Fundamentals of Financial Management (Concise 6th Edition)

236 Part 3 Financial Assets

U.S. Water’s return will be less than 5% or more than 15%. However, virtually any

return within these limits is possible.

The tighter (or more peaked) the probability distributions shown in Figure 8-3,

the more likely the actual outcome will be close to the expected value and, conse-

quently, the less likely the actual return will end up far below the expected return.

Thus, the tighter the probability distribution, the lower the risk. Since U.S. Water has a

relatively tight distribution, its actual return is likely to be closer to its 10% ex-

pected return than is true for Martin; so U.S. Water is less risky.^5

8-2b Measuring Stand-Alone Risk:

The Standard Deviation^6

It is useful to measure risk for comparative purposes, but risk can be de" ned and

measured in several ways. A common de" nition that is simple and is satisfactory

for our purpose is based on probability distributions such as those shown in Fig-

ure 8-3: The tighter the probability distribution of expected future returns, the smaller the

risk of a given investment. According to this de" nition, U.S. Water is less risky than

Martin Products because there is a smaller chance that the actual return of U.S.

Water will end up far below its expected return.

We can use the standard deviation (#, pronounced “sigma”) to quantify the

tightness of the probability distribution.^7 The smaller the standard deviation, the

tighter the probability distribution and, accordingly, the lower the risk. We calcu-

late Martin’s # in Table 8-2. We picked up Columns 1, 2, and 3 from Table 8-1. Then

in Column 4, we " nd the deviation of the return in each demand state from the ex-

pected return: Actual return – Expected 10% return. The deviations are squared

and shown in Column 5. Each squared deviation is then multiplied by the relevant

probability and shown in Column 6. The sum of the products in Column 6 is the

variance of the distribution. Finally, we " nd the square root of the variance—this is

(^5) In this example, we implicitly assume that the state of the economy is the only factor that a$ ects returns. In
reality, many factors, including labor, materials, and development costs, in" uence returns. This is discussed at
greater length in the chapters on capital budgeting.
(^6) This section is relatively technical, but it can be omitted without loss of continuity.
(^7) There are actually two types of standard deviations, one for complete distributions and one for situations that
involve only a sample. Di$ erent formulas and notations are used. Also, the standard deviation should be modi! ed
if the distribution is not normal, or bell-shaped. Since our purpose is simply to get the general idea across, we
leave the re! nements to advanced! nance and statistics courses.
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Economy,
Which
A"ects
Demand
(1)
Strong
Normal
Weak
Rate of
Return
If This
Demand
Occurs
(3)
Deviation
Squared
(5)
Squared
Deviation
x Prob.
(6)
Deviation:
Actual -
10%
Expected
Return
(4)
0.30
0.40
0.30
1.00
Probability
of This
Demand
Occurring
(2)
80%
10
-60
70%
0
-70
" = Variance:
Standard deviation = square root of variance: # =
Standard deviation expressed as a percentage: # =
0.2940
0.5422
0.4900
0.0000
0.4900
0.1470
0.0000
0.1470
54.22%
Tabl e 8 - 2 Calculating Martin Products’ Standard Deviation