In general, for $n$ assets, we can combine them to the overall return $\mu$ and risk $\sigma$: $$ \left{ \begin{array}{rcl} \mu &=& \boldsymbol{\mu}^T \mathbf{w}\ \sigma^2 &=& \mathbf{w}^T \Sigma \mathbf{w} \end{array} \right. $$ where $\mathbf{w}=[w_1, \dots, w_n]^T$, $\boldsymbol{\mu}=[\mu_1, \dots, \mu_n]^T$, and $\Sigma$ is the covariance matrix of these $n$ assets.

基本數學 Mean and variance for random variables $X$ and $Y$ DefinitionMean: $\mu_X = E(X)$ Variance: $\sigma_X^2 = V(X) \triangleq E((X-\mu_X)^2) = E(X^2)-\mu_X^2$ Covariance: $\sigma_{XY}=\sigma_{YX}=E((X-\mu_X)(Y-\mu_Y)) = E(XY)-\mu_X\mu_Y$ Basic formulas for a single variable $E(aX)=aE(X) \Rightarrow \mu_{aX}=a\mu_X$

In general, for $n$ assets, we can combine them to the overall return $\mu$ and risk $\sigma$: $$ \left{ \begin{array}{rcl} \mu &=& \boldsymbol{\mu}^T \mathbf{w}\ \sigma^2 &=& \mathbf{w}^T \Sigma \mathbf{w} \end{array} \right. $$ where $\mathbf{w}=[w_1, \dots, w_n]^T$, $\boldsymbol{\mu}=[\mu_1, \dots, \mu_n]^T$, and $\Sigma$ is the covariance matrix of these $n$ assets.

In general, for $n$ assets, we can combine them to the overall return $\mu$ and risk $\sigma$: $$ \left{ \begin{array}{rcl} \mu &=& \boldsymbol{\mu}^T \mathbf{w}\ \sigma^2 &=& \mathbf{w}^T \Sigma \mathbf{w} \end{array} \right. $$ where $\mathbf{w}=[w_1, \dots, w_n]^T$, $\boldsymbol{\mu}=[\mu_1, \dots, \mu_n]^T$, and $\Sigma$ is the covariance matrix of these $n$ assets.

Portfolio optimization The goal of portfolio optimization (PO) is to maximize returns and minimize the risk of a portfolio of assets according to modern portfolio theory (Markowitz, 1952) Given two assets as follows: $$ \left{ \begin{array}{l} \text{Asset 1: } \mu=\mu_1, \sigma^2=\sigma_1^2\ \text{Asset 2: } \mu=\mu_2, \sigma^2=\sigma_2^2 \end{array}

References Portfolio Optimization From Scratch Direct straightforward math derivation Good for code implementation Portfolio Optimization Methods With python code