234 Part 3 Financial Assets
(^3) It is completely unrealistic to think that any stock has no chance of a loss. Only in hypothetical examples could
this occur. To illustrate, the price of Countrywide Financial’s stock dropped from $45.26 to $4.43 in the 12 months
ending January 2008.
(^4) The expected return can also be calculated with an equation that does the same thing as the table:
Expected rate of return $ rˆ $ P 1 r 1 " P 2 r 2 "... " PNrN
8-1 $ ∑
i$ 1
N
Piri
The second form of the equation is a shorthand expression in which sigma ( ∑ ) means “sum up,” or add the values
of n factors. If i $ 1, then Piri $ P 1 r 1 ; if i $ 2, then Piri $ P 2 r 2 ; and so forth; until i $ N, the last possible outcome.
The symbol ∑
i$ 1
N
simply says, “Go through the following process: First, let i $ 1 and! nd the! rst product; then let
i $ 2 and! nd the second product; then continue until each individual product up to N has been found. Add
these individual products to! nd the expected rate of return.”
A B C D E F G
3 4 5 6 7 8 9
10
11
12
13
14
15
16
Economy,
Which
A"ects
Demand
(1)
Strong
Normal
Weak
Probability
of This
Demand
Occurring
(2)
Rate of
Return
If This
Demand
Occurs
(3)
Probability
of This
Demand
Occurring
(5)
Rate of
Return
If This
Demand
Occurs
(6)
Product
(2)!(3)
(4)
Product
(5)!(6)
(7)
Martin Products U.S. Water
0.30
0.40
0.30
1.00 10%
80%
10
-60
24%
4
-18
Expected return = Expected return = 10.0%
0.30
0.40
0.30
1.00
15%
10
5
4.5%
4.0
1.5
Tabl e 8 - 1 Probability Distributions and Expected Returns
Columns 3 and 6 show the returns for the two companies under each state of
the economy. Returns are relatively high when demand is strong and low when
demand is weak. Notice, though, that Martin’s rate of return could vary far more
widely than U.S. Water’s. Indeed, there is a fairly high probability that Martin’s
stock will suffer a 60% loss, while at worst, U.S. Water should have a 5% return.^3
Columns 4 and 7 show the products of the probabilities times the returns
under the different demand levels. When we sum these products, we obtain the
expected rates of return, rˆ “r-hat,” for the stocks. Both stocks have an expected re-
turn of 10%.^4
We can graph the data in Table 8-1 as we do in Figure 8-2. The height of each
bar indicates the probability that a given outcome will occur. The range of possible
returns for Martin is from !60% to "80%, and the expected return is 10%. The ex-
pected return for U.S. Water is also 10%, but its possible range (and thus maximum
loss) is much narrower.
In Figure 8-2, we assumed that only three economic states could occur: strong,
normal, and weak. Actually, the economy can range from a deep depression to a
fantastic boom; and there are an unlimited number of possibilities in between.
Suppose we had the time and patience to assign a probability to each possible level
of demand (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 8-1 except that it would have many more demand levels. This table could be
used to calculate expected rates of return as shown previously, and the probabili-
ties and outcomes could be represented by continuous curves such as those shown
in Figure 8-3. Here we changed the assumptions so that there is essentially no
chance that Martin’s return will be less than –60% or more than 80% or that
Expected Rate of
Return, rˆ
The rate of return expected
to be realized from an
investment; the weighted
average of the probability
distribution of possible
results.
Expected Rate of
Return, rˆ
The rate of return expected
to be realized from an
investment; the weighted
average of the probability
distribution of possible
results.