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# Pure Play Method: Derivation of Un-levering & Re-levering of Beta Formula

Updated: Jul 9, 2020

Derivation of un-levering and re-levering the beta

• According to CAPM, the expected return of a security is linearly related to its correlation with the return on the market portfolio

• This correlation is reflected by the betas of the securities

If we consider the expected returns of unlevered asset, equity and a risk free bond using CAPM method, the formulas will be as follows:

ru = rf + βu(rm – rf) – (1)

re = rf + βe(rm – rf) – (2)

rd = rf + βd(rm – rf) – (3)

We take the help of Modigliani & Miller’s 2nd Proposition (With Taxes) which states that as debt increases, equity becomes riskier, however WACC reduces due to tax shield.

Equation of M&M’s 2nd proposition is: re = ru + (1 – t) (D / E) (ru – rd) – (4)

Plug the return on un-levered asset and risk free bond, i.e. equation (1) & equation (3) in equation (4)

re = rf + βu(rm – rf) + (1 – t) (D / E) [rf + βu(rm – rf) – rf + βd(rm – rf)] - (5)

re = rf + βu(rm – rf) + (1 – t) (D / E)( (rm – rf) (βu - βd) - (6)

Rewriting return on equity, we get the 2nd CAPM equation mentioned above:

re = rf + [βu + (1 – t) (D / E) (βu – βd)] (rm – rf) - (7)

Equation number (7) is a version of CAPM in equation (2), where:

βe = βu + (1 – t) ) (D / E) (βu – βd) - (8)

Since rd is the return on a risk free bond, βd = 0, i.e. it is not correlated with market movements.

βe = βu (1+ (1 – t) (D/ E)) - (9) [Equation for Re-Levering Beta]

Rewriting the above equation:

βu = βe / [(1+ (1 – t) (D/ E)] - (10) [Equation for Un-Levering Beta]

Note:

In Equation (10), βe , tax rate, debt and equity are that of the comparable firm

In Equation (9), tax rate, debt and equity are that of the project.