A Bayesian Approach in Pricing Insurance under Stochastic Gompertz Model with Health Risk Factors
Abstract
This study aims to enhance the Gompertz mortality model by incorporating stochastic parameters and providing a Bayesian framework for pricing term life insurance based on policyholders’ survival and familial health history. The Vasicek model is used to describe the temporal evolution of these stochastic parameters, capturing their mean-reverting behavior, with its distribution at a fixed time point serving as the prior in the Bayesian framework. Bayes' rule is used to update these priors using observed survival and death outcomes along with familial health factors. Using illustrative data for 30-year-old policyholders, prior and posterior prices are computed across different health profiles, showing that posterior prices can increase or decrease relative to prior-based prices depending on the observed risk patterns. These results illustrate how incorporating observed survival and health information refines pricing and aligns prices with actual risk across policyholder profiles.
Full text article
References
B. Gompertz, “On the nature of the function expressive of the law of human mortality and on a new mode of determining life contingencies,” Philosophical Transactions of the Royal Society of London, vol. 115, pp. 513–585, 1825. https://doi.org/10.1098/rspl.1815.0271.
W. M. Makeham, “On the law of mortality and the construction of annuity tables,” Journal of the Institute of Actuaries, vol. 6, pp. 301–310, 1860. https://www.jstor.org/stable/41134925.
E. Pitacco, M. Denuit, S. Haberman, and A. Olivieri, Modelling longevity dynamics for pensions and annuity business. Oxford University Press, 2009. https://doi.org/10.1093/oso/9780199547272.001.0001.
M. Denuit, P. Devolder, and A. Goderniaux, “Securitization of longevity risk: Pricing survivor bonds with wang transform in the lee-carter framework,” Journal of Risk and Insurance, vol. 74, pp. 87–113, 2007. https://www.jstor.org/stable/4138426.
K. Antonio and E. A. Valdez, “Statistical concepts of a priori and a posteriori risk classification in insurance,” Advances in Statistical Analysis, vol. 96(2), pp. 187–224, 2012. https://doi.org/10.1007/s10182-011-0152-7.
O. Vasicek, “An equilibrium characterization of the term structure,” Journal of Financial Economics, vol. 5(2), pp. 177–188, 1977. https://doi.org/10.1016/0304-405X(77)90016-2.
E. Biffis, “Affine processes for dynamic mortality and actuarial valuations,” Insurance: Mathematics and Economics, vol. 37(3), pp. 443–468, 2005. https://doi.org/10.1016/j.insmatheco.2005.05.003.
B. P. Carlin and T. A. Louis, Bayesian Methods for Data Analysis. CRC Press, 3rd ed., 2009. https://books.google.com.ph/books?id=GTJUt8fcFx8C.
K. Barigou, P.-O. Goffard, S. Loisel, and Y. Salhi, “Bayesian model averaging for mortality forecasting using leave-future-out validation,” International Journal of Forecasting, vol. 39(2), pp. 674–690, 2023. https://doi.org/10.1016/j.ijforecast.2022.01.011.
R. D. Lee and L. R. Carter, “Modeling and forecasting u.s. mortality,” Journal of the American Statistical Association, vol. 87(419), pp. 659–671, 1992. https://doi.org/10.1080/01621459.1992.10475265.
A. E. Renshaw and S. Haberman, “A cohort-based extension to the lee–carter model for mortality reduction factors,” Insurance: Mathematics and Economics, vol. 38(3), pp. 556– 570, 2006. https://doi.org/10.1016/j.insmatheco.2005.12.001.
S. A. Klugman, H. H. Panjer, and G. E. Willmot, Loss Models: From Data to Decisions. John Wiley & Sons, Inc., 3rd ed., 2008. https://doi.org/10.1017/S1748499500000531.
T. Mikosch, Elementary stochastic calculus with finance in view. World scientific, 1998. http://lib.ysu.am/open_books/417513.pdf.
S. Ross, A First Course in Probability. London, United Kingdom: Pearson Education, Inc., 9th, illustrated edition, 2014. https://chengzhaoxi.xyz/download/pdf/book/A-First-Course-in-Probability.pdf.
S. E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models, vol. 11 of Springer Finance Textbooks. New York: Springer, 2004. https://www.scribd.com/document/397066094/Steven-E-Shreve-Stochastic-calculus-for-finance-II-Continuous-time-models-pdf.
A. I. Yashin, I. A. Iachine, and A. Z. Begun, “Mortality modeling: A review,” Mathematical Population Studies, vol. 8(4), pp. 305–332, 2000. https://doi.org/10.1080/08898480009525489.
R. Ambagaspitiya and C. Ford, “Calibrating interest rate models,” Society of Actuaries Research Report, October 2023. https://www.soa.org/globalassets/assets/files/resources/research-report/2023/interest-rate-model-calibration-study.pdf.
Y. Li and P. Yeh, “An interpretation of the moore–penrose generalized inverse of a singular fisher information matrix,” IEEE Transactions on Signal Processing, vol. 60(10), pp. 5532– 5536, 2012. https://doi.org/10.1109/TSP.2012.2208105.
J. James and N. Webber, Interest Rate Modeling. John Wiley & Sons, 2000. https://api.semanticscholar.org/CorpusID:153632252.
E. V. Slud, Actuarial Mathematics and Life-Table Statistics. University of Maryland, College Park, 2001. https://www.math.umd.edu/~slud/s470/BookChaps/01Book.pdf
Authors
Copyright (c) 2026 Journal of the Indonesian Mathematical Society
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Article Details
