The assessment of insurance risk premium with a simulation study using Archimedean copulas

Abstract

This study aims to analyze the effects of risk-averse and risk-seeking attitudes on premium adjustments in dependent risk groups. Using the risk aversion method, it examines how dependence structures modeled by different Archimedean copula families affect the premium strategies of insurance companies. The study combines theoretical derivations with simulation-supported analyses and presents concrete examples to illustrate the impact of risk attitudes on premiums. In the first stage, bivariate risk aversion coefficients and risk premiums were theoretically determined for Archimedean copula families. Subsequently, a simulation study based on different Spearman correlation coefficients was conducted to numerically evaluate the effects of dependence on premium levels and risk behaviors. The findings show that dependence plays a significant role in shaping insurers’ risk behavior. Specifically, it was concluded that risk-averse companies should set higher premiums, whereas risk-seeking companies may prefer lower premium levels.

Keywords

• Dependence
• Archimedean copulas
• risk aversion attitude
• risk premium
• insurance

References

1. Abbas, A.E., Multiattribute utility copulas, Oper. Res., 57(6) (2009), 1367-1383.
2. Abbas, A.E., Sun, Z., Archimedean utility copulas with polynomial generating functions, Decis. Anal., 16(3) (2019), 218-237.
3. Al-Shomrani, A. A., New bivariate family of distributions based on any copula function: Statistical properties, Heliyon, 9(4) (2023).
4. Ansari, J., Rockel, M., Dependence properties of bivariate copula families, Depend. Model., 12(1) (2024), 20240002.
5. Barakat, H.M., Alawady, M.A., Husseiny, I.A., Nagy, M., Mansi, A.H., Mohamed, M.O., Bivariate Epanechnikov- exponential distribution: statistical properties, reliability measures, and applications to computer science data, AIMS Mathematics, 9(11) (2024), 32299-32327.
6. Benson, S., Burroughs, R., Ladyzhets, V., Mohr, J., Shemyakin, A., Walczak, D., Zhang, H., Copula models of economic capital for life insurance companies, N. Am. Actuar. J., 58 (2020), 32-54.
7. Cardin, M., Ferretti, P., Bivariate Risk Aversion and Concordance Aversion: Similarities and Differences, Working paper, Department of Applied Mathematics, Univ. Ca’Foscari Venice, (2004), 27-35.
8. Cherubini, U., Luciano, E., Vecchiato, W., Copula Methods in Finance, England: John Wiley & Sons, 2004.

9. Courbage, C., Rey, B., Precautionary saving in the presence of other risks, Econ. Theory, 32 (2007), 417-424.
10. Denuit, M., Cornet, A., Multilife premium calculation with dependent future lifetimes, J. Actuar. Pract., 7 (1999), 147-171.
11. Denuit, M., Dhaene, J., Le Bailly de Tilleghem, C. Teghem, S., Measuring the impact of dependence among insured lifelengths, Belgian Actuarial Bulletin, 1(1) (2001), 18-39.
12. Dewick, P.R., Liu, S., Copula modelling to analyse financial data, J. Risk Financ. Manag., 15(3) (2022), 104.
13. Duncan, G.T., A matrix measure of multivariate local risk aversion, Econometrica: J. Econ. Soc., 45(4) (1977), 895-903.
14. Durukan, K., Orkcu, H., Kara, E.K., Risk premium for dependent risks using utility copulas and risk aversion. Istatistik: Journal of The Turkish Statistical Association, 12(1) (2019), 1-12.
15. Durukan K., Kızılok Kara E., Örkcü H.H., The Importance of Risk Aversion Measure for Insurance Companies under Dependent Risks: Risk Premium Calculations with a Simulation Study In:Management and Finance Studies, Kalay, F., Unvan Yüksel, A. (Eds.), ISBN:978-2-38236-156-6, Livre de Lyon, France (2021), 199-215.
16. Emamverdi, G., Karimi, M.S., Firouzi, M., Emdadi, F., An investigation about joint life policy’s premium using Copula; The Case study of an insurance company in Iran, Asia. Jour. Rese. Busi. Econ. and Manag., 4(10) (2014), 222-242.
17. Genest, C., MacKay, J., The joy of copulas: Bivariate distributions with uniform marginals, The American Statistician, 40(4) (1986), 280-283.
18. Genest, C., Rivest, L.P., Statistical inference procedures for bivariate Archimedean copulas, J. Am. Stat. Assoc., 88(423) (1993), 1034-1043.
19. Ghosh, I., Marques, F., Chakraborty, S., Bivariate binomial conditionals distributions with positive and negative correlations: A statistical study, arXiv preprint arXiv:2301.03087 (2023).
20. Goodwin, B.K., Hungerford, A., Copula-based models of systemic risk in US agriculture: Implications for crop insurance and reinsurance contracts, Am. J. Agr. Econ., 97(3) (2015), 879-896.
21. Goovaerts, M., Linders, D., Van Weert, K., Tank, F., On the interplay between distortion, mean value and Haezendonck-Goovaerts risk measures, Insur. Math. Econ., 51(1) (2012), 1018.
22. Härdle W.K., Okhrin, O., Okhrin, Y., Modeling dependencies in finance using copulae, 2008.
23. Hofert, M., Kojadinovic, I., Maechler, M. and Yan, J., Package: ’copula: Multivariate Dependence with Copula’, R package version: 1.1-3, (2023).
24. Hsieh, M.H., Tsai, C. J., Wang, J. L., Mortality risk management under the factor copula framework-With applications to insurance policy pools, N. Am. Actuar. J., 25(sup1) (2021), S119-S131.
25. Joe, H., Multivariate models and multivariate dependence concepts, CRC press, 1997.
26. Kara, E.K., Kemaloglu, S.A., Portfolio optimization of dynamic copula models for dependent financial data using change point approach, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 65(2) (2016), 175-188.
27. Kara, E.K., The earthquake risk analysis based on copula models for Turkey, Sigma J. Eng. Nat. Sci., 35(2) (2017), 187-200.
28. Kara, E.K., On Actuarial Premiums for Joint Last Survivor Life Insurance based On Asymmetric Dependent Lifetimes, Current Academic Studies in Science Mathematics Science-II, (2021), 33-47.
29. Kara, E.K., On the Impact of Asymmetric Dependence in the Actuarial Pricing of Joint Life Insurance Policies, Sains Malays., 51(11) (2022), 3807-3817.
30. Kara, E.K., Kemaloğlu, S.A., & Evkaya, O., Analysis of asymmetric financial data with directional dependence measures, Hacet. J. Math. Stat., (2023), 1-24.
31. Kara, E.K., Kemaloğlu, S.A., Modeling asymmetrically dependent automobile bodily injury claim data using Khoudraji copulas, Sigma J. Eng. Nat. Sci., 42(4) (2024), 1-11.
32. Kayalar, D.E., Küçüközmen, C.C., Selcuk-Kestel, A.S., The impact of crude oil prices on financial market indicators: copula approach, Energy Econ., 61 (2017), 162-173.
33. Kemaloglu, S.A., Kara, E.K., Modeling dependent financial assets by dynamic copula and portfolio optimization based on CVaR, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 64(1) (2015), 1-13.
34. Kettler, P.C., Utility Copulas. Oslo: Department of Mathematics, University of Oslo, Pure mathematics, 2007. http://urn. nb. no/URN: NBN: no-8076.
35. Kresta, A., Application of GARCH-copula model in portfolio optimization, Fin. Assets and Inves., 6(2) (2015), 7-20.
36. Kularatne, T.D., Li J., Pitt, D., On the use of Archimedean copulas for insurance modelling, Ann. Actuar. Sci., 15(1) (2021), 57-81.
37. Külekci, B.Y., Korn, R., Selcuk-Kestel, A.S., Ruin probability for heavy-tailed and dependent losses under reinsurance strategies. Math. Comput. Simul., 226 (2024), 118-138.
38. Lai, C.D., Xie, M., A new family of positive quadrant dependent bivariate distributions, Stat. Prob. Lett., 46(4) (2000), 359-364.
39. Lai, L.H., Risk aversion effect on the insurance premium in correlated lines, J. Stat. Manag. Syst., 14(1) (2011), 101-110.
40. Lai, L.H., Statistical premium in correlated losses of insurance, Econ. Model., 49 (2015), 248-253.
41. Lewellen, K., Financing decisions when managers are risk averse, J. Financ. Econ., 82(3) (2006), 551-589.
42. Li, X., Fang, R., A new family of bivariate copulas generated by univariate distributions, Journal of Data Science, 10(1) (2012), 1-17.
43. Li, J., Liu, D., Wang, J., Risk aversion with two risks: A theoretical extension, J. Math. Econ., 63 (2016), 100-105.
44. Messaoud, S.B., Aloui, C., Measuring risk of portfolio: GARCH-copula model, J. Econ. Integr., (2015), 172-205.
45. Nelsen, R.B., An Introduction to Copulas (Second Edition), New York: Springer, 2006.
46. Patton, A., Copula methods for forecasting multivariate time series, Handbook of Economic Forecasting, 2012.
47. Outreville, J.F., Risk aversion, risk behavior, and demand for insurance: A survey, J. Insur. Iss., (2014), 158-186.
48. Pourkhanali, A., Kim, J.M., Tafakori, L.,Fard, F.A., Measuring systemic risk using vine-copula, Econ. Model., 53 (2016), 63-74.
49. Sengupta, J.K., Multivariate risk aversion with applications, Mathematical Modelling, 4(4) (1983), 307-322.
50. Sklar, A., Functions de repartition an dimensions et leurs marges, Publications de l’Institut Statistique de l’Université de Paris, 8 (1959), 229-231.
51. Thomas, P.J., Measuring risk-aversion: The challenge, Measurement, 79 (2016), 285-301.
52. Ünözkan, H., Yılmaz, M., Construction of continuous bivariate distribution by transmuting dependent distribution, Cumhuriyet Sci. J., 40(4) (2019a), 860-866.
53. Ünözkan, H., Yılmaz, M., A new method for generating distributions: An application to flow data, Int. J. Stat. App., 9(3) (2019b), 92-99.
54. Ünözkan, H., Yılmaz, M., A new approach to bivariate transmutation: construction of continuous bivariate distribution under negative dependency, Cumhuriyet Sci. J., 41(4) (2020), 938-943.
55. Ünözkan, H., Yılmaz, M., A new transmutation: conditional copula with exponential distribution, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 72(2) (2023), 397-406.
56. Yılmaz, M., Ünözkan, H., On bivariate extension of the univariate transmuted distribution family, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 70(2) (2021), 1055-1064