JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2020, Vol. 55 ›› Issue (9): 81-88.doi: 10.6040/j.issn.1671-9352.0.2019.700

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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

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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

CLC Number: 

  • F224.7

Table 1

Expected discounted function values under one-point distribution"

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

Table 2

Expected discounted function values under two-point distribution"

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

Table 3

Expected discounted function values under uniform distribution on [0, θ]"

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
1 段白鸽, 栾杰. 长寿风险对寿险和年金产品准备金评估的对冲效应研究:基于中国人身保险业经验生命表两次修订视角[J]. 保险研究, 2017, (4): 15- 34.
DUAN Baige , LUAN Jie . Longevity risk's hedging effect on the reserve valuation of life insurance and annuity products: based on the two revisions of Chinese life insurance mortality table[J]. Insurance Studies, 2017, (4): 15- 34.
2 HANBALI H , DENUIT M , DHAENE J , et al. A dynamic equivalence principle for systematic longevity risk management[J]. Insurance: Mathematics and Economics, 2019, 86 (1): 158- 167.
3 DAHL M . Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts[J]. Insurance: Mathematics and Economics, 2004, 35 (1): 113- 136.
4 DAHL M , MØLLER T . Valuation and hedging of life insurance liabilities with systematic mortality risk[J]. Insurance: Mathematics and Economic, 2006, 39 (2): 193- 217.
5 NEVES C R , MIGON H S . Bayesian graduation of mortality rates: an application to reserve[J]. Insurance: Mathematics and Economics, 2007, 40 (3): 424- 434.
6 东明, 郭亚军, 杨怀东. 半连续型寿险保单组的准备金精算模型[J]. 数量经济技术经济研究, 2004, (8): 143- 148.
DONG Ming , GUO Yajun , YANG Huaidong . Reserve actuarial model of semi-continuous life insurance portfolio[J]. The Journal of Quantitative & Technical Economics, 2004, (8): 143- 148.
7 贾念念, 贾长青. 随机利率下的寿险责任准备金与风险分析[J]. 哈尔滨工程大学学报, 2009, 30 (8): 962- 966.
JIA Niannian , JIA Changqing . Life insurance reserve and risk analysis with stochastic interest rates[J]. Journal of Harbin Engineering University, 2009, 30 (8): 962- 966.
8 王彬. 一类随机利率模型下的年金精算现值[J]. 应用数学与计算数学学报, 2007, 21 (1): 77- 88.
WANG Bin . Actuarial present values of some annuities under random rates of interest[J]. Communication on Applied Mathematics and Computation, 2007, 21 (1): 77- 88.
9 CAI J , DICKSON D C M . Ruin probabilities with a Markov chain interest model[J]. Insurance: Mathematics and Economics, 2004, 35 (3): 513- 525.
10 NOLDE N , PARKER G . Stochastic analysis of life insurance surplus[J]. Insurance: Mathematics and Economics, 2014, 56 (1): 1- 13.
11 CHEN Li , LIN Luyao , LU Yi , et al. Analysis of survivorship life insurance portfolios with stochastic rates of return[J]. Insurance: Mathematics and Economics, 2017, 75 (1): 16- 31.
12 BEEKMAN J A , FUELING C P . Interest and mortality randomness in some annuities[J]. Insurance: Mathematics and Economics, 1990, 9 (2): 185- 196.
13 BEEKMAN J A , FUELING C P . Extra randomness in certain annuity models[J]. Insurance: Mathematics and Economics, 1991, 10 (4): 275- 287.
14 HOEDEMAKRS T , DARKIEWICZ G J , GOOVAERTS M . Approximations for life annuity contracts in a stochastic financial environment[J]. Insurance: Mathematics and Economics, 2005, 37 (2): 239- 269.
15 DUFRESNE D . Stochastic life annuities[J]. North American Actuarial Journal, 2007, 11 (1): 136- 157.
doi: 10.1080/10920277.2007.10597441
16 ZHAO Xia , ZHANG Bo . Pricing perpetual options with stochastic discount interest[J]. Quality and Quantity, 2012, 46 (1): 341- 349.
17 刘凌云, 汪荣明. 一类随机利率下的增额寿险模型[J]. 应用概率统计, 2001, 17 (3): 283- 290.
LIU Lingyun , WANG Rongming . A class of models about increasing payments life insurance under the stochastic interest[J]. Chinese Journal of Applied Probability and Statistics, 2001, 17 (3): 283- 290.
18 PARKER G . Stochastic analysis of portfolio of endowment insurance policies[J]. Scandinavian Actuarial Journal, 1994, 77 (2): 119- 130.
19 PARKER G . Stochastic analysis of the interaction between investment and insurance risks[J]. North American Actuarial Journal, 1997, 1 (2): 55- 84.
doi: 10.1080/10920277.1997.10595604
20 COSSETTE H , DENUIT M , DHAENE J , et al. Stochastic approximations of present value functions[J]. Bulletin of the Swiss Association of Actuaries, 2001, 1 (1): 15- 28.
21 VASICEK O . An equilibrium characterization of the term structure[J]. Journal of Financial Economics, 1977, 5 (2): 177- 188.
22 COX J C , INGERSOLL J E J , ROSS S A . A theory of the term structure of interest rate[J]. Econometrica, 1985, 53 (2): 385- 408.
23 HULL J , WHITE A . Pricing interest rate derivative securities[J]. Review of Financial Studies, 1990, 3 (4): 573- 592.
doi: 10.1093/rfs/3.4.573
24 HEATH D , JARROW R , MORTON A . Bond pricing and the term structure of interest rates: a new methodology[J]. Econometrica, 1992, 60 (1): 77- 105.
25 SINGOR S N , GRZELAK L A , CORNELIS W O . Pricing inflation products with stochastic volatility and stochastic interest rates[J]. Insurance: Mathematics and Economics, 2013, 52 (2): 286- 299.
26 GUAN Guohui , LIANG Zongxia . Optimal management of DC pension plan in a stochastic interest rate and stochastic volatility framework[J]. Insurance: Mathematics and Economics, 2014, 57 (1): 58- 66.
27 郑苏晋, 韩雪, 张乐然. Vasicek模型下寿险公司利率风险最低资本研究[J]. 保险研究, 2017, (3): 26- 38.
ZHENG Sujin , HAN Xue , ZHANG Leran . The research on minimum capital of interest rate risk of life insurance company based on Vasicek model[J]. Insurance Studies, 2017, (3): 26- 38.
28 DENG Guohe . Pricing American put option on zero-coupon bond in a jump-extended CIR model[J]. Communications in Nonlinear Science and Numerical Simulation, 2015, 22 (1/2/3): 186- 196.
29 LI L , MENDOZA A R , MITCHELL D . Analytical representations for the basic affine jump diffusion[J]. Operations Research Letters, 2016, 44 (1): 121- 128.
30 LI Shilong, YIN Chuancun, ZHAO Xia, et al. Stochastic interest model based on compound Poisson process and applications in actuarial science[J/OL]. Mathematical Problems in Engineering, 2017[2019-10-01]. https: //doi.org/10.1155/2017/3472319.
31 LI Shi , ZHAO Xia , YIN Chuancun , et al. Stochastic interest model driven by compound Poisson process and Brownian motion with applications in life contingencies[J]. Quantitative Finance and Economics, 2018, 2 (1): 731- 745.
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