Portfolio Theory
Dr. Andrea Rigamonti
andrea.rigamonti@econ.muni.cz
Lecture 2
Content:
• Interest rates
• Valuing bonds
Interest rates
Effective annual rate (EAR): also known as effective
annual yield (EAY) or annual percentage yield (APY), it
is the actual amount of interest that will be earned at
the end of one year.
Discount Rate Period Conversion. We can convert a
discount rate of R for one period to an equivalent
discount rate for n periods using the following formula:
Equivalent n-Period Discount Rate = (1 + R)n – 1
• n>1 to compute a rate over more than one period
• n<1 to compute a rate over a fraction of a period
Interest rates
Example:
The EAR is 5%. What is the monthly rate?
(1 + 0.05) 1/12
−1 = 0.004074 = 0.4047%
The monthly rate is 0.5% What is the yearly rate?
(1 + 0.005) 12
−1 = 0.061678 = 6.1678%
Taxes reduce the amount of interests the investor can keep.
Given a tax τ, the After-Tax Interest Rate is:
𝑅 − 𝜏𝑅 = 𝑅(1 − 𝜏)
Interest rates
Nominal interest rate (R): the rate at which the money will
grow if invested for a certain period
Real interest rate (Rr): the rate of growth of the purchasing
power, after adjusting for the inflation rate i.
𝐺𝑟𝑜𝑤𝑡ℎ 𝑖𝑛 𝑃𝑢𝑟𝑐ℎ𝑎𝑠𝑖𝑛𝑔 𝑃𝑜𝑤𝑒𝑟 = 1 + 𝑅 𝑟 =
1 + 𝑅
1 + 𝑖
Nominal interest rate tends to move with inflation.
Interest rates affect firms’ incentive to raise capital and invest
Interest rates
Term structure of interest rates: the relationship between
the investment term and the interest rate.
We can plot this relationship on a graph called yield curve.
Present Value of a Cash Flow Stream Using a Term
Structure of Discount Rates:
𝑃𝑉 = ෍
𝑛=1
𝑁
𝐶 𝑛
(1 + 𝑅 𝑛) 𝑛
Interest rates
Interest Rate Determination: the Federal Reserve
determines very short-term interest rates through its
influence on the federal funds rate( the rate at which banks
can borrow cash reserves on an overnight basis).
The other interest rates are set on the market.
Usually, the yield curve is increasing (longer term rates
higher than short term rates).
A sharply increasing curve indicates expectations that
interest rates will increase in the future.
A decreasing (inverted) curve indicates expectations of
decline interest rates. Usually this precedes a recession.
Valuing bonds
Bond certificate: a legal document describing the amounts
and dates of all the payments to be made.
Maturity date of a bond: the date on which the final
payment is made to the bondholder.
Term of a bond: the amount of time between the bond’s
issuance and its maturity.
Coupons: the promised interest payments of a bond.
Valuing bonds
Principal (or face value) of a bond: the amount paid to
the bondholder at maturity (i.e. at the expiration date).
Coupon rate: the amount of each coupon payment, i.e.
the rate of interest paid by the bond.
By convention, the coupon rate is expressed as an Annual
Percentage Rate (APR), i.e., as the amount of interests
earned in one year without compounding.
So the amount of each coupon payment, CPN, is:
𝐶𝑃𝑁 =
𝐶𝑜𝑢𝑝𝑜𝑛 𝑅𝑎𝑡𝑒 ∗ 𝐹𝑎𝑐𝑒 𝑉𝑎𝑙𝑢𝑒
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝐶𝑜𝑢𝑝𝑜𝑛 𝑃𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑝𝑒𝑟 𝑌𝑒𝑎𝑟
Valuing bonds
Zero-coupon bond: a coupon that does not make coupon
payments. The only cash payment the investor receives is
the face value of the bond on the maturity date.
Zero-coupon bonds trade at a discount (a price lower
than the face value), also called sub-par price. So they are
also called pure discount bonds.
Yield to maturity (YTM): the discount rate that sets the
present value of the promised bond payments equal to
the current market price of the bond.
Valuing bonds
Yield to maturity for a zero-coupon bond: the return you
will earn as an investor from holding the bond to maturity
and receiving the promised face value payment.
The YTM for a zero-coupon bond with n periods to
maturity, current price P, and face value FV solves
𝑃 =
𝐹𝑉
(1 + 𝑌𝑇𝑀 𝑛) 𝑛
Valuing bonds
Yield to Maturity of an n-Year Zero-Coupon Bond:
𝑌𝑇𝑀 𝑛 =
𝐹𝑉
𝑃
1/𝑛
− 1
𝑌𝑇𝑀 𝑛 is the per-period rate of return for holding the
bond from today until maturity on date n.
For the law of one price, the risk-free interest rate over a
certain period has to be equal to the 𝑌𝑇𝑀 𝑛 of a risk-free
zero coupon bond with maturity equal to that period.
Valuing bonds
Coupon bonds: bonds that pay investors their face value at
maturity, and also make regular coupon interest payments.
Yield to maturity of a coupon bond: the interest rate 𝑦 that
solves the following equation (assuming the first coupon will
be paid one period from now), where CPN is the coupon:
𝑃 = 𝐶𝑃𝑁 ∗
1
𝑦
1 −
1
1 + 𝑦 𝑁
+
𝐹𝑉
1 + 𝑦 𝑁
𝑦 is a rate per coupon interval. We can convert it to an APR
by multiply it by the number of coupons per year.
Valuing bonds
Coupon bonds may trade:
• at a discount (or below par);
• at a premium (a price greater than their face value, also
said above par);
• at par (a price equal to their face value).
The market price of a bond changes over time because:
• it gets closer to its maturity date
• changes in market interest rates affect its yield to maturity
Valuing bonds
The price of a zero coupon bond will simply rise as it gets
closer to maturity.
The price of a coupon bond will gradually drop if it trades at
a premium, or rise if it trades at discount. Moreover, as each
coupon is paid, its price drops by the amount of the coupon.
Ultimately, the prices of all bonds approach the bonds’ face
value when the bonds mature and their last coupon is paid.
Valuing bonds
Bond price over time if the yield to maturity remains at 5%
(i.e., the investor earns a 5% return):
Valuing bonds
In reality, the yield does not stay the same, but reacts to
changes in interest rates (and in the bond’s risk level).
A higher yield to maturity implies a higher discount rate for a
bond’s remaining cash flows, reducing their present value
and hence the bond’s price.
As interest rates and bond yields rise, bond prices will fall,
and vice versa.
Bond prices fluctuate due to unpredictable changes in their
yield while converging to their face value over time.
In the next slide, an example of a 30-year zero-coupon bond.
Valuing bonds
Valuing bonds
Duration: the sensitivity of a bond’s price to changes in
interest rates. Can be expressed in two ways:
• Macaulay duration:
𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦𝐷 = ෍
𝑡=1
𝑛
𝑃𝑉(𝐶𝑡)
𝑃
𝑡
where 𝑡 is the time in years till the cash flow is received
• Modified duration:
𝑀𝑜𝑑𝑖𝑓𝑖𝑒𝑑𝐷 =
𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦𝐷
1 +
𝑌𝑇𝑀
𝑛
where 𝑛 is the number of coupon periods per year
Valuing bonds
The Macaulay duration is an estimate of the number of
years required to repay the bond’s price.
Macaulay duration is a value-weighted maturity. Do not
confuse it with the maturity (they are both expressed in
years but their value is different)!
Modified duration measures the change in the price of a
bond given a 1% change in the interest rates.
Valuing bonds
For the law of one price, we can replicate the cash flows of a
coupon bond using zero-coupon bonds, obtaining the price of
the coupon bond from the zero coupon bonds portfolio price.
Alternatively, we can use the zero-coupon bond yields. The
price of a coupon bond must equal the present value of its
coupon payments and face value discounted at the
competitive market interest rates:
𝑃 =
𝐶𝑃𝑁
1 + 𝑌𝑇𝑀1
+
𝐶𝑃𝑁
1 + 𝑌𝑇𝑀2
2
+ ⋯ +
𝐶𝑃𝑁 + 𝐹𝑉
1 + 𝑌𝑇𝑀 𝑛
𝑛
where CPN is the coupon, 𝑌𝑇𝑀 𝑛 is the yield to maturity of a
zero-coupon bond that matures at the same time as the nth
coupon payment, and FV is the face value of the bond.
Valuing bonds
Corporate bonds: bonds issued by corporations. The issuer
may default, i.e. it might not pay back the full amount
promised in the bond prospectus.
Credit risk of the bond: the risk of default. Investors pay less
for bonds with credit risk than they would for an otherwise
identical default-free bond, because the cash flows are not
known with certainty.
The yield to maturity for a bond is computed using the
promised cash flows the yield of a risky bond is higher
than that of an otherwise identical default-free bond, and it
exceeds the expected return of investing in the bond.
Valuing bonds
For a risky bond, a high yield to maturity does not
necessarily imply a high expected return.
The creditworthiness of risky bonds is estimated by rating
agencies. Bonds with low default risk are called
investment-grade bonds. The others are called
speculative bonds, junk bonds, or high-yield bonds.
Default spread (or credit spread): the difference between
the yields of two bonds with different credit ratings.
Valuing bonds
Bond yield curves in 2015:
Valuing bonds
Sovereign bonds: bonds issued by national governments,
many of which are not considered default free, and can
therefore have high yields and low prices.
Countries might print money to repay debt in order to avoid
default, therefore investors consider inflation expectations
when determining the yield they are willing to accept.
Sometimes the required inflation rate would be too
extreme, or there might be political reasons to not do it, so
defaults still sometimes occur.