equals the ratio of GM™s market value to the market value of all listed stocks.

Such strategies are called indexing. The investor chooses a portfolio with all the stocks in

a broad market index such as the Standard & Poor™s 500 index. The rate of return on the port-

folio then replicates the return on the index. Indexing has become an extremely popular strat-

egy for passive investors. We call the capital allocation line provided by one-month T-bills

capital market and a broad index of common stocks the capital market line (CML). That is, a passive strat-

line egy based on stocks and bills generates an investment opportunity set that is represented by

the CML.

The capital allocation

line using the market

index portfolio as the

Historical Evidence on the Capital Market Line

risky asset.

Can we use past data to help forecast the risk-return trade-off offered by the CML? The notion

that one can use historical returns to forecast the future seems straightforward but actually is

somewhat problematic. On one hand, you wish to use all available data to obtain a large sam-

ple. But when using long time series, old data may no longer be representative of future cir-

cumstances. Another reason for weeding out subperiods is that some past events simply may

be too improbable to be given equal weight with results from other periods. Do the data we

have pose this problem?

Table 5.5 breaks the 76-year period, 1926“2001 into four subperiods and shows the risk

premium, standard deviation, and reward-to-variability ratio for each subperiod. That ratio is

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Essentials of Investments, and Prologue Companies, 2003

Fifth Edition

Triumph of the Optimists

As a whole, the last 7 decades have been very kind Table 5.1 would seem to indicate. First, results from the

to U.S. equity investors. Stock investments have out- first 25 years of the last century (which included the

performed investments in safe Treasury bills by more first World War) were less favorable to stocks. Second,

than 8% per year. The real rate of return averaged U.S. returns have been better than that of most other

more than 9%, implying an expected doubling of countries, and so a more representative value for the

the real value of the investment portfolio about every historical risk premium may be lower than the U.S. ex-

8 years! perience. Finally, the sample that is amenable to his-

Is this experience representative? A new book by torical analysis suffers from a self-selection problem.

three professors at the London Business School, Elroy Only those markets that have survived to be studied

Dimson, Paul Marsh, and Mike Staunton, extends the can be included in the analysis. This leaves out coun-

U.S. evidence to other countries and to longer time tries such as Russia or China, whose markets were shut

periods. Their conclusion is given in the book™s title, down during communist rule, and whose results if

Triumph of the Optimists*: in every country in their included would surely bring down the average perfor-

study (which included markets in North America, Eu- mance of equity investments. Nevertheless, there is

rope, Asia, and Africa), the investment optimists”those powerful evidence of a risk premium that shows its

who bet on the economy by investing in stocks rather force everywhere the authors looked.

than bonds or bills”were vindicated. Over the long

haul, stocks beat bonds everywhere. *Elroy Dimson, Paul Marsh, Mike Staunton, Triumph of the Optimists:

On the other hand, the equity risk premium is prob- 101 Years of Global Investment Returns. Princeton University Press,

ably not as large as the post-1926 evidence from Princeton, N.J.: 2002.

the slope of the CML based on the subperiod data. Indeed, the differences across subperiods

are quite striking.

The most plausible explanation for the variation in subperiod returns is based on the

observation that the standard deviation of returns is quite large in all subperiods. If we take

the 76-year standard deviation of 20.3% as representative and assume that returns in one year

are nearly uncorrelated with those in other years (the evidence suggests that any correlation

across years is small), then the standard deviation of our estimate of the mean return in any of

our 19-year subperiods will be 20.3/ 19 4.7% , which is fairly large. This means that in

approximately one out of three cases, a 19-year average will deviate by 4.7% or more from the

true mean. Applying this insight to the data in Table 5.5 tells us that we cannot reject with any

confidence the possibility that the true mean is similar in all subperiods! In other words, the

“noise” in the data is so large that we simply cannot make reliable inferences from average re-

turns in any subperiod. The variation in returns across subperiods may simply reflect statisti-

cal variation, and we have to reconcile ourselves to the fact that the market return and the

reward-to-variability ratio for passive (as well as active!) strategies is simply very hard to

predict.

The instability of average excess return on stocks over the 19-year subperiods in Table 5.5

also calls into question the precision of the 76-year average excess return (8.64%) as an esti-

mate of the risk premium on stocks looking into the future. In fact, there has been consider-

able recent debate among financial economists about the “true” equity risk premium, with an

emerging consensus that the historical average is an unrealistically high estimate of the future

risk premium. This argument is based on several factors: the use of longer time periods in

157

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Essentials of Investments, and Prologue Companies, 2003

Fifth Edition

158 Part TWO Portfolio Theory

which equity returns are examined; a broad range of countries rather than just the U.S. in

which excess returns are computed (Dimson, Marsh, and Staunton, 2001); direct surveys of

financial executives about their expectations for stock market returns (Graham and Harvey,

2001); and inferences from stock market data about investor expectations (Jagannathan,

McGrattan, and Scherbina, 2000; Fama and French, 2002). The nearby box discusses some of

this evidence.

Costs and Benefits of Passive Investing

How reasonable is it for an investor to pursue a passive strategy? We cannot answer such a

question definitively without comparing passive strategy results to the costs and benefits ac-

cruing to an active portfolio strategy. Some issues are worth considering, however.

First, the alternative active strategy entails costs. Whether you choose to invest your own

valuable time to acquire the information needed to generate an optimal active portfolio of

risky assets or whether you delegate the task to a professional who will charge a fee, con-

structing an active portfolio is more expensive than constructing a passive one. The passive

portfolio requires only small commissions on purchases of U.S. T-bills (or zero commissions

if you purchase bills directly from the government) and management fees to a mutual fund

company that offers a market index fund to the public. An index fund has the lowest operating

expenses of all mutual stock funds because it requires minimal effort.

A second argument supporting a passive strategy is the free-rider benefit. If you assume

there are many active, knowledgeable investors who quickly bid up prices of undervalued as-

sets and offer down overvalued assets (by selling), you have to conclude that most of the time

most assets will be fairly priced. Therefore, a well-diversified portfolio of common stock will

be a reasonably fair buy, and the passive strategy may not be inferior to that of the average ac-

tive investor. We will expand on this insight and provide a more comprehensive analysis of the

relative success of passive strategies in Chapter 8.

To summarize, a passive strategy involves investment in two passive portfolios: virtually

risk-free short-term T-bills (or a money market fund) and a fund of common stocks that mim-

ics a broad market index. Recall that the capital allocation line representing such a strategy is

called the capital market line. Using Table 5.5, we see that using 1926 to 2001 data, the pas-

sive risky portfolio has offered an average excess return of 8.6% with a standard deviation of

20.7%, resulting in a reward-to-variability ratio of 0.42.

SUMMARY • Investors face a trade-off between risk and expected return. Historical data confirm our

intuition that assets with low degrees of risk provide lower returns on average than do

those of higher risk.

• Shifting funds from the risky portfolio to the risk-free asset is the simplest way to reduce

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risk. Another method involves diversification of the risky portfolio. We take up

diversification in later chapters.

• U.S. T-bills provide a perfectly risk-free asset in nominal terms only. Nevertheless, the

standard deviation of real rates on short-term T-bills is small compared to that of assets

such as long-term bonds and common stocks, so for the purpose of our analysis, we

consider T-bills the risk-free asset. Besides T-bills, money market funds hold short-term,

safe obligations such as commercial paper and CDs. These entail some default risk but

relatively little compared to most other risky assets. For convenience, we often refer to

money market funds as risk-free assets.

• A risky investment portfolio (referred to here as the risky asset) can be characterized by its

reward-to-variability ratio. This ratio is the slope of the capital allocation line (CAL), the

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Essentials of Investments, and Prologue Companies, 2003

Fifth Edition

159

5 Risk and Return: Past and Prologue

line connecting the risk-free asset to the risky asset. All combinations of the risky and risk-

free asset lie on this line. Investors would prefer a steeper sloping CAL, because that

means higher expected returns for any level of risk. If the borrowing rate is greater than

the lending rate, the CAL will be “kinked” at the point corresponding to an investment of

100% of the complete portfolio in the risky asset.

• An investor™s preferred choice among the portfolios on the capital allocation line will

depend on risk aversion. Risk-averse investors will weight their complete portfolios more

heavily toward Treasury bills. Risk-tolerant investors will hold higher proportions of their

complete portfolios in the risky asset.

• The capital market line is the capital allocation line that results from using a passive

investment strategy that treats a market index portfolio, such as the Standard &

Poor™s 500, as the risky asset. Passive strategies are low-cost ways of obtaining

well-diversified portfolios with performance that will reflect that of the broad stock

market.

KEY

arithmetic average, 133 expected return, 136 reward-to-variability

TERMS

asset allocation, 148 geometric average, 133 ratio, 152

capital allocation line, 152 holding-period return, 132 risk aversion, 138

capital market line, 156 inflation rate, 147 risk-free rate, 137

complete portfolio, 149 nominal interest rate, 147 risk premium, 137

dollar-weighted average passive strategy, 156 scenario analysis, 136

return, 134 probability distribution, 136 standard deviation, 136

excess return, 138 real interest rate, 147 variance, 136

PROBLEM

1. A portfolio of nondividend-paying stocks earned a geometric mean return of

SETS

5.0% between January 1, 1996, and December 31, 2002. The arithmetic mean

return for the same period was 6.0 %. If the market value of the portfolio at the

beginning of 1996 was $100,000, what was the market value of the portfolio at

the end of 2002?

2. Which of the following statements about the standard deviation is/are true? A standard

deviation:

i. Is the square root of the variance.

ii. Is denominated in the same units as the original data.

iii. Can be a positive or a negative number.

3. Which of the following statements reflects the importance of the asset allocation

decision to the investment process? The asset allocation decision:

a. Helps the investor decide on realistic investment goals.

b. Identifies the specific securities to include in a portfolio.

c. Determines most of the portfolio™s returns and volatility over time.

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d. Creates a standard by which to establish an appropriate investment time

horizon.

4. Look at Table 5.2 in the text. Suppose you now revise your expectations regarding the

stock market as follows:

State of the

Economy Probability HPR

Boom 0.3 44%

Normal growth 0.4 14

Recession 0.3 16

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

Essentials of Investments, and Prologue Companies, 2003

Fifth Edition

160 Part TWO Portfolio Theory

Use Equations 5.3“5.5 to compute the mean and standard deviation of the HPR on

stocks. Compare your revised parameters with the ones in the text.

5. The stock of Business Adventures sells for $40 a share. Its likely dividend payout

and end-of-year price depend on the state of the economy by the end of the year as

follows:

Dividend Stock Price

Boom $2.00 $50

Normal economy 1.00 43

Recession .50 34

a. Calculate the expected holding-period return and standard deviation of the holding-

period return. All three scenarios are equally likely.

b. Calculate the expected return and standard deviation of a portfolio invested

half in Business Adventures and half in Treasury bills. The return on bills

is 4%.

Use the following data in answering questions 6, 7, and 8.

Utility Formula Data

Expected Standard

Investment Return E(r) Deviation

1 .12 .30

2 .15 .50

3 .21 .16

4 .24 .21

2

U E(r) 1

„2A where A 4

6. Based on the utility formula above, which investment would you select if you were risk

averse with A 4?

a. 1

b. 2

c. 3

d. 4

7. Based on the utility formula above, which investment would you select if you were risk

neutral?

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