The Present Value of a Series of Cashflows: Convergence in a Random Environment

Abstract

The present paper considers the present value, Z(t) , of a series of cashflows up to some time t. More specifically, the cashflows and the interest rate process will often be stochastic and not necessarily independent of one another or through time. We discuss under what circumstances Z(t) will converge almost surely to some finite value as t →∞. This problem has previously been considered by Dufresne (1990) who provided a sufficient condition for almost sure convergence of Z(t) (the Root Test) and then proceeded to consider some specific examples of such processes. Here, we develop Dufresne's work and show that the sufficient condition for convergence can be proved to hold for quite a general class of model which includes the growing number of Office Models with stochastic cashflows.