A Quantitative Revelation in Equity Valuation: The P/E Ratio is a Degenerate Case of the Potential Payback Period (PPP): Why Settle for the Limited when a Comprehensive Model Exists?

This article introduces the Potential Payback Period (PPP) as a generalized, risk- and growth-adjusted metric for equity valuation. In contrast to the traditional Price-to-Earnings (P/E) ratio, which assumes static earnings and neglects the time value of money, the PPP accounts for expected earnings growth and discount rates, thus providing a more dynamic and realistic measure of a stock’s investment appeal. By applying L’Hôpital’s Rule to the PPP formula, we demonstrate that the P/E ratio emerges as a limiting or degenerate case of PPP in two scenarios: (1) when the earnings growth rate converges to the discount rate, and (2) when both growth and discount rates are zero—an idealized static world. This theoretical result recontextualizes the P/E ratio within a broader, mathematically grounded framework and offers significant implications for valuation theory and portfolio management. PPP serves not merely as a substitute for P/E, but as its logical and quantitative extension.