系统方差为：?2?eP?=wA2?2?eA?+wB2?2?eB??wf2?2?ef??
0.302?0.302?0.452?0.402?0.252?02=4.05%
其中?2?eA?，?2?eB?是股票A和B的非系统方差，?2?ef?是国库券的非系统方差，
12值为零。因此，资产组合的标准残差为：??eP???4.05%??20.12%
22整个资产组合的方差为：?P2=0.78?0.22?4.05%?6.99%，标准差为：26.45%
??
8. 基于当前的红利收益和预期的增长率，股票A和股票B的期望收益率分别为11%和14%，股票A的贝塔值为0.8，股票B的贝塔值为1.5。当前国库券的收益率为6%，标准普尔500股票的期望收益率为12%。股票A的年度标准差为10%，股票B的年度标准差为11%。 a. 如果投资者目前持有充分分散化的资产组合，投资者愿意增加哪种股票的持有量？ b. 如果投资者只能投资于债券与这两种股票中的一种，投资者会如何选择？请用图标或定量分析说明股票的吸引力所在。
答：a. 指数模型：E(rA)-rf??A??A(E(rM)?rf)] 对于股票A，其阿尔法值为：
?A?E(rA)-[rf??A(E(rM)?rf)]?11%?[6%?0.8?(12%?6%)]?0.2%
同理可得股票B的阿尔法值为：?B?14%?[6%?1.5?(12%?6%)]??1% 将股票A加入充分多样化的资产组合是不错的选择，而且持有股票B的空头也是合理的。 b. 每只股票的夏普比率为：
11%?6%?0.5010%
14%?6%SB??0.7311%SA?当选择只投资于国库券或者这两只股票中的一只时，股票B是相对更优的选择。 下面是9-14题的数据，假设对股票A,B的指数模型是根据以下结果按照超额收益估计的结果：
RA?3%?0.7RM?eA
RB??2%?1.2RM?eB
22?M?20%;RA?0.2;RB?0.12
9每种股票的标准差是多少？ 见书??2A22?A?M2RA=0.098，标准差31.30%，B类似。
10计算每种股票的方差中的系统风险和企业的特有风险。 10系统风险?2A?=0.0196，特有风险=?????2M2A2A2M22?A?MR2A22-?A?M=0.0784，B类似。
11这两种股票之间的协方差和相关系数是多少？
211cov(rA,rB)??A?B?M=0.0336
?AB?cov(rA,rB)?A?B=0.155
12每种股票与市场指数间的协方差各是多少？
222212cov(=0.028,cov(=0.048 rA,rM)??A?M?M??A?MrB,rM)??B?M?M??B?M13如果把60%的资金投入到股票A，40%投入到股票B，重做9-12题 13求?P?(x1?1?x2?2?2x1x2?12)22221/2=35.81%
?P?x1?1?x2?2=0.90
222?2(eP)??P??P?M?0.0958
2cov(rP,rM)??M?M?0.90?0.22?0.036
14如果50%的资金按第13题比例投资，30%投入到市场指数，20%投资于短期国库券，重做13题。 14.
注意国库券的方差为0，它和任何资产的协方差是0，因此投资组合Q：
222?Q?w2P?P?wM?M?2?wP?wM?Cov(rP,rM)??1/2?(0.5?1,282.08)?(0.3?400)?(2?0.5?0.3?360)?22?1/2
?21.55%?Q?wP?P?wM?M?(0.5?0.90)?(0.3?1)?0?0.75
222?2(eQ)??Q??Q?M?464.52?(0.752?400)?239.52
Cov(rQ,rM)??Q?2M?0.75?400?300
15假设投资组合经理根据宏观和微观预测，得到如下一个输入表： 微观预测 资产 股票1 股票2 股票3 股票4 宏观和微观预测 资产 短期国库券 消极投资组合 期望收益% 8 16 标准差% 0 23 期望收益% 20 18 17 12 贝塔 1.3 1.8 0.7 1.0 残值标准差% 58 71 60 55 1计算各股票的期望超额收益、阿尔法值、残值方差是多少？ 2构建最优投资组合。
3最优风险投资组合的夏普值？积极投资组合对它的贡献是多少？
4假设投资者的风险厌恶系数为A=2.8，对短期国库券和消极股票组合投资比例是多少？
15答案15. a.
Alpha (?)
??i = E(ri )– [rf + ?i(E(rM )– rf ) ]
? = 20% – [8% + 1.3(16% – 8%)] = 1.6%
1
Expected excess return
E(ri ) – rf 20% – 8% = 12% 18% – 8% = 10% 17% – 8% = 9% 12% – 8% = 4%
? = 18% – [8% + 1.8(16% – 8%)] = – 4.4%
2
? = 17% – [8% + 0.7(16% – 8%)] = 3.4%
3
? = 12% – [8% + 1.0(16% – 8%)] = – 4.0%
4
Stocks A and C have positive alphas, whereas stocks B and D have negative alphas.
The residual variances are:
?2(e1 ) = 582 = 3,364 ?2(e2) = 712 = 5,041
?2(e3) = 602 = 3,600 ?2(e4) = 552 = 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:
1 2 3 4 Total ? ?2(e)0.000476 –0.000873 0.000944 –0.001322 –0.000775 ? / ?2(e) ?? ? ?2(e)–0.6142 1.1265 –1.2181 1.7058 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:
? A= [–0.6142 ? 1.6] + [1.1265 ? (– 4.4)] – [1.2181 ? 3.4] + [1.7058 ? (– 4.0)] = –16.90%
???= [–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(eA) = [(–0.6142)2 ? 3364] + [1.12652 ? 5041] + [(–1.2181)2 ? 3600] + [1.70582 ? 3025]
= 21,809.6 ?? eA ? = 147.68%
Here, again, the levered position in stock 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:
?/?2(ep)?16.90/21,809.6w0????0.05124 2[E(rM)?rf]/?M8/232The negative position is justified for the reason stated earlier. The adjustment for beta is:
w*A?w0?0.05124???0.0486
1?(1??A)w01?(1?2.08)(?0.05124)Since w*A 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 = ? /??e)= –16.90/147.68 = –0.1144 A2 = 0.0131
Hence, the square of Sharpe’s measure (S) of the optimized risky portfolio is:
?8?S?S?A????0.0131?0.1341
?23?22M22S = 0.3662
Compare this to the market’s Sharpe measure:
SM = 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:
?P = wM ?M + wA ? ?A = 1.0486 + [(–0.0486) ? 2.08] = 0.95
E(RP) = ? P + ?PE(RM) = [(–0.0486) ? (–16.90%)] + (0.95 ? 8%) = 8.42%
222?2)2?(?0.04862)?21,809.6?528.94 P??P?M??(eP)?(0.95?23???P?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
0.01?2.8?232In contrast, with a passive strategy:
y?