A Collective Risk Model for Claims Reserve Distribution

Stochastic models for outstanding claims valuation have been recently developed with the aim to obtain either a variability coefficient or the probability distribution of the reserve, useful for the assessment of the Reserve Risk capital requirement. Different methodologies, like Bootstrapping or Bayesian methods, allow to determine the prediction error and the claims reserve distribution under some assumptions based on the data figured out in the run-off triangle. The International Actuarial Association (IAA) proposed a different way to analyse out-standing claim reserve, focused on a paper by Meyers, Klinker and Lalonde where the overall reserve of a single LoB (without any distinction among different accident years) is described as a compound mixed poisson process, by a similar approach used for aggregate amount of claims when premium risk is to be estimated. In the present paper we extend this approach assuming that each single cell of the lower part of the run-off triangle follows a Compound Poisson Process (either Pure or Mixed). This assumption leads, under independence constraint, to obtain quite easily the exact moments of the reserve distribution only through the knowledge of the characteristics of the two main random variables (number and claim size of future payments). Furthermore, Monte Carlo methods allow to simulate outstanding claims distributions for each accident year, for both the overall reserve until complete run-off and the next calendar year only (in case of a one-year time horizon as prescribed in Solvency II). Model s parameters are calibrated from observed data and through a deterministic model (in the present paper an average cost method is used, namely Fisher-Lange method) based on the separate estimate of number of claims to be paid and future average costs for each cell of the lower triangle to be estimated. Finally, we analyze the one-year reserve risk in the perspective adopted by Solvency II. In particular a simulation approach, now usually called re-reserving , is here applied with the target to estimate the variability of claims development result and the percentiles, in order to quantify the capital requirement for the reserve risk. Main results will be compared using different stochastic reserving models, analysing the effect on both capital requirement for only reserve risk and risk margin.