Mathematics

Estimation of loss reserves based on a hierarchical bayesian model

  • ZHANG Yi ,
  • LIU Zhi-qiang ,
  • ZOU Si-si ,
  • WEN Li-min
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  • 1. School of Statistics, Jiangxi University of Finance and Economics, Nanchang 330013, China;
    2. Department of Statistics, Jiangxi Normal University, Nanchang 330022, China

Received date: 2017-09-29

  Online published: 2019-01-24

Abstract

Traditional loss reserve models are mainly based on aggregate loss triangles, in which the entries are obtained by summations of individual loss data. The summation procedures inevitably cause wastage of information contained in the raw individual data. Though this method is simple, it results in a larger error in the estimate of loss reserves. Individual loss reserve models emerging in recent years have failed to consider dependencies between policies. This article assumes the existence of certain common random effects between policies in the same accident year. Thus, a hierarchical bayesian model is built for individual data loss reserves. Using the ideas of credibility theory, we get the credibility estimate of loss reserves in each accident year, and thus the total reserve. In addition, the estimation of structural parameters and development factors are discussed. And the statistical properties are derived for those estimators of structural parameters. Finally, a numerical example is given to show the calculations with our estimators, and simulations are done to compare the mean square of reserve estimator between an individual loss model and an aggregate data model.

Cite this article

ZHANG Yi , LIU Zhi-qiang , ZOU Si-si , WEN Li-min . Estimation of loss reserves based on a hierarchical bayesian model[J]. Journal of East China Normal University(Natural Science), 2019 , 2019(1) : 13 -23 . DOI: 10.3969/j.issn.1000-5641.2019.01.002

References

[1] ARJAS E. The claims reserving problem in non-life insurance:Some structural ideas[J]. Astin Bulletin, 1989, 19(2):139-152.
[2] 张连增, 段白鸽. 未决赔款准备金的评估的随机性Munich链梯法[J]. 数理统计与管理, 2012, 31(5):880-897.
[3] BORNHUETTER R L, FERGUSON R E. The actuary and IBNR[J]. Proceedings of the casualty actuarial society, 1972, 6:181-195.
[4] VERRALL R J. A Bayesian generalized linear model for the Bornhuetter-Ferguson method of claims reserving[J]. North American Actuarial Journal. 2004, 8(3):67-89.
[5] 章溢, 温利民, 王江峰, 等. 随机B-F准备金模型中事故年索赔均值的信度估计[J]. 应用数学学报, 2016, 39(2):306-320.
[6] 俞雪梨, 温利民. 已报告未决赔款准备金的线性预估方法[J]. 统计与决策, 2010, 20:152-155.
[7] ZHAO X B, ZHOU X, WANG J L. Semiparametric model for prediction of individual claim loss reserving[J]. Insurance:Mathematics and Economics, 2009, 45(1):1-8.
[8] ZHAO X, ZHOU X. Applying copula models to individual claim loss reserving methods[J]. Insurance:Mathematics and Economics, 2010, 46:290-299.
[9] DHAENE J, GOOVAERTS M J. Dependency of risks and stop-loss order[J]. Astin Bulletin, 1996, 26:201-212.
[10] DHAENE J, DENNIT M, GOOVAERTS M J, et al. The concept of comonotonicity in actuarial science and finance:Theory[J]. Insurance:Mathematics and Economics, 2002, 31:3-33.
[11] 郑丹, 章溢, 温利民. 具有时间变化效应的信度模型[J]. 江西师范大学学报(自然科学版), 2012, 36(3):249-252.
[12] WEN L, WU, ZHOU X. The credibility premiums for models with dependence induced by common effects[J]. Insurance:Mathematics and Economics, 2009, 44:19-25.
[13] WEN L, FANG J, MEI G, et al. Optimal Linear estimation of random parameters in hierarchical random effect linear model[J]. Journal of Systems Science and Complexity. 2015, 28:1058-1069.
[14] BÜHLMANN H, GISLER A. Course in Credibility Theory and its Applications[M]. Amsterdam:Springer, 2005.
[15] 方婧, 章溢, 温利民. 聚合风险模型下的信度估计[J]. 江西师范大学学报(自然科学报), 2012, 36(6):607-611.
[16] WÜTHRICH M V, MERZ M. Stochastic Claims Reserving Methods in Insurance[M]. England:John Wiley & Sons, 2008.
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