【笔记】基于斜率最大的资产组合优化

基于《现代投资组合理论与投资分析 9th》；

对基于斜率的资产组合优化的推导，包含MATLAB代码；

基础定义

$$\begin{aligned} x_i&:分配给项目i的比率\\ \bar{r}_i&:项目i的平均回报率\\ r_f&:无风险回报率\\ \sigma_i&: 项目i的风险\\ p&:投资组合 \end{aligned}$$ $$\begin{aligned} \bar{r}_p &= \sum_{i=1}^n x_i\bar{r}_i\\ \sigma_p^2 &= \sum_{i=1}^n x_i^2\sigma_i^2 + \sum_{i=1}^n\sum_{j=1\\j\ne i}^nx_ix_j\sigma_{ij} \end{aligned}$$

推导过程

基于斜率最大，得出目标函数：

$$\begin{aligned} MAX\ \theta &= \frac{\bar{r}_p-r_f}{\sigma_p}\\ \sum_{i=1}^nx_i&=1 \end{aligned}$$

带入基础定义得：

$$\theta = \frac{\sum_{i=1}^n x_i(\bar{r}_i-r_f)}{\sqrt{\sum_{i=1}^n x_i^2\sigma_i^2 + \sum_{i=1}^n\sum_{j=1\\j\ne i}^nx_ix_j\sigma_{ij}}}$$

求最大值即对所有x求导等于0：

$$\begin{aligned} Set \ \lambda &= \sum_{i=1}^n x_i^2\sigma_i^2 + \sum_{i=1}^n\sum_{j=1\\j\ne i}^nx_ix_j\sigma_{ij}\\ \frac{d\theta}{dx_i}&=\frac{(\bar{r}_i-r_f)\lambda^{\frac{1}{2}}-(\sum_{i=1}^n x_i(\bar{r}_i-r_f))·\frac{1}{2}\lambda^{-\frac{1}{2}}·(2x_i\sigma_i^2+2\sum_{j=1\\j\ne i}^nx_j\sigma_{ij})}{\lambda}=0\\ &\Rightarrow (\bar{r}_i-r_f)-(\sum_{i=1}^n x_i(\bar{r}_i-r_f))·\lambda^{-1}·(x_i\sigma_i^2+\sum_{j=1\\j\ne i}^nx_j\sigma_{ij})=0\\ &\Rightarrow (\bar{r}_i-r_f)-(\bar{r}_p-r_f)·\lambda^{-1}·(x_i\sigma_i^2+\sum_{j=1\\j\ne i}^nx_j\sigma_{ij})=0\\ &\Rightarrow (\bar{r}_i-r_f)-\lambda_{new}·(\sum_{j=1}^nx_j\sigma_{ij})=0\\ &\Rightarrow (\bar{r}_i-r_f)=\lambda_{new}·(\sum_{j=1}^nx_j\sigma_{ij})\\ &Set\ \ \lambda_{new}x_i=z_i \\ &\Rightarrow (\bar{r}_i-r_f)=\sum_{j=1}^nz_j\sigma_{ij} \end{aligned}$$

当用于矩阵求解时：

$$R=\left[\begin{matrix}\bar{r}_1\\\bar{r}_2\\\vdots\\ \bar{r}_n\\\end{matrix}\right] ,Z=\left[\begin{matrix}z_1\\z_2\\\vdots\\z_n\\\end{matrix}\right]$$ $$H=\left[\begin{matrix} \sigma_1^2&\sigma_{12}&\cdots&\sigma_{1n}\\ \sigma_{21}&\sigma_2^2&\cdots&\sigma_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ \sigma_{n1}&\sigma_{n2}&\cdots&\sigma_n^2\\ \end{matrix} \right]$$ $$\begin{aligned} R-r_f&=HZ\\ Z &= H^{-1}(R-r_f)\\ x_i &= \frac{z_i}{\sum_{i=1}^nz_i} \end{aligned}$$

---

【MATLAB 代码】

function [X] = portfolio_optimization(R,H,rf)
% Z = H^(-1)(R-rf)
n = length(R);
X = zeros(n,1);
Z = inv(H)*(R-rf);
Z_sum = sum(Z);
for i = 1:1:n
    X(i)=Z(i)/Z_sum;
end
end