Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-12T04:23:32.542Z Has data issue: false hasContentIssue false

Modelling Claims Run-Off with Reversible Jump Markov Chain Monte Carlo Methods

Published online by Cambridge University Press:  09 August 2013

Richard Verrall
Affiliation:
Cass Business School, City University, London, E-Mail: r.j.verrall@city.ac.uk
Ola Hössjer
Affiliation:
Dept. of Mathematics, Stockholm University
Susanna Björkwall
Affiliation:
Dept. of Mathematics, Stockholm University

Abstract

In this paper we describe a new approach to modelling the development of claims run-off triangles. This method replaces the usual ad hoc practical process of extrapolating a development pattern to obtain tail factors with an objective procedure. An example is given, illustrating the results in a practical context, and the WinBUGS code is supplied.

Type
Research Article
Copyright
Copyright © International Actuarial Association 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Björkwall, S., Hössjer, O., Ohlsson, E. and Verrall, R. (2011) A generalized linear model with smoothing effects for claims reserving. Insurance, Mathematics and Economics, 49, 2737.Google Scholar
Brooks, S. and Gelman, A. (1998) General Methods for Monitoring Convergence of Iterative Simulations. Journal of Computational and Graphical Statistics, 7(4), 434455.Google Scholar
Brooks, S.P. and Roberts, G.O. (1998) Convergence Assessment Techniques for Markov Chain Monte Carlo. Statistics and Computing, 8(4), 319335.Google Scholar
Congdon, P. (2006). Bayesian Statistical Modelling, John Wiley.Google Scholar
Cowles, M.K. and Carlin, B.P. (1996) Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review. Journal of the American Statistical Association, 91, 883904.Google Scholar
de Alba, E. (2002) Bayesian Estimation of Outstanding Claim Reserves. North American Actuarial Journal, 6, 120.CrossRefGoogle Scholar
England, P.D. and Verrall, R.J. (2002) Stochastic Claims Reserving in General Insurance (with discussion). British Actuarial Journal, 8, 443544.CrossRefGoogle Scholar
England, P.D. and Verrall, R.J. (2006) Predictive Distributions of Outstanding Liabilities in General Insurance. Annals of Actuarial Science, 1, 221270.Google Scholar
England, P.D., Wüthrich, M.V. and Verrall, R.J. (2010) Bayesian Overdispersed Poisson Model and the Bornhuetter-Ferguson Claims Reserving Method. Pre-print.Google Scholar
Fan, Y., Peters, G.W. and Sisson, S.A. (2009) Automating and evaluating reversible jump MCMC proposal. Statistics and Computing 19, 409421.Google Scholar
Geman, S. and Geman, D. (1984) Stochastic Relaxation, Gibbs Distributions and the Bayesian Restoration of Images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721741.Google Scholar
Gelfand, A.E. and Smith, A.F.M. (1990) Sampling-Based Approaches to Calculating Marginal Densities. Journal of the American Statistical Association, 85, 398409.CrossRefGoogle Scholar
Gelman, A., Carlin, J.B., Stern, H.S. and Rubin, D.B. (1995) Bayesian Data Analysis. Chapman and Hall, London.CrossRefGoogle Scholar
Gelman, A. and Rubin, D.B. (1992) Inference from Iterative Simulation Using Multiple Sequences. Statistical Science, 7, 457511.Google Scholar
Geweke, J. (1991) Evaluating the Accuracy of Sampling Based Approaches to the Calculation of Posterior Moments. Federal Reserve Bank of Minneapolis. Research Department Staff Report 148.Google Scholar
Gilks, W.R., Richardson, S. and Spiegelhalter, D.J. (1996) Markov Chain Monte Carlo in Practice. Chapman and Hall, London.Google Scholar
Green, P.J. (1995) Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination. Biometrika, 82, 711732.CrossRefGoogle Scholar
Hoeting, J.A., Madigan, D., Raftery, A.E. and Volinsky, C.T. (1999) Bayesian Model Averaging: A Tutorial. Statistical Science, 14(4), 382417.Google Scholar
Katsis, A. and Ntzoufras, I. (2005) Testing Hypotheses for the Distribution of Insurance Claim Counts Using the Gibbs Sampler. Journal of Computational Methods in Sciences and Engineering, 5, 201214.Google Scholar
Lunn, D.J., Best, N. and Whittaker, J.C. (2009) Generic Reversible Jump MCMC Using Graphical Models. Statistics and Computing, 19, 395408.Google Scholar
Lunn, D.J., Thomas, A., Best, N. and Spiegelhalter, D. (2000) WinBUGS — a Bayesian Modelling Framework: Concepts, Structure, and Extensibility. Statistics and Computing, 10, 325337.Google Scholar
Makov, U. (2001) Principal Applications of Bayesian Methods in Actuarial Science: A Perspective. North American Actuarial Journal, 5, 5360.Google Scholar
Nevat, I., Peters, G.W. and Yuan, J. (2009) Channel Estimation in OFDM Systems with Unknown Power Delay Profile Using Transdimensional MCMC via Stochastic Approximation. 69th Vehicular Technology Conference, VTC, Spring 2009, 16.Google Scholar
Ntzoufras, I. (2009) Bayesian Modeling Using WinBUGS, John Wiley.Google Scholar
Ntzoufras, I. and Dellaportas, P. (2002) Bayesian Modelling of Outstanding Liabilities Incorporating Claim Count Uncertainty (with discussion). North American Actuarial Journal, 6, 113128.Google Scholar
Ntzoufras, I., Katsis, A. and Karlis, D. (2005) Bayesian Assessment of the Distribution of Insurance Claim Counts Using Reversible Jump MCMC. North American Actuarial Journal, 9, 90108.Google Scholar
Peters, G.W., Shevchenko, M.V. and Wüthrich, P.V. (2009) Model Uncertainty in Claims Reserving within Tweedie's Compound Poisson Models. Astin Bulletin, 39(1), 133.Google Scholar
Renshaw, A.E. and Verrall, R.J. (1998) A Stochastic Model Underlying the Chain Ladder Technique. British Actuarial Journal, 4 (IV), 903923.Google Scholar
Roberts, G.O. and Rosenthal, J.S. (2007) Coupling and Ergodicity of Adaptive Markov Chain Monte Carlo Algorithms. Journal of Applied Probability, 44(2), 458475.Google Scholar
Roberts, G.O. and Rosenthal, J.S. (2009) Examples of Adaptive MCMC. Journal of Computational and Graphical Statistics, 18(2), 349367.Google Scholar
Scollnik, D.P.M. (2001) Actuarial Modelling with MCMC and BUGS. North American Actuarial Journal, 5(2), 96125.CrossRefGoogle Scholar
Scollnik, D.P.M. (2002) Implementation of Four Models for Outstanding Liabilities in WinBUGS: A discussion of a paper by Ntzoufras and Dellaportas, North American Actuarial Journal, 6, 128136.Google Scholar
Sisson, S.A. (2005) Transdimensional Markov Chains. Journal of the American Statistical Association, 100(471), 10771089.Google Scholar
Smith, B.J. (2007) boa: An R Package for MCMC Output Convergence Assessment and Posterior Inference. Journal of Statistical Software, 21(11), 137.Google Scholar
Taylor, G.C. and Ashe, F.R. (1983) Second Moments of Estimates of Outstanding Claims. Journal of Econometrics, 23, 3761.CrossRefGoogle Scholar
Verrall, R.J. (2007) Obtaining Predictive Distributions for Reserves Which Incorporate Expert Opinion. Variance (Casualty Actuarial Society Journal), 1, 5380.Google Scholar
Verrall, R.J. and Wüthrich, M.V. (2010) Reversible Jump Markov Chain Monte Carlo Method for Parameter Reduction in Claims Reserving.Google Scholar
Wüthrich, M.V. (2007) Using a Bayesian Approach for Claims Reserving. Variance, 1(2), 292301.Google Scholar
Wüthrich, M.V. and Merz, M. (2008) Stochastic Claims Reserving in Insurance. John Wiley.Google Scholar