Modigliani & Miller with taxes: how is this equation derived?

Question

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.

Answer 1

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.