【推导】Modern Portfolio Theory

构造拉格朗日函数公式研究马科维茨投资模型；

目前包含：

1. 求最小 sigma_p；

2. 最优组合下每个资产的权重；

3. 加入 risk free security 后的权重；

前情提要

原模型等价于固定预期收益率为常数、权重之和为1的限制条件下求最小方差的线性规划问题如下：

$$\begin{align} 目标函数：&min\sum_{i=1}^n\sum_{j=1}^nw_iw_jCov(r_i,r_j)\\ &s.t.\begin{cases} \sum_{i=1}^nr_iw_i=E(R)\\ \sum_{i=1}^nw_i=1 \end{cases}\\ \end{align}$$

Part 1：

构造的拉格朗日函数公式如下：

$$\mathcal{L}(w,\lambda_1,\lambda_2)=\sum_{i=1}^n\sum_{j=1}^nw_iw_jCov(r_i,r_j) -\lambda_1(\sum_{i=1}^nr_iw_i-E(R))-\lambda_2(\sum_{i=1}^nw_i-1)$$

对$w_i,\lambda_1,\lambda_2$分别求导取0，满足条件如下:

$$\begin{aligned} \frac{\partial \mathcal{L}}{\partial w_i}&=\sum_{j=1}^nw_jCov(r_i,r_j) -\lambda_1r_i-\lambda_2=0\\ \frac{\partial \mathcal{L}}{\partial \lambda_1}&=\sum_{i=1}^nr_iw_i-E(R)=0\\ \frac{\partial \mathcal{L}}{\partial \lambda_2}&=\sum_{i=1}^nw_i-1=0\\ \end{aligned}$$

整合$w_j => w,r_i => R,Cov(r_i,r_j)=>\Sigma$

$$\begin{cases} &2\Sigma w -\lambda_1R-\lambda_21_n=0\\ &w^TR-E(R)=0\\ &e^T1_n-1=0 \end{cases}$$

假设收益率协方差可逆

$$w =\frac{1}{2}\Sigma^{-1}(\lambda_1R+\lambda_21_n) \tag{1}$$ $$E(R)=w^TR \tag{2}$$ $$e^T1_n=1 \tag{3}$$

将(2),(3)代入(1)

$$\begin{cases} E(R)=\frac{1}{2}(\lambda_1R^T\Sigma^{-1}R+\lambda_21_n^T\Sigma^{-1}R)\\ 1 = \frac{1}{2}(\lambda_1R^T\Sigma^{-1}1_n+\lambda_21_n^T\Sigma^{-1}1_n) \end{cases}$$

转为矩阵格式

$$\begin{bmatrix}E(R)\\1\end{bmatrix} =\frac{1}{2}\begin{bmatrix}R^T\Sigma^{-1}R&1_n^T\Sigma^{-1}R\\R^T\Sigma^{-1}1_n&1_n^T\Sigma^{-1}1_n\end{bmatrix} \begin{bmatrix}\lambda_1\\\lambda_2\end{bmatrix}\tag{4}$$

由于目标函数$=>min\quad\sigma_p^2$

$$\begin{aligned} \sigma_p^2&=w^T\Sigma w\\ &=\frac{1}{4}(\lambda_1R^T\Sigma^{-1}+\lambda_21_n^T\Sigma^{-1})\Sigma\Sigma^{-1}(\lambda_1R+\lambda_21_n) \\ &=\frac{1}{4}(\lambda_1^2R^T\Sigma^{-1}R+2\lambda_1\lambda_2R^T\Sigma^{-1}1_n+\lambda_2^21_n^T\Sigma^{-1}1_n)\\ &=\frac{1}{4}\begin{bmatrix}\lambda_1&\lambda_2 \end{bmatrix}\begin{bmatrix}R^T\Sigma^{-1}R&1_n^T\Sigma^{-1}R\\R^T\Sigma^{-1}1_n&1_n^T\Sigma^{-1}1_n\end{bmatrix} \begin{bmatrix}\lambda_1\\\lambda_2\end{bmatrix} \end{aligned}\tag{5}$$

联立(4),(5)得

$$\begin{bmatrix}E(R)&1\end{bmatrix}\begin{bmatrix}R^T\Sigma^{-1}R&1_n^T\Sigma^{-1}R\\R^T\Sigma^{-1}1_n&1_n^T\Sigma^{-1}1_n\end{bmatrix}^{-1}\begin{bmatrix}E(R)\\1\end{bmatrix}=\sigma_p^2$$

设 $a = R^T\Sigma^{-1}R,b = R^T\Sigma^{-1}1_n,c = 1_n^T\Sigma^{-1}1_n$ , 化简得

$$\sigma_p^2=\frac{cE(R)^2-2bE(R)+a}{ac-b^2}$$

function [P_return] = optimal_portfolio_e(mu,Sigma,sigma)
n = size(Sigma,1); 
V = Sigma^(-1); 
e = ones(n,1); 
A = mu'*V*mu;
B = e'*V*mu;
C = e'*V*e;
P_return = fzero(@(x) C*x^2-2*B*x+A-sigma^2*(A*C-B^2),1);
end

Part 2：

从矩阵形式得到

$$\begin{align} \begin{bmatrix}2\\2E(R)\end{bmatrix} &=\begin{bmatrix}1_n^T\Sigma^{-1}R&1_n^T\Sigma^{-1}1_n\\ R^T\Sigma^{-1}R&R^T\Sigma^{-1}1_n\end{bmatrix} \begin{bmatrix}\lambda_1\\ \lambda_2\end{bmatrix}\\ \begin{bmatrix}\lambda_1\\ \lambda_2\end{bmatrix} &=\begin{bmatrix}1_n^T\Sigma^{-1}R&1_n^T\Sigma^{-1}1_n\\R^T\Sigma^{-1}R&R^T\Sigma^{-1}1_n\end{bmatrix}^{-1}\begin{bmatrix}2\\2E(R)\end{bmatrix} \end{align}$$

代入$w$得

$$w = \begin{bmatrix} \Sigma^{-1}R&\Sigma^{-1}1_n\end{bmatrix}\begin{bmatrix}1_n^T\Sigma^{-1}R&1_n^T\Sigma^{-1}1_n\\R^T\Sigma^{-1}R&R^T\Sigma^{-1}1_n\end{bmatrix}^{-1}\begin{bmatrix}1\\E(R)\end{bmatrix}$$

求得$w^T\Sigma w$

Part 3：

加入收益率是 $r_f$ 的 risk free security 之后

$$\begin{align} &目标函数：min\quad w^T\Sigma w\\ &s.t.\quad (1-\sum_{i=1}^n)r_f+w^TR=E(R) \end{align}$$

Solution

$$\sigma_p^2=\frac{(E(R)-r_f)^2}{(R-r_f1_n)^T\Sigma^{-1}(R-r_f1_n)}$$ $$\xi=\frac{\sigma_p^2}{E(R)-r_f}$$ $$w=\xi\Sigma^{-1}(E(R)-r_f1_n)$$

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MATLAB代码：

function[w] = optimal_portfolio_1(R,Sigma,ER)
% no risk-free asset
n = size(Sigma,1); % the number of risky shares
V = Sigma^(-1); %inverse of  variance-covariance matrix
e = ones(n,1); % unit vector, column
A = e'*V*R;
B = R'*V*R;
C = e'*V*e;
K = [A C;  B A];
LM = K^(-1)*[1; ER];
w = [V*R, V*e]*LM; % optimal weight

function[w] = optimal_portfolio_2(mu,Sigma,rf,p)
% with risk free asset
n = size(Sigma,1); % the number of risky shares
V = Sigma^(-1); %inverse of  variance-covariance matrix
e = ones(n,1); % unit vector, column

sigma2min = (p - rf)^2/((mu-rf*e)'*V*(mu-rf*e));
xi = sigma2min/(p - rf);% tangency

w = xi*V*(mu-rf*e); % optimal weight

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Reference

1. 投资组合理论学习之马科维茨投资模型（一）
2. 矩阵求导公式的数学推导（矩阵求导——基础篇）