Unlevering Equity Beta with the Hamada Relationship
Summary
The document explains how to remove the effect of financial leverage from an equity beta and how to re-lever an asset beta for a different debt-to-equity ratio. Under the Hamada relationship, the unlevered beta is the levered beta divided by one plus the after-tax debt-to-equity ratio. The discussion corrects an attempted calculation that used addition and misplaced inputs, and distinguishes the levered beta from the unlevered beta in the numerical example.
This method assumes debt has zero market risk. If debt has a nonzero beta and that beta can be estimated, the relationship needs an adjustment for debt risk. The document also notes that market values of debt and equity are relevant to the ratio. It presents a practical simplifying assumption used when debt beta is difficult to estimate, rather than a universal formula that applies regardless of financing conditions.
Key ideas
- Unlevered beta removes the effect of a firm's debt financing from its equity beta.
- Under Hamada's relationship, divide levered beta by one plus the after-tax debt-to-equity ratio.
- Re-lever an asset beta by multiplying it by one plus the after-tax debt-to-equity ratio.
- The basic relationship assumes debt has zero market risk; a nonzero debt beta requires an adjustment.
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Full text
# How to calculate unlevered beta # How to calculate unlevered beta I have derived a firm's cost of equity using the WACC formula (see here), which means that the cost of equity has factored in the firms' debt (i.e. levered beta) and now I need to calculate the firm's unlevered beta. Here is my solution thus far, please let me know if I am on the right track. Formula to calculate unlevered beta: ``` βL = βU + [1 + (1 - t)(d/e)] Where: βL = the firm's beta with leverage = 1.5 βU = the firm's beta with no leverage t = the corporate tax rate = 40% d/e = the firms debt/equity ratio = 35/65 ``` Calculations ``` 1.1 = βU + [1 + (1 - 0.40)(35/65)] 1.1 = βU + [1 + (0.6)(0.538461538461538)] 1.1 = βU + [1 + (0.6)(0.538461538461538)] 1.1 = βU + 1.323077 βU = 1.323077 - 1.1 βU = 0.223077 ``` ### UPDATE I had some errors above, which were pointed out in the answer below. Here is the updated question (which I think is now correct). Revised Formula to calculate unlevered beta: ``` βU = βL * [1 / (1 + (1 - t)(d/e))] Where: βL = the firm's beta with leverage = 1.5 βU = the firm's beta with no leverage t = the corporate tax rate = 40% d/e = the firms debt/equity ratio = 35/65 ``` Revised Calculations ``` βU = 1.5 * [1 / (1 + (1 - 0.40)(35/65)) ] βU = 1.5 * [1 / 1.323077] βU = 1.5 * 0.755814 βU = 1.133721 ``` ## Answer by jeff m (score 1) https://quant.stackexchange.com/a/7877 Your formula is adding where you should be multiplying, and you plugged your inputs into the wrong places (your levered Beta notably). In any case, the process for un-levering/re-levering the beta goes like so: Step 1: Find benchmark company/asset/project Beta. Step 2: Un-lever the benchmark Beta: Unlevered Beta = Levered Beta * (1 / ( 1 + (1 - t)*D/E)) Step 3: Re-lever the beta with your company/projects D/E Ratio: Un-levered Beta * (1 + (1-t)*D/E) ## Answer by anusha (score 1) https://quant.stackexchange.com/a/14971 Unlevered Beta (Beta asset) = Levered Beta / 1+(1-tax) Debt/Equity Similarly , Levered Beta (Beta equity) = Unlevered Beta * 1+ (1-tax) Debt /Equity ## Answer by airstrike (score 1) https://quant.stackexchange.com/a/18079 It depends. If, and only if, you assume that debt carries a market risk of exactly 0, you may use Hamada's equation to easily go from levered to unlevered beta. Let $\theta = D/E$ - $\beta^L = \beta^U \times(1+(1-\tau)\times\theta)$ - $\beta^U = \beta^L \div(1+(1-\tau)\times\theta)$ Where $\tau$ is the tax rate, and $D$ and $E$ are the firm's market value of debt and equity. In practice, a lot of people use that just because it is hard to estimate debt betas. If you dislike that simplifying assumption, and if you have a way to estimate a debt beta, then the correct equation is: - $\beta^L = \beta^U \times(1+(1-\tau)\times\theta) \space – \space \beta_d\times(1-\tau)\times\theta$
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