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The CBD mortality indexes: modeling and applications. (English) Zbl 1412.91037

Summary: Most extrapolative stochastic mortality models are constructed in a similar manner. Specifically, when they are fitted to historical data, one or more series of time-varying parameters are identified. By extrapolating these parameters to the future, we can obtain a forecast of death probabilities and consequently cash flows arising from life contingent liabilities. In this article, we first argue that, among various time-varying model parameters, those encompassed in the Cairns-Blake-Dowd (CBD) model (also known as Model M5) are most suitably used as indexes to indicate levels of longevity risk at different time points. We then investigate how these indexes can be jointly modeled with a more general class of multivariate time-series models, instead of a simple random walk that takes no account of cross-correlations. Finally, we study the joint prediction region for the mortality indexes. Such a region, as we demonstrate, can serve as a graphical longevity risk metric, allowing practitioners to compare the longevity risk exposures of different portfolios readily.

MSC:

91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
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[1] Bartlett, M. S., Further Aspects of the Theory of Multiple Regression, Proceedings of the Cambridge Philosophical Society, 34, 33-40, (1938) · Zbl 0018.15803
[2] Blake, D.; Cairns, A. J. G.; Dowd, K., Longevity Risk and the Grim Reaper’s Toxic Tail: The Survivor Fan Charts, Insurance: Mathematics and Economics, 42, 1062-1066, (2008) · Zbl 1141.91485
[3] Box, G. E. P.; Jenkins, G. M., Time Series Analysis: Forecasting and Control., (1976), San Francisco: Holden-Day, San Francisco · Zbl 0363.62069
[4] Cairns, A. J. G., Modelling and Management of Longevity Risk: Approximations to Survival Functions and Dynamic Hedging, Insurance: Mathematics and Economics, 49, 438-453, (2011) · Zbl 1230.91068
[5] 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, 687-718, (2006)
[6] 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, 1-35, (2009)
[7] Cairns, A. J. G.; Blake, D.; Dowd, K.; Coughlan, G. D.; Khalaf-Allah, M., Bayesian Stochastic Mortality Modelling for Two Populations, ASTIN Bulletin, 41, 29-55, (2011)
[8] Chan, W. S.; Cheung, S. H.; Wu, K. H., On Exact Joint Forecast Regions for Vector Autoregressive Models, Journal of Applied Statistics, 26, 35-44, (1999) · Zbl 1072.62638
[9] Cheshire, J., Lives on the Line: Mapping Life Expectancy along the London Tube Network, Environment and Planning A, 44, 1525-1528, (2012)
[10] Coughlan, G.; Barrieu, P. M.; Albertini, L., Longevity Risk Transfer: Indices and Capital Market Solutions, The Handbook of Insurance Linked Securities, 261-282, (2009), London: Wiley, London
[11] Dowd, K.; Cairns, A. J. G.; Blake, D.; Coughlan, G. D.; Epstein, D.; Khalaf-Allah, M., Backtesting Stochastic Mortality Models: An ExPost Evaluation of Multi-Period-Ahead Density Forecasts, North American Actuarial Journal, 14, 281-298, (2010)
[12] Dowd, K.; Blake, D.; Cairns, A. J. G., Facing Up to Uncertain Life Expectancy: The Longevity Fan Charts, Demography, 47, 67-78, (2010)
[13] University of California, Berkeley (USA), and Max Planck Institute of Demographic Research (Germany)
[14] Li, S. H.; Chan, W. S., Outlier Analysis and Mortality Forecasting: The United Kingdom and Scandinavian Countries, Scandinavian Actuarial Journal, 3, 187-211, (2005) · Zbl 1092.91050
[15] Li, S. H.; Chan, W. S., The Lee-Carter Model for Forecasting Mortality, Revisited, North American Actuarial Journal, 11, 68-89, (2007)
[16] Li, J. S.-H.; Chan, W. S.; Cheung, S. H., Structural Changes in the Lee-Carter Mortality Indexes: Detection and Implications, North American Actuarial Journal, 15, 13-31, (2011)
[17] Li, J. S.-H.; Luo, A., Key q-Duration: A Framework for Hedging Longevity Risk, ASTIN Bulletin, 42, 413-452, (2012) · Zbl 1277.91089
[18] Lütkepohl, H., Introduction to Multiple Time Series Analysis, (1991), Berlin: Springer, Berlin
[19] Reinsel, G. C., Elements of Multivariate Time Series Analysis., (1997), Berlin: Springer, Berlin · Zbl 0873.62086
[20] Sims, C. A., Macroeconomics and Reality, Econometrica, 48, 1-48, (1980)
[21] Sweeting, P. J., Longevity Indices and Pension Fund Risk, (2010)
[22] Sweeting, P. J., A Trend-Change Extension of the Cairns-Blake-Dowd Model, Annals of Actuarial Science, 5, 143-162, (2011)
[23] Tiao, G. C.; Box, G. E. P., Modelling Multiple Time Series with Applications, Journal of the American Statistical Association, 76, 802-816, (1981) · Zbl 0483.62074
[24] Tsay, R. S., Analysis of Financial Time Series., (2010), HobokenNJ: John Wiley & Sons, HobokenNJ · Zbl 1209.91004
[25] Wei, W. W. S., Time Series Analysis: Univariate and Multivariate Method., (2006), New York: addison Wesley/Pearson, New York · Zbl 1170.62362
[26] Wilson, G. T., The Estimation of Parameters in Multivariate Time Series Models, Journal of the Royal Statistical Society, B35, 76-85, (1973) · Zbl 0259.62074
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