Computing the aggregate loss distribution based on numerical inversion of the compound empirical characteristic function of frequency and severity

TL;DR

This paper introduces a non-parametric method for computing the aggregate loss distribution by numerically inverting empirical characteristic functions, applicable in actuarial risk analysis and adaptable to semi-parametric models.

Contribution

It presents an efficient, purely non-parametric approach for evaluating aggregate loss distributions using empirical CFs, with potential for extensions to semi-parametric models.

Findings

Effective numerical inversion of CFs for ALD evaluation

Application demonstrated on Danish fire loss data

Method suitable for calculating risk measures like VaR

Abstract

A non-parametric method for evaluation of the aggregate loss distribution (ALD) by combining and numerically inverting the empirical characteristic functions (CFs) is presented and illustrated. This approach to evaluate ALD is based on purely non-parametric considerations, i.e., based on the empirical CFs of frequency and severity of the claims in the actuarial risk applications. This approach can be, however, naturally generalized to a more complex semi-parametric modeling approach, e.g., by incorporating the generalized Pareto distribution fit of the severity distribution heavy tails, and/or by considering the weighted mixture of the parametric CFs (used to model the expert knowledge) and the empirical CFs (used to incorporate the knowledge based on the historical data - internal and/or external). Here we present a simple and yet efficient method and algorithms for numerical inversion of the CF, suitable for evaluation of the ALDs and the associated measures of interest important for applications, as, e.g., the value at risk (VaR). The presented approach is based on combination of the Gil-Pelaez inversion formulae for deriving the probability distribution (PDF and CDF) from the compound (empirical) CF and the trapezoidal rule used for numerical integration. The applicability of the suggested approach is illustrated by analysis of a well know insurance dataset, the Danish fire loss data.