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
great question, I was wondering the same thing my self.
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.
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.
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)
THE EQUATION IS NOT WRONG, just remember to use the correct expression to V, similar to the one used to derive the unlevered beta V = Dx(1-t) + E
Let’s derive this formula using V = Dx(1-t) + E
Ro = (D/V)xRd(1-t) + (E/V)xRe
Ro - (DxRd(1-t)) / V = (E/V)xRe
Taking LCM and cutting both denominators (V)
VRo - DRd(1-t) = ERe
(VxRo)/E - (DxRd(1-t))/E= Re
Here we go with V = D(1.t)+E, Substituting V with D(1-t)+E
((D(1-t)+E)Ro)/E - (DxRd(1-t))/E = Re
Rewrite it
(D(1-t)/E x Ro) + ERo/E - D/E x Rd(1-t) = Re
(D(1-t)/E x Ro) + ERo/E - D/E x Rd(1-t) = Re
(D(1-t)/E)Ro + Ro - (D/E)Rd(1-t) = Re
Ro + (D(1-t)/E)Ro - (D/E)Rd(1-t) = Re
Taking D/E as common
Ro + (D/E) (Ro - Rd)(1-t) = Re
Rewrite it
Re = Ro + (Ro - Rd)(1-t)D/E
It’s derived in the correct way as Modiggliani Miller proposed first!
In EOC 26 Q3 they ask you to determine the cost of equity capital. Straight away I used this formula:
re = ro + (ro-rd)(1-t)(D/E)
And i got it wrong. They calculated re as the missing variable in the WACC equation (you were given the other variables).
How do we know what formula to use to calculate cost of equity capital???
In case we use the explanation given by rafaelespano, then every time we take value of a firm, do we take after tax debt or is this applicable only to MM theory? I’m asking this because when we calculate WACC we take total value of debt to find V and not after tax value.
The formula is a simplification and only works when ro = wacc if all equity-financed. In the excercise that you are using as an example there is no clue if the company is full equity financed.
See EOC 26 Q10 and you’ll see that they give you the wacc when firm is all equity- financed
Remember that you are looking for an after tax-cost of something weighted against total costs when we are talking about Modigliani Miller. Free Cash Flows are calculated before taking into account all costs so you have to evaluate all costs using proxy variables, such as a weighted average of total costs in an after-tax basis.
On the other hand, when you calculate the value of a firm you are not looking for the cost it takes for raising capital but for the equivalent cash value of its assets, so V = Total Net Claims = Debt + Equity - Cash and Equivalents.