Tag: Discount Rate

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Operating Leverage

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.

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Discount Rate vs. Capitalization Rate

Joachim Kuczynski, 27 November 2024

The valuation of investments and projects (or assets in general) is based on shifting and adjusting cash flows on the time axis. A shift backwards in time is called discounting. A shift forwards in time is called capitalization. Many analysts use the same rate for discounting and capitalization when valuing a project. But can you really shift the cash flows back and forth on the time axis at the same rate? The answer to this is a clear no.

The discount rate of a cash flow is based on its inherent risk (volatility). In general, it does not depend on the risk preferences of the capital providers, when their investment portfolio is diversified sufficiently. The capitalization rate of a cash flow, on the other hand, is based on the risk preferences and portfolio diversification of the company (or its capital providers). The capitalization rate does not depent on the inherent risk of the initial cash flow. Once it is available for capitalization, it does not matter under what circumstances it came about. Discount rates and capitalization rates are fundamentally different. Taking the same rate in calculations is a fundamental, logical error.

There are key figures for investment valuation that include a recapitalization of returning cash flows. A well-known example is the modified internal rate of return or the Baldwin rate. It is usually assumed that both discounting and reinvestment are carried out using the same rate, the WACC. The WACC is based on the return expectations of the capital providers. However, these do not necessarily have to correspond to the return opportunities of the investing company. Additional rates (“risk premiums”) are often added to the discount rate. This is intended to take currency or location risks into account, for example. If you now use the same rate for capitalization, you assume an increased reinvestment rate for the company. This makes no sense and increases the error in the investment calculation.

In summary, we can state that equating discount rate and capitalization rate is fundamentally wrong. Many formulas are simplified by equating these two rates. But it is simply wrong, anyway. The results of the calculation are no longer valid doing that. However, correct and meaningful results of the investment calculation are the basis for correct investment decisions. So take care and do not simplify what cannot be simplified. Economic reality is complex, and their calculations can be too.

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