WACC with Tax Shield - investment analysis

In this post I want to provide a derivation of the discount rate that includes savings because of interest tax shield. Further I can show a general expression for tax shields implementation, wherein this well known WACC formula is only a special case:

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

The formula includes “ $-t$ ” that comes from tax shield savings. $D$ and $E$ stand for debt and equity of the firm, $r_D$ and $r_E$ are the required return rates for debt and equity, $t$ is the marginal tax rate.

General case

We consider one time period starting at $t_0$ and ending at time $t_1$ . In $t_1$ we have a cash flow excluding tax shield of $C_1$ and an absolute tax shield value of $T_1$ . $r$ is the discount rate without tax shield. We are searching a discount rate $r^*$ that allows us to discount the cash flow excluding tax shield but including the tax shield effect in the present value in $t_0$ . Hence we have to adapt the discount rate. The discounted value of the cash flows in $t_0$ has to be the same for both discount rates:

$\frac{C_1+T_1}{1+r}=\frac{C_1}{1+r^{*}}$

That leads to a general relationship of $r^{*}$ and $r$ :

$r^{*}=\frac{C_1r-T_1}{C_1+T_1}$

That expression allows us to include a cash flow $T_1$ into the discount rate and to discount the cash flows in $t_1$ excluding $T_1$ with the adapted discount rate $r^*$ . It enables us to calculate the discount rate $r^*$ from the cash flows $C_1$ and $T_1$ in $t_1$ . You do not require any values from $t_0$ . Especially there is no need to have knowledge about the capital structure of the company. But for sure the capital structure is required to calculate $T_1$ in most cases.

Including capital structure

Next we want to consider the capital structure in $t_0$ . The discounted asset value consists of debt $D_0$ and equity $E_0$ . The discounted value of the cash flows must be the sum of debt and equity. With $\left( D_0 + E_0 \right) \left( 1+r \right) = C_1 + T_1$ we obtain:

$r^*=r-\frac{T_1}{D_0+E_0}$

Setting $r$ as weighted average return rates of debt and equity without tax shield we get:

$r^*=\frac{D_0}{D_0+E_0}r_{D}+\frac{E_0}{D_0+E_0}r_{E} -\frac{T_1}{D_0+E_0}$

Famous WACC after taxes

In most cases the tax shield is the interests paid on $D_0$ times the marginal tax rate $t$ . That means $T_1=D_0 r_D t$ . Hence we get the well known expression for $r^*$ :

$r^*=\frac{D_0}{D_0+E_0}r_{D}\left( 1-t \right)+\frac{E_0}{D_0+E_0}r_{E}$

This is the discount rate or “WACC” after taxes which is quoted in most books. But take care! It is only valid, if really the complete amount of interests paid can be deducted from taxes. Sometimes the company does not have enough profit to deduct all interest payments. In other cases the amount of tax deduction is limited by some constraints. In these two cases the previous formula does not work any more. The equation also shows that the capital structure in $t_0$ is important and not the capital structure in $t_1$ .

Maximum constraint of tax shield

If the company has for example a maximum for the tax shield $T_1^{\text{max}}$ , maybe a maximum share of EBITDA in $t_1$ , we obtain another expression:

$r^*=\frac{D_0}{D_0+E_0}r_{D}+\frac{E_0}{D_0+E_0}r_{E} -\frac{ \min \left( D_0 r_D t , T_1^{\text{max}} \right)}{D_0+E_0}$

Or if the company has tax shield savings from other periods in $t_1$ , the equation is not valid, too.

In my point of view, the Adjusted Present Value (APV) approach is much better than the WACC approach for the implementation of tax shield. Each time period has clear tax shield amounts. And in complex cases you do not have to adapt the WACC in each time period.