Reading: Unlevering and Relevering Betas

We start again with the relevant components of the balance sheet at market value as well as the associated risk measures (in blue).

Since this is a balance sheet, we know that the overall risk of the left-hand side (V_U and DTS) must correspond to the overall risk of the right-hand side (D and E). Consequently:

\( \beta_A \times V_U + \beta_D \times DTS = \beta_D \times D + \beta_E \times E \).

We can rearrange this expression to find the unlevered beta (β_A):

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

From the balance sheet above we know that:
 

\( V_U = D - DTS + E \).

Consequently, we can rewrite the expression for β_A as:

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

In words: The overall risk of the firm's assets (β_A) is, again, the weighted average of the risks of its debt (β_D) and equity (β_E). Under the assumption that the DTS falls in the risk class of the firm's debt, this is the appropriate expression to unlever the equity beta. The difference to the previous approach is that the weight of the debt in the balance sheet at market values is computed net of the debt tax savings (D-DTS). 

To estimate the relevered beta under the target capital structure (*), we solve the above expression for β_E and apply the relevant values under the target capital structure. After some rearrangements, we find:

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

Again, the only notable difference to the previous section is that we reflect the firm's debt net of tax savings. Consequently, to implement this approach, we have to estimate one additional item, namely the firm's debt tax shield.

To do so, most analysts assume that the firm's debt outstanding (D, in currency) is constant and that the future interest tax savings can be modelled with a simple level perpetuity. Using the notation from the previous chapter: 

• Annual interest expenses = k_D × D 
• Annual interest tax savings = interest expenses × τ = k_D × D × τ
• Present value of future tax savings (DTS):
   
  \( DTS = \frac{k_D \times D \times \tau }{k_D} = D \times \tau \).

Under this simplifying assumption, the debt tax shield therefore corresponds to the amount of debt outstanding (D) times the firm's tax rate (τ). If we plug this expression for DTS in the above equation to estimate the unlevered beta (β_U), we find:

 
\( \beta_A = \beta_D \times \frac{D \times (1-\tau) }{D \times (1-\tau)+E} + \beta_E \times \frac{E}{D \times (1-\tau)+E} \).

And the relevered beta under the target capital structure is:

 
\( \beta_E^* = \beta_A + (\beta_A - \beta_D^*) \times \big(\frac{D \times (1-\tau)}{E}\big)^* \).

Again, most textbooks and practitioners make the simplifying assumption that the beta of debt (β_D) is zero. If we set β_D = 0, the above expression for the unlevered beta simplifies to:

\( \beta_{A, simplified} = \frac{\beta_E}{1+(1-\tau) \times \frac{D}{E}} \).

And the expression for the relevered beta under the target capital structure simplifies to:

\( \beta_{E, simplified}^* = \beta_A \times \bigg(1+(1-\tau) \times \big(\frac{D}{E}\big)^*\bigg) \).

Example: Let us consider a firm with very similar characteristics to the one before. The only notable difference is that this firm now pursues a constant debt level:

• Current net debt level (D) = 2'000
• Current equity value (E)  = 8'000
• Current equity beta (β_E) = 1.5
• Current debt beta (β_D) = 0.4
• Target debt level = 6'000
• Corporate tax rate = 30%

We want to find the risk of the firm's equity under the new capital structure (β_E*) and, later, the WACC.

The first step is to compute the unlevered beta (β_A) under the current financing policy. Using the equations from above, we find:

\( \beta_A = \beta_D \times \frac{D \times (1-\tau) }{D \times (1-\tau)+E} + \beta_E \times \frac{E}{D \times (1-\tau)+E} \)

\( \beta_A = 0.4 \times \frac{2'000 \times (1-0.3) }{2'000 \times (1-0.3)+8'000} + 1.5 \times \frac{8'000}{2'000 \times (1-0.3)+8'000} = 1.34\).

This beta we can now re-lever to the firm's target capital structure. To do so, we also need to estimate:

• The new amount of debt outstanding (D*)
• The debt beta under the new capital structure (β_D*)
• The DTS under the new capital structure (DTS*)
• The equity value under the new capital structure (E*).

According to our assumptions from above, the firm increases debt outstanding from 2'000 to 6'000. Consequently, D* is 6'000. Moreover, let us continue to make the assumption from the previous chapter that the beta of debt increases from 0.4 to 0.6 under the new financing policy (β_D*).

Now what about the change in the DTS and the value of the equity?

If the firm increases net debt  from 2'000 to 6'000, the DTS increases by 1'200 from 600 (= 0.3 × 2'000) to 1'800 (= 0.3 × 6'000). More generally speaking, the change in the DTS can be computed as:

Change DTS = (New Net Debt - Old Net Debt) × tax rate

Why is this important? Because the increase in the DTS by 1'200 is an additional source of value that goes to the firm's shareholders. Put differently, because the firm saves taxes with the new financing policy, it generates more money for its shareholders.

What's the value of the firm's equity value under the new financing policy?

If we assume that the proceeds from debt financing will be paid out to the shareholders as a dividend (or in a share repurchase), the value of the firm's equity after the dividend payout (E*) will be:

E* = E + Change DTS - Dividend payments = 8'000 + 1'200 - 4'000 = 5'200.

So we have all the ingredients to estimate the relevered beta.

• Unlevered beta (β_A) = 1.34
• Target debt level (D*) = 6'000
• Target beta of debt (β_D*) = 0.6
• Target equity value (E*) = 5'200
• Corporate tax rate (τ) = 30%

Using the full equations, this implies an relevered beta (β_E*) of 1.93:

\( \beta_E^* = \beta_A + (\beta_A - \beta_D^*) \times \big(\frac{D \times (1-\tau)}{E}\big)^* \).

\( \beta_E^* = 1.34 + (1.34 - 0.6) \times \frac{6'000 \times (1-0.3)}{5'200} = 1.93 \).

In contrast, if we use the simplified approach and set β_D = 0, the result is as follows. The unlevered beta is 1.28 instead of 1.34: 

\( \beta_{A, simplified} = \frac{\beta_E}{1+(1-\tau) \times \frac{D}{E}} \).

\( \beta_{A, simplified} = \frac{1.5}{1+(1-0.3) \times \frac{2'000}{8'000}} = 1.28 \).

And the relevered beta with the simplified version is 2.31 instead of 1.93: 

 
\( \beta_{E, simplified}^* = \beta_A \times \bigg(1+(1-\tau) \times \big(\frac{D}{E}\big)^*\bigg) \).

\( \beta_{E, simplified}^* = 1.28 \times \big(1+(1-0.3) \times \frac{6'000}{5'200}\big) = 2.31\).

The conclusion is the same as in the previous chapter. While the simplified version is computationally less challenging, we see that the resulting relevered beta differs substantially from the value we get when using the "correct" equations. We should therefore stick to the correct equations and ignore the simplified approach to unlever and relever the betas. 

For your convenience, the provided excel file also allows you to unlever and relever the cost of capital for the case where the DTS falls into the same risk category as the firm's debt.