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A neural network extension of the Lee–Carter model to multiple populations

Published online by Cambridge University Press:  28 June 2019

Ronald Richman  and
Mario V. Wüthrich
Ronald Richman*
Affiliation:
Actuarial Department, AIG South Africa, Johannesburg, Gauteng 2196, South Africa
Mario V. Wüthrich
Affiliation:
RiskLab, Department of Mathematics, ETH Zurich, 8092, Zurich, Switzerland
*
*Corresponding author. Email: ron@ronaldrichman.co.za
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Abstract

The Lee–Carter (LC) model is a basic approach to forecasting mortality rates of a single population. Although extensions of the LC model to forecasting rates for multiple populations have recently been proposed, the structure of these extended models is hard to justify and the models are often difficult to calibrate, relying on customised optimisation schemes. Based on the paradigm of representation learning, we extend the LCmodel to multiple populations using neural networks, which automatically select an optimal model structure. We fit this model to mortality rates since 1950 for all countries in the Human Mortality Database and observe that the out-of-sample forecasting performance of the model is highly competitive.

Keywords

Mortality forecastingLee-Carter modelmultiple populationsneural networks
Type
Paper
Information
Annals of Actuarial Science , Volume 15 , Issue 2 , July 2021 , pp. 346 - 366
Copyright
© Institute and Faculty of Actuaries 2019

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References

Abadi, M., Barham, P., Chen, J., Chen, Z., Davis, A., Dean, J., Devin, M., Ghemawat, S., Irving, G., Isard, M., Kudlur, M., Levenberg, J., Monga, R., Moore, S., Murray, D.G., Steiner, B., Tucker, P., Vasudevan, V., Warden, P., Wicke, M., Yu, Y. & Zhang, X. (2016). TensorFlow: a system for large-scale machine learning. In Proceedings of the 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI ’16) (pp. 265–283).Google Scholar
Allaire, J.J. & Chollet, F. (2018). R interface to Keras. RStudio, Google.Google Scholar
Bengio, Y., Courville, A. & Vincent, P. (2013). Representation learning: a review and new perspectives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(8), 1798–1828.CrossRefGoogle ScholarPubMed
Bengio, Y., Ducharme, R., Vincent, P. & Jauvin, C. (2003). A neural probabilistic language model. Journal of Machine Learning Research, 3(2), 1137–1155.Google Scholar
Brouhns, N., Denuit, M. & Vermunt, J.K. (2002). A Poisson log-bilinear regression approach to the construction of projected lifetables. Insurance: Mathematics and Economics, 31(3), 373–393.Google Scholar
Cairns, A.J.G., Blake, D. & Dowd, K. (2006). A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration. Journal of Risk & Insurance, 73(4), 687–718.CrossRefGoogle Scholar
Chen, R.Y. & Millossovich, P. (2018). Sex-specific mortality forecasting for UK countries: a coherent approach. European Actuarial Journal, 8(1), 69–95.CrossRefGoogle ScholarPubMed
Chollet, F. (2015). Keras: the Python deep learning library.Google Scholar
Currie, I.D. (2016). On fitting generalized linear and non-linear models of mortality. Scandinavian Actuarial Journal, 2016(4), 356–383.CrossRefGoogle Scholar
Danesi, I.L., Haberman, S. & Millossovich, P. (2015). Forecasting mortality in sub-populations using Lee-Carter type models: a comparison. Insurance: Mathematics and Economics, 62, 151–161.Google Scholar
Dowd, K., Cairns, A.J., Blake, D., Coughlan, G.D., Epstein, D. & Khalaf-Allah, M. (2010). Backtesting stochastic mortality models: an ex post evaluation of multiperiod-ahead density forecasts. North American Actuarial Journal, 14(3), 281–298.CrossRefGoogle Scholar
Enchev, V., Kleinow, T. & Cairns, A.J.G. (2017). Multi-population mortality models: fitting, forecasting and comparisons. Scandinavian Actuarial Journal, 2017(4), 319–342.CrossRefGoogle Scholar
Efron, B. & Hastie, T. (2016). Computer Age Statistical Inference (Vol. 5). Cambridge, United Kingdom, Cambridge University Press.CrossRefGoogle Scholar
Guo, C. & Berkhahn, F. (2015). Entity embeddings of categorical variables. arXiv, arXiv:1604.06737.Google Scholar
He, K., Zhang, X., Ren, S. & Sun, J. (2015). Deep residual learning for image recognition. CoRR, abs/1512.03385.Google Scholar
Hinton, G., Srivastava, N., Krizhevsky, A., Sutskever, I. & Salakhutdinov, R. (2012). Improving neural networks by preventing co-adaptation of feature detectors. arXiv, arXiv:1207.0580.Google Scholar
Huang, G., Liu, Z. & Weinberger, K.Q. (2016). Densely connected convolutional networks. CoRR, abs/1608.06993.Google Scholar
Hyndman, R.J., Athanasopoulos, G., Razbash, S., Schmidt, D., Zhou, Z., Khan, Y., Bergmeir, C. & Wang, E. (2015). Forecast: forecasting functions for time series and linear models. R package.Google Scholar
Ioffe, S. & Szegedy, C. (2015). Batch normalization: accelerating deep network training by reducing internal covariate shift. CoRR, abs/1502.03167.Google Scholar
Kingma, D.P. & Ba, J. (2014). Adam: a method for stochastic optimization. arXiv, arXiv:1412.6980.Google Scholar
Kleinow, T. (2015). A common age effect model for the mortality of multiple populations. Insurance: Mathematics and Economics, 63, 147–152.Google Scholar
Lee, R.D. & Carter, L.R. (1992). Modeling and forecasting US mortality. Journal of the American Statistical Association, 87(419), 659–671.Google Scholar
Li, N. & Lee, R. (2005). Coherent mortality forecasts for a group of populations: an extension of the Lee-Carter method. Demography, 42(3), 575–594.CrossRefGoogle ScholarPubMed
Mikolov, T., Sutskever, I., Chen, K., Corrado, G. & Dean, J. (2013). Distributed representations of words and phrases and their compositionality. In Conference Proceedings of Neural Information Processing Systems, Electronic Publisher. (pp. 3111–3119).Google Scholar
Nair, V. & Hinton, G. (2010). Rectified linear units improve restricted Boltzmann machines. In Proceedings of the 27th International Conference on Machine Learning (pp. 807–814).Google Scholar
Office for National Statistics, United Kingdom (2018). Changing trends in mortality: an international comparison: 2000 to 2016.Google Scholar
Richman, R. (2018). AI in actuarial science. SSRN Manuscript ID 3218082. Version July 24, 2018.CrossRefGoogle Scholar
Rumelhart, D., Hinton, G. & Williams, R. (1986). Learning representations by back-propagating errors. Nature, 323(6088), 533.CrossRefGoogle Scholar
Turner, H. & Firth, D. (2007). Generalized nonlinear models in R: an overview of the gnm package. R package.Google Scholar
Villegas, A.M., Haberman, S., Kaishev, V.K. & Millossovich, P. (2017). A comparative study of two-population models for the assessment of basis risk in longevity hedges. ASTIN Bulletin, 47(3), 631–679.CrossRefGoogle Scholar
Wilmoth, J.R. & Shkolnikov, V. (2010). Human Mortality Database. University of California.Google Scholar
Wüthrich, M.V. & Buser, C. (2016). Data analytics for non-life insurance pricing. SSRN Manuscript ID 2870308. Version February 5, 2019.Google Scholar