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1973 40.54 14.75 1.40 2.90 6.91 8.71
1974 29.74 26.40 5.53 6.03 7.93 12.34
1975 69.54 37.26 8.50 6.79 5.80 6.94
1976 54.81 23.98 11.07 14.20 5.06 4.86
1977 22.02 7.26 0.90 1.12 5.10 6.70
1978 22.29 6.50 4.16 0.32 7.15 9.02
1979 43.99 18.77 9.02 4.29 10.45 13.29
1980 35.34 32.48 13.17 0.83 11.57 12.52
1981 7.79 4.98 3.61 6.09 14.95 8.92
1982 27.44 22.09 6.52 33.39 10.71 3.83
1983 34.49 22.37 0.53 5.44 8.85 3.79
1984 14.02 6.46 15.29 14.46 10.02 3.95
1985 28.21 32.00 32.68 23.65 7.83 3.80
1986 3.40 18.40 23.96 17.22 6.18 1.10
1987 13.95 5.34 2.65 1.68 5.50 4.43
1988 21.72 16.86 8.40 6.63 6.44 4.42
1989 8.37 31.34 19.49 14.82 8.32 4.65
1990 27.08 3.20 7.13 9.05 7.86 6.11
1991 50.24 30.66 18.39 16.67 5.65 3.06
1992 27.84 7.71 7.79 7.25 3.54 2.90
1993 20.30 9.87 15.48 12.02 2.97 2.75
1994 3.34 1.29 7.18 4.42 3.91 2.67
1995 33.21 37.71 31.67 18.07 5.58 2.54
1996 16.50 23.07 0.81 3.99 5.50 3.32
1997 22.40 33.17 15.08 7.69 5.32 1.70
1998 2.50 28.53 13.02 8.62 5.11 1.61
1999 21.26 21.04 8.74 0.41 4.80 2.68
2000 3.02 9.10 20.27 10.26 5.85 3.39
2001 1.03 11.89 4.21 8.16 4.09 1.67
Rate of
Return Statistics
Geometric average 12.19 10.51 5.23 5.12 3.80 3.06
Arithmetic average 18.29 12.49 5.53 5.30 3.85 3.15
Standard deviation 39.28 20.30 8.18 6.33 3.25 4.40
Minimum 52.71 45.56 8.74 5.81 1.59 10.27
Maximum 187.82 54.56 32.68 33.39 14.95 18.13
Excess
Return Statistics
Average 14.44 8.64 1.68 1.45
Standard deviation 39.98 20.70 7.94 5.73
Minimum 55.55 46.52 13.54 10.74
Maximum 187.75 54.49 26.09 22.68


Sources: Inflation data: Bureau of Labor Statistics; security return data for 1926“1995: Center for Research in Security Prices; security return data for 1996“2001:
returns on appropriate index portfolios (large stocks, S&P 500; small stocks, Russell 2000; long-term/intermediate T-bonds, Lehman Bros.; T-bills, Salomon
Smith Barney).
Bodie’Kane’Marcus: II. Portfolio Theory 5. Risk and Return: Past © The McGraw’Hill
Essentials of Investments, and Prologue Companies, 2003
Fifth Edition




142 Part TWO Portfolio Theory


month to month, the total rate of return on these T-bills is riskless only for 30-day holding
periods.3 The last column gives the annual inflation rate as measured by the rate of change in
the Consumer Price Index.
Figure 5.1 summarizes some of the data in Table 5.3. The first column gives the geometric
averages of the historical rates of return on each asset class; this figure thus represents the
compound rate of growth in the value of an investment in these assets. The second column
shows the arithmetic averages. The last column is the variability of asset returns, as measured
by standard deviation. The historical results are consistent with the risk-return trade-off:
Riskier assets have provided higher average returns, and historical risk premiums are consid-
erable, commensurate with their greater risk.


The dollar import of the risk premiums suggested by Table 5.3 can be illustrated with a simple
example. Consider two investors with $1,000 as of December 31, 1991. One invests in the small-
5.4 EXAMPLE stock portfolio, and the other in T-bills. Both investors reinvest all income from their portfolios and
liquidate their investments on December 31, 2001. Using the rates from Table 5.3 we have:
The Risk
Premium and
Small Stocks T-Bills
Growth of
Wealth December 31, 1991 $1,000 $1,000
December 31, 2001 $3,271 $1,577


The value of the portfolio in 2001 is obtained by multiplying the initial $1,000 investment in
1991 by 1 plus the rate of return (expressed as a decimal) for each year. The results show
an increase of 227.1% in the value of the small-stock portfolio and an increase of 57.7%
for T-bills.
We also can calculate the geometric average return over this period. For small stocks, the
geometric average, rG, over this 10-year period is defined by:
rG)10
(1 3.271
rG (3.271)1/10
1 1.1258
rG .1258 12.58%
You can confirm that the geometric average return of T-bills was 4.66%. Therefore, the 10-year
period 1991“2001 was more favorable to both small stocks and T-bills than the 1926“2001 pe-
riod. The geometric average return on small stocks in the most recent 10-year period was
12.58% versus an average of 12.19% in the longer period; the average return on bills was
4.66% in the recent period versus 3.80% in the longer period. In both periods, there was a con-
siderable risk premium to be earned from the risky investment.




The third statistic reported in Figure 5.1 is the standard deviation. The higher the standard
deviation, the more volatile the HPR. The standard deviation reported in Figure 5.1, however,
is based on historical data rather than forecasts of future scenarios, as in Equation 5.5. To cal-
culate standard deviation from historical data, we treat each year™s outcome as one possible
scenario in a scenario analysis. Each historical outcome is taken as equally likely and given a
“probability” of 1/n.
The formula for historical variance is thus similar to Equation 5.5, but instead of using de-
viations of returns around mean returns based on the scenario analysis, we use deviations from

3
The few negative returns in this column, all dating from before World War II, reflect periods where, in the absence
of T-bills, returns on government securities with about 30-day maturity were used. However, these securities included
options to be exchanged for other securities, thus increasing their price and lowering their yield relative to what a sim-
ple T-bill would have offered.
Bodie’Kane’Marcus: II. Portfolio Theory 5. Risk and Return: Past © The McGraw’Hill
Essentials of Investments, and Prologue Companies, 2003
Fifth Edition




143
5 Risk and Return: Past and Prologue




Small company stocks
50
Geometric Mean 12.19%
18.29% 40
Arithmetic Mean
Standard Distribution 39.28%
30
20
10

0
“90% “60% “30% 0% 30% 60% 90%

Large company stocks
50
Geometric Mean 10.51%
12.49% 40
Arithmetic Mean
Standard Distribution 20.30%
30

20
10

0
“90% “60% “30% 0% 30% 60% 90%

Long-term gov™t bonds
50
Geometric Mean 5.23%
40
Arithmetic Mean 5.53%
Standard Distribution 8.18%
30
20

10

0
“90% “60% “30% 0% 30% 60% 90%

Treasury bills
50
Geometric Mean 3.80%
40
Arithmetic Mean 3.85%
Standard Distribution 3.25%
30

20

10
0
“90% “60% “30% 0% 30% 60% 90%




F I G U R E 5.1
Frequency distribution of annual HPRs, 1926“2001. Each bar shows the number of years that the rate of return of the
particular market fell within a specified range.



average returns during the sample period. This procedure results in one minor complication.
When we use the sample average return r in place of the mean return, E(r), we must modify
the average of the squared deviations for what statisticians call a “lost degree of freedom.” The
Bodie’Kane’Marcus: II. Portfolio Theory 5. Risk and Return: Past © The McGraw’Hill
Essentials of Investments, and Prologue Companies, 2003
Fifth Edition




144 Part TWO Portfolio Theory


n
modification is easy: Multiply the average value of the squared deviations by n 1 . The for-
mula for variance based on historical data is thus:
n
2
Sample average of squared deviations from average return
n 1
r)2
n 1
n n
(ri ¯
r)2
(ri (5.8)
n 1 n 1
n
i 1 i 1

When you are using large samples and n is large, the modification is unimportant, since
n/(n 1) is close to 1.0 and 1/(n 1) is close to 1/n.

To illustrate how to calculate average returns and standard deviations from historical data, let™s
compute these statistics for the returns on the S&P 500 portfolio using the five years of data in
5.5 EXAMPLE Table 5.3 from 1988“1992. The average return is 16.7%, computed by dividing the sum of col-
umn (1) below, by the number of observations. In column (2), we take the deviation of each
Historical Means
year™s return from the 16.7% average return. In column (3), we calculate the squared devia-
and Standard
tion. The variance is, from equation 5.8, the sum of the five squared deviations divided by (5 1).
Deviations
The standard deviation is the square root of the variance. If you input the column of rates into
a spreadsheet, the “Average” and “StdDev” functions will give you the statistics directly.

(2)
(1) Deviation from (3)
Rate of Average Squared
Year Return Return Deviation

1988 16.9% 0.2% 0.0
1989 31.3 14.6 213.2
1990 3.2 19.9 396.0
1991 30.7 14.0 196.0
1992 7.7 9.0 81.0

Total 83.4% 886.2

Average rate of return 83.4/5 16.7
1
Variance 886.2 221.6
5 1
Standard deviation 221.6 14.9%


Figure 5.2 presents a graphic representation of the relative variabilities of the annual HPR
for three different asset classes: large stocks, long-term T-bonds, and T-bills. We have plotted
the three time series on the same set of axes to demonstrate clearly that the annual HPR on
stocks is the most variable series. The standard deviation of large-stock returns has been
20.30% compared to 8.18% for long-term government bonds and 3.25% for bills. Here is evi-
dence of the risk-return trade-off that characterizes security markets: The markets with the
highest average returns are also the most volatile.
An all-stock portfolio with a standard deviation of 20.3% would represent a very volatile
investment. For example, if stock returns are normally distributed with a standard deviation
of 20.3% and an expected rate of return of 12.5% (the historical average), then in roughly
one year out of three, returns will be less than 12.5 20.3 7.8%, or greater than
12.5 20.3 32.8%.

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