Error estimates for De Vylder type approximations in ruin theory

TL;DR

This paper provides a rigorous mathematical analysis of the accuracy of De Vylder type approximations in ruin theory, including error estimates for ruin probabilities and moments, and highlights their limitations in certain applications.

Contribution

It offers the first theoretical error bounds for De Vylder approximations of any order, extending their analysis beyond heuristic and numerical evidence.

Findings

Error estimates depend on safety loading, initial reserve, and approximation order.

De Vylder approximations are accurate for ruin probability but less so for moments of deficit and surplus.

Numerical examples confirm the theoretical error bounds and reveal paradoxical inaccuracies.

Abstract

Due to its practical use, De Vylder's approximation of the ruin probability has been one of the most popular approximations in ruin theory and its application to insurance. Surprisingly, only heuristic and numerical evidence has supported it, to some extent. Finding a mathematical estimate for its accuracy has remained an open problem, going from the original paper by De Vylder (1978) through an attempt of justification by Grandell (2000). The present paper consists of a mathematical and critical treatment of the problem. We more generally consider De Vylder type approximations of any order k, based on fitting the k first moments of the classical risk reserve process. Moreover, we not only deal with the ruin probability, but also with the moments of the time of ruin, of the deficit at ruin and of the surplus before ruin. We estimate the approximation errors in terms of the safety loading coefficient, the initial reserve and the approximation order. We show their different behaviours, and the extent to which each relative error remains small or blows up, so that one has to be careful when using this approximation. Our estimates are confirmed by numerical examples. Besides, it turns out that De Vylder type approximations become paradoxically inaccurate when applied to the moments of the deficit at ruin and of the surplus before ruin.