<p>I have encountered the following example in the context of cost of capital:</p>
<blockquote>
<p>A firm is considering investing into a project that has the same risk profile as the rest of the firm's projects. It may finance it with 100% equity, 75% equity or 50% equity. The firm can take a loan at 6% interest rate. The risk free rate is 3% and market's risk premium is 6%. The firm's beta is 1.5. We will ignore corporate tax. What will be the WACC in the three cases (100% equity, 75% equity and 50% equity)?</p>
</blockquote>
<p>The suggested answer uses the usual WACC equation without tax:
<span class="math-container">$$
R_{\text{WACC, no tax}}=\frac{E}{E+D}R_E+\frac{D}{E+D}R_D,
$$</span>
plugs in <span class="math-container">$\frac{E}{E+D}=1$</span> or <span class="math-container">$\frac{E}{E+D}=0.75$</span> or <span class="math-container">$\frac{E}{E+D}=0.5$</span> (and correspondingly <span class="math-container">$\frac{D}{E+D}=0$</span> or <span class="math-container">$\frac{D}{E+D}=0.25$</span> or <span class="math-container">$\frac{D}{E+D}=0.5$</span>) and gets a different number in each of the three cases.</p>
<p>However, I feel this goes against the <a href="https://en.wikipedia.org/wiki/Modigliani%E2%80%93Miller_theorem" rel="nofollow noreferrer">Modigliani-Miller theorem</a>. <strong>Question:</strong> Should the WACC not be the same regardless of the financing mix, as long as (1) we ignore corporate tax and (2) the investment project under consideration is the same? (And as @AKdemy mentions in a comment, should <span class="math-container">$R_E$</span> not vary with the level of leverage?)</p>
<p>I suppose the answer may depend on how big the new project is relative to the firm's portfolio of existing projects. For concreteness, let us assume the new project is very small, so the firm's capital structure does not change noticeably regardless of how it chooses to finance the new project. Then it would make sense to me if <span class="math-container">$R_{\text{WACC, no tax}}$</span> were determined by the firm's overall capital structure (on which the example is silent).</p>

CC BY-SA 4.0
2022-10-19T09:37:04.233Z
Another related question that has not been answered: https://economics.stackexchange.com/questions/53109
Richard Hardy
CC BY-SA 4.0
2022-10-19T11:48:38.110Z
[Wikipedia MM Theorem](https://en.wikipedia.org/wiki/Modigliani%E2%80%93Miller_theorem#Without_taxes)- $R_E$ is a function of the debt-to-equity ratio. A higher debt-to-equity ratio leads to a higher required return on equity, because of the higher risk involved for equity-holders in a company with debt. Insofar, you cannot simply set your $R_E$ to a fixed number.
AKdemy
CC BY-SA 4.0
2022-10-19T12:03:38.417Z
@AKdemy, indeed, and that must be the second problem with this example (a problem I did not immediately recognize). Could you say anything more about the rest of my question?
Richard Hardy
<corporate-finance><cost><capital>
Does this example involving WACC make sense?
Richard Hardy
# Does this example involving WACC make sense?

(Asked by: username_0 on 2022-10-19)

I have encountered the following example in the context of cost of capital:

> A firm is considering investing into a project that has the same risk profile as the rest of the firm's projects. It may finance it with 100% equity, 75% equity or 50% equity. The firm can take a loan at 6% interest rate. The risk free rate is 3% and market's risk premium is 6%. The firm's beta is 1.5. We will ignore corporate tax. What will be the WACC in the three cases (100% equity, 75% equity and 50% equity)?

The suggested answer uses the usual WACC equation without tax:
$$
R\_{\text{WACC, no tax}}=\frac{E}{E+D}R\_E+\frac{D}{E+D}R\_D,
$$
plugs in $\frac{E}{E+D}=1$ or $\frac{E}{E+D}=0.75$ or $\frac{E}{E+D}=0.5$ (and correspondingly $\frac{D}{E+D}=0$ or $\frac{D}{E+D}=0.25$ or $\frac{D}{E+D}=0.5$) and gets a different number in each of the three cases.

However, I feel this goes against the [Modigliani-Miller theorem](https://en.wikipedia.org/wiki/Modigliani%E2%80%93Miller_theorem). **Question:** Should the WACC not be the same regardless of the financing mix, as long as (1) we ignore corporate tax and (2) the investment project under consideration is the same? (And as @AKdemy mentions in a comment, should $R\_E$ not vary with the level of leverage?)

I suppose the answer may depend on how big the new project is relative to the firm's portfolio of existing projects. For concreteness, let us assume the new project is very small, so the firm's capital structure does not change noticeably regardless of how it chooses to finance the new project. Then it would make sense to me if $R\_{\text{WACC, no tax}}$ were determined by the firm's overall capital structure (on which the example is silent).

**Tags:** corporate-finance, cost, capital
> **Comments:**
> **username_0**: Another related question that has not been answered: https://economics.stackexchange.com/questions/53109  
> **username_1**: [Wikipedia MM Theorem](https://en.wikipedia.org/wiki/Modigliani%E2%80%93Miller_theorem#Without_taxes)- $R_E$ is a function of the debt-to-equity ratio. A higher debt-to-equity ratio leads to a higher required return on equity, because of the higher risk involved for equity-holders in a company with debt. Insofar, you cannot simply set your $R_E$ to a fixed number.  
> **username_0**: @AKdemy, indeed, and that must be the second problem with this example (a problem I did not immediately recognize). Could you say anything more about the rest of my question?  

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## 0 answers

