Stochastic Loss Reserving with the Collective Risk Model

By Glenn G. Meyers

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Abstract

This paper presents a Bayesian stochastic loss reserve model with the following features:

1. The model for expected loss payments depends upon unknown parameters that determine the expected loss ratio for each accident year and the expected payment for each settlement lag.
2. The distribution of outcomes is given by the collective risk model in which the expected claim severity increases with the settlement lag. The claim count distribution is given by a Poisson distribution with its mean determined by dividing the expected loss by the expected claim severity.
3. The parameter sets that describe the posterior distribution of the parameters in (1) above are calculated with the Metropolis-Hastings algorithm.
4. For each parameter set generated by the Metropolis- Hastings algorithm in (3), the predicted distribution of outcomes is calculated using a Fast Fourier Transform (FFT). The Bayesian predictive distribution of outcomes is a mixture of the distributions of outcomes over all the parameter sets produced by the Metropolis-Hastings algorithm.

KEYWORDS: Reserving methods, reserve variability, uncertainty and ranges, collective risk model, Fourier methods, Bayesian estimation

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Citation

Meyers, Glenn G., "Stochastic Loss Reserving with the Collective Risk Model," Variance 3:2, 2009, pp. 239-269.

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Variance (ISSN 1940-6452) is a peer-reviewed journal published by the Casualty Actuarial Society to disseminate work of interest to casualty actuaries worldwide. The focus of Variance is original practical and theoretical research in casualty actuarial science. Significant survey or similar articles are also considered for publication. Membership in the Casualty Actuarial Society is not a prerequisite for submitting papers to the journal and submissions by non-CAS members is encouraged.