×

A more meaningful parameterization of the Lee-Carter model. (English) Zbl 1452.91267

Summary: A new Lee-Carter model parameterization is introduced with two advantages. First, the Lee-Carter parameters are normalized such that they have a direct and intuitive interpretation, comparable across populations. Second, the model is stated in terms of the “needed-exposure” (NE). The NE is the number required in order to get one expected death and is closely related to the “needed-to-treat” measure used to communicate risks and benefits of medical treatments. In the new parameterization, time parameters are directly interpretable as an overall across-age NE. Age parameters are interpretable as age-specific elasticities: percentage changes in the NE at a particular age in response to a percent change in the overall NE. A similar approach can be used to confer interpretability on parameters of other mortality models.

MSC:

91G05 Actuarial mathematics
91D20 Mathematical geography and demography

Software:

Human Mortality
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Altman, D. G., Confidence intervals for the number needed to treat, BMJ: Br. Med. J., 317, 7168, 1309-1312 (1998)
[2] Beutner, E.; Reese, S.; Urbain, J.-P., Identifiability issues of age-period and age-period-cohort models of the Lee-Carter type, Insurance Math. Econom., 75, 117-125 (2017) · Zbl 1394.91188
[3] Booth, H.; Maindonald, J.; Smith, L., Applying Lee-Carter under conditions of variable mortality decline, Popul. Stud., 56, 3, 325-336 (2002)
[4] Booth, H.; Tickle, L., Mortality modelling and forecasting: A review of methods, Ann. Actuar. Sci., 3, 3-43 (2008)
[5] Brouhns, N.; Denuit, M.; Vermunt, J. K., A Poisson log-bilinear regression approach to the construction of projected lifetables, Insurance Math. Econom., 31, 3, 373-393 (2002) · Zbl 1074.62524
[6] Cairns, A. J.; Blake, D.; Dowd, K., Modelling and management of mortality risk: A review, Scand. Actuar. J., 2008, 2-3, 79-113 (2008) · Zbl 1224.91048
[7] Cairns, A. J.; Blake, D.; Dowd, K.; Coughlan, G. D.; Epstein, D.; Khalaf-Allah, M., Mortality density forecasts: An analysis of six stochastic mortality models, Insurance Math. Econom., 48, 3, 355-367 (2011)
[8] Camerer, C. F.; Kunreuther, H., Decision processes for low probability events: Policy implications, J. Policy Anal. Manage., 8, 4, 565-592 (1989)
[9] Cook, R. J.; Sackett, D. L., The number needed to treat: A clinically useful measure of treatment effect, BMJ: Br. Med. J., 310, 6977, 452-454 (1995)
[10] de Jong, P.; Tickle, L., Extending Lee-Carter mortality forecasting, Math. Popul. Stud., 13, 1, 1-18 (2006) · Zbl 1151.91742
[11] Deng, Y.; Brockett, P. L.; MacMinn, R. D., Longevity/mortality risk modeling and securities pricing, J. Risk Insurance, 79, 3, 697-721 (2012)
[12] Fung, M. C.; Peters, G. W.; Shevchenko, P. V., A unified approach to mortality modelling using state-space framework: characterisation, identification, estimation and forecasting, Ann. Actuar. Sci., 11, 2, 343-389 (2017)
[13] Girosi, F.; King, G., Understanding the Lee-Carter mortality forecasting method (2007), (Accessed 8 September 2019)
[14] Glei, D. A.; Horiuchi, S., The narrowing sex differential in life expectancy in high-income populations: Effects of differences in the age pattern of mortality, Popul. Stud., 61, 2, 141-159 (2007)
[15] Hinde, A., Demographic Methods (1998), Hodder Arnold Publication: Hodder Arnold Publication London, UK
[16] Hollmann, F. W.; Mulder, T. J.; Kallan, J. E., Methodology & Assumptions for the Population Projections of the United States: 1999 to 2010 (1999), US Department of Commerce, Bureau of the Census, Population Division, Population Projections Branch
[17] Human Mortality Database, F. W., University of California, Berkeley (USA), and max Planck institute for demographic research (Germany) (2019), (Acessed 10 September 2019)
[18] Hunt, A., Blake, D., 2015. Identifiability in age/period mortality models. Pensions Institute, Cass Business School, Discussion Paper PI-1508.
[19] Hyndman, R. J.; Booth, H.; Yasmeen, F., Coherent mortality forecasting: The product-ratio method with functional time series models, Demography, 50, 1, 261-283 (2013)
[20] Janssen, F., Advances in mortality forecasting: Introduction, Genus, 74, 21, 1-12 (2018)
[21] Janssen, F.; van Wissen, L.; Kunst, A., Including the smoking epidemic in internationally coherent mortality projections, Demography, 50, 4, 1341-1362 (2013)
[22] Kannisto, V.; Lauritsen, J.; Thatcher, A. R.; Vaupel, J. W., Reductions in mortality at advanced ages: Several decades of evidence from 27 countries, Popul. Dev. Rev., 20, 4, 793-810 (1994)
[23] Kunreuther, H.; Novemsky, N.; Kahneman, D., Making low probabilities useful, J. Risk Uncertain., 23, 2, 103-120 (2001) · Zbl 0996.91527
[24] Laupacis, A.; Sackett, D. L.; Roberts, R. S., An assessment of clinically useful measures of the consequences of treatment, New Engl. J. Med., 318, 26, 1728-1733 (1988)
[25] Lee, R. D.; Carter, L. R., Modeling and forecasting U.S. mortality, J. Amer. Statist. Assoc., 87, 419, 659-671 (1992) · Zbl 1351.62186
[26] Lee, R.; Miller, T., Evaluating the performance of the Lee-Carter method for forecasting mortality, Demography, 38, 4, 537-549 (2001)
[27] Li, J., A Poisson common factor model for projecting mortality and life expectancy jointly for females and males, Popul. Stud., 67, 1, 111-126 (2013)
[28] Li, J. S.-H.; Hardy, M. R., Measuring basis risk in longevity hedges, N. Am. Actuar. J., 15, 2, 177-200 (2011) · Zbl 1228.91042
[29] Li, N.; Lee, R., Coherent mortality forecasts for a group of populations: An extension of the Lee-Carter method, Demography, 42, 3, 575-594 (2005)
[30] Li, J.; Li, J. S.-H.; Tan, C. I.; Tickle, L., Assessing basis risk in index-based longevity swap transactions, Ann. Actuar. Sci., 13, 1, 166-197 (2019)
[31] Liu, Q.; Ling, C.; Li, D.; Peng, L., Bias-corrected inference for a modified Lee-Carter mortality model, Astin Bull., 49, 2, 433-455 (2019) · Zbl 1410.91277
[32] Liu, Q.; Ling, C.; Peng, L., Statistical inference for Lee-Carter mortality model and corresponding forecasts, N. Am. Actuar. J., 23, 3, 335-363 (2019) · Zbl 1426.91227
[33] Nielsen, B.; Nielsen, J. P., Identification and forecasting in mortality models, Sci. World J., 2014 (2014) · Zbl 1328.62575
[34] Niu, G.; Melenberg, B., Trends in mortality decrease and economic growth, Demography, 51, 5, 1755-1773 (2014)
[35] Plat, R., Stochastic portfolio specific mortality and the quantification of mortality basis risk, Insurance Math. Econom., 45, 1, 123-132 (2009) · Zbl 1231.91226
[36] Pollard, J. H., On the decomposition of changes in expectation of life and differentials in life expectancy, Demography, 25, 2, 265-276 (1988)
[37] Renshaw, A.; Haberman, S., A cohort-based extension to the Lee-Carter model for mortality reduction factors, Insurance Math. Econom., 38, 3, 556-570 (2006) · Zbl 1168.91418
[38] Reyna, V. F.; Brainerd, C. J., Numeracy, ratio bias, and denominator neglect in judgments of risk and probability, Learn. Individ. Differ., 18, 1, 89-107 (2008)
[39] Saver, J. L.; Lewis, R. J., Number needed to treat: Conveying the likelihood of a therapeutic effect, JAMA, 321, 8, 798-799 (2019)
[40] Schoen, R., The geometric mean of the age-specific death rates as a summary index of mortality, Demography, 7, 3, 317-324 (1970)
[41] Seligman, B.; Greenberg, G.; Tuljapurkar, S., Convergence in male and female life expectancy: Direction, age pattern, and causes, Demogr. Res., 34, 38, 1063-1074 (2016)
[42] Shang, H. L.; Hyndman, R. J., Grouped functional time series forecasting: An application to age-specific mortality rates, J. Comput. Graph. Stat., 26, 2, 330-343 (2017)
[43] Smith, D., Formal Demography, The Springer Series on Demographic Methods and Population Analysis (2013), Springer US
[44] Stoeldraijer, L.; van Duin, C.; van Wissen, L.; Janssen, F., Impact of different mortality forecasting methods and explicit assumptions on projected future life expectancy: The case of the Netherlands, Demogr. Res., 29, 323-353 (2013)
[45] Stone, E. R.; Yates, J.; Parker, A. M., Risk communication: Absolute versus relative expressions of low-probability risks, Organ. Behav. Hum. Decis. Process., 60, 3, 387-408 (1994)
[46] Thorslund, M.; Wastesson, J.; Agahi, N.; Lagergren, M.; Parker, M., The rise and fall of women’s advantage: A comparison of national trends in life expectancy at age 65 years, Eur. J. Ageing, 10, 4, 271-277 (2013)
[47] Tramèr, M. R.; Walder, B., Number needed to treat (or harm), World J. Surg., 29, 5, 576-581 (2005)
[48] Tuljapurkar, S.; Li, N.; Boe, C., A universal pattern of mortality decline in the G7 countries, Nature, 405, 6788, 789-792 (2000)
[49] Weinstein, N. D.; Kolb, K.; Goldstein, B. D., Using time intervals between expected events to communicate risk magnitudes., Risk Anal.:Int. J., 16, 3, 305-308 (1996)
[50] Wilmoth, J., Demography of longevity: Past, present, and future trends, Exp. Geront., 35, 9-10, 1111-1129 (2000)
[51] Wong-Fupuy, C.; Haberman, S., Projecting mortality trends: Recent developments in the United Kingdom and the United States, N. Am. Actuar. J., 8, 2, 56-83 (2004) · Zbl 1085.62517
[52] Wunsch, G., Introduction to Demographic Analysis: Principles and Methods (2012), Springer US · Zbl 0434.92013
[53] Yang, S. S.; Wang, C.-W., Pricing and securitization of multi-country longevity risk with mortality dependence, Insurance Math. Econom., 52, 2, 157-169 (2013) · Zbl 1284.91556
[54] Yerushalmy, J., A mortality index for use in place of the age-adjusted death rate*, Am. J. Public Health Nat. Health, 41, 8 part 1, 907-922 (1951)
[55] Zhou, R.; Wang, Y.; Kaufhold, K.; Li, J. S.-H.; Tan, K. S., Modeling period effects in multi-population mortality models: applications to solvency II, N. Am. Actuar. J., 18, 1, 150-167 (2014) · Zbl 1412.91060
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.