Chapter 8 - Index Models
12. Note that the correlation is the square root of R2:ρ?R2
Cov(rA,rM)???A?M?0.201/2?31.30?20?280Cov(rB,rM)???B?M?0.121/2?69.28?20?480
13. For portfolio P we can compute:
σP = [(0.62 × 980) + (0.42 × 4800) + (2 × 0.4 × 0.6 × 336)]1/2 = [1282.08]1/2 = 35.81% βP = (0.6 × 0.7) + (0.4 × 1.2) = 0.90
22σ2(eP)?σ2?(0.902?400)?958.08 P?βPσM?1282.08Cov(rP,rM ) = βPσ2 400=360 M=0.90 ×
This same result can also be attained using the covariances of the individual stocks with the market:
Cov(rP,rM ) = Cov(0.6rA + 0.4rB, rM ) = 0.6 × Cov(rA, rM ) + 0.4 × Cov(rB,rM )
= (0.6 × 280) + (0.4 × 480) = 360
14. Note that the variance of T-bills is zero, and the covariance of T-bills with any asset
is zero. Therefore, for portfolio Q:
222σQ?wPσP?wMσ2M?2?wP?wM?Cov(rP,rM)??1/2?(0.52?1,282.08)?(0.32?400)?(2?0.5?0.3?360)??1/2
?21.55%?Q?wP?P?wM?M?(0.5?0.90)?(0.3?1)?(0.20?0)?0.75
222σ2(eQ)?σQ?βQσM?464.52?(0.752?400)?239.52
Cov(rQ,rM)?βQσ2M?0.75?400?300
15. a.
Beta Books adjusts beta by taking the sample estimate of beta and averaging it with 1.0, using the weights of 2/3 and 1/3, as follows:
adjusted beta = [(2/3) × 1.24] + [(1/3) × 1.0] = 1.16
b.
If you use your current estimate of beta to be βt–1 = 1.24, then
βt = 0.3 + (0.7 × 1.24) = 1.168
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Chapter 8 - Index Models
16. For Stock A:
?A?rA?[rf??A?(rM?rf)]?.11?[.06?0.8?(.12?.06)]?0.2%
For stock B:
?B?rB?[rf??B?(rM?rf)]?.14?[.06?1.5?(.12?.06)]??1%
Stock A would be a good addition to a well-diversified portfolio. A short position in Stock B may be desirable. 17. a.
Alpha (α)
αi = ri – [rf + βi × (rM – rf ) ]
αA = 20% – [8% + 1.3 × (16% – 8%)] = 1.6% αB = 18% – [8% + 1.8 × (16% – 8%)] = – 4.4% αC = 17% – [8% + 0.7 × (16% – 8%)] = 3.4% αD = 12% – [8% + 1.0 × (16% – 8%)] = – 4.0%
Expected excess return
E(ri ) – rf 20% – 8% = 12% 18% – 8% = 10% 17% – 8% = 9% 12% – 8% = 4%
Stocks A and C have positive alphas, whereas stocks B and D have negative alphas.
The residual variances are:
?2(eA ) = 582 = 3,364 ?2(eB) = 712 = 5,041 ?2(eC) = 602 = 3,600 ?2(eD) = 552 = 3,025
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Chapter 8 - Index Models
b.
To construct the optimal risky portfolio, we first determine the optimal active portfolio. Using the Treynor-Black technique, we construct the active portfolio:
A B C D Total a ?2(e)0.000476 –0.000873 0.000944 –0.001322 –0.000775 a / ?2(e) Sa / ?2(e)–0.6142 1.1265 –1.2181 1.7058 1.0000 Be unconcerned with the negative weights of the positive α stocks—the entire active position 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.
?2(e) = [(–0.6142)2×3364] + [1.12652×5041] + [(–1.2181)2×3600] + [1.70582×3025]
= 21,809.6 ? (e) = 147.68%
The levered position in B [with high ?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:
w0??/?2(e)2[E(rM)?rf]/?M??.1690/21,809.6??0.05124 2.08/23The negative position is justified for the reason stated earlier. The adjustment for beta is:
w*?w0?0.05124???0.0486
1?(1?β)w01?(1?2.08)(?0.05124)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
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Chapter 8 - Index Models
c.
To calculate the Sharpe ratio 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 =
? = –16.90/147.68 = –0.1144 ?(e)A2 = 0.0131
Hence, the square of the Sharpe ratio (S) of the optimized risky portfolio is:
?8?2S2?SM?A2????0.0131?0.1341
?23?S = 0.3662
Compare this to the market’s Sharpe ratio:
SM = 8/23 = 0.3478 ? A difference of: 0.0184
The only moderate improvement in performance results from only a small position taken in the active portfolio A 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: βP = wM + (wA × βA ) = 1.0486 + [(–0.0486) ? 2.08] = 0.95
E(RP) = αP + βPE(RM) = [(–0.0486) ? (–16.90%)] + (0.95 × 8%) = 8.42%
222σ2)2?(?0.04862)?21,809.6?528.94 P?βPσM?σ(eP)?(0.95?232??σP?23.00%
Since A = 2.8, the optimal position in this portfolio is:
y?8.42?0.5685
0.01?2.8?528.948?0.5401?A difference of: 0.0284 20.01?2.8?23In contrast, with a passive strategy:
y?The final positions are (M may include some of stocks A through D): 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 (subject to rounding error) 100.00%
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Chapter 8 - Index Models
18. 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 A C 1.6 3.4 2?(e) 3,364 3,600 a 2 ?(e)0.000476 0.000944 0.001420 a / ?2(e) Sa / ?2(e)0.3352 0.6648 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
?2(e) = (0.33522 × 3,364) + (0.66482 × 3,600) = 1,969.03 σ(e) = 44.37%
The weight in the active portfolio is:
α/σ2(e)2.80/1,969.03w0???0.0940 2E(RM)/σ28/23MAdjusting for beta:
w*?w00.094??0.0931
1?(1??)w01?[(1?0.90)?0.094]The information ratio of the active portfolio is:
A??2.80??0.0631 ?(e)44.372Hence, the square of the Sharpe ratio is:
?8?S????0.06312?0.1250
?23?2Therefore: S = 0.3535
The market’s Sharpe ratio is: SM = 0.3478
When short sales are allowed (Problem 17), the manager’s Sharpe ratio is higher (0.3662). The reduction in the Sharpe ratio is the cost of the short sale restriction.
The characteristics of the optimal risky portfolio are:
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