Hostname: page-component-7c8c6479df-ph5wq Total loading time: 0 Render date: 2024-03-29T08:58:39.528Z Has data issue: false hasContentIssue false

Bayesian Stochastic Mortality Modelling for Two Populations

Published online by Cambridge University Press:  09 August 2013

Andrew J.G. Cairns
Affiliation:
Maxwell Institute for Mathematical Sciences, and, Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh, EH14 4AS, UK., E-Mail: A.Cairns@ma.hw.ac.uk
David Blake
Affiliation:
Pensions Institute, Cass Business School, City University, 106 Bunhill Row, London, EC1Y 8TZ, UK.
Kevin Dowd
Affiliation:
Pensions Institute, Cass Business School, City University, 106 Bunhill Row, London, EC1Y 8TZ, UK.
Guy D. Coughlan
Affiliation:
Pension Advisory Group, JP Morgan Chase Bank, 125 London Wall, London, EC2Y 5AJ, UK.
Marwa Khalaf-Allah
Affiliation:
Pension Advisory Group, JP Morgan Chase Bank, 125 London Wall, London, EC2Y 5AJ, UK.

Abstract

This paper introduces a new framework for modelling the joint development over time of mortality rates in a pair of related populations with the primary aim of producing consistent mortality forecasts for the two populations. The primary aim is achieved by combining a number of recent and novel developments in stochastic mortality modelling, but these, additionally, provide us with a number of side benefits and insights for stochastic mortality modelling. By way of example, we propose an Age-Period-Cohort model which incorporates a mean-reverting stochastic spread that allows for different trends in mortality improvement rates in the short-run, but parallel improvements in the long run. Second, we fit the model using a Bayesian framework that allows us to combine estimation of the unobservable state variables and the parameters of the stochastic processes driving them into a single procedure. Key benefits of this include dampening down of the impact of Poisson variation in death counts, full allowance for paramater uncertainty, and the flexibility to deal with missing data. The framework is designed for large populations coupled with a small sub-population and is applied to the England & Wales national and Continuous Mortality Investigation assured lives males populations. We compare and contrast results based on the two-population approach with single-population results.

Type
Research Article
Copyright
Copyright © International Actuarial Association 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Biatat, V.D. and Currie, I.D. (2010) Joint models for classification and comparison of mortality in different countries. Proceedings of 25rd International Workshop on Statistical Modelling, Glasgow, 8994.Google Scholar
Blake, D., Cairns, A.J.G. and Dowd, K. (2006) Living with mortality: longevity bonds and other mortality-linked securities. British Actuarial Journal, 12: 153197.CrossRefGoogle Scholar
Booth, H., Maindonald, J. and Smith, L. (2002a) Applying Lee-Carter under conditions of variable mortality decline. Population Studies, 56: 325336.Google Scholar
Booth, H., Maindonald, J. and Smith, L. (2002b) Age-time interactions in mortality projection: Applying Lee-Carter to Australia. Working Papers in Demography, The Australian National University.Google Scholar
Booth, H., Hyndman, R.J., Tickle, L. and de Jong, P. (2006) Lee-Carter mortality forecasting: A multi-country comparison of variants and extensions. Demographic Research, 15: 289310.Google Scholar
Booth, H. and Tickle, L. (2008) Mortality modelling and forecasting: A review of methods. Annals of Actuarial Science, 3: 343.Google Scholar
Bray, I. (2002) Application of Markov chain Monte Carlo methods to projecting cancer incidence and mortality. Applied Statistics, 51: 151164.Google Scholar
Brouhns, N., Denuit, M. and Vermunt J.K. (2002) A Poisson log-bilinear regression approach to the construction of projected life tables. Insurance: Mathematics and Economics, 31: 373393.Google Scholar
Cairns, A.J.G., Blake, D. and Dowd, K. (2006a) Pricing death: Frameworks for the valuation and securitization of mortality risk. ASTIN Bulletin, 36: 79120.CrossRefGoogle Scholar
Cairns, A.J.G., Blake, D. and Dowd, K. (2006b) A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. Journal of Risk and Insurance, 73: 687718.Google Scholar
Cairns, A.J.G., Blake, D. and Dowd, K. (2008) Modelling and management of mortality risk: A review. Scandinavian Actuarial Journal, 2008(2-3): 79113.Google Scholar
Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D., Ong, A. and Balevich, I. (2009) A quantitative comparison of stochastic mortality models using data from England & Wales and the United States. North American Actuarial Journal, 13: 135.Google Scholar
Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2011a) Mortality density forecasts: An analysis of six stochastic mortality models. Insurance: Mathematics and Economics, 48: 355367.Google Scholar
Cairns, A.J.G., Blake, D., Dowd, K. and Coughlan, G.D. (2011b) Longevity hedge effectiveness: A decomposition. Working paper, Heriot-Watt University.Google Scholar
Carter, L. and Lee, R.D. (1992) Modelling and forecasting US sex differentials in mortality. International Journal of Forecasting, 8: 393411.Google Scholar
Coughlan, G., Epstein, D., Ong, A., Sinha, A., Hevia-Portocarrero, J., Gingrich, E., Khalaf-Allah, M. and Joseph, P. (2007a) Life Metrics: A toolkit for measuring and managing longevity and mortality risks. Technical document. Available at www.lifemetrics.com.Google Scholar
Coughlan, G., Epstein, D., Sinha, A. and Honig, P. (2007b) q-Forwards: Derivatives for transferring longevity and mortality risk. Available at www.lifemetrics.com.Google Scholar
Coughlan, G.D. (2009. Longevity risk transfer: Indices and capital market solutions. In Barrieu, P.M. and Albertini, L. (eds), The Handbook of Insurance Linked Securities, Wiley, London.Google Scholar
Coughlan, G.D., Khalaf-Allah, M., Ye, Y., Kumar, S., Cairns, A.J.G., Blake, D. and Dowd, K. (2011) Longevity hedging 101: A framework for longevity basis risk analysis and hedge effectiveness. To appear in North American Actuarial Journal.Google Scholar
Currie, I.D., Durban, M. and Eilers, P.H.C. (2004) Smoothing and forecasting mortality rates. Statistical Modelling, 4: 279298.CrossRefGoogle Scholar
Czado, C., Delwarde, A. and Denuit, M. (2005) Bayesian Poisson log-bilinear mortality projections. Insurance: Mathematics and Economics, 36: 260284.Google Scholar
Dahl, M., Melchior, M. and Møller, T. (2008) On systematic mortality risk and risk minimisation with survivor swaps. Scandinavian Actuarial Journal, 2008(2-3): 114146.CrossRefGoogle Scholar
Dahl, M., Glar, S. and Møller, T. (2011) Mixed dynamic and static risk minimization with an application to survivor swaps. To appear in European Actuarial Journal.Google Scholar
Dowd, K., Cairns, A.J.G., Blake, D., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2010a). Evaluating the goodness of fit of stochastic mortality models. Insurance: Mathematics and Economics, 47: 255265.Google Scholar
Dowd, K., Cairns, A.J.G., Blake, D., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2010b). Backtesting stochastic mortality models: An ex-post evaluation of multi-period-ahead density forecasts. North American Actuarial Journal, 14: 281298.Google Scholar
Gilks, W.R., Richardson, S. and Spiegelhalter, S.J. (1996) Markov chain Monte Carlo in Practice. Chapman and Hall, New York.Google Scholar
Girosi, F. and King, G. (2008) Demographic Forecasting. Princeton University Press, Princeton.CrossRefGoogle Scholar
Hyndman, R.J. and Ullah, M.S. (2007) Robust forecasting of mortality and fertility rates: A functional data approach. Computational Statistics and Data Analysis, 51: 49424956.Google Scholar
Jacobsen, R., Keiding, N. and Lynge, E. (2002) Long-term mortality trends behind low life expectancy of Danish women. J. Epidemiol. Community Health, 56: 205208.CrossRefGoogle ScholarPubMed
Jarner, S.F. and Kryger, E.M. (2011) Modelling adult mortality in small populations: The SAINT model. To appear in ASTIN Bulletin.Google Scholar
Kogure, A., Kitsukawa, K. and Kurachi, Y. (2009) A Bayesian comparison of models for changing mortalities toward evaluating longevity risk in Japan. Asia Pacific Journal of Risk and Insurance, 3(2: 121.Google Scholar
Kogure, A. and Kurachi, Y. (2010) A Bayesian approach to pricing longevity risk based on risk-neutral predictive distributions. Insurance: Mathematics and Economics, 46: 162172.Google Scholar
Li, J.S.-H. and Hardy, M.R. (2009) Measuring basis risk involved in longevity hedges. Working paper, University of Waterloo.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: 575594.CrossRefGoogle ScholarPubMed
Li, N., Lee, R. and Tuljapurkar, S. (2004) Using the Lee-Carter method to forecast mortality for populations with limited data. International Statistical Review, 72: 1936.Google Scholar
Li, J.S.-H., Hardy, M.R. and Tan, K.S. (2009) Uncertainty in mortality forecasting: An extension to the classic Lee-Carter approach. ASTIN Bulletin, 39: 137164.Google Scholar
Loeys, J., Panigirtzoglou, N. and Ribeiro, R.M. (2007) Longevity: A market in the making. Available at www.lifemetrics.com.Google Scholar
Macdonald, A.S., Cairns, A.J.G., Gwilt, P.L. and Miller, K.A., (1998) An international comparison of recent trends in population mortality. British Actuarial Journal 4: 3141.Google Scholar
Oeppen, J. and Vaupel, J.W. (2002) Broken limits to life expectancy. Science, 296: 10291030.CrossRefGoogle ScholarPubMed
Olivieri, A. and Pitacco, E. (2009) Stochastic mortality: the impact on target capital. ASTIN Bulletin, 39: 541563.CrossRefGoogle Scholar
Osmond, C. (1985) Using age, period and cohort models to estimate future mortality rates. International Journal of Epidemiology, 14: 124129.Google Scholar
Osmond, C. and Gardner, M.J. (1982) Age, period and cohort models applied to cancer mortality rates. Statistics in Medicine, 1: 245259.CrossRefGoogle ScholarPubMed
Pedroza, C. (2006) A Bayesian forecasting model: Predicting U.S. male mortality. Biostatistics, 7: 530550.Google Scholar
Pitacco, E., Denuit, M., Haberman, S. and Olivieri, A. (2009) Modelling longevity dynamics for pensions and annuity business. Oxford University Press, Oxford.Google Scholar
Plat, R. (2009) Stochastic portfolio specific mortality and the quantification of mortality basis risk. Insurance: Mathematics and Economics, 45: 123132.Google Scholar
Reichmuth, W. and Sarferaz, S. (2008) Bayesian demographic modelling and forecasting: An application to US mortality. SFB 649 Discussion paper 2008-052.Google Scholar
Renshaw, A.E. and Haberman, S. (2003) Lee-Carter mortality forecasting with age-specific enhancement. Insurance: Mathematics and Economics, 33: 255272.Google Scholar
Renshaw, A.E. and Haberman, S. (2006) A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insurance: Mathematics and Economics, 38: 556570.Google Scholar
Tuljapurkar, S., Li, N. and Boe, C. (2000) A universal pattern of mortality change in G7 countries. Nature 405: 789792.Google Scholar