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# 23 8 6 16 r r e e w 2 2 m f m 2 σ σ α the negative

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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
Since A = 2.8, the optimal position in this portfolio is: 5685 . 0 94 . 528 8 . 2 01 . 0 42 . 8 y = × × = In contrast, with a passive strategy: 5401 . 0 23 8 . 2 01 . 0 8 y 2 = × × = This is a difference of: 0.0284 The final positions of the complete portfolio are: Bills 1 – 0.5685 = 43.15% M 0.5685 × l.0486 = 59.61% A 0.5685 × (–0.0486) × (–0.6142) = 1.70% B 0.5685 × (–0.0486) × 1.1265 = – 3.11% C 0.5685 × (–0.0486) × (–1.2181) = 3.37% D 0.5685 × (–0.0486) × 1.7058 = – 4.71% 100.00% [sum is subject to rounding error] Note that M may include positive proportions of stocks A through D. 19. 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: α σ 2 (e) α σ 2 (e) α / σ 2 (e) Σα / σ 2 (e) A 1.6 3,364 0.000476 0.3352 C 3.4 3,600 0.000944 0.6648 0.001420 1.0000 The forecast for the active portfolio is: α = (0.3352 × 1.6) + (0.6648 × 3.4) = 2.80% β = (0.3352 × 1.3) + (0.6648 × 0.7) = 0.90 σ 2 (e) = (0.3352 2 × 3,364) + (0.6648 2 × 3,600) = 1,969.03 σ( e ) = 44.37% 8-10

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The weight in the active portfolio is: 0940 . 0 23 / 8 03 . 969 , 1 / 80 . 2 / ) R ( E ) e ( / w 2 2 M M 2 0 = = σ σ α = Adjusting for beta: 0931 . 0 ] 094 . 0 ) 90 . 0 1 [( 1 094 . 0 w ) 1 ( 1 w * w 0 0 = × + = β + = The information ratio of the active portfolio is: A = α / σ (e) =2.80/44.37 = 0.0631 Hence, the square of Sharpe’s measure is: S 2 = (8/23) 2 + 0.0631 2 = 0.1250 Therefore: S = 0.3535 The market’s Sharpe measure is: S M = 0.3478 When short sales are allowed (Problem 18), the manager’s Sharpe measure is
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23 8 6 16 r r E e w 2 2 M f M 2 σ σ α The negative

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