<p>Note that with taxes, the value of the company is</p>
<p>{ the value of the company as if it had no debt } + { the value of the tax shield }:
<span class="math-container">$$
V_L=V_U+T_C D;
$$</span>
here, <span class="math-container">$V_L$</span> is the value of a levered company and <span class="math-container">$V_U$</span> is the value of an  unlevered company. Now, from the ownership perspective, this is split between equity and debt,
<span class="math-container">$$
V_U+T_C D=E+D. \tag{1}
$$</span>
The required cash flows corresponding to both sides of the equation are
<span class="math-container">$$
V_U R_U+T_C D R_D = E R_E + D R_D.
$$</span>
Divide both sides by <span class="math-container">$E$</span> to obtain
<span class="math-container">$$
\frac{V_U}{E} R_U+T_C \frac{D}{E} R_D = R_E + \frac{D}{E} R_D.
$$</span>
Rearrange to find out that
<span class="math-container">\begin{aligned}
R_E &amp;= \frac{V_U}{E} R_U+T_C \frac{D}{E} R_D - \frac{D}{E} R_D \\
    &amp;= \frac{V_U}{E} R_U - (1-T_C) \frac{D}{E} R_D.
\end{aligned}</span>
Express <span class="math-container">$V_U$</span> using <span class="math-container">$(1)$</span> as <span class="math-container">$V_U=E+(1-T_C)D$</span> and substitute in the equation above to obtain
<span class="math-container">\begin{aligned}
R_E &amp;= \frac{E+(1-T_C)D}{E} R_U - (1-T_C) \frac{D}{E} R_D \\
    &amp;= \left(1+(1-T_C)\frac{D}{E}\right)R_U - (1-T_C) \frac{D}{E} R_D \\
    &amp;= R_U + (R_U-R_D)\cdot (1-T_C)\cdot \frac{D}{E}
\end{aligned}</span>
which is what we needed.</p>

CC BY-SA 4.0
2022-08-25T16:34:41.697Z
2022-10-21T08:20:54.637Z
2022-10-21T08:20:54.637Z
Richard Hardy
<p>I am looking at Modigliani &amp; Miller Proposition II with corporate taxes. According to Hillier et al. &quot;Fundamentals of Corporate Finance&quot; (3rd ed., 2017) (<a href="https://www.mheducation.co.uk/fundamentals-of-corporate-finance-4e-9781526848628-emea-group" rel="nofollow noreferrer">here</a> is a link to a slightly different edition),</p>
<blockquote>
<p>M&amp;M Proposition II with corporate taxes states that the cost of equity is
<span class="math-container">$$
R_E=R_U+(R_U-R_D)\cdot \frac{D}{E}\cdot (1-T_C) \tag{15.4}
$$</span></p>
</blockquote>
<p>where <span class="math-container">$R_E$</span> is the cost of equity, <span class="math-container">$R_U$</span> is the cost of capital a firm would have if it had no debt (unlevered cost of capital), <span class="math-container">$R_D$</span> is the cost of debt, <span class="math-container">$D$</span> is debt, <span class="math-container">$E$</span> is equity and <span class="math-container">$T_C$</span> is the corporate tax rate.</p>
<p><strong>How is this equation derived?</strong> It is intuitive to me that <span class="math-container">$-R_D\cdot\frac{D}{E}$</span> is multiplied with <span class="math-container">$1-T_C$</span> but not that <span class="math-container">$R_U\cdot\frac{D}{E}$</span> is multiplied with the same thing. Or actually, the presence and nature of <span class="math-container">$R_U$</span> is probably what is confusing me.</p>

CC BY-SA 4.0
2022-08-25T12:24:43.377Z
Related question: [Understanding Modigliani & Miller: different graphs in different textbooks](https://economics.stackexchange.com/questions/52457).
Richard Hardy
<finance><corporate-finance><capital-structure>
Modigliani & Miller with taxes: how is this equation derived?
Richard Hardy
# Modigliani & Miller with taxes: how is this equation derived?

(Asked by: username_0 on 2022-08-24)

I am looking at Modigliani & Miller Proposition II with corporate taxes. According to Hillier et al. "Fundamentals of Corporate Finance" (3rd ed., 2017) ([here](https://www.mheducation.co.uk/fundamentals-of-corporate-finance-4e-9781526848628-emea-group) is a link to a slightly different edition),

> M&M Proposition II with corporate taxes states that the cost of equity is
> $$
> R\_E=R\_U+(R\_U-R\_D)\cdot \frac{D}{E}\cdot (1-T\_C) \tag{15.4}
> $$

where $R\_E$ is the cost of equity, $R\_U$ is the cost of capital a firm would have if it had no debt (unlevered cost of capital), $R\_D$ is the cost of debt, $D$ is debt, $E$ is equity and $T\_C$ is the corporate tax rate.

**How is this equation derived?** It is intuitive to me that $-R\_D\cdot\frac{D}{E}$ is multiplied with $1-T\_C$ but not that $R\_U\cdot\frac{D}{E}$ is multiplied with the same thing. Or actually, the presence and nature of $R\_U$ is probably what is confusing me.

**Tags:** finance, corporate-finance, capital-structure
> **Comments:**
> **username_0**: Related question: [Understanding Modigliani & Miller: different graphs in different textbooks](https://economics.stackexchange.com/questions/52457).  

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## 1 answers

### Answer by username_0 on 2022-08-25. Score: 4 | Accepted answer

Note that with taxes, the value of the company is

{ the value of the company as if it had no debt } + { the value of the tax shield }:
$$
V\_L=V\_U+T\_C D;
$$
here, $V\_L$ is the value of a levered company and $V\_U$ is the value of an unlevered company. Now, from the ownership perspective, this is split between equity and debt,
$$
V\_U+T\_C D=E+D. \tag{1}
$$
The required cash flows corresponding to both sides of the equation are
$$
V\_U R\_U+T\_C D R\_D = E R\_E + D R\_D.
$$
Divide both sides by $E$ to obtain
$$
\frac{V\_U}{E} R\_U+T\_C \frac{D}{E} R\_D = R\_E + \frac{D}{E} R\_D.
$$
Rearrange to find out that
\begin{aligned}
R\_E &= \frac{V\_U}{E} R\_U+T\_C \frac{D}{E} R\_D - \frac{D}{E} R\_D \\
&= \frac{V\_U}{E} R\_U - (1-T\_C) \frac{D}{E} R\_D.
\end{aligned}
Express $V\_U$ using $(1)$ as $V\_U=E+(1-T\_C)D$ and substitute in the equation above to obtain
\begin{aligned}
R\_E &= \frac{E+(1-T\_C)D}{E} R\_U - (1-T\_C) \frac{D}{E} R\_D \\
&= \left(1+(1-T\_C)\frac{D}{E}\right)R\_U - (1-T\_C) \frac{D}{E} R\_D \\
&= R\_U + (R\_U-R\_D)\cdot (1-T\_C)\cdot \frac{D}{E}
\end{aligned}
which is what we needed.


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