CP

the two distributions were normal, there would be a 68.26 percent probability that

Martin’s actual return would be in the range of 15 65.84 percent, or from 50.84 to

80.84 percent. For U.S. Water, the 68.26 percent range is 15 3.87 percent, or from

11.13 to 18.87 percent. With such a small , there is only a small probability that U.S.

Water’s return would be significantly less than expected, so the stock is not very risky.

For the average firm listed on the New York Stock Exchange, has generally been in

the range of 35 to 40 percent in recent years.

Using Historical Data to Measure Risk

In the previous example, we described the procedure for finding the mean and stan-

dard deviation when the data are in the form of a known probability distribution. If

only sample returns data over some past period are available, the standard deviation of

returns can be estimated using this formula:

(3-3a)

Here (“r bar t”) denotes the past realized rate of return in Period t, and is the

average annual return earned during the last n years. Here is an example:

Year

2000 15%

2001  5

2002 20

The historical is often used as an estimate of the future . Much less often, and gen-

erally incorrectly, for some past period is used as an estimate ofˆr, the expected

future return. Because past variability is likely to be repeated, S may be a good esti-

mate of future risk. But it is much less reasonable to expect that the past levelof return

(which could have been as high as 100% or as low as 50%) is the best expectation

of what investors think will happen in the future.^5

Measuring Stand-Alone Risk: The Coefficient of Variation

If a choice has to be made between two investments that have the same expected re-

turns but different standard deviations, most people would choose the one with the

lower standard deviation and, therefore, the lower risk. Similarly, given a choice be-

tween two investments with the same risk (standard deviation) but different expected

rAvg

B

350

2

13.2%.

Estimated  (or S)


B

(1510)^2 ( 5 10)^2 (2010)^2

3  1

rAvg

(15 5 20)

3

10.0%.

rt

rt rAvg

Estimated 
S


R

a

n

t
 1

(rtrAvg)^2

n 1

110 CHAPTER 3 Risk and Return

(^5) Equation 3-3a is built into all financial calculators, and it is very easy to use. We simply enter the rates of re-
turn and press the key marked S (or Sx) to get the standard deviation. Note, though, that calculators have no
built-in formula for finding S where unequal probabilities are involved; there you must go through the process
outlined in Table 3-3 and Equation 3-3. The same situation holds for computer spreadsheet programs.

108 Risk and Return