SUPPLEMENT
C: RISK
AND
UNCERTAINTY
IN IN
CAPISUPPLEMENT
C: RISK
AND
UNCERTAINTY
CAPITAL BUDGETING
TAL BUDGETING
A probability is a number between 0 and 1 which describes the likelihood that an event
will take place. A probability of zero is assigned to an event which has no chance of
occurring, while a probability of 1 denotes absolute certainty that an event will occur. An
outcome with a 90% chance of occurring is assigned a probability of .9 indicating that it
is expected to occur in 9 out of 10 times.
Assume that Lockwood Company in Vancouver is assessing which of two 5-year
$12,000 investment projects to undertake. Lockwood uses expected value as its decision
criterion. The cost of capital is estimated at 10%.
The analysis in Exhibit SC-1 shows that expected returns for both project A and project B are $10,800 per year. The expected return of each project can be thought of as a
return of $10,800 by the present value of an annuity factor of 3.791 and subtracting the
initial investment of five-year period annuity. The net present value can now be calculated
by multiplying the expected $12,000 ($10,800 3.791 – $12,000 = $28,943).
A limitation of using expected returns as a benchmark when deciding among competing projects is that it fails to account for the risk preferences of individual managers. A
full discussion is beyond the scope of this book, but one should be cognizant that management’s attitude toward risk may impact on the selector process. Risk takers may be
willing to forgo a project with a high expected return for another project with a lower
expected return but with some probability of reaching a higher maximum.
LEARNING OBJECTIVE 1
Measure risk in assessing
capital budgeting projects.
A Statistical Measure of Risk
One way of measuring the risk of capital budgeting projects is to determine the extent to
which actual returns deviate from the expected returns by calculating the standard deviation. The standard deviation of a project’s return is the square root of the sum of the
average squared deviations of actual returns from the expected return (EV) of the project.
The formula for calculating the standard deviation is as follows:
√(E – E) P
N
=
i
2
Standard deviation A statistical
tool used by the management
accountant to determine the
amount of variation in a project’s
actual returns from the project’s
expected returns.
i
i=i
where = standard deviation
N = number of possible outcomes
Ei = the value of the ith possible outcome
= the expected returns
E
Pi = probabilities that the ith outcome will occur
Probable Outcome
Net Cash
Flows
Project A:
Pessimistic..................................
Most likely..................................
Optimistic ..................................
$ 4,000
10,000
20,000
.20
.60
.20
1.00
$
$
.20
.60
.20
1.00
$
Project B:
Pessimistic..................................
Most likely..................................
Optimistic ..................................
–0–
11,000
21,000
Probability
Probable
Return
800
6,000
4,000
$10,800
–0–
6,600
4,200
$10,800
EXHIBIT SC–1
SC-2
Supplement C Risk and Uncertainty in Capital Budgeting
Statisticians have determined that for normal probability distributions, 68% of the outcomes are plus or minus one standard deviation from the expected value; 95% are plus
or minus two standard deviations; and 99% of all outcomes lie between plus or minus
three standard deviations away from the expected value. A normal probability distribution has one-half of the values in the distribution above the expected value and the other
half below the expected value. Most statistics textbooks illustrate the normal distribution graphically as a bell-shaped curve and also provide tables that give statistically calculated probabilities that are associated with various deviations from the expected
values. Exhibit SC-2 shows the calculation of the standard deviation of projects A and
B for Lockwood Company.
Although projects A and B have equal expected values of $10,800, it is clear from
the forgoing analysis that there is more variation in the returns of project B. Standard
deviation, a common statistical measure of variation, is higher for project B than it is
for project A.
Coefficient of variation A
relative measure of a project’s
dispersion calculated as the ratio
of project’s standard deviation to
its expected value.
Coefficient of Variation
The coefficient of variation (CV) is a measure of the relative risk of an investment project. It is calculated by dividing a project’s standard deviation by its expected value. For
Lockwood Company, the CV’s for projects A and B are as follows:
CV for project A = $5,154 = .48
$10,800
CV for project B = $6,645 = .62
$10,800
The larger standard deviation is as a percentage of the expected value, the higher the
risk is judged to be. Using CV as the decision criterion, project B is clearly riskier than
project A. This is consistent with the conclusion reached by using the standard deviation
alone. So long as the expected values of the projects are equal, the coefficient of variation
does not add any additional information. When the expected values differ, however, the
CV is a better measure of risk than the absolute measure that is provided by the standard
deviation.
EXHIBIT SC–1
Project A
Ei
E
(Ei – E
)
(In
thousands
of dollars)
(Ei – E
)2
Pessimistic
Most likely
Optimistic
$ 4,000
10,000
20,000
$10,800
10,800
10,800
$(6,800)
(800)
9,200
$46,240
640
84,640
Pi
(In
thousands
of dollars)
(Ei – E
)2Pi
.2
$ 9,248
.6
384
.2
16,928
Variance = $26,560
Standard deviation = vari
an
ce
= $26,5
60,0
00 = $5,154.00
Project A
Ei
E
(Ei – E
)
(In
thousands
of dollars)
(Ei – E
)2
Pessimistic
Most likely
Optimistic
$ –0–
11,000
21,000
$10,800
10,800
10,800
$(10,800)
200
10,200
$166,640
40
104,040
Pi
(In
thousands
of dollars)
(Ei – E
)2Pi
0.20
$23,328
0.60
24
0.20
20,808
Variance = $44,160
Standard deviation = vari
an
ce
= $44,1
60,0
00 = $6,645.30
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Supplement C Risk and Uncertainty in Capital Budgeting
SC-3
Example
To assist in choosing between two mutually exclusive investment projects (the selection of one
precludes the selection of the other), the following information is available for Calgary Equipment
Company:
Expected cash returns ...............................
Standard deviation (SD) ............................
Coefficient of variation ..............................
Project C
Project D
$20,000
$ 8,000
.40
$40,000
$12,000
.30
On the basis of standard deviation alone the management of Calgary Equipment
Company may be tempted to declare project D to be riskier than project C. The absolute
measure of risk provided by the standard deviation can lead to an error in the selection
process when the expected returns differ between the two projects. Using the coefficient
of variation to measure the relative risk of projects, it is apparent that project C is riskier
than project D.
From the perspective of an individual firm, it may be possible to reduce overall risk
by selecting individual projects with returns that vary negatively with the firm’s existing
projects. Correlation is a statistical technique that measures the relationship of the
returns of the project to those of another. A correlation of +1 indicates a perfect positive
relationship. This means, for example, that when returns of project X are high, returns on
project Y are also very high. When returns on project X are low, returns on project Y are
low as well. A correlation coefficient of –1 indicates a perfect negative correlation meaning, for example, that when returns are high for project X, they are low for project Y and
vice versa. Most correlation coefficients fall somewhere in between +1 and –1. The essential point for the manager to bear in mind is that is may be less risky to select projects with
returns that vary inversely with those of existing projects, so that a downturn for one project does not impact negatively on the returns on another project. In other words, by combining projects with returns that correlate negatively with the firm’s existing portfolio of
projects, the overall variability of risk for the firm may be reduced. In practice, these concepts are very hard to apply. In addition, there is strong theoretical support for choosing
projects on their individual merits because the residual owners of the firm (common
shareholders) can do their own diversifying by acquiring shares of a variety of companies.
Sensitivity Analysis
It is apparent that many estimates are required in the capital budgeting process. Managers may find it useful to prepare a what-if, or sensitivity, analysis to determine how
sensitive the net present value or internal rate of return of a project is to changes in these
estimates. Although sensitivity analysis does not actually quantify risk, it can provide
management with useful insights into the effects of changes in such input variables such
as cash flow estimates and discount rates. The more the net present value or internal rate
of return changes in response to changes in input variables, the riskier the project is
perceived to be.
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Correlation A number ranging
from +1 to –1 that is a statistical
measure of the degree of
association of the returns of one
project to those of another.
Sensitivity Analysis An analysis
of the effect that changes in a
project’s input variables may
have on its net present value or
internal rate of return.
SC-4
Supplement C Risk and Uncertainty in Capital Budgeting
PROBLEMS
PROBLEMS
PROBLEM SC–1 Risk and Net Present Value Analysis
Kentville Ltd. is evaluating two mutually exclusive investment proposals. Tony Atkinson, the firm’s
chief management accountant, has developed the following estimates for each project. The projects
have estimated lifespans of 15 years and the firm’s cost of capital is 12%.
Investment outlay.............................
Net cash inflows ...............................
Pessimistic...................................
Most likely..................................
Optimistic ..................................
Probability
Cash Flows
A
B
1.0
$15,000
$15,000
500
3,500
4,500
2,000
3,300
3,900
.15
.70
.15
Required:
1. Calculate the net present value of each project for each probability.
2. Calculate the expected net present value for each alternative.
3. Calculate the standard deviation and coefficient of variation for each project.
4. Which project would you recommend? Comment on the risk/return of these two projects.
PROBLEM SC–2 Expected Value; Standard Deviation; Coefficient of Variation
Liu Company of Burnaby, British Columbia, is evaluating the expected returns of two 5-year projects. These projects are not mutually exclusive. The expected returns are expressed in thousands
of dollars.
Year
X
Y
20X1
20X2
20X3
20X4
20X5
$ 4
6
7
3
10
$10
6
4
9
6
Required:
1. Calculate the expected value for each project.
2. Calculate the standard deviation and coefficient of variation for each project.
3. By observation, does there appear to be any correlation between the returns of the two
projects?
4. As a management accountant consulting for Liu Company, suggest how the company may
diversify its risk. Support your comments.
PROBLEM SC–3 Risk and net Present Value Analysis
Wolfville Ltd. is evaluating two mutually exclusive investment proposals. Edgar Scott, the firm’s
chief management accountant, has developed the following estimates for each project. The projects
have estimated lifespans of 10 years and the firm’s cost of capital is 14%.
Investment outlay.............................
Net cash inflows ...............................
Pessimistic...................................
Most likely..................................
Optimistic ..................................
Probability
Cash Flows
A
B
1.0
$20,000
$20,000
0.2
0.6
0.2
1,000
5,000
6,000
3,000
4,500
5,500
Required:
1. Calculate the net present value of each project for each probability.
2. Calculate the expected net present value for each alternative.
3. Calculate the standard deviation and coefficient of variation for each project.
4. Which project would you recommend? Comment on the risk/return of these two projects.
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