Stochastic Upper Bounds for Present Value Functions

Abstract

ABSTRACT In most practical cases, it is impossible to find an explicit expression for the distribution function of the present value of a sequence of cashflows that are discounted using a stochastic return process. In this article, the authors present an easily computable approximation for this distribution function. The approximation is a distribution function which is, in the sense of convex order, an upper bound for the original distribution function. Explicit examples are given for pricing stochastic annuities with a stochastic return process, for more general stochastic cash flows, as well as for pricing Asian options. Numerical results seem to indicate that the approximation will often be close to the original function. INTRODUCTION In several financial-actuarial problems, one is faced with the determination of the distribution function of random variables of the form V = [[[sigma].sup.n].sub.i=1] [[alpha].sub.i][e.sup.-[X.sub.i]] where [[alpha].sub.i] (i = 1,...,n] represents the deterministic cash flow at time i and [e.sup.-[X.sub.i]] (i = 1,...,n) is the stochastic discount factor for a payment made at time i. Hence, the random variable V can be interpreted as the present value at time 0 of a sequence of default-free payments at times 1,2,... , n. In an actuarial context, such random variables are used for describing the present value of the cash flow of an insurance portfolio [see, e.g., Dufresne (1990)]. They are also useful for the determination of incurred-but-not-reported (IBNR) reserves [see Goovaerts and Redant (1998)]. Each [[alpha].sub.i] has to be interpreted as an amount that has to be paid at time i. Equivalently, it can be said that there is an income equal to - [[alpha].sub.i] at time i. The random variable V will be called the loss variable; i.e., the present value of all future (deterministic) payments. Now assume that the distribution functions of the random variables [X.sub.i](i = 1,...,n) are known. One could assume, for example, that they are normally distributed. In reality, the random variables [X.sub.i] will certainly not be mutually independent. This means that besides the distribution functions of the [X.sub.i], the dependency structure of the random vector ([X.sub.1],...,[X.sub.n]) will also have to be taken into account in order to determine the distribution function of the loss variable V. Unfortunately, an expression for the distribution function of V is not available or is difficult to obtain in most cases. Thus, the authors shall introduce a technique that will enable bounds on V to be found. In actuarial literature, it is a common feature to replace a loss variable by a loss variable, which has a simpler structure, making it easier to determine the distribution function [see, e.g., Goovaerts, Kaas, Van Heerwaarden, and Bauwelinckx (1986)]. In order to clarify what the authors mean with a less favorable risk, the convex order will be used [see, e.g., Shaked and Shanthikumar (1994)]. Let V and W be two random variables (losses) such that E[[phi](V)] [less than or equal to] E[[phi](W)] for all convex functions [phi]: R [right arrow] R, provided the expectations exist. Then V is said to be smaller than W in the convex order (denoted as V [[less than or equal to].sub.cx] W). Roughly speaking, convex functions are functions that take on their largest values in the tails. Therefore V [[less than or equal to].sub.cx] W means that W is more likely than V to take on extreme values. Instead of saying that V is smaller than Win the convex order, it is often said that -V dominates -W in the sense of second degree stochastic dominance [see, e.g., Huang and Litzenberger (1988)]. In terms of utility theory, V [[less than or equal to].sub.cx] W means that the loss V is preferred to the loss W by all risk-averse decision makers, i.e., E[u (-V)] [greater than or equal to] E[u (-W)] for all concave utility functions u. …