After tax cost of Debt and Equity

I believe this has been discussed before… CFAI book, last problem (10 on Leverage chapter) Suppose Cost of capital for Gadget company is 10%. If Gadget company has a capital structure 50% debt and 50% equity, its before tax cost of debt is 5%, marginal tax rate is 20%, its cost of equity is closest to: a. 10% b. 12% c.14% d.16% answer given is: re = ra + (ra -rd) (D/E) (1-t) = .10 + (.10-.05) (1) (.8) = 14% However if we went from first principals: ra = 0.1 = .5 (.05) (.8) + .5re so re = (.1 - .02) / .5 = 16% I think the error lies in the calculation of the formula the CFAI has derived: ra = D/(D+E) * rd* + E/(D+E) re rd* = rd (1-t) (After tax cost of debt) So rearranging and solving for re re = ra + D/E ( ra - rd*) = ra + D/E ( ra - rd(1-t)) which gives the same answer of 16% however their formula is : re = ra + (ra -rd) (D/E) (1-t) where they have taken the entire (1-t) factor outside. Please NOTE THAT THERE IS AN ERRATA POSTED Study Session 11, Reading 49: On pp. 163-164, delete problems 8 through 10. These problems are covered in the “optional” segment and should not be assigned. If doing these problems, in the solution for problem 10 (p. A-7), the formula used to derive re is incorrect. The corrected formula is re = ra + [(ra – rd*)(D/E)], where rd* = rd (1 – t). The correct answer to problem 10. This is a rather Long Post… but just to bring out the Errata to everyone. Regards CP

I get the same answer i.e. 16. Thanks for this.

could someone remind me how to calculate cost of retained earnings? thanks

I arrive at 16% as well

and 16 is the right answer… per the errata. (that did not get typed up before)…