# Modigliani-Miller & Unlevered Betas

In 1958 Franco Modigliani and Merton Miller wrote a nobel-prize winning paper describing how leverage effects the value of companies. In this post we’ll discuss some of their results and apply it to the derivation of a well-known formula used to de-lever equity betas.

Suppose there exist two companies that own exactly the same assets and produce exactly the same stream of uncertain earnings. Let one company be completely unlevered, i.e. completely financed with equity, and the other company be partially financed by debt. Denote the annual earnings before interest and tax (or EBIT) of both companies by the random variables  $X_1, X_2, X_3, \ldots$ that are governed by some probability distributions. We assume that the value of each firm is only a function of its stream of after-tax EBIT (or EBIAT). However, since interest payments are tax-deductible, the value of each firm will depend on its leverage.

First suppose that the company is unlevered, i.e. has no debt and is fully financed with equity. In this case, on year $i$, the company produces after-tax EBIT of $X_i(1-\tau)$ where $\tau$ is the corporate tax rate for this company. The value of the company can be written as

$\displaystyle V_U = \mathbb{E}\left[\frac{X_1(1-\tau)}{(1+r_U)} + \frac{X_2(1-\tau)}{(1+r_U)^2} + \cdots \right]$

where $r_U$ is some discount factor appropriate for the risk associated with the earnings stream.

Note that we are discounting earnings and not cash. We assume that earnings convert into cash quickly and that management acts in the best interest of the shareholders when reinvesting earnings instead of paying cash dividends to equity holders.

Now consider the case where the company is partially financed with debt. Let $D$ be the market value of the debt and $r_d$ is the interest paid on debt or the cost of debt. On year $i$, this company will produce earnings after tax and before interest of $\displaystyle (X_i-r_dD)(1-\tau)+r_dD=X_i(1-\tau)+r_d\tau D$

Notice that we can separate these earnings into earnings identical to the unlevered firm $X_i(1-\tau)$, and fixed earnings $r_d\tau D$ from the tax benefit of debt financing. In fact, we can replicate these earnings exactly by holding a portfolio of the unlevered company and $\tau D$ amount of the debt of the company. Therefore, the value of the levered company must be exactly equal to the value of the portfolio containing the unlevered company and $\tau D$ debt.

$V_L = V_U+\tau D$

### Unlevered Betas

Now we will use the relationship derived above to show how the beta of a company’s equity to the market changes with its leverage. Let $E$ represent the value of the equity of a levered company (share price or market cap), and let $M$ be the level of a market index. By CAPM, we know that

$\displaystyle \frac{\Delta E}{E}=\beta_L \frac{\Delta M}{M} + \epsilon$

Where $\epsilon$ is a a random with $\epsilon \sim \mathcal{N}(0,\sigma^2)$. Here, $\beta_L$ is said to be the levered beta of the equity. Now suppose that the company had no leverage and denote the hypothetical value of the unlevered firm as $V_U$. Then we can write

$\displaystyle \frac{\Delta V_U}{V_U}=\beta_U \frac{\Delta M}{M} + \epsilon$

where $\beta_U$ is the unlevered beta of the company. Solving for $\frac{\Delta M}{M}$ in each and setting those equal to each other, we get

$\displaystyle \frac{1}{\beta_L}\left(\frac{\Delta E}{E} -\epsilon\right)=\frac{1}{\beta_U}\left(\frac{\Delta V_U}{V_U}-\epsilon\right)$

From above, we know that $V_L =D+E=V_U+\tau D$ so we can replace $V_U$ with $E+(1-\tau)D$. If we assume $D$ is constant and does not fluctuate with the market, this also implies that $\Delta E = \Delta V_U$.

$\displaystyle \frac{1}{\beta_L}\left(\frac{\Delta E}{E} -\epsilon\right)=\frac{1}{\beta_U}\left(\frac{\Delta E}{E+(1-\tau)D}-\epsilon\right)$

Taking expectations

$\displaystyle \frac{1}{\beta_L}\left(\frac{\mathbb{E}[\Delta E]}{E} \right)=\frac{1}{\beta_U}\left(\frac{\mathbb{E}[\Delta E]}{E+(1-\tau)D}\right)$

$\displaystyle \beta_U=\frac{E}{E+(1-\tau)D}\beta_L=\frac{\beta_L}{1+(1-\tau)\frac{D}{E}}$

## One comment

1. xi says:

thank you, it’s very helpful

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