# 2012 mock afternoon Q28

The answer the CFAI gives is that you must first solve for cost of equity (unlevered) with a special formula:

r(0) = [r(e) + r(d) * (1 - t) * D/E] / [1 + (1 - t) * D/E]

and then use that as an input into the MM cost of equity formula (p 106, vol 2).

r(e) = r(0) + (r(0) - r(d)) * (1 - t) * D/E

I’ve searched a while in the textbook as to where the CFAI got this r(0) formula, but it is not found. How was one supposed to deduce the answer? Was one supposed to find the r(0) formula just using algebra with the r(e) formula… is it recommended to memorize the formula the CFAI gives in the answer?

it’s an algebric manipulation of the bottom formula. I think.

I tried doing the algebra, but stopped because it was taking too long (though it does seem to be just a algebraic manipulation as you said)… but I think I may have an intuition behind the formula. It kind of looks like the formula to delever beta, so the formula is basically, I think, WACC delevered for the after-tax D/E ratio.

the unlevered cost of equity for this example has D=E or D/E = 1, so if you solve

re = ro + (ro-rd)(1-t)D/E with D/E = 1 then you have

re = ro + ro (1-t) - rd(1-t).

re + rd (1-t) = ro (1+1-t) = ro (2-t) and then divide out

re/(2-t) + rd(1-t)/(2-t) = ro

you use that formula to compute the unlevered cost of equity. once you find the ro, you can use the target D/E of the company and solve for re using my first equation.

it is similar unlevering the beta of a company.