Improving the forecast of longevity by combining models. (English) Zbl 1410.91253

Summary: Mortality is a dynamic process whose future evolution over time poses important challenges for life insurance, pension funds, public policy, and fiscal planning. In this paper, we propose two contributions: (1) a new dynamic corrective methodology of the predictive accuracy of the existing mortality projection models, by modeling a measure of their fitting errors as a Cox-Ingersoll-Ross process and; (2) various out-of-sample validation methods. Besides the usual static method, we develop a dynamic one allowing us to catch the change in behavior of the underlying data. For our numerical application, we choose the Cairns-Blake-Dowd (or M5) model. Using the Italian and French females mortality data and implementing the backtesting procedure, we empirically test the ex-post forecasting performance of the CBD model both for itself (CBD) and corrected by the CIR process (mCBD). We focus on age 65, but we show results for a wide range of ages, also much younger, and for cohort data. On the basis of average measures of forecasting errors and information criteria, we show that the mCBD model is parsimonious and provides better results in terms of predictive accuracy than the CBD model itself.


91B30 Risk theory, insurance (MSC2010)
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[1] Alho, J. M.; Spencer., B. D., Error model for Official Mortality Forecasts, Journal of the American Statistical Association, 85, 411, 609-16, (1990)
[2] (2013)
[3] Booth, H.; Tickle., L., Mortality modelling and forecasting: A review of methods, Annals of Actuarial Science, 3, 1-2, 3-43, (2008)
[4] Box, G. E. P., Science and statistics, Journal of the American Statistical Association, 71, 791-99, (1976) · Zbl 0335.62002
[5] Burnham, K. P.; Anderson., D. R., Multimodel inference: Understanding AIC and BIC in model selection, Sociological Methods and Research, 33, 2, 261-304, (2004)
[6] Cairns, A. J. G.; Blake, D.; Dowd, K., A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration, Journal of Risk and Insurance, 73, 4, 355-67, (2006)
[7] Cairns, A. J. G.; Blake, D.; Dowd, K.; Coughlan, G. D.; Epstein, D.; Ong, A.; Balevich., I., 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, (2009)
[8] Cox, J. C.; Ingersoll, J. E.; Ross., S. A., A theory of the term structure of interest rates, Econometrica, 53, 2, 385-407, (1985) · Zbl 1274.91447
[9] Currie, I. D., On fitting generalized linear and non-linear models of mortality, Scandinavian Actuarial Journal, 2016, 4, 356-83, (2016) · Zbl 1401.91123
[10] Dacorogna, M.; Kratz, M., Living in a stochastic world and managing complex risks, (2015)
[11] Debón, A.; Martínez-Ruiz, F.; Montes., F., A geostatistical approach for dynamic life tables: The effect of mortality on remaining lifetime and annuities, Insurance: Mathematics and Economics, 47, 327-36, (2010) · Zbl 1231.91173
[12] Deprez, P.; Shevchenko, P. V.; Wüthrich., M. V., Machine learning techniques for mortality modeling, European Actuarial Journal, 7, 337-52, (2017) · Zbl 1405.91254
[13] Di Lorenzo, E.; Sibillo, M.; Tessitore., G., A stochastic proportional hazard model for the force of mortality, Journal of Forecasting, 25, 529-36, (2006)
[14] Dowd, K.; Cairns, A. J. G.; Blake, D.; Coughlan, G. D.; Epstein, D.; Khalaf-Allah., M., Backtesting stochastic mortality models: An ex post evaluation of multiperiod-ahead density forecasts, North American Actuarial Journal, 14, 3, 281-98, (2013)
[15] Engle, R. F.; Granger., C. W. J., Co-integration and error correction: Representation, estimation and testing, Econometrica, 55, 2, 251-76, (1987) · Zbl 0613.62140
[16] (2012)
[17] Monte Carlo methods in financial engineering, (2003), New York, NY: Springer, New York, NY
[18] Glei, D. A., (2015)
[19] Haberman, S.; Renshaw., A., A comparative study of parametric mortality projection models, Insurance: Mathematics and Economics, 48, 35-55, (2011)
[20] (2016)
[21] Kass, R. E.; Raftery., A. E., Bayes factors, Journal of the American Statistical Association, 90, 430, 773-95, (1995) · Zbl 0846.62028
[22] Kladivko, K., (2012)
[23] Meese, R. A.; Rogoff., K., Empirical exchange rate models of the seventies: Do they fit out, of sample? Journal of International Economics, 14, 3-24, (1983)
[24] Olivieri, A.; Pitacco, E., Introduction to insurance mathematics: Technical and financial features of risk transfers, (2010), Switzerland: Springer, Switzerland
[25] Park, Y.; Choi, J. W.; Kim., H.-Y., Forecasting cause-age specific mortality using two random processes, Journal of the American Statistical Association, 101, 474, 472-83, (2006) · Zbl 1119.62377
[26] Pitacco, E., Matematica e tecnica attuariale delle assicurazioni sulla durata di vita, (2010), Italy: Lint, Italy
[27] Pitacco, E.; Denuit, M.; Haberman, S.; Olivieri, A., Modelling longevity dynamics for pensions and annuity business, (2009), New York, NY: Oxford University Press, New York, NY · Zbl 1163.91005
[28] Raftery, A. E., Bayes factors and BIC — Comment on a critique of the Bayesian information criterion for model selection, Sociological Methods and Research, 27, 3, 411-27, (1999)
[29] (2002)
[30] Villegas, A. M.; Millossovich, P.; Kaishev, V. K., (2016)
[31] Villegas, A. M.; Millossovich, P.; Kaishev, V. K., (2016)
[32] Weakliem, D. L., A critique of the Bayesian information criterion for model selection, Sociological Methods and Research, 27, 359-97, (1999)
[33] Willmott, C. J.; Matsuura., K., Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance, Climate Research, 30, 1, 79-82, (2005)
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