投资学第10版习题答案08.docx

1、投资学第10版习题答案08投资学第10版习题答案08CHAPTER 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 est

2、imation errors when implementing the procedure. The disadvantage of the index model arises from the models 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

3、in favor of an actively 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 for w 0 (equation 8.20) and w* (equation 8.21). Other things held equal,

4、w 0 is smaller the greater the residual variance of a candidate asset for inclusion in the portfolio. Further, we see that regardless of beta, when w 0 decreases, so does w*. Therefore, other things equal, the greater the residual variance of an asset, the smaller its position in the optimal risky p

5、ortfolio. That is, increased firm-specific risk reduces the extent to which an active investor will be willing to depart from an indexed portfolio.4. The total risk premium equals: + ( Market risk premium). We call alpha a nonmarket return premium because it is the portion of the return premium that

6、 is independent of market performance.The 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 alph

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

8、stimates of meansn = 60 estimates of variancesestimates of covariancesTherefore, in total: estimatesb. In a single index model: ri - rf = i + i (r M rf ) + e i Equivalently, using excess returns: R i = i + i R M + e i The variance of the rate of return can be decomposed into the components:(l) The v

9、ariance due to the common market factor: (2) The variance due to firm specific unanticipated events: In this model: The number of parameter estimates is:n = 60 estimates of the mean E(ri )n = 60 estimates of the sensitivity coefficient i n = 60 estimates of the firm-specific variance 2(ei )1 estimat

10、e of the market mean E(rM )1 estimate of the market varianceTherefore, 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:6. a. The standard deviation of each individual stock

11、is given by: Since A = 0.8, B = 1.2, (eA ) = 30%, (eB ) = 40%, and M = 22%, we get:A = (0.82 222 + 302 )1/2 = 34.78%B = (1.22 222 + 402 )1/2 = 47.93%b. The expected rate of return on a portfolio is the weighted average of the expected returns of the individual securities:E(rP ) = wA E(rA ) + wB E(rB

12、 ) + wf rf E(rP ) = (0.30 13%) + (0.45 18%) + (0.25 8%) = 14%The beta of a portfolio is similarly a weighted average of the betas of the individual securities:P = wA A + wB B + wf f P = (0.30 0.8) + (0.45 1.2) + (0.25 0.0) = 0.78The variance of this portfolio is:whereis the systematic component andi

13、s the nonsystematic component. Since the residuals (ei ) are uncorrelated, the nonsystematic variance is:= (0.302 302 ) + (0.452 402 ) + (0.252 0) = 405where 2(eA ) and 2(eB ) are the firm-specific (nonsystematic) variances of Stocks A and B, and 2(e f ), the nonsystematic variance of T-bills, is ze

14、ro. The residual standard deviation of the portfolio is thus:(eP ) = (405)1/2 = 20.12%The total variance of the portfolio is then:The total standard deviation is 26.45%.7. a. The two figures depict the stocks security characteristic lines (SCL). Stock A has higher firm-specific risk because the devi

15、ations 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 risk. The SCL for Stock B is steeper; hence Stock Bs systematic risk

16、is greater.c. The R2 (or squared correlation coefficient) of the SCL is the ratio of the explained variance of the stocks return to total variance, and the total variance is the sum of the explained variance plus the unexplained variance (the stocks residual variance):Since the explained variance fo

17、r Stock B is greater than for Stock A (the explained variance is, which is greater since its beta is higher), and its residual variance is smaller, its R2 is higher than Stock As. d. Alpha is the intercept of the SCL with the expected return axis. Stock A has a small positive alpha whereas Stock B h

18、as a negative alpha; hence, Stock As alpha is larger.e. The correlation coefficient is simply the square root of R2, so Stock Bs correlation with the market is higher.8. a. Firm-specific risk is measured by the residual standard deviation. Thus, stock A has more firm-specific risk: 10.3% 9.1%b. Mark

19、et risk is measured by beta, the slope coefficient of the regression. A has a larger beta coefficient: 1.2 0.8c. R2 measures the fraction of total variance of return explained by the market return. As R2 is larger than Bs: 0.576 0.436d. Rewriting the SCL equation in terms of total return (r) rather

20、than excess return (R): The intercept is now equal to:Since rf = 6%, the intercept would be: 9. The standard deviation of each stock can be derived from the following equation for R2:Therefore:For stock B:10. The systematic risk for A is:The firm-specific risk of A (the residual variance) is the dif

21、ference between As total risk and its systematic risk:980 196 = 784The systematic risk for B is:Bs 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):The correlation coefficient between

22、the returns of A and B is:12. Note that the correlation is the square root of R2:13. For portfolio P we can compute:P = (0.62 980) + (0.42 4800) + (2 0.4 0.6 336)1/2 = 1282.081/2 = 35.81%P = (0.6 0.7) + (0.4 1.2) = 0.90Cov(rP,rM ) = P=0.90 400=360This same result can also be attained using the covar

23、iances of the individual stocks with the market:Cov(rP,rM ) = Cov(0.6rA + 0.4rB, rM ) = 0.6 Cov(rA, rM ) + 0.4 Cov(rB,rM ) = (0.6 280) + (0.4 480) = 36014. Note that the variance of T-bills is zero, and the covariance of T-bills with any asset is zero. Therefore, for portfolio Q:15. a. Beta Books ad

24、justs beta by taking the sample estimate of beta and averaging it with 1.0, using the weights of 2/3 and 1/3, as follows:adjusted beta = (2/3) 1.24 + (1/3) 1.0 = 1.16b. If you use your current estimate of beta to be t1 = 1.24, thent = 0.3 + (0.7 1.24) = 1.16816. For Stock A:For stock B:Stock A would

25、 be a good addition to a well-diversified portfolio. A short position in Stock B may be desirable.17. a.Alpha ()Expected excess returni = ri rf + i (rM rf ) E(ri ) rf A = 20% 8% + 1.3 (16% 8%) = 1.6%20% 8% = 12%B = 18% 8% + 1.8 (16% 8%) = 4.4%18% 8% = 10%C = 17% 8% + 0.7 (16% 8%) = 3.4%17% 8% = 9%D

26、= 12% 8% + 1.0 (16% 8%) = 4.0%12% 8% = 4%Stocks A and C have positive alphas, whereas stocks B and D have negative alphas.The residual variances are:2(eA ) = 582 = 3,3642(eB) = 712 = 5,0412(eC) = 602 = 3,6002(eD) = 552 = 3,025b. To construct the optimal risky portfolio, we first determine the optima

27、l active portfolio. Using the Treynor-Black technique, we construct the active portfolio:A0.0004760.6142B0.0008731.1265C0.0009441.2181D0.0013221.7058Total0.0007751.0000Be unconcerned with the negative weights of the positive stocksthe entire active position will be negative, returning everything to

28、good order.With these weights, the forecast for the active portfolio is: = 0.6142 1.6 + 1.1265 ( 4.4) 1.2181 3.4 + 1.7058 ( 4.0) = 16.90% = 0.6142 1.3 + 1.1265 1.8 1.2181 0.70 + 1.7058 1 = 2.08The high beta (higher than any individual beta) results from the short positions in the relatively low beta

29、 stocks and the long positions in the relatively high beta stocks. 2(e) = (0.6142)23364 + 1.126525041 + (1.2181)23600 + 1.705823025 = 21,809.6 (e) = 147.68%The levered position in B with high 2(e) overcomes the diversification effect and results in a high residual standard deviation. The optimal ris

30、ky portfolio has a proportion w* in the active portfolio, computed as follows:The negative position is justified for the reason stated earlier.The adjustment for beta is:Since w* is negative, the result is a positive position in stocks with positive alphas and a negative position in stocks with nega

31、tive alphas. The position in the index portfolio is:1 (0.0486) = 1.0486c. To calculate the Sharpe ratio for the optimal risky portfolio, we compute the information ratio for the active portfolio and Sharpes measure for the market portfolio. The information ratio for the active portfolio is computed as follows:A = = 16.90/147.68 = 0.1144A2 = 0.0