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

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.

• Aug 25, 2022 at 12:24

{ 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.