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《山东大学学报(理学版)》 ›› 2020, Vol. 55 ›› Issue (9): 81-88.doi: 10.6040/j.issn.1671-9352.0.2019.700

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带跳的Vasicek利率模型下的寿险净保费责任准备金

宋春燕2,李世龙3,*()   

  1. 1. 山东财经大学会计学院,山东 济南 250014
    2. 山东财经大学数学与数量经济学院,山东 济南 250014
    3. 山东财经大学保险学院,山东 济南 250014
  • 收稿日期:2019-10-14 出版日期:2020-09-20 发布日期:2020-09-17
  • 通讯作者: 李世龙 E-mail:lishl@sdufe.edu.cn
  • 作者简介:张彦国(1974—),男,硕士,讲师,研究方向为保险会计. E-mail:zhangsan@sina.com.cn
  • 基金资助:
    国家自然科学基金资助项目(71671104);国家社会科学基金重点资助项目(16AZD019);山东省高校科技计划资助项目(J15LI01)

Net premium reserve in life insurance under Vasicek model with jumps

Chun-yan SONG2,Shi-long LI3,*()   

  1. 1. School of Account, Shandong University of Finance and Economics, Jinan 250014, Shandong, China
    2. School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, Shandong, China
    3. School of Insurance, Shandong University of Finance and Economics, Jinan 250014, Shandong, China
  • Received:2019-10-14 Online:2020-09-20 Published:2020-09-17
  • Contact: Shi-long LI E-mail:lishl@sdufe.edu.cn

摘要:

基于市场利率的随机跳跃波动特征,利用复合Poisson过程和Ornstein-Uhlenbeck过程分别刻画利率的随机跳跃性和随机连续变化性,并将二者进行耦合构建具有随机跳跃性的利息力函数,得到一类带Poisson跳的Vasicek利率模型。研究在该利率模型下的累积利息力函数和货币期望折扣函数的数学表达形式,给出相应的数值分析,并基于此进一步研究了寿险产品净保费准备金的测算问题。

关键词: 随机利率模型, 复合Poisson过程, Ornstein-Uhlenbeck过程, 寿险, 净保费责任准备金

Abstract:

Considering the characteristics of random volatility with jumps of market interest rates, both compound Poisson process and Ornstein-Uhlenbeck process are utilized to describe the stochastic jumps and random continuous changes of interest rates respectively. A Vasicek interest model with Poisson jumps is obtained by coupling the two kinds of stochastic processes. The mathematical expressions of the cumulative interest force function and the expected discount function of money under the model are studied. At the same time, the corresponding numerical analysis is given. Based on this interest model, the calculation of the net premium reserves of life insurance products is further studied

Key words: stochastic interest model, compound Poisson process, Ornstein-Uhlenbeck process, life insurance, net premium reserve

中图分类号: 

  • F224.7

表1

F(z)为单点分布时的期望折扣函数值"

α期望折扣函数值
p=0.5 p=0.5 p=0.5 p=0.6 p=0.6 p=0.6
σ=0.005 σ=0.005 σ=0.01 σ=0.005 σ=0.005 σ=0.01
β=0.10 β=0.15 β=0.15 β=0.10 β=0.15 β=0.20
0.002 8 0.673 488 0.673 124 0.676 283 0.636 807 0.636 463 0.638 502
0.002 6 0.673 245 0.672 882 0.676 039 0.639 129 0.638 784 0.640 831
0.002 4 0.673 021 0.672 657 0.675 814 0.641 478 0.641 131 0.643 185
0.002 2 0.672 815 0.672 451 0.675 607 0.643 851 0.643 504 0.645 565
0.002 0 0.672 626 0.672 263 0.675 416 0.646 251 0.645 902 0.647 972

表2

F(z)服从两点分布时的期望折扣函数值"

α1 α2 q期望折扣函数值
p=0.5 p=0.5 p=0.5 p=0.6 p=0.6 p=0.6
σ=0.005 σ=0.005 σ=0.01 σ=0.005 σ=0.005 σ=0.01
β=0.10 β=0.15 β=0.20 β=0.10 β=0.15 β=0.20
0.001 0.003 0.4 0.673 030 0.672 666 0.674 821 0.644 057 0.643 709 0.645 771
0.001 0.003 0.5 0.672 850 0.672 487 0.674 642 0.646 466 0.646 117 0.648 187
0.001 0.003 0.6 0.672 671 0.672 308 0.674 462 0.648 884 0.648 534 0.650 612
0.001 0.004 0.4 0.673 973 0.673 609 0.675 767 0.637 264 0.636 919 0.638 960
0.001 0.004 0.5 0.673 636 0.673 272 0.675 429 0.640 779 0.640 433 0.642 485
0.001 0.004 0.6 0.673 299 0.672 935 0.675 092 0.644 314 0.643 965 0.646 029
0.002 0.003 0.4 0.673 299 0.672 935 0.675 091 0.639 180 0.638 835 0.640 882
0.002 0.003 0.5 0.673 187 0.672 823 0.674 979 0.640 353 0.640 007 0.642 058
0.002 0.003 0.6 0.673 075 0.672 711 0.674 866 0.641 529 0.641 182 0.643 236
0.002 0.004 0.4 0.674 243 0.673 878 0.676 037 0.632 439 0.632 097 0.634 122
0.002 0.004 0.5 0.673 973 0.673 609 0.675 767 0.634 720 0.634 377 0.636 410
0.002 0.004 0.6 0.673 703 0.673 339 0.675 497 0.637 001 0.636 665 0.638 705

表3

F(z)服从[0, θ]上均匀分布时的期望折扣函数值"

θ期望折扣函数值
p=0.5 p=0.5 p=0.5 p=0.6 p=0.6 p=0.6
σ=0.005 σ=0.005 σ=0.01 σ=0.005 σ=0.005 σ=0.01
β=0.10 β=0.15 β=0.20 β=0.10 β=0.15 β=0.15
0.004 0 0.672 925 0.672 562 0.674 716 0.646 538 0.646 188 0.639 449
0.003 5 0.672 645 0.672 281 0.674 435 0.649 508 0.649 157 0.641 782
0.003 0 0.672 402 0.672 039 0.674 192 0.652 529 0.652 176 0.644 140
0.002 5 0.672 197 0.671 833 0.673 986 0.655 600 0.655 245 0.646 523
0.002 0 0.672 029 0.671 665 0.673 817 0.658 721 0.658 365 0.648 933
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