Modigliani and Miller Proposition 2 (with tax) formula

I would also like to know how the formula is rearranged to derive Kei.

It is easier to rearrange when you actually have numbers in it!!!
But here goes 🙂

Ke = Kei + (1 – T)(Kei – Kd) Vd/Ve
Multiply everything by Ve:
Ke Ve = Kei Ve + Vd (1 – T) (Kei – Kd)
Multiply both terms in the last bracket by Vd(1-T)
Ke Ve = Kei Ve + Vd(1-T)Kei – Vd(1-T)Kd
Add Vd(1-T)Kd to both sides
KeVe + KdVd(1-T) = KeiVe + Kei Vd(1-T) = Kei ( Ve + Vd (1-T))
Divide both sides by (Ve + Vd (1-T)) (and write the equation the other way round)

Kei = (KeVe + KdVd(1-T)) / (Ve + Vd(1-T))

🙂

0.12 = Kei + 0.7 (Kei – 0.04) 0.25
0.12 = Kei + 0.175 (Kei – 0.04)
0.12 = Kei + 0.175Kei – 0.007
0.127 = 1.175Kei
Kei = 0.127/1.175 = 0.1081 or 10.81%

You are welcome 🙂

First, it is best to express the % as a decimal.

So…..

0.358 = Kei + (1-0.24)(Kei – 4.6)(65/35)

0.348 = Kei + 0.76 (Kei – 4.6) 1.9429

0.348 = Kei + 1.463 (Kei – 4.6) (1.463 = 0.76 x 1.9429)

0.348 = Kei + 1.463 Kei – 6.730 (6.730 = 1.463 x 4.6)

0.348 = 2.463 Kei – 6.730

add 6.730 to both sides:

7.079 = 2.463 Kei

Divide both sides by 2.463

Kei = 7.079/2.463 = 2.874

This would be a ridiculous answer – it is 287.4% !!!

Check my workings, but I don’t think I made a mistake. I think that maybe you have copied the figures in the question wrongly 🙂

But I explain all of this (with an example) in my free lectures !!!

1. The cost of debt is the actual pre-tax cost of debt.

2. There is no Ve/Vd in the formula, it is Vd/Ve. These market values themselves are not subject to tax, but the effect of tax is dealt with in the second term of the formula (where there is (1-t).