I'm confused about how the formula for re was derived. When rewrite the rwacc formula to solve for re, I get the same results but my formula is worded very differently.

EXAMPLE 14.5 Reducing Leverage and the Cost of Capital Problem NRG Energy, Inc. (NRG) is an energy company with a market debt-equity ratio of 2. Suppose its current debt cost of capital is 6%, and its equity cost of capital is 12%. Suppose also that if NRG issues equity and uses the proceeds to repay its junior debt and reduce its debt-equity ratio to 1, it will lower its debt cost of capital to 5.5%. With perfect capital markets, what effect will this transaction have on NRG's equity cost of capital and WACC? What would happen if NRG issues even more equity and pays off its debt completely? How would these alternative capital structures affect NRG's enterprise value? Solution We can calculate NRG's initial WACC and unlevered cost of capital using Eqs. 14.6 and 14.7: Twat = ru E D 2 E + D TE E + D 1 + 2 (12%) + 1 + 2 (6%) = 8% Given NRG's unlevered cost of capital of 8%, we can use Eq. 14.5 to calculate NRG's equity cost of capital after the reduction in leverage: TE = ru D E (ru - ID) = 8% + (8% - 5.5%) = 10.5% The reduction in leverage will cause NRG's equity cost of capital to fall to 10.5%. Note, though, that with perfect capital markets, NRG's WACC remains unchanged at 8% = 2 (10.5%) + 2(5.5%), and there is no net gain from this transaction. If NRG pays off its debt completely, it will be unlevered. Thus, its equity cost of capital will equal its WACC and unlevered cost of capital of 8%. In either scenario, NRG's WACC and free cash flows remain unchanged. Thus, with perfect capital markets, its enterprise value will not be affected by these different capital structure choices