3
$\begingroup$

I am looking at Modigliani & Miller Proposition II with corporate taxes. According to Hillier et al. "Fundamentals of Corporate Finance" (3rd ed., 2017) (here 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.

$\endgroup$
1

1 Answer 1

4
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.