On the accuracy of phase-type approximations of heavy-tailed risk models

TL;DR

This paper investigates the number of phases needed in phase-type approximations to accurately estimate ruin probabilities in heavy-tailed risk models, providing error bounds and algorithms for specific claim distributions.

Contribution

It determines the required phases for desired accuracy, develops an algorithm for hyperexponential approximation, and compares with heavy traffic and heavy tail methods.

Findings

Derived error bounds for phase-type approximations.

Established the number of phases needed for specific accuracy.

Compared phase-type approximation with other methods.

Abstract

Numerical evaluation of ruin probabilities in the classical risk model is an important problem. If claim sizes are heavy-tailed, then such evaluations are challenging. To overcome this, an attractive way is to approximate the claim sizes with a phase-type distribution. What is not clear though is how many phases are enough in order to achieve a specific accuracy in the approximation of the ruin probability. The goals of this paper are to investigate the number of phases required so that we can achieve a pre-specified accuracy for the ruin probability and to provide error bounds. Also, in the special case of a completely monotone claim size distribution we develop an algorithm to estimate the ruin probability by approximating the excess claim size distribution with a hyperexponential one. Finally, we compare our approximation with the heavy traffic and heavy tail approximations.