222 PART 2 Important Financial Concepts
- The formula that is commonly used to find the standard deviation of returns, k, in a situation in which allout-
comes are known andtheir related probabilities are assumed equal, is
k
(5.3a)
where nis the number of observations. Equation 5.3 is emphasized in this chapter because returns and related prob-
abilities are often available.n
j 1 (kjk)2
n 1TABLE 5.4 Expected Values of Returns for
Assets A and B
Weighted value
Possible Probability Returns [(1)(2)]
outcomes (1) (2) (3)Asset A
Pessimistic .25 13% 3.25%
Most likely .50 15 7.50Optimistic . (^2) (^5) (^17) (^4) . (^2) (^5)
Total 1
.
0
0
Expected return 1
5
.
0
0
%
Asset B
Pessimistic .25 7% 1.75%
Most likely .50 15 7.50
Optimistic . (^2) (^5) (^23) (^5) . (^7) (^5)
Total 1
.
0
0
Expected return 1
5
.
0
0
%
EXAMPLE The expected values of returns for Norman Company’s assets A and B are pre-
sented in Table 5.4. Column 1 gives the Prj’s and column 2 gives the kj’s. In each
case nequals 3. The expected value for each asset’s return is 15%.
The expression for the standard deviation of returns,k, is^10
k
n
j 1
(kjk)^2 Prj (5.3)
In general, the higher the standard deviation, the greater the risk.
EXAMPLE Table 5.5 presents the standard deviations for Norman Company’s assets A and
B, based on the earlier data. The standard deviation for asset A is 1.41%, and the
standard deviation for asset B is 5.66%. The higher risk of asset B is clearly
reflected in its higher standard deviation.
Historical Returns and Risk We can now use the standard deviation as a
measure of risk to assess the historical (1926–2000) investment return data in
Table 5.2. Table 5.6 repeats the historical returns and shows the standard devia-
tions associated with each of them. A close relationship can be seen between the
investment returns and the standard deviations: Investments with higher returns
have higher standard deviations. Because higher standard deviations are associ-
ated with greater risk, the historical data confirm the existence of a positive rela-