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Bayesian over-dispersed Poisson model and the Bornhuetter & Ferguson claims reserving method

Published online by Cambridge University Press:  27 February 2012

Peter D. England ,
Richard J. Verrall  and
Mario V. Wüthrich
Peter D. England
Affiliation:
Towers Watson, London, UK
Richard J. Verrall
Affiliation:
Cass Business School, City University, London, UK
Mario V. Wüthrich*
Affiliation:
ETH Zurich, RiskLab, Department of Mathematics, Switzerland
*
*Correspondence to: Mario V. Wüthrich, ETH Zurich, RiskLab, Department of Mathematics, 8092 Zurich, Switzerland. E-mail: mario.wuethrich@math.ethz.ch
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Abstract

We consider the Bayesian over-dispersed Poisson (ODP) model for claims reserving in general insurance. We choose two different types of prior distributions for the parameters and then study the different Bayesian predictors. This study leads, on the one hand, to the classical chain ladder predictor and, on the other hand, to Bornhuetter & Ferguson predictors. We highlight (either analytically or numerically) how these predictors are obtained and how their prediction uncertainty can be determined.

Keywords

Claims reservingOver-dispersed Poisson modelBornhuetter and Ferguson methodChain ladder methodPrediction uncertaintyMean square error of predictionMarkov chain Monte Carlo method
Type
Papers
Information
Annals of Actuarial Science , Volume 6 , Issue 2 , 23 August 2012 , pp. 258 - 283
Copyright
Copyright © Institute and Faculty of Actuaries 2012

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References

Alai, D.H., Merz, M., Wüthrich, M.V. (2009). Mean square error of prediction in the Bornhuetter-Ferguson claims reserving method. Annals of Actuarial Science, 4/1, 7–31.CrossRefGoogle Scholar
Asmussen, S., Glynn, P.W. (2007). Stochastic Simulation. Springer.Google Scholar
Bornhuetter, R.L., Ferguson, R.E. (1972). The actuary and IBNR. Proceedings CAS, LIX, 181–195.Google Scholar
England, P.D., Verrall, R.J. (2002). Stochastic claims reserving in general insurance. British Actuarial Journal, 8/3, 443–518.Google Scholar
England, P.D., Verrall, R.J. (2006). Predictive distributions of outstanding liabilities in general insurance. Annals of Actuarial Science, 1/2, 221–270.Google Scholar
Gilks, W.R., Richardson, S., Spiegelhalter, D.J. (1996). Markov Chain Monte Carlo in Practice. Chapman & Hall.Google Scholar
Hastings, W.K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109.CrossRefGoogle Scholar
Mack, T. (1991). A simple parametric model for rating automobile insurance or estimating IBNR claims reserves. ASTIN Bulletin, 21/1, 93–109.Google Scholar
Mack, T. (1993). Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin, 23/2, 213–225.Google Scholar
Mack, T. (2008). The prediction error of Bornhuetter/Ferguson. ASTIN Bulletin, 38/1, 87–103.CrossRefGoogle Scholar
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E. (1953). Equation of state calculations by fast computing machines. Journal Chemical Physics, 21/6, 1087–1092.CrossRefGoogle Scholar
Saluz, A., Gisler, A., Wüthrich, M.V (2011). Development pattern and prediction error for the stochastic Bornhuetter-Ferguson claims reserving model. ASTIN Bulletin, 41/2, 279–313.Google Scholar
Scollnik, D.P.M. (2001). Actuarial modeling with MCMC and BUGS. North American Actuarial Journal, 5/2, 96–125.Google Scholar
Smith, R.L. (1998). Bayesian and frequentist approach to parametric predictive inference. In:Bayesian statistics, 6, Bernardo, J.M., Berger, J.O., Dawid, A.P. & Smith, A.F.M. (eds.) Oxford University Press, 589–612.Google Scholar
Spiegelhalter, D.J., Thomas, A., Best, N.G., Gilks, W.R. (1995). BUGS: Bayesian Inference Using Gibbs Sampling, Version 0.5. MRC Biostatistics Unit, Cambridge.Google Scholar
Spiegelhalter, D.J., Best, N.G., Carlin, B.P., van der Linde, A. (2002). Bayesian measures of model complexity and fit (with discussion). Journal Royal Statistical Society, 64, 583–639.Google Scholar
Verrall, R.J. (2004). A Bayesian generalized linear model for the Bornhuetter-Ferguson method of claims reserving. North American Actuarial Journal, 8/3, 67–89.CrossRefGoogle Scholar
Wüthrich, M.V., Merz, M. (2008). Stochastic Claims Reserving Methods in Insurance. Wiley.Google Scholar