WACC, Return Rates & Betas with Debt - Investment and Project Valuation

Joachim Kuczynski, 02 April 2021

In this post I want to summarize some interesting results concerning equity return rates , betas and WACC of a levered company. Regarding the market value balance sheet of a firm we can state that the value of the unlevered firm VU plus the present value of the tax shield VTS must be the same as the sum of levered equity E and debt D:

$VU+VTS=E+D$

Further the rates of return on each side of the balance sheet are the weighted average of the component rates of return:

$r_A\frac{VU}{VU+VTS}+r_{TS}\frac{VTS}{VU+VTS}=r_E\frac{E}{E+D}+r_D\frac{D}{E+D}$

Substituting VU in the rate of return expression we get a general form of the equity return rate:

$r_E=r_A+\left(r_A-r_D\right)\frac{D}{E}-\left(r_A-r_{TS}\right)\frac{VTS}{E}$

Consequently the general form of CAPM beta is given by:

$\beta_E=\beta_A+\left(\beta_A-\beta_D\right)\frac{D}{E}-\left(\beta_A-\beta_{TS}\right)\frac{VTS}{E}$

The WACC is defined as the weighted average of equity and debt return rates including tax shield at corporate income tax rate $T_C$ . If the tax shield savings are proportional to the taxes paid (see WACC with Tax Shield), the WACC is given by:

$WACC=r_E\frac{E}{E+D}+r_D\left(1-T_C\right)\frac{D}{E+D}$

Substituting the equity return rate we get a general form of the WACC:

$WACC=r_A\left(1-\frac{VTS}{V}\right)-r_DT_C\frac{D}{V}+r_{TS}\frac{VTS}{V}$

$r_A$ , $r_D$ , $r_{TS}$ are the return rates of the unlevered asset, debt and tax shield. V and VTS are the values of the levered firm and the tax shield. D is the amount of debt, V is the value of the levered firm, namely the sum of equity and debt.

Modigliani and Miller: Constant debt value
If the firm keeps its dept value D constant, there are no specific market risks concerning the tax shield. Therefore we can set the tax shield discount rate $r_{TS}$ equal to the debt discount rate, $r_D$ . The tax shield present value with constant debt D is:

$VTS=\sum_{j=1}^{\infty}{DT_C\left(\frac{1}{1+r_D}\right)}^j=\frac{DT_C}{r_D}$

Hence we get simplified expressions for equity return, equity beta and WACC:

$r_E=r_A+\left(r_A-r_D\right)\frac{D}{E}\left(1-T_C\right)$

$\beta_E=\beta_A+\left(\beta_A-\beta_D\right)\frac{D}{E}\left(1-T_C\right)$

$WACC=r_A\left(1-T_C\frac{D}{E+D}\right)$

Assuming that debt interest rate does not depend on the market return rate (CAPM) we can set $\beta_D=0$ . Hence we get the well-known Hamada equation for levered beta:

$\beta_E=\beta_A\left(1+\frac{D}{E}\left(1-T_C\right)\right)$

It is important to realize that Hamada’s equation is only valid if the value of debt is kept constant over time.

Harris and Pringle: Constant leverage ratio
Constant leverage ratio means that debt value is proportional to the value of the unlevered firm. According to Harris and Pringle that results in $r_{TS}=r_A$ .

$r_E=r_A+\left(r_A-r_D\right)\frac{D}{E}$

$\beta_E=\beta_A+\left(\beta_A-\beta_D\right)\frac{D}{E}$

$WACC=r_A-r_DT_C\frac{D}{E+D}$

But we have to take care. Miles and Ezzell, Arzac and Glosten have shown that you have a tax shield discount rate of $r_D$ in the first period, and of $r_A$ in the following periods to have a constant leverage ratio over time. The premise of $r_{TS}=r_A$ does not hold.

Miles and Ezzell
With a perpetuity growing rate g of debt and discounting in the first period with $r_D$ instead of $r_A$ we obtain:

${VTS}^{ME}=\frac{Dr_DT_C\left(1+r_A\right)}{\left(r_A-g\right)\left(1+r_D\right)}$

Harris and Pringle
Taking the formula of Miles and Ezzell and setting $r_TS$ equal to $r_A$ in the first period, we get the a simplified expression for $VTS$ :

${VTS}^{HP}=\frac{Dr_DT_C}{\left(r_A-g\right)}$

General debt ratio
If the amount of leverage is flexible and not constant or growing with a constant growth rate over time, the previous formulas do not work. In this case you have to use the APV method, in which you calculate the tax shield in each time period seperately.