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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:

Revised estimate = 12% + [(1 × 2%) + (0.5 × 3%)] = 15.5%

Note that the IP estimate is computed as: 1 × (5% - 3%), and the IR estimate is computed as: 0.5 × (8% - 5%).

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 factors that correlate with uncertainty in consumption and investment opportunities. GDP, the inflation rate, and interest rates are among the factors that can be expected to determine risk premiums. In particular, industrial production (IP) is a good

indicator of changes in the business cycle. Thus, IP is a candidate for a factor that is highly correlated with uncertainties that have to do with investment and consumption opportunities in the economy.

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 explain returns using a reasonably limited number of explanatory variables (i.e., systematic factors such as unemployment levels, GDP, and oil prices). 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:

RP1 = [E(r1 ) ? rf ] and RP2 = [E(r2 ) ? rf ]

In order to do so, we solve the following system of two equations with two unknowns:

.31 = .06 + (1.5 × RP1 ) + (2.0 × RP2 ) .27 = .06 + (2.2 × RP1 ) + [(–0.2) × RP2 ] The solution to this set of equations is

RP1 = 10% and RP2 = 5%

Thus, the expected return-beta relationship is

E(rP ) = 6% + (βP1 × 10%) + (βP2 × 5%)

CHAPTER 10: ARBITRAGE PRICING THEORY AND MULTIFACTOR MODELS OF RISK AND RETURN

10-1

McGraw-Hill Education.

Chapter 10 - Arbitrage Pricing Theory and Multifactor Models of Risk and Return

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 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:

E(rG ) = (0.5 × 12%) + (0.5 × 6%) = 9% βG = (0.5 × 1.2) + (0.5 × 0%) = 0.6

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

rG – rE =[9% + (0.6 × F)] ? [8% + (0.6 × F)] = 1% That is, 1% of the funds (long or short) in each portfolio.

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):

12% = rf + (1.2 × RP) 9% = rf + (0.8 × RP) Solving these equations, we obtain

rf = 3% and RP = 7.5%

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. Denoting the systematic market factor as RM , the expected dollar return is (noting that the expectation of nonsystematic risk, e, is zero):

$1,000,000 × [0.02 + (1.0 × RM )] ? $1,000,000 × [(–0.02) + (1.0 × RM )] = $1,000,000 × 0.04 = $40,000

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 out. (Notice that the terms involving RM sum to zero.) Thus, the systematic component of total risk is also zero. The variance of the analyst’s profit is not zero, however, since this portfolio is not well diversified.

10-2

McGraw-Hill Education.

Chapter 10 - Arbitrage Pricing Theory and Multifactor Models of Risk and Return

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 exposure is zero, but firm-specific risk has not been fully diversified. The variance of dollar returns from the positions in the 20 stocks is

20 × [(100,000 × 0.30)2] = 18,000,000,000 The standard deviation of dollar returns is $134,164.

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

50 × [(40,000 × 0.30)2] = 7,200,000,000 The standard deviation of dollar returns is $84,853.

Similarly, if n = 100 stocks (50 stocks long and 50 stocks short), the investor will have a $20,000 position in each stock, and the variance of dollar returns is

100 × [(20,000 × 0.30)2] = 3,600,000,000 The standard deviation of dollar returns is $60,000.

Notice that, when the number of stocks increases by a factor of 5 (i.e., from 20 to 100), standard deviation decreases by a factor of 5= 2.23607 (from $134,164 to $60,000).

2σ2?β2σ2?σ(e) M222σ2?(0.8?20)?25?881 A222σ2?(1.0?20)?10?500 Ba.

2σC?(1.22?202)?202?976

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: Well-diversifiedσ2256AWell-diversifiedσ2B400

2Well-diversifiedσC576

The mean will equal that of the individual (identical) stocks.

10-3

McGraw-Hill Education.

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