广义加权损失函数下未决赔款准备金的信度估计
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摘要
为了使得估计的准备金不依赖于先验分布的具体形式,在贝叶斯链梯模型中,采用信度理论的思想,在广义加权损失函数下得到链梯因子的信度估计,建立了案均赔款法下的未决赔款准备金模型.最后,给出保险公司的实际例子,将得到的信度估计与经典链梯法和随机链梯法估计进行了比较.结论显示,方法对未决赔款准备金是有效的.
We consider the chain ladder reserving method in a Bayesian set up, which allows for combining individual claims development data with portfolio information and avoids choosing the prior estimates. This paper presents Bayes chain ladder model which is based on payments per claim incurred. The ideas from credibility theory are used and credibility estimates of chain ladder factors are derived under generalized weighted loss function. Furthermore, we get estimates of reserves. Finally, practical examples of insurance company are given and differences are compared among our credibility estimates obtained, chain ladder estimates and stochastic chain ladder estimates. The conclusions show that credibility claims reserving which is based on the generalized weighted loss function is valid.
We consider the chain ladder reserving method in a Bayesian set up, which allows for combining individual claims development data with portfolio information and avoids choosing the prior estimates. This paper presents Bayes chain ladder model which is based on payments per claim incurred. The ideas from credibility theory are used and credibility estimates of chain ladder factors are derived under generalized weighted loss function. Furthermore, we get estimates of reserves. Finally, practical examples of insurance company are given and differences are compared among our credibility estimates obtained, chain ladder estimates and stochastic chain ladder estimates. The conclusions show that credibility claims reserving which is based on the generalized weighted loss function is valid.
引文
[1] Wiithrich M V, Merz M. Stochastic claims reserving methods in insurance[M]. Ann Statist, 1999,27:1519-1535.
[2] Gerber H U. Credibility for esscher premium[J]. Bulletin of the Swiss Association of Actuaries,1980, 3:307-312.
[3] Furman E, Zitikis R. Weighted premium calculation principals[J]. Insurance:Mathematics and Eco-nomics, 2008, 42(1):459-465.
[4]温利民,梅国平.新型广义加权保费原理下风险保费的信度估计[J].应用数学学报,2013, 36(2):257-268.
[5] Jia Xiong, Limin Wen. Nonparametric estimation of Kamp premium principle[J]..Statistical and Application, 2013, 2:70-75.
[6] Vylder D. Estimation of IBNR claims by credibility theory[J]. Insurance:Mathematics and Economics, 1982, 1:35-40.
[7]Gisler A, W(u|¨)thrich M V. Credibility for the chain ladder reserving method[J]. Astin Bulletin, 2008,38(2):565-600.
[8] Biihlmann H, Moriconi F. Credibility claims reserving with stochastic diagonal effects[J]. Astin Bulletin, 2015, 45(2):309-353.
[9]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.
[10]陈晓,张连增.未决赔款准备金估计的Munich链梯法及其优化[J].统计与决策,2010(2):27-30.
[11] Mack T. Distribution-free calculation of the standard error of chain ladder reserve estimates[J].ASTIN Bulletin, 1993, 23(2):213-225.
[12] Gisler A. The estimation error in the chain ladder reserving method:bayesian approach[J]. ASTIN Bulletin, 2006, 36(2):554-565.
[13]张连增,段白鸽.未决赔款准备金评估的Mack模型及其预测均方误差的实现[J].统计与决策.2011,13:20-23.
[14] Alba E D. Bayesian estimation of outstanding claim reserres[J]. North American Actuarial Journal,2002, 6(4):1-20.
[15] Biihlmann H, Gisler A. A Course in Credibility Theory and Its Applications[M]. Springer, 2005, 18:77-117.
[16] Jong P D, Zehnwirth B. Claims reserving, state-space models and the kalman filter[J]. Journal of the Institute of Actuaries, 1983, 110(1):157-181.
[2] Gerber H U. Credibility for esscher premium[J]. Bulletin of the Swiss Association of Actuaries,1980, 3:307-312.
[3] Furman E, Zitikis R. Weighted premium calculation principals[J]. Insurance:Mathematics and Eco-nomics, 2008, 42(1):459-465.
[4]温利民,梅国平.新型广义加权保费原理下风险保费的信度估计[J].应用数学学报,2013, 36(2):257-268.
[5] Jia Xiong, Limin Wen. Nonparametric estimation of Kamp premium principle[J]..Statistical and Application, 2013, 2:70-75.
[6] Vylder D. Estimation of IBNR claims by credibility theory[J]. Insurance:Mathematics and Economics, 1982, 1:35-40.
[7]Gisler A, W(u|¨)thrich M V. Credibility for the chain ladder reserving method[J]. Astin Bulletin, 2008,38(2):565-600.
[8] Biihlmann H, Moriconi F. Credibility claims reserving with stochastic diagonal effects[J]. Astin Bulletin, 2015, 45(2):309-353.
[9]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.
[10]陈晓,张连增.未决赔款准备金估计的Munich链梯法及其优化[J].统计与决策,2010(2):27-30.
[11] Mack T. Distribution-free calculation of the standard error of chain ladder reserve estimates[J].ASTIN Bulletin, 1993, 23(2):213-225.
[12] Gisler A. The estimation error in the chain ladder reserving method:bayesian approach[J]. ASTIN Bulletin, 2006, 36(2):554-565.
[13]张连增,段白鸽.未决赔款准备金评估的Mack模型及其预测均方误差的实现[J].统计与决策.2011,13:20-23.
[14] Alba E D. Bayesian estimation of outstanding claim reserres[J]. North American Actuarial Journal,2002, 6(4):1-20.
[15] Biihlmann H, Gisler A. A Course in Credibility Theory and Its Applications[M]. Springer, 2005, 18:77-117.
[16] Jong P D, Zehnwirth B. Claims reserving, state-space models and the kalman filter[J]. Journal of the Institute of Actuaries, 1983, 110(1):157-181.