Joachim Kuczynski, 22 October 2025

Introduction

Equity investors (shareholders) are mainly interested in the returns of their investments and the associated risk. On the one hand the returns depend on the market development. On the other hand there are company characteristics like fixed costs and debt that influence the returns and leverage the market dependency. In this post I try to figure out these leverage effects based on P&L and balance sheet figures. In previous articles I had a look at the leverage effects from the cash flow and NPV perspective. But in reality P&L and balance sheet figures are better available and an investigation from their point of view seems to be rational.

Leverage of Net Income

Net income can be considered as valid figure for an equity development. Keeping debt constant, net income increases equity. With the abbriviations P … 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:

$P=S-F-V-A-T-I+X$

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 $T=(S-F-V-A ) r_t$ . Substituting that in the previous expression we obtain:

$P=(S-F-V-A)(1-r_t)-Dr_D+Dr_Dr_t$

Summarizing the terms leads to:

$P=(S-F-V-A-Dr_D)(1-r_t)$

The return rate of a sum is the weighted average of its components’ return rates, hence we get:

$Pr_P=(Sr_S-Fr_F-Vr_V-Ar_A-Dr_D)(1-r_t)$

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:

$P\beta_P=(S\beta_S-F\beta_F-V\beta_V-A\beta_A-\beta_DDr_D)(1-r_t)$

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

$P\beta_P=\left( S-V \right) \left( 1-r_t) \beta_S$

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

$\beta_P=\frac{\left( S-V \right) }{\left( S-F-V-A -Dr_D\right)}\beta_S$

Rearranging the terms gives:

$\beta_P=\left( 1+ \frac{ F+A+Dr_D }{ S-F-V-A -Dr_D}\right)\beta_S$

Next we want to have a look at this proportional factor and derive it with respect to $F$ :

$\frac{\partial }{\partial F}\left(\frac{\beta_P}{\beta_S} \right)=\frac{ S - V }{ \left( S-F-V-A -Dr_D \right)^2}\ge 0$

We see that the derivation is always $\ge 0$ , if sales $S$ are higher than variable expenses $V$ . That means that higher fixed expenses always increase $\beta_P$ at a given sales beta $\beta_S$ , if sales $S$ are higher than variable expenses $V$ . By the way, increasing abbreviation and amortization has the same effect as increasing fixed expenses.

Deriving with respect to $D$ leads us to:

$\frac{\partial }{\partial D}\left(\frac{\beta_P}{\beta_S} \right)=r_D\frac{ S - V }{ \left( S-F-V-A -Dr_D \right)^2}\ge 0$

We see that the derivation is always $\ge 0$ , if sales are higher than variable expenses. That means that higher debt increases $\beta_P$ at a given sales beta $\beta_S$ . Further we realize that is proportional to the derivation in respect to F:

$\frac{\partial }{\partial D}\left(\frac{\beta_P}{\beta_S} \right)=r_D\frac{\partial }{\partial F}\left(\frac{\beta_P}{\beta_S} \right)\ge 0$

$r_D$ is much smaller than 1 in ordinary cases. Thus the dependency on fixed expenses is much higher than on debt.

Leverage of Return on Equity (ROE)

Return on equity is the indicator that measures relative equity increase best. Return on equity is defined by:

$\text{ROE}=\frac{\text{net income}}{\text{equity}}=\frac{P}{E}$

In the previous section we had a look at the beta of net income. Now we want to link the beta of ROE to the beta of net income. Having a look at the basic beta definition with the bilinearity of covariance provides this relationship:

$\beta_{\text{"ROE"}}=\beta_{\frac{P}{E}}=\frac{cov(r_\frac{P}{E}, r_m)}{var(r_m)}=\frac{\frac{1}{E}cov(r_P, r_m)}{var(r_m)}=\frac{1}{E}\beta_P$

We assume a given enterprise value $Z$ , hence $E=Z-D$ .

$\beta_{\frac{P}{E}}=\frac{1}{Z-D}\beta_P$

Substituting with the previous results in the article we get:

$\beta_{\frac{P}{E}}=\left( 1+ \frac{ F+A+Dr_D }{ S-F-V-A -Dr_D}\right)\frac{1}{Z-D}\beta_S$

This is the relationship of ROE beta and net income beta. It depends on fixed expenses $F$ as well as on debt $D$ . We derive in respect to $F$ and $D$ to investigate this relatioship.

The first derivation in respect to $F$ is:

$\frac{\partial }{\partial F}\left(\frac{\beta_{\frac{P}{E}}}{\beta_S} \right)=\frac{ S - V }{ \left( S-F-V-A -Dr_D \right)^2}\frac{1}{Z-D}$

We realize that the derivation is always $\ge 0$ , if sales are higher than variable expenses. That means that higher fixed expenses increase the ROE beta at a given sales beta $\beta_S$ . That means the higher fixed expenses the higher the dependency of ROE to the market return rate and the higher the risk premium for the equity investors, see Capital Asset Pricing Model (CAPM). A higher risk premium means that shareholders require a higher minimum return rate to invest in the company.

The first derivation in respect to $D$ is:

$\frac{\partial }{\partial D}\left(\frac{\beta_{\frac{P}{E}}}{\beta_S} \right)=\frac{r_D\frac{ A + Dr_D + F}{\left( S-F-V-A - Dr_D \right)^2}+\frac{r_D}{S-F-V-A-Dr_D}}{Z-D}+$