An all-equity firm has a 9% cost of equity. It recapitalises to 40% debt at market value, borrowing at 5% pre-tax with a 21% tax rate. What is the new cost of equity, and what happens to the weighted average cost of capital?

An all-equity firm has a 9% cost of equity. It recapitalises to 40% debt at market value, borrowing at 5% pre-tax with a 21% tax rate. What is the new cost of equity, and what happens to the weighted average cost of capital?

Approach: Apply Modigliani-Miller proposition two with taxes to relever the cost of equity, recompute the weighted average, then check it against the tax shield form of the same result.

11.11%. Proposition two with taxes gives r_E = r_U + (D/E)*(r_U - r_D)*(1 - T), so 9% + (0.4/0.6)*(9% - 5%)*0.79 is 9% + 2.11%, or 11.11%. The weighted average cost of capital is 0.6 * 11.11% + 0.4 * 5% * 0.79, which is 6.66% + 1.58%, or 8.24%, and that matches the tax shield form r_U * (1 - T*D/V), evaluated as 9% times (1 - 0.21 * 0.4). So relevering raises the cost of equity by 211 basis points while the weighted average falls by 76 basis points, and the entire fall is the tax shield on the debt. With no taxes the two effects cancel exactly and the weighted average stays at 9%, which is the content of Modigliani-Miller.

Follow-up: At 70% debt the rating falls and pre-tax debt costs 9%. What happens to the weighted average cost of capital and why does the tax shield form stop working?

Key concepts: Modigliani-Miller, cost of equity, tax shield, relevering.