3. The Cost of Capital

3.2. Firms with Target Debt Ratios

Now let us look at the same considerations for a company that pursues a financing policy with a target debt ratio (e.g., 40% of firm value). As we have argued before, the appropriate discount rate for the interest tax savings of such a firm is the overall cost of assets, kA.

The following figure summarizes the relevant costs of capital for the different sources of value of such a company:

 

cost of capital ka

 

As we know from before, the total expected return generated by the left side of the balance sheet must correspond to the total expected return that the providers of capital claim:

 

\( V_U \times k_A + DTS \times k_A = D \times k_D + E \times k_E \)

 

Consequently:

 

\( (V_U + DTS) \times k_A = D \times k_D + E \times k_E \)

 

Because VU + DTS = VL = D + E, we can also write:

 

\( (D + E) \times k_A = D \times k_D + E \times k_E \)

 

Solving this expression for kA yields:

 

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

 

Solving the expression for kE yields:

 

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

 

These are the very same equation as the ones that we have derived in the course section that looked at a world without taxes. 

 

To practice these considerations, let us go back to our original example and modify it slightly: Let us assume that the firm still borrows 2'500 today, but that it will adjust the amount of debt outstanding going forward to maintain a constant debt ratio. What are the value implications of this change in financing policy?

  • Because the future tax savings are less certain, the appropriate discount rate is kA.
  • With annual expected interest tax savings of 87.5 and a kA of 12%, the present value of the DTS is 87.5/0.12 = 729.17.
  • Given an unlevered firm value (VU) of 3'250 and an amount of debt of 2'500, the value of the firm's equity under the revised financing policy is:

    E = VU + DTS - D = 3'250 + 729.17 - 2'500 =  1'479.17.

 

Consequently, the firm's cost of equity is 15.38%:

 

\( k_E = k_A + (k_A - k_D) \times \frac{D}{E} = 0.12 + (0.12-0.10) \times \frac{2'500}{1'479.17} \) = 15.38%.

 

To verify these computations, remember that the annual payment to shareholders is 227.5, according to the firm's income statement. If we capitalize these cash flows at a cost of equity of 15.38%, the value of the firm's equity is indeed 1'479.17:

  

Equity value = \( \frac{\text{Equity cash flow}}{\text{Cost of equity}} = \frac{227.5}{0.1538} \) = 1'479.17.