Derivation of r(e) from WACC

Hi guys!

The cost of equity is given as: r(e) = r(o) + [r(o) - r(d)] [D/E]

The above equation is said to be derived from the WACC: r(wacc) = [D/V] r(d) + [E/V] r(e)

I can’t quite get my head on how it is derived. Can someone help I’ve a feeling the derivation has been shown before some place.

Thanks and happy studying to all those preparing for the exams now!

r(w) = r(d)*D/V + r(e)*E/V

Multiply through by V/E

r(w)*V/E = r(d)*D/E + r(e)

Rearrange to isolate r(e)

r(e) = r(w)*V/E - r(d)*D/E

Remember that V = D+E

r(e) = r(w)*(D+E)/E - r(d)*D/E

r(e) = r(w)*D/E + r(w)*E/E - r(d)*D/E

r(e) = r(w)*D/E + r(w) - r(d)*D/E

r(e) = r(w) + (D/E)*[r(w) - r(d)]

Hi Kevndc, thanks for the working.

Just to understand how is it that r(w) = r(o)?

If I’m not mistaken r(o) is the unlevered cost of capital while r(w) is then WACC which is the total cost of capital?


r(WACC) = r(o) for MM Prop I.

Thanks kevndc sorry I couldn’t find that in my textbook.

How is it that r(wacc) can equal r(o) for MM Prop I when clearly there debt and leverage included in WACC and r(o) is defined as the unlevered cost of capital?

p.s the LOS doesn’t mention calculating any of these formulas for the MM reading.

Feel like I have come across problems in the TT and in mocks that require you to do the calcs in the MM Props