Bayesian Poisson log-bilinear models for mortality projections with multiple populations. (English) Zbl 1329.91111

Summary: Life insurers, pension funds, health care providers and social security institutions face increasing expenses due to continuing improvements of mortality rates. The actuarial and demographic literature has introduced a myriad of (deterministic and stochastic) models to forecast mortality rates of single populations. This paper presents a Bayesian analysis of two related multi-population mortality models of log-bilinear type, designed for two or more populations. Using a larger set of data, multi-population mortality models allow joint modelling and projection of mortality rates by identifying characteristics shared by all sub-populations as well as sub-population specific effects on mortality. This is important when modeling and forecasting mortality of males and females, regions within a country and when dealing with index-based longevity hedges. Our first model is inspired by the two factor Lee-Carter model of A. E. Renshaw and S. Haberman [Insur. Math. Econ. 33, No. 2, 255–272 (2003; Zbl 1103.91371)] and the common factor model of L. R. Carter and R. D. Lee [“Modeling and forecasting US sex differentials in mortality”, Int. J. Forecast. 8, No. 3, 393–411 (1992; doi:10.1016/0169-2070(92)90055-E)]. The second model is the augmented common factor model of N. Li and R. Lee [“Coherent mortality forecasts for a group of populations: an extension of the Lee-Carter method”, Demography 42, No. 3, 575–594 (2005; doi:10.1353/dem.2005.0021)]. This paper approaches both models in a statistical way, using a Poisson distribution for the number of deaths at a certain age and in a certain time period. Moreover, we use Bayesian statistics to calibrate the models and to produce mortality forecasts. We develop the technicalities necessary for Markov Chain Monte Carlo (MCMC) simulations and provide software implementation (in R) for the models discussed in the paper. Key benefits of this approach are multiple. We jointly calibrate the Poisson likelihood for the number of deaths and the times series models imposed on the time dependent parameters, we enable full allowance for parameter uncertainty and we are able to handle missing data as well as small sample populations. We compare and contrast results from both models to the results obtained with a frequentist single population approach and a least squares estimation of the augmented common factor model.


91D20 Mathematical geography and demography
91B30 Risk theory, insurance (MSC2010)
62F15 Bayesian inference
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B84 Economic time series analysis
91G60 Numerical methods (including Monte Carlo methods)


Zbl 1103.91371


R; BayesDA; GMRFLib
Full Text: DOI Link


[1] Abel G, Bijak J, Forster J, Raymer J, Smith P (2010) What do Bayesian methods offer population forecasters? ESRC Centre for Population Change, Working Paper, 6: http://www.cpc.ac.uk/publications/cpc_working_papers/pdf/2010_WP6_What_do_Bayesian_Methods_Offer_Population_Forecasters_Abel_et_al.pdf
[2] Barrieu, P; Bensusan, H; Karoui, NE; Hillairet, C; Loisel, S; Ravanelli, C; Salhi, Y, Understanding, modelling and managing longevity risk : key issues and main challenges, Scand Actu J, 3, 203-231, (2012) · Zbl 1277.91073
[3] Bray, I, Application of Markov chain Monte Carlo methods to projecting cancer incidence and mortality, Appl Stat, 3, 3-43, (2002)
[4] Brouhns, N; Denuit, M; Vermunt, J, A Poisson log-bilinear regression approach to the construction of projected life tables, Insur Math Eco, 31, 373-393, (2002) · Zbl 1074.62524
[5] Cairns, A; Blake, D; Dowd, K, A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration, J Risk Insur, 73, 687-718, (2006)
[6] Cairns, A; Blake, D; Dowd, K; Coughlan, G; Epstein, D; Ong, A; Balevich, I, A quantitative comparison of stochastic mortality models using data from england and wales and the united states, North Am Actuar J, 13, 1-35, (2009)
[7] Cairns, A; Blake, D; Dowd, K; Coughlan, G; Khalaf-Allah, M, Bayesian stochastic mortality modelling for two populations, ASTIN Bull, 41, 29-59, (2011)
[8] Carter, L; Lee, R, Forecasting demographic components: modeling and forecasting US sex differentials in mortality, Int J Forecast, 8, 393-411, (1992)
[9] Congdon P (2006) Bayesian statistical modelling. Wiley, Oxford · Zbl 1193.62034
[10] Czado, C; Delwarde, A; Denuit, M, Bayesian Poisson log-bilinear mortality projections, Insur Math Eco, 36, 260-284, (2005) · Zbl 1110.62142
[11] Danesi, I; Haberman, S; Millossovich, P, Forecasting mortality in subpopulations using Lee-Carter type models: a comparison, Insur Math Eco, 62, 151-161, (2015) · Zbl 1318.91109
[12] Dellaportas, P; Smith, A; Stavropoulos, P, Bayesian analysis of mortality data, J Royal Stat Soc Series A (Stat Soc), 164, 275-291, (2001) · Zbl 1002.91504
[13] Denuit, M; Goderniaux, A-C, Closing and projecting life tables using log-linear models, Bull Swiss Assoc Actuar, 1, 29-48, (2005) · Zbl 1333.62251
[14] Dowd, K; Blake, D; Cairns, A; Coughlan, G; Khalaf-Allah, M, A gravity model of mortality rates for two related populations, North Am Actua J, 15, 334-356, (2011) · Zbl 1228.91032
[15] Fushimi T, Kogure A (2014) A Bayesian approach to longevity derivative pricing under stochastic interest rates with a two-factor Lee-Carter model. ARIA 2014 Annual Meeting: http://www.aria.org/Annual_Meeting/2014/2014_Accepted_Papers/4C/Fushimi · Zbl 1348.91162
[16] Gelman A, Carlin J, Stern H, Dunson D, Vehtari A, Rubin D (2013) Bayesian data analysis. Chapman & Hall/CRC Texts in Statistical Science, New York · Zbl 1279.62004
[17] Girosi F, King G (2008) Demographic forecasting. Princeton University Press, Princeton
[18] Haberman, S; Renshaw, A, A comparative study of parametric mortality models, Insur Math Eco, 48, 35-55, (2011)
[19] Hatzopoulos, P; Haberman, S, Common mortality modeling and coherent forecast: an empirical analysis of worldwide mortality data, Insur Math Eco, 52, 320-337, (2013) · Zbl 1284.91238
[20] Hyndman, R; Booth, H; Yasmeen, F, Coherent mortality forecasting: the product-ratio method with functional time series models, Demography, 50, 261-283, (2013)
[21] Hyndman, R; Ullah, M, Robust forecasting of mortality and fertility rates: a functional data approach, Comp Stat Data Anal, 51, 4942-4956, (2007) · Zbl 1162.62434
[22] Jarner, S; Kryger, E, Modelling adult mortality in small populations: the SAINT model, ASTIN Bull, 41, 377-418, (2011) · Zbl 1239.91128
[23] Kogure, A; Yoshiyuki, K, A Bayesian approach to pricing longevity risk based on risk-neutral predictive distributions, Insur Math Eco, 46, 162-172, (2010) · Zbl 1231.91438
[24] Koninklijk Actuarieel Genootschap (2014) Prognosetafel AG2014. http://www.ag-ai.nl
[25] Lee, R; Carter, L, Modeling and forecasting the time series of US mortality, J Am Stat Assoc, 87, 659-671, (1992)
[26] Li, H; Waegenaere, AD; Melenberg, B, The choice of sample size for mortality forecasting: A Bayesian learning approach, Insur Math Eco, 63, 153-168, (2015) · Zbl 1348.91162
[27] Li, J, A Poisson common factor model for projecting mortality and life expectancy jointly for females and males, Popul Studies J Demogr, 67, 111-126, (2013)
[28] Li, J; Hardy, M, Measuring basis risk in longevity hedges, North Am Actuar J, 15, 177-200, (2011) · Zbl 1228.91042
[29] Li, N; Lee, R, Coherent mortality forecasts for a group of populations: an extension of the Lee-Carter method, Demography, 42, 575-594, (2005)
[30] Mardia K, Kent J, Bibby J (2008) Multivariate analysis. Academic Press, Amsterdam · Zbl 0432.62029
[31] Mavros G, Cairns A, Kleinow T, Streftaris G (2014) A parsimonious approach to stochastic mortality modelling with dependent residuals. Working paper: http://www.macs.hw.ac.uk/andrewc/papers/MavrosEtAl2014.pdf · Zbl 1414.91219
[32] Niu, G; Melenberg, B, Trends in mortality decrease and economic growth, Demography, 51, 1755-1773, (2014) · Zbl 1314.42025
[33] Njenga C, Sherris M (2011) Modeling mortality with a Bayesian vector autoregression. Working Paper: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1776532
[34] Pedroza, C, A Bayesian forecasting model: predicting U.S. male mortality, Biostatistics, 7, 530-550, (2006) · Zbl 1170.62397
[35] Pitacco E, Denuit M, Haberman S, Olivieri A (2009) Modeling longevity dynamics for pensions and annuity business. Oxford University Press, London · Zbl 1163.91005
[36] Plat, R, On stochastic mortality models, Insur Math Eco, 45, 393-404, (2009) · Zbl 1231.91227
[37] Raftery, A; Chunn, J; Gerland, P; Sevcikova, H, Bayesian probabilistic projections of life expectancy for all countries, Demography, 50, 777-801, (2013)
[38] Reichmuth W, Sarferaz S (2008) Bayesian demographic modelling and forecasting: An application to US mortality. SFB 649 Discussion Paper 2008-2052: http://edoc.hu-berlin.de/series/sfb-649-papers/2008-52/PDF/52.pdf · Zbl 1103.91371
[39] Renshaw, A; Haberman, S, Lee-Carter mortality forecasting with age specific enhancement, Insur Math Eco, 33, 255-272, (2003) · Zbl 1103.91371
[40] Renshaw, A; Haberman, S, A cohort-based extension to the Lee-Carter model for mortality reduction factors, Insur Math Eco, 38, 556-570, (2006) · Zbl 1168.91418
[41] Riebler, A; Held, L, The analysis of heterogeneous time trends in multivariate age-period-cohort models, Biostatistics, 11, 57-69, (2010)
[42] Riebler, A; Held, L; Rue, H, Estimation and extrapolation of time trends in registry data: borrowing strength from related populations, Ann Appl Stat, 6, 304-333, (2012) · Zbl 1235.62030
[43] Rue H, Held L (2005) Gaussian Markov Random fields: theory and applications, vol 104 of Monographs on Statistics and Applied Probability. Chapman and Hall, London · Zbl 1093.60003
[44] Stoeldraijer L, van Duin C, Janssen F (2013a) Bevolkingsprognose 2012-2060: model en veronderstellingen betreffende de sterfte. Bevolkingstrends, Juli:1-27
[45] Stoeldraijer, L; Duin, C; Wissen, L; Janssen, F, Impact of different mortality forecasting methods and explicit assumptions on projected future life expectancy: the case of The Netherlands, Demography, 29, 323-354, (2013)
[46] Van Berkum F, Antonio K, Vellekoop M (2014) The impact of multiple structural changes on mortality predictions. Scand Actuar J. doi:10.1080/03461238.2014.987807 · Zbl 1401.91221
[47] Wan C, Berschi L, Yang Y (2013) Coherent mortality forecasting for small populations: an application to Swiss mortality data. Working Paper: http://www.actuaries.org/lyon2013/papers/AFIR_Wan_Bertschi_Yang.pdf
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.