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A non-parametric method for evaluation of the aggregate loss distribution(ALD) by combining and numerically inverting the empirical characteristicfunctions (CFs) is presented and illustrated. This approach to evaluate ALD isbased on purely non-parametric considerations, i.e., based on the empirical CFsof frequency and severity of the claims in the actuarial risk applications.This approach can be, however, naturally generalized to a more complexsemi-parametric modeling approach, e.g., by incorporating the generalizedPareto distribution fit of the severity distribution heavy tails, and/or byconsidering the weighted mixture of the parametric CFs (used to model theexpert knowledge) and the empirical CFs (used to incorporate the knowledgebased on the historical data - internal and/or external). Here we present asimple and yet efficient method and algorithms for numerical inversion of theCF, suitable for evaluation of the ALDs and the associated measures of interestimportant for applications, as, e.g., the value at risk (VaR). The presentedapproach is based on combination of the Gil-Pelaez inversion formulae forderiving the probability distribution (PDF and CDF) from the compound(empirical) CF and the trapezoidal rule used for numerical integration. Theapplicability of the suggested approach is illustrated by analysis of a wellknow insurance dataset, the Danish fire loss data.

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