CAPTER 7: TE TEORY OF ACTIVE PORTFOLIO ANAGEENT 1. a. Define R r r f Note that e compute the estimates of standard deviation using 4 degrees of freedom (i.e., e divide the sum of the squared deviations from the mean by 4 despite the fact that e have 5 observations), since deviations are taken from the sample mean, not the theoretical population mean. E(R B ) 11.16% B 1.4% E(R U ) 8.4% U 14.85% Risk neutral investors ould prefer the Bull Fund because its performance suggests a higher mean. b. Using the reard-to-volatility (Sharpe) measure: S S E(rB ) r E(R ) 11.16 1.4 f B B B B E(rU ) rf E(R ) 8.4 14.85 U U U U.554.567 The data suggest that the Unicorn Fund dominates for a risk averse investor. c. The decision rule for the proportion to be invested in the risky asset is given by the folloing formula: E(r) r y f E(R).1A This value of y maximizes a mean-variance utility function of the form: U E(r).5A For utility functions of this form, Sharpe s measure is the appropriate criterion for the selection of optimal risky portfolios. An investor ith A 3 ould invest the folloing fraction in Unicorn: 8.4.1 3 14.85 y U 1.77 Note that the investor seeks to borro in order to invest in Unicorn. In that case, his portfolio risk premium and standard deviation ould be: E(r P ) r f 1.77 8.4% 1.7% P 1.77 14.85% 18.9% 7-1
The investor s utility level ould be: U(P) r f + 1.7 (.5 3 18.9 ) r f + 5.36 If borroing is not alloed, investing 1% in Unicorn ould lead to: E(r P ) r f 8.4% P 14.85% U(P) r f + 8.4 (.5 3 14.85 ) r f + 5.11 Note that, if Bull must be chosen, then: 11.16.1 3 1.4 y B.846 E(r P ) r f.846 11.16% 9.% P.846 1.4% 17.51% U(P) r f + 9. (.5 3 17.51 ) r f + 4.6 Thus, even ith a borroing restriction, Unicorn (ith the loer mean) is still superior to Bull.. Write the Black-Scholes formula from Chapter 1 as: C S N(d 1 ) PV(X)N(d ) In this application, here e express the value of timing per dollar of assets, e use S 1. for the value of the stock. The present value of the exercise price is also equal to 1. Note that: d 1 ln(s / X) + (r + / )T and d d1 T T When S PV(X) and T 1, the formula for d 1 reduces to: d 1 / The formula for d becomes: d / Therefore: C N(/) N( /) Finally, recall that: N( x) 1 N(x) Therefore, e can rite the value of the call as: C N(/) [1 N(/)] N(/) 1 Since.55, the value of the option is: C N(.75) 1 Interpolating from the standard normal table in Chapter 1: 7-
.75 C.58 + (.516.58) 1 1.. ence the added value of a perfect timing strategy is.% per month.. 3. a. Using the relative frequencies to estimate the conditional probabilities P 1 and P for timers A and B, e find: Timer A Timer B P 1 78/135.58 86/135.64 P 57/9.6 5/9.54 P* P 1 + P..18 The data suggest that timer A is the better forecaster. b. Using the folloing equation to value the imperfect timing services of Timer A and Timer B: C(P * ) C(P 1 + P 1) C A (P * ).%..44% per month C B (P * ).%.18.4% per month Timer A s added value is greater by 4 basis points per month. 4. a. Alpha (α) Expected excess return α i r i [r f + β i (r r f ) ] E(r i ) r f α A % [8% + 1.3(16% 8%)] 1.6% % 8% 1% α B 18% [8% + 1.8(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 A and C have positive alphas, hereas stocks B and D have negative alphas. The residual variances are: (e A ) 58 3,364 (e B ) 71 5,41 (e C ) 6 3,6 (e D ) 55 3,5 7-3
b. To construct the optimal risky portfolio, e first determine the optimal active portfolio. Using the Treynor-Black technique, e construct the active portfolio: α (e) α / (e) Σα / (e) A.476.614 B.873 1.165 C.944 1.181 D.13 1.758 Total.775 1. Do not be concerned that the positive alpha stocks have negative eights and vice versa. We ill see that the entire position in the active portfolio ill be negative, returning everything to good order. With these eights, the forecast for the active portfolio is: α [.614 1.6] + [1.165 ( 4.4)] [1.181 3.4] + [1.758 ( 4.)] 16.9% β [.614 1.3] + [1.165 1.8] [1.181.7] + [1.758 1].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] + [1.165 541] + [( 1.181) 36] + [1.758 35] 1,89.6 (e) 147.68% ere, again, the levered position in stock B [ith high (e)] overcomes the diversification effect, and results in a high residual standard deviation. The optimal risky portfolio has a proportion * in the active portfolio, computed as follos: α / (e) 16.9 / 1,89.6.514 [E(r ) r ]/ 8/ 3 f The negative position is justified for the reason stated earlier. The adjustment for beta is:.514 *.486 (1 β) (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 7-4
the index portfolio is: 1 (.486) 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: A α /(e) 16.9/147.68.1144 A.131 ence, the square of Sharpe s measure (S) of the optimized risky portfolio is: S S S.366 + A 8 3 +.131.1341 Compare this to the market s Sharpe measure: S 8/3.3478 The difference is:.184 Note that the only-moderate improvement in performance results from the fact that only a small position is taken in the active portfolio A because of its large residual variance. We calculate the odigliani-squared ( ) measure, as follos: E(r P * ) r f + S P 8% + (.366 3%) 16.43% E(r P * ) E(r ) 16.43% 16%.43% d. To calculate the exact makeup of the complete portfolio, e first compute `the mean excess return of the optimal risky portfolio and its variance. The risky portfolio beta is given by: β P + ( A β A ) 1.486 + [(.486).8].95 E(R P ) α P + β P E(R ) [(.486) ( 16.9%)] + (.95 8%) 8.4% (.486 ) 1,89.6) 58. 94 P βp + (e P ) (.95 3) + P 3.% Since A.8, the optimal position in this portfolio is: 8.4 y.5685.1.8 58.94 In contrast, ith a passive strategy: 7-5
8 y.1.8 3 This is a difference of:.84. 541 The final positions of the complete portfolio are: Bills 1.5685 43.15%.5685 l.486 59.61% A.5685 (.486) (.614) 1.7% B.5685 (.486) 1.165 3.11% C.5685 (.486) ( 1.181) 3.37% D.5685 (.486) 1.758 4.71% 1.% [sum is subject to rounding error] Note that may include positive proportions of stocks A through D. 5. 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 A and C: α (e) α (e) α / (e) Σα / (e) A 1.6 3,364.476.335 C 3.4 3,6.944.6648.14 1. The forecast for the active portfolio is: α (.335 1.6) + (.6648 3.4).8% β (.335 1.3) + (.6648.7).9 (e) (.335 3,364) + (.6648 3,6) 1,969.3 (e) 44.37% The eight in the active portfolio is: α / (e) E(R ) /.8 /1,969.3 8/ 3 Adjusting for beta: * (1 β).94.94.931 [(1.9).94] The information ratio of the active portfolio is: 7-6
A α /(e).8/44.37.631 ence, the square of Sharpe s measure is: S (8/3) +.631.15 Therefore: S.3535 The market s Sharpe measure is: S.3478 When short sales are alloed (Problem 4), the manager s Sharpe measure is higher (.366). The reduction in the Sharpe measure is the cost of the short sale restriction. We calculate the odigliani-squared or measure as follos: E(r P * ) r f + S P 8% + (.3535 3%) 16.135% E(r P * ) E(r ) 16.135% 16%.135% When short sales are alloed:.43% The characteristics of the optimal risky portfolio are: β P + A β A (1.931) + (.931.9).99 E(R P ) α P + β P E(R ) (.931.8%) + (.99 8%) 8.18% P βp + (e P ) (.99 3) + (.931 1,969.3) 535.54 P 3.14% With A.8, the optimal position in this portfolio is: 8.18 y.5455.1.8 535.54 The final positions in each asset are: Bills 1.5455 45.45%.5455 (1.931) 49.47% A.5455.931.335 1.7% C.5455.931.6648 3.38% 1.% b. The mean and variance of the optimized complete portfolios in the unconstrained and short-sales constrained cases, and for the passive strategy are: E(R C ) Unconstrained.5685 8.4 4.79.5685 58.94 17.95 7-7 C
Constrained.5455 8.18 4.46.5455 535.54 159.36 Passive.541 8. 4.3.541 59. 154.31 The utility levels belo are computed using the formula: Unconstrained 8 + 4.79 (.5.8 17.95) 1.4 Constrained 8 + 4.46 (.5.8 159.36) 1.3 Passive 8 + 4.3 (.5.8 154.31) 1.16 E(r ).5A C C 6. a. The optimal passive portfolio is obtained from equation (8.7) in Chapter 8: Optimal Risky Portfolios: E(R ) E(R + E(R ) ) E(R [E(R )Cov(r, r ) + E(R ) )]Cov(r, r ) E(R ) 8%, E(R ) % and Cov(r, r ) ρ.6 3 18 48.4 ( 8 18 ) ( 48.4) 1.797 (8 18 ) + ( 3 ) [(8 + ) 48.4].797 If short sales are not alloed, portfolio ould have to be omitted from the passive portfolio because is negative. b. With short sales alloed: E(R passive ) (1.797 8%) + [(.797) %] 1.78% passive (1.797 3) + [(.797) 18] + [ 1.797 (.797) 48.4] 1,.54 passive 34.68% Sharpe s measure in this case is: S passive 1.78/34.68.3685 The market s Sharpe measure is: S 8/3.3478 c. The improvement in utility for the expanded model of and versus a portfolio of alone, for A.8, is: 1.78 y.3796.1.8 1,.54 7-8
Therefore: U passive 8 + (1.78.3796) (.5.8.3796 1,.54) 1.43 This result is greater than U passive 1.16 from Problem 5. 7. The first step is to find the beta of the stocks relative to the optimized passive portfolio. For any stock i, the covariance ith a portfolio is the sum of the covariances ith the portfolio components, accounting for the eights of the components. Thus: β Therefore: β β β β i Cov(ri,r passive passive ) β i i passive + β ( 1. 1.797 3 ) + ( 1.8 (.797) 18 ). 561 A 1,.54 ( 1.4 1.797 3 ) + ( 1.1 (.797) 18 ). 875 B 1,.54 (.5 1.797 3 ) + ( 1.5 (.797) 18 ). 731 C 1,.54 ( 1. 1.797 3 ) + (. (.797) 18 ). 7476 D 1,.54 The alphas relative to the optimized portfolio are: α i E(r i ) r f [β i, passive E(R passive )] α A 8 (.561 1.78) 4.8% α B 18 8 (.875 1.78) 1.1% α C 17 8 (.731 1.78) 8.7% α D 1 8 (.7476 1.78) 5.55% The residual variances are no obtained from: i i ( β ) ( e,passive) i: passive here: β + e ) from Problem 4. i ( i passive (e A ) (1.3 3) + 58 (.561 34.68) 3,878.1 (e B ) (1.8 3) + 71 (.875 34.68) 5,843.59 7-9
(e C ) (.7 3) + 6 (.731 34.68) 3,85.78 (e D ) (1. 3) + 55 (.7476 34.68),881.8 From this point, the procedure is identical to that of Problem 6: The active portfolio parameters are: α (e) α / (e A ) Σα / (e A ) A.143 1.189 B.19.1574 C.95 1.717 D.196 1.5787 Total.1 1. α (1.189 4.8) + [(.1574) ( 1.1)] + (1.717 8.7) + [( 1.5787) ( 5.55)] 7.71% β (1.189.561) + [(.1574).875] + (1.717.731) + [( 1.5787).7476).619 (e) (1.189 3,878.1) + [(.1574) 5,843.59] + (1.717 3,85.78) + [( 1.5787) 881.8],714.3 The proportions in the overall risky portfolio can no be determined: α / (e) E(R ) / 7.71/,714.3 1.78/1,.54 passive passive * (1 β).1148.1148.968 [(1 +.619).1148] a. Sharpe s measure for the optimal risky portfolio is S S passive α + (e).3685 S.4118 compared to S passive.3685 7.71 +.1696,714.3 Therefore, the difference in the Sharpe measure is:.433 b. The beta of the optimal risky portfolio is: β P *β A + (1 *) [.968 (.619)] +.93.8433 The mean excess return of this portfolio is: 7-1
E(R) (.968 7.71%) + (.8433 1.78%) 13.46% The variance and standard deviation are:.8433 1,.54 +.968,714.3 1,68.3 3.68% Therefore, the position in the optimal risky portfolio ould be: 13.46 y.451.1.8 1,68.3 The utility value for this portfolio is: U 8 + (.451 13.46) (.5.8.451 1,68.3) 11.3 This value is superior to all previous alternatives. 8. If short sales are not alloed, then the passive portfolio reverts to, and the solution mimics the solution to Problem 5. 9. All 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:.3 16.9% 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: α / (e) E(r r ) / 5.7 / 1,89.6 8/ 3 f.1537 The negative position is justified for the reason given earlier. The adjustment for beta is: * (1 β).1537.151 [(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) 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: A α /(e) 5.7/147.68.343 and A.118 7-11
ence, the square of Sharpe s measure of the optimized risky portfolio is: S S + A (8/3) +.118.1 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 7-1