# 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

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Pages 53 and 54 of Volume 4 of the CFA level curriculum. I'm not sure where you're getting your formula but my book states $\beta_{\textrm{asset}} = \beta_{\textrm{equity}} \frac{1}{1+(1-t){\frac{D}{E}}}$. –  Bob Jansen Apr 29 '13 at 20:42
@BobJansen yes thank you, Bob. I had the formula wrong to begin with. –  Ben Apr 29 '13 at 21:11
If and only if the Beta of debt is zero. –  André Terra Jun 1 at 3:09

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)

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thank you! this is great. Will make amendments to original question to incorporate your answers... cheers –  Ben Apr 29 '13 at 21:02

Unlevered Beta (Beta asset) = Levered Beta / 1+(1-tax) Debt/Equity

Similarly , Levered Beta (Beta equity) = Unlevered Beta * 1+ (1-tax) Debt /Equity

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These formulas assume debt carries a market risk of zero, which is a very simplifying (but sometimes inevitable) assumption. –  André Terra May 31 at 19:09

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 $\theta = D/E$

• $\beta^L = \beta^U *(1+(1-\tau) \theta)$
• $\beta^U = \beta^L \div(1+(1-\tau)\theta)$

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:

• $\beta^L = \beta^U *(1+(1-\tau)\theta) \space – \space \beta_d(1-\tau)\theta$
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For a quick source to this information, please see page 9 (printed page 71) of Prof. Aswath Damodaran's class slides at people.stern.nyu.edu/adamodar/pdfiles/eqnotes/discrate2.pdf –  André Terra May 31 at 19:08