Hostname: page-component-77f85d65b8-8v9h9 Total loading time: 0 Render date: 2026-04-18T19:00:00.276Z Has data issue: false hasContentIssue false
  • English
  • Français

Predictive Distributions of Outstanding Liabilities in General Insurance

Published online by Cambridge University Press:  10 May 2011

P. D. England  and
R. J. Verrall
P. D. England
Affiliation:
EMB Consultancy, Saddlers Court, 64-74 East Street, Epsom KT17 1HB, U.K., Email: peter.england@emb.co.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper extends the methods introduced in England & Verrall (2002), and shows how predictive distributions of outstanding liabilities in general insurance can be obtained using bootstrap or Bayesian techniques for clearly defined statistical models. A general procedure for bootstrapping is described, by extending the methods introduced in England & Verrall (1999), England (2002) and Pinheiro et al. (2003). The analogous Bayesian estimation procedure is implemented using Markov-chain Monte Carlo methods, where the models are constructed as Bayesian generalised linear models using the approach described by Dellaportas & Smith (1993). In particular, this paper describes a way of obtaining a predictive distribution from recursive claims reserving models, including the well known model introduced by Mack (1993). Mack's model is useful, since it can be used with data sets which exhibit negative incremental amounts. The techniques are illustrated with examples, and the resulting predictive distributions from both the bootstrap and Bayesian methods are compared.

Keywords

BayesianBootstrapChain LadderDynamic Financial AnalysisGeneralised Linear ModelMarkov Chain Monte CarloReserving RiskStochastic Reserving

Information

Type
Papers
Information
Annals of Actuarial Science , Volume 1 , Issue 2 , September 2006 , pp. 221 - 270
Copyright
Copyright © Institute and Faculty of Actuaries 2006

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.)

Article purchase

Temporarily unavailable

References

Albert, J.H. & Pepple, P.A. (1989). A Bayesian approach to some overdispersion models. Canadian Journal of Statistics, 17(3), 333–344.CrossRefGoogle Scholar
Ashe, F.R. (1986). An essay at measuring the variance of estimates of outstanding claim payments. ASTIN Bulletin, 16S, 99–113.CrossRefGoogle Scholar
Brickman, S., Barlow, C., Boulter, A., English, A., Furber, L., Ibeson, D., Lowe, L., Pater, R. & Tomlinson, D. (1993). Variance in claims reserving. Proceedings of the 1993 General Insurance Convention, Institute of Actuaries and Faculty of Actuaries.Google Scholar
Congdon, P. (2003). Applied Bayesian modelling. Wiley, Chichester.CrossRefGoogle Scholar
De Alba, E. (2002). Bayesian estimation of outstanding claim reserves. North American Actuarial Journal, 6(4), 1–20.CrossRefGoogle Scholar
Dellaportas, P. & Smith, A.F.M. (1993). Bayesian inference for generalized linear and proportional hazards models via Gibbs sampling. Applied Statistics, 42(3), 443–459.CrossRefGoogle Scholar
Dey, D.K., Ghosh, S.K. & Mallick, B.K. (2000). Generalized linear models: a bayesian perspective. Marcel Dekker.CrossRefGoogle Scholar
Dey, D.K. & Ravishanker, N. (2000). Bayesian approaches for overdispersion in generalised linear models. In generalized linear models: a Bayesian perspective. Edited by Dey, D.K., Ghosh, S.K. & Mallick, B.K.. Marcel Dekker.CrossRefGoogle Scholar
Efron, B. & Tibshirani, R.J. (1993). An Introduction to the bootstrap. Chapman and Hall, London.CrossRefGoogle Scholar
England, P.D. (2002). Addendum to ‘Analytic and bootstrap estimates of prediction errors in claims reserving’. Insurance: Mathematics and Economics, 31, 461–466.Google Scholar
England, P.D. & Verrall, R.J. (1999). Analytic and bootstrap estimates of prediction errors in claims reserving. Insurance: Mathematics and Economics, 25, 281–293.Google Scholar
England, P.D. & Verrall, R.J. (2002). Stochastic claims reserving in general insurance (with discussion). British Actuarial Journal, 8, 443–544.CrossRefGoogle Scholar
Gelfand, A.E. & Ghosh, M. (2000). Generalised linear models: a Bayesian view. In Generalized linear models: a Bayesian perspective. Edited by Dey, D.K., Ghosh, S.K. & Mallick, B.K.. Marcel Dekker.Google Scholar
Gilks, W.R. & Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling. Applied Statistics, 41(2), 337–348.CrossRefGoogle Scholar
Gilks, W.R., Best, N.G. & Tan, K.K.C. (1995). Adaptive rejection metropolis sampling within Gibbs sampling. Applied Statistics, 44(4), 455–472.CrossRefGoogle Scholar
Haastrup, S. & Arjas, E. (1996). Claims reserving in continuous time; a non-parametric Bayesian approach. ASTIN Bulletin, 26(2), 139–164.CrossRefGoogle Scholar
Igloo Professional with ExtrEMB (2005). Igloo Professional with ExtrEMB v2.3.1. EMB Software Ltd, Epsom U.K.Google Scholar
Kaas, R., Goovaerts, M., Dhaene, M. & Denuit, M. (2002). Modern actuarial risk theory. Kluwer.Google Scholar
Klugman, S.A. (1992). Bayesian statistics in actuarial science. Kluwer (Boston).CrossRefGoogle Scholar
Lowe, J. (1994). A practical guide to measuring reserve variability using: bootstrapping, operational time and a distribution free approach. Proceedings of the 1994 General Insurance Convention, Institute of Actuaries and Faculty of Actuaries.Google Scholar
Mack, T. (1991). A simple parametric model for rating automobile insurance or estimating IBNR claims reserves. ASTIN Bulletin, 22(1), 93–109.CrossRefGoogle Scholar
Mack, T. (1993). Distribution-free calculation of the standard error of chain-ladder reserve estimates. ASTIN Bulletin, 23, 213–225.CrossRefGoogle Scholar
Makov, U.E. (2001). Principal applications of Bayesian methods in actuarial science: a perspective. North American Actuarial Journal, 5(4), 53–73.CrossRefGoogle Scholar
Makov, U.E., Smith, A.F.M. & Liu, Y.-H. (1996). Bayesian methods in actuarial science. The Statistician, 45(4), 503–515.CrossRefGoogle Scholar
McCullagh, P. & Nelder, J. (1989). Generalized linear models (2nd edition). Chapman and Hall, London.CrossRefGoogle Scholar
Moulton, L.H. & Zeger, S.L. (1991). Bootstrapping generalized linear models. Computational Statistics and Data Analysis, 11, 53–63.CrossRefGoogle Scholar
Ntzoufras, I. & Dellaportas, P. (2002). Bayesian modelling of outstanding liabilities incorporating claim count uncertainty. North American Actuarial Journal, 6(1), 113–128.CrossRefGoogle Scholar
Pinheiro, P.J.R., Andrade, E., Silva, J.M. & Centeno, M.L.C. (2003). Bootstrap methodology in claim reserving. Journal of Risk and Insurance, 70(4), 701–714.CrossRefGoogle Scholar
Renshaw, A.E. (1994). Claims reserving by joint modelling. Actuarial Research Paper No. 72, Department of Actuarial Science and Statistics, City University, London, EC1V 0HB.Google Scholar
Renshaw, A.E. & Verrall, R.J. (1998). A stochastic model underlying the chain ladder technique. British Actuarial Journal, 4, 903–923.CrossRefGoogle Scholar
Scollnik, D.P.M. (2001). Actuarial modelling with MCMC and BUGS. North American Actuarial Journal, 5(2), 96–125.CrossRefGoogle Scholar
Spiegelhalter, D.J., Thomas, A., Best, N.G. & Gilks, W.R. (1996). BUGS 0.5: Bayesian inference using Gibbs sampling. MRC Biostatistics Unit, Cambridge, UK.Google Scholar
Taylor, G.C. (1988). Regression models in claims analysis: theory. Proceedings of the Casualty Actuarial Society, 74, 354–383.Google Scholar
Taylor, G.C. (2000). Loss reserving: an actuarial perspective. Kluwer.CrossRefGoogle Scholar
Taylor, G.C. & Ashe, F.R. (1983). Second moments of estimates of outstanding claims. Journal of Econometrics, 23(1), 37–61.CrossRefGoogle Scholar
Verrall, R.J. (2000). An investigation into stochastic claims reserving models and the chain-ladder technique. Insurance: Mathematics and Economics, 26, 91–99.Google Scholar
Verrall, R.J. (2004). A Bayesian generalized linear model for the Bornhuetter-Ferguson method of claims reserving. North American Actuarial Journal, 8, 67–89,CrossRefGoogle Scholar
Verrall, R.J. & England, P.D. (2005). Incorporating expert opinion into a stochastic model for the chain-ladder technique: Insurance: Mathematics and Economics, 37, 355–370.Google Scholar