博迪第八版投资学第十章课后习题答案.doc

1、Chapter 10 - Arbitrage Pricing Theory and Multifactor Models of Risk and ReturnCHAPTER 10: ARBITRAGE PRICING THEORYAND MULTIFACTOR MODELS OF RISK AND RETURNPROBLEM SETS1.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 une

2、xpected 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

3、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

4、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

5、 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(rp) = rf + bP1 E(r1 ) - rf + bP2 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) -

6、rf In order to do so, we solve the following system of two equations with two unknowns:31 = 6 + (1.5 RP1) + (2.0 RP2)27 = 6 + (2.2 RP1) + (0.2) RP2The solution to this set of equations is:RP1 = 10% and RP2 = 5%Thus, the expected return-beta relationship is:E(rP) = 6% + (bP1 10%) + (bP2 5%)5.The expe

7、cted 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 = 5For Portfolio E, the ratio is lower at: (8 6)/0.6 = 3.33This implies that an arbitrage opportunity exists. For instance, you can create a Portfolio G w

8、ith beta equal to 0.6 (the same as Es) 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%bG = (0.5 1.2) + (0.5 0) = 0.6Comparing Portfolio G to Portfolio E, G has the same beta and higher return. Therefo

9、re, 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.6.Substituting the portfolio returns and betas in the expected

10、 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%7.a.Shorting an equally-weighted portfolio of the ten negative-alpha stocks and i

11、nvesting the proceeds in an equally-weighted portfolio of the ten 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 non-systematic risk, e, is zero)

12、:$1,000,000 0.02 + (1.0 RM ) - $1,000,000 (0.02) + (1.0 RM ) = $1,000,000 0.04 = $40,000The 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.) Thu

13、s, the systematic component of total risk is also zero. The variance of the analysts profit is not zero, however, since this portfolio is not well diversified.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.

14、 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 = 18,000,000,000The standard deviation of dollar returns is $134,164.b.If n = 50 stocks (25 stocks long and 25 stocks short),

15、the investor will have a $40,000 position in each stock, and the variance of dollar returns is:50 (40,000 0.30)2 = 7,200,000,000The 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 = 3,600,000,000The 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 = 2.23607 (f