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E(rP) rf P[E(rM) rf ]
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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Essentials of Investments, and Arbitrage Pricing Companies, 2003
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.
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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

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