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Beta - Unlevered and Levered

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Beta is a measure of market risk.

Unlevered Firm u

If a firm has no debt, it's all equity-financed and thus its equity's beta βE\beta_{E} equals its asset's beta βA\beta_{A}. This beta is also the unlevered beta, βunlevered\beta_{\text{unlevered}}, since it's unaffected by leverage. The unlevered beta measures the market risk exposure of the firm's shareholders. Let's call this firm uu, Hence, we have:

βunlevered=βEu=βAu\begin{equation} \beta_{\text{unlevered}}=\beta_E^u=\beta_A^u \end{equation}

This equality says that in an unlevered firm, the unlevered beta equals its equity beta and its asset beta.

Levered Firm l

If the same firm is partly financed by debt, let's call it firm ll. The asset of the levered firm ll is financed by both equity and debt, and hence the asset's market risk is from both equity and debt. The asset's beta is a weighted average of its equity beta and debt beta.

βAl=EE+D(1t)βEl+D(1t)E+D(1t)βDl\begin{equation} \beta_A^l = \frac{E}{E+D(1-t)} \beta_E^l + \frac{D(1-t)}{E+D(1-t)} \beta_D^l \end{equation}

This part is not very hard to understand. The beta of a portfolio is the weighted average beta of its constituents. If you believe that debt beta is zero since the value of debt may not be affected by the equity market, then βDl=0\beta_D^l=0 and the equation (2) can be simplified to:

βAl=EE+D(1t)βEl=11+DE(1t)βEl\begin{align} \beta_A^l &= \frac{E}{E+D(1-t)} \beta_E^l \newline &= \frac{1}{1+\frac{D}{E}(1-t)} \beta_E^l \end{align}

However, this firm's shareholders are now more exposed to the market risk than before, because leverage increases the variation in the payoff to shareholders. This means the equity's beta of this levered firm is higher than the equity's beta of the unlevered firm, i.e. βEl>βEu\beta_E^l>\beta_E^u.

Note that, the levered beta βlevered\beta_{\text{levered}} that we talk about refers to βEl\beta_E^l, which is the equity beta of the levered firm ll.

Unlevered vs Levered

On the other hand, firm uu and firm ll differ only in capital structure whilst both have the same asset. Let's say we have a portfolio of firm uu's asset and the other portfolio of firm ll's asset, then these two portfolios should have the same expected return and market risk exposure.1 This means the two portfolios have the same beta, implying:

βAu=βAl\begin{equation}\beta_A^u = \beta_A^l \end{equation}

If we substitue in the definition of unlevered and levered beta (equation (1) and (4)):

βunlevered=11+DE(1t)βlevered\begin{equation} \beta_{\text{unlevered}} = \frac{1}{1+\frac{D}{E}(1-t)} \beta_{\text{levered}} \end{equation}

or

βlevered=(1+DE(1t))βunlevered\begin{equation} \beta_{\text{levered}} = \left( 1+\frac{D}{E}(1-t) \right) \beta_{\text{unlevered}} \end{equation}

This is the formula that we use to lever and unlever beta.2

Further Clarification

The equity beta of a firm with debts is levered. To remove the impact of leverage on shareholders' market risk exposure, we need to unlever this beta in order to get the unlevered beta. This unlevered beta is also called the asset beta.

Note that the asset beta is a syncronym for unlevered beta. It is not, however, the asset's beta βAl\beta_A^l when the firm is leveraged as in equation (2) to (4). This convention is confusing indeed, so throughout this post, I'm using asset's beta to refer to the beta of a portfolio of all securities (debt and equity) of the levered firm.

Notations

Footnotes

  1. Modigliani-Miller theorem states that the capital structure should not affect a firm's value.

  2. This eq.(7) is also named Hamada Equation, where we assumed a zero debt beta. It draws on the Modigliani-Miller theorem on capital structure, and appeared in Prof. Robert Hamada's paper "The Effect of the Firm's Capital Structure on the Systematic Risk of Common Stocks" in the Journal of Finance in 1972.