New Properties of the Old Present Value Operator

Advances in computing technology have greatly enhanced methods for numerical calculations of present value and related measures such as duration and convexity. Nevertheless, closed form solutions continue to play an important role both in the classroom and in the real world. For example, it is well known that if r is the rate of discount and if C1 denotes the value in period 1 for a cash flow that grows at constant percentage rate, g, then the present value of the future cash flow can be represented as C1 / (r – g). Yet how many students or practitioners, and dare we ask how many finance professors, are aware that the duration of a perpetual cash flow that grows at a uniform geometric rate can be represented as (1 r) / (r –g) ? For that matter, how widely is it known that a simple closed form solution exists for the present value of a cash flow that exhibits cyclical variation over time or a cash flow that grows by a constant dollar amount each period rather than by a constant percentage amount? The objective of this paper is to demonstrate that these results, and countless others, can be derived from one simple but previously under developed property of the traditional present value operator.