How do I calculate the weighted variance, $\sigma^2$, of a set of $N$ random variables considering their correlation $\rho$? [duplicate]

Question

In a finance textbook of mine, there is an equation for calculating the variance $\sigma^2$ of a portfolio of two risky assets (i.e. random variables) $X$ and $Y$ by considering the correlation $\rho$ of the assets. I do not know if this equation is generalized, and no derivation is provided.

The textbook equation: $\sigma^2 = x^2_1\sigma^2_1 + x^2_2 \sigma^2_2 + 2x_1x_2\rho \sigma_1 \sigma_2$

$x_1$ and $x_2$ in the equation refer to the assets' weight in the portfolio.

So, how can I calculate the variance for $N$ number of weighted random variables also consider their correlation?

Thank you for your help!

Answer 1

From the fact that your question stems from a finance textbook, I assume that you are dealing with Markowitz's portfolio theory. In most books, the author just focuses on the 2-asset case, but this can easily be generalized to $n$ risky assets by using matrix algebra.

Formally, let $\boldsymbol{\omega}=(\omega_1,\dots,\omega_n)^\top$ be a vector of portfolio weights and let $\boldsymbol{R}=(R_1,\dots,R_n)^\top$ be the vector of returns. I.e., $R_i$ is the return of asset $i$, which is a random variable, an $\omega_i$ is the corresponding weight, which is a scalar. The covariance matrix of $\boldsymbol{R}$ is given by: \begin{align} \boldsymbol{\Sigma}&= \begin{pmatrix} Var(R_1) & Cov(R_1,R_2) & \dots &Cov(R_1,R_n) \\ Cov(R_2,R_1) & Var(R_2) & \dots &Cov(R_2,R_n) \\ \vdots & \vdots & \ddots & \vdots \\ Cov(R_n,R_1) & Cov(R_n,R_2) & \dots & Var(R_n) \end{pmatrix} =\begin{pmatrix} \sigma_1^2 & \sigma_{12} & \dots & \sigma_{1n} \\ \sigma_{21} & \sigma_2^2 & \dots &\sigma_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ \sigma_{n1} & \sigma_{n2} & \dots & \sigma_n^2 \end{pmatrix} \end{align} Then, $$ R_{\boldsymbol{\omega}}=\boldsymbol{\omega}^\top\boldsymbol{R}=\sum_{i=1}^n\omega_iR_i $$ is the return of your portfolio. Basically, you want to calculate the variance of $R_{\boldsymbol{\omega}}$ and the easiest way to do this is as follows: The expected return of the portfolio is given by: \begin{align} E(R_{\boldsymbol{\omega}})=E(\boldsymbol{\omega}^\top \boldsymbol{R})=\boldsymbol{\omega}^\top E(\boldsymbol{R})=\boldsymbol{\omega}^\top \boldsymbol{\mu} \end{align} The variance $\sigma_{\boldsymbol{\omega}}^2$ of the portfolio return is given by: \begin{align*} \sigma_{\boldsymbol{\omega}}^2&=E([R_{\boldsymbol{\omega}}-E(R_{\boldsymbol{\omega}})][R_{\boldsymbol{\omega}}-E(R_{\boldsymbol{\omega}})]^\top) \\ &=E([\boldsymbol{\omega}^\top\boldsymbol{R}-\boldsymbol{\omega}^\top \boldsymbol{\mu}][\boldsymbol{\omega}^\top\boldsymbol{R}-\boldsymbol{\omega}^\top \boldsymbol{\mu}]^\top) \\ &=E(\boldsymbol{\omega}^\top[\boldsymbol{R}-\boldsymbol{\mu}][\boldsymbol{\omega}^\top[\boldsymbol{R}-\boldsymbol{\mu}]]^\top) \\ &=E(\boldsymbol{\omega}^\top[\boldsymbol{R}-\boldsymbol{\mu}][\boldsymbol{R}-\boldsymbol{\mu}]^\top\boldsymbol{\omega}) \\ &=\boldsymbol{\omega}^\top E([\boldsymbol{R}-\boldsymbol{\mu}][\boldsymbol{R}-\boldsymbol{\mu}]^\top)\boldsymbol{\omega} \\ &= \boldsymbol{\omega}^\top \boldsymbol{\Sigma}\boldsymbol{\omega}\\ &=\sum_{i=1}^n\sum_{i=1}^n\omega_i\omega_jCov(R_i,R_j) \end{align*} Thus, to implement this in code, just specify a vector of weights, take your covariance matrix and calculate $\boldsymbol{\omega}^\top \boldsymbol{\Sigma}\boldsymbol{\omega}$.

I hope that this answers your question, if not, do not hesitate to ask further questions.

Answer 2

It's nothing but the application of variance of linear combination of random variables:

$$\operatorname{Var}\left(\sum_{i=1}^n a_iX_i\right) =\sum_{i=1}^n a_i^2\operatorname{Var}(X_i)+ 2\mathop{\sum_{i=1}^n\sum_{j=1}^n} \limits_{i<j} a_ia_j\operatorname{Cov}(X_i,X_j).$$ The proof can be seen here.
The present case is for $n =2.$ Also $\operatorname{Cov}(X_1, X_2) = \rho\sigma_1\sigma_2.$