博迪第八版投资学第十章课后习题答案Word格式.doc

1、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） - 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 solu

2、tion 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 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 Po

3、rtfolio 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 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（r

4、G ） = （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. 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.

5、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 （rf ） and the factor risk premium （RP）:12 = rf + （1.2 RP）9 = rf +

6、（0.8 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 investing the proceeds in an equally-weighted portfolio of the ten positive-alpha stocks eliminates the market exposure and creates a zero-investment portfolio

7、. Denoting the systematic market factor as RM , the expected dollar return is （noting that the expectation of non-systematic 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 i

8、s 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 analysts profit is not zero, however, since this portfolio is not well diversified

9、.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

10、（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）, 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 doll

11、ar 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 o

12、f stocks increases by a factor of 5 （i.e., from 20 to 100）, standard deviation 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

13、since the non-systematic risk will approach zero with large n. 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

14、.a.A long position in a portfolio （P） comprised of Portfolios A and B will offer an expected return-beta tradeoff lying on a straight line between points A and B. Therefore, we can choose weights such that bP = bC but with expected return higher than that of Portfolio C. Hence, combining P with a sh

15、ort position in C will create 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 b2 must be zero in order to preclude arbitrage opportunities.10.a.E（r） = 6 + （1.2 6） + （0.5 8） + （0.3 3） = 18.

16、1%b. Surprises in the 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 is

17、:required E（r） = 6 + （1 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

18、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 a

19、 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: inflati

20、on 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 latter

21、 are likely to command a risk premium in the capital markets.13.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 encounter securities with larger residual variances. The starting point is to

22、 determine the practical limit on the portfolio residual standard deviation, s（eP）, that still qualifies as a well-diversified portfolio. A reasonable approach is to compare s2（eP） to the market variance, or equivalently, to compare s（eP） to the market standard deviation. Suppose we do not allow s（e

23、P） to exceed psM, where p is a small decimal fraction, for example, 0.05; then, the smaller the value we choose for p, the more stringent our criterion for defining how diversified a well-diversified portfolio must be.Now construct a portfolio of n securities with weights w1, w2,wn, so that Swi =1.

24、The portfolio residual variance is: s2（eP） = Sw12s2（ei）To meet our practical definition of sufficiently diversified, we require this residual variance to be less than （psM）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 h

25、ighest possible value allowed under the assumptions of the problem: s2（ei） = ns2MIn that case: s2（eP） = Swi2 nsM2Now apply the constraint: Swi2 n sM2 （psM）2This requires that: nSwi2 p2Or, equivalently, that: Swi2 p2/nA relatively easy way to generate a set of well-diversified portfolios is to use po

26、rtfolio weights that follow a geometric progression, since the computations then become relatively straightforward. Choose w1 and a common factor q for the geometric progression such that q 1. Therefore, the weight on each stock is a fraction q of the weight on the previous stock in the series. Then

27、 the sum of n terms is:Swi = w1（1 qn）/（1 q） = 1or:w1 = （1 q）/（1 qn）The sum of the n squared weights is similarly obtained from w12 and a common geometric progression factor of q2. Therefore:Swi2 = w12（1 q2n）/（1 q 2）Substituting for w1 from above, we obtain:Swi2 = （1 q）2/（1 qn）2 （1 q2n）/（1 q 2）For sufficient diversification, we choose q so that:For example, continue to assume that p = 0.05 and n = 1,000. If we chooseq = 0.9