<p>The first equation can be written as:</p>

<p>$$ r_E(Levered) = \frac{E+D}{E}r_E(Unlevered) - \frac{D}{E}r_D $$</p>

<p>Then, isolating the unlevered return gives:</p>

<p>$$ r_E(Unlevered) = \frac{E}{E+D}r_E(Levered) + \frac{D}{E+D}r_D$$</p>

<p>And this is the WACC.</p>

CC BY-SA 3.0
2015-01-08T02:38:44.477Z
2015-01-08T02:38:44.477Z
Bryce
<p>In the Wikipedia article on the <a href="https://en.wikipedia.org/wiki/Modigliani%E2%80%93Miller_theorem#Without_taxes" rel="nofollow noreferrer">Modigliani--Miller theorem</a>, it states two propositions. (It gives the cases of with and without taxes. Here I'll just focus on the case without taxes.)
The first proposition is that the value of an unlevered firm is the same as a levered firm. Given the assumptions, this is clear from the discussion:</p>

<blockquote>
  <p>To see why this should be true, suppose an investor is considering buying one of the two firms U or L. Instead of purchasing the shares of the levered firm L, he could purchase the shares of firm U and borrow the same amount of money B that firm L does. The eventual returns to either of these investments would be the same. Therefore the price of L must be the same as the price of U minus the money borrowed B, which is the value of L's debt.</p>
</blockquote>

<p>However, here I am asking about "Proposition II:"</p>

<blockquote>
  <p>$$r_E(Levered) = r_E(Unlevered) + \frac DE (r_E(Unlevered) - r_D),$$
  where</p>
  
  <ul>
  <li>$r_E$ ''is the required rate of return on equity, or cost of equity,''</li>
  <li>$r_D$ ''is the required rate of return on borrowings, or cost of debt,''</li>
  <li>and $\frac{D}{E}$ ''is the debt-to-equity ratio.''</li>
  </ul>
</blockquote>

<p>The article states that the "formula is derived from the theory of weighted average cost of capital (WACC)." (See a related question <a href="https://economics.stackexchange.com/questions/2932/deriving-and-explaining-the-weighted-cost-of-capital">here</a>.) My question is this: how can we arrive at this result from WACC?</p>

CC BY-SA 4.0
2022-10-31T16:24:26.950Z
And here is the more intricate case with corporate tax: https://economics.stackexchange.com/questions/52465
Richard Hardy
<finance><financial-economics><corporate-finance>
Deriving the Modigliani--Miller Theorem
jmbejara
# Deriving the Modigliani--Miller Theorem

(Asked by: username_0 on 2015-01-07)

In the Wikipedia article on the [Modigliani--Miller theorem](https://en.wikipedia.org/wiki/Modigliani%E2%80%93Miller_theorem#Without_taxes), it states two propositions. (It gives the cases of with and without taxes. Here I'll just focus on the case without taxes.)
The first proposition is that the value of an unlevered firm is the same as a levered firm. Given the assumptions, this is clear from the discussion:

> To see why this should be true, suppose an investor is considering buying one of the two firms U or L. Instead of purchasing the shares of the levered firm L, he could purchase the shares of firm U and borrow the same amount of money B that firm L does. The eventual returns to either of these investments would be the same. Therefore the price of L must be the same as the price of U minus the money borrowed B, which is the value of L's debt.

However, here I am asking about "Proposition II:"

> $$r\_E(Levered) = r\_E(Unlevered) + \frac DE (r\_E(Unlevered) - r\_D),$$
> where
>
> * $r\_E$ ''is the required rate of return on equity, or cost of equity,''
> * $r\_D$ ''is the required rate of return on borrowings, or cost of debt,''
> * and $\frac{D}{E}$ ''is the debt-to-equity ratio.''

The article states that the "formula is derived from the theory of weighted average cost of capital (WACC)." (See a related question [here](https://economics.stackexchange.com/questions/2932/deriving-and-explaining-the-weighted-cost-of-capital).) My question is this: how can we arrive at this result from WACC?

**Tags:** finance, financial-economics, corporate-finance
> **Comments:**
> **username_1**: And here is the more intricate case with corporate tax: https://economics.stackexchange.com/questions/52465  

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## 1 answers

### Answer by username_2 on 2015-01-08. Score: 6 | Accepted answer

The first equation can be written as:

$$ r\_E(Levered) = \frac{E+D}{E}r\_E(Unlevered) - \frac{D}{E}r\_D $$

Then, isolating the unlevered return gives:

$$ r\_E(Unlevered) = \frac{E}{E+D}r\_E(Levered) + \frac{D}{E+D}r\_D$$

And this is the WACC.


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