# 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.

To know more about M&M’s 2nd proposition, __click here__

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.