Title: Evaluating compound sums through Panjer's formula and discretization of continuous random distributions
Authors: Alessandro Barbiero - Università degli Studi di Milano (Italy) [presenting]
Abstract: The accurate evaluation of compound sums is an important task in actuarial science and operational risk management. The total claims amount that a non-life insurance company has to pay in a specific period of time can be modeled as $S_N=X_1+...+X_N$, with $N$ being the number of occurring claims and $X_i$ the $i-$th claim size, $i=1,...,N$; the $X_i$ are assumed to be iid random variables, typically continuous, and $N$ is assumed to be independent of the $X_i$. While determining the first integer moments of $S_N$ (when they exist) is easy, the evaluation of the whole distribution based on convolution is, in general, not analytically viable and is computationally demanding if addressed numerically. Approximations to the Gaussian and translated Gamma distributions can be employed. Alternatively, a recursive approach for the determination of the single probabilities of $S_N$ is available (Panjer's recursive formula); however, a special type of distribution for the number of claims $N$ and, more importantly, for the claim size $X_i$ is required, which limits the range of application in practice. We show how discretizing the (continuous) claim size $X_i$ on a lattice, according to some criterion for determining the relevant probabilities, and then applying Panjer's formula, can lead to acceptable and computationally feasible approximations of the distribution of $S_N$.