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# Below the market average β of 10 the three estimates

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below the market average β of 1.0. The three estimates of XYZ’s β vary significantly among the three sources, ranging as high as 1.45 for the weekly data over the most recent two years. One could infer that XYZ’s β for the future might be well above 1.0, meaning it might have somewhat greater systematic risk than was 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 will add more to total volatility. 13. For Stock A: α A = r A [ r f + β A (r M r f )] = 11 [6 +0.8(12 6)] = 0.2% For stock B: α B = 14 [6 + 1.5(12 6)] = 1% Stock A would be a good addition to a well-diversified portfolio. A short position in Stock B may be desirable.

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8-7 14. The R 2 of the regression is: 0.70 2 = 0.49 Therefore, 51% of total variance is unexplained by the market; this is nonsystematic risk. 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
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

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below the market average β of 10 The three estimates of...

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