2.8 KiB
组合S定义
总收益定义为
[
R_P = 1 + i_F + r_P
]
其中 ( i_F ) 是无风险利率，( r_P ) 是超额收益。
无风险资产的总收益是
[
R_F = 1 + i_F
]
组合 ( S ) 是如下问题的最优解：
[
\min E[R_P^2]
]
其中 ( P ) 可包含风险资产与无风险资产。
构造组合 ( P(w) )
取任意组合 ( P )，构造
[
P(w) : \quad R_{P(w)} = R_S + w (R_P - R_S)
]
定义
[
g_P(w) = E[R_{P(w)}^2]
]
代入得：
[ g_P(w) = E[R_S^2] + 2w E[R_S(R_P - R_S)] + w^2 E[(R_P - R_S)^2] ]
一阶条件
因为 ( S ) 是最小值点，考虑 ( w=0 ) 对应组合 ( S )，应有
[
g_P'(0) = 0
]
计算导数：
[
g_P'(w) = 2 E[R_S(R_P - R_S)] + 2w E[(R_P - R_S)^2]
]
在 ( w=0 ) 时：
[
g_P'(0) = 2 E[R_S(R_P - R_S)] = 0
]
所以
[
E[R_S(R_P - R_S)] = 0 \quad \text{对任意组合 } P
]
即
[ E[R_S R_P] = E[R_S^2] \tag{1} ]
展开协方差形式
[
E[R_S R_P] = \text{Cov}(R_S, R_P) + E[R_S] E[R_P]
]
由 (1) 得：
[ \text{Cov}(R_S, R_P) + E[R_S] E[R_P] = E[R_S^2] \tag{2} ]
对无风险组合 ( F ) 应用 (2)
对 ( P = F )（无风险组合），( R_F = 1 + i_F ) 是常数，所以
[
\text{Cov}(R_S, R_F) = 0
]
于是 (2) 给出：
[
0 + E[R_S] R_F = E[R_S^2]
]
即
[ E[R_S^2] = E[R_S] R_F \tag{3} ]
回到一般组合 ( P )
由 (2)：
[
\text{Cov}(R_S, R_P) + E[R_S] E[R_P] = E[R_S^2]
]
用 (3) 替换 ( E[R_S^2] )：
[
\text{Cov}(R_S, R_P) + E[R_S] E[R_P] = E[R_S] R_F
]
所以
[ \text{Cov}(R_S, R_P) = E[R_S] (R_F - E[R_P]) \tag{4} ]
用超额收益表示
超额收益 ( r_P = R_P - R_F )，所以
[
E[R_P] = R_F + E[r_P]
]
代入 (4) 右端：
[ \text{Cov}(R_S, R_P) = E[R_S] (R_F - (R_F + E[r_P])) = - E[R_S] E[r_P] \tag{5} ]
又
[
R_S = R_F + r_S, \quad R_P = R_F + r_P
]
所以
[
\text{Cov}(R_S, R_P) = \text{Cov}(R_F + r_S, R_F + r_P) = \text{Cov}(r_S, r_P)
]
因为 ( R_F ) 是常数。
于是 (5) 变为：
[ \text{Cov}(r_S, r_P) = - E[R_S] E[r_P] \tag{6} ]
解出 ( E[r_P] )
由 (6)：
[
E[r_P] = \frac{-1}{E[R_S]} \text{Cov}(r_P, r_S)
]
记
[
\phi = \frac{-1}{E[R_S]} = \frac{-1}{E[1 + i_F + r_S]}
]
则
[ E[r_P] = \phi , \text{Cov}(r_P, r_S) \tag{8A-28} ]
证明 (8A-29) 第二个等式
由 (3) 得
[
E[R_S] R_F = E[R_S^2] = \sigma_S^2 + (E[R_S])^2
]
所以
[
\sigma_S^2 = E[R_S] R_F - (E[R_S])^2 = E[R_S] (R_F - E[R_S])
]
定义 ( f_S = E[r_S] = E[R_S] - R_F )，则
[
R_F - E[R_S] = - f_S
]
于是
[
\sigma_S^2 = E[R_S] (-f_S)
]
即
[
E[R_S] = - \frac{\sigma_S^2}{f_S}
]
因此
[
\phi = \frac{-1}{E[R_S]} = \frac{-1}{-\sigma_S^2 / f_S} = \frac{f_S}{\sigma_S^2}
]
得证。