Modigliani Miller for the cost of equity with tax

[PierreCFA]

In the Study session 7 (Corporate Finance), reading 22 (Capital Structure), the formula Modigliani Miller for the cost of equity with tax is

r_e=r₀+(r₀-r_d)(1-tax)*D/E (equation 9 in CFA Curriculum book3 page 99)

with r_e : cost of equity, r_d : cost of debt , tax: tax rate , r₀: cost of capital for company financed only by equity, D: the market value of debt, E: the market value of Equity , V = D+E

But I don’t know how to get it. From the equation 3 page 96, r_WACC =r₀ . I replace r_WACC by r₀ in the equation 8 page 98: r_WACC=(D/V)r_d(1-tax) + (E/V)r_e , I get:

r₀ = (D/V)r_d(1-tax) + (E/V)r_e

=> (D+E)/E * r₀ = (D/E)r_d(1-tax) + r_e

=> r₀ +(D/E) *( r₀-r_d(1-tax)) = r_e

where am I wrong?

Thank you in advance!

[BenjaminGraham]

I think there is quite a bit of mathematical manipulation when it comes to deriving this formula. I remember someone saying Basit from Wiley mentioned in one of his videos that the derivation for this formula is not straight forward i.e. you have to do something unusual to get the answer. I do not know what that is though.

[PierreCFA]

About what Basit (Wiley) said, could you give me more information? In fact, I don’t know whether the (right) formula (the equation 9) is not straight forward

because the (right) proof is complicated

or

because we don’t have enough input so that it is impossible to prove the equation 9.

In addition, I just don’t know where is the error in my proof of the formula.

[BenjaminGraham]

I think the reason is the first one.

The equation 9 formula is the correct one I believe for finding the cost of equity given the inputs.

As for the derivation of the formula, all I know is that it is some kind of mathematical manipulation of the equation that is fairly tedious.

also, in this equation,[r₀ = (D/V)r_d(1-tax) + (E/V)r_e ] r₀ is the cost of capital assuming 0 debt. So it is basically the cost of equity when there is no debt in the company. So you do not need r_d to calculate it. It can be found by using:

[EBIT(1-tax)/(Value of unleveraged firm)]

Not sure if that made total sense. Pardon me.

[PierreCFA]

I’m not sure whether supposing r0 = rWACC (for obtaining the equation [r0 = (D/V)rd(1-tax) + (E/V)re]) is the error of my proof because this assumption is used for MM proposition without tax.

But thank you very much for the information. Even the right proof is not straight forward, I hope being able to know it before exam (learning by heart a formula is really not easy )

[BenjaminGraham]

Knowing the formula and how its variables function is more important because I don’t think the derivation is part of the LOS so knowing the formula and how its variables function is more important.

[self_employed]

Indeed something is amiss.

I also get different formula for r_e using derivation from WACC.

[Nukular]

I know I am a bit late to the party (and congrats to you all on passing level II), but I think I can help here. Consider a leveraged firm (subscript L) and an unleveraged firm (subscript U), which are identical in every other aspect (i.e. identical EBIT, identical tax rate). Then we can write the value (V) of the two firms as

V_U = FCFF_U / r₀ = EBIT*(1-t) / r₀

V_L = FCFF_L / WACC with WACC = r_e*E/V_L+r_d*D/V_L

where I have used that the value of the levered firm is equal to the value of debt plus the value of equity (V_L = D+E). Now take the V_L equation from above and multiply by WACC to obtain formular (A):

r_e*E+r_d*D = FCFF_L (A)

At this point you have to eliminate the FCFF_L out of the equation. The difference between the unlevered FCFF_U and the levered one is the tax shield effect of debt:

FCFF_L = FCFF_U+t*Interest = FCFF_U+t*D*r_d (B)

Discounting FCFF_L at the appropriate discount rate (WACC) gives us the value of the levered firm. Alternatively we can discount the two cashflows at the right side of the above equation at their appropriate discount rates to get:

V_L = FCFF_L / WACC

V_L = FCFF_U / r₀ + t*D*r_d / r_d = V_u + t*D or (alternatively) solving for t*D

t*D = V_L - V_U or (alternatively) solving for V_u

V_u = V_L - t*D = D + E - t*D

This can then be used to rewrite the levered cashflow (B) as

FCFF_L = FCFF_U+t*D*r_d = r₀*V_u + r_D*(V_L - V_U) = r₀*V_U + r_D*(D + E - V_U) = (r₀ - r_d)*V_U + r_D*(D + E) = (r₀ - r_d)*(D+E-t*D)+ r_d*(D + E)

We can now insert into (A) to obtain:

r_e*E+r_d*D = (r₀ - r_d)*(D + E - t*D)+ r_D*(D + E)

r_e*E = (r₀ - r_d)*(D + E - t*D)+ r_d*E

r_e = (r₀ - r_d)*(D/E + 1 - t*D/E) + r_d

r_e = r₀ + (r₀ - r_d)*(D/E)*(1-t)

Which is the formular you were looking for. Hope that was not too confusing

As far as the initial solution of PierreCFA is concerned:

The problem with your way of doing it is that neither FCFF nor firm values for the levered and unlevered firm are identical in the case with taxes. The assumption r₀ = r_WACC does only hold in the case without taxes, because then the cashflows and values of the levered and unlevered firm are identical. And if both cashflows and value are identical so has to be the discount rate. This is no longer true with taxes. Without taxes WACC = const. whereas with taxes WACC is a function of the amount of debt in the capital structure.

[keep_running]

Hello,

So just to clarify, r0 does not equal to WACC?

[Nukular]

r₀ is the cost of equity for an unlevered firm. This is equal to WACC if the firm is unlevered, but in general it is not.

The MM proposition without taxes is a special case. In this case WACC does not depend on capital structure. And for an unlevered firm r₀ = WACC. Thus in this case r₀ = WACC holds for any capital structure.

[S2000magician]

@keep\_running

Hello,

So just to clarify, r0 does not equal to WACC?

r₀ = WACC only when the firm is financed 100% with equity, or when t = 0%.