Reading: Estimating the Cost of Capital

The final issue in the computation of the cost of capital concerns taxes: 

• From the point of view of corporate taxation, debt and equity are treated differently.
   
• Payments to debtholders in the form of interest payments constitute tax-deductible corporate expenses and, therefore lower the firm's tax bill.
   
• In contrast, payments to shareholders in the form of dividends or buyback are not considered expenses and, therefore, leave the firm's tax bill unaffected.
     

Example: 

Let us assume a company has debt of 100'000 outstanding (D), on which it pays an interest rate equal to the cost of debt of 5%. The (marginal) corporate tax rate (\(\tau_C\)) is 21% on the firm's pre-tax income. What are the firm's cost of debt financing after taxes?

Two things happen:

• First, the firm pays interest of 5% on the notional amount of 100'000, namely 5'000:
     
  Interest expenses = D × k_D = 100'000 × 0.05 = 5'000.
     
• Second, these interest expenses lower the firm's pre-tax income by 5'000! Therefore, because of the interest expenses, the firm can lower it's tax bill by 1'050, which is the reduction in the taxable income times the (marginal) corporate tax rate (\(\tau_C\)):
    
  Tax savings = Interest expenses × \(\tau_C\) = 5'000 × 0.21 = 1'050.

After taxes, the overall cost of borrowing therefore is 3'950, i.e., the interest expenses net of the tax savings:

After-tax cost of borrowing ($) = Interest expenses − Tax savings = 5'000 − 1'050 = 3'950.

In words: Because interest expenses are tax deductible, the actual after-tax cost of borrowing is lower than the interest rate paid to the bank. The reason is that for each dollar sent to the bank, the firm receives a tax credit of 21 cents from the tax authorities. After taxes, a dollar in interest payments therefore only costs 0.79 cents!

We can therefore write:

• Pre-tax cost of debt = \(k_D\)
   
• After-tax cost of debt = \(\bf{k_D \times (1-\tau_C)}\)

In the example above, the firm's after-tax cost of debt therefore is 3.95%:

After-tax cost of debt = \(k_D \times (1-\tau_C) = 0.05 \times (1-0.21) = 0.0395 = 3.95\%\)

The Weighted Average Cost of Capital

The preceding considerations imply that corporate taxes lower the firm's cost of borrowing. For the firm, the relevant cost is not the pre-tax cost of debt but the after-tax cost of debt \(k_D \times (1-\tau_C)\)). 

Before, we have concluded that the overall cost of capital (\(k_A\)) is:
 

\( k_A = k_D \times \frac{D}{D+E} + k_E \times \frac{E}{D+E} \)
 

From the discussion above, we now know that k_A is, in fact, the pre-tax overall cost of capital, since it ignores the tax savings associated with the firm's interest expenses. This is why k_A is also called the unlevered cost of capital. 

In contrast, if we are interested in the after-tax cost of capital, we should adjust the borrowing cost accordingly. The result is the so-called Weighted Average Cost of Capital (WACC), which is the same expression as above, with the sole difference that the pre-tax cost of debt (\(k_D\)) is replaced with the after-tax cost of debt (\(k_D \times (1-\tau_C)\)):
 

\( \bf{WACC = k_D \times (1-\tau_C) \times \frac{D}{D+E} + k_E \times \frac{E}{D+E}} \)

The WACC is the most commonly used discount rate in firm and project valuation. It factually indicates the rate of return a firm has to generate on its assets, on average, to satisfy the return expectations of the firm's providers of capital (debt plus equity), after taking into consideration corporate tax effects.

It can be shown fairly easily that when we combine the equations above, we can write the WACC also as follows:

\( \bf{WACC=k_A - k_D \times \tau_C \times \frac{D}{D+E}}\)

Consequently, to compute the after tax cost of capital, we can start with the pre-tax overall cost of capital and then subtract the overall tax effect of debt financing on the cost of capital.

Example

Let's go back one last time to the case of Amazon. There, we have estimated the following elements so far:

Now let us assume that the firm's marginal tax rate (\(\tau_C\)) is 21%, i.e., the corporate tax rate in the U.S. With this information, we can compute the firm's WACC of 7.9%:

 \( WACC = k_D \times (1-\tau_C) \times \frac{D}{D+E} + k_E \times \frac{E}{D+E} \) \(= 0.032\times 0.79 \times 0.1 + 0.0853 \times 0.9 = 0.0790 = 7.93\%\)  

Alternatively, we can use the other expression and compute the WACC based on the pre-tax cost of capital (\(k_A\)):

\( \bf{WACC=k_A - k_D \times \tau_C \times \frac{D}{D+E}}\) \(=0.0800 - 0.032 \times 0.21 \times 0.1 = 0.0793 = 7.93\%\)

Consequently, our estimates imply that the firm's after-tax cost of capital is 7.93%.

Unlike the overall cost of capital, \(k_A\), which solely reflects the firm's business risk, the WACC also incorporates any corporate tax effects of financing. When capitalizing cash flows with the WACC, we therefore directly include these side-effects of financing in the cost of capital.

The purpose of this section was to provide a condensed introduction to the idea of the cost of capital and how firms can estimate the appropriate discount rates to value their projects. The Online Calculator Cost of Capital provides a simple tool to quickly compute the cost of capital (WACC) using the equations above. A more profound discussion of this topic is presented in the module Cost of Capital and Valuation.