2. DTS has Same Risk as the Firm's Assets

2.3. Unlevering and Relevering the Cost of Capital

How does this now compare to our previous approach of unlevering and relevering the cost of capital instead of the betas? The section "Estimating the WACC in Specific Valuation Situations" has discussed this approach in all details. Here, we simply want to show that the results are fully consistent.

 

We have seen that the unlevered cost of capital, kA, can be estimated as follows:

 

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

   

We have also seen that the cost of equity under the target capital structure (relevered cost of capital, kE*) is:

  

\( k_E=k_A+(k_A-k_D^*) \times \big(\frac{D}{E}\big)^* \).

 

Finally, we have derived the WACC expressions as:

  

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

  

Let us now go to our example from before to compute the WACC under the new capital structure. Here is the information that we have gathered over the last few pages:

  • Risk-free rate of return RF = 2%
  • Market risk premium MRP = 5%
  • Current equity beta βE = 1.5
  • Current debt ratio (D/(D+E)) = 20%
  • Current cost of debt kD = 4%

 

With this information, we can compute the current cost of equity as:

  

kE = RF + βE × MRP = 0.02 + 1.5 × 0.05 = 9.5%.

  

Consequently, the unlevered cost of capital (kA) is 8.4%: 

   

\( k_A=k_D \times \frac{D}{D+E}+k_E \times \frac{E}{D+E} = 0.04 \times 0.2 + 0.095 \times 0.8 \) = 8.4%.

  

Note that we obtain the same result when plugging the asset beta of 1.28 from the previous section in the CAPM:

 

kA = RF + βA × MRP = 0.02 + 1.28 × 0.05 = 8.4%.

 

Now we can study the implications of the new capital structure on the firm's financing costs:

  • Target debt ratio D/(D+E)* = 0.6
  • New borrowing cost kD* = 5% 
  • Corporate tax rate = 30%

  

Consequently, the cost of equity under the target capital structure (kE*) is 13.5%: 

  

\( k_E=k_A+(k_A-k_D^*) \times \big(\frac{D}{E}\big)^* = 0.084 + (0.084-0.05) \times \frac{0.6}{0.4} \) = 13.5%.

  

This is the same value as we have derived before. Similarly, the firm's WACC under the target capital structure is 7.5%:

 

WACC = \( k_D^* \times(1- \tau_C ) \times \big(\frac{D}{D+E}\big)^*+k_E^* \times \big(\frac{E}{D+E}\big)^* = 0.05 \times(1- 0.3 ) \times 0.6+0.135 \times 0.4 \) = 7.5%.

  

Which, again, is the same result as before.

 

In sum, we can estimate the firm's cost of capital under a new target capital structure using either one of the following approaches:

  • Unlever and relever the betas
  • Unlever and relever the cost of capital.

Correctly implemented, the two approaches lead to the exact same result. Which approach to use is therefore a questions of the preferences of the analyst. In many situations, however, it would seem to be easier to directly unlever and relever the cost of capital.