CHAPTER 10: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND
RETURN
CHAPTER 10: ARBITRAGE PRICING THEORY AND
MULTIFACTOR MODELS OF RISK AND RETURN
PROBLEM SETS
1. The revised estimate of the expected rate of return on the stock would be the old
estimate plus the sum of the products of the unexpected change in each factor times
the respective sensitivity coefficient:
2. The APT factors must correlate with major sources of uncertainty, i.e., sources of
uncertainty that are of concern to many investors. Researchers should investigate
3. Any pattern of returns can be explained if we are free to choose an indefinitely large
number of explanatory factors. If a theory of asset pricing is to have value, it must
4. Equation 10.11 applies here:
We need to find the risk premium (RP) for each of the two factors:
In order to do so, we solve the following system of two equations with two unknowns:
The solution to this set of equations is
Thus, the expected return-beta relationship is
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CHAPTER 10: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND
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5. The expected return for portfolio F equals the risk-free rate since its beta equals 0.
This implies that an arbitrage opportunity exists. For instance, you can create a
portfolio G with beta equal to 0.6 (the same as E’s) by combining portfolio A and
portfolio F in equal weights. The expected return and beta for portfolio G are then:
Comparing portfolio G to portfolio E, G has the same beta and higher return.
Therefore, an arbitrage opportunity exists by buying portfolio G and selling an
equal amount of portfolio E. The profit for this arbitrage will be
6. Substituting the portfolio returns and betas in the expected return-beta relationship,
we obtain two equations with two unknowns, the risk-free rate (rf) and the factor
risk premium (RP):
7. a. Shorting an equally weighted portfolio of the ten negative-alpha stocks and
investing the proceeds in an equally-weighted portfolio of the 10 positive-alpha
The sensitivity of the payoff of this portfolio to the market factor is zero
because the exposures of the positive alpha and negative alpha stocks cancel
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CHAPTER 10: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND
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For n = 20 stocks (i.e., long 10 stocks and short 10 stocks) the investor will
have a $100,000 position (either long or short) in each stock. Net market
b. If n = 50 stocks (25 stocks long and 25 stocks short), the investor will have a
$40,000 position in each stock, and the variance of dollar returns is
The standard deviation of dollar returns is $84,853.
The standard deviation of dollar returns is $60,000.
5
8. a.
)(σσβσ 2222 e
M
88125)208.0(σ 2222
A
50010)200.1(σ
2222
B
97620)202.1(σ
2222
C
b. If there are an infinite number of assets with identical characteristics, then a
well-diversified portfolio of each type will have only systematic risk since the
nonsystematic risk will approach zero with large n. Each variance is simply β2
× market variance:
2
2
2
Well-diversifiedσ 256
Well-diversifiedσ 400
Well-diversifiedσ 576
A
B
C
;
;
;
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CHAPTER 10: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND
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CHAPTER 10: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND
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c. There is no arbitrage opportunity because the well-diversified portfolios all
9. a. A long position in a portfolio (P) composed of portfolios A and B will offer an
expected return-beta trade-off lying on a straight line between points A and B.
b. The argument in part (a) leads to the proposition that the coefficient of β2 must
be zero in order to preclude arbitrage opportunities.
b. Surprises in the macroeconomic factors will result in surprises in the return of
the stock:
Unexpected return from macro factors =
11. The APT required (i.e., equilibrium) rate of return on the stock based on rf and the
factor betas is
According to the equation for the return on the stock, the actually expected return
12. The first two factors seem promising with respect to the likely impact on the firm’s
cost of capital. Both are macro factors that would elicit hedging demands across
broad sectors of investors. The third factor, while important to Pork Products, is a
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CHAPTER 10: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND
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to
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CHAPTER 10: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND
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13. The formula is
( ) 0.04 1.25 0.08 1.5 0.02 .17 17%E r
CHAPTER 10: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND
RETURN
17. The maximum residual variance is tied to the number of securities (n) in the
portfolio because, as we increase the number of securities, we are more likely to
encounter securities with larger residual variances. The starting point is to
determine the practical limit on the portfolio residual standard deviation, (eP), that
Now construct a portfolio of n securities with weights w1, w2,…,wn, so that wi =1.
The portfolio residual variance is 2(eP) = w122(ei)
To meet our practical definition of sufficiently diversified, we require this residual
variance to be less than (pM)2. A sure and simple way to proceed is to assume the
A relatively easy way to generate a set of well-diversified portfolios is to use portfolio
weights that follow a geometric progression, since the computations then become
The sum of the n squared weights is similarly obtained from w12 and a common
geometric progression factor of q2. Therefore
Substituting for w1 from above, we obtain
For sufficient diversification, we choose q so that wi2 ≤ p2/n
For example, continue to assume that p = 0.05 and n = 1,000. If we choose
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CHAPTER 10: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND
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In this case, w1 is about 15 times wn. Despite this significant departure from equal
weighting, this portfolio is nevertheless well diversified. Any value of q between
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CHAPTER 10: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND
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18. a. Assume a single-factor economy, with a factor risk premium EM and a (large)
set of well-diversified portfolios with beta P. Suppose we create a portfolio Z
by allocating the portion w to portfolio P and (1 – w) to the market portfolio
M. The rate of return on portfolio Z is:
Portfolio Z is riskless if we choose w so that Z = 0. This requires that:
Substitute this value for w in the expression for RZ:
Taking expectations we have:
This is the SML for well-diversified portfolios.
b. The same argument can be used to show that, in a three-factor model with
factor risk premiums EM, E1 and E2, in order to avoid arbitrage, we must have:
This is the SML for a three-factor economy.
19. a. The Fama-French (FF) three-factor model holds that one of the factors driving
returns is firm size. An index with returns highly correlated with firm size
(i.e., firm capitalization) that captures this factor is SMB (small minus big),
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CHAPTER 10: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND
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b. There seems to be no reason for the merged firm to underperform the returns of
the component companies, assuming that the component firms were unrelated
c. There seems to be no reason for the merger to affect total market capitalization,
d. The model predicts that firm size affects average returns so that, if two firms
e. This question appears to point to a flaw in the FF model. Therefore, the question
revolves around the behavior of returns for a portfolio of small firms, compared
Perhaps the reason the size factor seems to help explain stock returns is that,
when small firms become large, the characteristics of their fortunes (and hence
CFA PROBLEMS
2. b. Since portfolio X has = 1.0, then X is the market portfolio and E(RM) =16%.
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CHAPTER 10: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND
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6. c. Investors will take on as large a position as possible only if the mispricing
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