Reading: Unlevering and Relevering Betas

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

 

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

 

Let us go back to the previous example and estimate the WACC under the new capital structure. To do so, we need a few additional pieces of information.

• The risk-free rate of return as well as the expected market risk premium to implement the CAPM;
• The firm's marginal corporate tax rate (τ).

 

To illustrate, let us assume the following:

• Risk-free rate of return (R_F) = 2%
• Market risk premium (MRP) = 5%
• Marginal tax rate (τ) = 30%.

    

We can first use this information to better understand the firm's borrowing costs. We do not generally have "betas" in mind when thinking about debt financing. The more typical approach is to assess the firm's credit risk and then try to quantify the credit spread that is associated with that risk. As it turns out, these two approaches are logically and economically equivalent. 

For our hypothetical firm, we have assumed a debt beta of 0.4 under the current capital structure and a debt beta of 0.6 under the target capital structure. Given our assumption of a market risk premium of 5%, these betas translate into credit spreads of 2% (= 0.4 × 5%) and 3% (= 0.6 × 5%), respectively. Given a risk-free rate of 2%, the associated borrowing costs are 4% (= 2% + 2%) under the current capital structure and 5% (= 2% + 3%) under the target capital structure. 

More generally, speaking, while we find it hard to think about debt betas, we can easily find the implied debt beta from the cost of debt wit the following expression:
 

\( \beta_{D, simplied} = \frac{k_D - R_F}{MRP} \). 

  

Let us now turn to the WACC estimation. With the available information, we can estimate the relevant ingredients of the WACC expression. Let us first estimate the resulting WACC when we use the "correct" equations from the previous page:

• Cost of equity, k_E* = R_F + β_E* × MRP = 0.02 + 2.3 × 0.05 = 13.5%
• Cost of debt, k_D* = RF + β_D* × MRP = 0.02 + 0.6 × 0.05 = 5.0%
• Target debt ratio, (D/(D+E))* = 60%
• Target equity ratio, (E/(D+E))* = 40%
• Marginal tax rate, τ = 30%.

 

Consequently, the firm's WACC is 7.5%:

 

WACC_simplified = \( k_D^* \times(1- \tau_C ) \times \big(\frac{D}{D+E}\big)^*+k_E^* \times \big(\frac{E}{D+E}\big)^* = 0.05 \times (1 - 0.3) \times 0.6 + 0.170 \times 0.4 \) = 8.9%.

 

Put differently, the simplifying assumptions we made when unlevering and relevering the equity beta lead to a substantial overestimation of the firm's WACC (8.9% vs. 7.5%). This bias is systematic:

• If a firm increases its debt ratio, the simplifying equations will systematically overestimate the firm's cost of capital.
• In contrast, if the firm decreases its debt ratio, the simplifying equations will systematically underestimate the firm's cost of capital.

This bias can be avoided easily: Use the full equation from before (or the provided excel file).

  

Taken together, the recommended procedure is as follows:

1) Estimate the unlevered beta using the following equation:

\( \beta_A = \beta_D \times \frac{D}{D+E} + \beta_E \times \frac{E}{D+E} \).

  

2) If you do not "know" the current β_D, extract it from the current cost of debt as follows: 

\( \beta_{D, simplied} = \frac{k_D - R_F}{MRP} \). 

 

3) Estimate β_E* under the target capital structure (relevered beta):

\( \beta_E^* = \beta_A + (\beta_A - \beta_D^*) \times \big(\frac{D}{E}\big)^* \)

  

4) Think about how the new financing policy will affect the cost of borrowing. If the cost of borrowing change, use the expression from (2) to extract the new beta of debt (β_D*).

 

5) Use the betas from (3) and (4) to estimate the respective costs of capital using the CAPM:

k_j = R_F + β_j × MRP

 

6) Use these values to estimate the firm's WACC under the target capital structure.

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