Application of discounted cash flows
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Course Information
• Dagmar Linnertová
• Dep. of Finance, 408
• Dagmar.linnertova@mail.muni.cz
• Application of discounted CF
• Statistical concepts and market returns
• Probability concepts
• Common Probability distributions
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1. Introduction
• Capital budgeting is the allocation of funds to long-lived capital
projects.
• A capital project is a long-term investment (in tangible assets).
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Conventional and nonconventional cash flows
Conventional Cash Flow (CF) Patterns
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–CF +CF +CF +CF +CF +CF
–CF –CF +CF +CF +CF +CF
–CF +CF +CF +CF +CF
Conventional and nonconventional cash flows
Nonconventional Cash Flow Patterns
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–CF +CF +CF +CF +CF –CF
–CF +CF –CF +CF +CF +CF
–CF –CF +CF +CF +CF –CF
Independent vs. mutually
exclusive projects
• When evaluating more than one project at a time, it is important to
identify whether the projects are independent or mutually exclusive
• This makes a difference when selecting the tools to evaluate the projects.
• Independent projects are projects in which the acceptance of one
project does not preclude the acceptance of the other(s).
• Mutually exclusive projects are projects in which the acceptance of
one project precludes the acceptance of another or others.
Copyright © 2013 CFA Institute 6
Investment decision criteria
Net Present Value (NPV)
Internal Rate of Return (IRR)
Payback Period
Discounted Payback Period
Average Accounting Rate of Return (AAR)
Profitability Index (PI)
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Net present Value
The net present value is the present value of all incremental cash flows,
discounted to the present, less the initial outlay:
NPV =
CFt
(1+r)
t
n
t=1 − Outlay (2-1)
Or, reflecting the outlay as CF0,
NPV =
CFt
(1+r)
t
n
t=0 (2-2)
where
CFt = After-tax cash flow at time t
r = Required rate of return for the investment
Outlay = Investment cash flow at time zero
If NPV > 0:
• Invest: Capital project adds value
If NPV < 0:
• Do not invest: Capital project destroys value
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Example: NPV
Consider the Hoofdstad Project, which requires an investment of $1
billion initially, with subsequent cash flows of $200 million, $300
million, $400 million, and $500 million. We can characterize the
project with the following end-of-year cash flows:
What is the net present value of the Hoofdstad Project if the required
rate of return of this project is 5%?
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Period
Cash Flow
(millions)
0 –$1,000
1 200
2 300
3 400
4 500
Example: NPV
Time Line
Solving for the NPV:
NPV = –$1,000 +
$200
1 + 0.05 1
+
$300
1 + 0.05 2
+
$400
1 + 0.05 3
+
$500
1 + 0.05 4
NPV = −$1,000 + $190.48 + $272.11 + $345.54 + $411.35
NPV = $219.47 million
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0 1 2 3 4
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–$1,000 $200 $300 $400 $500
NPV Profile: Hoofdstad Capital project
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-$200
-$100
$0
$100
$200
$300
$400
$500
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%
NPV
(millions)
Required Rate of Return
NPV Profile: Hoofdstad Capital project
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$400
$361
$323
$287
$253
$219
$188
$157
$127
$99
$72
$46
$20
–$4
–$28
–$50
–$72
–$93
–$114
–$133
–$152
-$200
-$100
$0
$100
$200
$300
$400
$500
0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%
NPV
(millions)
Required Rate of Return
Internal rate of return
The internal rate of return is the rate of return on a project.
• The internal rate of return is the rate of return that results in NPV = 0.
CFt
(1 + IRR)
t
n
t=1 − Outlay = 0 (2-3)
Or, reflecting the outlay as CF0,
CFt
(1 + IRR)
t
n
t=0 = 0 (2-4)
If IRR > r (required rate of return):
• Invest: Capital project adds value
If IRR < r:
• Do not invest: Capital project destroys value
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Example: IRR
Consider the Hoofdstad Project that we used to demonstrate the NPV
calculation:
The IRR is the rate that solves the following:
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Period
Cash Flow
(millions)
0 –$1,000
1 200
2 300
3 400
4 500
$0 = −$1,000 +
$200
1 + IRR 1 +
$300
1 + IRR 2 +
$400
1 + IRR 3 +
$500
1 + IRR 4
A note on solving for IRR
• The IRR is the rate that causes the NPV to be equal to zero.
• The problem is that we cannot solve directly for IRR, but rather must
either iterate (trying different values of IRR until the NPV is zero) or
use a financial calculator or spreadsheet program to solve for IRR.
• In this example, IRR = 12.826%:
• Or linear interpolation
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$0 = −$1,000 +
$200
1 + 0.12826 1 +
$300
1 + 0.12826 2 +
$400
1 + 0.12826 3 +
$500
1 + 0.12826 4
Example: IRR
• Initial investment 1.000.000.000
• Perpetual CFs 100.000.000
• Using IRR accept the project or not if your required rate of return is
8% p.a.
• Using IRR accept the project or not if your required rate of return is
15% p.a.
Copyright © 2013 CFA Institute 16
Payback Period
• The payback period is the length of time it takes to recover the initial cash outlay
of a project from future incremental cash flows.
• In the Hoofdstad Project example, the payback occurs in the last year, Year 4:
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Period
Cash Flow
(millions)
Accumulated
Cash flows
0 –$1,000 –$1,000
1 200 –$800
2 300 –$500
3 400 –$100
4 500 +400
Payback Period: Ignoring Cash Flows
For example, the payback period for both Project X and Project Y is three years,
even through Project X provides more value through its Year 4 cash flow:
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Year
Project X
Cash Flows
Project Y
Cash Flows
0 –£100 –£100
1 £20 £20
2 £50 £50
3 £45 £45
4 £60 £0
Discounted Payback Period
• The discounted payback period is the length of time it takes for
the cumulative discounted cash flows to equal the initial outlay.
• In other words, it is the length of time for the project to reach NPV = 0.
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Example: Discounted Payback Period
Consider the example of Projects X and Y. Both projects have a discounted payback
period close to three years. Project X actually adds more value but is not distinguished
from Project Y using this approach.
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Cash Flows
Discounted
Cash Flows
Accumulated
Discounted
Cash Flows
Year Project X Project Y Project X Project Y Project X Project Y
0 –£100.00 –£100.00 –£100.00 –£100.00 –£100.00 –£100.00
1 20.00 20.00 19.05 19.05 –80.95 –80.95
2 50.00 50.00 45.35 45.35 –35.60 –35.60
3 45.00 45.00 38.87 38.87 3.27 3.27
4 60.00 0.00 49.36 0.00 52.63 3.27
Profitability index
The profitability index (PI) is the ratio of the present value of future
cash flows to the initial outlay:
PI =
Present value of future cash flows
Initial investment
= 1 +
NPV
Initial investment
(2-5)
If PI > 1.0:
• Invest
• Capital project adds value
If PI < 0:
• Do not invest
• Capital project destroys value
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Example: PI
In the Hoofdstad Project, with a required rate of return of 5%,
the present value of the future cash flows is $1,219.47. Therefore, the PI is:
PI =
$1,219.47
$1,000.00
= 1.219
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Period
Cash Flow
(millions)
0 -$1,000
1 200
2 300
3 400
4 500
NPV vs. IRR
• If projects are independent, the decision to invest in one does not
preclude investment in the other.
• NPV and IRR will yield the same investment decisions.
• Projects are mutually exclusive if the selection of one project precludes
the selection of another project  project selection is determined by
rank.
• NPV and IRR may give different ranks when
• The projects have different scales (sizes)
• The timing of the cash flows differs
• If projects have different ranks  use NPV.
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NPV vs. IRR
Focus On: Calculations
• Consider Project C with the following cash flows:
• The NPV is $28,600.26.
• The IRR is 24.42%.
• If the projects are independent, you accept all three.
• If the projects are mutually exclusive, you accept Project A even though it has
the smallest IRR.
• If Projects B and C are mutually exclusive, you accept Project C.
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t = 0
t = 1 t = 2
–$90,000
Hurdle rate = 10%
r = 10%
t = 3
$30,000
$40,000
$45,000
$40,000
t = 4
Project A Project B Project C
NPV $29,872.52 $27,783.12 $28,600.26
IRR 21.84% 25.62% 24.42%
Decision Accept Accept Accept
IRR Challenges
IRR is a very appealing measure because it is intuitive; we all
understand (or think we do) rates of return.
• Unfortunately, IRR has several shortcomings.
• We will only realize the IRR as calculated if we
a) can reinvest all the project cash flows at that IRR, and
b) hold the investment to maturity.
• IRR and NPV can give different rankings when
• The scale of the projects being compared is different
• The timing of the cash flows is different
• Conclusion: NPV should be preferred to IRR.
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Decision at various required
rates of return
Project P Project Q Decision
NPV @ 0% $42 $32 Accept P, Reject Q
NPV @ 4% $21 $20 Accept P, Reject Q
NPV @ 6% $12 $14 Reject P, Accept Q
NPV @ 10% –$3 $5 Reject P, Accept Q
NPV @ 14% –$16 –$4 Reject P, Reject Q
IRR 9.16% 12.11%
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NPV Profiles: Project P and Project Q
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-$30
-$20
-$10
$0
$10
$20
$30
$40
$50
0% 2% 4% 6% 8% 10% 12% 14%
NPV
Required Rate of Return
NPV of Project P NPV of Project Q
Example NPV and IRR
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The multiple IRR problem
• If cash flows change sign more than once during the life of the
project, there may be more than one rate that can force the present
value of the cash flows to be equal to zero.
• This scenario is called the “multiple IRR problem.”
• In other words, there is no unique IRR if the cash flows are nonconventional.
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Example: The multiple IRR problem
Consider the fluctuating capital project with the following end of year
cash flows, in millions:
What is the IRR of this project?
30
Year Cash Flow
0 –€550
1 €490
2 €490
3 €490
4 –€940
Example: The Multiple IRR Problem
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-€120
-€100
-€80
-€60
-€40
-€20
€0
€20
€40
0% 8% 16% 24% 32% 40% 48% 56% 64%
NPV
(millions)
Required Rate of Return
IRR = 2.856%
IRR = 34.249%