考虑奖金饥渴症的奖惩系统定价研究
摘要
在奖惩系统中,作为一个理性的投保人,当事故损失较小时,往往不向保险公司索赔以获得来年保费的折扣奖励,投保人与保险公司的这一类行为博弈即称为奖金饥渴症。奖金饥渴症是机动车辆保险奖惩系统中一种特有的现象,将导致大量低风险投保人往高折扣保费等级聚集,危害保险公司财务状况；截断投保人的真实损失分布,给保险公司的费率厘定带来风险。本文即在奖惩系统框架下,在假设投保人都存在奖金饥渴症的基础之上,给出保险公司根据报告索赔次数和索赔额得到的先验的最优定价方案,并且对其他为避免BMS这一缺陷的方法展开讨论。
本文提出的定价方法是通过多次迭代来完成的,因为在实际中,每一次保险公司给出不同的定价,投保人均会有对应的与前一次不同的最优索赔策略,这是一个重复博弈的过程。本文实现了Lemaire在参考文献[8]中提出的无限时域动态规划的算法,并证明在满足一定条件下,最优策略解的存在唯一性,以此计算理性投保人追逐奖励时采取的最优索赔策略；另一方面,本文基于我国2007年颁布的统一商业车险BMS系统的转移规则,在保险公司费率厘定平均误差最小的意义下,考虑投保人的最优索赔策略,给出最优定价方法,多次迭代,最终给出最优定价保费。
In BMS, when the loss is small, a rational policyholder will not claim so as to get the reward from next year's premium. This behavior between the policyholders and the insurance company is called hunger for bonus. Hunger for bonus is a specific phenomenon in BMS of automobiles insurance. It will cause the aggregation of the low risk policyholders in high discount levels, and damage the financial stability of the company. It will also truncate the true loss distribution so that bring the risk to the pricing assumptions. This article will propose an optimal priori pricing method based on the claim amount and claim count under the assumption that all the policyholders are all rational within a BMS framework. In Chapter 5, some other ways to consider the hunger for bonus will be provided for the discussion.
The pricing method proposed in this paper is accomplished by iterations. In reality, the policyholder will take different strategies according to the different tariffs, so the pricing process is repeated games. This article has realized the infinite time-domain dynamic programming algorithm presented in references [5], proved the existence and uniqueness of strategy under certain conditions, and calculated the optimal claim strategy for a rational policyholder on this account. On the other hand, based on the transition rules issued in 2007 for all the commercial automobile insurances' BMS, this paper has provided an iteration optimal pricing method in the sense of the minimum mean pricing errors.
本文提出的定价方法是通过多次迭代来完成的,因为在实际中,每一次保险公司给出不同的定价,投保人均会有对应的与前一次不同的最优索赔策略,这是一个重复博弈的过程。本文实现了Lemaire在参考文献[8]中提出的无限时域动态规划的算法,并证明在满足一定条件下,最优策略解的存在唯一性,以此计算理性投保人追逐奖励时采取的最优索赔策略；另一方面,本文基于我国2007年颁布的统一商业车险BMS系统的转移规则,在保险公司费率厘定平均误差最小的意义下,考虑投保人的最优索赔策略,给出最优定价方法,多次迭代,最终给出最优定价保费。
In BMS, when the loss is small, a rational policyholder will not claim so as to get the reward from next year's premium. This behavior between the policyholders and the insurance company is called hunger for bonus. Hunger for bonus is a specific phenomenon in BMS of automobiles insurance. It will cause the aggregation of the low risk policyholders in high discount levels, and damage the financial stability of the company. It will also truncate the true loss distribution so that bring the risk to the pricing assumptions. This article will propose an optimal priori pricing method based on the claim amount and claim count under the assumption that all the policyholders are all rational within a BMS framework. In Chapter 5, some other ways to consider the hunger for bonus will be provided for the discussion.
The pricing method proposed in this paper is accomplished by iterations. In reality, the policyholder will take different strategies according to the different tariffs, so the pricing process is repeated games. This article has realized the infinite time-domain dynamic programming algorithm presented in references [5], proved the existence and uniqueness of strategy under certain conditions, and calculated the optimal claim strategy for a rational policyholder on this account. On the other hand, based on the transition rules issued in 2007 for all the commercial automobile insurances' BMS, this paper has provided an iteration optimal pricing method in the sense of the minimum mean pricing errors.
引文
[1]Baione, F., Levantesi, S. and Menzietti, M. (2002) The Development of an Optimal Bonus-Malus System in a Competitive Market[J], ASTIN Bulletin 32(1), 159-169.
[2]Box, G.E.P. and Tiao, GC. (1973) Bayesian Inference in Statistical Analysis[M], Addison-Wesley, Reading, MA.
[3]Buhlmann H. Experience Rating and Credibility[J]. Astin Bulletin,1967, (4):199-207.
[4]Dionne, G. and C. Vanasse. A Generalization of Automobile Insurance Rating Models:the Negative binomial distribution with a regression component[J]. ASTIN Bulletin,1989,19(2):199-212.
[5]Frangos, N. E. and S. D. Vrontos. Designing of optimal Bonus-Malus Systems with a frequency and a severity component on an individual basis in automobile insurance[J]. ASTIN Bulletin,2001,31(1):1-22.
[6]Gerber H. On additive premium calculation principles [J]. ASTIN Bulletin,1974,7(3):215-222.
[7]Lemaire, J. How to Define a Bonus-Malus System with an Exponential Utility Function[J]. Astin Bulletin,1979,10(3):274-282.
[8]Lemaire, J. Bonus-Malus Systems in automobile insurance[M]. Dordrecht: Kluwer Academic Publishers,1995.
[9]Lemaire, J.,H. Zi.(1994). "High Deductibles Instead of Bonus-Malus:Can it work?" ASTIN Bulletin,24,75-86.
[10]Loimaranta,K(1972), "Some asymptotic properties of bonus systems' Astin Bulletin, Vol.6, pp.233-245.
[11]Lemaire, J. (1985) Automobile Insurance. Actuarial Models[M]. Kluwer-Nijhoff Publishing, Dordrecht.
[12]Kemeny, J.G. and SNELL, J.L. (1976) Finite Markov Chains. Springer-Verlag, Berlin.
[13]Holtan, J. (1994) "Bonus Made Easy". ASTIN Bulletin,24,61-74
[14]Lemaire, J. (1998) Bonus-Malus Systems:the European and Asian Approach to Merit-Rating. [J] North American Actuarial Journal 2(1),26-47.
[15]Pinquet,J. Designing optimal Bonus-Malus Systems from different types of claims[J]. ASTIN Bulletin,1998,28(2):205-220.
[16]Norberg, R. (1976). A credibility theory for automobile bonus system. Scandinavian Actuarial Journal,92-107.
[17]S. Pitrebois, J. F. Walhin and M. Denuit. Bonus Malus Systems with Varying Deductibles [J]. ASTIN Bulletin,2005, Vol.35, No.1:261-274.
[18]Subramanian, K. Bonus-Malus Systems in a competitive Environment[J]. North American Actuarial Journal,1998,2(1):38-44.
[19]Vandebroek, M. (1993). "Bonus-Malus System or Partial Coverage to Oppose Moral Hazard Problems? " Insurance:Mathematics and Economics,13,1-5.
[20]Walhin, J.F and Paris,J. The true claim amount and frequency distributions within a Bonus-Mains Systems[J]. ASTIN Bulletin,2000,30(2):391-403.
[21]Walhin, J.F. and Paris, J. (2001). The practical replacement of a bonus-malus system[J], ASTIN Bulletin 31,317-335
[22]孟生旺,袁卫.汽车保险的精算模型及其应用[J].数理统计与管理,2001,第3期.
[23]夏东,谢皓宇,肖宇谷.我国汽车保险奖惩系统变免赔额的实证研究[J].现代商业,2008年第2期,pp287-288
[24]郭丹丹.汽车保险奖惩系统及在我国的应用[D].2006
[25]孟生旺.中国汽车保险奖惩系统的实证研究[J].精算通讯,2004,4(4)：37-43.
[26]王奕渲,秦焱.一个考虑了索赔事故责任比例的最优奖惩系统[J].湖南大学学报(自然科学版),2004,31(6)：110-112.
[27]朱少杰.基于行为博弈的机动车辆保险奖惩系统的研究[D].2007
[28]孟生旺,袁卫.汽车保险中的BMS[J].应用概率统计,1999,15(1)：48-52.
[29]叶芳,崔晔.BMS下四种最优保费尺度的比较[J].统计与决策,2007,第9期：19-22.
[30]王弈渲,周叔子.一种基于索赔次数和索赔额的奖惩系统[J]湖南大学学报(自然科学版),2002,29(6)：31-33.
[31]温利民,韩天雄.多级无赔款优待系统的定价[J].应用概率统计,2005,21(3)：261-266.
[2]Box, G.E.P. and Tiao, GC. (1973) Bayesian Inference in Statistical Analysis[M], Addison-Wesley, Reading, MA.
[3]Buhlmann H. Experience Rating and Credibility[J]. Astin Bulletin,1967, (4):199-207.
[4]Dionne, G. and C. Vanasse. A Generalization of Automobile Insurance Rating Models:the Negative binomial distribution with a regression component[J]. ASTIN Bulletin,1989,19(2):199-212.
[5]Frangos, N. E. and S. D. Vrontos. Designing of optimal Bonus-Malus Systems with a frequency and a severity component on an individual basis in automobile insurance[J]. ASTIN Bulletin,2001,31(1):1-22.
[6]Gerber H. On additive premium calculation principles [J]. ASTIN Bulletin,1974,7(3):215-222.
[7]Lemaire, J. How to Define a Bonus-Malus System with an Exponential Utility Function[J]. Astin Bulletin,1979,10(3):274-282.
[8]Lemaire, J. Bonus-Malus Systems in automobile insurance[M]. Dordrecht: Kluwer Academic Publishers,1995.
[9]Lemaire, J.,H. Zi.(1994). "High Deductibles Instead of Bonus-Malus:Can it work?" ASTIN Bulletin,24,75-86.
[10]Loimaranta,K(1972), "Some asymptotic properties of bonus systems' Astin Bulletin, Vol.6, pp.233-245.
[11]Lemaire, J. (1985) Automobile Insurance. Actuarial Models[M]. Kluwer-Nijhoff Publishing, Dordrecht.
[12]Kemeny, J.G. and SNELL, J.L. (1976) Finite Markov Chains. Springer-Verlag, Berlin.
[13]Holtan, J. (1994) "Bonus Made Easy". ASTIN Bulletin,24,61-74
[14]Lemaire, J. (1998) Bonus-Malus Systems:the European and Asian Approach to Merit-Rating. [J] North American Actuarial Journal 2(1),26-47.
[15]Pinquet,J. Designing optimal Bonus-Malus Systems from different types of claims[J]. ASTIN Bulletin,1998,28(2):205-220.
[16]Norberg, R. (1976). A credibility theory for automobile bonus system. Scandinavian Actuarial Journal,92-107.
[17]S. Pitrebois, J. F. Walhin and M. Denuit. Bonus Malus Systems with Varying Deductibles [J]. ASTIN Bulletin,2005, Vol.35, No.1:261-274.
[18]Subramanian, K. Bonus-Malus Systems in a competitive Environment[J]. North American Actuarial Journal,1998,2(1):38-44.
[19]Vandebroek, M. (1993). "Bonus-Malus System or Partial Coverage to Oppose Moral Hazard Problems? " Insurance:Mathematics and Economics,13,1-5.
[20]Walhin, J.F and Paris,J. The true claim amount and frequency distributions within a Bonus-Mains Systems[J]. ASTIN Bulletin,2000,30(2):391-403.
[21]Walhin, J.F. and Paris, J. (2001). The practical replacement of a bonus-malus system[J], ASTIN Bulletin 31,317-335
[22]孟生旺,袁卫.汽车保险的精算模型及其应用[J].数理统计与管理,2001,第3期.
[23]夏东,谢皓宇,肖宇谷.我国汽车保险奖惩系统变免赔额的实证研究[J].现代商业,2008年第2期,pp287-288
[24]郭丹丹.汽车保险奖惩系统及在我国的应用[D].2006
[25]孟生旺.中国汽车保险奖惩系统的实证研究[J].精算通讯,2004,4(4)：37-43.
[26]王奕渲,秦焱.一个考虑了索赔事故责任比例的最优奖惩系统[J].湖南大学学报(自然科学版),2004,31(6)：110-112.
[27]朱少杰.基于行为博弈的机动车辆保险奖惩系统的研究[D].2007
[28]孟生旺,袁卫.汽车保险中的BMS[J].应用概率统计,1999,15(1)：48-52.
[29]叶芳,崔晔.BMS下四种最优保费尺度的比较[J].统计与决策,2007,第9期：19-22.
[30]王弈渲,周叔子.一种基于索赔次数和索赔额的奖惩系统[J]湖南大学学报(自然科学版),2002,29(6)：31-33.
[31]温利民,韩天雄.多级无赔款优待系统的定价[J].应用概率统计,2005,21(3)：261-266.