beta - Strategic Investments

Joachim Kuczynski, 12 September 2025

Beta ( $\beta$ ) measures the sensitivity (covariance) of an asset’s return to the market return. If you have multiple assets in a portfolio, the betas don’t simply add up. Rather they are weighted by their values in the portfolio. This is what I want to prove in this post.

A portfolio beta $\beta_p$ is defined by:

$\beta_p=\frac{cov(r_p,r_m)}{var(r_m)}$

$r_m$ is the market return rate. The return rate of a portfolio, $r_p$ , is the weighted average of its components’ return rates:

$r_p=\sum_{i=1}^{n}\alpha_ir_i$

$\alpha_i$ stands for the relative value share of asset $i$ in the portfolio. The covariance of two random variables is bilinear. Hence we get:

$cov(r_p, r_m)=cov(\sum_{i=1}^{n}\alpha_ir_i,r_m)=\sum_{i=1}^{n}\alpha_icov(r_p, r_m)$

Inserting that into the definition of $\beta_p$ leads to the final result:

$\beta_p=\frac{\sum_{i=1}^{n}\alpha_icov(r_p, r_m)}{var(r_m)}=\sum_{i=1}^{n}\alpha_i\beta_i$

This proves that $\beta_p$ of a portfolio with $n$ assets is the weighted average of its components, weighted by their relative value share in the portfolio.

Tag: beta

Portfolio Beta