After-Tax Discount Rate

In this post I want do derive the after-tax discount rate from the before-tax discount rate. “Before tax” means that the tax shield is not considered in the discount rate. It does not mean that the tax expenses (without tax shield) are not considered in the free cash flow. The tax expenses (without tax shield) are a part of the free cash flow in the before-tax and in the after-tax discount rate. For further information have a look at my other post WACC with Tax Shield. Abbreviations:

r … before-tax discount rate
r^{*} … after-tax discount rate
L … rate of debt to sum of equity E and debt D, L=D/(E+D)
r_D … debt interest rate
r_E … equity interest rate
t … marginal corporate tax rate

We assume that the values of r_D, r_E and L are known. Then the before-tax discount rate is:

    \[r=\left( 1-L \right)r_{E}+Lr_D\]


Rearranging the above to solve for r_{E} we have:

    \[r_{E}=\frac{r-Lr_D}{1-L}\]


The after-tax discount rate at a constant leverage rate is:

    \[r^{*}=\left( 1-L \right)r_{E}+L\left( 1-t \right)r_D\]


This is the famous equation most financial analysts might know. The factor “-t” comes from the tax shield and decreases the discount rate. Hence the discount rate after taxes is lower than the return rate before taxes. But you have to take care. This after-tax formula is only valid if the leverage rate L remains constant. Additionally it assumes that the total amount of tax expenses can be deducted by tax shield. If these two premises are not true, the previous formula does not work and you have to an analyze the topic with the adjusted present value (APV) approach. For a general view see this post. By substituting r_{E} we get:

    \[r^{*}= r -Ltr_D\]


This formula can be useful, because you do not have to know the equity return rate to calculate the after-tax return rate. But have in mind that this is only valid, if the leverage ratio is constant and the total tax shield amount can really be deducted from the tax expenses.

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