Archives

Archives
Review Articles

Stochastic loss reserving using individual information model with over-dispersed Poisson

Zhigao Wang ,

School of Statistics, East China Normal University, Shanghai, People’s Republic of China

Xianyi Wu ,

School of Statistics, East China Normal University, Shanghai, People’s Republic of China

Chunjuan Qiu

School of Statistics, East China Normal University, Shanghai, People’s Republic of China

Pages 114-128 | Received 20 Aug. 2020, Accepted 01 Mar. 2021, Published online: 19 Mar. 2021,
  • Abstract
  • Full Article
  • References
  • Citations

For stochastic loss reserving, we propose an individual information model (IIM) which accommodates not only individual/micro data consisting of incurring times, reporting developments, settlement developments as well as payments of individual claims but also heterogeneity among policies. We give over-dispersed Poisson assumption about the moments of reporting developments and payments of every individual claims. Model estimation is conducted under quasi-likelihood theory. Analytic expressions are derived for the expectation and variance of outstanding liabilities, given historical observations. We utilise conditional mean square error of prediction (MSEP) to measure the accuracy of loss reserving and also theoretically prove that when risk portfolio size is large enough, IIM shows a higher prediction accuracy than individual/micro data model (IDM) in predicting the outstanding liabilities, if the heterogeneity indeed influences claims developments and otherwise IIM asymptotically equivalent to IDM. Some simulations are conducted to investigate the conditional MSEPs for IIM and IDM. A real data analysisis performed basing on real observations in health insurance.

References

  • Alai, D. H., Merz, M., & Wüthrich, M. V. (2009). Mean square error of prediction in the Bornhuetter–Ferguson claims reserving method. Annals of Actuarial Science, 4(1), 7–31. https://doi.org/10.1017/S1748499500000580
  • Alai, D. H., Merz, M., & Wüthrich, M. V. (2010). Prediction uncertainty in the Bornhuetter–Ferguson claims reserving method: Revisited. Annals of Actuarial Science, 5(1), 7–7. https://doi.org/10.1017/S1748499510000023
  • Antonio, K., & Plat, R. (2014). Micro-level stochastic loss reserving for general insurance. Scandinavian Actuarial Journal, 2014(7), 649–669. https://doi.org/10.1080/03461238.2012.755938
  • Arjas, E. (1989). The claims reserving problem in non-life insurance: Some structural ideas. ASTIN Bulletin: The Journal of the IAA, 19(2), 139–152. https://doi.org/10.2143/AST.19.2.2014905
  • Crevecoeur, J., Antonio, K., & Verbelen, R. (2019). Modeling the number of hidden events subject to observation delay. European Journal of Operational Research, 277(3), 930–944. https://doi.org/10.1016/j.ejor.2019.02.044
  • England, P. D., & Verrall, R. J. (2002). Stochastic claims reserving in general insurance. British Actuarial Journal, 8(3), 443–544. https://doi.org/10.1017/S1357321700003809
  • Huang, J., Qiu, C., & Wu, X. (2015). Stochastic loss reserving in discrete time: Individual vs. aggregate data models. Communications in Statistics-Theory and Methods, 44(10), 2180–2206. https://doi.org/10.1080/03610926.2014.976473
  • Huang, J., Qiu, C., Wu, X., & Zhou, X. (2015). An individual loss reserving model with independent reporting and settlement. Insurance: Mathematics and Economics, 64(1), 232–245. https://doi.org/10.1016/j.insmatheco.2015.05.010
  • Huang, J., Wu, X., & Zhou, X. (2016). Asymptotic behaviors of stochastic reserving: Aggregate versus individual models. European Journal of Operational Research, 249(2), 657–666. https://doi.org/10.1016/j.ejor.2015.09.039
  • Lindholm, M., Lindskog, F., & Wahl, F. (2020). Estimation of conditional mean squared error of prediction for claims reserving. Annals of Actuarial Science, 14(1), 93–128. https://doi.org/10.1017/S174849951900006X
  • Mack, T. (1993). Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin: The Journal of the IAA, 23(2), 213–225. https://doi.org/10.2143/AST.23.2.2005092
  • Mack, T. (2000). Credible claims reserves: The Benktander method. ASTIN Bulletin: The Journal of the IAA, 30(2), 333–347. https://doi.org/10.2143/AST.30.2.504639
  • McCullagh, P., & Nelder, J. A. (1989). Generalized linear models (2nd ed). Chapman and Hall.
  • Norberg, R. (1993). Prediction of outstanding liabilities in non-life insurance 1. ASTIN Bulletin: The Journal of the IAA, 23(1), 95–115. https://doi.org/10.2143/AST.23.1.2005103
  • Norberg, R. (1999). Prediction of outstanding liabilities II. Model variations and extensions. ASTIN Bulletin: The Journal of the IAA, 29(1), 5–25. https://doi.org/10.2143/AST.29.1.504603
  • Pigeon, M., Antonio, K., & Denuit, M. (2013). Individual loss reserving with the multivariate skew normal framework. ASTIN Bulletin: The Journal of the IAA, 43(3), 399–428. https://doi.org/10.1017/asb.2013.20
  • Pigeon, M., Antonio, K., & Denuit, M. (2014). Individual loss reserving using paid-incurred data. Insurance: Mathematics and Economics, 58(2), 121–131. https://doi.org/10.1016/j.insmatheco.2014.06.012
  • Van der Vaart, A. W. (2000). Asymptotic statistics. Cambridge University Press.
  • Verrall, R. J., Nielsen, J. P., & Jessen, A. H. (2010). Prediction of RBNS and IBNR claims using claim amounts and claim counts. Astin Bulletin, 40(2), 871–887. https://doi.org/10.2143/AST.40.2.2061139
  • Wahl, F. (2019). Explicit moments for a class of micro-models in non-life insurance. Insurance: Mathematics and Economics, 89(7), 140–156. https://doi.org/10.1016/j.insmatheco.2019.10.001
  • Wahl, F., Lindholm, M., & Verrall, R. (2019). The collective reserving model. Insurance: Mathematics and Economics, 87(7), 34–50. https://doi.org/10.1016/j.insmatheco.2019.04.003
  • Wüthrich, M. V., & Merz, M. (2008). Stochastic claims reserving methods in insurance. John Wiley and Sons.
  • Yu, X., & He, R. (2016). Individual claims reserving models based on marked Cox processes. Chinese Journal of Applied Probability and Statistics 32(2), 201-219. https://doi.org/http://aps.ecnu.edu.cn/EN/Y2016/V32/I2/201
  • Zhao, X., & Zhou, X. (2010). Applying copula models to individual claim loss reserving methods. Insurance: Mathematics and Economics, 46(2), 290–299. https://doi.org/10.1016/j.insmatheco.2009.11.001

To cite this article: Zhigao Wang, Xianyi Wu & Chunjuan Qiu (2021): Stochastic loss reserving
using individual information model with over-dispersed Poisson, Statistical Theory and Related
Fields, DOI: 10.1080/24754269.2021.1898181
To link to this article: https://doi.org/10.1080/24754269.2021.1898181

Authors

About the Journal

Links

3663 N.Zhongshan Rd., Shanghai 200062  Tel.: 86-21-62233333