ActivePortfolioManagement/P193 命题3/命题3.md

### 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。