Equivalence of the Apv, Wacc and Flows to Equity Approaches to Firm Valuation

We show that the three valuation methods (if used correctly) always yield the same result. The most striking result of this paper is that for a firm growing at a rate g, the Net Present Value of the tax shield due to interest payments (in the APV approach) must be calculated as follows: NPV OF INTEREST TAX SHIELDS = D Ku T / (Ku - g) T = Corporate tax rate; Ku = Cost of unlevered equity; D = Value of debt in period 0. It seems that this formula considers that the debt has a cost of Ku, and discounts these flows also at Ku, but this is not the case. The Net Present Value of interest tax shields is not (and this is the main error in previous papers about this topic) the NPV of a unique flow, but the difference of two NPVs of two flows with different risk: the NPV of the taxes paid in the unlevered firm and the NPV of taxes paid in the levered firm. Our formula is the difference of the two NPV. Obviously, the flow of taxes paid in the levered firm is smaller, but riskier than the flow of taxes paid in the unlevered firm. We show the equivalence of the three approaches to firm valuation for perpetuities, then for growing companies (at a constant rate g) and, finally, for any company