MM Prop 2 and WACC

Starting with WACC for pure equity company:


= Ra

= (D/D+E) Rd + (E/E+D) Re

Where Ra is average return from assets, Rd and Rd are return on debt and equity respectively.

MM I states Ra is invariant to financing, thus Re adjusts by just enough to keep WACC constant when debt levels change

WACC w corporate taxes

= Ra

= (D/D+E) Rd (1-tc) + (E/E+D) Re (2)

only difference now is the addition of (1-tc) as a multiplier to transform Rd into effective Rd

My question is that when I rearrange (2) to find Re I get:

Re = Ra + D/E [ Ra- (1-Tc) Rd )

which is clearly different from MM II’s formula:

Re = Ra + D/E ( Ra - Rd ) (1-Tc)

am I doing anything wrong? The formulas don’t tally.

There’s a LOT of algebra between D/V(Rd)(1-TR) +E/V(Re) to get to the final product.

You can look up proofs online but it’s waste of time. Just memorize the final solution.