投资学第10版习题答案

1、CHAPTER 8: INDEX MODELSPROBLEM SETS1. The advantage of the index model, compared to the Markowitz procedure, is the vastly reduced number of estimates required. In addition, the large number of estimates required for the Markowitz procedure can result in large aggregate estimation errors when implem

2、enting the procedure. The disadvantage of the index model arises from the model ' s assumption that return residuals are uncorrelated. This assumption will be incorrect if the index used omits a significant risk factor.2. The trade-off entailed in departing from pure indexing in favor of an acti

3、vely managed portfolio is between the probability (or the possibility) of superior performance against the certainty of additional management fees.3. The answer to this question can be seen from the formulas forw0(equation and w* (equation . Otherthings held equal,w0 issmaller the greater the residu

4、al variance of a candidate asset for inclusion in the portfolio. Further, we see that regardless of beta, when w0 decreases, so does w Therefore, other things equal, the greater the residual variance of an asset, the smaller its position in the optimal risky portfolio. That is, increased firm-specif

5、ic risk reduces the extent to which an active investor will be willing to depart from an indexed portfolio.4. The total risk premium equals:+ ( x Market risk premium). Wecall alpha a nonmarket return premium because it is the portion of the return premium that is independent of market performance.Th

6、e Sharpe ratio indicates that a higher alpha makes a security more desirable. Alpha, the numerator of the Sharpe ratio, is a fixed number that is not affected by the standard deviation of returns, the denominator of the Sharpe ratio. Hence, an increase in alpha increases the Sharpe ratio. Since the

7、portfolio alpha is the portfolio- weighted average of the securities ' alphas, then, holding all other parameters fixed, an increase in a security alpha results in an increase in the portfolio Sharpe ratio.5. a. To optimize this portfolio one would need:n = 60 estimates of meansn = 60 estimates

8、of variances2 1,770 estimates of covariances2Therefore, in total:n2 3n1,890 estimates2b.In a single index model:门 rt = a i + 0 i (r m r f) + e iEquivalently, using excess returns:R = a i + p i Rm + e iThe variance of the rate of return can be decomposed into the components:(1) The variance due to th

9、e common market factor:(2) The variance due to firm specific unanticipated events:小)In this model:Cov(ri, rj)Bi 生 bThe number of parameter estimates is:n = 60 estimates of the mean E( r i )n = 60 estimates of the sensitivity coefficientB in = 60 estimates of the firm-specific variance6 2( ej )1 esti

10、mate of the market meanE( r m)21 estimate of the market variance MTherefore, in total, 182 estimates.The single index model reduces the total number of required estimates from 1,890 to 182. In general, the number of parameter estimates is reduced from:n2 3n2to(3n2)6. a. The standard deviation of eac

11、h individual stock is given by:2221/2q ft m(ei)Since 0 a = ,0 b = , (r(eA)= 30%, (r(eB)= 40%, and (tm = 22%,we get:6 A = x 222 + 30 2 )1/2 = %(TB = X 222 + 40 2 )1/2 = %b. The expected rate of return on a portfolio is the weighted average of the expected returns of the individual securities:E(r p )

12、= w a x E(r a ) + w b x E(r b ) + w f 乂 r fE(rp ) = x 13%) + x 18%) + x 8%) = 14%The beta of a portfolio is similarly a weighted average of the betas of the individual securities:P P = w A x 0A + w B X 0B + w f X 0f 0P = x + X + X =The variance of this portfolio is:2PM22(eP )where 凉 / is the systema

13、tic component and2 (ep) is thenonsystematic component. Since the residuals (ei ) areuncorrelated, the nonsystematic variance is:2222222(eP ) wA(eA) wB(eB) wf(ef)= x 302 ) + x 402 ) + 义 0) = 405where o- 2( eA ) and 6 2( eB) are the firm-specific (nonsystematic) variances of Stocks A and B, and 6 2(e

14、f ), the nonsystematic variance of T-bills, is zero. The residual standard deviation of the portfolio is thus:(r(eP ) = (405) 1/2 = %The total variance of the portfolio is then:4(0.782 222) 405 699.47The total standard deviation is %.7. a. The two figures depict the stocks security characteri stic l

15、ines (SCL). Stock A has higher firm-specific risk because the deviations of the observations from the SCL are larger for Stock A than for Stock B. Deviations are measured by the vertical distance of each observation from the SCL.b. Beta is the slope of the SCL, which is the measure of systematic ris

16、k. The SCL for Stock B is steeper; hence Stock B s systematic risk is greater.c. The R2 (or squared correlation coefficient) of the SCL is theratio of the explained variance of the stock' s return to totalvariance, and the total variance is the sum of the explainedvariance plus the unexplained v

17、ariance (the stock' s residualvariance):Since the explained variance for Stock B is greater than forStock A (the explained variance is 重吊，which is greatersince its beta is higher), and its residual variance 2 (eB) is smaller, itsR2 is higher than Stock A ' s.d. Alpha is the intercept of the

18、SCL with the expected return axis. Stock A has a small positive alpha whereas Stock B has a negative alpha; hence, Stock A ' s alpha is larger.e. The correlation coefficient is simply the square root ofR2, soStock B ' s correlation with the market is h igher.8. a. Firm-specific risk is measu

19、red by the residual standard deviation. Thus, stock A has more firm-specific risk: % > %b. Market risk is measured by beta, the slope coefficient of the regression. A has a larger beta coefficient: >c. R2 measures the fraction of total variance of return explainedby the market return. A s R2 i

20、s larger than B ' s: >d. Rewriting the SCL equation in terms of total return (r) ratherthan excess return (R):a rf(mrf)arf (1)mThe intercept is now equal to:rf (1) 1% rf (1 1.2)Since r f = 6%, the intercept would be:1% 6%(1 1.2) 1% 1.2%0.2%9.The standard deviation of each stock can be derived

21、 from the following equation forR2:R2Explained varianceTotal varianceTherefore:2bA oM0.72 2026A 2-Ra0.20oa 31.30%For stock B:_2_ 21.2200.1269.28%4,80010.The systematic risk for A is:A M 0.702 202196The firm-specific risk of A (the residual variance) is the difference between A ' s total risk and

22、 its systematic risk:980 - 196 = 784The systematic risk for B is:M 1.202 202576B' s firm -specific risk (residual variance) is:4,800 576 = 4,22411.The covariance between the returns of A and B is (since the residuals are assumed to be uncorrelated):2Cov(rA,rB)限饱 函 0.70 1.20 400 336The correlatio

23、n coefficient between the returns of A and B is:PabCovQaJb)336 0.155 31.30 69.2812. Note that the correlation is the square root ofR： p JR2一 ,、1/2_Cov(rA,M )am0.2031.3020280_1/2 _Cov(rB,rM)B M0.1269.282048013. For portfolioP we can compute:(TP = x 980) + x 4800) + (2 x x x 336) 1/2 = 口 1/2 = %B P =

24、X + X =/ (ep)O-P僚 bM1282.08 (0.902 400)958.08Cov(rp,r m) = B p £ = x 400=360This same result can also be attained using the covariances of the individual stocks with the market:Cov(rp,r m) = Cov + , rM ) = x Cov(rA,m) + x Cov(rB,m)= x 280) + x 480) = 36014. Note that the variance of T-bills is

25、zero, and the covariance of T- bills with any asset is zero. Therefore, for portfolio Q:22221/2(TqWp bpWM OM2 Wp WMCov(rp,rM )221/2(0.51,282.08) (0.3400) (2 0.5 0.3 360)21.55%Qwp pwM M (0.5 0.90) (0.3 1) (0.20 0) 0.75o2 (eQ)oQ够 oM464.52 (0.752 400) 239.522Cov(rQ,rM)Bq 0M0.75 400 30015. a. Beta Books

26、 adjusts beta by taking the sample estimate of beta and averaging it with , using the weights of 2/3 and 1/3, as follows:adjusted beta = (2/3) x + (1/3) x =b. If you use your current estimate of beta to beB t 1 = , thenB t = + x =16.For Stock A:AArfA(Mrf).11.060.8(.12.06)0.2%For stock B:BrbrfB(rmrf)

27、.14.061.5(.12.06)1%Stock A would be a good addition to a well-diversified portfolio. A short position in Stock B may be desirable.17.a.Alpha ( a )a i = r i - rf + 储 x (m- r f)aA= 20% - 8% + a B = 18% - 8% + -%a C = 17% - 8% + a D = 12% - 8% + 一%x (16% - 8%) = %x (16% - 8%)=x (16% - 8%) = %X (16% - 8

28、%)=Expected excessreturnE(ri ) rf20% - 8% = 12%18% - 8% = 10%17% - 8% = 9%12% 8% = 4%Stocks A and C have positive alphas, whereas stocks B and D have negative alphas.The residual variances are:2(eA ) = 58 2 = 3,364 2(eB) = 71 2 = 5,041 2(eC) = 60 2 = 3,600 2(eD) = 55 2 = 3,025b. To construct the opt

29、imal risky portfolio, we first determine the optimal active portfolio. Using the Treynor-Black technique, we construct the active portfolio:Be unconcerned with the negative weights of the positiveastocks the entire active position will be negative, returning everything to good order.With these weigh

30、ts, the forecast for the active portfolio is:X (- - X + X (-x - x + x 1=The high beta (higher than any individual beta) results from the short positions in the relatively low beta stocks and the long positions in the relatively high beta stocks.2(e) = (-2X3364 + X5041 + (- 2X 3600 + X 3025=21,(e) =

31、%The levered position in B with high2( e) overcomes thediversification effect and results in a high residual standard deviation. The optimal risky portfolio has a proportionw* inthe active portfolio, computed as follows:/ 2(e)E(m) rf/ M.1690/21,809.6.08/ 2320.05124The negative position is justified

32、for the reason stated earlier.The adjustment for beta is:0.0486*wo0.05124w* 1 (1B)w01 (1 2.08)( 0.05124)Since w is negative, the result is a positive position in stocks with positive alphas and a negative position in stocks with negative alphas. The position in the index portfolio is:1 (一二c.To calcu

33、late the Sharpe ratio for the optimal risky portfolio, we compute the information ratio for the active portfolio and Sharpe' s measure for the market portfolio . The information ratio for the active portfolio is computed as follows:A =(e)A2 =Hence, the square of the Sharpe ratio (S) of the optim

34、ized risky portfolio is:2S2 SM a2-80.0131 0.134123S =Compare this to the market ' s Sharpe ratio:Sm = 8/23 =A difference of:The only moderate improvement in performance results from only a small position taken in the active portfolioA because of itslarge residual variance.d. To calculate the mak

35、eup of the complete portfolio, first compute the beta, the mean excess return, and the variance of the optimal risky portfolio:0p = w + (WxX 0a) = + (-=E( R) =ap + p pE(RM) = ( - %) +x 8%) =%O-P BpbM 02(ep)(0.9523)2( 0.04862)21,809.6528.94即 23.00%Since A = , the optimal position in this portfolio is

36、:8.42y 0.01 2.8 528.940.5685In contrast, with a passive strategy:y 0.01 2.8 2320.5401 A difference of:The final positions are (M may include some of stocks A throughD):Bills1 - =%M=A(-(-二B(-二C(-(-二D(-二(subject to rounding error)18. a. If a manager is not allowed to sell short, he will not include st

37、ocks with negative alphas in his portfolio, so he will consider only A and C:A2(e)aa / 彳e) Sa /2(e)2(e)A3,364C3,600The forecast for the active portfolio is:B = x + X =2(e) = x 3,364) + x 3,600) = 1, (r(e) = %The weight in the active portfolio is:woa/ o2(e)2e(Rm )/ 小2.80/1,969.032 8/2320.0940Adjustin

38、g for beta:w*w。1 (1 )w 00.0941 (1 0.90) 0.0940.0931The information ratio of the active portfolio is:A -2180- 0.0631(e)44.37Hence, the square of the Sharpe ratio is:S22320.063120.1250Therefore: S =The market ' s Sharpe ratio is: Sm =When short sales are allowed (Problem 17), the manager ' sSh

39、arpe ratio is higher . The reduction in the Sharpe ratio is the cost of the short sale restriction.The characteristics of the optimal risky portfolio are:WmWaE(Rp)p222PPMp 23.14%(1 0.0931) (0.0931 0.9) 0.99p E(Rm) (0.0931 2.8%) (0.99 8%) 8.18% 2(eP) (0.99 23)2 (0.09312 1969.03) 535.54With A = , the

40、optimal position in this portfolio is:8.180.01 2.8 535.540.5455The final positions in each asset are:Bills1 - =%M(1=A=C=b. The mean and variance of the optimized complete portfolios in the unconstrained and short-sales constrained cases, and for the passive strategy are:UnconstrainedConstrainedPassi

41、veE(R)X % =X % =X % =2久X =X =X =The utility levels below are computed using the formula:E(rC) 0.005A 4Unconstrained8% + % -xX=%Constrained8% + % -xX=%Passive8% + % XX=%19. All alphas are reduced to times their values in the original case. Therefore, the relative weights of each security in the activ

42、e portfolio are unchanged, but the alpha of the active portfolio is only times its previous value:x % =%The investor will take a smaller position in the active portfolio.The optimal risky portfolio has a proportionw in the activeportfolio as follows:/ 2(e)L，、 ， 2E(rMrf ) / M00507/21,809.60.08/ 2320.

43、01537The negative position is justified for the reason given earlier.The adjustment for beta is:0.0151*Wo0.01537w* 1 (1B)w01 (1 2.08) ( 0.01537)Since W is negative, the result is a positive position in stocks with positive alphas and a negative position in stocks with negative alphas. The position i

44、n the index portfolio is: 1一(一To calculate the Sharpe ratio for the optimal risky portfolio wecompute the information ratio for the active portfolio and the Sharperatio for the market portfolio. The information ratio of theactiveportfolio is times its previous value:A =(e)5.07 二147.68Hence, the squa

45、re of the Sharpe ratio of the portfolio is:S2 = S2M + A = (8%/23%) 2 + =optimized riskyCompare this to the marketThe difference is:'s Sharpe ratio:Sm = -8%23%Note that the reduction of the forecast alphas by a factor of reduced the squared information ratio and the improvement in the squared Sha

46、rpe ratio by a factor of:20. If each of the alpha forecasts is doubled, then the alpha of theactive portfolio will also double. Other things equal, theinformation ratio (IR) of the active portfolio also doubles. Thesquare of the Sharpe ratio for the optimized portfolio (S-square)equals the square of

47、 the Sharpe ratio for the market index (SM-square) plus the square of the information ratio. Since theinformation ratio has doubled, its square quadruples. Therefore:S-square = SMsquare + ( 4 x IR)Compared to the previous S-square, the difference is: 3 IRNow you can embark on the calculations to ver

48、ify this result.CFA PROBLEMS1. The regression results provide quantitative measures of return and risk based on monthly returns over the five-year period.B for ABC was , considerably less than the average stocks 0 of .This indicates that, when the S&P 500 rose or fell by 1 percentage poin t, ABC

49、 s return on average rose or fell by only percentage point. Therefore, ABC s systematic risk (or market risk) was low relative to the typical value for stocks.ABC s alpha (theintercept of the regression) was%, indicating that when themarket return was 0%, the average return on ABC was %. ABC s unsystematic risk (or residual risk), as measured byo- (e), was %.For ABC, R2 was , indicating closeness of fit to the linear regression greater than the value for a typical stock.B for XYZ was somewhat higher, at , indicating XYZ ' s return pattern was very s