Hence, we arrive at the same expected return“beta relationship as the CAPM without any

assumption about either investor preferences or access to the all-inclusive (and elusive) mar-

ket portfolio.

The APT and the CAPM

Why did we need so many restrictive assumptions to derive the CAPM when the APT seems

to arrive at the expected return“beta relationship with seemingly fewer and less objectionable

assumptions? The answer is simple: The APT applies only to well-diversified portfolios. Ab-

sence of riskless arbitrage alone cannot guarantee that, in equilibrium, the expected

return“beta relationship will hold for any and all assets.

With additional effort, however, one can use the APT to show that the relationship must

hold approximately even for individual assets. The essence of the proof is that if the expected

return“beta relationship were violated by many individual securities, it would be virtually im-

possible for all well-diversified portfolios to satisfy the relationship. So the relationship must

almost surely hold true for individual securities.

We say “almost” because, according to the APT, there is no guarantee that all individual

assets will lie on the SML. If only a few securities violated the SML, their effect on well-

diversified portfolios could conceivably be offsetting. In this sense, it is possible that the SML

relationship is violated for single securities. If many securities violate the expected return“beta

relationship, however, the relationship will no longer hold for well-diversified portfolios com-

prising these securities, and arbitrage opportunities will be available.

The APT serves many of the same functions as the CAPM. It gives us a benchmark for fair

rates of return that can be used for capital budgeting, security evaluation, or investment per-

formance evaluation. Moreover, the APT highlights the crucial distinction between nondiver-

sifiable risk (systematic or factor risk) that requires a reward in the form of a risk premium and

diversifiable risk that does not.

The bottom line is that neither of these theories dominates the other. The APT is more gen-

eral in that it gets us to the expected return“beta relationship without requiring many of the un-

realistic assumptions of the CAPM, particularly the reliance on the market portfolio. The latter

improves the prospects for testing the APT. But the CAPM is more general in that it applies

to all assets without reservation. The good news is that both theories agree on the expected

return“beta relationship.

It is worth noting that because past tests of the expected return“beta relationship examined

the rates of return on highly diversified portfolios, they actually came closer to testing the APT

than the CAPM. Thus, it appears that econometric concerns, too, favor the APT.

Multifactor Generalization of the APT and CAPM

We™ve assumed all along that there is only one systematic factor affecting stock returns. This

assumption may be too simplistic. It is easy to think of several factors that might affect stock

returns: business cycles, interest rate fluctuations, inflation rates, oil prices, and so on. Pre-

sumably, exposure to any of these factors singly or together will affect a stock™s perceived

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Fifth Edition Theory

248 Part TWO Portfolio Theory

riskiness and appropriate expected rate of return. We can use a multifactor version of the APT

to accommodate these multiple sources of risk.

Suppose we generalize the single-factor model expressed in Equation 7.5 to a two-factor

model:

Ri i1RM1 i2RM2 ei (7.7)

i

where RM1 and RM2 are the excess returns on portfolios that represent the two systematic fac-

tors. Factor 1 might be, for example, unanticipated changes in industrial production, while fac-

tor 2 might represent unanticipated changes in short-term interest rates. We assume again that

there are many securities available with any combination of betas. This implies that we can

form well-diversified factor portfolios, that is, portfolios that have a beta of 1.0 on one factor

factor portfolio

and a beta of zero on all others. Thus, a factor portfolio with a beta of 1.0 on the first factor

A well-diversified

will have a rate of return of RM1; a factor portfolio with a beta of 1.0 on the second factor will

portfolio constructed

have a rate of return of RM2; and so on. Factor portfolios can serve as the benchmark portfo-

to have a beta of 1.0

on one factor and a lios for a multifactor generalization of the security market line relationship.

beta of zero on any Suppose the two-factor portfolios, here called portfolios 1 and 2, have expected returns

other factor.

E(r1) 10% and E(r2) 12%. Suppose further that the risk-free rate is 4%. The risk premium

on the first factor portfolio is therefore 6%, while that on the second factor portfolio is 8%.

Now consider an arbitrary well-diversified portfolio (A), with beta on the first factor,

0.5, and on the second factor, A2 0.75. The multifactor APT states that the portfolio

A1

risk premium must equal the sum of the risk premiums required as compensation to investors

for each source of systematic risk. The risk premium attributable to risk factor 1 is the port-

folio™s exposure to factor 1, A1, times the risk premium earned on the first factor portfolio,

E(r1) rf . Therefore, the portion of portfolio A™s risk premium that is compensation for its

exposure to the first risk factor is A1[E(r1) rf ] 0.5 (10% 4%) 3%, while the risk pre-

mium attributable to risk factor 2 is A2[E(r2) rf ] 0.75 (12% 4%) 6%. The total risk

premium on the portfolio, therefore, should be 3 6 9%, and the total return on the port-

folio should be 13%.

4% Risk-free rate

3% Risk premium for exposure to factor 1

6% Risk premium for exposure to factor 2

13% Total expected return

To generalize this argument, note that the factor exposure of any portfolio P is given by

its betas, P1 and P2. A competing portfolio, Q, can be formed from factor portfolios with

the following weights: P1 in the first factor portfolio; P2 in the second factor portfolio; and

1 P2 in T-bills. By construction, Q will have betas equal to those of portfolio P

P2

and an expected return of

E(rQ) P1E(r1) P2E(r2) (1 P2)rf

P1

rf P1[E(r1) rf ] P2[E(r2) rf ] (7.8)

Using our numbers,

E(rQ) 4 .5 (10 4) .75 (12 4) 13%

Because portfolio Q has precisely the same exposures as portfolio A to the two sources of risk,

their expected returns also ought to be equal. So portfolio A also ought to have an expected

return of 13%.

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Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

EXCE L Applications www.mhhe.com/bkm

> Estimating the Index Model

The spreadsheet below (available at www.mhhe.com/bkm) also contains monthly returns for

the stocks that comprise the Dow Jones Industrial Average. The spreadsheet contains workbooks

that show raw returns, risk premiums, correlation coefficients, and beta coefficients for the stocks

that are in the DJIA. The security characteristic lines are estimated with five years of monthly

returns.

A B C D E F G H I

1 SUMMARY OUTPUT AXP

2

Regression Statistics

3

4 Multiple R 0.69288601

5 R Square 0.48009103

6 Adjusted R Square 0.47112708

7 Standard Error 0.05887426

8 Observations 60

9

10 ANOVA

df SS MS F Significance F

11

12 Regression 1 0.185641557 0.1856416 53.55799 8.55186E-10

13 Residual 58 0.201038358 0.0034662

14 Total 59 0.386679915

15

Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%

16

17 Intercept 0.01181687 0.00776211 1.522379 0.133348 0.003720666 0.027354414 0.0037207 0.02735441

18 X Variable 1 1.20877413 0.165170705 7.3183324 8.55E-10 0.878149288 1.539398969 0.87814929 1.53939897

Suppose, however, that the expected return on portfolio A is 12% rather than 13%. This re-

turn would give rise to an arbitrage opportunity. Form a portfolio from the factor portfolios

with the same betas as portfolio A. This requires weights of 0.5 on the first factor portfolio,

0.75 on the second portfolio, and 0.25 on the risk-free asset. This portfolio has exactly the

same factor betas as portfolio A: a beta of 0.5 on the first factor because of its 0.5 weight on

the first factor portfolio and a beta of 0.75 on the second factor.

Now invest $1 in portfolio Q and sell (short) $1 in portfolio A. Your net investment is zero,

but your expected dollar profit is positive and equal to

$1 E(rQ) $1 E(rA) $1 .13 $1 .12 $.01.

Moreover, your net position is riskless. Your exposure to each risk factor cancels out because

you are long $1 in portfolio Q and short $1 in portfolio A, and both of these well-diversified

portfolios have exactly the same factor betas. Thus, if portfolio A™s expected return differs

from that of portfolio Q™s, you can earn positive risk-free profits on a zero net investment

position. This is an arbitrage opportunity.

Hence, any well-diversified portfolio with betas P1 and P2 must have the return given in

Equation 7.8 if arbitrage opportunities are to be ruled out. A comparison of Equations 7.2 and

7.8 shows that 7.8 is simply a generalization of the one-factor SML.

Finally, extension of the multifactor SML of Equation 7.8 to individual assets is precisely

the same as for the one-factor APT. Equation 7.8 cannot be satisfied by every well-diversified

portfolio unless it is satisfied by virtually every security taken individually. Equation 7.8 thus

represents the multifactor SML for an economy with multiple sources of risk.

249

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Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

250 Part TWO Portfolio Theory

The generalized APT must be qualified with respect to individual assets just as in the sin-

gle-factor case. A multifactor CAPM would, at the cost of additional assumptions, apply to

any and all individual assets. As we have seen, the result will be a security market equation (a

multidimensional SML) that is identical to that of the multifactor APT.

>

6. Using the factor portfolios just considered, find the fair rate of return on a security

Concept

with 1 0.2 and 2 1.4.

CHECK

SUMMARY • The CAPM assumes investors are rational, single-period planners who agree on a common

input list from security analysis and seek mean-variance optimal portfolios.

• The CAPM assumes ideal security markets in the sense that: (a) markets are large and

investors are price takers, (b) there are no taxes or transaction costs, (c) all risky assets are

publicly traded, and (d) any amount can be borrowed and lent at a fixed, risk-free rate.

• These assumptions mean that all investors will hold identical risky portfolios. The CAPM

implies that, in equilibrium, the market portfolio is the unique mean-variance efficient

tangency portfolio, which indicates that a passive strategy is efficient.

• The market portfolio is a value-weighted portfolio. Each security is held in a proportion

equal to its market value divided by the total market value of all securities. The risk

2

premium on the market portfolio is proportional to its variance, M, and to the risk

aversion of the average investor.

• The CAPM implies that the risk premium on any individual asset or portfolio is the

product of the risk premium of the market portfolio and the asset™s beta. The security

market line shows the return demanded by investors as a function of the beta of their

investment. This expected return is a benchmark for evaluating investment performance.

• In a single-index security market, once an index is specified, a security beta can be

estimated from a regression of the security™s excess return on the index™s excess return.

This regression line is called the security characteristic line (SCL). The intercept of the

SCL, called alpha, represents the average excess return on the security when the index

excess return is zero. The CAPM implies that alphas should be zero.

• An arbitrage opportunity arises when the disparity between two or more security prices

enables investors to construct a zero net investment portfolio that will yield a sure profit.

Rational investors will want to take infinitely large positions in arbitrage portfolios

regardless of their degree of risk aversion.

• The presence of arbitrage opportunities and the resulting volume of trades will create

pressure on security prices that will persist until prices reach levels that preclude arbitrage.

Only a few investors need to become aware of arbitrage opportunities to trigger this

process because of the large volume of trades in which they will engage.

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• When securities are priced so that there are no arbitrage opportunities, the market satisfies

the no-arbitrage condition. Price relationships that satisfy the no-arbitrage condition are

important because we expect them to hold in real-world markets.

• Portfolios are called well diversified if they include a large number of securities in such

proportions that the residual or diversifiable risk of the portfolio is negligible.

• In a single-factor security market, all well-diversified portfolios must satisfy the expected

return“beta relationship of the SML in order to satisfy the no-arbitrage condition.

• If all well-diversified portfolios satisfy the expected return“beta relationship, then all but a

small number of securities also must satisfy this relationship.

Bodie’Kane’Marcus: II. Portfolio Theory 7. Capital Asset Pricing © The McGraw’Hill

Essentials of Investments, and Arbitrage Pricing Companies, 2003

Fifth Edition Theory

251

7 Capital Asset Pricing and Arbitrage Pricing Theory

• The APT implies the same expected return“beta relationship as the CAPM, yet does not

require that all investors be mean-variance optimizers. The price of this generality is that

the APT does not guarantee this relationship for all securities at all times.

• A multifactor APT generalizes the single-factor model to accommodate several sources of

systematic risk.

KEY

alpha, 227 expected return“beta security market line

TERMS

arbitrage, 242 relationship, 225 (SML), 226

arbitrage pricing theory factor portfolio, 248 well-diversified

(APT), 244 market portfolio, 221 portfolio, 245

capital asset pricing model mutual fund theorem, 223 zero-investment