Penentuan Premi Tunggal Bersih pada Reversionary Annuity untuk Pasangan Suami Istri dengan Model Frank’s Copula

Furlo Gilbert Godfrey; I Gusti Putu Purnaba; Ruhiyat Ruhiyat

doi:10.34312/jjom.v4i2.13952

Penentuan Premi Tunggal Bersih pada Reversionary Annuity untuk Pasangan Suami Istri dengan Model Frank’s Copula

Furlo Gilbert Godfrey, I Gusti Putu Purnaba, Ruhiyat Ruhiyat

Abstract

One of the multi-life annuity products is a reversionary annuity, a life annuity product for two insured people. The annuity payment for this product will begin after one of the insured specified in the contract dies first until the other insured dies as well. The calculation of the annuity premium is usually done by assuming independence between the random variable of remaining life-times of the insured parties. However, this is not relevant to the actual situation because the husband and wife are interrelated with their lives. This study considered this relationship when modelling the joint distribution of the remaining life-times between husband and wife. Frank’s copula was used to model the joint distribution of the remaining life-times of husband and wife. It was built from marginal distribution, which was assumed to follow the mortality value in the 2019 Indonesian Mortality Table IV. The mortality value at non-integer ages was assumed to follow a uniform distribution (uniform distribution of death). Furthermore, the joint survival distribution modelled by copula was constructed exactly when husband and wife are married. This study also considered the net single premium of reversionary annuity for several beneficiary cases. In general, the results of the calculation of the net single premium with Frank’s copula produced a lower value than the calculation of the assumption of independence for all beneficiary cases. In addition, the purchase time of an annuity also affects the net single premium, which increases up to a certain point of marriage age and then decreases thereafter.

Keywords

Frank’s Copula; Married Couple; Net Single Premium; Reversionary Annuity; Uniform Distribution of Death

Full Text:

PDF

References

H. U. Gerber, Life Insurance Mathematics. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997, doi: http://dx.doi.org/10.1007/978-3-662-03460-6.

E. W. Frees, J. Carriere, and E. Valdez, “Annuity Valuation with Dependent Mortality,” The Journal of Risk and Insurance, vol. 63, no. 2, p. 229, jun 1996, doi: http://dx.doi.org/10.2307/253744.

M. Denuit and A. Cornet, “Multilife Premium Calculation with Dependent Future Lifetimes,” Journal of Actuarial Practice, vol. 7, pp. 147–180, 1999.

M. Young, “The Mortality of Widowers,” The Lancet, vol. 282, no. 7305, pp. 454–457, aug 1963, doi: http://dx.doi.org/10.1016/S0140-6736(63)92193-7.

C. M. Parkes, B. Benjamin, and R. G. Fitzgerald, “Broken Heart: A Statistical Study of Increased Mortality among Widowers,” BMJ, vol. 1, no. 5646, pp. 740–743, mar 1969, doi: http://dx.doi.org/10.1136/bmj.1.5646.740.

A. W. Ward, “Mortality of bereavement.” BMJ, vol. 1, no. 6011, pp. 700–702, mar 1976, doi: http://dx.doi.org/10.1136/bmj.1.6011.700.

C. Jagger and C. J. Sutton, “Death after marital bereavement-is the risk increased?” Statistics in Medicine, vol. 10, no. 3, pp. 395–404, mar 1991, doi: http://dx.doi.org/10.1002/sim.4780100311.

F. Elwert and N. A. Christakis, “The Effect of Widowhood on Mortality by the Causes of Death of Both Spouses,” American Journal of Public Health, vol. 98, no. 11, pp. 2092–2098, nov 2008, doi: http://dx.doi.org/10.2105/AJPH.2007.114348.

P. Margus, “Generalized Frasier Claim Rates Under Survivorship Life Insurance Policies,” North American Actuarial Journal, vol. 6, no. 2, pp. 76–94, apr 2002, doi: http://dx.doi.org/10.1080/10920277.2002.10596045.

H. Youn and A. Shemyakin, “Statistical Aspects of Joint Life Insurance Pricing,” in Proceedings of Amer. Stat. Assoc. University of St.Thomas: University of St.Thomas, 1999, pp. 34–38.

E. Luciano, J. Spreeuw, and E. Vigna, “Spouses’ Dependence across Generations and Pricing Impact on Reversionary Annuities,” Risks, vol. 4, no. 2, p. 16, may 2016, doi: http://dx.doi.org/10.3390/risks4020016.

J. F. Carriere, “Bivariate Survival Models for Coupled Lives,” Scandinavian Actuarial Journal, vol. 2000, no. 1, pp. 17–32, jan 2000, doi: http://dx.doi.org/10.1080/034612300750066700.

R. J. Cunningham, T. N. Herzog, and R. L. London, Models for Quantifying Risk, 5th ed. Winsted (CT): ACTEX Publications, Inc., 2012.

R. B. Nelsen, An Introduction to Copulas, 2nd ed., ser. Springer Series in Statistics. New York, NY: Springer New York, 2006, doi: http://dx.doi.org/10.1007/0-387-28678-0.

J. De¸bicka, S. Heilpern, and A. Marciniuk, “Application of Copulas to Modelling of Marriage Reverse Annuity Contract,” Prague Economic Papers, vol. 29, no. 4, pp. 445–468, aug 2020, doi: http://dx.doi.org/10.18267/j.pep.745.

N. Bowers, H. Gerber, and J. Hickman, Actuarial Mathematics, 2nd ed. Schaumburg (IL): The Society of Actuaries, 1997.

DOI: https://doi.org/10.34312/jjom.v4i2.13952