Principles of Managerial
Finance
9th Edition
Chapter 10
Risk & Refinements
in Capital Budgeting
Learning Objectives
• Understand the importance of explicitly recognizing
risk in the analysis of capital budgeting projects.
• Discuss breakeven cash flow, sensitivity and scenario
analysis, and simulation as behavioral approaches for
dealing with risk, and the unique risks facing
multinational companies.
• Describe the two basic risk-adjustment techniques in
terms of NPV and the procedures for applying the
certainty equivalent (CE) approach.
Learning Objectives
• Review the use of risk-adjusted discount rates
(RADRs), portfolio effects, and the practical aspects of
RADRs relative to CEs.
• Recognize the problem caused by unequal-lived
mutually exclusive projects and the use of annualized
net present values (ANPVs) to resolve it.
• Explain the objective of capital rationing and the two
basic approaches to project selection under it.
Behavioral Approaches for Dealing with Risk
• In the context of the capital budgeting projects
discussed in this chapter, risk results almost entirely
from the uncertainty about future cash inflows
because the initial cash outflow is generally known.
• These risks result from a variety of factors including
uncertainty about future revenues, expenditures and
taxes.
• Therefore, to asses the risk of a potential project, the
analyst needs to evaluate the riskiness of the cash
inflows.
Behavioral Approaches for Dealing with Risk
Sensitivity Analysis
Treadwell Tire has a 10% cost of capital and is
considering investing in one of two mutually exclusive
projects A or B. Each project has a $10,000 initial cost
and a useful life of 15 years.
As financial manager, you have provided pessimistic,
most-likely, and optimistic estimates of the equal annual
cash inflows for each project as shown in the following
table.
Behavioral Approaches for Dealing with Risk
Sensitivity
Analysis
Behavioral Approaches for Dealing with Risk
Simulation
• Simulation is a statistically-based behavioral approach
that applies predetermined probability distributions
and random numbers to estimate risky outcomes.
• Figure 10.1 presents a flowchart of the simulation of
the NPV of a project.
• The use of computers has made the use of simulation
economically feasible, and the resulting output
provides an excellent basis for decision-making.
Behavioral Approaches for Dealing with Risk
Simulation
Behavioral Approaches for Dealing with Risk
International Risk Consideration
• Exchange rate risk is the risk that an unexpected
change in the exchange rate will reduce NPV of a
project’s cash flows.
• In the short term, much of this risk can be hedged by
using financial instruments such as foreign currency
futures and options.
• Long-term exchange rate risk can best be minimized
by financing the project in whole or in part in the local
currency.
Behavioral Approaches for Dealing with Risk
International Risk Considerations
• Political risk is much harder to protect against once a
project is implemented.
• A foreign government can block repatriation of profits
and even seize the firm’s assets.
• Accounting for these risks can be accomplished by
adjusting the rate used to discount cash flows -- or
better -- by adjusting the project’s cash flows.
Behavioral Approaches for Dealing with Risk
International Risk Considerations
• Since a great deal of cross-border trade among MNCs
takes place between subsidiaries, it is also important
to determine the net incremental impact of a project’s
cash flows overall.
• As a result, it is important to approach international
capital projects from a strategic viewpoint rather than
from a strictly financial
如：搶市場灘頭堡，確保
perspective. 原料來源，即使是個NPV
似乎<0的跨國project
Risk-Adjustment Techniques
Certainty Equivalents
Bennett Company is currently evaluating two
projects, A and B.
The firm’s cost of capital is 10% and the initial
investment and operating cash flows are shown
on the following slide.
Risk-Adjustment Techniques
Certainty Equivalents
Bennett Company
Project's A and B
(10% cost of Captial)
Year
0
NPV
Project A
$
Project B
(42,000) $
(45,000)
1
14,000
28,000
2
14,000
12,000
3
14,000
10,000
4
14,000
10,000
5
14,000
10,000
$11,071
$10,924
Risk-Adjustment Techniques
Certainty Equivalents
Assume that it is determined that Project A is
actually more risky than B.
To adjust for this risk, you decide to apply
certainty equivalents (CEs) to the cash flows,
where CEs represent the percentage of the cash
flows that you would be satisfied to receive for
certain rather than the original (possible) cash
flows.
Risk-Adjustment Techniques
Certainty Equivalents
Bennett Company
Certainty Equivalents Applied to Project A
(Risk-free rate = 6%)
Present
Certain
Year
0
Project A
CE
$ (42,000) 1.00
Cash flows
$ (42,000)
Value
PVIF
1.0000 $ (42,000)
1
14,000
0.90
$
12,600
0.9434
11,887
2
14,000
0.90
$
12,600
0.8900
11,214
3
14,000
0.80
$
11,200
0.8396
9,404
4
14,000
0.70
$
9,800
0.7921
7,763
5
14,000
0.60
$
8,400
0.7473
6,277
Net Present Value
$
4,544
Risk-Adjustment Techniques
Certainty Equivalents
Bennett Company
Certainty Equivalents Applied to Project B
(Risk-free rate = 6%)
Certain
Year
0
Project B
CE
$ (45,000) 1.00
Cash flows
Present
PVIF
Value
$ (45,000)
1.0000 $ (45,000)
1
28,000
1.00
$
28,000
0.9434
26,415
2
12,000
0.90
$
10,800
0.8900
9,612
3
10,000
0.90
$
9,000
0.8396
7,557
4
10,000
0.80
$
8,000
0.7921
6,337
5
10,000
0.70
$
7,000
0.7473
5,231
和project A相比較
Net Present Value
$ 10,151
Risk-Adjustment Techniques
Risk-Adjusted Discount Rates
Bennett Company also wishes to apply the RiskAdjusted Discount Rate (RADR) approach to
determine whether to implement Project A or B.
To do so, Bennett has developed the following
Risk Index to assist them in their endeavor.
Risk-Adjustment Techniques
Risk-Adjusted Discount Rates
Required
Risk
Return
Index
(RADR)
0.0
6%
0.2
7%
0.4
8%
0.6
9%
0.8
10%
1.0
11%
1.2
12%
1.4
13%
1.6
14%
1.8
15%
2.0
16%
Risk-Adjustment Techniques
Risk-Adjusted Discount Rates
Project B has been assigned a Risk Index Value
of 1.0 (average risk) with a RADR of 11%, and
Project A has been assigned a Risk Index Value
of 1.6 (above average risk) with a RADR of 14%.
These rates are then applied as the discount
rates to the two projects to determine NPV as
shown on the following slide.
Risk-Adjustment Techniques
Risk-Adjusted Discount Rates
Bennett Company
Risk Adjusted Discount Rate Applied to Project A
(RADR = 14%)
Present
Year
0
Project A
$
PVIF
Value
(42,000)
1.0000
1
14,000
0.8772
12,281
2
14,000
0.7695
10,773
3
14,000
0.6750
9,450
4
14,000
0.5921
8,289
5
14,000
0.5194
7,271
Net Present Value
$
$
(42,000)
6,063
Risk-Adjustment Techniques
“理論上”也可用
CAPM來尋找
project的RADR：
Risk-Adjusted Discount Rates
Bennett Company
Risk Adjusted Discount Rate Applied to Project B
E (ri )  rf   i [ E (rm )  rf ]
IRR
rf
(RADR = 11%)
SML
IRR
Present
Year
β
若project的IRR落在
SML上方，則
accept the project，
因為其NPV>0
若project的IRR落在
SML下方，則reject
the project，因為其
NPV<0
0
Project B
$
PVIF
Value
(45,000)
1.0000
1
28,000
0.9009
25,225
2
12,000
0.8116
9,739
3
10,000
0.7312
7,312
4
10,000
0.6587
6,587
5
10,000
0.5935
5,935
Net Present Value
$
$
(45,000)
9,798
Risk-Adjustment Techniques
Portfolio Effects
• As noted in Chapter 6, individual investors must hold
diversified portfolios because they are not rewarded
for assuming diversifiable risk.
• Because business firms can be viewed as portfolios of
assets, it would seem that it is also important that they
too hold diversified portfolios.
• Surprisingly, however, empirical evidence suggests
that firm value is not affected by diversification.
• In other words, diversification is not normally rewarded
and therefore is generally not necessary.
Risk-Adjustment Techniques
Portfolio Effects
• It turns out that firms are not rewarded for
diversification because investors can do so
themselves.
• An investor can diversify more readily, easily, and
costlessly simply by holding portfolios of stocks.
Risk-Adjustment Techniques
CE Versus RADR in Practice
• In general, CEs are the theoretically preferred
approach for project risk adjustment because they
separately adjust for risk and time.
• CEs first eliminate risk from the cash flows and then
discount the certain cash flows at a risk-free rate.
• RADRs on the other hand, have a major theoretical
problem: they combine the risk and time adjustments
in a single discount rate adjustment.
Risk-Adjustment Techniques
CE Versus RADR in Practice
• Because of the mathematics of discounting, the RADR
approach implicitly assumes that risk is an increasing
function of time.
• However, because of the complexity in developing
CEs, RADRs are more often used in practice.
• More specifically, firms often establish a number of
risk classes, with an RADR assigned to each.
• Projects are then placed in the appropriate risk class
and the corresponding RADR is then applied.
Capital Budgeting Refinements
Comparing Projects With Unequal Lives
• If projects are independent, comparing projects with
unequal lives is not critical.
• But when unequal-lived projects are mutually
exclusive, the impact of differing lives must be
considered because they do not provide service over
comparable time periods.
• This is particularly important when continuing service
is needed from the projects under consideration.
Capital Budgeting Refinements
Comparing Projects With Unequal Lives
The AT Company, a regional cable-TV firm, is evaluating
two projects, X and Y. The projects’ cash flows and
resulting NPVs at a cost of capital of 10% is given below.
Project X
Year
Project Y
Cash Flow s
0
$
1
$
28,000
$
35,000
2
$
33,000
$
30,000
3
$
38,000
$
25,000
4
$
-
$
20,000
5
$
-
$
15,000
6
$
-
$
10,000
NPV
(70,000) $
$11,277
(85,000)
$19,013
Capital Budgeting Refinements
Comparing Projects With Unequal Lives
The AT Company, a regional cable-TV firm, is evaluating
two projects, X and Y. The projects’ cash flows and
resulting NPVs at a cost of capital of 10% is given below.
Ignoring the difference in their useful lives, both projects
are acceptable (have positive NPVs). Furthermore, if the
projects were mutually exclusive, project Y would be
preferred over project X. However, it is important to
recognize that at the end of its 3 year life, project Y must
be replaced, or renewed.
Although a number of approaches are available for
dealing with unequal lives, we will present the most
efficient technique -- the annualized NPV approach.
Capital Budgeting Refinements
Comparing Projects With Unequal Lives
Annualized NPV (ANPV)
The ANPV approach converts the NPV of unequal-lived
projects into an equivalent (in NPV terms) annual amount
that can be used to select the best project.
1. Calculate the NPV of each project over its live using the
appropriate cost of capital.
2. Divide the NPV of each positive NPV project by the
PVIFA at the given cost of capital and the project’s live
to get the ANPV for each project.
3. Select the project with the highest ANPV.
Capital Budgeting Refinements
Comparing Projects With Unequal Lives
Annualized NPV (ANPV)
1. Calculate the NPV for projects X and Y at 10%.
NPVX = $11,277; NPVY = $19,013.
2. Calculate the ANPV for Projects X and Y.
ANPVX = $11,277/PVIFA10%,3 years = $4,534
ANPVY = $19,013/PVIFA10%,6 years = $4,366
3. Choose the project with the higher ANPV.
Pick project X.
Capital Rationing
• Firm’s often operate under conditions of capital
rationing -- they have more acceptable independent
projects than they can fund.
• In theory, capital rationing should not exist -- firms
should accept all projects that have positive NPVs.
• However, research has found that management
internally imposes capital expenditure constraints to
avoid what it deems to be “excessive” levels of new
financing, particularly debt.
• Thus, the objective of capital rationing is to select the
group of projects within the firm’s budget that provides
the highest overall NPV
Capital Rationing
Example
Gould Company Investment Proposals
k=10%
Project
Initial Investm ent
IRR
PV of Inflow s
$
80,000
12%
$
B
70,000
20%
112,000
42,000
C
100,000
16%
145,000
45,000
D
40,000
8%
36,000
(4,000)
E
60,000
15%
79,000
19,000
F
110,000
11%
126,500
16,500
A
100,000
NPV
$
20,000
Capital Rationing
若capital constraint
為$250,000，則選B
、C、E。
IRR Approach
Gould Proposals
(Ranked by IRR)
Initial Investment
Project
IRR
B
20%
C
16%
100,000
E
15%
60,000
A
12%
80,000
F
11%
110,000
D
8%
40,000
$
70,000
Capital Rationing
IRR Approach
Assume the firm’s
cost of capital
is 10% and has
a maximum of
$250,000 available
for investment.
Ranking the
projects according
to IRR, the
optimal set of
projects for
Gould is B, C,
and E.
Gould Proposals
(Cum ulative Investm ent)
Initial
Cum ulative
Investm ent
Project
IRR
Investm ent
B
20%
$ 70,000
C
16%
100,000
170,000
E
15%
60,000
230,000
A
12%
80,000
310,000
F
11%
110,000
420,000
D
8%
40,000
460,000
$
70,000
Capital Rationing
IRR Approach
If we ration
capital using the
IRR approach
and maintain the
rankings provided
by IRR, the total
PV of inflows and
NPV would be
$336,000 and
$106,000
respectively.
Gould Company Investment Proposals
(Ranked by IRR)
PV of
Initial
Inflows
Investment
Project
IRR
B
20%
$ 112,000 $
C
16%
145,000
100,000
45,000
E
15%
79,000
60,000
19,000
Totals
$ 336,000 $
70,000 $
NPV
42,000
230,000 $ 106,000
Capital Rationing
NPV Approach
However, if we
rank them such
that NPV is
maximized, then
we can use our
entire budget and
raise the PV of
inflows and NPV to
$357,000 and
$107,000
respectively.
若選C、B、A而非前面的B、C、E，則可剛好
用盡所有的資金$250,000，且可提升NPV至
$107,000
Gould Company Investment Proposals
(Ranked by NPV)
PV of
Initial
Inflow s
Investment
Project
IRR
B
20%
$ 112,000 $
C
16%
145,000
100,000
(1) 45,000
A
12%
100,000
80,000
(3) 20,000
Totals
$ 357,000 $
NPV
70,000 $(2) 42,000
250,000 $ 107,000
注意：B、C、E、A、F的IRR皆大於k=10%