ñòð. 122 
CHAP. 91 CAPITAL BUDGETING UNDER RISK
Once certainty equivalent coefficients are obtained, they are multiplied by the original cash flow to
obtain the equivalent certain cash flow. Then, the acceptorreject decision is made, using the normal
capital budgeting criteria. The riskfree rate of return is used as the discount rate under the NPV method
and as the cutoff rate under the IRR method.
EXAMPLE 9.4 XYZ,Inc., with a 14 percent cost of capital after taxes is considering a project with an expected
life of 4 years. The project requires an initial certain cash outlay of $50,000.The expected cash inflows and certainty
equivalent coefficients are as follows:
AfterTax Cash Flow ($)
Year Certainty Equivalent coefficient
10,000 0.95
1
2 0.80
15,000
20,000
3 0.70
4 25,000 0.60
The riskfree rate of return is 5 percent; compute the NPV and IRR.
The equivalent certain cash inflows are obtained as follows:
AftepTax Certainty Equivalent Equivalent Certain
Cash Inflow ($)
Year Cash I d o w ($) Coefficient PV at 5% PV ($)
0.9524 9,048
9,500
10,000 0.95
1
10,884
0.9070
12,000
15,000 0.80
2
14,000 0.8638 12,093
3 20,000 0.70
0.8227 12,341
15,000
0.60
4 25,000
44,366
NPV = $44,366  $50,000 = $5,634
By trial and error, we obtain 4 percent as the IRR. Therefore, the project should be rejected, since (1)
NPV = 5,634, which is negative and/or (2) IRR = 4 percent is less than the riskfree rate of 5 percent.
Simulation
This risk analysis method is frequently called the Monte Carlo simulation. It requires that a
probability distribution be constructed for each of the important variables affecting the projectâ€™s cash
flows. Since a computer is used to generate many results using random numbers, project simulation is
expensive.
Sensitivity Analysis
Forecasts of many calculated NPVs under various alternative functions are compared to see how
sensitive NPV is to changing conditions. It may be found that a certain variable or group of variables,
once their assumptions are changed or relaxed, drastically alters the NPV. This results in a much riskier
asset than was originally forecast.
Decision â€™Ikees
Some firms use decision trees (probability trees) to evaluate the risk of capital budgeting proposals.
A decision tree is a graphical method of showing the sequence of possible outcomes. A capital budgeting
tree would show the cash flows and NPV of the project under different possible circumstances. The
decision tree method has the following advantages: (1) It visually lays out all the possible outcomes of
the proposed project and makes management aware of the adverse possibilities, and (2) the conditional
nature of successive yearsâ€™ cash flows can be expressly depicted. The primary disadvantage is that most
problems are too complex to permit a yearbyyear depiction. For example, for a 3year project with
three possible outcomes following each year, there are 27 paths For a 10year project (again with three
possible outcomes following each year) there will be about 60,000 paths.
CAPITAL BUDGETING UNDER RISK [CHAP. 9
258
EXAMPLE 9.5 A firm has an opportunity to invest in a machine which will last 2 years, initially cost $125,000,
and has the following estimated possible aftertax cash inflow pattern: In year 1, there is a 40 percent chance that
the aftertax cash inflow will be $45,000, a 25 percent chance that it will be $65,000, and a 35 percent chance that
it will be $90,000. In year 2, the aftertax cash inflow possibilities depend on the cash inflow that occurs in year 1;
that is, the year 2 aftertax cash inflows are Conditional probabilities. Assume that the firmâ€™s aftertax cost of capital
is 12 percent. The estimated conditional aftertax cash inflows (ATCI) and probabilities are given below.
If ATCIl = $90,000
If ATCIl = $65,O00
If ATCIl= !$4S,O00
Probability Probability Probability
ATCIz ($) ATCIz ($) ATCIZ($)
0.3 80,000 90,000 0.1
0.2
30,000
0.4 90,000 0.6 100,000 0.8
60,000
110,000
0.3 100,000 0.2 0.1
90,000
Then the decision tree which shows the possible aftertax cash inflow in each year, including the conditional nature
of the year 2 cash inflow and its probabilities, can be depicted as follows:
Expected
Joint
lime 1 NPV
NPV at 12% Probability
Time 0 lime 2
$7,309
0.12Ob
$60,905â€˜
5,919
0.160
$36,995
1,570
0.120
$13,085
160
0.050
$ 3,195
$ 80,000
716
0.150
$ 4,775
637
0.050
$12,745
949
0.035
$27,100
0.280 9,820
$35,070
1SO6
$43,040 0.035
$1,330
1,000

$30,000
$45,000
 (1 + 0.12)2  $125,000
â€œNPV = PV  I =
+
(1 + 0.12)
+ ˜ ˜ , ˜ ) ( P V I F $125,000
= $45,000(PVIF12%,1) $ I˜,˜˜)
= $45,000(0.893) + $30,000(0.797) $125,000 = $40,185 + $23,910  $125,000 = $60,905
ñòð. 122 