Increased life expectancy in developed countries has led researchers to pay more attention to mortality projection to anticipate changes in mortality rates. Following the scheme proposed in Deprez et al. (Eur Actuar J 7(2):337–352, 2017) and extended by Levantesi and Pizzorusso (Risks 7(1):26, 2019), we propose a novel approach based on the combination of random forest and two-dimensional P-spline, allowing for accurate mortality forecasting. This approach firstly provides a diagnosis of the limits of the Lee–Carter mortality model through the application of the random forest estimator to the ratio between the observed deaths and their estimated values given by a certain model, while the two-dimensional P-spline are used to smooth and project the random forest estimator in the forecasting phase. Further considerations are devoted to assessing the demographic consistency of the results. The model accuracy is evaluated by an out-of-sample test. Finally, we analyze the impact of our model on the pricing of q-forward contracts. All the analyses have been carried out on several countries by using data from the Human Mortality Database and considering the Lee–Carter model.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Barrieu PM, Veraart LAM (2016) Pricing q-forward contracts: an evaluation of estimation window and pricing method under different mortality models. Scand Actuar J 2:146–166
Breiman L (1996) Bagging predictors. Mach Learn 24(2):123–140
Breiman L (2001) Random forests. Mach Learn 45(1):5–32
Breiman L, Friedman J, Stone CJ, Olshen RA (1984) Classification and regression trees. Chapman and Hall/CRC
Brouhns N, Denuit M, Vermunt JK (2002) A Poisson log-bilinear regression approach to the construction of projected lifetables. Insur Math Econ 31(3):373–393
Cairns AJG, Blake D, Dowd K (2008) Modelling and management of mortality risk: a review. Scand Actuar J 2–3:79–113 Pensions Institute Discussion Paper No. PI-0814
Camarda CG (2012) MortalitySmooth: an R package for smoothing Poisson counts with P-splines. J Stat Softw 50(1):1–24. http://cran.r-project.org/package=MortalitySmooth
Currie ID, Durban M (2002) Flexible smoothing with P-splines: a unified approach. Stat Model 2:333–49
Currie ID, Durban M, Eilers PHC (2004) Smoothing and forecasting mortality rates. Stat Model 4:279–298
Currie ID, Durban M, Eilers PHC (2006) Generalized linear array models with applications to multidimensional smoothing. J R Stat Soc B 68:259–280
D’Amato V, Piscopo G, Russolillo M (2011) The mortality of the Italian population: smoothing techniques on the Lee–Carter model. Ann Appl Stat 5(2A):705–724
De Boor C (1978) A Practical guide to splines. Springer, New York
Deprez P, Shevchenko PV, Wüthrich M (2017) Machine learning techniques for mortality modeling. Eur Actuar J 7(2):337–352
Eilers PHC, Marx BD (1996) Flexible smoothing with b-splines and penalties. Stat Sci 11:89–102
Eilers PHC, Marx BD (2002) Multivariate calibration with temperature interaction using two-dimensional penalized signal regression. Chemom Intell Lab Syst 66:159–174
Eilers PHC, Currie ID, Durban M (2006) Fast and compact smoothing on large multidimensional grids. Comput Stat Data Anal 50:61–76
Eilers PHC, Marx BD (2010) Splines, knots, and penalties. Wiley Interdiscip Rev Comput Stat 2:637–653
Girosi F, King G (2008) Demographic forecasting. Princeton University Press, Princeton
Hainaut D (2018) A neural-network analyzer for mortality forecast. Astin Bull 48(2):481–508
James G, Witten D, Hastie T, Tibshirani R (2017) An introduction to statistical learning: with applications in R. Springer texts in statistics. Springer, Berlin. ISBN-10: 1461471370
Lee RD, Carter RL (1992) Modeling and forecasting US mortality. J Am Stat Assoc 87(419):659–671
Lee R, Miller T (2001) Evaluating the performance of the Lee–Carter method for forecasting mortality. Demography 38(4):537–549
Levantesi S, Menzietti M (2017) Maximum market price of longevity risk under solvency regimes: the case of solvency II. Risks 5(2):29
Levantesi S, Pizzorusso V (2019) Application of machine learning to mortality modeling and forecasting. Risks 7(1):26
Li N, Lee R, Gerland P (2013) Extending the Lee–Carter method to model the rotation of age patterns of mortality decline for long-term projections. Demography 50(6):2037–2051
Liaw A (2018) Package randomforest. https://cran.r-project.org/web/packages/randomForest/randomForest.pdf
Loeys J, Panigirtzoglou N, Ribeiro R (2007) Longevity: a market in the making. J.P. Morgan’s Global Market Strategy, London
Loh WY (2011) Classification and regression trees. Wiley Interdiscip Rev Data Min Knowl Discov 1:14–23
Morgan JN, Sonquist JA (1963) Problems in the analysis of survey data, and a proposal. J Am Stat Assoc 58:415–434
Nigri A, Levantesi S, Marino M, Scognamiglio S, Perla F (2019) A deep learning integrated Lee–Carter model. Risks 7(1):33
O’Sullivan F (1986) A statistical perspective on ill-posed inverse problems (with discussion). Stat Sci 1:505–527
O’Sullivan F (1988) Fast computation of fully automated logdensity and log-hazard estimators. SIAM J Sci Stat Comput 9:363–379
Piscopo G (2017) Dynamic evolving neuro-fuzzy inference system for mortality prediction. Int J Eng Res Appl 7(3):26–29
Piscopo G (2018a) AR dynamic evolving neuro-fuzzy inference system for mortality data. In: Skiadas CH, Skiadas C (eds) Demography and health issues. Population aging, mortality and data analysis. Springer, Berlin
Piscopo G (2018b) A comparative analysis of neuro fuzzy inference systems for mortality prediction. In: Corazza M, Durbán M, Grané A, Perna C, Sibillo M (eds) Mathematical and statistical methods for actuarial sciences and finance. Springer, Berlin
Quinlan JR (1986) Induction of decision trees. Mach Learn 1:81–106
Richman R, Wüthrich M (2018) A neural network extension of the Lee–Carter model to multiple populations. SSRN manuscript, ID 3270877
Ruppert D (2002) Selecting the number of knots for penalized splines. J Comput Graph Stat 11:735–57
Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464
The Life and Longevity Markets Association (2010) Technical note: q-forward. http://www.llma.org
Villegas AM, Kaishev VK, Millossovich P (2015) Stmomo: an r package for stochastic mortality modelling. J Stat Softw 84(3). https://cran.r-project.org/web/packages/StMoMo/vignettes/StMoMoVignette.pdf
Zeddouk F, Devolder P (2019) Pricing of longevity derivatives and cost of capital. Risks 7:41
Conflict of interest
The authors declare that they have no conflict of interest.
This article does not contain any studies with human participants performed by any of the authors.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by Philippe de Peretti.
About this article
Cite this article
Levantesi, S., Nigri, A. A random forest algorithm to improve the Lee–Carter mortality forecasting: impact on q-forward. Soft Comput 24, 8553–8567 (2020). https://doi.org/10.1007/s00500-019-04427-z
- Machine learning
- Two-dimensional P-spline