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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 impleme
2、nting 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 in favor of an actively mana
3、ged 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, w 0 is smaller the greater th
4、e 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 portfolio. That is, increased
5、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: a + (b Market risk premium). We call alpha a nonmarket return premium because it is the portion of the return premium that is independent of market p
6、erformance.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 alpha increases the Sharpe rati
7、o. 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 estimates of meansn = 60 estim
8、ates 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 variance due to the common mark
9、et 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 estimate of the market mean E(rM )1 es
10、timate 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 is given by: Since A = 0.8, B = 1
11、.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 ) + wf rf E(rP ) = (0.30 13%) + (
12、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 andis the nonsystematic component. Sin
13、ce 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 zero. The residual standard deviatio
14、n 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 deviations of the observations from the
15、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 is greater.c. The R2 (or squared corr
16、elation 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 for Stock B is greater than for Stock A
17、 (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 has a negative alpha; hence, Stock As a
18、lpha 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.Market risk is measured by beta, the slope coe
19、fficient 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 than excess return (R): The intercept is now
20、 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 difference between As total risk and its systemat
21、ic 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 the returns of A and B is:12.Note that the corr
22、elation 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 covariances of the individual stocks with the market:C
23、ov(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 adjusts beta by taking the sample estimate of beta and
24、 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 be a good addition to a well-diversified portfolio.
25、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 = 12% 8% + 1.0 (16% 8%) = 4.0%12% 8% = 4%Stocks A and
26、C have positive alphas, whereas stocks B and D have negative alphas.The residual variances are:s2(eA ) = 582 = 3,364s2(eB) = 712 = 5,041s2(eC) = 602 = 3,600s2(eD) = 552 = 3,025b.To construct the optimal risky portfolio, we first determine the optimal active portfolio. Using the Treynor-Black techniq
27、ue, 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 good order.With these weights, the forecast for the
28、 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 stocks and the long positions in the relatively hi
29、gh beta stocks. s2(e) = (0.6142)23364 + 1.126525041 + (1.2181)23600 + 1.705823025 = 21,809.6s (e) = 147.68%The levered position in B with high s2(e) overcomes the diversification effect and results in a high residual standard deviation. The optimal risky portfolio has a proportion w* in the active p
30、ortfolio, 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 negative alphas. The position in the index portfolio
31、 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 = 0.0131
32、Hence, the square of the Sharpe ratio (S) of the optimized risky portfolio is:S = 0.3662Compare this to the markets Sharpe ratio:SM = 8/23 = 0.3478 A difference of: 0.0184The only moderate improvement in performance results from only a small position taken in the active portfolio A because of its la
33、rge residual variance.d.To calculate the makeup of the complete portfolio, first compute the beta, the mean excess return, and the variance of the optimal risky portfolio:P = wM + (wA A ) = 1.0486 + (0.0486) 2.08 = 0.95E(RP) = P + PE(RM) = (0.0486) (16.90%) + (0.95 8%) = 8.42%Since A = 2.8, the opti
34、mal position in this portfolio is:In contrast, with a passive strategy:A difference of: 0.0284The final positions are (M may include some of stocks A through D):Bills1 0.5685 =43.15%M0.5685 l.0486 =59.61A0.5685 (0.0486) (0.6142) =1.70B0.5685 (0.0486) 1.1265 = 3.11C0.5685 (0.0486) (1.2181) =3.37D0.56
35、85 (0.0486) 1.7058 = 4.71(subject to rounding error)100.00%18.a.If a manager is not allowed to sell short, he will not include stocks with negative alphas in his portfolio, so he will consider only A and C:s2(e) A1.63,3640.0004760.3352C3.43,6000.0009440.66480.0014201.0000The forecast for the active
36、portfolio is: = (0.3352 1.6) + (0.6648 3.4) = 2.80% = (0.3352 1.3) + (0.6648 0.7) = 0.90s2(e) = (0.33522 3,364) + (0.66482 3,600) = 1,969.03(e) = 44.37%The weight in the active portfolio is:Adjusting for beta:The information ratio of the active portfolio is:Hence, the square of the Sharpe ratio is:T
37、herefore: S = 0.3535The markets Sharpe ratio is: SM = 0.3478When short sales are allowed (Problem 17), the managers Sharpe ratio is higher (0.3662). The reduction in the Sharpe ratio is the cost of the short sale restriction.The characteristics of the optimal risky portfolio are:With A = 2.8, the op
38、timal position in this portfolio is:The final positions in each asset are: Bills1 0.5455 =45.45%M0.5455 (1 - 0.0931) =49.47A0.5455 0.0931 0.3352 =1.70C0.5455 0.0931 0.6648 =3.38100.00b.The mean and variance of the optimized complete portfolios in the unconstrained and short-sales constrained cases,
39、and for the passive strategy are:E(RC )Unconstrained0.5685 8.42% = 4.790.56852 528.94 = 170.95Constrained0.5455 8.18% = 4.460.54552 535.54 = 159.36Passive0.5401 8.00% = 4.320.54012 529.00 = 154.31The utility levels below are computed using the formula: Unconstrained8% + 4.79% (0.005 2.8 170.95) = 10
40、.40%Constrained8% + 4.46% (0.005 2.8 159.36) = 10.23%Passive8% + 4.32% (0.005 2.8 154.31) = 10.16%19.All alphas are reduced to 0.3 times their values in the original case. Therefore, the relative weights of each security in the active portfolio are unchanged, but the alpha of the active portfolio is
41、 only 0.3 times its previous value: 0.3 -16.90% = -5.07%The investor will take a smaller position in the active portfolio. The optimal risky portfolio has a proportion w* in the active portfolio as follows:The negative position is justified for the reason given earlier.The adjustment for beta is:Sin
42、ce 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 (0.0151) = 1.0151To calculate the Sharpe ratio for the optimal risky portfolio we compute the information ratio for th
43、e active portfolio and the Sharpe ratio for the market portfolio. The information ratio of the active portfolio is 0.3 times its previous value:A = = 0.0343 and A2 =0.00118Hence, the square of the Sharpe ratio of the optimized risky portfolio is:S2 = S2M + A2 = (8%/23%)2 + 0.00118 = 0.1222S = 0.3495
44、Compare this to the markets Sharpe ratio: SM = = 0.3478The difference is: 0.0017Note that the reduction of the forecast alphas by a factor of 0.3 reduced the squared information ratio and the improvement in the squared Sharpe ratio by a factor of:0.32 = 0.0920.If each of the alpha forecasts is doubl
45、ed, then the alpha of the active portfolio will also double. Other things equal, the information ratio (IR) of the active portfolio also doubles. The square of the Sharpe ratio for the optimized portfolio (S-square) equals the square of the Sharpe ratio for the market index (SM-square) plus the squa
46、re of the information ratio. Since the information ratio has doubled, its square quadruples. Therefore: S-square = SM-square + (4 IR)Compared to the previous S-square, the difference is: 3IRNow you can embark on the calculations to verify this result.CFA PROBLEMS1.The regression results provide quantitative measures of return and risk base