Using parametric bootstrap to introduce and manage uncertainty: replicated loaded insurance life tables. (English) Zbl 1426.91231

Summary: Insurance companies develop loaded life tables to protect themselves against deviations, for example, in the number of expected deaths or in the (residual) expectation of life of their insured. In doing so, however, the single random vector of experience crude death rates from which loaded tables are constructed is treated as deterministic or, at best, as a single realization of an underlying stochastic process, omitting the fact that it is estimated and subject to error and uncertainty. This can result in serious consequences for the insurer. To solve this problem, we follow the example of other researchers and propose a method to replicate loaded life tables using parametric bootstrap. We focus on estimating period-loaded life tables from company portfolios, where the sizes of the exposed-to-risk populations are significantly smaller than those of general populations. If we have a set of \(B\) loaded life tables, the average behavior and some extreme values can be computed and subsequently used in managing premiums or reserves. This article offers life insurers a simple way of incorporating the experience uncertainty in actuarial tasks (for example, in pricing) by comparing the limits of the confidence intervals obtained between parametric bootstrap and classical approaches (such as limit theorems).


91G05 Actuarial mathematics
62P05 Applications of statistics to actuarial sciences and financial mathematics
62F40 Bootstrap, jackknife and other resampling methods
Full Text: DOI


[1] American Academy of Actuaries, Report of the American Academy of Actuaries’ Commissioners Standard Ordinary Task Force, paper presented at the National Association of Insurance Commissioners’ Life and Health Actuarial Task Force (2002)
[2] American Academy of Actuaries, Report on the 2017 CSO and 2017 CSO Preferred Structure Table Development (2015)
[3] American Academy of Actuaries, The Pension Protection Act: Successes, shortcomings, and opportunities for improvement (2018)
[4] Ayuso, M.; Corrales, H.; Guillén, M.; Pérez‐Marín, A. M.; Rojo, J. L., Estadística actuarial vida (2007), Barcelona: Edicions Universitat de Barcelona, Barcelona
[5] Beer, J., Dealing with uncertainty in population forecasting, Statistics Netherlands, 1-40 (2000)
[6] Benjamin, B.; Pollard, J. H., The Analysis of Mortality and other Actuarial Statistics (1993), Oxford: Institute of Actuaries and the Faculty of Actuaries, Oxford
[7] Bijak, J., English Life Tables: 17 Methodology (2015)
[8] Blätter DGVFM, Herleitung der Sterbetafel DAV 2008 T für Lebensversicherungen mit Todesfallcharakter, DAV-Unterarbeitsgruppe Todesfallrisiko, 30, 189-224 (2009)
[9] Boletín Oficial del Estado, 4 Dirección General del Seguro y Fondos de Pensiones. Resolución del 6 de julio de 2012, 2012-07-21, BOE, 174, Sec. I., 52491 (2012)
[10] Brouhns, N.; Denuit, M., Risque de longévité et rentes viagères II. Tables de mortalitè prospectives pour la population belge, Belgian Actuarial Bulletin, 2, 1, 50-63 (2002)
[11] Brouhns, N.; Denuit, M.; Keilegom., I. V., Bootstrapping the Poisson log-bilinear model for mortality forecasting, Scandinavian Actuarial Journal, 3, 212-24 (2005) · Zbl 1092.91038
[12] Brouhns, N.; Denuit, M.; Vermunt., J. K., A Poisson log-bilinear regression approach to the construction of projected lifetables, Insurance: Mathematics & Economics, 31, 3, 373-93 (2002) · Zbl 1074.62524
[13] Brouhns, N.; Denuit, M.; Vermunt, J. K., Measuring the longevity risk in mortality projections, Bulletin of the Swiss Association of Actuaries, 105-30 (2002) · Zbl 1187.62158
[14] Chernick, M. R., Bootstrap methods: A Guide for Practitioners and Researchers (2008), Hoboken, NJ: John Wiley & Sons, Hoboken, NJ · Zbl 1136.62029
[15] Chiang, C. L., The Life Tables and its Applications (1984), Malabar, FL: R. E. Krieger, Malabar, FL
[16] Copas, J. B.; Haberman., S., Non-parametric graduation using kernel methods, Journal of the Institute of Actuaries, 110, 1, 135-56 (1983)
[17] 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
[18] Debón, A.; Montes, F.; Puig., F., Modelling and forecasting mortality in Spain, European Journal of Operational Research, 189, 3, 624-37 (2008) · Zbl 1142.62419
[19] Debón, A.; Montes, F.; Sala, R., A comparison of parametric models for mortality graduation, Application to mortality data for the Valencia Region (Spain). SORT: Statistics and Operation Research Transitions, 29, 2, 269-88 (2005) · Zbl 1274.62691
[20] Debón, A.; Montes, F.; Sala., R., A comparison of models for dynamic life tables. Application to mortality data from the Valencia Region (Spain), Lifetime Data Analysis, 12, 2, 223-44 (2006) · Zbl 1134.62369
[21] Delwarde, A.; Denuit, M.; Partrat., C., Negative binomial version of the Lee-Carter model for mortality forecasting, Applied Stochastic Models in Business and Industry, 23, 385-401 (2007) · Zbl 1150.91426
[22] Efron, B.; Tibshirani, R., An Introduction to the Bootstrap (1993), London: Chapman & Hall/CRC, London · Zbl 0835.62038
[23] Forfar, D. O.; Mccutcheon, M. A.; Wilkie., M. A., On graduation by mathematical formula, Journal of the Institute of Actuaries, 115, I, 1-149 (1988)
[24] Gaille, S.; Sherris., M., Modelling mortality with common stochastic long-run trends, Geneva Papers on Risk and Insurance—Issues and Practice, 36, 595-621 (2011)
[25] Garg, M. L.; Rao, R. B.; Redmond., C. K., Maximum-likelihood estimation of the parameters of the Gompertz survival function, Journal of the Royal Statistical Society: Series C (Applied Statistics), 19, 2, 152-59 (1970)
[26] Gavin, J.; Haberman, S.; Verrall., R., Moving weighted average graduation using kernel estimation, Insurance: Mathematics and Economics, 12, 113-26 (1993) · Zbl 0778.62096
[27] Gavin, J.; Haberman, S.; Verrall., R., On the choice of bandwidth for kernel graduation, Journal of the Institute of Actuaries, 121, 1, 119-34 (1994)
[28] Grothendieck, G., nls2: Non-linear regression with brute force. R package version 0.2 (2013)
[29] Haberman, S.; Renshaw., A., On age-period-cohort parametric mortality rate projections, Insurance: Mathematics and Economics, 45, 255-70 (2009) · Zbl 1231.91195
[30] Han, P. K. J.; Klein, W.; Neeraj., K. A., Varieties of uncertainty in health care: A conceptual taxonomy, Medical Decision Making, 31, 828-38 (2011)
[31] Helligman, L. M. A.; Pollard., J. H., The age pattern of mortality, Journal of the Institute of Actuaries, 107, I, 49-82 (1980)
[32] Hernandez-Flores, C. N.; Artiles-Romero, J.; Saavedra-Santana., P., Estimation of the population spectrum with replicated time series, Computational Statistics & Data Analysis, 30, 3, 271-80 (1999) · Zbl 1042.62605
[33] Human Mortality Database, University of California, Berkeley (USA), and Max Planck Institute for Demographic Research (Germany) (2013)
[34] ICEA (2012)
[35] Karlis, D.; Kostaki., A., Bootstrap techniques for mortality models, Biometrical Journal, 44, 7, 850-66 (2002)
[36] Koissi, M. C.; Shapiro, A. F.; Högnäs., G., Evaluating and extending the Lee-Carter model for mortality forecasting: Bootstrap confidence interval, Insurance: Mathematics and Economics, 38, 1-20 (2006) · Zbl 1098.62138
[37] Lazar, D.; Denuit., M. M., A multivariate time series approach to projected life tables, Applied Stochastic Models in Business and Industry, 25, 806-23 (2009) · Zbl 1224.91069
[38] Lee, R.; Carter., L., Modelling and forecasting U.S, mortality. Journal of the American Statistical Association, 87, 659-71 (1992) · Zbl 1351.62186
[39] Li, H.; Lu., Y., A Bayesian non-parametric model for small population mortality, Scandinavian Actuarial Journal, 2018, 7, 605-28 (2018) · Zbl 1416.91204
[40] Lledó, J.; Pavía, J. M.; Morillas., F. G., Assessing implicit hypothesis in life table construction, Scandinavian Actuarial Journal, 2017, 6, 495-518 (2017) · Zbl 1402.91207
[41] Lledó, J.; Pavía, J. M.; Morillas, F. G.; Corazza, M., Mathematical and Statistical Methods for Actuarial Sciences and Finance, The level of mortality in insured populations, 449-54 (2018), New York: Springer, New York
[42] Nocon, A. S.; Scott., W. F., An extension of the Whittaker-Henderson method of graduation, Scandinavian Actuarial Journal, 2011, 70-79 (2011) · Zbl 1277.62216
[43] Pavía, J. M., Testing goodness-of-fit with the kernel density estimator: GoFKernel, Journal of Statistical Software, 66, CS1, 1-27 (2015)
[44] Pavía Miralles, J. M.; Escuder Vallés, R., El proceso estocástico de muerte. Diferentes estrategias para la elaboración de tablas recargadas. Análisis de sensibilidad, Estadística Española, 153, 253-74 (2003)
[45] Prieto, E.; Fernández, M. J., Tablas de mortalidad de la población española de 1950 a 1990. Tabla proyectada del año 2000 (1994), Madrid: UNESPA, Madrid
[46] R Core Team, R: A language and environment for statistical computing (2013), Vienna: R Foundation for Statistical Computing, Vienna
[47] Ramlau-Hansen, H., Smoothing counting process intensities by means of kernel functions, Annals of Statistics, 11, 2, 453-66 (1983) · Zbl 0514.62050
[48] Regan, H. M.; Colyvan, M.; Burgman., M. A., A taxonomy and treatment of uncertainty for ecology and conservation biology, Ecological Society of America, 12, 2, 618-28 (2002)
[49] Rempala, G. A.; Szatzschneider., K., Bootstrapping parametric models of mortality, Scandinavian Actuarial Journal, 2004, 53-78 (2004) · Zbl 1129.91335
[50] Renshaw, A. E.; Haberman, S.; Hatzopoulos., P., The modelling of recent mortality trends in United Kingdom male assured lives, British Actuarial Journal, 2, 2, 449-77 (1996)
[51] Renshaw, A. E.; Haberman, S.; Hatzapoulos., P., On the duality of assumptions underpinning the construction of life tables, ASTIN Bulletin, 27, 1, 5-22 (1997)
[52] Sakamoto, Y.; Ishiguro, M.; Kitagawa, G., Akaike Information Criterion Statistics (1986), Dordrecht: Dordrecht, D. Reidel · Zbl 0608.62006
[53] Sithole, T. Z.; Haberman, S.; Verrall., R. J., An investigation into parametric models for mortality projections, with applications to immediate annuitants’ and life office pensioners’ data, Insurance: Mathematics and Economics, 27, 3, 285-312 (2000) · Zbl 1055.62555
[54] Su, S.; Sherris., M., Heterogeneity of Australian population mortality and implications for a viable life annuity market, Insurance: Mathematics and Economics, 51, 322-32 (2012)
[55] Villegas, A. M.; Haberman., S., On the modeling and forecasting of socioeconomic mortality differentials: An application to deprivation and mortality in England, North American Actuarial Journal, 18, 1, 168-93 (2014) · Zbl 1412.91057
[56] Wong-Fupuy, C.; Haberman., S., Projecting mortality trends: Recent developments in the United Kingdom and the United States, North American Actuarial Journal, 8, 2, 56-83 (2004) · Zbl 1085.62517
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.