The First Difference Property of the Present Value Operator

Abstract

This paper identifies a fundamental relationship between the present value of a given cash flow and the present value of the period by period change in that cash flow. The new relationship is shown to be highly useful for the identification of analytic expressions for present value and related measures such as duration and convexity; expressions that continue to play an instructive role by helping to relate the quantitative outcomes of numerical calculations to the driving forces behind those calculations. This new method applies only simple arithmetic operations and avoids the use of differential calculus and advanced series summation in order to derive these analytic results. We apply the method to a variety of nontraditional cash flows, including cash flows with linear growth or decay, cash flows that are subject to different tax effects for dividends and capital gain, and cash flows that are projected to exhibit cyclical variation over time.