Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T07:40:24.416Z Has data issue: false hasContentIssue false
  • English
  • Français

STOCHASTIC CLAIMS RESERVING VIA A BAYESIAN SPLINE MODEL WITH RANDOM LOSS RATIO EFFECTS

Published online by Cambridge University Press:  13 July 2017

Guangyuan Gao  and
Shengwang Meng
Guangyuan Gao*
Affiliation:
Renmin University of China, Center for Applied Statistics and School of Statistics, Room 1006, Mingde Building, 59 Zhongguancun Avenue, Haidian District, Beijing, 100872, China
Shengwang Meng
Affiliation:
Renmin University of China, Center for Applied Statistics and School of Statistics, 59 Zhongguancun Avenue, Haidian District, Beijing, 100872, China E-Mail: mengshw@ruc.edu.cn
*
E-Mail: guangyuan.gao@anu.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a Bayesian spline model which uses a natural cubic B-spline basis with knots placed at every development period to estimate the unpaid claims. Analogous to the smoothing parameter in a smoothing spline, shrinkage priors are assumed for the coefficients of basis functions. The accident period effect is modeled as a random effect, which facilitate the prediction in a new accident period. For model inference, we use Stan to implement the no-U-turn sampler, an automatically tuned Hamiltonian Monte Carlo. The proposed model is applied to the workers' compensation insurance data in the United States. The lower triangle data is used to validate the model.

Keywords

Stochastic claims reservingtail factornatural cubic splineHamiltonian Monte CarloBayesian modeling
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 1 , January 2018 , pp. 55 - 88
Copyright
Copyright © Astin Bulletin 2017 

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

Antonio, K. and Beirlant, J. (2008) Issues in claims reserving and credibility: A semiparametric approach with mixed models. Journal of Risk and Insurance, 75, 643–676.CrossRefGoogle Scholar
Armagan, A., Dunson, D.B. and Lee, J. (2013) Generalized double Pareto shrinkage. Statistica Sinica, 23, 119–143.Google ScholarPubMed
Betancourt, M. (2017) A conceptual introduction to Hamiltonian Monte Carlo. Preprint, arXiv:1701.02434.Google Scholar
Bishop, C.M. (2006) Pattern Recognition and Machine Learning. New York: Springer.Google Scholar
Bornhütter, R.L. and Ferguson, R. (1972) The actuary and IBNR. Proceedings of the Casualty Actuarial Society, 59, 181–195.Google Scholar
Clark, D.R. (2003) LDF curve-fitting and stochastic reserving: A maximum likelihood approach. Casualty Actuarial Society Forum (Fall), 41–92.Google Scholar
Crainiceanu, C.M., Ruppert, D. and Wand, M.P. (2005) Bayesian analysis for penalized spline regression using WinBUGS. Journal of Statistical Software, 14, 1–14.CrossRefGoogle Scholar
Duane, S., Kennedy, A.D., Pendleton, B.J. and Roweth, D. (1987) Hybrid Monte Carlo. Physics Letters B, 195, 216–222.CrossRefGoogle Scholar
England, P.D. and Verrall, R.J. (2001) A flexible framework for stochastic claims reserving. Proceedings of the Casualty Actuarial Society, 88, 1–38.Google Scholar
Friedland, J. (2010) Estimating unpaid claims using basic techniques. Casualty Actuarial Society Study Notes. Available at http://www.casact.org/library/studynotes/Friedland_estimating.pdf.Google Scholar
Gelman, A., Carlin, J.B., Stern, H.S. and Rubin, D.B. (2014) Bayesian Data Analysis, 3rd edition. Boca Raton: Chapman & Hall.Google Scholar
Gelman, A., Lee, D. and Guo, J. (2015) Stan: A probabilistic programming language for Bayesian inference and optimization. Journal of Educational and Behavioral Statistics, 40, 530–543.CrossRefGoogle Scholar
Hastie, T.J. and Tibshirani, R.J. (1990) Generalized Additive Models. New York: Chapman & Hall.Google Scholar
Hastie, T.J., Tibshirani, R.J. and Friedman, J.H. (2009) The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd edition. New York: Springer.CrossRefGoogle Scholar
Hastings, W.K. (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109.CrossRefGoogle Scholar
Homan, M.D. and Gelman, A. (2014) The no-u-turn sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo. The Journal of Machine Learning Research, 15, 1593–1623.Google Scholar
James, G., Witten, D., Hastie, T. and Tibshirani, R. (2013) An Introduction to Statistical Learning. New York: Springer.CrossRefGoogle 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, 325–337.CrossRefGoogle 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. (1999) The standard error of chain-ladder reserve estimates, recursive calculation and inclusion of a tail factor. ASTIN Bulletin, 29, 361–366.CrossRefGoogle Scholar
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. and Teller, E. (1953) Equation of state calculations by fast computing machines. Journal of Chemical Physics, 21, 1087–1092.CrossRefGoogle Scholar
Meyers, G. (2015) Stochastic loss reserving using Bayesian MCMC models. CAS Monograph Series, 1, 1–64.Google Scholar
Neal, R.M. (1994) An improved acceptance procedure for the hybrid Monte Carlo algorithm. Journal of Computational Physics, 111, 194–203.CrossRefGoogle Scholar
Neal, R.M. (2011) MCMC using Hamiltonian dynamics. Handbook of Markov Chain Monte Carlo (eds. Brooks, S., Gelman, A., Jones, G., and Meng, X.L.), pp. 113–160. Boca Raton, FL: CRC Press.CrossRefGoogle Scholar
Park, T. and Casella, G. (2008) The Bayesian lasso. Journal of the American Statistical Association, 103, 681–686.CrossRefGoogle Scholar
Roberts, G.O., Gelman, A. and Gilks, W.R. (1997) Weak convergence and optimal scaling of random walk metropolis algorithms. Annals of Applied Probability, 7, 110–120.Google Scholar
Ruppert, D., Wand, M.P. and Carroll, R.J. (2003) Semiparametric Regression. New York: Cambridge University Press.CrossRefGoogle Scholar
Stan Development Team (2016) RStan: The R interface to Stan. http://mc-stan.org.Google Scholar
Taylor, G. (2000) Loss Reserving: An Actuarial Perspective. Huebner International Series on Risk, Insurance and Economic Security. Boston: Kluwer Academic Publishers.CrossRefGoogle Scholar
Verrall, R., Hössjer, O. and Björkwall, S. (2012) Modelling claims run-off with reversible jump Markov chain Monte Carlo methods. ASTIN Bulletin, 42, 35–58.Google Scholar
Verrall, R.J. (1996) Claims reserving and generalised additive models. Insurance: Mathematics and Economics, 19, 31–43.Google Scholar
Verrall, R.J. and Wüthrich, M.V. (2012) Reversible jump Markov chain Monte Carlo method for parameter reduction in claims reserving. North American Actuarial Journal, 16, 240–259.CrossRefGoogle Scholar
Wood, S. (2006) Generalized Additive Models: An Introduction with R. New York: Chapman & Hall.CrossRefGoogle Scholar
Wright, T.S. (1990) A stochastic method for claims reserving in general insurance. Journal of the Institute of Actuaries, 117, 677–731.CrossRefGoogle Scholar
Wüthrich, M.V. and Merz, M. (2008) Stochastic Claims Reserving Methods in Insurance. Chichester: John Wiley & Sons.Google Scholar
Wüthrich, M.V. and Merz, M. (2015) Stochastic claims reserving manual: Advances in dynamic modeling. SSRN Manuscript ID 2649057.CrossRefGoogle Scholar
Zhang, Y., Dukic, V. and Guszcza, J. (2012) A Bayesian non-linear model for forecasting insurance loss payments. Journal of the Royal Statistical Society A, 175, 637–656.CrossRefGoogle Scholar
Zhang, Y.W. and Dukic, V. (2013) Predicting multivariate insurance loss payments under the Bayesian copula framework. Journal of Risk and Insurance, 80, 891–919.CrossRefGoogle Scholar