投资学10版习题答案10.docx

1、投资学10版习题答案10CHAPTER 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 unexpected change in each factor times the respective sensitivity coe

2、fficient: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%).2. The APT factors must correlate with major sources of uncertainty, i.e., sources of uncertainty that are of concern to many investors.

3、 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 cha

4、nges 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 f

5、actors. 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).4. Equation 10.11 applies here:E(rp) = rf + P1 E(r1 ) rf + P2 E(r2 ) rfWe need to find

6、 the risk premium (RP) for each of the two factors:RP1 = E(r1 ) rf and RP2 = E(r2 ) rfIn 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 isRP1 = 10% and RP2 =

7、5%Thus, the expected return-beta relationship isE(rP) = 6% + (P1 10%) + (P2 5%)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 = 5For portfolio E, the ratio is lower at (8 6)/0.6 = 3.33This impl

8、ies that an arbitrage opportunity exists. For instance, you can create a portfolio G with 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%G = (0.5 1.2) + (0.5 0%) = 0.

9、6Comparingportfolio 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 berG rE =9% + (0.6 F) 8% + (0.6 F) = 1%That is, 1% of the funds (long or short)

10、 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 (rf) and the factor risk premium (RP):12% = rf + (1.2 RP)9% = rf + (0.8 RP)Solving these equations, we obtainrf = 3% and RP = 7.5

11、%7. a. Shorting an equallyweighted portfolio of the ten negative-alpha stocks and investing the proceeds in an equally-weighted portfolio of the 10positive-alpha stocks eliminates the market exposure and creates a zero-investment portfolio. Denoting the systematic market factor as RM, the expected d

12、ollar 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 = $40,000The sensitivity of the payoff of this portfolio to the market factor is zero because the exposures of the positive alpha and negative alph

13、a 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 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 in

14、vestor 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 is20 (100,000 0.30)2 = 18,000,000,000The standard deviation of dollar retu

15、rns is $134,164.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 is50 (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 an

16、d 50 stocks short), the investor will have a $20,000 position in each stock, and the variance of dollar returns is100 (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 d

17、eviation decreases by a factor of = 2.23607 (from $134,164 to $60,000).8. a. 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 va

18、riance is simply 2 market variance:The mean will equal that of the individual (identical) stocks.c. There is no arbitrage opportunity because the well-diversified portfolios all plot on the security market line (SML). Because they are fairly priced, there is no arbitrage.9. a. A long position in a p

19、ortfolio (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. Therefore, we can choose weights such that P = C but with expected return higher than that of portfolio C. Hence, combining P with a short position in C will crea

20、te an arbitrage portfolio with zero investment, zero beta, and positive rate of return.b. The argument in part (a) leads to the proposition that the coefficient of 2 must be zero in order to preclude arbitrage opportunities.10. a. E(r) = 6% + (1.2 6%) + (0.5 8%) + (0.3 3%) = 18.1%b.Surprises in the

21、macroeconomic factors will result in surprises in the return of the stock:Unexpected return from macro factors = 1.2 (4% 5%) + 0.5 (6% 3%) + 0.3 (0% 2%) = 0.3%E(r) =18.1% 0.3% = 17.8%11. The APT required (i.e., equilibrium) rate of return on the stock based on rf and the factor betas isRequired E(r)

22、 = 6% + (1 6%) + (0.5 2%) + (0.75 4%) = 16%According to the equation for the return on the stock, the actually expected return on the stock is 15% (because the expected surprises on all factors are zero by definition). Because the actually expected return based on risk is less than the equilibrium r

23、eturn, we conclude that the stock is overpriced.12. The first two factors seem promising with respect to the likely impact on the firms 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

24、a poor choice for a multifactor SML because the price of hogs is of minor importance to most investors and is therefore highly unlikely to be a priced risk factor. Better choices would focus on variables that investors in aggregate might find more important to their welfare. Examples include: inflat

25、ion uncertainty, short-term interest-rate risk, energy price risk, or exchange rate risk. The important point here is that, in specifying a multifactor SML, we not confuse risk factors that are important to a particular investor with factors that are important to investors in general; only the latte

26、r are likely to command a risk premium in the capital markets.13. The formula is 14. If and based on the sensitivities to real GDP (0.75) and inflation (1.25), McCracken would calculate the expected return for the Orb Large Cap Fund to be: Therefore, Kwons fundamental analysis estimate is congruent

27、with McCrackens APT estimate. If we assume that both Kwon and McCrackens estimates on the return of Orbs Large Cap Fund are accurate, then no arbitrage profit is possible.15. In order to eliminate inflation, the following three equations must be solved simultaneously, where the GDP sensitivity will

28、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. Here,x represents Orbs High Growth Fund, y represents Large Cap Fund and z represents Utility Fund. Using algebraic manipulation will yield wx = wy

29、= 1.6 and wz = -2.2.16. 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 positively correlated with inflation to preserve value, and less correlated with the variable growth of GDP. T

30、hus, Stiles is wrong. McCracken is correct in that supply side macroeconomic policies are generally designed to increase output at a minimum of inflationary pressure. Increased output would mean higher GDP, which in turn would increase returns of a fund positively correlated with GDP.17. The maximum

31、 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

32、, (eP), that still qualifies as a well-diversified portfolio. A reasonable approach is to compare 2(eP) to the market variance, or equivalently, to compare (eP) to the market standard deviation. Suppose we do not allow (eP) to exceed p M, where p is a small decimal fraction, for example, 0.05; then, the smaller the value we choose forp, the more stringe