CHAPTER 8: INDEX MODELS

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Stanley McDowell
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1 CHTER 8: INDEX ODELS CHTER 8: INDEX ODELS ROBLE SETS 1. The advantage of the index model, compared to the arkoitz procedure, is the vastly reduced number of estimates required. In addition, the large number of estimates required for the arkoitz procedure can result in large aggregate estimation errors hen implementing the procedure. The disadvantage of the index model arises from the model s assumption that return residuals are uncorrelated. This assumption ill be incorrect if the index used omits a significant risk factor.. The trade-off entailed in departing from pure indexing in favor of an actively managed portfolio is beteen the probability (or the possibility) of superior performance against the certainty of additional management fees. 3. The anser to this question can be seen from the formulas for (equation 8.) and * (equation 8.1). Other things held equal, is smaller the greater the residual variance of a candidate asset for inclusion in the portfolio. Further, e see that regardless of beta, hen decreases, so does *. 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-specific risk reduces the extent to hich an active investor ill be illing 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 is independent of market performance. The Sharpe ratio indicates that a higher alpha makes a security more desirable. lpha, 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 portfolio alpha is the portfolio-eighted 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 ould need: n = 6 estimates of means n = 6 estimates of variances n n 1,77 estimates of covariances n 3n Therefore, in total: 1, 89 estimates b. In a single index model: r i r f = α i + β i (r r f ) + e i 8-1

2 CHTER 8: INDEX ODELS Equivalently, using excess returns: R i = α i + β i R + e i The variance of the rate of return can be decomposed into the components: (l) The variance due to the common market factor: i () The variance due to firm specific unanticipated events: e ) In this model: Cov (r,r i j) The number of parameter estimates is: n = 6 estimates of the mean E(r i ) i n = 6 estimates of the sensitivity coefficient β i j n = 6 estimates of the firm-specific variance σ (e i ) 1 estimate of the market mean E(r ) 1 estimate of the market variance Therefore, in total, 18 estimates. The single index model reduces the total number of required estimates from 1,89 to 18. In general, the number of parameter estimates is reduced from: n 3n to (3n ) ( i 6. a. The standard deviation of each individual stock is given by: [ (e 1/ i i i )] Since β =.8, β B = 1., σ(e ) = 3%, σ(e B ) = 4%, and σ = %, e get: σ = ( ) 1/ = 34.78% σ B = ( ) 1/ = 47.93% b. The expected rate of return on a portfolio is the eighted average of the expected returns of the individual securities: E(r ) = E(r ) + B E(r B ) + f r f E(r ) = (.3 13%) + (.45 18%) + (.5 8%) = 14% The beta of a portfolio is similarly a eighted average of the betas of the individual securities: β = β + B β B + f β f β = (.3.8) + (.45 1.) + (.5.) =.78 The variance of this portfolio is: ( e ) here ( e ) is the nonsystematic component. Since the residuals (e i ) are uncorrelated, the non-systematic variance is: 8-

3 CHTER 8: INDEX ODELS ( e ) ( e ) ( e ) ( e ) B B f f = (.3 3 ) + (.45 4 ) + (.5 ) = 45 here σ (e ) and σ (e B ) are the firm-specific (nonsystematic) variances of Stocks and B, and σ (e f ), the nonsystematic variance of T-bills, is zero. The residual standard deviation of the portfolio is thus: σ(e ) = (45) 1/ =.1% The total variance of the portfolio is then: (.78 ) change to The total standard deviation is 6.41%. 7. a. The to figures depict the stocks security characteristic lines (SCL). Stock has higher firm-specific risk because the deviations of the observations from the SCL are larger for Stock 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, hich is the measure of systematic risk. The SCL for Stock B is steeper; hence Stock B s systematic risk is greater. c. The R (or squared correlation coefficient) of the SCL is the ratio 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): R β i σ β i σ σ (e ) i Since the explained variance for Stock B is greater than for Stock (the explained variance is, hich is greater since its beta is higher), and its residual variance B ( e B ) is smaller, its R is higher than Stock s. d. lpha is the intercept of the SCL ith the expected return axis. Stock has a small positive alpha hereas Stock B has a negative alpha; hence, Stock s alpha is larger. e. The correlation coefficient is simply the square root of R, so Stock B s correlation ith the market is higher. 8. a. Firm-specific risk is measured by the residual standard deviation. Thus, stock has more firm-specific risk: 1.3% > 9.1% b. arket risk is measured by beta, the slope coefficient of the regression. has a larger beta coefficient: 1. >.8 c. R measures the fraction of total variance of return explained by the market return. s R is larger than B s:.576 >.436 d. Reriting the SCL equation in terms of total return (r) rather than excess return (R): 8-3

4 CHTER 8: INDEX ODELS r r ( r r ) f f r r (1 ) r f The intercept is no equal to: r (1 ) 1% r (1 1.) f f Since r f = 6%, the intercept ould be: 1% 6%(1 1.) 1% 1.%.% 9. The standard deviation of each stock can be derived from the folloing equation for R : R i Therefore: i i For stock B: B B R 31.3% % Explained variance Total variance.7. 4, The systematic risk for is: The firm-specific risk of (the residual variance) is the difference beteen s total risk and its systematic risk: = 784 The systematic risk for B is: B B s firm-specific risk (residual variance) is: = The covariance beteen the returns of and B is (since the residuals are assumed to be uncorrelated): Cov(r,r ) B B The correlation coefficient beteen the returns of and B is: B Cov(r,rB ) B 8-4

5 CHTER 8: INDEX ODELS 1. Note that the correlation is the square root of R : R Cov r r 1/ (, ) Cov r r 1/ ( B, ) B For portfolio e can compute: σ = [(.6 98) + (.4 48) + ( )] 1/ = [18.8] 1/ = 35.81% β = (.6.7) + (.4 1.) =.9 σ (e ) σ β σ 18.8 (.9 Cov(r,r ) = β σ =.9 4=36 4) This same result can also be attained using the covariances of the individual stocks ith the market: Cov(r,r ) = Cov(.6r +.4r B, r ) =.6 Cov(r, r ) +.4 Cov(r B,r ) = (.6 8) + (.4 48) = Note that the variance of T-bills is zero, and the covariance of T-bills ith any asset is zero. Therefore, for portfolio : 1/ Cov(r,r ) (.5 1,8.8) (.3 4) ( ) 1/ 1.55% (.5.9) (.31) (. ).75 (e ) (.75 4) 39.5 Cov(r,r ) a. Beta Books adjusts beta by taking the sample estimate of beta and averaging it ith 1., using the eights of /3 and 1/3, as follos: adjusted beta = [(/3) 1.4] + [(1/3) 1.] = 1.16 b. If you use your current estimate of beta to be β t 1 = 1.4, then 16. For Stock : β t =.3 + (.7 1.4) = r [ r ( r r )].11 [.6.8 (.1.6)].% For stock B: f f r [ r ( r r )].14 [ (.1.6)] 1% B B f B f 8-5

6 CHTER 8: INDEX ODELS Stock ould be a good addition to a ell-diversified portfolio. short position in Stock B may be desirable. 17. a. lpha (α) Expected excess return α i = r i [r f + β i (r r f ) ] E(r i ) r f α = % [8% (16% 8%)] = 1.6% % 8% = 1% α B = 18% [8% (16% 8%)] = 4.4% 18% 8% = 1% α C = 17% [8% +.7 (16% 8%)] = 3.4% 17% 8% = 9% α D = 1% [8% + 1. (16% 8%)] = 4.% 1% 8% = 4% Stocks and C have positive alphas, hereas stocks B and D have negative alphas. The residual variances are: (e ) = 58 = 3,364 (e B ) = 71 = 5,41 (e C ) = 6 = 3,6 (e D ) = 55 = 3,5 b. To construct the optimal risky portfolio, e first determine the optimal active portfolio. Using the Treynor-Black technique, e construct the active portfolio: a (e) a / (e) Sa / (e) B C D Total Be unconcerned ith the negative eights of the positive α stocks the entire active position ill be negative, returning everything to good order. With these eights, the forecast for the active portfolio is: α = [ ] + [1.165 ( 4.4)] [ ] + [1.758 ( 4.)] = 16.9% β = [ ] + [ ] [ ] + [ ] =.8 The high beta (higher than any individual beta) results from the short positions in the relatively lo beta stocks and the long positions in the relatively high beta stocks. (e) = [(.614) 3364] + [ ] + [( 1.181) 36] + [ ] = 1,89.6 (e) = % The levered position in B [ith high (e)] overcomes the diversification effect, and results 8-6

7 CHTER 8: INDEX ODELS in a high residual standard deviation. The optimal risky portfolio has a proportion * in the active portfolio, computed as follos:.514 / ( e).169 / 1,89.6 [ E( r ) rf ] /.8 / 3 The negative position is justified for the reason stated earlier. The adjustment for beta is: * 1 (1 ) (1.8)(.514) Since * is negative, the result is a positive position in stocks ith positive alphas and a negative position in stocks ith negative alphas. The position in the index portfolio is: 1 (.486) = c. To calculate Sharpe s measure for the optimal risky portfolio, e 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 follos: = = 16.9/ =.1144 () e =.131 Hence, the square of Sharpe s measure (S) of the optimized risky portfolio is: S S S =.366 Compare this to the market s Sharpe measure: S = 8/3 =.3478 difference of:.184 The only-moderate improvement in performance results from only a small position taken in the active portfolio because of its large 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: β = + ( β ) = [(.486).8] =.95 E(R ) = α + β E(R ) = [(.486) ( 16.9%)] + (.95 8%) = 8.4% 3.% (e ) (.95 3) (.486 ) 1, Since =.8, the optimal position in this portfolio is: 8.4 y In contrast, ith a passive strategy: 8-7

8 CHTER 8: INDEX ODELS 8 y difference of:.84 The final positions are ( may include some of stocks through D): Bills = 43.15%.5685 l.486 = 59.61%.5685 (.486) (.614) = 1.7% B.5685 (.486) = 3.11% C.5685 (.486) ( 1.181) = 3.37% D.5685 (.486) = 4.71% (subject to rounding error) 1.% 18. a. If a manager is not alloed to sell short he ill not include stocks ith negative alphas in his portfolio, so he ill consider only and C: Α (e) 8-8 a (e) a / (e) Sa / (e) 1.6 3, C 3.4 3, The forecast for the active portfolio is: α = ( ) + ( ) =.8% β = ( ) + ( ) =.9 (e) = (.335 3,364) + ( ,6) = 1,969.3 σ(e) = 44.37% The eight in the active portfolio is: / (e) E(R ) /.8 /1, / 3 djusting for beta: * 1 (1 ) [(1.9).94] The information ratio of the active portfolio is: ( e) Hence, the square of Sharpe s measure is: S Therefore: S =.3535

9 CHTER 8: INDEX ODELS The market s Sharpe measure is: S =.3478 When short sales are alloed (roblem 17), the manager s Sharpe measure is higher (.366). The reduction in the Sharpe measure is the cost of the short sale restriction. The characteristics of the optimal risky portfolio are: (1.931) (.931.9).99 E( R ) E( R ) (.931.8%) (.99 8%) 8.18% ( e ) (.99 3) ( ) % With =.8, the optimal position in this portfolio is: 8.18 y The final positions in each asset are: Bills = 45.45%.5455 (1.931) = 49.47% = 1.7% C = 3.38% 1.% b. The mean and variance of the optimized complete portfolios in the unconstrained and shortsales constrained cases, and for the passive strategy are: E(R C ) C Unconstrained % = = Constrained % = = assive % = = The utility levels belo are computed using the formula: Unconstrained 8% % ( ) = 1.4% Constrained 8% % ( ) = 1.3% assive 8% + 4.3% ( ) = 1.16% 8-9 E(r ) ll alphas are reduced to.3 times their values in the original case. Therefore, the relative eights of each security in the active portfolio are unchanged, but the alpha of the active portfolio is only.3 times its previous value: % = 5.7% The investor ill take a smaller position in the active portfolio. The optimal risky portfolio has a proportion * in the active portfolio as follos: / ( e).57 / 1,89.6 E( r rf ) /.8 / The negative position is justified for the reason given earlier. The adjustment for beta is: C C

10 CHTER 8: INDEX ODELS * 1 (1 ) [(1.8) (.1537)] Since * is negative, the result is a positive position in stocks ith positive alphas and a negative position in stocks ith negative alphas. The position in the index portfolio is: 1 (.151) = To calculate Sharpe s measure for the optimal risky portfolio e compute the information ratio for the active portfolio and Sharpe s measure for the market portfolio. The information ratio of the active portfolio is.3 times its previous value: = 5.7 =.343 and =.118 ( e) Hence, the square of Sharpe s measure of the optimized risky portfolio is: S = S + = (8%/3%) =.1 S =.3495 Compare this to the market s Sharpe measure: S = 8% 3% =.3478 The difference is:.17 Note that the reduction of the forecast alphas by a factor of.3 reduced the squared information ratio and the improvement in the squared Sharpe ratio by a factor of:.3 =.9. If each of the alpha forecasts is doubled, then the alpha of the active portfolio ill 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 (S-square) plus the square of the information ratio. Since the information ratio has doubled, its square quadruples. Therefore: S-square = S-square + (4 IR) Compared to the previous S-square, the difference is: 3IR No you can embark on the calculations to verify this result. 8-1

11 CHTER 8: INDEX ODELS CF ROBLES 1. The regression results provide quantitative measures of return and risk based on monthly returns over the five-year period. β for BC as.6, considerably less than the average stock s β of 1.. This indicates that, hen the S& 5 rose or fell by 1 percentage point, BC s return on average rose or fell by only.6 percentage point. Therefore, BC s systematic risk (or market risk) as lo relative to the typical value for stocks. BC s alpha (the intercept of the regression) as 3.%, indicating that hen the market return as %, the average return on BC as 3.%. BC s unsystematic risk (or residual risk), as measured by σ(e), as 13.%. For BC, R as.35, indicating closeness of fit to the linear regression greater than the value for a typical stock. β for XYZ as somehat higher, at.97, indicating XYZ s return pattern as very similar to the β for the market index. Therefore, XYZ stock had average systematic risk for the period examined. lpha for XYZ as positive and quite large, indicating a return of 7.3%, on average, for XYZ independent of market return. Residual risk as 1.45%, half again as much as BC s, indicating a ider scatter of observations around the regression line for XYZ. Correspondingly, the fit of the regression model as considerably less than that of BC, consistent ith an R of only.17. The effects of including one or the other of these stocks in a diversified portfolio may be quite different. If it can be assumed that both stocks betas ill remain stable over time, then there is a large difference in systematic risk level. The betas obtained from the to brokerage houses may help the analyst dra inferences for the future. The three estimates of BC s β are similar, regardless of the sample period of the underlying data. The range of these estimates is.6 to.71, ell belo the market average β of 1.. The three estimates of XYZ s β vary significantly among the three sources, ranging as high as 1.45 for the eekly data over the most recent to years. One could infer that XYZ s β for the future might be ell above 1., meaning it might have somehat greater systematic risk than as implied by the monthly regression for the five-year period. These stocks appear to have significantly different systematic risk characteristics. If these stocks are added to a diversified portfolio, XYZ ill add more to total volatility.. The R of the regression is:.7 =.49 Therefore, 51% of total variance is unexplained by the market; this is nonsystematic risk = 3 + (11 3) = d. 5. b. 8-11

CHAPTER 8: INDEX MODELS

CHAPTER 8: INDEX MODELS Chapter 8 - Index odels CHATER 8: INDEX ODELS ROBLE SETS 1. The advantage of the index model, compared to the arkowitz procedure, is the vastly reduced number of estimates required. In addition, the large