Project WACC and Company WACC

Joachim Kuczynski, 14 November 2025

This article shows how to derive the WACC (weighted average cost of capital) of a project from the WACC of a company (or asset portfolio in general). Further on company and portfolio have the same meaning. Let us assume a company or portfolio with $n$ projects. The relative share of project $i$ in the portfolio is $\alpha_i$ . The WACC of project $i$ is $\omega_i$ . The cost of capital of the portfolio, $\omega_p$ , is given by:

(1) $\begin{equation*}\omega_p=\frac{\sum_{i=1}^{n}V_i\omega_i}{\sum_{j=1}^{n}V_j}=\sum_{i=1}^{n}\frac{V_i}{\sum_{j=1}^{n}V_j}\omega_i=\sum_{i=1}^{n}\alpha_i\omega_i\end{equation*}$

That means that the WACC of a portfolio is the weighted average of the components’ WACC.

Let us have a look at the variance of the portfolio WACC:

(2) $\begin{equation*}\mathrm{var}\left( \omega_p \right)=\mathrm{cov}\left( \omega_p,\omega_p \right)=\mathrm{cov}\left(\sum_{i=1}^{n} \alpha_i\omega_i, \omega_p \right)\end{equation*}$

The covariance is bilinear, we get:

(3) $\begin{equation*}\mathrm{var}\left( \omega_p \right)=\sum_{i=1}^{n}\alpha_i\mathrm{cov}\left( \omega_i, \omega_p \right)\end{equation*}$

The correlation of an asset’s WACC $\omega_i$ with the portfolio’s WACC $\omega_p$ is defined by:

(4) $\begin{equation*}\mathrm{corr}\left( \omega_i,\omega_p \right):=\frac{\mathrm{cov}\left( \omega_i,\omega_p \right)}{\sigma\left( \omega_i \right)\sigma\left( \omega_p \right)}\end{equation*}$

$\sigma (\omega_i)$ and $\sigma (\omega_p)$ are the standard deviations of $\omega_i$ and $\omega_p$ . Substituting covariance by correlation we get:

(5) $\begin{equation*}\mathrm{var}\left( \omega_p \right) =\sum_{i=1}^{n}\alpha_i \sigma \left( \omega_i \right)\sigma\left( \omega_p \right)\mathrm{corr}\left( \omega_i, \omega_p \right)\end{equation*}$

Dividing by the standard deviation $\sigma\left( \omega_p \right)$ leads us to the standard deviation of the portfolio:

(6) $\begin{equation*}\sigma\left( \omega_p \right) =\sum_{i=1}^{n}\alpha_i \sigma \left( \omega_i \right)\mathrm{corr}\left( \omega_i, \omega_p \right)\end{equation*}$

That means that the incremental risk contribution of each project to the risk of the portfolio is $\sigma\left( \omega_i \right)\mathrm{corr}\left( \omega_i, \omega_p \right)$ and not just the weighted sum of its projects’ risks. Hence we can e.g. reduce the WACC of the portfolio by adding a project with negative correlation to the portfolio.\

Instead of including a new project to the portfolio the company can also increase the return of the portfolio by increasing the risk of the portfolio. This reward-to-volatility ratio of the tangential portfolio is given by the Sharpe Ratio:

(7) $\begin{equation*}\frac{\mathbb{E}\left( \omega_p\right)-r_D}{\sigma \left( \omega_p \right)}\end{equation*}$

$\mathbb{E}\left( \omega_p\right)$ is the expected value of $\omega_p$ and $r_D$ is the after-tax debt interest rate. The company wants to invest in the new project, if the additional WACC of this project is lower than an investment in the existing portfolio with the same risk changes. Hence we obtain the requirement to invest in the new project:

(8) $\begin{equation*}\mathbb{E}\left( \omega_i \right)-r_D < \sigma\left( \omega_i \right)\mathrm{corr}\left( \omega_i,\omega_p \right)\frac{\mathbb{E}\left( \omega_p\right)-r_D}{\sigma\left( \omega_p \right)}\end{equation*}$

With that we can define the sensitivity $\beta_i^p$ of the new project to the existing portfolio:

(9) $\begin{equation*}\beta_i^p:=\frac{\sigma\left( \omega_i \right)\mathrm{corr}\left( \omega_i,\omega_p \right)}{\sigma\left( \omega_p \right)}\end{equation*}$

Substituting with $\beta_i^p$ the requirement for the new investment becomes the well-known equation:

(10) $\begin{equation*}\mathbb{E}\left( \omega_i \right) < r_D+\beta_i^p\left( \mathbb{E}\left( \omega_p\right)-r_D \right)\end{equation*}$

With that we can define a maximum annual WACC of the additional project. This is the maximum WACC at which a company would decide to invest in the project.

(11) $\begin{equation*}\mathbb{E}\left( \omega_i \right)^{max} = r_D+\beta_i^p\left( \mathbb{E}\left( \omega_p\right)-r_D \right)\end{equation*}$

Let us assume that the new project and the company are correlated completely, which means $\mathrm{corr}\left( \omega_i,\omega_p \right)=1$ . In this case we get:

(12) $\begin{equation*}\mathbb{E}\left( \omega_i \right)^{max} = r_D+\frac{\sigma\left( \omega_i \right)}{\sigma\left( \omega_p \right)}\left( \mathbb{E}\left( \omega_p\right)-r_D \right)\end{equation*}$

The quotient $\sigma\left( \omega_i \right)/\sigma\left( \omega_p \right)$ is called relative risk factor of project $i$ . The risk premium of the company WACC is adapted by the quotient of the standard deviations of the project and the company. The higher the risk of the project the higher the required WACC of the project to become a new part of the company portfolio. If the risk of the project and the risk of the company are the same, the WACC of the project and the company are the same, too. The standard deviation can be generated by Monte Carlo Simulation or approximated by best/worst case scenarios. Another approach is to estimate the standard deviations by management. Instead of comparing the project to the whole company portfolio we can also compare it to a benchmark project. This benckmark project is considered to be representative for the company.

04/11/202510/11/2025

CAPM Beta and Market Return Rate

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.

01/11/202512/11/2025

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, 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 $\beta_E$ :

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

With this $\beta_E$ we can calculate the new equity return rate $r_E$ :

(18) $\begin{equation*}r_E=r_f+\beta_E\left( r_m-r_f \right)\end{equation*}$

After the releveraging process we get the releveraged WACC with the appropriate $r_E$ :

(19) $\begin{equation*}\text{WACC}=dr_D\left( 1-r_t \right)+(1-d)r_E\end{equation*}$

12/09/202521/10/2025

Portfolio Beta

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.