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 $r_{m}$ 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.

Joachim Kuczynski, 01 November 2025

Net Income

With the abbreviations $\mu$ … net income, S … sales, F … fix expenses, V … variable expenses, A … depreciation and amortization, T … taxes, I … interests for debt and X … tax shield we can define net income by:

(1) $\begin{equation*}\mu=S-F-V-A-T-I+X\end{equation*}$

Let $r_D$ be the interest rate for debt and $r_t$ be the incremental tax rate. We assume that we get full tax shield of $X=Dr_Dr_t$ . Taxes $T$ are paid on EBIT, that means $T=(S-F-V-A ) r_t$ . $Z$ =equity+debt is the asset or enterprise value and d is the debt ratio, $d=D/Z$ . Substituting that in the previous expression we obtain:

(2) $\begin{equation*}\mu=(S-F-V-A)(1-r_t)-dZr_D+dZr_Dr_t\end{equation*}$

Summarizing the terms leads to:

(3) $\begin{equation*}\mu=(S-F-V-A-dZr_D)(1-r_t)\end{equation*}$

Beta of a weighted sum is the weighted sum of the components’ betas, shown in the post “Portfolio Beta”. Thus we get an equation for the betas:

(4) $\begin{equation*}\mu\beta_\mu=(S\beta_S-F\beta_F-V\beta_V-A\beta_A-dZr_D\beta_D)(1-r_t)\end{equation*}$

We assume that fix expenses (F), depreciation, amortization (A) and debt (D) have no correlation to the market return rate, $\beta_F=0$ , $\beta_A=0$ , $\beta_D=0$ . Variable expenses should have the same correlation to market development as sales, that gives $\beta_V=\beta_S$ . We obtain:

(5) $\begin{equation*}\mu \beta_\mu=\left( S-V \right) \left( 1-r_t \right) \beta_S\end{equation*}$

Substituting $\mu=\left( S-F-V-A -dZr_D \right) \left( 1-r_t \right)$ leads to:

(6) $\begin{equation*}\beta_\mu= \frac{ S-V }{ S-F-V-A -dZr_D }\beta_S\end{equation*}$

Rearranging the terms shows:

(7) $\begin{equation*}\beta_\mu=\left( 1+ \frac{ F+A+dZr_D }{ S-F-V-A -dZr_D } \right) \beta_S\end{equation*}$

Return on Equity

Return on equity measures relative equity increase. It is defined by:

(8) $\begin{equation*}\text{ROE}=\frac{\text{net income}}{\text{equity}}=\frac{\mu}{(1-d)Z}\end{equation*}$

With the substitution of $\mu$ we obtain:

(9) $\begin{equation*}\text{ROE}=\frac{\left( S-F-V-A-dZr_D \right)(1-r_t) }{(1-d)Z}\end{equation*}$

We want to link the beta of ROE to the beta of net income. We take the definition of $\beta$ with its bilinearity of covariance and get:

(10) $\begin{equation*}\beta_{\text{ROE}}=\frac{cov(r_\frac{\mu}{E}, r_m)}{var(r_m)}=\frac{\frac{1}{E}cov(r_\mu, r_m)}{var(r_m)}=\frac{1}{E}\beta_\mu=\frac{1}{(1-d)Z}\beta_\mu\end{equation*}$

Hence we get $\beta_{\text{ROE}}$ as function of $\beta_S$ :

(11) $\begin{equation*}\beta_{\text{ROE}}=\left(\frac{ S-V }{ S-F-V-A -dZr_D}\right)\frac{1}{(1-d)Z}\beta_S\end{equation*}$

(12) $\begin{equation*}\beta_{\text{ROE}}=\left( 1+ \frac{ F+A+dZr_D }{ S-F-V-A -dZr_D}\right)\frac{1}{(1-d)Z}\beta_S\end{equation*}$

This is the relationship of the ROE beta and the sales beta. It depends on several variables, but especially on fixed expenses $F+A+dZr_D$ .

Releveraging Equity Return Rate and WACC

An equity investor wants to know which return rate market provides at the same risk (volatility) level as our investigated investment. The WACC after taxes of the appropriate market portfolio is given by:

(13) $\begin{equation*}\text{WACC'}=d'r_D'\left( 1-r_t' \right)+(1-d')r_{E'}\end{equation*}$

I indicate all market portfolio parameters with a line on the top. $r_{E'}$ ist the required return rate of an incremental equity investor. This is the return rate of the completely diversified market portfolio of the appropriate industry segment. The Capital Asset Pricing Model (CAPM) states that $r_{E'}$ can be approximated by a linear function of $\beta_{E'}$ by a given market return rate $r_m$ :

(14) $\begin{equation*}r_{E'}=r_f+\beta_{E'}\left( r_m-r_f \right); \beta_{E'}=\frac{cov(r_{E'},r_m)}{var(r_m)}\end{equation*}$

$\beta_{E'}$ is the average equity $\beta$ of an market portfolio representing the investigated investment. It is based on an averaged financial and operating leverage of the market portfolio. If these parameters do not match capital and cost structure of the considered investment, you have to “releverage” $\beta_{E'}$ .

Operating leverage: We have a look at the relationship of $\beta_{E'}$ and $\beta_{S'}$ :

(15) $\begin{equation*}\beta_{\text{E'}}=\left(\frac{ S-V' }{ S'-F'-V'-A' -d'Z'r_D'}\right)\frac{1}{(1-d')Z'}\beta_{S'}\end{equation*}$

$\beta_{E'}$ is provided by an official data collection in most cases. With the other data from the market portfolio we can calculate $\beta_{S'}$ :

(16) $\begin{equation*}\beta_{\text{S'}}=\left(\frac{ S'-F'-V'-A' -d'Z'r_{D'} }{S'-V'}\right)(1-d')Z'\beta_{E'}\end{equation*}$

$\beta_{S'}$ is the linear approximated change of sales because of a change in market return rate $r_m$ . $\beta_{S'}$ does not depend on $F$ , $V$ , $Z$ , $D$ , $d$ and $r_D$ . $\beta_{S'}$ is the same for all combinations of these parameters, that means $\beta_S=\beta_{S'}$ . Now we take the parameters of the investment, to which we want to adjusted $\beta_E$ . With the $\beta_{S'}$ from the previous equation we obtain the adjusted $\beta_E$ :