Stand-Alone Risk 109- Square each deviation, then multiply the result by the probability of occurrence for
its related outcome, and then sum these products to obtain the varianceof the prob-
ability distribution as shown in Columns 2 and 3 of the table:
(3-2)- Finally, find the square root of the variance to obtain the standard deviation:
(3-3)Thus, the standard deviation is essentially a weighted average of the deviations from
the expected value, and it provides an idea of how far above or below the expected
value the actual value is likely to be. Martin’s standard deviation is seen in Table 3-3 to
be 65.84%. Using these same procedures, we find U.S. Water’s standard deviation
to be 3.87 percent. Martin Products has the larger standard deviation, which indicates
a greater variation of returns and thus a greater chance that the expected return will
not be realized. Therefore, Martin Products is a riskier investment than U.S. Water
when held alone.
If a probability distribution is normal, the actualreturn will be within 1 standard
deviation of the expectedreturn 68.26 percent of the time. Figure 3-3 illustrates
this point, and it also shows the situation for 2 and 3 . For Martin Products,
r 15% and 65.84%, whereasr ˆ15% and 3.87% for U.S. Water. Thus, ifStandard deviation (^) Ba
n
i 1
(ri ˆr)^2 Pi.
Variance^2 a
n
i 1
(rir)ˆ^2 Pi.
FIGURE 3-3 Probability Ranges for a Normal Distribution
–3σ –2σ –1σ ˆr +1σ +2σ +3σ
68.26%
95.46%
99.74%
Notes:
a. The area under the normal curve always equals 1.0, or 100 percent. Thus, the areas under any pair of normal
curves drawn on the same scale, whether they are peaked or flat, must be equal.
b .Half of the area under a normal curve is to the left of the mean, indicating that there is a 50 percent probability
that the actual outcome will be less than the mean, and half is to the right of r, indicating a 50 percent probabil-ˆ
ity that it will be greater than the mean.
c .Of the area under the curve, 68 .26 percent is within 1 of the mean, indicating that the probability is 68.26
percent that the actual outcome will be within the range r ˆ 1 to r ˆ 1 .
d .Procedures exist for finding the probability of other ranges. These procedures are covered in statistics courses.
e .For a normal distribution, the larger the value of , the greater the probability that the actual outcome will vary
widely from, and hence perhaps be far below, the expected, or most likely, outcome. Since the probability of
having the actual result turn out to be far below the expected result is one definition of risk, and since mea-
sures this probability, we can use as a measure of risk.This definition may not be a good one, however, if we
are dealing with an asset held in a diversified portfolio. This point is covered later in the chapter.
For more discussion of
probability distributions,
see the Chapter 3 Web
Extension on the textbook’s
web site at http://ehrhardt.
swcollege.com.