Adjusting Levered Beta and Discount Rates as Debt Ratios Change

Summary

The document starts from the Hamada relationship between unlevered beta, the tax rate, and the debt-to-equity ratio. It asks how to apply that relationship when a company's leverage is expected to decline over time as debt is repaid, rather than remain constant. The author proposes calculating a period-specific levered beta from each period's debt-to-equity ratio and asks whether each period's cash flow should then be discounted using its corresponding beta.

The post also points to the interaction between financing choices, free cash flows, and valuation: debt repayment changes cash available to equity holders, while lower leverage changes systematic risk. However, it does not derive the appropriate discount rate or establish that substituting a time-varying Hamada beta directly into a valuation is valid. The question therefore surfaces a modeling issue but leaves unresolved the distinction between valuing enterprise cash flows and equity cash flows, as well as the assumptions needed for a consistent valuation.

Key ideas

• The Hamada relationship links levered beta to unlevered beta and the debt-to-equity ratio after tax effects.
• If leverage changes over time, applying the relationship period by period produces changing levered beta estimates.
• Debt repayment affects both cash flows and the risk associated with equity.
• A valuation must use discount rates consistent with the cash flows being valued, an issue the document raises but does not resolve.

Tags

# Levered beta with changing equity/debt ratios


# Levered beta with changing equity/debt ratios












I know how to calculate a bottom up levered beta for a privately held and not publicly traded company with Hamada (Proof of Hamada's Formula (Relationship between levered and unlevered beta)) and the help of http://people.stern.nyu.edu/adamodar/New_Home_Page/TenQs/TenQsBottomupBetas.htm.

$$\beta_{L}=\beta_U×(1+(1-t)\times D/E)$$

However, all the explanations I see in the literature assume that the debt ratio $\theta = D/E$ is a constant. Only sometimes it is casually mentioned that the betas need to be adjusted when D/E changes, but this is never formalised.

What happens if I do a valuation of a company that plans to pay off all debt so that for the future periods $t = 1,2,...$ : $\theta_1 = 0.5$, $\theta_2 = 0.25$, $\theta_3 = 0$. Do we have a time series of betas?

$$\beta_{L,t}=\beta_U×(1+(1-t)\times \theta_t)$$

...and do we need to discount future cash flows in each period by its own levered beta? In $t = 1,2$ the free cash flows will be much lower since the debt is paid off but for $t \geq3: \beta_{L,t}=\beta_U$. The value of the business should be much higher because free cash flows are hihger and the discount factor is smaller. Am I right?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.