Modigliani and Miller Formula

[Rukus520]

Why is the Modigliani and Miller Proposition 2 formula with taxes “r_e = r_o + (r_o - r_d) * (1 - t) * (D/E)” and not r_e = r_o + (r_o - r_d * (1 - t)) * (D/E)? If you take the WACC formula and solve for r_e, you get the latter, not the former:
WACC = r_o = r_e * E/(E+D) + r_d * (1-t) * D/(E+D)
r_o * (E+D) = r_e * E + r_d * (1-t) * D
r_o * E + r_o * D = r_e * E + r_d * (1-t) * D
r_o + r * D/E - r_d * (1-t) * D/E = r_e
r_e = r_o + [ r_o - r_d * (1-t) ] * D/E

 

[KevPei]

great question, I was wondering the same thing my self.

 

[QuangHoang]

Which suggests that MM prop 2 doesn’t come from the WACC formula? I believe this formula has been discussed here previously, just look for the old thread.
However, intuitively, I believe the WACC formula is one way only, i.e. you calculate your WACC from r_d and r_e, not the other way around. If it’s two way, then a company will never gain from changing its capital structure.

 

[KevPei]

well.. intuitively I can make sense of it.. the problem i’m having is the mathematical derivation of MM prop 2 with taxes.. like op demonstrated, when you actually try to isolate re, you’d get r_e = r_o + [ r_o - r_d * (1-t) ] * D/E instead of r_e = r_o + (r_o - r_d) * (1 - t) * (D/E) in the CFAI text. And the ladder just simply cant be algebraically manipulated into the latter.

 

[QuangHoang]

The important thing missed out here is that in arriving at r_e = r_o + [r_o - r_d*(1-t)] * D/E, the OP has assumed that WACC = r _o.
Starting from WACC = r_e * E/(E+D) + r_d * (1-t) * D/(E+D), what we have is r_e = WACC + [WACC - r_d*(1-t)] * D/E not r_e = r_o + [r_o - r_d*(1-t)] * D/E.
r_e will only equal r_o + [r_o - r_d*(1-t)] * D/E if WACC = r_o, which is only true when D = 0. Remember, your WACC is not a given number, it’s a calculated number based on two (sometimes three) inputs: r_d and r_e (and cost of preferred equity).
PS: that’s just my reasoning, if you want to get to the bottom of this, then the CFAI text does refer to the Financial theory and Corporate policy (2005)