# Why is PV(tax shield) calculated using cost of debt capital for discounting?

I understand that (interest payment)×(corporation tax) is the cash flow saving (assumed to go on in perpetuity) and it can be written as (debt)×(cost of debt capital)×(corporation tax). But why is it discounted with cost of debt capital? If it is a cash outflow saving on the whole operation of the company, why isn't is discounted with WACC?

• I would be willing to attempt a detailed answer which goes through when and when not a tax shield is appropriate, but first can you please define WACC? Are you referring simply to the empirical cost of capital using market values for invested capital? Or are you referring to the WACC as implied by the CAPM? Mar 31 '17 at 16:38

We are comparing two situations: (1) an all equity firm, versus (2) the same firm which has decided to do a predefined amount of borrowing. Because the tax shields arise as a result of the borrowing decision (i.e. they are a potential advantage of borrowing) they are discounted at the borrowing rate. It is treated almost as a separate project: to borrow money for the purpose of saving on future taxes.

(Note: here we are assuming a firm with flexibility about whether it borrows or not and on what time frame. In the case of a highly indebted LBO, where the repayment schedule of the debt depends on the future earnings of the firm, it may make more sense to discount the tax shields at the WACC of the whole firm. See The APV approach to Leveraged Buyouts).

• It answered my question perfectly, thank you! Apr 1 '17 at 14:09

Your thought process on valuing the tax savings to the whole enterprise at the cost of capital makes perfect sense to me. While I get the thought process on tax savings being funded by debt, the cash flows due to tax savings do not flow back to debt, but rather to the enterprise whether or not the firm is highly leveraged.

I think a simple scenario approached from two different angles which results in the same answer provides clarification as to why one would choose to discount tax deferment like any other cash flow.

Assume you know the following information about a firm:

$V_{E,0} =$ the net present value of equity

$V_{t,0} =$ the net present value of tax deferment

$C_\tau =$ cash flows (net of cash spending) at time $\tau$

$D_\tau =$ tax deductible depreciation at time $\tau$

$I_\tau =$ interest on debt

$r_E =$ the required return on equity

$r_D =$ cost of debt

$r_c =$ WACC

$r_t =$ nominal income tax rate

$r_e =$ effective tax rate

One approach of a DCF which does not require one to value a debt shield would proceed as follows:

(1) $V_{E,0} = \sum_{t=0}^T \frac{C_{\tau}-I_{\tau}}{(1+r_c)^{\tau}}(1-r_e)$

where:

$r_c$ is recovered by: $\omega_E*r_E + \omega_D*r_D * (1-r_t)$

and $r_e$ is recovered by: $Max[0,\frac{(C_{\tau}-D_{\tau}-I_{\tau})}{C_\tau}*r_t]$

Under such an approach, cash taxes are treated like any other cash flow since the tax saving are embedded within effective tax rate.

A more common approach is to value the equity net of taxes, but before tax saving, and then add back tax savings due to interests payments and depreciation (some of this is changing under the new U.S. tax laws going into effect which limit the deductions of interest deductions).

(2) $V_{E,0} = (\sum_{t=0}^T \frac{C_{\tau}-I_{\tau}}{(1+r_c)^{\tau}}(1-r_t)) + V_{t,0}$

where:

$V_{t,0}$ is recovered by: $(\sum_{t=0}^T \frac{C_{\tau}-D_{\tau}-I_{\tau}}{(1+r_c)^t}*r_t)$

In either case, one recovers the same value of equity. Example (1) shows that by treating cash taxes as any other normal cash flow, discounting at the WACC is a sound approach.

I know this question has been answered but I am giving a complete derivation to assist others as well as serve as my notes.

We will take a firm with a free cash flow $\mathcal F$ every year perpetually. This is a simplification that can easily be generalized to variable payments.

Unlevered Firm

The earnings by the stockholders are reduced by taxes. Thus the value of equity is

$$E_U = \frac{\mathcal F}{r^U_E} \tag{1}$$

where $r^U_E$ is the cost of equity for unlevered firm. If we use the CAPM model we can write this as

$$r^U_E = r_f + \beta_U r_M \tag{2}$$

where $r_M$ is the market premia.

Levered Firm

For a levered firm the earnings are divided into those for equity holders and debt holders. Valuing the equity part as above we get

$$E_L = \frac{\mathcal F-r_D D (1-t)}{r^L_E} \tag{3}$$

and trivially the cost of debt is

$$D= \frac{r_D D}{r_D} = D \tag{4}$$

Using CAPM again we have

$$r^L_E = r_f + \beta_L r_M \tag{5}$$

and we can write the rate of interest on the loan as

$$r_D = r_f + \beta_D r_M \tag{6}$$

Relation between Unlevered and Levered

The earnings are independent of the level of leverage so writing the terms equal to $\mathcal F$ we have

$$r_f \big[E_U - E_L - D(1-t) \big] + r_M \big[\beta_U E_U - \beta_L E_L - \beta_D D (1-t) \big]=0 \tag{7}$$

Since $r_f$ and $r_M$ are independent, their coefficients must vanish independently as well.

Thus we have

$$E_U =E_L + D(1-t) \tag{8}$$

and the relation between the $\beta$s

$$\beta_U E_U = \beta_L E_L + \beta_D D(1-t) \tag{9}$$

With the firm value as the total of equity and debt we get

$$V_L = E_L + D \\ =E_U + Dt$$ $$=V_U + Dt \tag{10}$$

Showing that tax shield has value $Dt$.

Corrolary

A corrolary of (9) is

$$r^L_E = r^U_E \frac{E_U}{E_L} - \frac{D}{E_L} r_D (1-t) \tag{11.1}$$

which using (8) can be written as

$$r^L_E = r^U_E + ( r^U_E - r_D) \frac{D}{E_L} (1-t) \tag{11.2}$$

Another interesting concept is $R_{wacc}$ defined (although not explicitly) as the discount rate that when applied to free cash flow will give the appropriate firm value

$$\mathcal F = R^L_{WACC} V_L = R^U_{WACC} V_U$$

So clearly $R^U_{WACC} = r^U_E$ and we also have $R^L_{WACC} = \frac{V_U}{V_L} r^U_E$ giving

$$R^L_{WACC} = \frac{1 + \frac{D}{E_L} (1-t)}{1+\frac{D}{E_L}} R^U_{WACC} \tag{12.1}$$

which can be written using (11) as

$$R^L_{WACC} = \frac{1}{1+ \frac{D}{E_L}} r^L_E + \frac{\frac{D}{E_L}(1-t)}{1+ \frac{D}{E_L}} r_D \tag{12.2}$$

Note that in the last term all values are levered ones so the subscript can be dropped and that's how it is usually written in textbooks.