Exploring Multivariate Copula Models and Fuzzy Interest Rates in Assessing Family Annuity Products

Kurnia Novita Sari; Ady Febrisutisyanto; Randi Deautama; Nursiti Azirah; Pida Mahani

doi:10.31764/jtam.v8i2.17467

Exploring Multivariate Copula Models and Fuzzy Interest Rates in Assessing Family Annuity Products

Kurnia Novita Sari, Ady Febrisutisyanto, Randi Deautama, Nursiti Azirah, Pida Mahani

Abstract

This research explores the development of a reversionary annuity product transformed into a family annuity covering three individuals: husband, wife, and children. The innovative design of this product considers the sequencing of annuity payments post-participant's demise, aiming to mitigate the risk of parents' death impacting their children. Recognizing the inadequacy of assuming independence among individuals in premium calculations, the study employs a multivariate Archimedean Copula model to account for interdependence. The primary objective is to compute the survival single-life function for each individual taken from TMI IV 2009. Then the copula model is implemented with Clayton and Frank copulas at varying Kendall’s tau values (0.25, 0.5, and 0.75). Meanwhile, the interest rates are modeled using the BI-7-day (reverse) rate with a Triangular Fuzzy α-cut. The findings reveal that increasing Kendall’s tau values lead to higher pure premiums, and notably, the Frank Copula model yields higher premium values than the Clayton Copula model. This research contributes valuable insights into the actuarial assessment of family annuity products, shedding light on the significance of considering dependencies among individuals for more accurate premium calculations.

Keywords

Clayton; Copula; Family Annuity; Frank; Kendall’s tau; Premium.

Full Text:

DOWNLOAD [PDF]

References

Aalaei, M. (2022). Using Fuzzy Interest Rates for Uncertainty Modelling in Enhanced Annuities Pricing. International Journal of Mathematical Modelling & Computations, 12(04), 265–274. https://doi.org/10.30495/ijm2c.2023.1968679.1262

Aggarwal, A. G., & Sharma, A. (2018). An Innovative B2C E-commerce Websites Selection using the ME-OWA and Fuzzy AHP. Proceedings of the First International Conference on Information Technology and Knowledge Management, 14(1), 13–19. https://doi.org/10.15439/2017km37

Arias, L. A. S., & Cirillo, P. (2021). Joint and survivor annuity valuation with a bivariate reinforced urn process. Insurance: Mathematics and Economics, 99(1), 174–189. https://doi.org/10.1016/j.insmatheco.2021.04.004

Baláž, V. (2023). Indulgence, Self-Control, and Annuity Preferences: Annuity Choices by Members of the Slovak-Funded Private Pension Pillar. Social Sciences, 12(3), 190–207. https://doi.org/10.3390/socsci12030190

Blanner, C., Mejldal, A., Prina, A. M., Munk-Jørgensen, P., Ersbøll, A. K., & Andersen, K. (2020). Widowhood and mortality: A Danish nationwide register-based cohort study. Epidemiology and Psychiatric Sciences, 29(1), 1–11. https://doi.org/10.1017/S2045796020000591

Cabrignac, O., Charpentier, A., & Gallic, E. (2020). Modeling Joint Lives within Families.

Chen, G., & Matkin, D. S. T. (2017). Actuarial Inputs and the Valuation Of Public Pension Liabilities and Contribution Requirements: A Simulation Approach. Public Budgeting and Finance, 37(1), 68–87. https://doi.org/10.1111/pbaf.12154

De Andreas-Sanchez, J., & Puchades, L. G. (2017). Some computational results for the fuzzy random value of life actuarial liabilities. Iranian Journal of Fuzzy Systems, 14(4), 1–25. https://doi.org/10.22111/IJFS.2017.3323

Dickson, D. C. M., Hardy, M. R., & Waters, H. R. (2009). Actuarial Mathematics for Life Contingent Risks. Cambridge University Press. https://doi.org/10.1017/CBO9780511800146

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), 326–337. https://doi.org/10.1017/S1748499518000076

Elwert, F., & Christakis, N. A. (2008). The Effect of Widowhood on Mortality by the Causes of Death of Both Spouses. American Journal of Public Health, 98(11), 2092–2098. https://doi.org/10.2105/AJPH.2007.114348

Godfrey, F. G., Purnaba, I. G. P., & Ruhiyat, R. (2022). Penentuan Premi Tunggal Bersih pada Reversionary Annuity untuk Pasangan Suami Istri dengan Model Frank’s Copula. Jambura Journal of Mathematics, 4(2), 277–295. https://doi.org/10.34312/jjom.v4i2.13952

Henshaw, K., Constantinescu, C., & Pamen, O. M. (2020). Stochastic mortality modelling for dependent coupled lives. Risks, 8(1), 1–28. https://doi.org/10.3390/risks8010017

Li, J., Vestergaard, M., Cnattingius, S., Gissler, M., Bech, B. H., Obel, C., & Olsen, J. (2014). Mortality after Parental Death in Childhood: A Nationwide Cohort Study from Three Nordic Countries. PLoS Medicine, 11(7), 1–14. https://doi.org/10.1371/journal.pmed.1001679

Lu, Y. (2017). Broken-heart, Common life, Heterogeneity: Analyzing the Spousal Mortality Dependence. ASTIN Bulletin, 47(3), 837–874. https://doi.org/10.1017/asb.2017.8

Luciano, E., Spreeuw, J., & Vigna, E. (2016). Spouses’ dependence across generations and pricing impact on reversionary annuities. Risks, 4(2), 1–18. https://doi.org/10.3390/risks4020016

Mircea, I., & Covrig, M. (2015). A Discrete Time Insurance Model with Reinvested Surplus and a Fuzzy Number Interest Rate. Procedia Economics and Finance, 32(15), 1005–1011. https://doi.org/10.1016/s2212-5671(15)01561-0

Nelsen, R. B. (2006). An Introduction to Copulas (Second). Springer Series in Statistics, Springer Series in Statistics, NY: Springer New York.

Noor, N. M. M., Retnowardhani, A., Abd, M. L., & Saman, M. Y. M. (2013). Crime Forecasting using ARIMA Model and Fuzzy Alpha-cut. Journal of Applied Sciences, 13(1), 167–172. https://doi.org/10.3923/jas.2013.167.172

Roelfs, D. J., Shor, E., Curreli, M., Clemow, L., Burg, M. M., & Schawrtz, J. E. (2012). Widowhood and Morality: A Meta-Analysis and Meta- Regressiona. Demography, 49(2), 1575–1606. https://doi.org/10.1007/s13524-012-0096-x

Sanders, L., & Melenberg, B. (2016). Estimating the joint survival probabilities of married individuals. Insurance: Mathematics and Economics, 67(1), 88–106. https://doi.org/10.1016/j.insmatheco.2015.12.006

Sari, K. N., Deautama, R., & Febrisutisyanto, A. (2023). the Benefits of Family Annuity Calculation With Vine’S Copula and Fuzzy Interest Rate. BAREKENG: Jurnal Ilmu Matematika Dan Terapan, 17(4), 2461–2470. https://doi.org/10.30598/barekengvol17iss4pp2461-2470

Spreeuw, J., & Owadally, I. (2013). Investigating the Broken-Heart Effect: a Model for Short-Term Dependence between the Remaining Lifetimes of Joint Lives. Annals of Actuarial Science, 7(2), 236–257. https://doi.org/10.1017/s1748499512000292

Uribarri, A., Nunez-Gil, I. J., Conty, D. A., Vedia, O., Almendro-Delia, M., Cambra, A. D., Martin-Garcia, A. C., Barrionuevo-Sanchez, M., Martınez-Selles, M., Raposeiras-Roubın, S., Guillen, M., Garcia Acuna, J. M., Matute-Blanco, L., Linares Vicente, J. A., Grande Flecha, A. S., Andres, M., Perez-Castellanos, A., & Lopez-Pais, J. (2019). Short-and long-term prognosis of patients with takotsubo syndrome based on different triggers: Importance of the physical nature. Journal of the American Heart Association, 8(24), 1–12. https://doi.org/10.1161/JAHA.119.013701

Vakamudi, M. (2016). ‘Broken-heart syndrome’… Be aware.. Indian Journal of Anaesthesia, 60(3), 155–156. https://doi.org/10.4103/0019-5049.177863

Walter, O., Patrick, W., Ottieno, J., & Carolyne, O. (2021). Positive Stable Frailty Approach in the Construction of Dependence Life-Tables. Open Journal of Statistics, 11(04), 506–523. https://doi.org/10.4236/ojs.2021.114032

DOI: https://doi.org/10.31764/jtam.v8i2.17467