CP

calculate expected rates of return as shown previously, and the probabilities and out-

comes could be approximated by continuous curves such as those presented in Figure

3-2. Here we have changed the assumptions so that there is essentially a zero proba-

bility that Martin Products’ return will be less than 70 percent or more than 100

percent, or that U.S. Water’s return will be less than 10 percent or more than 20 per-

cent, but virtually any return within these limits is possible.

The tighter, or more peaked, the probability distribution, the more likely it is that the ac-

tual outcome will be close to the expected value, and, consequently, the less likely it is that the

actual return will end up far below the expected return. Thus, the tighter the probability dis-

tribution, the lower the risk assigned to a stock.Since U.S. Water has a relatively tight

probability distribution, its actual returnis likely to be closer to its 15 percent expected

returnthan is that of Martin Products.

Measuring Stand-Alone Risk: The Standard Deviation

Risk is a difficult concept to grasp, and a great deal of controversy has surrounded at-

tempts to define and measure it. However, a common definition, and one that is satis-

factory for many purposes, is stated in terms of probability distributions such as those

presented in Figure 3-2: The tighter the probability distribution of expected future returns,

the smaller the risk of a given investment.According to this definition, U.S. Water is less

risky than Martin Products because there is a smaller chance that its actual return will

end up far below its expected return.

To be most useful, any measure of risk should have a definite value—we need a

measure of the tightness of the probability distribution. One such measure is the stan-

dard deviation,the symbol for which is 
, pronounced “sigma.” The smaller the

standard deviation, the tighter the probability distribution, and, accordingly, the lower

Stand-Alone Risk 107

FIGURE 3-1 Probability Distributions of Martin Products’ and U.S. Water’s Rates of Return

Probability of

Occurrence

a. Martin Products

–70 0 15 100

0.4

0.3

0.2

0.1

Probability of

Occurrence

b. U.S. Water

Rate of Return

(%)

Rate of Return

(%)

010 15 20

0.4

0.3

0.2

0.1

Expected Rate

of Return

Expected Rate

of Return

Risk and Return 105