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, 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 $F+A+dZr_D$ .

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

Operating Leverage

Net Income

Return on Equity

Portfolio Beta