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
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
4. Equation 10.11 applies here:
E(rp ) = rf + βP1 [E(r1 ) − rf ] + βP2 [E(r2 ) – rf ]
We need to find the risk premium (RP) for each of the two factors:
Chapter 10 – Arbitrage Pricing Theory and Multifactor Models of Risk and Return
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5. The expected return for portfolio F equals the risk-free rate since its beta equals 0.
For portfolio A, the ratio of risk premium to beta is (12 − 6)/1.2 = 5
For portfolio E, the ratio is lower at (8 – 6)/0.6 = 3.33
This implies that an arbitrage opportunity exists. For instance, you can create a
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
stocks eliminates the market exposure and creates a zero-investment portfolio.
Chapter 10 – Arbitrage Pricing Theory and Multifactor Models of Risk and Return
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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
8. a.
)(σσβσ 2222 e
M+=
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:
Chapter 10 – Arbitrage Pricing Theory and Multifactor Models of Risk and Return
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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.
10. a. E(r) = 6% + (1.2 × 6%) + (0.5 × 8%) + (0.3 × 3%) = 18.1%
11. The APT required (i.e., equilibrium) rate of return on the stock based on rf and the
factor betas is
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
Chapter 10 – Arbitrage Pricing Theory and Multifactor Models of Risk and Return
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13. The formula is
( ) 0.04 1.25 0.08 1.5 0.02 .17 17%Er = + + = =
14. If
4%
r=
and based on the sensitivities to real GDP (0.75) and inflation (1.25),
15. In order to eliminate inflation, the following three equations must be solved
simultaneously, where the GDP sensitivity will equal 1 in the first equation,
inflation sensitivity will equal 0 in the second equation and the sum of the weights
must equal 1 in the third equation.
16. Since retirees living off a steady income would be hurt by inflation, this portfolio
would not be appropriate for them. Retirees would want a portfolio with a 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
Chapter 10 – Arbitrage Pricing Theory and Multifactor Models of Risk and Return
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
worst, that is, assume that the residual variance of each security is the highest
The sum of the n squared weights is similarly obtained from w12 and a common
geometric progression factor of q2. Therefore
wi2 = w12(1– q2n)/(1– q 2)
Substituting for w1 from above, we obtain
Chapter 10 – Arbitrage Pricing Theory and Multifactor Models of Risk and Return
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0.9973 and 1.0 results in a well-diversified portfolio. As q gets closer to 1, the
portfolio approaches equal weighting.
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:
factor risk premiums EM, E1 and E2, in order to avoid arbitrage, we must have:
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),
Chapter 10 – Arbitrage Pricing Theory and Multifactor Models of Risk and Return
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b. This question appears to point to a flaw in the FF model. The model predicts
that firm size affects average returns so that, if two firms merge into a larger
firm, then the FF model predicts lower average returns for the merged firm.
However, there seems to be no reason for the merged firm to underperform
CFA PROBLEMS
1. a. This statement is incorrect. The CAPM requires a mean-variance efficient
market portfolio, but APT does not.
3. d.
Chapter 10 – Arbitrage Pricing Theory and Multifactor Models of Risk and Return
4. c.