# Modigliani and Miller Proposition problem

In one of the exerices for the cfa books it ask "Suppose the weighted average cost of capital of the Gadget Company is 10 percent. If Gadget has a capital strcuture of 50 percent and 50 percent equity, a before-tax cost of debt 5 percent and a marginal tax rate of 20 percent, then its cost of equity capital is closest to:

a) 12 percent

b) 14 percent

c)16 percent

Can anyone explain why is it not answer b. The correct answer in the books is C. Thanks

Cost of capital = 10% on average

50% comes from debt, which is calculated on an after-tax basis, so 5% * (1 - 20%) = 4%

50% * 4% + 50% * X = 10%

50% * X = 10% - 50% * 4%

50% * X = 10% - 2%

50% * X = 8%

100% * X = 16%

100% = 1

X = 16%

Basic algebra, although I provided every step imaginable.

WACC=.10=(.5)(.05)(1-.2)+(.5)(Cost of Eq)

Solve for Cost of Eq.

isn’t the forumula re= r0+(r0-rd)(1-t)(d/e)?

or that forumala doesnt apply to this equation

D

Check the formula isn’t it Re = R0 + (R0 - Rd(1-t)) x (D/E)?

You can also use this. But you don’t have ro (100% all-equity rate)

This is a straight L1 WACC question.

To use MM II with taxes Re=Ro+(D/E)*(Ro-Rd)*(1-T) you’d need the Debt / Equity ratio which you don’t have. Note also that Re here is marginal cost of equity capital

Under MM Prop II with taxes the cost of equity is a linear function of the debt to equity ratio.

50% debt, 50% equity

D/E = 1

this formula: re= r0+(r0-rd)(1-t)(d/e) is wrong.

right formula is re= r0+(r0-rd(1-t))(d/e)

I also saw the same formular (Re= Ro + D/E*(1-t)*(R0-Rd)) in the shweser note.

So, is shweser note also wrong?

I think it is not!

You are wronging!

Please renember that, R0 is the cost of all equity company.

WACC is cost of equity and debt firm.

WACC = D/VxRd(1-t) + E/VxRe

10 = 0.5 x 5(1-20%) + 0.5xRe

10 = 0.5 x 4 + 0.5xRe

10 = 2 + 0.5xRe

10-2 = 0.5 x Re

8 = 0.5xRe

16 = Re

Solving it the other way

Re = Ro + (Ro - Rd(1-t)) x D/E

Re = 10 + (10 - 5(1-20%) x 0.5/0.5

Re = 10 + (10 - 4) x 1

Re = 10 + 6

Re = 16

Let’s derive this formula

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

We know V = D+E, Substituting V with D+E

((D+E)Ro)/E - (DxRd(1-t))/E = Re

Rewrite it

(D/E x Ro) + ERo/E - D/E x Rd(1-t) = Re

(D/E x Ro) + ERo/E - D/E x Rd(1-t) = Re

(D/E)Ro + Ro - (D/E)Rd(1-t) = Re

Ro + (D/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!

@CFA1310 you first need to multply (1-t) with Rd and then subtract it from Ro to reach to Re

I am agree with u mohammad.belaal

please report it to CFA institute.

I don’t understand why MM is needed. Aaronhotchner provided a correct and detailed answer.

THE EQUATION IS NOT WRONG, I’m going to use mohammad.bellal’s demonstration but using 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!

The issue with the problem is weather the capital structure is D / (D+E) = 50% or D / [D(1-t)+E] = 50%

My thoughts in order to answer the problem assuiming that re = 16% is correct, are below:

In order to have an answer equal to C: re=16, we need that D/E = 1.5 (16% = 10% + (10%-5%)*(1-0.2)*1.5)

So with D/E = 1.5 and t = 0,2 , D/[D*(1-t) +E] will be equal to 0.681818, see how I found it:

If you divide the expresion D/[D*(1-t) +E] by E in both the numerator and denominator you will have (D/E) / [(D/E)*(1-t) +1], and by substituting with D/E = 1.5 and t = 0.2 I arrived to D/ D(1-T)+E = 0.681818

The thing that freaks me out is that there should be a mistake in my calculation or in the way I propose the problem as the solution doesn’t seem to be ok:

WACC = [D / (D(1-t)+E)] x rd x (1-t) + [E / (D(1-t)+E) x re]

10% = 0.681818 x 5% x (1-0.2) + [50% x re]

reordering

re = (10% - [0.681818 x 5% x (1-0.2)] ) / 50%

re = 14.5455%

Would someone help me find a solution without telling me that the formula proposed by MM is wrong and we should change 50 years of papers (and the cfa curriculum too)?

tks

For future reference.

If you have taxes, WACC changes because of debt. So you cannot use (Wacc with debt) = Wacc (without debt)

But we can use Value of company.

V = E + D

Also V = V(u) + Tax shield = Ebit*(1-T)/Ro + D*T

Also, E = (Ebit - interests - taxes)/Re = (EBIT-RdD)(1-T)/Re So, Ebit = ERe/(1-T) + RdD

So,

E+D = Ebit*(1-T)/Ro + D*T (just place de Ebit formula above here, and you will get the result.)

Re = Ro + (Ro - Rd)(1-t)*D/E