.
4. Accounts payable is often called "interest-free financing." As such, explain why a company would choose to pay the amount owed on its purchases of inventory 50 days early. Base your answer on these facts:
■ The annualized cost of forgoing an early payment discount is approximately 16 percent.
■ The company's cost of borrowing short-term on a bank line of credit is 9 percent.
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Risk Management and the Financial Manager
Figure 20.1 Financial managers must consider prudent ways to manage economic volatility and the risk it poses to a company, (credit: modification of "Risk text on Dollar banknotes" by Marco Verch/flickr CC BY 2.0)
Chapter Outline
20.1 The Importance of Risk Management
20.2 Commodity Price Risk
20.3 Exchange Rates and Risk
20.4 Interest Rate Risk
J
Why It Matters
Each year, American Airlines consumes approximately four billion gallons of jet fuel. In the spring of 2018, jet fuel prices rose from an average of $2.07 per gallon to a price of $2.19 per gallon.2 A $0.12-per-gallon increase in the price of jet fuel may not seem significant, but on an annualized basis, a price increase of this magnitude would increase the company's jet fuel bill by approximately $500 million.
That added cost cuts into the profits of the company, leaving less money available to provide a return to the company's investors. Rising costs could even cause the business to become unprofitable and close, causing many employees to lose their jobs. The financial managers of American Airlines are not able to control the price of jet fuel. However, they must be aware of the risk that price volatility poses to the company and consider prudent ways to manage this risk.
1 American Airlines. American Airlines 2019 Environmental Data. Accessed July 8, 2021. https://www.americanairlines.in/content/ images/customer-service/about-us/corporate-governance/aag-2019-environmental-data.pdf
2 S&P Global. "Plattsjet Fuel." S&P Global Platts. Accessed July 8, 2021. https://www.spglobal.com/platts/en/oil/refined-products/ jetfuel*
600 20 • Risk Management and the Financial Manager
20.1
The Importance of Risk Management
Learning Outcomes
By the end of this section, you will be able to:
■ Describe risk in the context of financial management.
■ Explain how risk can impact firm value.
■ Distinguish between hedging and speculating.
What Is Risk?
The job of the financial manager is to maximize the value of the firm for the owners, or shareholders, of the company. The three major areas of focus for the financial manager are the size, the timing, and the riskiness of the cash flows of the company. Broadly, the financial manager should work to
■ increase cash coming into the company and decrease cash going out of the company;
■ speed up cash coming into the company and slow down cash going out of the company; and
■ decrease the riskiness of both money coming in and money going out of the company.
The first item in this list is obvious. The more revenue a company has, the more profitable it will be. Businesspeople talk about "top line" growth when discussing this objective because revenue appears at the top of the company's income statement. Also, the lower the company's expenses, the more profitable the company will be. When businesspeople talk about the "bottom line," they are focused on what will happen to a company's net income. The net income appears at the bottom of the income statement and reflects the amount of revenue left over after all of the company's expenses have been paid.
The second item in the list—the speed at which money enters and exits the company—has been addressed throughout this book. One of the basic principles of finance is the time value of money—the idea that a dollar received today is more valuable than a dollar received tomorrow. Many of the topics explored in this book revolve around the issue of the time value of money.
The focus of this chapter is on the third item in the list: risk. In finance, risk is defined as uncertainty. Risk occurs because you cannot predict the future. Compared to other business decisions, financial decisions are generally associated with contracts in which the parties of the contract fulfill their obligations at different points in time. If you choose to purchase a loaf of bread, you pay the baker for the bread as you receive the bread; no future obligation arises for either you or the baker because of this purchase. If you choose to buy a bond, you pay the issuer of the bond money today, and in return, the issuer promises to pay you money in the future. The value of this bond depends on the likelihood that the promise will be fulfilled.
Because financial agreements often represent promises of future payment, they entail risk. Even if the party that is promising to make a payment in the future is ethical and has every intention of honoring the promise, things can happen that can make it impossible for them to do so. Thus, much of financial management hinges on managing this risk.
Risk and Firm Value
You would expect the managers of Starbucks Corporation to know a lot about coffee. They must also know a lot about risk. It is not surprising that the term coffee appears in the text of the company's 2020 annual report 179 times, given that the company's core business is coffee. It might be surprising, however, that the term risk appears in the report 99 times.3 Given that the text of the annual report is less than 100 pages long, the word risk appears, on average, more than once per page.
Starbucks faces a number of different types of risk. In 2020, corporations experienced an unprecedented risk because of COVID-19. Coffee shops were forced to remain closed as communities experienced government-
3 Starbucks. Starbucks Fiscal 2020 Annual Report. Seattle: Starbucks Corporation, 2020. https://s22.q4cdn.com/869488222/files/ doc_financials/2020/ar/2020-Starbucks-Annual-Report.pdf
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20.1 • The Importance of Risk Management 601
mandated lockdowns. Locations that were able to service customers through drive-up windows were not immune to declining revenue due to the pandemic. As fewer people gathered in the workplace, Starbucks experienced a declining number of to-go orders from meeting attendees. In addition, Starbucks locations faced the risk of illness spreading as baristas gathered in their buildings to fill to-go orders.
While COVID-19 brought discussions of risk to the forefront of everyday conversations, risk was an important focus of companies such as Starbucks before the pandemic began. (The term risk appeared in the company's 2019 annual report 82 times.4) Starbucks's business model revolves around turning coffee beans into a pleasurable drink. Anything that impacts the company's ability to procure coffee beans, produce a drink, and sell that drink to the customer will impact the company's profitability.
The investors in the company have allowed Starbucks to use its capital to lease storefronts, purchase espresso machines, and obtain all of the assets necessary for the company to operate. Debt holders expect interest to be paid and their principal to be returned. Stockholders expect a return on their investment. Because investors are risk averse, the riskier they perceive the cash flows they will receive from the business to be, the higher the expected return they will require to let the company use their money. This required return is a cost of doing business. Thus, the riskier the cash flows of a company, the higher the cost of obtaining capital. As any cost of operating a business increases, the value of the firm declines.
Starbucks
The most recent annual report for Starbucks Corp., along with the reports from recent years, is available on the company's investor relations website under the Financial Data section (https://0penstax.0rg/r/ Financial Data section). Go to the most recent annual report for the company. Search for the word risk in the annual report, and read the discussions surrounding this topic. Note the major types of risk the company discusses. Pay attention to the types of risk that Starbucks categorizes as uncontrollable and which types of risk the company attempts to mitigate.
In the following sections, you will learn about some of the types of risk that firms commonly face. You will also learn about ways in which firms can reduce their exposure to these risks. When firms take actions to reduce their exposures to risk, they are said to be hedging. Firms hedge to try to protect themselves from losses. Thus, in finance, hedging is a risk management tool.
Certain strategies are commonly used by firms to hedge risk, which is part of corporate financial management. Many of these same strategies can be used by economic players who wish to speculate. Speculating occurs when someone bets on a future outcome. It involves trying to predict the future and profit off of that prediction, knowing that there is some risk that an incorrect prediction will lead to a loss. Speculators bet on the future direction of an asset price. Thus, speculation involves directional bets.
If you are concerned that the price of hand sanitizer is going to rise because people are concerned about a new virus and you purchase a few extra bottles to keep on your shelf "just in case," you are hedging. If you see this situation as a business opportunity and purchase bottles of hand sanitizer, hoping that you can sell them on eBay in a few weeks at twice what you paid for them, you are speculating.
In the popular press, you will often hear of some of the strategies in this chapter discussed in terms of people using them to speculate. In upper-level finance courses, these strategies are discussed in more depth, including how they might be used to speculate. In this chapter, however, the focus is on the perspective of a financial manager using these strategies to manage risk.
4 Starbucks. Starbucks Fiscal 2019 Annual Report. Seattle: Starbucks Corporation, 2019. https://s22.q4cdn.com/869488222/files/ doc_financials/2019/2019-Annual-Report.pdf
602 20 • Risk Management and the Financial Manager
20.2
Commodity Price Risk
Learning Outcomes
By the end of this section, you will be able to:
■ Describe commodity price risk.
■ Explain the use of long-term contracts as a hedge.
■ Explain the use vertical integration as a hedge.
■ Explain the use of futures contracts as a hedge.
One of the most significant risks that many companies face arises from normal business operations. Companies purchase raw materials to produce the products and provide the services they sell. A change in the market price of these raw materials can significantly impact the profitability of a company.
For example, Starbucks must purchase coffee beans in order to make its coffee drinks. The price of coffee beans is highly volatile. Sample prices of a pound of Arabica coffee beans over the past couple of decades are shown in Table 20.1. Over this period, the price of coffee beans ranged from a low of $0.52 per pound in the summer of 2002 to a high of over $3.00 per pound in the spring of 2011. The costs, and thus the profits, of Starbucks will vary greatly depending on if the company is paying less than $1.00 per pound for coffee or if it is paying three times that much.
Date Price per Pound ($)
January 1, 2000 1.09 January 1, 2004 0.74 January 1, 2008 1.39 January 1,2012 2.41 January 1, 2016 1.46 January 1, 2020 1.50
Table 20.1 Price of Coffee in Select Years,
Long-Term Contracts
One method of hedging the risk of volatile input prices is for a firm to enter into long-term contracts with its suppliers. Starbucks, for example, could enter into an agreement with a coffee farmer to purchase a particular quantity of coffee beans at a predetermined price over the next several years.
These long-term contracts can benefit both the buyer and the seller. The buyer is concerned that rising commodity prices will increase its cost of goods sold. The seller, however, is concerned that falling commodity prices will mean lower revenue. By entering into a long-term contract, the buyer is able to lock in a price for its raw materials and the seller is able to lock in its sales price. Thus, both parties are able to reduce uncertainty.
While long-term contracts reduce uncertainty about the commodity price, and thus reduce risk, there are several possible disadvantages to these types of contracts. First, both parties are exposed to the risk that the other party may default and fail to live up to the terms of the contract. Second, these contracts cannot be entered into anonymously; the parties to the contract know each other's identity. This lack of anonymity may have strategic disadvantages for some firms. Third, the value of this contract cannot be easily determined, making it difficult to track gains and losses. Fourth, canceling the contract may be difficult or even impossible.
5 Data from International Monetary Fund. "Global Price of Coffee, Other Mild Arabica (PCOFFOTMUSDM)." FRED. Federal Reserve Bank of St. Louis, accessed August 6, 2021. https://fred.stlouisfed.org/series/PCOFFOTMUSDM
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20.2 • Commodity Price Risk 603
Vertical Integration
A common method of handling the risk associated with volatile input prices is vertical integration, which involves the merger of a company and its supplier. For Starbucks, a vertical integration would involve Starbucks owning a coffee bean farm. If the price of coffee beans rises, the firm's costs increase and the supplier's revenues rise. The two companies can offset these risks by merging.
Although vertical integration can reduce commodity price risk, it is not a perfect hedge. Starbucks may decrease its commodity price risk by purchasing a coffee farm, but that action may expose it to other risks, such as land ownership and employment risk.
Futures Contracts
Another method of hedging commodity price risk is the use of a futures contract. A commodity futures contract is designed to avoid some of the disadvantages of entering into a long-term contract with a supplier. A futures contract is an agreement to trade an asset on some future date at a price locked in today. Futures exist for a range of commodities, including natural resources such as oil, natural gas, coal, silver, and gold and agricultural products such as soybeans, corn, wheat, rice, sugar, and cocoa.
Futures contracts are traded anonymously on an exchange; the market price is publicly observable, and the market is highly liquid. The company can get out of the contract at any time by selling it to a third party at the current market price.
A futures contract does not have the credit risk that a long-term contract has. Futures exchanges require traders to post margin when buying or selling commodities futures contracts. The margin, or collateral, serves as a guarantee that traders will honor their obligations. Additionally, through a procedure known as marking to market, cash flows are exchanged daily rather than only at the end of the contract. Because gains and losses are computed each day based on the change in the price of the futures contract, there is not the same risk as with a long-term contract that the counterparty to the contract will not be able to fulfill their obligation.
The CME Group
In 2007, the Chicago Mercantile Exchange merged with the Chicago Board of Trade to form CME Group Inc. CME Group provides trading in futures as well as other types of contracts that companies can use to hedge risk.
You can watch the video Getting Started with Your Broker (https://0penstax.0rg/r/ Getting Started with Your Broker) to learn how futures contracts for agricultural products such as coffee beans, corn, wheat, and soybeans are traded. You will also see other types of futures contracts traded, including futures for silver, crude oil, natural gas, Japanese yen, and Russian rubles.
604 20 • Risk Management and the Financial Manager
20.3
Exchange Rates and Risk
Learning Outcomes
By the end of this section, you will be able to:
■ Describe exchange rate risk.
■ Identify transaction, translation, and economic risks.
■ Describe a natural hedge.
■ Explain the use of forward contracts as a hedge.
■ List the characteristics of an option contract.
■ Describe the payoff to the holder and writer of a call option.
■ Describe the payoff to the holder and writer of a put option.
The managers of companies that operate in the global marketplace face additional complications when managing the riskiness of their cash flows compared to domestic companies. Managers must be aware of differing business climates and customs and operate under multiple legal systems. Often, business must be conducted in multiple languages. Geopolitical events can impact business relationships. In addition, the company may receive cash flows and make payments in multiple currencies.
Exchange Rates
The costs to companies are impacted when the prices of the raw materials they use change. Very little coffee is grown in the United States. This means that all of those coffee beans that Starbucks uses in its espresso machines in Seattle, New York, Miami, and Houston were bought from suppliers outside of the United States. Brazil is the largest coffee-producing country, exporting about one-third of the world's coffee.6 When a company purchases raw materials from a supplier in another country, the company needs not just money but the money that is used in that country to make the purchase. Thus, the company is concerned about the exchange rate, or the price of the foreign currency.
6
2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021
-Brazil / US Foreign Exchange Rate
Figure 20.2 Brazilian Reals to One US Dollar7
The currency used in Brazil is called the Brazilian real. Figure 20.2 shows how many Brazilian reals could be purchased for $1.00 from 2010 through the first quarter of 2021. In March 2021, 5.4377 Brazilian reals could be purchased for $1.00. This will often be written in the form of
6 Global Agricultural Information Network. Brazil: Coffee Annual 2019. GAIN Report No. BR19006. Washington, DC: USDA Foreign Agricultural Service, May 2019. https://apps.fas.usda.gov/newgainapi/api/report/ downloadreportbyfilename?filename=Coffee%20Annual_Sao%20Paulo%20ATO_Brazil_5-16-2019.pdf
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20.3 • Exchange Rates and Risk 605
USD 1 = BRL 5.4377
BRL is an abbreviation for Brazilian real, and USD is an abbreviation for the US dollar. This price is known as a currency exchange rate, or the rate at which you can exchange one currency for another currency.
If you know the price of $1.00 is 5.4377 Brazilian reals, you can easily find the price of Brazilian reals in US dollars. Simply divide both sides of the equation by 5.4377, or the price of the US dollar:
USD 1 = BRL 5.4377
USD 1 _ BRL 5.4377 5.4377 5.4377
USD 0.1839 = BRL 1
If you have US dollars and want to purchase Brazilian reals, it will cost you $0.1839 for each Brazilian real you want to buy.
The foreign exchange rate changes in response to demand for and supply of the currency. In early 2020, the exchange rate was USD 1 = BRL 4. In other words, $1 purchased fewer reals in early 2020 than in it did a year later. Because you receive more reals for each dollar in 2021 than you would have a year earlier, the dollar is said to have appreciated relative to the Brazilian real. Likewise, because it takes more Brazilian reals to purchase $1.00, the real is said to have depreciated relative to the US dollar.
Exchange Rate Risks
Starbucks, like other firms that are engaged in international business, faces currency exchange rate risk. Changes in exchange rates can impact a business in several ways. These risks are often classified as transaction, translation, or economic risk.
Transaction Risk
Transaction risk is the risk that the value of a business's expected receipts or expenses will change as a result of a change in currency exchange rates. If Starbucks agrees to pay a Brazilian coffee grower seven million Brazilian reals for an order of one million pounds of coffee beans, Starbucks will need to purchase Brazilian reals to pay the bill. How much it will cost Starbucks to purchase these Brazilian reals depends on the exchange rate at the time Starbucks makes the purchase.
In March 2021, with an exchange rate of USD 0.1839 = BRL 1, it would have cost Starbucks $0.1839 X 7,000,000 = $1,287,300 to purchase the reals needed to receive the one million pounds of coffee beans. If, however, Starbucks agreed in March to purchase the coffee beans several months later, in July, Starbucks would not have known then what the exchange rate would be when it came time to complete the transaction. Although Starbucks would have locked in a price of BRL 7,000,000 for one million pounds of coffee beans, it would not have known what the coffee beans would cost the company in terms of US dollars.
If the US dollar appreciated so that it cost less to purchase each Brazilian real in July, Starbucks would find that it was paying less than $1,287,300 for the coffee beans. For example, suppose the dollar appreciated so that the exchange rate was USD 0.1800 = BRL 1 in July 2021. Then the coffee beans would only cost Starbucks $0.1800 X 7,000,000 = $1,260,000.
On the other hand, if the US dollar depreciated and it cost more to purchase each Brazilian real, then Starbucks would find that its dollar cost for the coffee beans was higher than it expected. If the US dollar depreciated (and the Brazilian real appreciated) so that the exchange rate was USD 0.2000 = BRL 1 in July 2021, then the coffee beans would cost Starbucks $0.2000 X 7,000,000 = $1,000,400. This uncertainty regarding the dollar cost of the coffee beans Starbucks would purchase to make its lattes is an example of transaction risk.
7 Data from Board of Governors of the Federal Reserve System (US). "Brazil / US Foreign Exchange Rate (DEXBZUS)." FRED. Federal Reserve Bank of St. touis, accessed August 6, 2021. https://fred.stlouisfed.org/series/DEXBZUS
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A global company such as Starbucks has transaction risk not only because it is purchasing raw materials in foreign countries but also because it is selling its product—and thus collecting revenue—in foreign countries. Customers in Japan, for example, spend Japanese yen when they purchase a Starbucks cappuccino, coffee mug, or bag of coffee beans. Starbucks must then convert these Japanese yen to US dollars to pay the expenses that it incurs in the United States to produce and distribute these products.
The Japanese yen-US dollar foreign exchange rates from 2011 through the first quarter of 2021 are shown in Figure 20.3. In 2012, $1.00 could be purchased with fewer than 80 Japanese yen. In 2015, it took over 120 yen to purchase $1.00.
130
2011
2012 2013 2014 2015 2016 2017 2018 2019
2020 2021
—Japan / US Foreign Exchange Rate Figure 20.3 Japanese Yen to One US Dollar8
If a company is receiving yen from customers and paying expenses in dollars, the company is harmed when the yen depreciates relative to the dollar, meaning that the yen the company receives from its customers can be exchanged for fewer dollars. Conversely, when the yen appreciates, it takes fewer yen to purchase each dollar; this appreciation of the yen benefits companies with revenues in yen and expenses in dollars.
Projecting Sales in US Dollars
The managers of a firm think that the exchange rate of Japanese yen to US dollars will be JPY 100 = USD 1 next year. If the company thinks that it will have sales of 50 million yen next year, how much will it project these sales will be worth in dollars? What happens if the actual exchange rate over the next year is JPY 120 = USD 1?
Solution:
An exchange rate of JPY 100 = USD 1 implies that JPY 1 = USD 0.0100. If the company expects to have sales of JPY 50,000,000, it is expecting sales to be worth USD 500,000. If the exchange rate is really JPY 120 = USD 1, this is the same as JPY 1 = USD 0.0083. With this exchange rate, JPY 50,000,000 in sales would be equivalent to USD 415,000. The company would receive $85,000 less from these sales in Japan than it was expecting, even if it met its goals as far as the number of units of items sold.
8 Data from Board of Governors of the Federal Reserve System (US). "Japan / US Foreign Exchange Rate (DEXJPUS)." FRED. Federal Reserve Bank of St. Louis, accessed August 6, 2021. https://fred.stlouisfed.org/series/DEXJPUS
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20.3 • Exchange Rates and Risk 607
Translation Risk
In addition to the transaction risk, if Starbucks holds assets in a foreign country, it faces translation risk. Translation risk is an accounting risk. Starbucks might purchase a coffee plantation in Costa Rica for 120 million Costa Rican colones. This land is an asset for Starbucks, and as such, the value of it should appear on the company's balance sheet.
The balance sheet for Starbucks is created using US dollar values. Thus, the value of the coffee plantation has to be translated to dollars. Because exchange rates are volatile, the dollar value of the asset will vary depending on the day on which the translation takes place. If the exchange rate is 500 colones to the dollar, then this coffee plantation is an asset with a value of $240,000. If the Costa Rican colon depreciates to 600 colones to the dollar, then the asset has a value of only $200,000 when translated using this exchange rate.
Although it is the same piece of land with the same productive capacity, the value of the asset, as reported on the balance sheet, falls as the Costa Rican colon depreciates. This decrease in the value of the company's assets must be offset by a decrease in the stockholders' equity for the balance sheet to balance. The loss is due simply to changes in exchange rates and not the underlying profitability of the company.
Economic Risk
Economic risk is the risk that a change in exchange rates will impact a business's number of customers or its sales. Even a company that is not involved in international transactions can face this type of risk. Consider a company located in Mississippi that makes shirts using 100% US-grown cotton. All of the shirts are made in the United States and sold to retail outlets in the United States. Thus, all of the company's expenses and revenues are in US dollars, and the company holds no assets outside of the United States.
Although this firm has no financial transactions involving international currency, it can be impacted by changes in exchange rates. Suppose the US dollar strengthens relative to the Vietnamese dong. This will allow US retail outlets to purchase more Vietnamese dong, and thus more shirts from Vietnamese suppliers, for the same amount of US dollars. Because of this, the retail outlets experience a drop in the cost of procuring the Vietnamese shirts relative to the shirts produced by the firm in Mississippi. The Mississippi company will lose some of its customers to these Vietnamese producers simply because of a change in the exchange rate.
Hedging
Just as companies may practice hedging techniques to reduce their commodity risk exposure, they may choose to hedge to reduce their currency risk exposure. The types of futures contracts that we discussed earlier in this chapter exist for currencies as well as for commodities. A company that knows that it will need Korean won later this year to purchase raw materials from a South Korean supplier, for example, can purchase a futures contract for Korean won.
While futures contracts allow companies to lock in prices today for a future commitment, these contracts are not flexible enough to meet the risk management needs of all companies. Futures contracts are standardized contracts. This means that the contracts have set sizes and maturity dates. Futures contracts for Korean won, for example, have a contract size of 125 million won. A company that needs 200 million won later this year would need to either purchase one futures contract, hedging only a portion of its needs, or purchase two futures contracts, hedging more than it needs. Either way, the company has remaining currency risk.
In this next section, we will explore some additional hedging techniques.
Forward Contracts
Suppose a company needs access to 200 million Korean won on March 1. In addition to a specified contract size, currency futures contracts have specified days on which the contracts are settled. For most currency futures contracts, this occurs on the third Wednesday of the month. If the company needed 125 million Korean won (the basic contract size) on the third Wednesday of March (the standard settlement date), the futures
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contract could be useful. Because the company needs a different number of Korean won on a different date from those specified in the standard contract, the futures contract is not going to meet the specific risk management needs of the company.
Another type of contract, the forward contract, can be used by this company to meet its specific needs. A forward contract is simply a contractual agreement between two parties to exchange a specified amount of currencies at a future date. A company can approach its bank, for example, saying that it will need to purchase 200 million Korean won on March 1. The bank will quote a forward rate, which is a rate specified today for the sale of currency on a future date, and the company and the bank can enter into a forward contract to exchange dollars for 200 million Korean won at the quoted rate on March 1.
Because a forward contract is a contract between two parties, those two parties can specify the amount that will be traded and the date the trade will occur. This contract is similar to your agreeing with a hotel that you will arrive on March 1 and rent a room for three nights at $200 per night. You are agreeing today to show up at the hotel on a future (specified) date and pay the quoted price when you arrive. The hotel agrees to provide you the room on March 1 and cannot change the price of the room when you arrive. With a forward contract, you are also agreeing that you will indeed make the purchase and you cannot change your mind; so, using the hotel room analogy, this would mean that the hotel will definitely charge your credit card for the agreed-upon $200 per night on March 1.
The forward contract is an individualized contract between the buyer and the seller; they are both under a contractual obligation to honor the contract. Because this contract is not standardized like the futures contract (so that it can be traded on an exchange), it can be tailored to the needs of the two parties. While the forward contract has the advantage of being fine-tuned to meet the company's needs, it has a risk, known as counterparty risk, that the futures contract does not have. The forward contract is only as good as the promise of the counterparty. If the company enters into a forward contract to purchase 200 million Korean won on March 1 from its bank and the bank goes out of business before March 1, the company will not be able to make the exchange with a nonexistent bank. The exchanges on which futures contracts are traded guard the purchaser of a futures contract from this type of risk by guaranteeing the contract.
Natural Hedges
A hedge simply refers to a reduction in the risk or exposure that a company has to volatility and uncertainty. We have been focusing on how a company might use financial market instruments to hedge, but sometimes a company can use a natural hedge to mitigate risk. A natural hedge occurs when a business can offset its risk simply through its own operations. With a natural hedge, when a risk occurs that would decrease the value of a company, an offsetting event occurs within the firm that increases the value of the company.
As an example, consider a British-based travel agency. One of the major tours the company offers is a tour of Italy. The company arranges for transportation, lodging, meals, and sightseeing for Brits to visit the highlights of Rome, Florence, and Venice. Because the company charges customers in British pounds but must pay the bus companies, hotels, and other service providers in Italy in euros, the travel agency faces significant transaction exposure. If the value of the British pound depreciates after the company sets the price it will charge for the tour but before it pays the Italian suppliers, the company will be harmed. In fact, if the British pound depreciates by a great deal, the company could end up in a situation in which the British pounds it collects are not enough to purchase the euros it needs to pay its suppliers.
The company could create a natural hedge by offering tours of London to individuals living in the European Union. The travel agency could charge people who live in Germany, Italy, Spain, or any other country that has the euro as its currency for a travel package to London. Then the agency would pay British restaurants, tour guides, hotels, and bus companies in British pounds. This segment of the business also has currency risk. If the British pound depreciates, the company gains because the euros it collects from its EU customers will purchase more British pounds than before.
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20.3 • Exchange Rates and Risk 609
Thus, the company has created a situation in which if the British pound depreciates, the decrease in value of its tours of Italy is exactly offset by the increase in value of its tours of London. If the British pound appreciates, the opposite occurs: the company experiences a gain in its division that charges British pounds for tourists traveling to Italy and an offsetting loss in its division that charges euros for tourists traveling to London.
Options
A financial option gives the owner the right, but not the obligation, to purchase or sell an asset for a specified price at some future date. Options are considered derivative securities because the value of a derivative is derived from, or comes from, the value of another asset.
Options Terminology
Specific terminology is used in the finance industry to describe the details of an options contract. If the owner of an option decides to purchase or sell the asset according to the terms of the options contract, the owner is said to be exercising the option. The price the option holder pays if purchasing the asset or receives if selling the asset is known as the strike price or exercise price. The price the owner of the option paid for the option is known as the premium.
An option contract will have an expiration date. The most common kinds of options are American options,
which allow the holder to exercise the option at any time up to and including the expiration date. Holders of European options may exercise their options only on the expiration date. The labels American option and European option can be confusing as they have nothing to do with the location where the options are traded. Both American and European options are traded worldwide.
Option contracts are written for a variety of assets. The most common option contracts are options on shares of stock. Options are traded for US Treasury securities, currencies, gold, and oil. There are also options on agricultural products such as wheat, soybeans, cotton, and orange juice. Thus, options can be used by financial managers to hedge many types of risk, including currency risk, interest rate risk, and the risk that arises from fluctuations in the prices of raw materials.
Options are divided into two main categories, call options and put options. A call option gives the owner of the option the right, but not the obligation, to buy the underlying asset. A put option gives the owner the right, but not the obligation, to sell the underlying asset.
Call Options
If a Korean company knows that it will need pay a $100,000 bill to a US supplier in six months, it knows how many US dollars it will need to pay the bill. As a Korean company, however, its bank account is denominated in Korean won. In six months, it will need to use its Korean won to purchase 100,000 US dollars.
The company can determine how many Korean won it would take to purchase $100,000 today. If the current exchange rate is KWN 1,100 = USD 1, then it will need KWN 110,000,000 to pay the bill. The current exchange rate is known as the spot rate.
The company, however, does not need the US dollars for another six months. The company can purchase a call option, which is a contract that will allow it to purchase the needed US dollars in six months at a price stated in the contract. This allows the company to guarantee a price for dollars in six months, but it does not obligate the company to purchase the dollars at that price if it can find a better price when it needs the dollars in six months.
The price that is in the contract is called the strike price (exercise price). Suppose the company purchases a call contract for US dollars with a strike price of KWN 1,200/USD. While this contract would be for a set size, or a certain number of US dollars, we will talk about this transaction as if it were per one US dollar to highlight how options contracts work.
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The company must pay a price, known as the premium, to purchase this call option contract. For our example, let's assume the premium for the call option contract is KWN 50. In other words, the company has paid KWN 50 for the right to buy US dollars in six months for a price of KWN 1,200/USD.
In six months, the company makes a choice to either (1) pay the strike price of KWN 1,200/USD or (2) let the option expire. If the company chooses to pay the strike price and purchase the US dollars, it is exercising the option. How does the company choose which to do? It simply compares the strike price of KWN 1,200/USD to the market, or spot, exchange rate at the time the option is expiring.
If, six months from now, the spot exchange rate is KWN 1,150 = USD 1, it will be cheaper for the company to buy the US dollars it needs at the spot price than it would be to buy the dollars with the option. In fact, if the spot rate is anything below KWN 1,200 = USD 1, the company will not choose to exercise the option. If, however, the spot exchange rate in six months is KWN 1,300 = USD 1, the company will exercise the option and purchase each US dollar for only KWN 1,200.
The profitability, or the payoff, to the owner of a call option is represented by the chart in Figure 20.4 below. Possible spot prices are measured from left to right, and the financial gain or loss to the company of the option contract is measured vertically. If the spot price is anything less than KWN 1,200/USD, the option expires without being exercised. The company paid KWN 50 for something that ended up being worthless.
Figure 20.4 The Payoff to the Holder of a Call Option
If, in six months, the spot exchange rate is KWN 1,225 = USD 1, then the company will choose to exercise the option. The company will be saving KWN 25 for each dollar purchased, but the company originally paid 50 KWN for the contract. So, the company will be 25 KWN worse off than if it had never purchased the call option.
If the spot exchange rate is KWN 1,250 = USD 1, the company will be in exactly the same position having purchased and exercised the call option as it would have been if it had not purchased the option. At any spot price higher than KWN 1,250/USD, the firm will be in a better financial position, or will have a positive payoff, because it purchased the call option. The more the Korean won depreciates over the next six months, the higher the payoff to the firm of owning the call contract. Purchasing the call contract is a way that the company can protect itself from the currency exposure it faces.
For any transaction, there must be two parties—a buyer and a seller. For the company to have purchased the call option, another party must have sold the call option. The seller of a call option is called the option writer. Let's consider the potential benefits and risks to the writer of the call option.
When the company purchases the call option, it pays the premium to the writer. The writer of the option does not have a choice regarding whether the option will be exercised. The purchaser of the option has the right to make the choice; in essence, the writer of the option sold the right to make that decision to the purchasers of the call option.
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20.3 • Exchange Rates and Risk 611
Figure 20.5 shows the payoff to the writer of the call option. Recall that the buyer of the call option will let the option expire if the spot rate is less than KWN 1,200 = USD 1 when the call option matures in six months. If this occurs, the writer of the option collected the KWN 50 option premium when the contract was sold and then never hears from the purchaser again. This is what the writer of the option is hoping for; the writer of the call option profits when the options contract is not exercised
Figure 20.5 The Payoff to the Writer of a Call Option
If the spot rate is above KWN 1,200 = USD 1, then the holder of the option will choose to exercise the right to purchase the won at the option strike price. Then the writer of the option will be obligated to sell the Korean won at a price of KWN 1,200/USD. If the spot rate is KWN 1,250 = USD 1, the option writer will be obligated to sell the dollars for KWN 50 less than what they are worth; because the option writer was initially paid a KWN 50 premium for taking on that obligation, the option writer will just break even. For any exchange rate higher than KWN 1,250 = USD 1, the writer of the call option will have a loss.
The option contract is a zero-sum game. Any payoff the owner of the option receives is exactly equal to the loss the writer of the option has. Any loss the owner of the option has is exactly equal to the payoff the writer of the option receives.
Put Options
While the call option you just considered gives the owner the right to buy an underlying asset, the put option gives the owner to right to sell an underlying asset. Take, for example, an Indian company that has a contract to provide graphic artwork for a US company. The US company will pay the Indian company 200,000 US dollars in three months.
While the Indian company receives US dollars, it must pay its workers in Indian rupees. Because the company does not know what the spot exchange rate will be in three months, it faces transaction risk and may be interested in hedging this exposure using a put option.
The company knows that the current spot rate is INR 75 = USD 1, meaning that the company would be able to use $200,000 to purchase USD 200,000 X "^J5 = INR 15,000,000 if it possessed the $200,000 today. If the Indian rupee appreciates relative to the US dollar over the next three months, however, the company will receive fewer rupees when it makes the exchange; perhaps the company will not be able to purchase enough rupees to cover the wages of its employees.
Assume the company can purchase a put option that gives it the right to sell US dollars in three months at a strike price of INR 75/USD; the premium for this put option is INR 5. By purchasing this put option, the company is spending INR 5 to guarantee that it can sell its US dollars for rupees in three months at a price of INR 75/USD.
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If, in three months, when the company receives payment in US dollars, the spot exchange rate is higher than INR 75 = USD 1, the company will simply exchange the US dollars for rupees at that exchange rate, allowing the put option to expire without exercising it. The payoff to the company for the option is INR -5, the premium that was paid for the option that was never used (see Figure 20.6).
Figure 20.6 The Payoff to the Holder of a Put Option
If, however, in three months, the spot exchange rate is anything less than INR 75 = USD 1, then the
company will choose to exercise the option. If the spot rate is between INR 70 = USD 1 and
INR 75 = USD 1, the payoff for the option is negative. For example, if the spot exchange rate is
INR 72 = USD 1, the company will exercise the option and receive three more Indian rupees per dollar than
it would in the spot market. However, the company had to spend INR 5 for the option, so the payoff is INR -2.
At a spot exchange rate of INR 70 = USD 1, the company has a zero payoff; the benefit of exercising the
option, INR 5, is exactly equal to the price of purchasing the option, the premium of INR 5.
If, in three months, the spot exchange rate is anything below INR 70 = USD 1, the payoff of the put option is positive. At the theoretical extreme, if the USD became worthless and would purchase no rupees in the spot market when the company received the dollars, the company could exercise its option and receive INR 75/USD, and its payoff would be INR 70.
Now that we have considered the payoff to a purchaser of a put contract, let's consider the opposite side of the contract: the seller, or writer, of the put option. The writer of a put option is selling the right to sell dollars to the purchaser of the put option. The writer of the put option collects a premium for this. The writer of the put has no choice as to whether the put option will be exercised; the writer only has an obligation to honor the contract if the owner of the put option chooses to exercise it.
The owner of the option will choose to let the option expire if the spot exchange rate is anything above INR 75 = USD 1. If that is the case, the writer of the put option collects the INR 5 premium for writing the put, as shown by the horizontal line in Figure 20.7. This is what the writer of the put is hoping will occur.
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20.3 • Exchange Rates and Risk 613
Spot Rate
Figure 20.7 The Payoff to the Writer of a Put Option
The owner of the option will choose to exercise the option if the exchange rate is less than INR 75 = USD 1. If the spot exchange rate is between INR 70 = USD 1 and INR 75 = USD 1, the writer of the put option has a positive payoff. Although the writer must now purchase US dollars for a price higher than what the dollars are worth, the INR 5 premium that the writer received when entering into the position is more than enough to offset that loss.
If the spot exchange rate drops below INR 70 = USD 1, however, the writer of the put option is losing more than INR 5 when the option is exercised, leaving the writer with a negative payoff. In the extreme, the writer of the put will have to purchase worthless US dollars for INR 75/USD, resulting in a loss of INR 70.
Notice that the payoff to the writer of the put is the negative of the payoff to the holder of the put at every spot price. The highest payoff occurs to the writer of the put when the option is never exercised. In that instance, the payoff to the writer is the premium that the holder of the put paid when purchasing the option (see Figure 20.7).
Table 20.2 provides a summary of the positions that the parties who enter into options contract are in. Remember that the buyer of an option is always the one purchasing the right to do something. The seller or writer of an option is selling the right to make a decision; the seller has the obligation to fulfill the contract should the buyer of the option choose to exercise the option. The most the seller of an option can ever profit is by the premium that was paid for the option; this occurs when the option is not exercised.
Benefits Harm
Party to an Option Contract Right of the Party Obligation of the Party When Maximum Profit When Maximum Loss
Buyer of a call To buy Price of underlying rises Unlimited Price of underlying falls Premium paid
Seller of a call To sell Price of underlying falls Premium received Price of underlying rises Unlimited
Buyer of a put To sell Price of underlying falls Strike price minus premium Price of underlying rises Premium paid
Table 20.2 Summary of Option Contracts
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Ber íefits H arm
Party to an Option Contract Right of the Party Obligation of the Party When Maximum Profit When Maximum Loss
Seller of a put
To buy
Price of
underlying
rises
Premium received
Price of
underlying
falls
Strike price minus premium
Table 20.2 Summary of Option Contracts
20.4
Interest Rate Risk
Learning Outcomes
By the end of this section, you will be able to:
■ Describe interest rate risk.
■ Explain how a change in interest rates changes the value of cash flows.
■ Describe the use of an interest rate swap.
An interest rate is simply the price of borrowing money. Just as other prices are volatile, interest rates are also volatile. Just as volatility in other prices leads to uncertain cash flows for a company, volatility in interest rates can also lead to uncertain cash flows.
Measuring Interest Rate Risk
Suppose that a company is supposed to pay a bill of $1,000 in 10 years. The present value of this bill depends
$1 000
on the level of interest rates. If the interest rate is 5%, the present value of the bill is ■»■»»» = $613-9L If
(1 + 0.05)10
ffl AAA
the interest rate rises to 6%, the present value of the bill is ——-^ — $558.39. The increase in the interest
(1 + 0.06)10
rate by 1% causes the present value of the expected cash flow to fall by 613'9611J95158-39 = 0.0904 = 9.04%.
Interest rate risk can be highlighted by looking at bonds. Consider two $1,000 face value bonds with a 5% coupon rate, paid semiannually. One of the bonds matures in five years, and the other bond matures in 30 years. If the market interest rate is 5%, each of these bonds will sell for face value, or $1,000. If, instead, the market interest rate is 6%, the five-year bond will sell for $957.35 and the 30-year bond will sell for $861.62.
Notice that as the interest rate rises, the price of both of these bonds will fall. However, the price of the longer-term bond will fall by more than the price of the shorter-term bond. The longer-term bond price will fall by 1.38%; the shorter-term bond price will fall by only 0.43%.
Consider two additional $1,000 face value bonds. The difference is that these bonds have a 6% coupon rate, paid semiannually. If a bond has a 6% coupon rate and matures in five years, it will sell for $1,043.76 when the market interest rate is 5%. A 30-year bond that matures in 30 years and has a 6% coupon rate will sell for $1,154.54 when the market interest rate is 5%. However, if the interest rate in the economy is 6%, both of these bonds will sell for a price of $1,000. The price of the five-year bond will drop by 4.19%; the price of the 30-year bond will drop by 13.39%.
Calculating Bond Prices as the Interest Rates Changes
You are considering purchasing a $10,000 face value bond with a 4% coupon rate, paid semiannually, that matures in 20 years. If you require a 5% return to purchase this bond, what is the maximum price you
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20.4 • Interest Rate Risk 615
would be willing to pay for the bond? If, instead, you require an 8% return to purchase this bond, what is the maximum price you would be willing to pay for the bond?
Solution:
If the bond pays coupon interest semiannually, you will receive one-half of the face value of the bond
multiplied by the coupon rate every six months. So, you will receive 40 coupon payments of
$10,000 X 0.02 = $200. At maturity, you will receive one lump sum of the face value of the bond. Follow
the steps in Table 20.3 to calculate the price of the bond if you require a 5% return, using a financial
calculator.
Step Description Enter Display
1 Enter the number of coupon payments you will receive 40 N N = 40
2 Enter your semiannual required return 2.5 I/Y I/Y = 2.5
3 Enter the semiannual coupon payment 200 PMT PMT = 200
4 Enter the face value of the bond 10000 FV FV = 10,000
5 Calculate the present value CPT PV PV = -8,744.86
Table 20.3 Calculator Step to Price a Bond Requiring a 5% Return9
When your required yield is 5%, the most you would be willing to pay for this bond is $8,744.86.
To calculate the price of the bond if your required return is 8%, use the same process, replacing the I/YR in step 2 with 4 (see Table 20.4). All other variables remain the same because the characteristics of the bond have not changed.
Step Description Enter Display
1 Enter the number of coupon payments you will receive 40 N N = 40
2 Enter your semiannual required return 4 I/Y I/Y = 4
3 Enter the semiannual coupon payment 200 PMT PMT = 200
4 Enter the face value of the bond 10000 FV FV = 10,000
5 Calculate the present value CPT PV PV = -6,041.45
Table 20.4 Calculator Steps to Price a Bond Requiring an 8% Return
If your required return is 8% to invest in this bond, you will be willing to pay only $6,041.45 to purchase the bond.
Thus, if interest rates rise because of changing market conditions, the price of bonds will fall.
The sensitivity of bond prices to changes in the interest rate is known as interest rate risk. Duration is an important measure of interest rate risk that incorporates the maturity and coupon rate of a bond as well as the level of current market interest rates. Calculating duration is a complex topic that is beyond the scope of this introductory textbook, but it is useful to note that
■ the higher the duration of a bond, the more sensitive the price of the bond will be to interest rate
9 The specific financial calculator in these examples is the Texas Instruments BA II Plus™ Professional model, but you can use other financial calculators for these types of calculations.
616 20 • Risk Management and the Financial Manager
changes;
■ the duration of a bond will be higher when market yields are lower, all else being equal;
■ the duration of a bond will be higher the longer the maturity of the bond, all else being equal; and
■ the duration of a bond will be higher the lower the coupon rate on the bond, all else being equal.
Swap-Based Hedging
As the name suggests, a swap involves two parties agreeing to swap, or exchange, something. Generally, the two parties, known as counterparties, are swapping obligations to make specified payment streams.
To illustrate the basics of how an interest rate swap works, let's consider two hypothetical companies. Alpha and Beta. Alpha is a strong, well-established company with a AAA (triple-A) bond rating. This means that Alpha has the highest rating a company can have. With this high rating. Alpha can borrow at relatively low interest rates. Often, companies in this situation will borrow at a floating rate. This means that their interest rate goes up and down as interest rates in the overall economy vary. The floating rate will be tied to a benchmark rate that is widely quoted in the financial press. Historically, companies have often used the London Interbank Offered Rate (LIBOR) as the benchmark rate. Because published quotes for LIBOR will be phased out by 2023, firms are beginning to use alternative rates. As of yet, no single alternative has emerged as the most commonly used rate; therefore, LIBOR will be used in our example. Suppose that Alpha finds that it can borrow money at rate equal to LIBOR + 0.25%; thus, if LIBOR is 2.75%, the company will pay 3.0% to borrow. If the company wants to borrow at a long-term fixed rate, its cost of borrowing will be 5.0%.
LIBOR Transition
Although the basic principles of financial transactions remain the same over time, the particular financial instruments used change from time to time. Innovation, regulation, and technological advances lead to these changes in financial instruments. The use of LIBOR as a benchmark rate is winding down in the early 2020s. To find out more about this transition and how it impacts companies, visit the About LIBOR Transition (https://openstax.Org/r/About LIBOR Transition) website.
Beta has a BBB bond rating. Although this is considered a good, investment-grade rating, it is lower than the rating of Alpha. Because Beta is less creditworthy and a bit riskier than Alpha, it will have to pay a higher interest rate to borrow money. If Beta wants to borrow money at a floating rate, it will need to pay LIBOR + 0.75%. If LIBOR is 2.75%, Beta must pay 3.5% on its floating rate debt. In order for Beta to borrow at a long-term fixed rate, its cost of borrowing will be 6.75%.
Let's consider how these two companies can enter into a swap in which both parties benefit. Table 20.5 summarizes the situation and the rates at which Alpha and Beta can borrow. It also illustrates a way in which an interest rate swap can benefit both Alpha and Beta.
Alpha Beta
Bond rating AAA BBB
Floating rate LIBOR+ 0.25 LIBOR + 0.75
Fixed rate 5 6.75
Rate company chooses Fixed at 5.0 Floating at LIBOR + 0.75
Swap N/A N/A
Beta pays Alpha fixed rate 5.5 -5.5
Table 20.5 Example of a Swap Agreement
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20.4 • Interest Rate Risk 617
Alpha Beta
Alpha pays Beta floating rate -LIBOR +LIBOR
Payments and receipts -5.0 + 5.5 - LIBOR -(LIBOR + 0.75) - 5.5 + LIBOR
Net amount 0.5- LIBOR -6.25
Benefit 0.75 0.5
Table 20.5 Example of a Swap Agreement
Alpha borrows in the capital markets at a fixed rate of 5%. Beta chooses to borrow at a floating rate that equals LIBOR + 0.75%. Beta also agrees to pay Alpha a fixed rate of 5.5%. In essence. Beta is paying 5.5% to Alpha, 0.75% to its lender, and LIBOR to its lender.
In return. Alpha promises to pay Beta LIBOR. The exact amount that Alpha will pay to Beta fluctuates as LIBOR fluctuates. However, from Beta's perspective, the payment of LIBOR it receives from Alpha exactly offsets the payment of LIBOR it makes to its lender. When LIBOR increases, the rate of LIBOR + 0.75% that Beta is paying to its lender increases, but the LIBOR rate it receives from Alpha also increases. When LIBOR decreases. Beta receives less from Alpha, but it also pays less to its lender. Because the LIBOR it receives from Alpha is exactly equal to the LIBOR it pays to its lender. Beta's net amount of interest paid is 6.25%—the 5.5% it pays to Alpha plus the 0.75% it pays to its lender.
Alpha is in the position of paying 5.0% to its lender and LIBOR to Beta while receiving 5.5% from Beta. This means that Alpha's net interest paid is LIBOR - 0.5%. Alpha is said to have swapped its fixed interest rate for a floating rate. Because it is paying LIBOR - 0.5%, it will experience fluctuating interest rates; however, as a company with a AAA bond rating, it is a strong, creditworthy company that can withstand that interest rate exposure. It would have cost Alpha LIBOR + 0.25% to borrow the money from its lenders at a variable rate. By participating in this swap arrangement. Alpha has been able to lower its interest rate by 0.75%.
Through this swap arrangement. Beta has been able to fix its interest rate at 6.25% rather than having a variable rate. This predictability is a benefit for a company, especially one that is in a bit more precarious position as far as its creditworthiness and stability. The 6.25% Beta pays as a result of this arrangement is 0.5% below the 6.75% it would have paid if it simply borrowed from its lenders at a fixed rate.
618 20 •Summary
a Summary
20.1 The Importance of Risk Management
Risk arises due to uncertainty. The future is unpredictable. One job of the financial manager is to manage the risks of both cash inflows and cash outflows. Investors are risk-averse. The riskier a firm's cash flows are, the higher the rate of return investors require to provide capital to the company.
20.2 Commodity Price Risk
Companies do not know how much they will have to pay for raw materials in future months. The price of raw materials will change as economic conditions change, impacting a company's cost of goods sold and profits. Some ways that a company can hedge this risk are through vertical integration, long-term contracts, and futures contracts.
20.3 Exchange Rates and Risk
Exchange rates are unpredictable. This leads to transaction risk, translation risk, and economic risk as currency values change. A forward contract is an agreement between two parties to make an exchange at a particular rate on a given date in the future. Companies can use options to mitigate the risks. A call option gives the holder the right, but not the obligation, to purchase an underlying asset. A put option give the holder the right, but not the obligation, to sell an underlying asset.
20.4 Interest Rate Risk
When interest rates increase, the present value of future cash flows decreases. Duration is a measure of interest rate risk. A swap involves two parties agreeing to exchange something, often specified payment streams.
t Key Terms
American option an option that the holder can exercise at any time up to and including the exercise date appreciate when one unit of a currency will purchase more of a foreign currency than it did previously call option an option that gives the owner the right, but not the obligation, to buy the underlying asset at a
specified price on some future date depreciate when one unit of a currency will purchase less of a foreign currency than it did previously derivative a security that derives its value from another asset duration a measure of interest rate risk
economic risk the risk that a change in exchange rates will impact the number of customers a business has or its sales
European option an option that the holder can exercise only on the expiration date exchange rate the price of one currency in terms of another currency
exercise price (strike price) the price the option holder pays for the underlying asset when exercising an option
exercising choosing to purchase or sell the asset underlying a held option according to the terms of the
option contract expiration date the date an option contract expires
forward contract a contractual agreement between two parties to exchange a specified amount of assets on a specified future date
futures contract a standardized contract to trade an asset on some future date at a price locked in today hedging taking an action to reduce exposure to a risk
margin the collateral that must be posted to guarantee that a trader will honor a futures contract marking to market a procedure by which cash flows are exchanged daily for a futures contract, rather than at the end of the contract
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20- CFA Institute 619
natural hedge when a company offsets the risk that something will decrease in value by having a company
activity that would increase in value at the same time option an agreement that gives the owner the right, but not the obligation, to purchase or sell an asset at a
specified price on some future date option writer seller of a call or put option
premium the price a buyer of an option pays for the option contract
put option an option that gives the owner the right, but not the obligation, to sell the underlying asset at a
specified price on some future date speculating attempting to profit by betting on the uncertain future, knowing that a risk of loss is involved spot rate the current market exchange rate
strike price (exercise price) the price an option holder pays for the underlying asset when exercising the option
swap an agreement between two parties to exchange something, such as their obligations to make specified payment streams
transaction risk the risk that a change in exchange rates will impact the value of a business's expected receipts or expenses
translation risk the risk that a change in exchange rates will impact the value of items on a company's
financial statements vertical integration the merger of a company with its supplier
m CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session (https://openstax.Org/r/cfa-institute-Level-I-Study-Session). Reference with permission of CFA Institute.
a
Multiple Choice
1. Which of the following does a financial manager want to do to maximize the value of the firm?
a. Decrease the speed of money coming into the firm
b. Speed up cash going out and slow down cash coming in
c. Decrease the riskiness of cash inflows and cash outflows
d. Increase the volatility and speed of cash going out of the firm
2. In finance, risk is_.
a. the same thing as profit
b. ignored because it is inevitable
c. thought of as uncertainty or unpredictability
d. something that financial managers should strive to increase and maximize
3. American Jeans Corp. purchases a cotton farm. The cotton grown on the farm will be used to make denim cloth for the company's jeans. This is an example of_.
a. striking a price
b. vertical integration
c. a forward contract
d. an American option
4. The price that a holder of an option pays to buy the underlying asset when exercising a call option is known as_.
a. the strike price
b. the maturity price
620 20 • Multiple Choice
c. the exchange price
d. the underlying premium
5. Which of the following gives the holder the right, but not the obligation, to purchase an underlying asset?
a. A call option
b. A forward contract
c. A European put option
d. An American put option
6. An American option allows the holder to_.
a. exercise the option only on the expiration date
b. exercise the option at any time up to and including the expiration date
c. sell stocks, and a European option allows the holder to purchase stocks
d. purchase stocks, and a European option allows the holder to purchase bonds
7. The holder of a(n)_has the right to buy and the holder of a(n)_has the right to sell an
underlying asset.
a. call option; put option
b. put option; call option
c. American option; European option
d. European option; American option
8. The three main categories of foreign exchange risk a company faces are_.
a. economic risk, business risk, and exposure risk
b. exposure risk, fluctuation risk, and forward risk
c. transaction risk, translation risk, and economic risk
d. appreciation risk, depreciation risk, and duplication risk
9. In January, the exchange rate between the South Korean won and the US dollar was
KWN 1,100 = USD 1. Three months later, the exchange rate was KWN 1,200 = USD 1. This means that
a. the Korean won appreciated relative to the US dollar
b. the Korean won depreciated relative to the US dollar
c. the US dollar depreciated relative to the Korean won
d. both the Korean won and the US dollar appreciated
10. In January, the exchange rate between the South Korean won and the US dollar was
KWN 1,100 = USD 1. Three months later, the exchange rate was KWN 1,200 = USD 1. This means that_.
a. it will cost US companies more to purchase raw materials from South Korea
b. it will cost Korean companies more to purchase raw materials from the United States
c. US companies that sell their products in South Korea will find their revenue has increased
d. Korean companies that sell their products in the United States will find that their revenue has decreased
11. A foreign exchange forward contract_.
a. is a standardized contract that is inflexible
b. occurs when a company swaps its translation exposure for transaction exposure
c. is a contractual agreement between two parties to exchange a specified amount of currencies on a future date
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20 • Review Questions 621
d. states the date on which a trade will take place, but the price for the trade will be determined at the time the trade occurs
12. Which of the following is a measure of interest rate risk?
a. LIBOR
b. Duration
c. Translation exposure
d. Contract inflexibility
13. A swap occurs when_.
a. a company exchanges obligations with another company to make specified payment streams
b. a company purchases commodities from a company in another country, exposing it to both commodity and currency risk
c. a company chooses a local supplier over an international supplier to avoid currency exposure
d. a company chooses a foreign supplier so that its commodity risk will be offset by its currency risk
5, Review Questions
1. What is the difference between someone using a derivative security to hedge risk and someone using a derivative security to speculate?
2. Explain how vertical integration may be used as a method of hedging against commodity price risk.
3. What is the difference between a forward contract and a futures contract?
4. You are considering purchasing a call option to purchase Mexican pesos in three months with a strike price of MXN 20/USD. The premium for this call option is MXN 2. Show the payoff you will receive at various prices in a diagram.
5. You are considering writing a call option to purchase Mexican pesos in three months with a strike price of MXN 20/USD. The premium for this call option is MXN 2. Show the payoff you will receive at various prices in a diagram.
6. Why are options considered to be a "zero-sum game"?
0 Problems
1. The Olive Orchard is a US retail outlet for high-quality olive oils. One of the major suppliers of olive oil for the company is a farm in Greece. The Olive Orchard must pay the Greek farm 5.00 euros per liter of olive oil it purchases. The Olive Orchard would like to purchase 7,000 liters of the Greek farm's olive oil next year. Currently, it costs 0.900 euros to purchase 1 US dollar. If the exchange rate remains constant, how much will it cost the Olive Orchard (in US dollars) to purchase the 7,000 liters? If the exchange rate changes so that it costs 0.8599 euros to purchase 1 US dollar, how much will it cost to purchase the 7,000 liters of olive oil?
2. International Automobile Parts (IAP) holds a call option to purchase US dollars. The strike price on the call option isJPY 115/USD. IAP paidJPY 10 for the option. The spot price isJPY 120/USD, and the option expires today. Should IAP exercise the option? What is IAP's payoff?
3. Global Producers (GP) holds a put option to sell US dollars. The strike price on the put option isJPY 114/ USD. GP paidJPY 10 for the option. The spot price isJPY 120/USD, and the option expires today. Should GP exercise the option? What is GP's payoff?
622 20 • Video Activity
[►J Video Activity
Hedging at Southwest Airlines
Click to view content (https://0penstax.0rg/r/Single Biggest Risk)
Jet fuel costs represent a major expense for airlines. Southwest Airlines has been known as the most aggressive airline when it comes to hedging the risk of jet fuel cost volatility. In this interview, the CEO of Southwest Airlines, Gary Kelly, discusses crude oil prices in the spring of 2012 and the impact on Southwest Airlines.
1. How volatile are oil prices, and how large of an impact does that volatility have on the cost structure of an airline?
2. Gary Kelly states that he sees fuel prices as the largest single business risk Southwest Airlines faces and that hedging that risk has become more expensive. Why do you think it became more expensive for Southwest Airlines to hedge this risk in 2012?
BMW in the United States
Click to view content (https://0penstax.0rg/r/BMW US Natural Hedging)
While the name BMW may sound German, a significant amount of BMW's production occurs outside of Germany. Watch this video to learn about this international activity of BMW.
3. In the video, the potential car buyer is concerned about the impact of the value of the euro on the price of the BMW. Why, if he is paying for the car in US dollars, do you think that he is impacted by the currency exchange rate?
4. How do you think opening plants in the United States, and in other parts of the world, provides a currency hedge for BMW?
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Index 623
Index
Symbols Apple, Inc. 112.131.132.134. Brokers 22
"C&G" Credit Ratings 358 135. 136. 137.139.141.142. 147. Budget analyst 19
(ESG) 53 153. 154, 156. 157. 284. 450 Buffett 332. 370, 373
(GAAP) 156 appreciated 605 Bureau of Labor Statistics 77.
2014 Winter Olympics 421 arithmetic average return 453 217
3M 288 arithmetic mean 382 Bureau of Labor Statistics (BLS)
3M (MMM) 450 articles of incorporation 44 18,79
401k plans 9 articles of organization 44 business cycle 83
assets 134 Business Cycle Dating
A AT&T 514 Committee 300
AAPL (Apple) 468 audit committee (AC) 48 Business finance 8
accounting equation 105.134 B
Accounting Standards Update c
No. 2014-09 110 bad debt expense 581 C corporation 42
accounts receivable aging balance sheet 536 Cabela's 184
schedule 581 Bank of America Merrill Lynch call option 609
accounts receivable turnover 22 Call risk 302
ratio 168 Banking Act of 1935 10 capital 508
accrual basis 144 bankruptcies 580 capital asset pricing model
accrual-basis accounting 102 bar graph 400 (CAPM) 461
acid-test ratio 172 Bass Pro Shops 184 capital budgeting 479
after-tax cost of debt 510 benchmarking 574 capital employed 325
agency problem 50 Bennett 284 capital gain yield 450
Agency theory Berkshire Hathaway 332 capital gains 304
agent 50 Berkshire Hathaway (BRK) 373 capital investments 269
AIG 47 Bernard L. Madoff Investment capital market 27
Alibaba 172 Securities LLC 52 Capital structure 9. 508
allowance for doubtful best-fit linear regression model capitalize 112
accounts 581 427 CAPM 464
Alphabet (Google) 359 beta 432,463 Carnival Cruises 304
Amazon 21.43.180. 327,332. Big 5 Sporting Goods 54Q, 542 Carrying costs 584
341.359 bill of lading 576 cash basis JJ4.
Amazon (AMZN) 178 Billion Prices Project 78 cash budget 577
American Airlines 67, 507, 514. Billions 9 cash conversion cycle 570
515.599 Biogen 332.341 cash cycle 570
American Express 579 Bivariate data 403 cash deficit 551
American Institute of Certified BlackRock 356 cash discounts 576
Public Accountants (AICPA) 110 Blue Ribbon Companies cash flow 264
American options 609 Bluebonnet Industries 509 cash forecast 550
American Superconductor board of directors 44 cash rate 298
Corporation 169 bond call 306 cash ratio 123
amortization 323 bond laddering 305 cash surplus 551
AMZN (Amazon) 467 Bond price 287 cash-basis accounting 102
annual meeting 55 bond ratings 302 Cavallo 78
annual report 54 bond returns 369 CDs 354
annuity 231 Book value per share 181 Census Bureau 29
annuity due 234 Brazil 604 CEO 42
Apple 179.359.508 British 608 certificate of deposit (CD) 384
624 Index
ceteris paribus 68 CFO 14
Chapter 11 bankruptcy 121 Chicago Board of Trade 603 Chicago Mercantile Exchange
603
chief financial officer 14 chief financial officer (CFO) 47,
423
Chris's Landscaping 104 City of Chicago 285 Clear Lake Sporting Goods 129, 167, 536
Close (closely held) corporation
44
Cloud storage 12 CME Group Inc. 603 Coca-Cola 293.439.514,515 Commercial paper (CP) 26, 354 common stock 319 common-size 539 comparable company analysis (comps) 323
compensating balances 578 compounding interest 218 compounding period 218 comptroller 14 conference call 55 conflicts of interest 49 constant perpetuity 229 consumer price index (CPI) 22, 217, 357, 363,403 Consumer Price Index for All Urban Consumers: All Items (CPIAUCSL) 78 contra-asset 581 conversion price 528 conversion ratio 528 Convertible bonds 286, 528 core inflation index 77 corporate charter 44 Corporate Finance Institute 580 Corporate governance 47, 48 corporation 40 Correlation 418 Correlation analysis 417 correlation coefficient 412 Costa Rica 607 Coupon payment 284 Coupon rate 284
COVID-19 179. 300, 458. 495. 545. 565. 600
CPI Inflation Calculator 7_Z CPI, 403 credit 106 Credit analyst 19 credit period 576, 581 credit rating 572 credit risk 88,301 Credit Suisse 374 credit terms 580 Cuban 263
Current assets 135, 566 Current liabilities 136 current ratio 122 CVS 455.456
CVS Health Corp. (CVS) 454.456 CVS Pharmacy (CVS) 178
d
DAL 456.
Damodaran 185, 367. 368. 369.
371.372. 522
Danko 228
Data digitization 17
Data visualization 400
days' sales 167
Days' sales in inventory 121
dealers 22
debenture 358
debit 106
debt-to-assets ratio 174
debt-to-equity ratio 125
deep discount bonds 288
default 286
Default risk 13, 301
Delta Airlines 304
Delta Airlines (DAL) 451,455,
456
Demand 68 demand curve 68 Democrats 24
Department of Commerce 82 depreciated 605 Depreciation 114,131. 323 derivative 609 direct method 144 discount bond 296 discount period 576 discount rate 214, 235
discount window 355 discounted cash flow (DCF) 341 discounted payback period 489 Disney 450 Diversifiable risk 13 diversification 458 dividend 323 dividend discount model (DDM) 225
Dividend yield 323.450 dividends 138 DJIA 431
Domestic corporation 44 double-entry accounting 104 Dow 30 28. 370 Dowjones Company 569 Dow Jones Industrial Average (DJIA) 28, 370. 399. 431 Dun & Bradstreet 569,523 DuPont method 183 duration 301. 615 Duration risk 301
E
E-Trade 22 e-trail 12
earnings per share (EPS) 321 Earnings per share (EPS) 126 EBIT 324 EBITDA 133 EBITDA (earnings before interest, taxes, depreciation, and amortization) 122 Economic exposure 90 Economic risk 607 Economic value 30 Economics 23. EDGAR (Electronic Data Gathering, Analysis, and Retrieval system) 57 EDGAR system 58 effective annual rate (EAR) 451 effective interest rate 243 efficiency ratios 152 EMC Corporation 22Q empirical rule 399 Enron 51,303 Enterprise value (EV) 324 enterprise value (EV) multiples 323
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Index 625
equal annuity approach 493 Equifax 240
Equifax Small Business 573 equilibrium 72 equilibrium price 72 equity multiples 323 European options 609 European Union 608 Excel 203. 264. 273f 399.496. 553. 587
exchange rate 604 exchange-traded funds (ETFs) 9.354.356 exercising 609 expansion 83 expected value 397 expense 102 expense recognition 111 expenses 130 Experian 240 Experian Business 573 expiration date 609 exponential distribution 395 extreme values 383 ExxonMobil (XOM) 456
F
Facebook 21,327,341,359 Facebook(FB) 451 factoring 575 FASB (Financial Accounting Standards Board) 144 federal funds 26, 355 Federal Reserve 217,355, 355 Federal Reserve Bank of New York 354
Federal Reserve Economic Data (FRED) 91
Federal Reserve funds rate (federal funds rate) 288 Federal Reserve System (the Fed) 292 Fidelity 355
Fidelity Investments Inc. 459 Finance professor 19 Financial Accounting Standards Board (FASB) 102 Financial analyst 19,166 financial calculator 388.421 financial crisis of 2008 355
financial distress 521 Financial examiner 1_9 Financial Industry Regulatory Authority (FINRA) 10 financial instrument 196 financial intermediary 22 financial leverage 508 Financial manager 1_8 Financial markets and institutions 10 financial risk 196 financing activities 538 firm-specific risk 460 Fisher effect 217 Fitch 358
fixed-income securities 302 floating-rate bonds 287 floor planning 575 Florence 608 flotation costs 526 Foley 17 Ford 304
Ford Motor Company 579
forecast 535
Foreign corporation 44
Form10-K 55. 578
Fortune 49
Fortune 500 425.434
forward contract 608
FRED 91
free cash flow 147 Free cash flow (FCF) 148 frequency distribution 392 future value 267 future value (FV) 193 futures contract 603
G
GAAP 102
gains 104.130
Gates 370. 374
GDP 81
GDP deflator 77
general obligation (GO) bonds
Generally Accepted Accounting Principles (GAAP) 57.166. 583 geometric average return 452 geometric mean 384 Germany 608
Giving Pledge 370. 374 Glass-Steagall Act (1933) 10 global markets 57 Goodyear Tire and Rubber 514 Google 327. 332, 341 Gordon growth model 326 Governmental Accounting Standards Board (GASB) 102 Graham 373
graphic user interface (GUI) 406
Great Depression 10 Great Recession 367 Gross 369
gross domestic product 219 Gross domestic product (GDP) 80
gross working capital 566 growing perpetuity 229 growth rate 195, 326
H
Harding 24 hedging 601 Helu 353
Hewlett-Packard 42 histogram 4Q1 holding period percentage return 451 Houston 604
hybrid form of business 42 i
IBM 514 IEI 356
in inventory 167
income statement 130. 536
income statement (net income)
104
Indenture 357 index 93
indirect method 144 individual retirement accounts (IRAs) 9
inflation 76,78,215,356 Inflation risk 13 initial public offering 360 initial public offerings (IPOs) 21 Insurance underwriter 19 intangible assets 322
626 Index
intercept 431 interest 192 interest income 287 interest rate risk 301 interest tax shield 520 internal rate of return (IRR) 486 Internal Revenue Service 103 International Federation of Accountants (IFAC) 110 International Financial Reporting Standards (IFRS) 54. 57. 57.102 internet 121
interquartile range (IQR) 391 intrinsic value 326 Inventory turnover 170 investing activities 538 investment 192 investment grade 302 Investment relations associate 18
Investments 9 Investopedia 177 investor 41
investor relations (IR) 54 iPhone 25 Italy 608
J
Japan 92,94 Jensen's alpha 466 Johnson &Johnson 358 junk bonds 286 just-in-time inventory 572
K
KB Homes 514.515 Khan Academy 143 Korean 607 Kraft Heinz 514, 515 Kroger 514.515
l
law of demand 68 letter of credit 576 levered equity 517 liabilities 134 limited liability corporation (LLC) 42
limited liability partnership
(LLP) 42
linear correlation 420 linear regression 418 liquid asset 216 Liquidity 43.172. 566 Liquidity risk 301 Loan amortization 240 Loan officer 19 London 608
London Interbank Offered Rate (LIBOR) 288. 616 long-term asset HI loss 130 losses 104
lower-volatility investments 396 lumpsum 233 LUV 456
M
Macroeconomics 76 Malkiel 360
management discussion and analysis (MD&A) 157 margin 603
market capitalization 322 market risk premium 464 market value ratios 176 Market Watch 177 marketable securities 566 MarketWatch 464.514 marking to market 603 Massachusetts Institute of Technology 78 MasterCard 11. 579 Maturity date 285 McDonald's Corporation U9 McKinley Investment Management 465 McKinsey & Company 495 median 383
method of least squares 426
Mexican peso 89
Mexico 90
Miami 604
microeconomics 68
Microsoft 42.155. 358.359. 514
Microsoft Suite 17
Miller 517
mixed stream 264
MM Proposition I 517
mode 384
modified internal rate of return
(MIRR) 489
Modigliani 517
money market 25
money market investments 218
money market mutual funds
(MMMFs) 354
money market securities 25
money supply 217
MoneyWeek 284
Monster Beverage 341
Moody's 358
Morgan Stanley 22
mortgage bond 358
mortgage loan 240
municipal bonds ("munis") 27,
286
municipal bonds (munis) 357 mutually exclusive projects 491
n
NASDAQ 360
National Bureau of Economic
Research (NBER) 84
natural hedge 608
Negotiable certificates of deposit
(NCDs) 26.354
Net book value 322
net debt 516
net income 102,132
net present value (NPV) 482
Netterms 576
net working capital 567
Netflix 521
New York 604
New York Stock Exchange
(NYSE) 360
NFLX (Netflix) 468
Nike 21.404.418
nominal interest rate 87
Non-diversifiable risk 13
Non-stock corporation 44
noncash expenses 144
noncurrent assets 135
Noncurrent liabilities 136
Nonprofit corporation 44
NOPAT (net operating profit
after taxes) 177
normal distribution 394
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Index 627
0
Obama 24
Occidental Petroleum 304 operating activities 538 operating cycle 168, 574 opportunity cost 221. 577 option 609 option writer 610 Ordering costs 584 ordinary annuity 231 outliers 383
over-the-counter (OTC) market 359
owner's equity 142 p
P/ETTM 181
Par value 284
partnership 41
payback period 480
payday advance loan (PAL) 245
PayPal 11
Peloton 17.458
PepsiCo 49
percentiles 391
perpetuity 228.326
Personal financial advisor 19
Personal lines of credit 240
Political risk 13
Ponzi scheme 52
population data 382
portfolio 396.458
pound 609
PR Newswire 56
precautionary motive 577
precedent transaction analysis
(precedents) 323
prediction 425
preferred stock 228, 343, 523
premium 610
premium bond 296
present value 267
present value (PV) 195
press release 55
price-to-book (P/B) ratio 322
price-to-cash-flow (P/CF) ratio
323
price-to-earnings (P/E) ratio 320
price-to-sales (P/S) ratio 323 price/earnings (P/E) ratio 181 primary market 21. 360 prime rate 288 principal 50 pro forma 14, 544 probability distribution 397 producer price index (PPI) 77 Professional corporation 44 Profit margin 182 profitability index (PI) 488 Public benefit corporation 44 Public Company Accounting Oversight Board 47 put option 609
q
quarterly 54 quartiles 391 quick payment 576 quick ratio 172
r
R 438
R statistical analysis tool 406
rating agencies (bond rating
services) 302
ratios 568
real interest rate 87
real interest rates 217
reaIized return 305,450
recession 84
Refund anticipation loans
(RALs) 245
Regression analysis 417 Reinvestment risk 302 Renaissance Capital 361 replacement chain approach 492
report 54 Republicans 24 required rate of return 228 required return 326 residual 425 retained earnings 137 retirement planning 238 Return on equity 183 return on total assets 182 revenue 102.130 revenue bonds 357
revenue recognition 108 Revolving lines of credit (revolvers) 239 Rigobon 78
Risk Management Association
(RMA) 569
risk premium 462
risk-free rate 462
Robinhood 22
Rome 608
s
S corporation 42 S8