A study on life insurance premiums under asymmetric dependence using Canadian insurance data

Abstract

This study evaluates the impact of symmetric and asymmetric dependence on premium calculations for various annuity and life insurance products across different age groups. Initially, we determined the marginal survival probabilities for individual lifetimes at specific ages using the Gompertz mortality model. Subsequently, joint survival probabilities were derived, considering independent and dependent future lifetimes for individuals within a group. The dependency structure was examined using Archimedean copulas for symmetric models and Khoudraji copulas for asymmetric models, which are widely referenced in the literature. In addition, actuarial calculations were conducted using real data on dependent lifetimes sourced from a Canadian insurance company. The data set is divided into three different populations based on age differences between married couples: the entire population without considering age differences, the population where males are older, and the population where females are older. The symmetric and asymmetric dependence structures of these populations were determined using an asymmetry test. The best-fitting models were identified using maximum likelihood estimation and goodness-of-fit tests. Finally, actuarial calculations were performed on the data set. Our findings showed that there were no significant differences between symmetric and asymmetric premium calculations for the whole population. However, when the population is disaggregated by age, the asymmetry becomes evident in the data structures, which increases the differences in the premium calculations. For example, the Kho-Fr model selected for the population of older female exhibiting asymmetric dependency was generally found to produce higher premiums than the Gumbel model. These findings reveal the importance of determining the dependency structure and working with age-based sub-populations rather than treating the whole population as a homogenous structure in model selection.

Keywords

• Dependent lifetimes
• copula
• annuity
• insurance
• premium
• dependence
• asymmetry
• multiple life products
• actuarial calculations

References

1. [1] S. Arvidsson, Reducing asymmetric information with usage-based automobile insurance, Swedish National Road & Transport Research Institute (VTI), 2, 2011.
2. [2] S. Benson, R. Burroughs, V. Ladyzhets, J. Mohr, A. Shemyakin, D. Walczak and H. Zhang, Copula models of economic capital for life insurance companies, North American Actuarial Journal, 58, 32-54, 2020.
3. [3] M. I. Bhatti and H. Q. Do, Recent development in copula and its applications to the energy, forestry and environmental sciences, International Journal of Hydrogen Energy, 44, 19453-19473, 2019.
4. [4] N. L. Bowers, H. U. Gerber, J. C. Hickman, D. A. Jones and C. J. Nesbitt, Actuarial Mathematics, Society of Actuaries, 1997.
5. [5] J. F. Carriere, An investigation of the Gompertz law of mortality, Actuarial Research Clearing House, 2, 161-177, 1994.
6. [6] J. F. Carriere, Bivariate survival models for coupled lives, Scandinavian Actuarial Journal, 1,17-32, 2000.
7. [7] A. C. Callau Poduje, A. Belli, and U. Haberlandt, Dam risk assessment based on univariate versus bivariate statistical approaches: A case study for Argentina, Hydrological Sciences Journal, 59, 2216-2232, 2014.
8. [8] V. Y. Chang, K. M. Hung, K. C. Wang and S. Yang, Information asymmetry in reinsurance through various ceded contracts,Pacific-Basin Finance Journal, 84, 102302, 2024.

9. [9] A. Chen, M. Hinken and Y. Shen, Life reinsurance under perfect and asymmetric information,Scandinavian Actuarial Journal, 1-23, 2023.
10. [10] U. Cherubini, E. Luciano and W. Vecchiato, Copula methods in finance, John Wiley & Sons, 2004.
11. [11] A. Cohen, Asymmetric information and learning: Evidence from the automobile insurance market, Review of Economics and statistics, 87, 197-207, 2005.
12. [12] C. Czado, Analyzing dependent data with vine copulas, Lecture Notes in Statistics, Springer, 222, 2019.
13. [13] M. Denuit and A. Cornet, Multilife premium calculation with dependent future lifetimes, Journal of Actuarial Practice, 7, 147-171, 1999.
14. [14] M. Denuit, J. Dhaene, C. Le Bailly de Tilleghem and S. Teghem, Measuring the impact of dependence among insured lifelengths, Belgian Actuarial Bulletin, 1, 18-39, 2001.
15. [15] F. Dufresne, E. Hashorva, G. Ratovomirija and Y. Toukourou, On age difference in joint lifetime modelling with life insurance annuity applications, Annals of Actuarial Science, 12, 350-371, 2018.
16. [16] F. Durante, Construction of non-exchangeable bivariate distribution functions, Statist. Papers, 50, 383-391, 2009.
17. [17] G. Emamverdi, M. S. Karimi, M. Firouzi and F. Emdadi, An investigation about joint life policy’s premium using Copula; The Case study of an insurance company in Iran, Asian Journal of Research in Business Economics and Management, 4, 222-242, 2014.
18. [18] P. Erawati and M. Subhan, Penentuan Premi Asuransi Jiwa Berjangka Status Last Survivor Menggunakan Model GFGM-Type II Copula, Journal of Mathematics UNP, 7, 69-75, 2022.
19. [19] E. W. Frees, J. F. Carriere and E. Valdez, Annuity valuation with dependent mortality, Journal of Risk and Insurance, 63, 229-261, 1996.
20. [20] F. Gao, M. R. Powers and J. Wang, Decomposing asymmetric information in China’s automobile insurance market, Journal of Risk and Insurance, 84, 1269-1293, 2017.
21. [21] C. Genest and A. C. Favre, Everything you always wanted to know about copula modelling but were afraid to ask, Journal of Hydrologic Engineering, 12, 347-368, 2007.
22. [22] C. Genest, B. Rémillard and D.Beaudoin, Goodness-of-fit tests for copulas: a review and a power study, Insurance: Mathematics and Economics, 44, 199-213, 2009.
23. [23] C. Genest, J. Nelehová and J.F. Quessy, Tests of symmetry for bivariate copulas, Ann. Inst. Statist. Math., 64, 811-834, 2012.
24. [24] H. U. Gerber, Life Insurance Mathematics, Springer, Science & Business Media, 1997.
25. [25] I. Ghosh, D.Watts and S. Chakraborty, Modelling Bivariate Dependency in Insurance Data via Copula: A Brief Study, Journal of Risk and Financial Management, 15(8), 329, 2022.
26. [26] M. Hofert, I. Kojadinovic, M. Maechler and J. Yan, Package: “copula: Multivariate Dependence with Copula”, R package version: 1.0-0, 2020.
27. [27] P. Hougaard, Analysis of Multivariate Survival Data, Springer Science & Business Media, New York, 2000.
28. [28] M. H. Hsieh, C.J. Tsai and J. L. Wang, Mortality risk management under the factor copula framework With applications to insurance policy pools, North American Actuarial Journal, 25, 119-131, 2021.
29. [29] P. Jaworski, F. Durante, W. K. Hardle and T. Rychlik, Copula theory and its applications, Berlin: Springer, 198, 2010.
30. [30] H. Jeong and D. Dey, Application of a vine copula for multi-line insurance reserving, Risks, 8(4), 111, 2020.
31. [31] M. Ji, M. Hardy and J. S. H. Li, Markovian approaches to joint-life mortality, North American Actuarial Journal, 15, 357-376, 2011.
32. [32] E. K. Kara and O. Yıldız, Bivariate analysis of precipitation and runoff in the Hirfanli Dam Basin, Turkey, using copulas, Istatistik Journal of The Turkish Statistical Association, 7, 63-70,2014.
33. [33] S. A. Kemaloglu and E. K. Kara, Modelling dependent financial assets by dynamic copula and portfolio optimization based on CVaR, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 64, 1-13, 2015.
34. [34] E. K. Kara and S. A. Kemaloglu, Portfolio optimization of dynamic copula models for dependent financial data using change point approach, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 65, 175-188, 2016.
35. [35] E. K. Kara, The Earthquake Risk Analysis Based on Copula Models for Turkey, Sigma Journal of Engineering and Natural Sciences, 35(2), 187-200, 2017.
36. [36] E. K. Kara, On Actuarial Premiums for Joint Last Survivor Life Insurance based On Asymmetric Dependent Lifetimes, Current Academic Studies in Science and Mathematics Sciences-II, 33-47,2021.
37. [37] E. K. Kara, A study on modeling of lifetime with right-truncated composite lognormalpareto distribution: Actuarial premium calculations, Gazi University Journal of Science, 34(1), 272-288, 2021.
38. [38] E. K. Kara, On the Impact of Asymmetric Dependence in the Actuarial Pricing of Joint Life Insurance Policies, Sains Malaysiana, 51, 3807-3817, 2022.
39. [39] E. K. Kara, S.A. Kemaloglu and O. Evkaya, Analysis of asymmetric financial data with directional dependence measures, Hacettepe Journal of Mathematics and Statistics, 1-24, 2023.
40. [40] E. K. Kara and S. A. Kemaloglu, Modelling asymmetrically dependent automobile bodily injury claim data using Khoudraji copulas, Sigma Journal of Engineering and Natural Sciences, 42, 1-11,Doi: 10.14744/sigma.2023.00133, 2024.
41. [41] A. Khoudraji, Contributions a l’etude des copules et a la modelisation de valeurs extremes bivariees, PhD thesis, Universit ede Laval, Qu ebec (Canada),1995.
42. [42] J. M. Kim, Y. S. Jung, E. A. Sungur, K. H. Han, C. Park and I. A. Sohn, Copula method for modelling directional dependence of genes, BMC Bioinformatics, 9, 1-12, 2008.
43. [43] H. Kim, D. Kim, S. Im and J. W. Hardin, Evidence of asymmetric information in the automobile insurance market: Dichotomous versus multinomial measurement of insurance coverage,Journal of Risk and Insurance, 76, 343-366, 2009.
44. [44] S. Kumar, N. C. Sahu and P. Kumar, Insurance consumption and economic policy uncertainty in India: an analysis of asymmetric effects, The Singapore Economic Review, 69, 637-665, 2024.
45. [45] B. Y. Külekci, R. Korn and A. S. Selcuk-Kestel, Ruin probability for heavy-tailed and dependent losses under reinsurance strategies, Mathematics and Computers in Simulation, 226, 118-138, 2024.
46. [46] E. Liebscher, Construction of asymmetric multivariate copulas, Journal of Multivariate Analysis, 99(10), 2234-2250, 2008.
47. [47] Y. Lu, Broken-heart, common life, heterogeneity: analyzing the spousal mortality dependence, ASTIN Bulletin: The Journal of the IAA, 47, 1-38, 2017.
48. [48] E. Luciano, J. Spreeuw and E. Vigna, Modelling stochastic mortality for dependent lives, Insurance: Mathematics and Economics, 43, 234-244, 2008.
49. [49] E. Luciano, J. Spreeuw and E. Vigna, Spouses dependence across generations and pricing impact on reversionary annuities, Risks, 4, 16, 2016.
50. [50] P. Maeder, La construction des tables de mortalite du tarif collectif 1995 de l’UPAV, Insurance Mathematics and Economics, 3, 226, 1996.
51. [51] S. Maliar and B. Salanie, Testing for Asymmetric Information in Insurance with Deep Learning, arXiv preprint arXiv:2404.18207, 2024.
52. [52] W. O. Menge and J. W. Glover, An introduction to the mathematics of life insurance, Macmillan, 1938.
53. [53] A. U. Montero and J. Wagner, On potential information asymmetries in long-term care insurance: A simulation study using data from Switzerland, Insurance: Mathematics and Economics, 111, 230-241, 2023.
54. [54] R. B. Nelsen, An Introduction to Copulas, Springer Science & Business Media, 2006.
55. [55] R. Oh, J. Y. Ahn and W. Lee, On copula-based collective risk models: from elliptical copulas to vine copulas, Scandinavian Actuarial Journal, 2021(1), 1-33, 2021.
56. [56] C. M. Parkes, B. Benjamin and R. G. Fitzgerald, Broken heart: a statistical study of increased mortality among widowers, Journal of Occupational and Environmental Medicine, 12, 143, 1970.
57. [57] A. J. Patton, Modelling asymmetric exchange rate dependence,International economic review, 47, 527-556,2006.
58. [58] A. Patton, Copula methods for forecasting multivariate time series, Handbook of Economic Forecasting, 2012.
59. [59] S. C. Peng and C. S. Li, Bundled insurance coverage and asymmetric information: Claim patterns of automobile theft insurance in Taiwan, Pacific-Basin Finance Journal, 84, 102279, 2024.
60. [60] M. K. Refaie, N. S. Butt and E. I. Ali, A new probability distribution: properties, copulas and applications in medicine and engineering, Pakistan Journal of Statistics and Operation Research, 257-278, 2023.
61. [61] K. Saito, Testing for asymmetric information in the automobile insurance market under rate regulation, Journal of Risk and Insurance, 73, 335-356, 2006.
62. [62] P. Shi and E. W. Frees, Dependent loss reserving using copulas, ASTIN Bulletin: The Journal of the IAA, 41(2), 449-486, 2011.
63. [63] P. Shi and E. A. Valdez, A copula approach to test asymmetric information with applications to predictive modelling, Insurance: Mathematics and Economics, 49, 226-239, 2011.
64. [64] M. Sklar, Fonctions de répartition à n dimensions et leurs marges, In Annales de l’ISUP, 8, 229-231, 1959.
65. [65] M. Spindler, J. Winter and S. Hagmayer, Asymmetric information in the market for automobile insurance: Evidence from Germany, Journal of Risk and Insurance, 81, 781-801, 2014.
66. [66] J. Spreeuw and I. Owadally, Investigating the broken-heart effect: a model for shortterm dependence between the remaining lifetimes of joint lives, Annals of Actuarial Science, 7(2), 236-257, 2013.
67. [67] L. Yang and C. Czado, Twopart Dvine copula models for longitudinal insurance claim data, Scandinavian Journal of Statistics, 49(4), 1534-1561, 2022.
68. [68] H. Youn and A. Shemyakin, Statistical aspects of joint life insurance pricing, 1999 Proceedings of the Business and Statistics Section of the American Statistical Association, Baltimore, Maryland, 8-12 August,1999.