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).

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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.

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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%.