CAPM Beta and Market Return Rate - Strategic Investments

Joachim Kuczynski, 04 November 2025

This article is about an unusual sight on the CAPM beta. It provides a derivation from linear regression with the method of least squares. Often the definition of CAPM beta seems to be very abstract. But from the perspective of this article`s approach it might be clearer.

We assume a data set of market return rates $r_{m,i}$ and equity return rates $r_{E,i}$ . $i$ indicates the $i$ -th of $N$ data pairs. $\overline{r_m}$ and $\overline{r_E}$ are the arithmetic mean values of $r_{m,i}$ and $r_{E,i}$ . With the method of least squares we can make a linear regression that approximates the relationship with a linear function $r_E$ .

(1) $\begin{equation*}r_E=a+br_m\end{equation*}$

The slope $b$ of the linear regression function $r_E$ is given by:

(2) $\begin{equation*}b=\frac{\sum_{i=1}^{N}\left( r_{m,i}-\overline{r_m} \right)\left( r_{E,i}-\overline{r_E} \right)}{\sum_{i=1}^{N}\left( r_{m,i}-\overline{r_m} \right)^2}\end{equation*}$

The covariance of $r_m$ and $r_E$ is given by:

(3) $\begin{equation*}\text{cov}(r_E,r_m)=\frac{1}{N-1}\sum_{i=1}^{N}\left( r_{m,i}-\overline{r_m} \right)\left( r_{E,i}-\overline{r_E} \right)\end{equation*}$

The variance of $r_{m,i}$ is given by:

(4) $\begin{equation*}\text{var}(r_m)=\frac{1}{N-1}\sum_{i=1}^{N}\left( r_{m,i}-\overline{r_m} \right)^2\end{equation*}$

Substituting that in the expression for $b$ leads to:

(5) $\begin{equation*}b=\frac{\text{cov}(r_E,r_m)}{\text{var}(r_m)}\end{equation*}$

This is the same as beta ( $\beta_{r_E}$ ) in the Capital Asset Pricing Model (CAPM). That means that $\beta_{r_E}$ is the slope of the linear approximated relationship of equity return rate $r_E$ and market return rate $r_m$ using the method of least squares. In linear approximation we can state:

(6) $\begin{equation*}b=\beta_{r_E}=\frac{\text{cov}(r_E,r_m)}{\text{var}(r_m)}=\frac{\partial r_E}{\partial r_m}\end{equation*}$

In my point of view this is a very important point to know when doing investment analysis and using CAPM. It provides an additional interpretation of $\beta_{r_E}$ . This point of view is valid for all betas in general.