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 $\beta_{E}$ equals its asset’s beta $\beta_{A}$. This beta is also the **unlevered beta**, $\beta_{\text{unlevered}}$, since it’s unaffected by leverage. The unlevered beta measures the market risk exposure of the firm’s shareholders. Let’s call this firm $u$, Hence, we have:

\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 $l$. The asset of the levered firm $l$ 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.

\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}

$\beta_A^l$ measures the change in the return on a portfolio of all firm $l$’s securities (debt and equity) for each additional one percent change in the market return.

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 $\beta_D^l=0$ and the equation (2) can be simplified to:

$$ \begin{align} \beta_A^l &= \frac{E}{E+D(1-t)} \beta_E^l \newline &= \frac{1}{1+\frac{D}{E}(1-t)} \beta_E^l \end{align} $$

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. $\beta_E^l>\beta_E^u$.

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

## Unlevered vs Levered

On the other hand, firm $u$ and firm $l$ differ only in capital structure whilst both have the same asset. Let’s say we have a portfolio of firm $u$’s asset and the other portfolio of firm $l$’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:

$$\begin{equation}\beta_A^u = \beta_A^l \end{equation}$$

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

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

or

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

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 $\beta_A^l$ 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

- $\beta_E^u$: the equity’s beta of the unlevered firm
- $\beta_A^u$: the asset’s beta of the unlevered firm
- $\beta_E^l$: the equity’s beta of the levered firm
- $\beta_D^l$: the debt’s beta of the levered firm
- $\beta_A^l$: the asset’s beta of the levered firm
- $D$: the size of the firm’s debt
- $E$: the size of the firm’s equity
- $t$: the tax rate
- $\beta_{\text{unleverd}}$:
**unlevered beta**, the equity (asset) beta of the unlevered version of the firm - $\beta_{\text{leverd}}$:
**levered beta**, the equity beta of the levered version of the firm

- Modigliani-Miller theorem states that the capital structure should not affect a firm’s value.
^{[return]} - 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.^{[return]}