15 9 3 β 11 3 β 075 16 d 17 b 18 a alpha α

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15. 9 = 3 + β (11 3) β = 0.75 16. d. 17. b. 18. a. Alpha ( α ) Expected excess return α i = r i – [r f + β i (r M – r f ) ] E(r i ) – r f α A = 20% – [8% + 1.3(16% – 8%)] = 1.6% 20% – 8% = 12% α B = 18% – [8% + 1.8(16% – 8%)] = – 4.4% 18% – 8% = 10% α C = 17% – [8% + 0.7(16% – 8%)] = 3.4% 17% – 8% = 9% α D = 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: σ 2 (e A ) = 58 2 = 3,364 σ 2 (e B ) = 71 2 = 5,041 σ 2 (e C ) = 60 2 = 3,600 σ 2 (e D ) = 55 2 = 3,025
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b. To construct the optimal risky portfolio, we first determine the optimal active portfolio. Using the Treynor-Black technique, we construct the active portfolio: α σ 2 (e) α / σ 2 (e) Σα / σ 2 (e) A 0.000476 –0.6142 B –0.000873 1.1265 C 0.000944 –1.2181 D –0.001322 1.7058 Total –0.000775 1.0000 Do not be concerned that the positive alpha stocks have negative weights and vice versa. We will see that the entire position in the active portfolio will be negative, returning everything to good order. With these weights, the forecast for the active portfolio is: α = [–0.6142 × 1.6] + [1.1265 × (– 4.4)] – [1.2181 × 3.4] + [1.7058 × (– 4.0)] = –16.90% β = [–0.6142 × 1.3] + [1.1265 × 1.8] – [1.2181 × 0.70] + [1.7058 × 1] = 2.08 The 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 high beta stocks. σ 2 (e) = [(–0.6142) 2 × 3364] + [1.1265 2 × 5041] + [(–1.2181) 2 × 3600] + [1.7058 2 × 3025] = 21,809.6 σ( e ) = 147.68% Here, again, the levered position in stock B [with high σ 2 (e)] overcomes the diversification effect, and results in a high residual standard deviation. The optimal risky portfolio has a proportion w * in the active portfolio, computed as follows: 05124 . 0 23 / 8 6 . 809 , 21 / 90 . 16 / ] r ) r ( E [ ) e ( / w 2 2 M f M 2 0 = = σ σ α = The negative position is justified for the reason stated earlier. 8-8
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The adjustment for beta is: 0486 . 0 ) 05124 . 0 )( 08 . 2 1 ( 1 05124 . 0 w ) 1 ( 1 w * w 0 0 = + = β + = 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 is: 1 – (–0.0486) = 1.0486 c. To calculate Sharpe’s measure for the optimal risky portfolio, we 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 follows: A = α / σ( e)= –16.90/147.68 = –0.1144 A 2 = 0.0131 Hence, the square of Sharpe’s measure (S) of the optimized risky portfolio is: 1341 . 0 0131 . 0 23 8 A S S 2 2 2 M 2 = + = + = S = 0.3662 Compare this to the market’s Sharpe measure: S M = 8/23 = 0.3478 The difference is: 0.0184 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. d. To calculate the exact makeup of the complete portfolio, we first compute `the mean excess return of the optimal risky portfolio and its variance. The risky portfolio beta is given by: β P = w M + (w A × β A ) = 1.0486 + [(–0.0486) × 2.08] = 0.95 E(R P ) = α P + β P E(R M ) = [(–0.0486) × (–16.90%)] + (0.95 × 8%) = 8.42% ( ) 94 . 528 6 . 809 , 21 ) 0486 . 0 ( ) 23 95 . 0 ( ) e ( 2 2 P 2 2 M 2 P 2 P = × + × = σ + σ β = σ % 00 . 23 P = σ 8-9
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Since A = 2.8, the optimal position in this portfolio is: 5685 . 0
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