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

in this paper we propose an analytical solution to the circularity problem between value and cost of capital. our solution is derived starting from a central principle of finance that relates value today to value, cash flow, and the discount rate for next period. we present a general formulation without circularity for the equity value (e), cost of levered equity (ke), levered firm value (v), and the weighted average cost of capital (wacc). we furthermore compare the results obtained from these formulas with the results of the application of the adjusted present value approach (no circularity) and the iterative solution of circularity based upon the iteration feature of a spreadsheet, concluding that all methods yield exactly the same answer. the advantage of this solution is that it avoids problems such as using manual methods (i.e., the popular "rolling wacc") ignoring the circularity issue, setting a target leverage (usually constant) with the inconsistencies that result from it, the wrong use of book values, or attributing the discrepancies in values to rounding errors.