Prediction error for credible claims reserves: an h -likelihood approach
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Prediction error for credible claims reserves: an h-likelihood approach Patrizia Gigante • Liviana Picech • Luciano Sigalotti
Received: 11 October 2012 / Revised: 26 November 2012 / Accepted: 29 April 2013 Ó DAV / DGVFM 2013
Abstract In this paper we consider a hierarchical overdispersed Poisson-gamma model for claims reserving as a hierarchical generalized linear model, in which the h-likelihood approach is applied to estimate the parameters. The model allows us to take account of external data, e.g. external estimates of ultimate claims. Predictions and prediction errors of the claims reserves are evaluated. For each origin year, the estimated reserve can be seen as a credible claims reserve: a mixture of a Chain– ladder type and a Bornhuetter–Ferguson type claims reserve. Keywords Claims reserves Chain–ladder Bornhuetter–Ferguson Credible claims reserves Hierarchical generalized linear models h-likelihood Conditional mean square error of prediction
1 Introduction The Chain–ladder (CL) and the Bornhuetter–Ferguson (BF) are among the most used claims reserving methods and various studies have introduced stochastic versions to evaluate the prediction errors of estimated reserves (see e.g. Wu¨thrich and Merz [29]). Whereas the CL relies on the run-off data only, the BF also takes account of external estimates of ultimate claims. In many practical situations, mixtures of the two reserves for any origin year, called credible claims reserves, are of interest. P. Gigante L. Picech (&) University of Trieste, Piazzale Europa 1, 34127 Trieste, Italy e-mail: [email protected] P. Gigante e-mail: [email protected] L. Sigalotti University of Udine, Via Tomadini 30, 33100 Udine, Italy e-mail: [email protected]
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Credible claims reserves have been discussed in the actuarial literature for some time. The Benktander–Hovinen method (Benktander [3], reviewed in Mack [17]), sets the weights of the mixtures so that the higher the available information, the higher the weight assigned to the CL reserve. Alternatively, the weights can be determined according to various approaches such as the credibility approach (e.g. Neuhaus [22]); an optimality approach, in which the mean square error of prediction of the reserve estimator is minimized (e.g. Mack [17]); and the Bayesian approach (see Wu¨thrich and Merz [29] for an extensive review). In Arbenz and Salzmann [2] a distribution-free stochastic model allows for a weighted combination between a data-driven approach, resembling the CL, and an approach similar to the BF. The weights can be set according to an expert opinion. Wu¨thrich [27] shows that under Bayesian models with exponential dispersion family distributions and conjugate priors for the risk parameters, given the claims development pattern, the claims reserve can be estimated by an exact credibility formula as a mixture of the CL and BF reserves. In this paper we assume a hierarchical overdispersed Poisson model for the incremental payments with gamma distributed ri
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