Analytical solution to the circularity problem in the discounted cash flow valuation framework

Since the Modigliani and Miller 1958 seminal paper, there has been a problem posed by the fact that the discount rate to value cash flows depends on the value of thesecash flows. This gives raise to the circularity problem.In this paper we propose an analytical solution to this circularity problem. Our solution is derived starting from a central principle of finance which states that value today is equal to value of next period plus cash flow of next period, discounted with the discount rate of next period. We derive a general formulation for the equity value, E, at a given period and propose a general formula that depends on the value of equity and cash flow to equity for next period, the values of debt and tax savings, TS, at actual period, the discount rate for the TS in the next period, , the cost of debt, Kd, and the unlevered cost of equity, Ku. We then present this formula for two special cases: one for equal to Kd and another for equal to Ku. From the expression for E we derive the formulation with no circularity for total value that depends on free cash flow, FCF, and TS. We also derive the weighted average cost of capital, WACC, without circularity. However, we acknowledge that this formulation is less intuitive than the traditional textbook WACC formulation. We furthermore compare the results obtained using these formulas with the resultsusing the Adjusted Present Value approach (no circularity) and the iterative solution of circularity based upon the iteration feature of a spreadsheet. Our conclusion is that, as expected, all methods produce the same answer and the analyst should use the simplest method available, which means using the Adjusted PresentValue, APV, with the appropriate discount rate for the TS. When this discount rate is equal to Ku, the cost of unlevered equity, the APV method results in the capital cash flow, CCF, approach.