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Stochastic Claims Reserving in General Insurance

Published online by Cambridge University Press:  10 June 2011

P.D. England  and
R.J. Verrall
P.D. England
Affiliation:
EMB Consultancy, Saddlers Court, 64-74 East Street, Epsom, Surrey KT17 1HP, U.K. E-mail: [email protected]
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Abstract

This paper considers a wide range of stochastic reserving models for use in general insurance, beginning with stochastic models which reproduce the traditional chain-ladder reserve estimates. The models are extended to consider parametric curves and smoothing models for the shape of the development run-off, which allow extrapolation for the estimation of tail factors. The Bornhuetter-Ferguson technique is also considered, within a Bayesian framework, which allows expert opinion to be used to provide prior estimates of ultimate claims. The primary advantage of stochastic reserving models is the availability of measures of precision of reserve estimates, and in this respect, attention is focused on the root mean squared error of prediction (prediction error). Of greater interest is a full predictive distribution of possible reserve outcomes, and different methods of obtaining that distribution are described. The techniques are illustrated with examples throughout, and the wider issues discussed, in particular, the concept of a ‘best estimate’; reporting the variability of claims reserves; and use in dynamic financial analysis models.

Keywords

Stochastic ReservingGeneral InsuranceChain-LadderBornhuetter-FergusonGeneralised Linear ModelGeneralised Additive ModelBayesianBootstrapSimulationMarkov Chain Monte CarloDynamic Financial Analysis
Type
Sessional meetings: papers and abstracts of discussions
Information
British Actuarial Journal , Volume 8 , Issue 3 , 01 August 2002 , pp. 443 - 518
Copyright
Copyright © Institute and Faculty of Actuaries 2002

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References

Barnett, G. & Zehnwirth, B. (1998). Best estimates for reserves. Casualty Actuarial Society, Fall Forum.Google Scholar
Benktander, G. (1976). An approach to credibility in calculating IBNR for casualty excess reinsurance. Actuarial Review, April 1976, 7.Google Scholar
Bornheutter, R. L. & Ferguson, R. E. (1972). The actuary and IBNR. Proceedings of the Casualty Actuarial Society, LIX, 181–195.Google Scholar
Chambers, J. M. & Hastie, T. J. (1992). Statistical models in S. Chapman and Hall, London.Google Scholar
Christofides, S. (1990). Regression models based on log-incremental payments. Claims Reserving Manual, 2, Institute of Actuaries, London.Google Scholar
De Jong, P. & Zehnwirth, B. (1983). Claims reserving, state-space models and the Kalman filter. J.I.A. 110, 157–182.Google Scholar
Doray, L. G. (1996). UMVUE of the IBNR reserve in a lognormal linear regression model. Insurance: Mathematics and Economics, 18, 43–57.Google Scholar
Efron, B. & Tibshirani, R. J. (1993). An introduction to the bootstrap. Chapman and Hall.CrossRefGoogle Scholar
England, P. D. (2001). Addendum to ‘Analytic and bootstrap estimates of prediction errors in claims reserving’. Actuarial Research Paper No. 138, Department of Actuarial Science and Statistics, City University, London, EC1V 0HB.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. (2001). A flexible framework for stochastic claims reserving. Proceedings of the Casualty Actuarial Society (to appear).Google Scholar
Gibson, L. (2000). Reserving best estimates and ranges. Proceedings of the General Insurance Convention, 2, 185–187, Institute of Actuaries.Google Scholar
Green, P. J. & Silverman, B. W. (1994). Nonparametric regression and generalized linear models. Chapman and Hall, London.CrossRefGoogle Scholar
Haastrup, S. (1997). Some fully Bayesian micro models for claims reserving. PhD thesis, University of Copenhagen.Google Scholar
Haastrup, S. & Arjas, E. (1996). Claims reserving in continuous time: a nonparametric Bayesian approach. ASTIN Bulletin, 26, 139–164.CrossRefGoogle Scholar
Hastie, T. J. & Tibshirani, R. J. (1990). Generalized additive models. Chapman and Hall, London.Google Scholar
Historical Loss Development Study (1991). Reinsurance Association of America. Washington, D.C.Google Scholar
Kremer, E. (1982). IBNR claims and the two way model of ANOVA. Scandinavian Actuarial Journal, 47–55.CrossRefGoogle Scholar
Mack, T. (1991). A simple parametric model for rating automobile insurance or estimating IBNR claims reserves. ASTIN Bulletin, 22, 1, 93–109.Google Scholar
Mack, T. (1993). Distribution-free calculation of the standard error of chain-ladder reserve estimates. ASTIN Bulletin, 23, 213–225.CrossRefGoogle Scholar
Mack, T. (1994a). Which stochastic model is underlying the chain-ladder model? Insurance: Mathematics and Economics, 15, 133–138.Google Scholar
Mack, T. (1994b). Measuring the variability of chain-ladder reserve estimates. Casualty Actuarial Society, Spring Forum.Google Scholar
Mack, T. (2000). Credible claims reserves: the Benktander method. ASTIN Bulletin, 30, 333–347.CrossRefGoogle Scholar
Mack, T. & Venter, G. (2000). A comparison of stochastic models that reproduce chain-ladder reserve estimates. Insurance: Mathematics and Economics, 26, 101–107.Google Scholar
McCullagh, P. & Nelder, J. (1989). Generalised linear models, 2nd edition. Chapman and Hall.CrossRefGoogle Scholar
Murphy, D. M. (1994). Unbiased loss development factors. Proceedings of the Casualty Actuarial Society, LXXXI, 154–222.Google Scholar
Norberg, R. (1993). Prediction of outstanding liabilities in non-life insurance. ASTIN Bulletin, 23, 95–115.CrossRefGoogle Scholar
Norberg, R. (1999). Prediction of outstanding liabilities II. Model variations and extensions. ASTIN Bulletin, 29, 5–25.CrossRefGoogle Scholar
Pereira, F. C. (2000). Bayesian Markov chain Monte Carlo methods in general insurance. PhD thesis, City University, London.Google Scholar
Renshaw, A. E. (1989). Chain-ladder and interactive modelling (claims reserving and GLIM). J.I.A. 116, 559–587.Google Scholar
Renshaw, A. E. (1994a). Claims reserving by joint modelling. Actuarial Research Paper No. 72, Department of Actuarial Science and Statistics, City University, London.Google Scholar
Renshaw, A. E. (1994b). On the second moment properties and the implementation of certain GLIM based stochastic claims reserving models. Actuarial Research Paper No. 65, Department of Actuarial Science and Statistics, City University, London.Google Scholar
Renshaw, A. E. & Verrall, R. J. (1998). A stochastic model underlying the chain-ladder technique. B.A.J. 4, 903–923.Google Scholar
Schmidt, K. D. & Schnaus, A. (1996). An extension of Mack's model for the chain-ladder method. ASTIN Bulletin, 26, 247–262.CrossRefGoogle Scholar
Scollnik, D. P. M. (2001). Actuarial modeling with MCMC and BUGS. North American Actuarial Journal, 5 (2), 96–124.CrossRefGoogle Scholar
Sherman, R. E. (1984). Extrapolating, smoothing and interpolating development factors. Proceedings of the Casualty Actuarial Society, LXXI, 122–192.Google 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 U.K.Google Scholar
S-Plus (2001). S-PLUS 6 for Windows user's guide. Insightful Corporation, Seattle, WA.Google Scholar
Taylor, G. C. (2000). Loss reserving: an actuarial perspective. Kluwer Academic Publishers.CrossRefGoogle Scholar
Taylor, G. C. & Ashe, F. R. (1983). Second moments of estimates of outstanding claims. Journal of Econometrics, 23, 37–61.CrossRefGoogle Scholar
Venables, W. N. & Ripley, B. D. (1999). Modern applied statistics with S-Plus. 3rd edition. New-York: Springer-Verlag.CrossRefGoogle Scholar
Verrall, R. J. (1989). A state space representation of the chain-ladder linear model. J.I.A. 116, 589–610.Google Scholar
Verrall, R. J. (1990). Bayes and empirical Bayes estimation for the chain-ladder model. ASTIN Bulletin, 20, 2, 217–243.CrossRefGoogle Scholar
Verrall, R. J. (1991a). On the unbiased estimation of reserves from loglinear models. Insurance: Mathematics and Economics, 10, 1, 75–80.Google Scholar
Verrall, R. J. (1991b). Chain-ladder and maximum likelihood. J.I.A. 118, 489–499.Google Scholar
Verrall, R. J. (1994). A method for modelling varying run-off evolutions in claims reserving. ASTIN Bulletin, 24, 2, 325–332.CrossRefGoogle Scholar
Verrall, R. J. (1996). Claims reserving and generalised additive models. Insurance: Mathematics and Economics, 19, 31–43.Google 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. (2001). A Bayesian generalised linear model for the Bornhuetter-Ferguson method of claims reserving. Actuarial Research Paper No. 139, Department of Actuarial Science and Statistics, City University, London.Google Scholar
Verrall, R. J. & England, P. D. (2000). Comments on: ‘A comparison of stochastic models that reproduce chain-ladder reserve estimates', by Mack & Venter. Insurance: Mathematics And Economics, 26, 109–111.Google Scholar
Wang, S. (1995). Insurance pricing and increased limits ratemaking by proportional hazards transforms. Insurance: Mathematics And Economics, 17, 43–54.Google Scholar
Whittaker, E. T. (1923). On a new method of graduation. Proceedings of the Edinburgh Mathematical Society, 41, 63–75.CrossRefGoogle Scholar
Wright, T. S. (1990). A stochastic method for claims reserving in general insurance. J.I.A. 117, 677–731.Google Scholar
Wright, T. S. (1997). Probability distribution of outstanding liability from individual payments data. Claims Reserving Manual, 2, Institute of Actuaries, London.Google Scholar
Zehnwirth, B. (1985). ICRFS version 4 manual and users guide. Benhar Nominees Pty Ltd, Turramurra, NSW, Australia.Google Scholar
Zehnwirth, B. (1989). The chain-ladder technique — a stochastic model. Claims Reserving Manual, 2, Institute of Actuaries, London.Google Scholar