Chapter 8 Risk and Rates of Return 237the standard deviation, and it is shown at the bottom of Column 6 as a fraction and
a percentage.^8
The standard deviation is a measure of how far the actual return is likely to
deviate from the expected return. Martin’s standard deviation is 54.2%, so its
actual return is likely to be quite different from the expected 10%.^9 U.S. Water’s
standard deviation is 3.9%, so its actual return should be much closer to the ex-
pected return of 10%. The average publicly traded " rm’s # has been in the range
of 20% to 30% in recent years; so Martin is more risky than most stocks, while U.S.
Water is less risky.
8-2c Using Historical Data to Measure Risk^10
In the last section, we found the mean and standard deviation based on a subjective
probability distribution. If we had actual historical data instead, the standard devi-
ation of returns could be found as shown in Table 8-3.^11 Because past results are
often repeated in the future, the historical # is often used as an estimate of future
risk.^12 A key question that arises when historical data is used to forecast the future is
how far back in time should we go. Unfortunately, there is no simple answer. Using
a longer historical time series has the bene" t of giving more information, but some
of that information may be misleading if you believe that the level of risk in the
future is likely to be very different than the level of risk in the past.
Standard Deviation,
! (sigma)
A statistical measure of
the variability of a set of
observations.Standard Deviation,
! (sigma)
A statistical measure of
the variability of a set of
observations.(^8) This formula summarizes what we did in Table 8-2:
Standard deviation $ # $^ √
∑
i$ 1
N
(r (^) i! rˆ)^2 Pi^ 8-2
(^9) With a normal (bell-shaped) distribution, the actual return should be within one # about 68% of the time.
(^10) Again, this section is relatively technical, but it can be omitted without loss of continuity.
(^11) The 4 years of historical data are considered to be a “sample” of the full (but unknown) set of data, and the pro-
cedure used to! nd the standard deviation is di$ erent from the one used for probabilistic data. Here is the equa-
tion for sample data, and it is the basis for Table 8-3:
Estimated # $
√^
∑
t$ 1
N
(r - t! r-Avg)^2
___N! 1 8-2a
Here r-t (“r bar t”) denotes the past realized rate of return in Period t, and r-Avg is the average annual return earned
over the last N years.
(^12) The average return for the past period (10.3% in our example) may also be used as an estimate of future
returns, but this is problematic because the average historical return varies widely depending on the period
examined. In our example, if we went from 2005 to 2007, we would get a di$ erent average from the 10.3%. The
average historical return stabilizes with more years of data, but that brings into question whether data from many
years ago is still relevant today.
Tabl e 8 - 3 Finding! Based On Historical Data
A B C D E F
35
36
37
38
39
40
41
42
43
44
45
46
47
48
Year
(1)
2005
2006
2007
2008
30.0%
-10.0
-19.0
40.0
Average 10.3%
Return
(2)
Deviation
from
average
(3)
Squared
Deviation
(4)
19.8%
-20.3
-29.3
29.8
Variance = ": 25.4%
Variance/(N–1) = Variance/3:
Standard deviation = Square root of variance: # =
8.5%
3.9%
4.1
8.6
8.9
29.1%