A Two‐Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration
The authors would like to thank two anonymous referees for their very helpful observations and comments. AC wishes to thank the Isaac Newton Institute in Cambridge, where he was a visitor during the preparation of this paper. This research was conducted under research grants RES‐000‐27‐0014 and RES‐000‐23‐1036 from the UK Economic and Social Research Council.
Abstract
In this article, we consider the evolution of the post‐age‐60 mortality curve in the United Kingdom and its impact on the pricing of the risk associated with aggregate mortality improvements over time: so‐called longevity risk. We introduce a two‐factor stochastic model for the development of this curve through time. The first factor affects mortality‐rate dynamics at all ages in the same way, whereas the second factor affects mortality‐rate dynamics at higher ages much more than at lower ages. The article then examines the pricing of longevity bonds with different terms to maturity referenced to different cohorts. We find that longevity risk over relatively short time horizons is very low, but at horizons in excess of ten years it begins to pick up very rapidly.
A key component of the article is the proposal and development of a method for calculating the market risk‐adjusted price of a longevity bond. The proposed adjustment includes not just an allowance for the underlying stochastic mortality, but also makes an allowance for parameter risk. We utilize the pricing information contained in the November 2004 European Investment Bank longevity bond to make inferences about the likely market prices of the risks in the model. Based on these, we investigate how future issues might be priced to ensure an absence of arbitrage between bonds with different characteristics.
Citing Literature
Number of times cited according to CrossRef: 525
- Rachel Wingenbach, Jong-Min Kim, Hojin Jung, Living longer in high longevity risk, Journal of Demographic Economics, 10.1017/dem.2019.20, 86, 1, (47-86), (2020).
- Lydia Dutton, Athanasios A. Pantelous, Malgorzata Seklecka, The impact of economic growth in mortality modelling for selected OECD countries, Journal of Forecasting, 10.1002/for.2640, 39, 3, (533-550), (2020).
- Cary Chi-Liang Tsai, Adelaide Di Wu, Incorporating hierarchical credibility theory into modelling of multi-country mortality rates, Insurance: Mathematics and Economics, 10.1016/j.insmatheco.2020.01.001, (2020).
- Ugofilippo Basellini, Søren Kjærgaard, Carlo Giovanni Camarda, An age-at-death distribution approach to forecast cohort mortality, Insurance: Mathematics and Economics, 10.1016/j.insmatheco.2020.01.007, (2020).
- Sumaira Mubarik, Fang Wang, Muhammad Fawad, Yafeng Wang, Ishfaq Ahmad, Chuanhua Yu, Trends and Projections in Breast Cancer Mortality among four Asian countries (1990–2017): Evidence from five Stochastic Mortality Models, Scientific Reports, 10.1038/s41598-020-62393-1, 10, 1, (2020).
- Selin Özen, Şule Şahin, Transitory mortality jump modeling with renewal process and its impact on pricing of catastrophic bonds, Journal of Computational and Applied Mathematics, 10.1016/j.cam.2020.112829, (112829), (2020).
- Silvia Rizzi, Søren Kjærgaard, Marie-Pier Bergeron Boucher, Carlo Giovanni Camarda, Rune Lindahl-Jacobsen, James W. Vaupel, Killing off cohorts: Forecasting mortality of non-extinct cohorts with the penalized composite link model, International Journal of Forecasting, 10.1016/j.ijforecast.2020.03.003, (2020).
- Yuantao Xie, Xinzhu Zhang, Huijuan Lv, Xiaojing Guo, The New Fertility Policy and the Actuarial Balance of China Urban Employee Basic Endowment Insurance Fund Based on Stochastic Mortality Model, Mathematical Problems in Engineering, 10.1155/2020/3596384, 2020, (1-12), (2020).
- Liqun Diao, Yechao Meng, Chengguo Weng, A DSA Algorithm for Mortality Forecasting, SSRN Electronic Journal, 10.2139/ssrn.3537368, (2020).
- S⊘ren Kjærgaard, Yunus Emre Ergemen, Marie‐Pier Bergeron‐Boucher, Jim Oeppen, Malene Kallestrup‐Lamb, Longevity forecasting by socio‐economic groups using compositional data analysis, Journal of the Royal Statistical Society: Series A (Statistics in Society), 10.1111/rssa.12555, 183, 3, (1167-1187), (2020).
- Johnny Siu-Hang Li, Yanxin Liu, The heat wave model for constructing two-dimensional mortality improvement scales with measures of uncertainty, Insurance: Mathematics and Economics, 10.1016/j.insmatheco.2020.04.001, (2020).
- Fei Huang, Ross Maller, Xu Ning, Modelling life tables with advanced ages: An extreme value theory approach, Insurance: Mathematics and Economics, 10.1016/j.insmatheco.2020.04.004, (2020).
- Apostolos Bozikas, Georgios Pitselis, Incorporating crossed classification credibility into the Lee–Carter model for multi-population mortality data, Insurance: Mathematics and Economics, 10.1016/j.insmatheco.2020.06.005, 93, (353-368), (2020).
- Marie Gaille, Marco Araneda, Clément Dubost, Clémence Guillermain, Sarah Kaakai, Elise Ricadat, Nicolas Todd, Michael Rera, Ethical and social implications of approaching death prediction in humans - when the biology of ageing meets existential issues, BMC Medical Ethics, 10.1186/s12910-020-00502-5, 21, 1, (2020).
- Kyran Cupido, Petar Jevtić, Antonio Paez, Spatial patterns of mortality in the United States: A spatial filtering approach, Insurance: Mathematics and Economics, 10.1016/j.insmatheco.2020.08.003, (2020).
- Rebecca Graziani, Stochastic Population Forecasting: A Bayesian Approach Based on Evaluation by Experts, Developments in Demographic Forecasting, 10.1007/978-3-030-42472-5_2, (21-42), (2020).
- PENGJIE WANG, Athanasios A. Pantelous, Farshid Vahid, Multi-population Mortality Projection: The Augmented Common Factor Model with Structural Breaks, SSRN Electronic Journal, 10.2139/ssrn.3614333, (2020).
- Tzuling Lin, Cary Chi-liang Tsai, NATURAL HEDGES WITH IMMUNIZATION STRATEGIES OF MORTALITY AND INTEREST RATES, ASTIN Bulletin, 10.1017/asb.2019.38, (1-31), (2020).
- Pintao Lyu, Anja De Waegenaere, Bertrand Melenberg, A Multi-population Approach to Forecasting All-Cause Mortality Using Cause-of-Death Mortality Data, North American Actuarial Journal, 10.1080/10920277.2019.1662316, (1-36), (2020).
- Alexandre Boumezoued, Improving HMD Mortality Estimates with HFD Fertility Data, North American Actuarial Journal, 10.1080/10920277.2019.1672567, (1-25), (2020).
- Ming-Hua Hsieh, Chenghsien Jason Tsai, Jennifer L. Wang, Mortality Risk Management Under the Factor Copula Framework—With Applications to Insurance Policy Pools, North American Actuarial Journal, 10.1080/10920277.2019.1653201, (1-13), (2020).
- Jie Wen, Torsten Kleinow, Andrew J. G. Cairns, Trends in Canadian Mortality by Pension Level: Evidence from the CPP and QPP, North American Actuarial Journal, 10.1080/10920277.2019.1679190, (1-29), (2020).
- Andrew Hunt, David Blake, Forward Mortality Rates in Discrete Time I: Calibration and Securities Pricing, North American Actuarial Journal, 10.1080/10920277.2019.1649159, (1-26), (2020).
- Michael Sherris, Pengyu Wei, A Multi-state Model of Functional Disability and Health Status in the Presence of Systematic Trend and Uncertainty, North American Actuarial Journal, 10.1080/10920277.2019.1708755, (1-23), (2020).
- Andrew Hunt, David Blake, On the Structure and Classification of Mortality Models, North American Actuarial Journal, 10.1080/10920277.2019.1649156, (1-20), (2020).
- Mercedes Ayuso, Jorge M. Bravo, Robert Holzmann, Getting life expectancy estimates right for pension policy: period versus cohort approach, Journal of Pension Economics and Finance, 10.1017/S1474747220000050, (1-20), (2020).
- Valeria D’Amato, Susanna Levantesi, Massimiliano Menzietti, De-risking long-term care insurance, Soft Computing, 10.1007/s00500-019-04658-0, (2020).
- Johnny S.-H. Li, Rui Zhou, Yanxin Liu, George Graziani, R. Dale Hall, Jennifer Haid, Andrew Peterson, Laurence Pinzur, Drivers of Mortality Dynamics: Identifying Age/Period/Cohort Components of Historical U.S. Mortality Improvements, North American Actuarial Journal, 10.1080/10920277.2020.1716808, (1-23), (2020).
- Le Chang, Yanlin Shi, Dynamic modelling and coherent forecasting of mortality rates: a time-varying coefficient spatial-temporal autoregressive approach, Scandinavian Actuarial Journal, 10.1080/03461238.2020.1773523, (1-21), (2020).
- Han Lin Shang, Steven Haberman, Retiree Mortality Forecasting: A Partial Age-Range or a Full Age-Range Model?, Risks, 10.3390/risks8030069, 8, 3, (69), (2020).
- Fadoua Zeddouk, Pierre Devolder, Mean reversion in stochastic mortality: why and how?, European Actuarial Journal, 10.1007/s13385-020-00237-y, (2020).
- Jie Wen, Andrew J.G. Cairns, Torsten Kleinow, Fitting multi-population mortality models to socio-economic groups, Annals of Actuarial Science, 10.1017/S1748499520000184, (1-29), (2020).
- Jr-Wei Huang, Sharon S. Yang, Chuang-Chang Chang, Modeling Housing Price Dynamics and their Impact on the Cost of no-Negative-Equity-Guarantees for Equity Releasing Products, The Journal of Real Estate Finance and Economics, 10.1007/s11146-020-09776-3, (2020).
- Jackie Li, Uditha Balasooriya, Jia Liu, Using hierarchical Archimedean copulas for modelling mortality dependence and pricing mortality-linked securities, Annals of Actuarial Science, 10.1017/S1748499520000251, (1-14), (2020).
- Ronald Richman, AI in actuarial science – a review of recent advances – part 1, Annals of Actuarial Science, 10.1017/S1748499520000238, (1-23), (2020).
- Ronald Richman, AI in actuarial science – a review of recent advances – part 2, Annals of Actuarial Science, 10.1017/S174849952000024X, (1-29), (2020).
- David Blake, Andrew J. G. Cairns, Longevity risk and capital markets: the 2018–19 update, Annals of Actuarial Science, 10.1017/S1748499520000202, (1-43), (2020).
- Kenneth Q. Zhou, Johnny Siu-Hang Li, Asymmetry in mortality volatility and its implications on index-based longevity hedging, Annals of Actuarial Science, 10.1017/S174849952000010X, (1-24), (2020).
- Andrew Hunt, David Blake, Identifiability in age/period mortality models, Annals of Actuarial Science, 10.1017/S1748499520000111, (1-39), (2020).
- Iain D. Currie, Constraints, the identifiability problem and the forecasting of mortality, Annals of Actuarial Science, 10.1017/S1748499520000020, (1-30), (2020).
- Irina Gemmo, Ralph Rogalla, Jan-Hendrik Weinert, Optimal portfolio choice with tontines under systematic longevity risk, Annals of Actuarial Science, 10.1017/S1748499520000214, (1-14), (2020).
- Kevin Dowd, Andrew J. G. Cairns, David Blake, CBDX: a workhorse mortality model from the Cairns–Blake–Dowd family, Annals of Actuarial Science, 10.1017/S1748499520000159, (1-16), (2020).
- Annamaria Olivieri, Ermanno Pitacco, Linking annuity benefits to the longevity experience: alternative solutions, Annals of Actuarial Science, 10.1017/S1748499519000137, (1-22), (2020).
- Stephen J. Richards, Iain D. Currie, Torsten Kleinow, Gavin P. Ritchie, Longevity trend risk over limited time horizons, Annals of Actuarial Science, 10.1017/S174849952000007X, (1-16), (2020).
- Fabrice Balland, Alexandre Boumezoued, Laurent Devineau, Marine Habart, Tom Popa, Mortality data reliability in an internal model, Annals of Actuarial Science, 10.1017/S1748499520000081, (1-25), (2020).
- Miguel Santolino, The Lee-Carter quantile mortality model, Scandinavian Actuarial Journal, 10.1080/03461238.2019.1707109, (1-20), (2020).
- Michael Sherris, Yajing Xu, Jonathan Ziveyi, Cohort and value-based multi-country longevity risk management, Scandinavian Actuarial Journal, 10.1080/03461238.2019.1711450, (1-27), (2020).
- Peter Hatzopoulos, Aliki Sagianou, Introducing and Evaluating a New Multiple-Component Stochastic Mortality Model, North American Actuarial Journal, 10.1080/10920277.2019.1658606, (1-53), (2020).
- David Atance, Ana Debón, Eliseo Navarro, A Comparison of Forecasting Mortality Models Using Resampling Methods, Mathematics, 10.3390/math8091550, 8, 9, (1550), (2020).
- Andrea Nigri, Susanna Levantesi, Mario Marino, Life expectancy and lifespan disparity forecasting: a long short-term memory approach, Scandinavian Actuarial Journal, 10.1080/03461238.2020.1814855, (1-24), (2020).
- Erengul Dodd, Jonathan J. Forster, Jakub Bijak, Peter W. F. Smith, Stochastic modelling and projection of mortality improvements using a hybrid parametric/semi-parametric age–period–cohort model, Scandinavian Actuarial Journal, 10.1080/03461238.2020.1815238, (1-22), (2020).
- Yumo Dong, Fei Huang, Honglin Yu, Steven Haberman, Multi-population mortality forecasting using tensor decomposition, Scandinavian Actuarial Journal, 10.1080/03461238.2020.1740314, (1-22), (2020).
- Donatien Hainaut, Michel Denuit, WAVELET-BASED FEATURE EXTRACTION FOR MORTALITY PROJECTION, ASTIN Bulletin, 10.1017/asb.2020.18, (1-33), (2020).
- Uditha Balasooriya, Johnny Siu-Hang Li, Jackie Li, The Impact of Model Uncertainty on Index-Based Longevity Hedging and Measurement of Longevity Basis Risk, Risks, 10.3390/risks8030080, 8, 3, (80), (2020).
- David Atance, Alejandro Balbás, Eliseo Navarro, Constructing dynamic life tables with a single-factor model, Decisions in Economics and Finance, 10.1007/s10203-020-00308-5, (2020).
- Yajing Xu, Michael Sherris, Jonathan Ziveyi, Market Price of Longevity Risk for a Multi‐Cohort Mortality Model With Application to Longevity Bond Option Pricing, Journal of Risk and Insurance, 10.1111/jori.12273, 87, 3, (571-595), (2019).
- S. J. Richards, I. D. Currie, T. Kleinow, G. P. Ritchie, A stochastic implementation of the APCI model for mortality projections, British Actuarial Journal, 10.1017/S1357321718000260, 24, (2019).
- Kyran Cupido, Petar Jevtic, Antonio Paez, Spatial Patterns of Mortality in the United States: A Spatial Filtering Approach, SSRN Electronic Journal, 10.2139/ssrn.3359353, (2019).
- Ronald Richman, Annuities, Encyclopedia of Gerontology and Population Aging, 10.1007/978-3-319-69892-2, (1-8), (2019).
- Marco Bacci, Amedeo Caflisch, Andreas Vitalis, On the removal of initial state bias from simulation data, The Journal of Chemical Physics, 10.1063/1.5063556, 150, 10, (104105), (2019).
- Jennifer Alonso-García, Michael Sherris, One size fits all? Drawdown structures in Australia and The Netherlands, The Journal of the Economics of Ageing, 10.1016/j.jeoa.2018.07.002, 13, (14-27), (2019).
- Johnny Siu‐Hang Li, Kenneth Q. Zhou, Xiaobai Zhu, Wai‐Sum Chan, Felix Wai‐Hon Chan, A Bayesian approach to developing a stochastic mortality model for China, Journal of the Royal Statistical Society: Series A (Statistics in Society), 10.1111/rssa.12473, 182, 4, (1523-1560), (2019).
- Søren F. Jarner, Snorre Jallbjørn, Pitfalls and merits of cointegration-based mortality models, Insurance: Mathematics and Economics, 10.1016/j.insmatheco.2019.10.005, (2019).
- Kevin Dowd, Dean Buckner, David Blake, John Fry, The valuation of no-negative equity guarantees and equity release mortgages, Economics Letters, 10.1016/j.econlet.2019.108669, (108669), (2019).
- undefined Danardono, R S Utami, R Fitriani, undefined Zulaela, S H Kartiko, Semiparametric approaches for modelling mortality with application to university employee data, Journal of Physics: Conference Series, 10.1088/1742-6596/1306/1/012050, 1306, (012050), (2019).
- Sarah Krömer, Wolfgang Stummer, A New Toolkit for Mortality Data Analytics, Stochastic Models, Statistics and Their Applications, 10.1007/978-3-030-28665-1_30, (393-407), (2019).
- Quentin Guibert, Olivier Lopez, Pierrick Piette, Forecasting mortality rate improvements with a high-dimensional VAR, Insurance: Mathematics and Economics, 10.1016/j.insmatheco.2019.07.004, 88, (255-272), (2019).
- Ankush Agarwal, Christian-Oliver Ewald, Yongjie Wang, Hedging Longevity Risk in Defined Contribution Pension Schemes, SSRN Electronic Journal, 10.2139/ssrn.3539724, (2019).
- Andrew J. G. Cairns, Malene Kallestrup Lamb, Carsten Rosenskjold, David P. Blake, Kevin Dowd, Modelling Socio-Economic Differences in Mortality Using a New Affl uence Index, SSRN Electronic Journal, 10.2139/ssrn.3376527, (2019).
- Hong Li, Ken Seng Tan, Shripad Tuljapurkar, Wenjun Zhu, Gompertz Law Revisited: Forecasting Mortality with a Multi-factor Exponential Model, SSRN Electronic Journal, 10.2139/ssrn.3495369, (2019).
- Koon-Shing Kwong, Wai-Sum Chan, Johnny Siu-Hang Li, Actuarial Modeling and Analysis of the Hong Kong Life Annuity Scheme, Asia-Pacific Journal of Risk and Insurance, 10.1515/apjri-2018-0013, 0, 0, (2019).
- Franca Glenzer, Bertrand Achou, Annuities, long-term care insurance, and insurer solvency, The Geneva Papers on Risk and Insurance - Issues and Practice, 10.1057/s41288-019-00125-x, 44, 2, (252-276), (2019).
- D. Blake, A. J. G. Cairns, K. Dowd, A. R. Kessler, Still living with mortality: the longevity risk transfer market after one decade, British Actuarial Journal, 10.1017/S1357321718000314, 24, (2019).
- Man Fung, Katja Ignatieva, Michael Sherris, Managing Systematic Mortality Risk in Life Annuities: An Application of Longevity Derivatives, Risks, 10.3390/risks7010002, 7, 1, (2), (2019).
- Cary Chi-Liang Tsai, Ying Zhang, A multi-dimensional Bühlmann credibility approach to modeling multi-population mortality rates, Scandinavian Actuarial Journal, 10.1080/03461238.2018.1563911, (1-26), (2019).
- Yige Wang, Nan Zhang, Zhuo Jin, Tin Long Ho, Pricing longevity-linked derivatives using a stochastic mortality model, Communications in Statistics - Theory and Methods, 10.1080/03610926.2018.1563171, (1-20), (2019).
- Ugofilippo Basellini, Carlo Giovanni Camarda, Modelling and forecasting adult age-at-death distributions, Population Studies, 10.1080/00324728.2018.1545918, (1-20), (2019).
- Mariarosaria Coppola, Maria Russolillo, Rosaria Simone, An Indexation Mechanism for Retirement Age: Analysis of the Gender Gap, Risks, 10.3390/risks7010021, 7, 1, (21), (2019).
- Apostolos Bozikas, Georgios Pitselis, Credible Regression Approaches to Forecast Mortality for Populations with Limited Data, Risks, 10.3390/risks7010027, 7, 1, (27), (2019).
- Anthony Medford, James W. Vaupel, An introduction to gevistic regression mortality models, Scandinavian Actuarial Journal, 10.1080/03461238.2019.1586756, (1-17), (2019).
- Fabio Gómez, Jaime Londoño, Andrés Villegas, Valor Presente de las Pensiones en el Régimen de Prima Media de Colombia, Cuadernos de Economía, 10.15446/cuad.econ.v38n76.60002, 38, 76, (173-206), (2019).
- Fadoua Zeddouk, Pierre Devolder, Pricing of Longevity Derivatives and Cost of Capital, Risks, 10.3390/risks7020041, 7, 2, (41), (2019).
- Andrew J.G. Cairns, Malene Kallestrup-Lamb, Carsten Rosenskjold, David Blake, Kevin Dowd, MODELLING SOCIO-ECONOMIC DIFFERENCES IN MORTALITY USING A NEW AFFLUENCE INDEX, ASTIN Bulletin, 10.1017/asb.2019.14, (1-36), (2019).
- Ronald Richman, Mario V. Wüthrich, A neural network extension of the Lee–Cartermodel to multiple populations, Annals of Actuarial Science, 10.1017/S1748499519000071, (1-21), (2019).
- Han Li, Colin O’Hare, Mortality Forecasting: How Far Back Should We Look in Time?, Risks, 10.3390/risks7010022, 7, 1, (22), (2019).
- Jean-François Bégin, ECONOMIC SCENARIO GENERATOR AND PARAMETER UNCERTAINTY: A BAYESIAN APPROACH, ASTIN Bulletin, 10.1017/asb.2019.6, (1-38), (2019).
- Zhou Rui, MODELLING MORTALITY DEPENDENCE WITH REGIME-SWITCHING COPULAS, ASTIN Bulletin, 10.1017/asb.2019.10, (1-35), (2019).
- Farid Flici, Frédéric Planchet, Experience Prospective Life-Tables for the Algerian Retirees, Risks, 10.3390/risks7020038, 7, 2, (38), (2019).
- Kevin Dowd, Andrew J. G. Cairns, David Blake, Hedging Annuity Risks with the Age-Period-Cohort Two-Population Gravity Model, North American Actuarial Journal, 10.1080/10920277.2019.1652102, (1-12), (2019).
- Ralph Rogalla, Optimal Portfolio Choice in Retirement With Participating Life Annuities, North American Actuarial Journal, 10.1080/10920277.2019.1650284, (1-14), (2019).
- Karen C. Su, Jack C. Yue, A Synthesis Mortality Model for the Elderly, North American Actuarial Journal, 10.1080/10920277.2019.1651659, (1-25), (2019).
- Marie-Pier Bergeron-Boucher, Søren Kjærgaard, Jim Oeppen, James W. Vaupel, The impact of the choice of life table statistics when forecasting mortality, Demographic Research, 10.4054/DemRes.2019.41.43, 41, (1235-1268), (2019).
- Jack C. Yue, Hsin-Chung Wang, Tzu-Yu Wang, Using Graduation to Modify the Estimation of Lee–Carter Model for Small Populations, North American Actuarial Journal, 10.1080/10920277.2019.1650288, (1-11), (2019).
- David McCarthy, Po-Lin Wang, An Analysis of Period and Cohort Mortality Shocks in International Data, North American Actuarial Journal, 10.1080/10920277.2019.1650287, (1-25), (2019).
- Kenneth Q. Zhou, Johnny Siu-Hang Li, Longevity Greeks: What Do Insurers and Capital Market Investors Need to Know?, North American Actuarial Journal, 10.1080/10920277.2019.1650283, (1-31), (2019).
- Johnny Siu-Hang Li, Jackie Li, Uditha Balasooriya, Kenneth Q. Zhou, Constructing Out-of-the-Money Longevity Hedges Using Parametric Mortality Indexes, North American Actuarial Journal, 10.1080/10920277.2019.1650285, (1-32), (2019).
- Atsuyuki Kogure, Takahiro Fushimi, Shinichi Kamiya, Mortality Forecasts for Long-Term Care Subpopulations with Longevity Risk: A Bayesian Approach, North American Actuarial Journal, 10.1080/10920277.2019.1653202, (1-11), (2019).
- Yanxin Liu, Johnny Siu-Hang Li, An Efficient Method for Mitigating Longevity Value-at-Risk, North American Actuarial Journal, 10.1080/10920277.2019.1658607, (1-32), (2019).
- Tzuling Lin, Cary Chi-Liang Tsai, Hedging Mortality/Longevity Risks for Multiple Years, North American Actuarial Journal, 10.1080/10920277.2019.1625789, (1-23), (2019).
- Stephen J. Richards, A Hermite-spline model of post-retirement mortality, Scandinavian Actuarial Journal, 10.1080/03461238.2019.1642239, (1-18), (2019).
- See more
