Stable Laws and the Present Value of Fixed Cash-Flows
North American Actuarial Journal, Vol. 7, No. 4, pp. 32-43, 2003
20 Pages Posted: 2 Mar 2006
In the current contribution, we consider the present value of a series of fixed cash flows under stochastic interest rates. In order to model these interest rates, we don't use the common lognormal model, but stable laws, which better fit in with reality. For this present value, we want to derive a result about the distribution function. However, due to the dependencies between successive discounted payments, the calculation of an exact analytical distribution for the present value is impossible. Therefore, use is made of the methodology of comonotonic variables and the convex ordering of risks, introduced by the same authors in some previous papers. The present paper starts with a brief overview of properties and qualities of stable laws, and of the possible application of the concept of convex ordering to sums of risks - which is also the situation for a present value of future payments. Afterwards, it is shown how for the present value under investigation an approximation in the form of a convex upper bound can be derived. This upper bound has an easier structure than the original present value, and we derive elegant calculation formulas for the distribution of this bound. Finally, we provide some numerical examples, which illustrate the precision of the approximation. Due to the design of the present value and due to the construction of the upper bound, these illustrations show great promise concerning the accuracy of the approximation.
Keywords: cash flow, stochastic interest rates, stable laws, distribution,convex order
JEL Classification: C10, C63, E43
Suggested Citation: Suggested Citation