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For proposal A:
(A, A)вЂ™pr
Alp, 6) (A, A) ($)
Al($) Pl
300,000
0.3 300 1,000
1O O
,O
0
800
2,000 0.4 0
3,000 0.3 900 1,000 300,000
A = 2,000 u2= 600,000
254
CAPITAL BUDGETING UNDER RISK 255
CHAP. 91
Since a2= 600,000, a = 775. Thus
$775 0.39
x=$2,ooo=
For proposal B:
(&A) ($1 (&A)вЂ™ (AiA)вЂ™Pi
($)
Ai ($) Alp, ($)
Pi
1,950 3,802,500 1,400,750
500 0.3 150
81,000
202,500
0.4 800 450
2,000

5,000 1,500
0.3 2,550 6,502,500 2,601,000
o2= 3,822,750
2,450
Since u2= 3,822,750, u = $1,955. Thus
Therefore, proposal A is relatively less risky than proposal B, as indicated by the lower coefficient of variation.
9.3 RISK ANALYSIS IN CAPITAL BUDGETING
Since different investment projects involve different risks, it is important to incorporate risk into
the analysis of capital budgeting. There are several methods for incorporating risk, including:
1. Probability distributions
2. Riskadjusted discount rate
3. Certainty equivalent
4. Simulation
5. Sensitivity analysis
6. Decision trees (or probability trees)
Probability Distributions
Expected values of a probability distribution may be computed. Before any capital budgeting
method is applied, compute the expected cash inflows or, in some cases, the expected life of the
asset.
EXAMPLE 9.2 A firm is considering a $30,000 investment in equipment that will generate cash savings from
operating costs. The following estimates regarding cash savings and useful life, along with their respective
probabilities of occurrence, have been made:
Annual Cash Savings Useful Life
4 years 0.2
$ 6,000 0.2
0.5 5 years 0.6
$ 8,000
0.3 6 years 0.2
$10,000
Then, the expected annual saving is:
$6,000(0.2) = $1,200
$8,000(0.5) = 4,000
$10,000(0.3)= 3,000

$8,200
256 CAPITAL BUDGETING UNDER RISK [CHAP. 9
The expected useful life is:
4(0.2) = 0.8
5(0.6) = 3.0
6(0.2) = 
1.2
5 years
The expected NPV is computed as follows (assuming a 10 percent cost of capital):
PV  I = $8,200(PVIFA,,%,5) $30,000
NPV =
= $8,200(3.7908)  $30,000 = $31,085  $30,000 = $1,085
The expected IRR is computed as follows: By definition, at IRR,
I = PV
$30,000 = $ 8,200(PVIFAV,5)
which is about halfway between 10 percent and 12 percent in Appendix D, so that we can estimate the rate to be
11 percent. Therefore, the equipment should be purchased, since (1) NPV = $1,085, which is positive, and/or (2)
IRR = 11 percent, which is greater than the cost of capital of 10 percent.
RiskAdjusted Discount Rate
This method of risk analysis adjusts the cost of capital (or discount rate) upward as projects become
riskier. Therefore, by increasing the discount rate from 10percent to 15percent, the expected cash flow
from the investment must be relatively larger or the increased discount rate will generate a negative
NPV, and the proposed acquisition/investment would be turned down.
The use of the riskadjusted discount rate is based on the assumption that investors demand higher
returns for riskier projects. The expected cash flows are discounted at the riskadjusted discount rate and
then the usual capital budgeting criteria such as NPV and IRR are applied.
EXAMPLE 9 3 A firm is considering an investment project with an expected life of 3 years. It requires an initial
.
investment of $35,000. The firm estimates the following data in each of the next 3 years:
AfterTax Cash M o w Probability
$5,000 0.2
$10,000 0.3
$30,000 0.3
$50,000 0.2
Assuming a riskadjusted required rate of return (after taxes) of 20 percent is appropriate for the investment
projects of this level of risk, compute the riskadjusted NPV.
First,
+ $10,000(0.3) + $30,000(0.3) + $50,000(0.2) = $21,000
A $5,000(0.2)
=
The expected NPV $21,000(PVIFAm%,3) $35,000

=
$21,00(2.1065)  $35,000 = $44,237  $35,000 = $9,237
=
Certainty Equivalent
The certainty equivalent approach to risk analysis is drawn directly from the concept of utility theory.
This method forces the decision maker to specify at what point the firm is indifferent to the choice
between a certain sum of money and the expected value of a risky sum.
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