A Study on Modeling of Lifetime with Right-Truncated Composite Lognormal-Pareto Distribution: Actuarial Premium Calculations

The modeling of lifetime is important to compute actuarial quantities such as the premium on insurance and annuity products. De Moivre, Gompertz, and Makeham are laws of mortality frequently used in lifetime modeling. Composite distributions have also been used to model lifetime, recently. However, there are not many actuarial applications of these models in the literature. Therefore, the main aim of the study is to perform a case study that gives a comparison of marginal and composite models on premiums. For this purpose, firstly, it is aimed to achieve a new mortality function for a lifetime using composite distribution. The second aim is to analytically compute premiums for whole life and term life insurance products. Here, it is assumed that lifetime distribution is modeled with lognormal, Type 2 Pareto (Pareto) and composite lognormal-Pareto. Firstly, the right truncated distributions of the models were obtained under the consideration that the last age of death was 100. Afterwards, the survival and mortality functions were inferenced using Mathematica 10.2 for the right truncated models. Finally, premium coefficients were analytically presented for whole life and term life insurances in single and joint life statuses. The results show that there are significantly differences in these premium coefficients. It has been observed that the premium coefficients for the term life insurance were higher than the premium coefficients for whole life insurance. In addition, the premium coefficients of the insurances issued for the joint life were smaller than the premium coefficients for the single life.

Keywords:

Composite lifetime, Truncated distributions, Mortality-survival, Insurance premiums Single-joint life,

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[1] Klugman, S.A., Panjer, H.H. and Willmot, G.E., Loss models: from data to decisions (second edition), New York: John Wiley and Sons, (2004).
[2] Cooray, K. and Ananda, M.M., “Modeling actuarial data with a composite lognormal-Pareto model”, Scandinavian Actuarial Journal, (5): 321-334, (2005).
[3] Scollnik, D.P., “On composite lognormal-Pareto models”, Scandinavian Actuarial Journal, (1): 20-33, (2007).
[4] Dominicy, Y. and Sinner, C., Distributions and Composite Models for Size-Type Data. Advances in Statistical Methodologies and Their Application to Real Problems, 159. (2017).
[5] Lindeboom, M. and Van den Berg, G.J., “Heterogeneity in models for bivariate survival: the importance of the mixing distribution.”, Journal of the Royal Statistical Society: Series B (Methodological), 56(1): 49-60, (1994).
[6] Parner, E.T., “A composite likelihood approach to multivariate survival data”, Scandinavian Journal of Statistics, 28(2): 295-302, (2001).
[7] Teodorescu, S. and Vernic, R., “Some composite Exponential-Pareto models for actuarial prediction”, Romanian Journal of Economic Forecasting, 12(4): 82-100, (2009).
[8] Nadarajah, S. and Bakar, S.A., “New composite models for the Danish fire insurance data”, Scandinavian Actuarial Journal, (2): 180-187, (2014).
[9] Dickson, D.C., Hardy, M., Hardy, M.R. and Waters, H.R., Actuarial mathematics for life contingent risks, Cambridge University Press, (2013).
[10] Menge, W.O. and Glover, J.W., An introduction to the mathematics of life insurance, Macmillan, (1938).
[11] Melnikov, A. and Romaniuk, Y., “Evaluating the performance of Gompertz, Makeham and Lee–Carter mortality models for risk management with unit-linked contracts”, Insurance: Mathematics and Economics, 39(3): 310-329, (2006)
[12] Yörübulut, S. and Gebizlioglu, O.L., “Bivariate Pseudo–Gompertz distribution and concomitants of its order statistics”, Journal of Computational and Applied Mathematics, 247: 68-83 (2013).
[13] Montserrat, H.S., “The use of the Premium calculation principles in actuarial pricing based scenario in a coherent risk measure”, Journal of Applied Quantitative Methods, 34, (2014).
[14] Terzioğlu, M.K. and Sucu, M., “Gompertz–Makeham parameter estimations and valuation approaches: Turkish life insurance sector”, European Actuarial Journal, 5(2): 447-468, (2015).
[15] Zayanti, D.A., Kresnawati, E.S. and Megah, M., “Joint life insurance based on Gompertz assumptions and interest rate affected by the excange rate”, In Journal of Physics: Conference Series (Vol. 1282, No. 1, p. 012006). IOP Publishing., (2019).
[16] Riaman, S., Supian, S. and Bon, A.T., “Analysis of Determination of Adjusted Premium Reserves for Last Survivor Endowment Life Insurance Using the Gompertz Assumption”, IEOM Society International, 430-438, (2019).
[17] Handoyo, F., Riaman, R., Gusriani, N., Supian, S. and Subiyanto, S., “Joint Life Term Insurance Reserves Use the Retrospective Method Based on De Moivre Law”, World Scientific News, 128(2): 315-327, (2019).
[18] Büyükyazıcı, M. and Karagül, B.Z., “Optimal Reinsurance Minimizing the Absolute Value of the Difference between the Profits of the Insurer and the Reinsurer”, Gazi University Journal of Science, 33(2): (2020).
[19] Zhou, F., Yu, T., Du, R., Fan, G., Liu, Y., Liu, Z., ... and Guan, L., Clinical course and risk factors for mortality of adult inpatients with COVID-19 in Wuhan, China: a retrospective cohort study. The lancet, (2020).
[20] Covid, C.D.C. and Team, R., “Severe outcomes among patients with coronavirus disease 2019 (COVID-19)”—United States, February 12–March 16, 2020. MMWR Morb Mortal Wkly Rep, 69(12): 343-346, (2020).
[21] Mandıracıoğlu, A., “Demographic characteristics of the elderly population in Turkey and the world”, Ege Journal of Medicine, 49(3): 39-45, (2010).
[22] Foreman, K.J., Marquez, N., Dolgert, A., Fukutaki, K., Fullman, N., McGaughey, M., ... and Brown, J.C., “Forecasting life expectancy, years of life lost, and all-cause and cause-specific mortality for 250 causes of death: reference and alternative scenarios for 2016–40 for 195 countries and territories”, The Lancet, 392(10159): 2052-2090, (2018).
[23] Bengtsson, T. and Keilman, N., Old and new perspectives on mortality forecasting, Springer Nature, (2019).
[24] Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A. and Nesbitt, C.J., Actuarial Mathematics, The Society of Actuaries, New York, (1997).
[25] Gerber, H.U., Life Insurance Mathematics, Springer Science & Business Media, (1997).