An Extension of Collective Risk Model for Stochastic Claim Reserving (Initial version)

The evaluation of outstanding claims uncertainty plays a fundamental role in managing insurance companies; reserve risk is indeed an essential part of underwriting risks. In literature, stochastic methodologies for valuation of future loss payments have been built up with the aim to assess, as far as possible, the capital requirement for reserve risk, through the estimation of the probability distribution of risks involved. To this end, the International Actuarial Association (IAA) (see [13]) provided a method based on a paper of Meyers et al. [17] in which the overall reserve of a single LoB (without any distinction among different accident years) follows a compound mixed Poisson process, as used in premium risk to estimate aggregate loss distribution (see [1], [2], [12] and [19]). Some extensions are given in [20] and [21] where claims reserve is obtained assuming that each incremental payment is described by a compound mixed Poisson process. Exact mean and variance of the reserve distribution are also provided. We extend here previous approaches. We propose to describe each cell of the lower run-off triangle as a compound mixed Poisson process in which a structure variable is directly applied to the claim size. In this way, we catch parameter uncertainty on both frequency and severity distributions. Through this approach, we also take into account the dependency between payments of different development or accident years. Exact values of mean, variance and skewness of the reserve distribution are proved. Furthermore, Monte Carlo method allows to simulate the outstanding claims distribution for the overall reserve until complete run-off and the next calendar year only, in case of a one-year time horizon is required (e.g. for Solvency II purposes). Model’s parameters are calibrated by observed data and through the Frequency- Severity deterministic method which provides a separate estimation of number of claims and average costs. Finally, a detailed numerical analysis shows a comparison between the proposed methodology and the well-known bootstrapping Over Dispersed Poisson model.