3. DTS has Same Risk as the Firm's Debt

3.3. Unlevering and Relevering the Cost of Capital

Again, we can compare the two approaches of unlevering and relevering the betas vs. the cost of capital.

  

It can be shown that, under the assumption that the firm's DTS has the same risk class as the firm's debt outstanding, the unlevered cost of capital, kA, can be estimated as follows:

 

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

   

Moreover, it can be shown that the cost of equity under the target capital structure (relevered cost of capital, kE*) is:

  

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

 

Finally, we the WACC remains the same, namely:

   

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

Earlier in this chapter, we have already estimated the firm's cost of equity under the current capital structure, namely:

 

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

   

Consequently, the unlevered cost of capital (kA) is 8.7%: 

 

\( k_A=k_D \times \frac{D \times (1-\tau) }{(D \times (1-\tau)+E)}+k_E \times \frac{E}{(D\times (1-\tau)+E)} \)

\( k_A=0.04 \times \frac{2'000 \times (1-0.3) }{2'000 \times (1-0.3)+8'000}+0.095 \times \frac{8'000}{2'000\times (1-0.3)+8'000} \) = 8.7%.

    

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 + 1.34 × 0.05 = 8.7%.

 

Now we can study the implications of the new capital structure on the firm. Remember from our previous considerations that:

  • New debt level (D*) = 6'000
  • New equity value (E*) = 5'200
  • New cost of borrowing (kD*) = 5% 
  • Corporate tax rate = 30%

  

Consequently, the cost of equity under the target capital structure (kE*) is 11.65%: 

  

\( k_E=k_A+(k_A-k_D^*) \times \big(\frac{D\times(1-\tau)}{E}\big)^* = 0.087 + (0.087-0.05) \times \frac{6'000\times(1-0.3)}{5'200} \) = 11.65%.

   

This is the same value as we have derived before. Similarly, the firm's WACC under the target capital structure is 7.3%, namely:

  

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

WACC = \( 0.05 \times (1 - 0.3) \times \frac{6'000}{6'000 + 5'200} + 0.1165 \times \frac{5'200}{6'000 + 5'200} \) = 7.3%.

  

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.