P193 命题3

P193 命题3/命题3.md

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+
+### 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。
+