Abstract:In view of the obvious shortcomings of single factor Lee-Carter model where mortality rate improves at constant rate, this paper, using Bayesian Markov Chain Monte Carlo (MCMC) method, gives a study on the predictive power of two-factor Lee-Carter model with Chinese actual population data. The test results indicate that two-factor model can preferably capture the volatility of mortality improvement and significantly outperform the original model in terms of goodness of fit and deviance information criteria (DIC). Moreover, comparisons of annuity prices, statistic characteristics, risk measurements and capital requirements between the two models show that the two-factor model, characterized by sharp peak and heavy tail feature of annuity price kernel density figure, should work together with the TVaR, instead of the traditional Solvency Ⅱ and VaR principle, to better respond to the longevity risk of annuities caused by larger-than-expected mortality improvement.
[1] Lee R D, Carter L. Modeling and forecasting the time series of U.S. mortality[J]. Journal of the American Statistical Association, 1992, 87(14):659-675. [2] Brouhns N, Denuit M, Vermunt J K. A poisson log-bilinear regression approach to the construction of projected lifetables[J]. Insurance:Mathematics and Economics, 2002, 31(3):373-393. [3] Renshaw A E, Haberman S. A cohort-based extension to the Lee-Carter model for mortality reduction Factors[J]. Insurance:Mathematics and Economics, 2006, 38(3):556-570. [4] Cairns A J G, Blake D, Dowd K, et al. Bayesian stochastic mortality modelling for two populations[J]. ASTIN Bulletin, 2011, 41(1):29-59. [5] 王晓军, 任文东. 有限数据下Lee-Carter模型在人口死亡率预测中的应用[J]. 统计研究, 2012, 29(6):87-94.Wang X J, Ren W D. Application of Lee-Carter method in forecasting the mortality of Chinese population with limited data[J]. Statistical Research, 2012, 29(6):87-94. [6] Li J. An application of MCMC simulation in mortality projection for populations with limited data[J]. Demographic Research, 2014, 30:1-48. [7] 曾燕, 陈曦, 邓颖璐. 创新的动态人口死亡率预测及其应用[J]. 系统工程理论与实践, 2016, 36(7):1710-1718. Zeng Y, Chen X, Deng Y L. Innovative dynamic mortality rate prediction and implementation[J]. Systems Engineering-Theory & Practice, 2016, 36(7):1710-1718. [8] Renshaw A E, Haberman S. Lee-Carter mortality forecasting with age specific enhancement[J]. Insurance:Mathematics and Economics, 2003, 33(2):255-272. [9] Renshaw A E, Haberman S. Lee-Carter mortality forecasting:A parallel generalized linear modelling approach for England and Wales mortality projections[J]. Journal of the Royal Statistical Society, 2010, 52(1):119-137. [10] Renshaw A E, Haberman S. Lee-Carter mortality forecasting incorporating bivariate time series for England and Wales mortality projections[R]. City University Actuarial Research Paper 163, 2005. [11] Lazar D, Denuit M M. A multivariate time series approach to projected life tables[J]. Applied Stochastic Models in Business and Industry, 2009, 25:806-823. [12] Girosi F, King G. Demographic forecasting[M]. Cambridge:Cambridge University Press, 2006. [13] 吴恒煜, 朱福敏, 温金明, 等. 基于序贯贝叶斯参数学习的Lévy动态波动率模型研究[J]. 系统工程理论与实践, 2017, 37(3):556-569. Wu H Y, Zhu F M, Wen J M, et al. Sequential parameter learning for Lévy-driven volatility models[J]. Systems Engineering-Theory & Practice, 2017, 37(3):556-569. [14] Kogure A, Kitsukawa K, Kurachi Y. A Bayesian comparison of models for changing mortalities toward evaluating longevity risk in Japan[J]. Asia-Pacific Journal of Risk and Insurance, 2009, 3(2):1-21. [15] Czado C, Delwarde A, Denuit M. Bayesian poisson log-bilinear mortality projections[J]. Insurance:Mathematics and Economics, 2005, 36(3):260-284. [16] Coale A, Guo G. Revised regional model life tables at very low levels of mortality[J]. Population Index, 1989, 55(4):613-643. [17] Ntzoufras I. Bayesian modelling using WinBUGS[M]. Hoboken, New Jersey:John Wiley & Sons, Inc, 2009:117-120. [18] Bignozzi V, Tsanakas A. Parameter uncertainty and residual estimation risk[J]. Journal of Risk and Insurance, 2016, 83(4):949-978.