CHAPTER 8: INDEX MODELSPROBLEM SETS1. The advantage of the index model, compared to the Markowitzprocedure, is the vastly reduced number of estimates required. Inaddition, the large number of estimates required for the Markowitz procedure can result in large aggregate estimation errors whenimplementing the procedure. The disadvantage of the index modelarises from the model’s assumption that return residuals areuncorrelated. This assumption will be incorrect if the index usedomits a significant risk factor.2. The trade-off entailed in departing from pure indexing in favor ofan actively managed portfolio is between the probability (or thepossibility) of superior performance against the certainty ofadditional management fees.3. The answer to this question can be seen from the formulas for w0(equation and w* (equation . Other things held equal, w0 issmaller the greater the residual 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, thegreater the residual variance of an asset, the smaller its position in the optimal risky portfolio. That is, increased firm-specificrisk 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). Wecall alpha a nonmarket return premium because it is the portion of the return premium that is independent of market performance.The Sharpe ratio indicates that a higher alpha makes a securitymore desirable. Alpha, the numerator of the Sharpe ratio, is afixed number that is not affected by the standard deviation ofreturns, the denominator of the Sharpe ratio. Hence, an increase in alpha 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 security’s alpha results in an increase in the portfolio Sharpe ratio.5. a. To optimize this portfolio one would need:n = 60 estimates of means n = 60 estimates of variances770,122=-nn estimates of covariances Therefore, in total: 890,1232=+nn estimatesb. In a single index model: r ir f = α i + β i (r M – r f ) + e iEquivalently, using excess returns: R i = α i + β 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 market factor: 22M i σβ(2) The variance due to firm specific unanticipated events:)(σ2i e In this model: σββ),(Cov j i j i r r = The number of parameter estimates is:n = 60 estimates of the mean E (r i )n = 60 estimates of the sensitivity coefficient β i n = 60 estimates of the firm-specific variance σ2(e i ) 1 estimate of the market mean E (r M )1 estimate of the market variance 2M σTherefore, 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:)23( to 232+⎪⎪⎭⎫ ⎝⎛+n n n6. a. The standard deviation of each individual stock is given by:2/1222)](β[σi M i i e σσ+=Since βA = , βB = , σ(e A) = 30%, σ(e B) = 40%, and σM = 22%, we get:σA = × 222 + 302 )1/2 = %σB = × 222 + 402 )1/2 = %b. The expected rate of return on a portfolio is the weightedaverage of the expected returns of the individual securities:E(r P ) = w A × E(r A ) + w B × E(r B ) + w f × r f E(r P ) = × 13%) + × 18%) + × 8%) = 14%The beta of a portfolio is similarly a weighted average of the betas of the individual securities:βP = w A × βA + w B × βB + w f × β f βP = × + × + × = The variance of this portfolio is:)(σβσ2222P M P P e +=σwhere 22σβM P is the systematic component and )(2P e σis the nonsystematic component. Since the residuals (e i ) are uncorrelated, the nonsystematic variance is:2222222()()()()P A A B B f f e w e w e w e σσσσ=⨯+⨯+⨯= × 302 ) + × 402 ) + × 0) = 405where σ2(e A ) and σ2(e B ) are the firm-specific (nonsystematic)variances of Stocks A and B, and σ2(e f ), the nonsystematic variance of T-bills, is zero. The residual standard deviation of the portfolio is thus:σ(e P ) = (405)1/2 = %The total variance of the portfolio is then:47.699405)2278.0(σ222=+⨯=PThe total standard deviation is %.7. a. The two figures depict the stocks’ security characteri sticlines (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 ofsystematic risk. The SCL for Stock B is steeper; hence Stock B’s systematic risk is greater.c. The R 2(or squared correlation coefficient) of the SCL is theratio of the explained variance of the stock’s return to total variance, and the total variance is the sum of the explained variance plus the unexplained variance (the stock’s residual variance):)(σσβσβ222222i M i M i e R += Since the explained variance for Stock B is greater than forStock A (the explained variance is 22σβM B , which is greatersince its beta is higher), and its residual variance 2()B e σ is smaller, its R 2 is higher than Stock A’s.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 A’s alpha is larger.e. The correlation coefficient is simply the square root of R 2, soStock B’s correlation with the market is h igher.8. a. Firm-specific risk is measured by the residual standarddeviation. Thus, stock A has more firm-specific risk: % > %b. Market risk is measured by beta, the slope coefficient of theregression. A has a larger beta coefficient: >c. R 2 measures the fraction of total variance of return explainedby the market return. A’s R 2 is larger than B’s: >d. Rewriting the SCL equation in terms of total return (r ) ratherthan excess return (R ):()(1)A f M f A f Mr r r r r r r αβαββ-=+⨯-⇒=+⨯-+⨯The intercept is now equal to:(1)1%(1 1.2)f f r r αβ+⨯-=+⨯- 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 from thefollowing equation for R 2:==2222σσβiMi iR Explained variance Total variance Therefore:%30.31σ98020.0207.0σβσ222222==⨯==A A MA AR For stock B:%28.69σ800,412.0202.1σ222==⨯=B B10. The systematic risk for A is:22220.7020196A M βσ⨯=⨯=The firm-specific risk of A (the residual variance) is the difference between A’s total risk and its systematic risk:980 – 196 = 784The systematic risk for B is:22221.2020576B M βσ⨯=⨯=B’s firm -specific risk (residual variance) is:4,800 – 576 = 4,22411. The covariance between the returns of A and B is (since theresiduals are assumed to be uncorrelated):33640020.170.0σββ)(Cov 2=⨯⨯==M B A B A ,r rThe correlation coefficient between the returns of A and B is:155.028.6930.31336σσ),(Cov ρ=⨯==B A B A AB r r12. Note that the correlation is the square root of R 2:2ρR =1/2,1/2,()0.2031.3020280()0.1269.2820480A M A MB M B M Cov r r Cov r r ρσσρσσ==⨯⨯===⨯⨯=13. For portfolio P we can compute:σP = [ × 980) + × 4800) + (2 × × × 336)]1/2 = []1/2 = % βP = × + × =958.08400)(0.901282.08σβσ)(σ22222=⨯-=-=M P P P eCov(r P ,r M ) = βP 2σM = × 400=360This same result can also be attained using the covariances of the individual stocks with the market:Cov(r P ,r M ) = Cov + , r M ) = × Cov(r A , r M ) + × Cov(r B ,r M )= × 280) + × 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:[][]%55.21)3603.05.02()4003.0()08.282,15.0(),(Cov 2σσσ2/1222/12222=⨯⨯⨯+⨯+⨯=⨯⨯⨯++=M P M P M M P P Q r r w w w w(0.50.90)(0.31)(0.200)0.75Q P P M M w w βββ=+=⨯+⨯+⨯=52.239)40075.0(52.464σβσ)(σ22222=⨯-=-=M Q Q Q e30040075.0σβ),(Cov 2=⨯==M Q M Q r r15. a. Beta Books adjusts beta by taking the sample estimate of betaand averaging it with , using the weights of 2/3 and 1/3, as follows:adjusted beta = [(2/3) × ] + [(1/3) × ] =b. If you use your current estimate of beta to be βt –1 = , thenβt = + × =16. For Stock A:[()].11[.060.8(.12.06)]0.2%A A f A M f r r r r αβ=-+⨯-=-+⨯-= For stock B:[()].14[.06 1.5(.12.06)]1%B B f B M f r r r r αβ=-+⨯-=-+⨯-=-Stock A would be a good addition to a well-diversified portfolio. A short position in Stock B may be desirable. 17. a.Alpha (α)Expected excessreturn αi = r i – [r f + βi × (r M – r f ) ] E (r i ) – r f αA = 20% – [8% + × (16% – 8%)] = % 20% – 8% = 12% αB = 18% – [8% + × (16% – 8%)] = – %18% – 8% = 10% αC = 17% – [8% + × (16% – 8%)] = % 17% – 8% = 9% αD = 12% – [8% + × (16% – 8%)] = – %12% – 8% = 4%Stocks A and C have positive alphas, whereas stocks B and D have negative alphas. The residual variances are:2(e A ) = 582 = 3,364 2(e B ) = 712 = 5,041 2(e C ) = 602 = 3,600 2(e D ) = 552 = 3,025b. To construct the optimal risky portfolio, we first determine theoptimal active portfolio. Using the Treynor-Black technique, we construct the active portfolio: a 2(e) a / 2(e)Sa / 2(e)A –B –C –D –Total –Be unconcerned with the negative weights of the positive αstocks —the entire active position will be negative, returning everything to good order.With these weights, the forecast for the active portfolio is:α = [– × ] + [ × (– ] – [ × ] + [ × (– ]= –%β = [– × ] + [ × ] – [ × ] + [ × 1] =The high beta (higher than any individual beta) results fromthe short positions in the relatively low beta stocks and thelong positions in the relatively high beta stocks.2(e ) = [(–2×3364] + [×5041] + [(–2×3600] + [×3025]= 21,(e ) = %The levered position in B [with high 2(e )] overcomes thediversification effect and results in a high residual standard deviation. The optimal risky portfolio has a proportion w * inthe active portfolio, computed as follows:2022/().1690/21,809.60.05124[()]/.08/23M f M e w E r r ασσ-===-- The negative position is justified for the reason statedearlier.The adjustment for beta is:0486.0)05124.0)(08.21(105124.0)β1(1*00-=--+-=-+=w w w Since w * is negative, the result is a positive position instocks with positive alphas and a negative position in stockswith negative alphas. The position in the index portfolio is:1 – (– =c. To calculate the Sharpe ratio for the optimal risky portfolio, wecompute the information ratio for the active portfolio andSharpe’s measure for the market portfolio . The information ratio for the active portfolio is computed as follows:A = ()e ασ = – = – A 2 =Hence, the square of the Sharpe ratio (S) of the optimizedrisky portfolio is:1341.00131.02382222=+⎪⎭⎫ ⎝⎛=+=A S S M S =Compare this to the market’s Sharpe ratio:S M = 8/23 = A difference of:The only moderate improvement in performance results from only a small position taken in the active portfolio A because of its large residual variance.d. To calculate the makeup of the complete portfolio, firstcompute the beta, the mean excess return, and the variance ofthe optimal risky portfolio:βP = w M + (w A × βA ) = + [(– ] =E (R P ) = αP + βP E (R M ) = [(– (–%)] + × 8%) = %()94.5286.809,21)0486.0()2395.0()(σσβσ222222=⨯-+⨯=+=P M P P e %00.23σ=PSince A = , the optimal position in this portfolio is:5685.094.5288.201.042.8=⨯⨯=y In contrast, with a passive strategy: 5401.0238.201.082=⨯⨯=y A difference of:The final positions are (M may include some of stocks A throughD):Bills 1 – = %M = A (– (– = B (– = – C (– (– = D (– = –(subject to rounding error) %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:Α2(e )a 2(e) a / 2(e)Sa / 2(e) A 3,364C 3,600The forecast for the active portfolio is:α = × + × = %β = × + × =2(e) = × 3,364) + × 3,600) = 1,σ(e) = %The weight in the active portfolio is:0940.023/803.969,1/80.2σ/)()(σ/α2220===M M R E e w Adjusting for beta:0931.0]094.0)90.01[(1094.0w )1(1w *w 00=⨯-+=β-+= The information ratio of the active portfolio is:2.800.0631()44.37A e ασ=== Hence, the square of the Sharpe ratio is:22280.06310.125023S ⎛⎫=+= ⎪⎝⎭ Therefore: S =The market’s Sharpe ratio is: S M =When short sales are allowed (Problem 17), the manager’sSharpe ratio is higher . The reduction in the Sharpe ratio isthe cost of the short sale restriction.The characteristics of the optimal risky portfolio are:222222(10.0931)(0.09310.9)0.99()()(0.0931 2.8%)(0.998%)8.18%()(0.9923)(0.09311969.03)535.5423.14%P M A A P P P M P P M P P w w E R E R e ββαβσβσσσ=+⨯=-+⨯==+⨯=⨯+⨯==⨯+=⨯+⨯==With A = , the optimal position in this portfolio is:5455.054.5358.201.018.8y =⨯⨯= The final positions in each asset are:Bills1 – = % M (1 = A = C =b. The mean and variance of the optimized complete portfolios inthe unconstrained and short-sales constrained cases, and for the passive strategy are:E (R C ) 2σCUnconstrained× % = × = Constrained× % = × = Passive × % = × = The utility levels below are computed using the formula:2σ005.0)(C C A r E -Unconstrained 8% + % – × × = %Constrained8% + % – × × = % Passive 8% + % – × × = %19. All alphas are reduced to 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 only times its previous value: × % = %The investor will take a smaller position in the active portfolio. The optimal risky portfolio has a proportion w * in the activeportfolio as follows:2022/()0.0507/21,809.60.01537()/0.08/23M f M e w E r r ασσ-===-- The negative position is justified for the reason given earlier. The adjustment for beta is:0151.0)]01537.0()08.21[(101537.0)β1(1*00-=-⨯-+-=-+=w w w Since w * is negative, the result is a positive position in stocks with positive alphas and a negative position in stocks withnegative alphas. The position in the index portfolio is: 1 – (– =To calculate the Sharpe ratio for the optimal risky portfolio wecompute the information ratio for the active portfolio and the Sharpe ratio for the market portfolio. The information ratio of the active portfolio is times its previous value: A = 5.07()147.68e ασ-== – and A 2 = Hence, the square of the Sharpe ratio of the optimized riskyportfolio is:S 2 = S 2M + A 2 = (8%/23%)2 + =S =Compare this to the market’s Sharpe ratio: S M =8%23%= The difference is:Note that the reduction of the forecast alphas by a factor of reduced the squared information ratio and the improvement in the squared Sharpe 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, 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 square 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 andrisk based on monthly returns over the five-year period.βfor ABC was , considerably less than the average stock’s β of .This indicates that, when the S&P 500 rose or fell by 1 percentagepoin t, ABC’s return on average rose or fell by only percentagepoint. Therefore, ABC’s systematic risk (or market risk) was lowrelative 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’sunsystematic risk (or residual risk), as measured by σ(e), was %.For ABC, R2 was , indicating closeness of fit to the linearregression greater than the value for a typical stock.β for XYZ was somewhat higher, at , indicating XYZ’s returnpattern was very similar to the β for the market index. Therefore, XYZ stock had average systematic risk for the period examined.Alpha for XYZ was positive and quite large, indicating a returnof %, on average, for XYZ independent of market return. Residualrisk was %, half again as much as ABC’s, indicating a widerscatter of observations around the regression line for XYZ.Correspondingly, the fit of the regression model was considerablyless than that of ABC, consistent with an R2 of only .The effects of including one or the other of these stocks in adiversified portfolio may be quite different. If it can be assumedthat both stocks’ betas will remain stable over time, then thereis a large difference in systematic risk level. The betas obtainedfrom the two brokerage houses may help the analyst draw inferencesfor the future. The three estimates of ABC’s β are similar,regardless of the sample period of the underlying data. The rangeof these estimates is to , well below the market average β of .The three estimates of XYZ’s β vary significantly among the threesources, ranging as high as for the weekly data over the mostrecent two years. One could infer that XYZ’s β for the futuremight be well above , meaning it might have somewhat greatersystematic risk than was implied by the monthly regression for thefive-year period.These stocks appear to have significantly different systematic riskcharacteristics. If these stocks are added to a diversified portfolio,XYZ will add more to total volatility.2. The R2 of the regression is: =Therefore, 51% of total variance is unexplained by the market;this is nonsystematic risk.3. 9 = 3 + (11 3) =4. d.5. b.。