A Bühlmann credibility approach to modeling mortality rates. (English) Zbl 1414.91237

Summary: In this article, we first propose a Bühlmann nonparametric credibility approach to forecasting mortality rates, and we then compare forecasting performances between the proposed Bühlmann approach and the Lee-Carter/Cairns-Blake-Dowd (CBD) models. Empirical results based on mortality data for both genders of Japan, the United Kingdom, and the United States with two age spans, a wide range of fitting year spans, and three forecasting periods show that the credibility approach contributes to much better forecasting performances measured by the mean absolute percentage error. Moreover, we give an informative credibility interpretation regarding the average decrements of an individual time trend for age \(x\) and a group time trend for all ages, and we discuss the effects of the slope and intercept of the linear functions for the forecasted mortality rates under the proposed Bühlmann nonparametric credibility approach and the Lee-Carter/CBD models. Finally, we also estimate the parameters of the Bühlmann credibility approach in a semiparametric framework, and we provide a stochastic version of forecasting mortality rates for the Bühlmann credibility approach.


91B30 Risk theory, insurance (MSC2010)
62P05 Applications of statistics to actuarial sciences and financial mathematics
62M20 Inference from stochastic processes and prediction
Full Text: DOI


[1] Blake, D.; Burrows, W., Survivor Bonds: Helping to Hedge Mortality Risk, Journal of Risk and Insurance, 68, 339-348, (2001)
[2] Bühlmann, H., Experience Rating and Credibility, ASTIN Bulletin, 4, 199-207, (1967)
[3] Bühlmann, H.; Gisler, A., A Course in Credibility Theory and Its Applications, (2005), Amsterdam: Springer, Amsterdam · Zbl 1108.91001
[4] Bühlmann, H.; Straub, E., Glaubwürdigkeit für schadensätze, Bulletin of the Swiss Association of Actuaries, 70, 111-133, (1970) · Zbl 0197.46502
[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, 1-35, (2009)
[7] Cox, S. H.; Lin, Y.; Tian, R.; Zuluaga, L. F., Mortality Portfolio Risk Management, Journal of Risk and Insurance, 80, 4, 853-890, (2013)
[8] D’Amato, V.; Haberman, S.; Piscopo, G.; Russolillo, M., Modelling Dependent Data for Longevity Projections, Insurance: Mathematics and Economics, 51, 694-701, (2012) · Zbl 1285.91054
[9] Dornheim, H.; Brazauskas, V., Robust-Efficient Credibility Models with Heavy-Tailed Claims: A Mixed Linear Models Perspective, Insurance: Mathematics and Economics, 48, 72-84, (2011) · Zbl 1233.91142
[10] Dowd, K.; Blake, D.; Cairns, A. J. G.; Dawson, P., Survivor Swaps, Journal of Risk and Insurance, 73, 1-17, (2006)
[11] Frees, E. W., Multivariate Credibility for Aggregate Loss Models, North American Actuarial Journal, 7, 13-37, (2003) · Zbl 1084.62110
[12] Frees, E. W.; Wang, P., Credibility Using Copulas, North American Actuarial Journal, 9, 31-48, (2005) · Zbl 1085.62121
[13] Hachemeister, C. A.; Kahn, P. M., Credibility: Theory and Applications, Credibility for Regression Models with Application to Trend, 307-348, (1975), New York: Academic Press, New York
[14] Hári, N.; Waegenaere, A.; Melenberg, B.; Nijman, T. E., Estimating the Term Structure of Mortality, Insurance: Mathematics and Economics, 42, 492-504, (2008) · Zbl 1152.91585
[15] Jewell, W. S., Credible Means are Exact Bayesian for Exponential Families, ASTIN Bulletin, 8, 77-90, (1974)
[16] Jones, D.; Gerber, H., Credibility Formulas of the Updating Type, Transactions of the Society of Actuaries, 27, 31-52, (1975)
[17] Kaas, R.; Dannenburg, D.; Goovaerts, M., Exact Credibility for Weighted Observations, ASTIN Bulletin, 27, 287-295, (1997)
[18] Kaas, R.; Goovaerts, M.; Dhaene, J.; Denuit, M., Modern Actuarial Risk Theory—Using R, (2008), Heidelberg: Springer, Heidelberg · Zbl 1148.91027
[19] Kim, J. H. T.; Jeon, Y., Credibility Theory Based on Trimming, Insurance: Mathematics and Economics, 53, 36-47, (2013) · Zbl 1284.91245
[20] Klugman, S. A.; Panjer, H. H.; Willmot, G. E., Loss Models: From Data to Decisions, (2012), New York: John Wiley & Sons, New York · Zbl 1272.62002
[21] Landsman, Z.; Makov, U. E., Credibility Evaluation for the Exponential Dispersion Family, Insurance: Mathematics and Economics, 24, 23-29, (1999) · Zbl 0927.62109
[22] Lee, R. D.; Carter, L. R., Modelling and Forecasting U.S. Mortality, Journal of the American Statistical Association, 87, 659-671, (1992) · Zbl 1351.62186
[23] Li, J. S. H., Pricing Longevity Risk with the Parametric Bootstrap: A Maximum Entropy Approach, Insurance: Mathematics and Economics, 47, 176-186, (2010) · Zbl 1231.91441
[24] Li, J. S.; Hardy, M.; Tan, K. S., Uncertainty in Mortality Forecasting: An Extension to the Classical Lee-Carter Approach, ASTIN Bulletin, 39, 137-164, (2009) · Zbl 1203.91113
[25] Li, J. S. H.; Ng, A. C. Y., Canonical Valuation of Mortality-Linked Securities, Journal of Risk and Insurance, 78, 853-884, (2011)
[26] Lin, T.; Tsai, C. C. L., On the Mortality/Longevity Risk Hedging with Mortality Immunization, Insurance: Mathematics and Economics, 53, 3, 580-596, (2013) · Zbl 1290.91093
[27] Lin, T.; Tsai, C. C. L., Applications of Mortality Durations and Convexities in Natural Hedges, North American Actuarial Journal, 18, 3, 417-442, (2014)
[28] Lin, T.; Wang, C. W.; Tsai, C. C. L., Age Specific Copula-AR-GARCH Mortality Models, Insurance: Mathematics and Economics, 61, 110-124, (2015) · Zbl 1314.91143
[29] Longley-Cook, L., An Introduction to Credibility Theory, Proceedings of the Casualty Actuarial Society, 49, 194-221, (1962)
[30] Mitchell, D.; Brockett, P.; Mendoza-Arriaga, R.; Muthuraman, K., Modeling and Forecasting Mortality Rates, Insurance: Mathematics and Economics, 52, 275-285, (2013) · Zbl 1284.91259
[31] Mowbray, A., How Extensive a Payroll is Necessary to Give Dependable Pure Premium?, Proceedings of the Casualty Actuarial Society, 1, 24-30, (1914)
[32] Norberg, R., The Credibility Approach to Experience Rating, Scandinavian Actuarial Journal, 181-221, (1979) · Zbl 0424.62071
[33] Payandeh Najafabadi, A. T., A New Approach to Credibility Formula, Insurance: Mathematics and Economics, 46, 334-338, (2010) · Zbl 1231.91224
[34] Payandeh Najafabadi, A. T.; Hatami, H.; Najafabadi, M. O., A Maximum-Entropy Approach to the Linear Credibility Formula, Insurance: Mathematics and Economics, 51, 216-221, (2012) · Zbl 1284.91261
[35] Pitselis, G., Quantile Credibility Models, Insurance: Mathematics and Economics, 52, 477-489, (2013) · Zbl 1284.91265
[36] Plat, R., On Stochastic Mortality Modeling, Insurance: Mathematics and Economics, 45, 393-404, (2009) · Zbl 1231.91227
[37] Renshaw, A. E.; Haberman, S., A Cohort-Based Extension to the Lee-Carter Model for Mortality Reduction Factors, Insurance: Mathematics and Economics, 38, 556-570, (2006) · Zbl 1168.91418
[38] Wong, A.; Sherris, M.; Stevens, R., Natural Hedging Strategies for Life Insurers: Impact of Product Design and Risk Measure, Journal of Risk and Insurance 84: 153–175., (2017)
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