Sains Malaysiana 51(11)(2022): 3807-3817

http://doi.org/10.17576/jsm-2022-5111-24

 

On the Impact of Asymmetric Dependence in the Actuarial Pricing of Joint Life Insurance Policies

(Kesan Kebersandaran Asimetri dalam Harga Aktuari Polisi Insurans Hayat Tercantum)

 

EMEL KIZILOK KARA*

 

Department of Actuarial Sciences, Faculty of Arts and Sciences, Kirikkale University, Kirikkale, Turkey

 

Diserahkan: 1 April 2022/Diterima: 19 Julai 2022

 

Abstract

Multipopulation mortality modeling is a significant research problem in actuarial science. Mortality functions involving multiple lives are also essential to determine the pricing of premiums. Moreover, the lifetime models based on dependence and asymmetry are more realistic. Hence, this paper applies an asymmetric copula model, Generalized FGM (GFGM) to model the bivariate joint distribution of future lifetimes. Premiums of first-death life insurance products are calculated based on the proposed model and compared with independent and symmetrical models. The results display that asymmetry has a significant effect on premium calculations. Also, it is concluded that the lowest premiums are generally in asymmetric lifetime models. This paper also provides analytical examples for the proposed model with Gompertz’s marginal law.

 

Keywords: Asymmetric dependence; copula; insurance; joint life (first-death); premium 

 

Abstrak

Pemodelan mortaliti populasi berbilang merupakan permasalahan penyelidikan yang penting dalam bidang sains aktuari. Fungsi mortaliti yang melibatkan model hayat berbilang juga berperanan untuk menentukan harga premium. Selain itu, model masa-hayat berdasarkan kebersandaran dan asimetri adalah lebih realistik. Oleh itu, makalah ini menggunakan model kopula asimetri dan Generalized FGM (GFGM) untuk memodelkan taburan tercantum bivariat bagi jangka hayat masa hadapan. Premium bagi produk insurans hayat kematian-pertama dihitung berdasarkan model yang dicadangkan dan dibandingkan dengan model tak bersandar dan simetri. Keputusan menunjukkan bahawa asimetri mempunyai kesan yang signifikan ke atas pengiraan premium. Selain itu, dapat disimpulkan bahawa premium terendah kebiasaannya ditunjukkan dalam model masa hayat asimetri. Kajian ini juga menyediakan contoh analisis bagi model yang dicadangkan menggunakan marginal Gompertz.

 

Kata kunci: Hayat tercantum (kematian-pertama); insurans; kebersandaran asimetri; kopula; premium

 

RUJUKAN

Ang, A., Chen J. & Xing, Y. 2006. Downside risk. Review of Financial Studies 19: 1191-1239.

Bairamov, I. & Kotz, S. 2002. Dependence structure and symmetry of Huang–Kotz FGM distributions and their extensions. Metrika 56: 55-72.

Bairamov, I., Kotz, S. & Bekci, M. 2001. New generalized Farile-Gumbel-Morgenstern distributions and concomitants of order statistics. Journal of Applied Statistics 28(5): 521-536.

Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. & Nesbitt, C.J. 1997. Actuarial Mathematics. New York: The Society of Actuaries.

Bücher, A., Irresberger, F. & Weiss, G.N. 2017. Testing asymmetry in dependence with copula-coskewness. North American Actuarial Journal 21(2): 267-280.

Carriere, J.F. 2000. Bivariate survival models for coupled lives. Scandinavian Actuarial Journal 2000(1): 17-32.

Denuit, M. & Cornet, A. 1999. Multilife premium calculation with dependent future lifetimes. Journal of Actuarial Practice 7: 147-180.

Dickson, D.C., Hardy, M., Hardy, M.R & Waters, H.R. 2013. Actuarial Mathematics for Life Contingent Risks. Cambridge: Cambridge University Press.

Dufresne, F., Hashorva, E., Ratovomirija, G. & Toukourou, Y. 2018. On age difference in joint lifetime modelling with life insurance annuity applications. Annals of Actuarial Science 12(2): 350-371.

Frees, E.W., Carriere, J. & Valdez, E. 1996. Annuity valuation with dependent mortality. Journal of Risk and Insurance 63(2): 229-261.

Gerber, H.U. 1997. Life Insurance Mathematics. Springer, Science & Business Media.

Güven, B. & Kotz, S. 2008. Test of independence for generalized Farlie–Gumbel–Morgenstern distributions. Journal of Computational and Applied Mathematics 212(1): 102-111.

Harvey, C. & Siddique, A. 2000. Conditional skewness in asset pricing tests. Journal of Finance 55: 1263-1295.

Hsieh, M.H., Tsai, C.J. & Wang, J.L. 2020. Mortality risk management under the factor Copula framework-with applications to insurance policy pools. North American Actuarial Journal 25(Issue sup1: Longevity Risk and Capital Markets - Longevity 12 and Longevity 13): S119-S131.

Huang, J.S. & Kotz, S. 1999. Modifications of the Farlie–Gumbel–Morgenstern distributions. A tough hill to climb. Metrika 49: 135-145.

Jagger, C. & Sutton, C.J. 1991. Death after marital bereavement is the risk increased? Statistics in Medicine 10(3): 395-404.

Jung, Y.S., Kim, J.M. & Kim, J. 2008. New approach of directional dependence in exchange markets using generalized fgm copula function. Communications in Statistics-Simulation and Computation 37(4): 772-788.

Kara, E.K. 2021. Chapter III. On actuarial premiums for joint last survivor life insurance based on asymmetric dependent lifetimes. In Current Academic Studies in Science and Mathematics Sciences-II, edited by Yildiz, D.E. & Özkan, E.Y. Lyon, France: Livre de Lyon. pp. 33-47.

Lee, I., Lee, H. & Kim, H.T. 2014. Analysis of reserves in multiple life insurance using copula. Communications for Statistical Applications and Methods 21(1): 23-43.

Lu, Y. 2017. Broken-heart, common life, heterogeneity: Analyzing the spousal mortality dependence. ASTIN Bulletin: The Journal of the IAA 47(3): 837-874.

Luciano, E., Spreeuw, J. & Vigna, E. 2008. Modelling stochastic mortality for dependent lives. Insurance: Mathematics and Economics 43(2): 234-244.

Menge, W.O. & Glover, J.W. 1938. An Introduction to the Mathematics of Life Insurance. Macmillan.

Nelsen, R.B. 2007. An Introduction to Copulas. Springer, Science & Business Media.

Rodriguez-Lallena, J.A. & Úbeda-Flores, M. 2004. A new class of bivariate copulas. Statistics & Probability Letters 66(3): 315-325.

Shemyakin, A.E. & Youn, H. 2006. Copula models of joint last survivor analysis. Applied Stochastic Models in Business and Industry 22(2): 211-224.

Shubina, M. & Lee, M.L.T. 2004. On maximum attainable correlation and other measures of dependence for the Sarmanov family of bivariate distributions. Communications in Statistics-Theory and Methods 33(5): 1031-1052.

Sklar, M. 1959. Fonctions De Repartition an Dimensions Et Leurs Marges. Publ. Inst. Statist. Univ, Paris. 8: 229-231.

Uhm, D., Kim, J.M. & Jung, Y.S. 2012. Large asymmetry and directional dependence by using copula modeling to currency exchange rates. Model Assisted Statistics and Applications 7(4): 327-340.

Zhu, W., Tan, K.S. & Wang, C.W. 2017. Modeling multicountry longevity risk with mortality dependence: A Lévy subordinated hierarchical Archimedean copulas approach. Journal of Risk and Insurance 84(S1): 477-493.

 

*Pengarang untuk surat-menyurat; email: emel.kizilok@kku.edu.tr

 

 

 

 

   

sebelumnya