Intermediate Financial Management (with Thomson One)

Table 2-3
Calculating Sale.com’s Standard Deviation
r i ˆr (r i ˆr ) 2 (r i ˆr ) 2 P i
(1) (2) (3)
100 15 85 7,225 (7,225)(0.3) 2,167.5
15 15 0 0 (0)(0.4) 0.0
70 15 85 7,225 (7,225)(0.3) 2,167.5
Variance 2 4,335.0
Standard deviation 2 2 24,335 65.84
For Sale.com, we previously found ˆr 15%.
2. Subtract the expected rate of return (ˆr) from each possible outcome (r i ) to
obtain a set of deviations about ˆr as shown in Column 1 of Table 2-3:
Deviation i r i ˆr
3. Square each deviation, then multiply the result by the probability of occurrence
for its related outcome, and then sum these products to obtain the variance
of the probability distribution as shown in Columns 2 and 3 of the table:
Variance 2 a (ri rˆ) 2Pi | 2-2 |
4. Finally, find the square root of the variance to obtain the standard deviation:
Standard Deviation
B a (ri rˆ) 2Pi | 2-3 |
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. Sale.com’s standard deviation is
seen in Table 2-3 to be 65.84%. Using these same procedures, we find Basic
Foods’ standard deviation to be 19.36 percent. Sale.com has the larger standard
deviation, which indicates a greater variation of returns and thus a greater chance
that the actual return may be substantially lower than the expected return. Therefore,
Sale.com is a riskier investment than Basic Foods when held alone.
If a probability distribution is normal, the actual return will be within 1
standard deviation of the expected return 68.26 percent of the time. Figure 2-3
illustrates this point, and it also shows the situation for 2 and 3. For
Sale.com, ˆr 15% and 65.84%, whereas ˆr 15% and 19.36% for Basic
Foods. Thus, if the two distributions were normal, there would be a 68.26 percent
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Chapter 2 Risk and Return: Part I • 39