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5 Risk and Return: Past and Prologue

F I G U R E 5.2

50%

Rates of return on

stocks, bonds, and

Treasury bills,

30%

1926“2001

Source: Prepared from data

Rates of return

in Table 5.2.

10%

10%

Stocks

30% T-bonds

T-bills

50%

1926 1941 1956 1971 1986 2001

68.26%

95.44%

99.74%

0

“4σ “3σ “2σ “1σ +1σ +2σ +3σ +4σ

12.5

“68.7 “48.4 “28.1 “7.8 32.8 53.1 73.4 93.7

F I G U R E 5.3

The normal distribution with mean return 12.5% and standard deviation 20.3%

Figure 5.3 is a graph of the normal curve with mean 12.5% and standard deviation 20.3%.

The graph shows the theoretical probability of rates of return within various ranges given

these parameters. Now observe the frequency distributions in Figure 5.1. The resemblance to

the normal distribution is clear, and the variation in the dispersion of the frequency distribu-

tions across the different asset classes vividly illustrates the difference in standard deviation

and its implication for risk.

The similarity to a normal distribution allows us to use the historical average and standard

deviation to estimate probabilities for various scenarios. For example, we estimate the proba-

bility that the return on the large-stock portfolio will fall below zero in the next year at .27. This

is so because a rate of zero is 12.5 percentage points, or .6 standard deviations ( 12.5/20.3)

Bodie’Kane’Marcus: II. Portfolio Theory 5. Risk and Return: Past © The McGraw’Hill

Essentials of Investments, and Prologue Companies, 2003

Fifth Edition

146 Part TWO Portfolio Theory

Geometric Arithmetic Standard

TA B L E 5.4 Decile Mean Mean Deviation

Size-Decile Portfolios

1-Largest 10.3% 12.1% 19.05%

of the NYSE/

AMEX/NASDAQ 2 11.3 13.6 22.00

Summary Statistics of 3 11.6 14.2 23.94

Annual Returns, 4 11.5 14.6 26.25

1926“2000 5 11.8 15.2 27.07

6 11.8 15.5 28.15

7 11.6 15.7 30.43

8 11.7 16.6 34.21

9 11.8 17.4 37.13

10-Smallest 13.1 20.9 45.82

NYSE/AMEX/NASDAQ

Total Value Weighted Index 10.6% 12.6% 20.20%

below the mean. Using a table of the normal distribution, we find that the probability of a nor-

mal variable falling .6 or more standard deviations below its mean is .27.

The performance of the small-stock portfolio documented in the preceding figures and ta-

bles is striking. Table 5.4 shows average returns and standard deviations for NYSE portfolios

arranged by firm size. Average returns generally are higher as firm size declines. The data

clearly suggest that small firms have earned a substantial risk premium and therefore that firm

size seems to be an important proxy for risk. In later chapters we will further explore this phe-

nomenon and will see that the size effect can be further related to other attributes of the firm.

We should stress that variability of HPR in the past sometimes can be an unreliable guide

to risk, at least in the case of the risk-free asset. For an investor with a holding period of one

year, for example, a one-year T-bill is a riskless investment, at least in terms of its nominal re-

turn, which is known with certainty. However, the standard deviation of the one-year T-bill

rate estimated from historical data is not zero: This reflects year-by-year variation in expected

returns rather than fluctuations of actual returns around prior expectations.

>

Concept 4. Compute the average excess return on large company stocks (over the T-bill rate)

and the standard deviation for the years 1926“1934.

CHECK

5.4 INFL ATION AND REAL RATES OF RETURN

The historical rates of return we reviewed in the previous section were measured in dollars. A

10% annual rate of return, for example, means that your investment was worth 10% more at

the end of the year than it was at the beginning of the year. This does not necessarily mean,

however, that you could have bought 10% more goods and services with that money, for it is

possible that in the course of the year prices of goods also increased. If prices have changed,

the increase in your purchasing power will not equal the increase in your dollar wealth.

At any time, the prices of some goods may rise while the prices of other goods may fall; the

general trend in prices is measured by examining changes in the consumer price index, or CPI.

The CPI measures the cost of purchasing a bundle of goods that is considered representative of

the “consumption basket” of a typical urban family of four. Increases in the cost of this stan-

dardized consumption basket are indicative of a general trend toward higher prices. The infla-

inflation rate

tion rate, or the rate at which prices are rising, is measured as the rate of increase of the CPI.

The rate at which

Suppose the rate of inflation (the percentage change in the CPI, denoted by i) for the last

prices are rising,

year amounted to i 6%. This tells you the purchasing power of money is reduced by 6% a

measured as the rate

of increase of the CPI. year. The value of each dollar depreciates by 6% a year in terms of the goods it can buy.

Therefore, part of your interest earnings are offset by the reduction in the purchasing power of

Bodie’Kane’Marcus: II. Portfolio Theory 5. Risk and Return: Past © The McGraw’Hill

Essentials of Investments, and Prologue Companies, 2003

Fifth Edition

147

5 Risk and Return: Past and Prologue

the dollars you will receive at the end of the year. With a 10% interest rate, after you net out nominal

the 6% reduction in the purchasing power of money, you are left with a net increase in pur- interest rate

chasing power of about 4%. Thus, we need to distinguish between a nominal interest rate”

The interest

the growth rate of your money”and a real interest rate”the growth rate of your purchasing rate in terms

power. If we call R the nominal rate, r the real rate, and i the inflation rate, then we conclude of nominal

(not adjusted

r R i for purchasing

power) dollars.

In words, the real rate of interest is the nominal rate reduced by the loss of purchasing power

resulting from inflation.

real interest rate

In fact, the exact relationship between the real and nominal interest rate is given by

The excess

1 R of the interest rate

1 r

over the inflation

1 i

rate. The growth rate

In words, the growth factor of your purchasing power, 1 r, equals the growth factor of your of purchasing power

derived from an

money, 1 R, divided by the new price level that is 1 i times its value in the previous pe-

investment.

riod. The exact relationship can be rearranged to

R i

r

1 i

which shows that the approximate rule overstates the real rate by the factor 1 i.

For example, if the interest rate on a one-year CD is 8%, and you expect inflation to be 5%

over the coming year, then using the approximation formula, you expect the real rate to be

.08 .05

r 8% 5% 3%. Using the exact formula, the real rate is r .0286, or

1 .05

2.86%. Therefore, the approximation rule overstates the expected real rate by only 0.14 per-

centage points. The approximation rule is more accurate for small inflation rates and is per-

fectly exact for continuously compounded rates.

To summarize, in interpreting the historical returns on various asset classes presented in

Table 5.3, we must recognize that to obtain the real returns on these assets, we must reduce the

nominal returns by the inflation rate presented in the last column of the table. In fact, while the

return on a U.S. Treasury bill usually is considered to be riskless, this is true only with regard

to its nominal return. To infer the expected real rate of return on a Treasury bill, you must sub-

tract your estimate of the inflation rate over the coming period.

It is always possible to calculate the real rate after the fact. The inflation rate is published

by the Bureau of Labor Statistics. The future real rate, however, is unknown, and one has to

rely on expectations. In other words, because future inflation is risky, the real rate of return is

risky even if the nominal rate is risk-free.

The Equilibrium Nominal Rate of Interest

We™ve seen that the real rate of return on an asset is approximately equal to the nominal rate

minus the inflation rate. Because investors should be concerned with their real returns”the

increase in their purchasing power”we would expect that as inflation increases, investors will

demand higher nominal rates of return on their investments. This higher rate is necessary to

maintain the expected real return offered by an investment.

Irving Fisher (1930) argued that the nominal rate ought to increase one-for-one with in-

creases in the expected inflation rate. If we use the notation E(i) to denote the current expec-

tation of the inflation rate that will prevail over the coming period, then we can state the

so-called Fisher equation formally as

R r E(i)

Bodie’Kane’Marcus: II. Portfolio Theory 5. Risk and Return: Past © The McGraw’Hill

Essentials of Investments, and Prologue Companies, 2003

Fifth Edition

148 Part TWO Portfolio Theory

F I G U R E 5.4 16%

Inflation and interest

14%

rates, 1953“2001

T-bill rate

12%

10%

8%

6%

4%

Inflation rate

2%

0%

2%

1953 1958 1963 1968 1973 1978 1983 1988 1993 1998 2003

Suppose the real rate of interest is 2%, and the inflation rate is 4%, so that the nominal in-

terest rate is about 6%. If the rate of inflation increases to 5%, the nominal rate should climb

to roughly 7%. The increase in the nominal rate offsets the increase in the inflation rate, giv-

ing investors an unchanged growth of purchasing power at a 2% real rate. The evidence for the

Fisher equation is that periods of high inflation and high nominal rates generally coincide.

Figure 5.4 illustrates this fact.

>

5. a. Suppose the real interest rate is 3% per year, and the expected inflation rate is

Concept

8%. What is the nominal interest rate?

CHECK b. Suppose the expected inflation rate rises to 10%, but the real rate is un-

changed. What happens to the nominal interest rate?

5.5 ASSET ALLOCATION ACROSS RISKY AND