CP

(National Geographic (Little) Kids) #1
The expected rate of return calculation can also be expressed as an equation that
does the same thing as the payoff matrix table:^4

(3-1)

Here riis the ith possible outcome, Piis the probability of the ith outcome, and n is
the number of possible outcomes. Thus, r is a weighted average of the possible out-ˆ
comes (the rivalues), with each outcome’s weight being its probability of occurrence.
Using the data for Martin Products, we obtain its expected rate of return as follows:
ˆr P 1 (r 1 ) P 2 (r 2 ) P 3 (r 3 )
0.3(100%) 0.4(15%) 0.3(70%)
15%.
U.S. Water’s expected rate of return is also 15 percent:
ˆr 0.3(20%) 0.4(15%) 0.3(10%)
15%.
We can graph the rates of return to obtain a picture of the variability of possible out-
comes; this is shown in the Figure 3-1 bar charts. The height of each bar signifies the
probability that a given outcome will occur. The range of probable returns for Martin
Products is from 70 to 100 percent, with an expected return of 15 percent. The ex-
pected return for U.S. Water is also 15 percent, but its range is much narrower.
Thus far, we have assumed that only three situations can exist: strong, normal, and
weak demand. Actually, of course, demand could range from a deep depression to a
fantastic boom, and there are an unlimited number of possibilities in between. Sup-
pose we had the time and patience to assign a probability to each possible level of de-
mand (with the sum of the probabilities still equaling 1.0) and to assign a rate of return
to each stock for each level of demand. We would have a table similar to Table 3-1, ex-
cept that it would have many more entries in each column. This table could be used to

 a

n

i 1

Piri.

Expected rate of returnrˆP 1 r 1 P 2 r 2 Pnrn

106 CHAPTER 3 Risk and Return

TABLE 3-2 Calculation of Expected Rates of Return: Payoff Matrix

Martin Products U.S. Water
Demand for Probability Rate of Return Rate of Return
the Company’s of This Demand if This Demand Product: if This Demand Product:
Products Occurring Occurs (2) (3) Occurs (2) (5)
(1) (2) (3) (4) (5) (6)
Strong 0.3 100% 30% 20% 6%
Normal 0.4 15 6 15 6
Weak 0.3 (70) (21) 10 3

1.0 ˆr 15% ˆr 15%

(^4) The second form of the equation is simply a shorthand expression in which sigma () means “sum up,” or
add the values of n factors. If i 1, then PiriP 1 r 1 ; if i 2, then PiriP 2 r 2 ; and so on until i n, the
last possible outcome. The symbol in Equation 3-1 simply says, “Go through the following process:
First, let i 1 and find the first product; then let i 2 and find the second product; then continue until each
individual product up to i n has been found, and then add these individual products to find the expected
rate of return.”
a
n
i 1


104 Risk and Return
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