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A BAYESIAN JOINT MODEL FOR POPULATION AND PORTFOLIO-SPECIFIC MORTALITY

Published online by Cambridge University Press:  18 July 2017

Frank van Berkum ,
Katrien Antonio  and
Michel Vellekoop
Frank van Berkum*
Affiliation:
Faculty of Economics and Business, University of Amsterdam, Amsterdam, The Netherlands
Katrien Antonio
Affiliation:
Faculty of Economics and Business, University of Amsterdam, Amsterdam, The Netherlands, Faculty of Economics and Business, KU Leuven, Belgium
Michel Vellekoop
Affiliation:
Faculty of Economics and Business, University of Amsterdam, Amsterdam, The Netherlands
*
E-Mail: f.vanberkum@uva.nl
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Abstract

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Insurance companies and pension funds must value liabilities using mortality rates that are appropriate for their portfolio. These can only be estimated in a reliable way from a sufficiently large historical dataset for such portfolios, which is often not available. We overcome this problem by introducing a model to estimate portfolio-specific mortality simultaneously with population mortality. By using a Bayesian framework, we automatically generate the appropriate weighting for the limited statistical information in a given portfolio and the more extensive information that is available for the whole population. This allows us to separate parameter uncertainty from uncertainty due to the randomness in individual deaths for a given realization of mortality rates. When we apply our method to a dataset of assured lives in England and Wales, we find that different prior specifications for the portfolio-specific factors lead to significantly different posterior distributions for hazard rates. However, in short-term predictive distributions for future numbers of deaths, individual mortality risk turns out to be more important than parameter uncertainty in the portfolio-specific factors, both for large and for small portfolios.

Keywords

Bayesian analysisportfolio-specific mortalitymortality projectionvon Mises–Fisher priorsmoothing prior
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 47 , Issue 3 , September 2017 , pp. 681 - 713
Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - ND
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use or in order to create a derivative work.
Copyright
Copyright © Astin Bulletin 2017

References

Antoniadis, A., Grégoire, G. and McKeague, I. (2004) Bayesian estimation in single-index models. Statistica Sinica, 14, 1147–1164.Google Scholar
Antonio, K., Bardoutsos, A. and Ouburg, W. (2015) A Bayesian Poisson log-bilinear model for mortality projections with multiple populations. European Actuarial Journal, 5 (2), 245–281.Google Scholar
Antonio, K. and Zhang, Y. (2014) Nonlinear mixed models. In Predictive Modeling Applications in Actuarial Science (eds. Frees, E., Derrig, R. and Meyers, G.), volume 1, pp. 398–426. New York: Cambridge University Press.Google Scholar
Barrieu, P., Bensusan, H., Karoui, N.E., Hillairet, C., Loisel, S., Ravanelli, C. and Salhi, Y. (2012) Understanding, modelling and managing longevity risk: Key issues and main challenges. Scandinavian Actuarial Journal, 3, 203–231.Google Scholar
Brouhns, N., Denuit, M. and Vermunt, J. (2002) A Poisson log-bilinear regression approach to the construction of projected lifetables. Insurance: Mathematics and Economics, 31 (3), 373–393.Google Scholar
Cairns, A., Blake, D. and Dowd, K. (2006) A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. Journal of Risk and Insurance, 73 (4), 687–718.CrossRefGoogle Scholar
Cairns, A., Blake, D., Dowd, K., Coughlan, G., Epstein, D., Ong, A. and Balevich, I. (2009) A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. North American Actuarial Journal, 13 (1), 1–35.Google Scholar
Cairns, A., Blake, D., Dowd, K., Coughlan, G. and Khalaf-Allah, M. (2011) Bayesian stochastic mortality modelling for two populations. ASTIN Bulletin, 41 (1), 25–59.Google Scholar
Congdon, P. (2009) Life expectancies for small areas: A Bayesian random effects methodology. International Statistical Review, 77 (2), 222–240.Google Scholar
Czado, C., Delwarde, A. and Denuit, M. (2005) Bayesian Poisson log-bilinear mortality projections. Insurance: Mathematics and Economics, 36 (3), 260–284.Google Scholar
Denuit, M., Dhaene, J., Goovaerts, M. and Kaas, R. (2005) Actuarial Theory for Dependent Risks. Chichester, UK: John Wiley & Sons.Google Scholar
Denuit, M., Maréchal, X., Pitrebois, S. and Walhin, J.-F. (2007) Actuarial Modelling of Claim Counts. John Wiley & Sons, Ltd.Google Scholar
Dowd, K., Cairns, A., Blake, D., Coughlan, G. and Khalaf-Allah, M. (2011) A gravity model of mortality rates for two related populations. North American Actuarial Journal, 15 (2), 334–356.Google Scholar
Finkelstein, A. and Poterba, J. (2002) Selection effects in the United Kingdom individual annuities market. The Economic Journal, 112 (476), 28–50.Google Scholar
Fisher, R. (1953) Dispersion on a sphere. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 217 (1130), 295–305.Google Scholar
Gelman, A. (2006) Prior distributions for variance parameters in hierarchical models. Bayesian Analysis, 1 (3), 515–534.CrossRefGoogle Scholar
Gschlössl, S., Schoenmaekers, P. and Denuit, M. (2011) Risk classification in life insurance: Methodology and case study. European Actuarial Journal, 1, 23–41.Google Scholar
Haberman, S., Kaishev, V., Millossovich, P., Villegas, A., Baxter, S., Gaches, A., Gunnlaugsson, S. and Sison, M. (2014) Longevity basis risk: A methodology for assessing basis risk. Technical report, Institute and Faculty of Actuaries. Available online at: www.actuaries.org.uk/events/pages/sessional-research-programme.Google Scholar
Haberman, S. and Renshaw, A. (2011) A comparative study of parametric mortality projection models. Insurance: Mathematics and Economics, 48 (1), 35–55.Google Scholar
Hoem, J. (1973) Levels of error in population forecasts. Artikler fra Statistisk Sentralbyrå, 61, 1–46.Google Scholar
Hoff, P. (2009) Simulation of the matrix Bingham-von-Mises-Fisher distribution, with applications to multivariate and relational data. Journal of Computation and Graphical Statistics, 18 (3), 438–456.CrossRefGoogle Scholar
Kan, H. (2012) A Bayesian mortality forecasting framework for population and portfolio mortality. MSc thesis, University of Amsterdam, The Netherlands.Google Scholar
Kleinow, T. (2015) A common age effect model for the mortality of multiple populations. Insurance: Mathematics and Economics, 63, 147–152.Google Scholar
Lantz, P., House, J., Lepkowski, J., Williams, D., Mero, R. and Chen, J. (1998) Socioeconomic factors, health behaviors, and mortality: Results from a nationally representative prospective study of US adults. Journal of the American Medical Association, 279 (21), 1703–1708.Google Scholar
Lee, R. and Carter, L. (1992) Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87 (419), 659–671.Google Scholar
Li, J. (2014) An application of MCMC simulation in mortality projection for populations with limited data. Demographic Research, 30 (1), 1–48.Google Scholar
Li, N. and Lee, R. (2005) Coherent mortality forecasts for a group of populations: An extension of the Lee–Carter method. Demography, 42 (3), 575–594.Google Scholar
Olivieri, A. (2011) Stochastic mortality: Experience-based modeling and application issues consistent with Solvency 2. European Actuarial Journal, 1, S101–S125.Google Scholar
Pitacco, E., Denuit, M., Haberman, S. and Olivieri, A. (2009) Modelling Longevity Dynamics for Pensions and Annuity Business. New York: Oxford University Press.Google Scholar
Plat, R. (2009) Stochastic portfolio specific mortality and the quantification of mortality basis risk. Insurance: Mathematics and Economics, 45, 123–132.Google Scholar
Purcaru, O., Guillén, M. and Denuit, M. (2004) Linear credibility models based on time series for claim counts. Belgian Actuarial Bulletin, 4 (1), 62–74.Google Scholar
Renshaw, A. and Haberman, S. (2006) A cohort-based extension to the Lee–Carter model for mortality reduction factors. Insurance: Mathematics and Economics, 38 (3), 556–570.Google Scholar
Richards, S., Kaufhold, K. and Rosenbusch, S. (2013) Creating portfolio-specific mortality tables: A case study. European Actuarial Journal, 3, 295–319.Google Scholar
van Berkum, F., Antonio, K. and Vellekoop, M. (2016) The impact of multiple structural changes on mortality predictions. Scandinavian Actuarial Journal, 7, 581–603.Google Scholar
Villegas, A. and Haberman, S. (2014) On the modeling and forecasting of socioeconomic mortality differentials: An application to deprivation and mortality in England. North American Actuarial Journal, 18 (1), 168–193.CrossRefGoogle Scholar
von Mises, R. (1918) Über die ‘Ganzzahligkeit’ der Atomgewicht und verwandte Fragen. Physikalische Zeitschrift, 19, 490–500.Google Scholar
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