bkmsol-ch10-9e corrected 7.29.2010

# bkmsol-ch10-9e corrected 7.29.2010 - CHAPTER 10 ARBITRAGE...

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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% 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 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. 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 explain returns using a reasonably limited number of explanatory variables (i.e., systematic factors). 4. Equation 10.9 applies here: E(r p ) = r f + β P1 [E(r 1 ) - r f ] + β P2 [E(r 2 ) – r f ] We need to find the risk premium (RP) for each of the two factors: RP 1 = [E(r 1 ) - r f ] and RP 2 = [E(r 2 ) - r f ] In order to do so, we solve the following system of two equations with two unknowns: .31 = .06 + (1.5 × RP 1 ) + (2.0 × RP 2 ) .27 = .06 + (2.2 × RP 1 ) + [(–0.2) × RP 2 ] The solution to this set of equations is: RP 1 = 10% and RP 2 = 5% Thus, the expected return-beta relationship is: E(r P ) = 6% + (β P1 × 10%) + (β P2 × 5%) 10-1

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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 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(r G ) = (0.5 × 12%) + (0.5 × 6%) = 9% β G = (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: r G – r E =[9% + (0.6 × F)] - [8% + (0.6 × F)] = 1% That is, 1% of the funds (long or short) in each portfolio. 6. Substituting the portfolio returns and betas in the expected return-beta relationship, we obtain two equations with two unknowns, the risk-free rate (r f ) and the factor risk premium (RP): 12% = r f + (1.2 × RP) 9% = r f + (0.8 × RP) Solving these equations, we obtain: r f = 3% and RP = 7.5% 7. a.Shorting an equally-weighted portfolio of the ten negative-alpha stocks and investing the proceeds in an equally-weighted portfolio of the ten positive- alpha stocks eliminates the market exposure and creates a zero-investment portfolio.
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