Abstract
This chapter reviews the history, development, and application of mathematical mortality models. It consists of four sections. The first section focuses on parametric mortality models for single individuals. It starts with the innovation involved in creating a mathematical model of mortality, the importance, and the application of mortality tables, followed by the mathematical evolution of mortality models. It ends with the most recent model formulation – the stochastic mortality model. Next, the second section introduces parametric joint mortality models for coupled lives. It covers both the deterministic and stochastic model settings. Joint mortality models for coupled lives have great potential in the insurance industry for pricing joint and last survivor life insurance and annuity products (including pension products). As shown in previous studies, health condition and the expected life length of an individual can be significantly impacted by his or her significant partner, so consideration of bivariate mortality models can have significant practical importance. The third section summarizes nonparametric mortality models, for both individual and coupled lives. Compared with parametric models, the nonparametric ones are free of distributional assumptions and accordingly have a wider and more robust applications when understanding and parametrically modeling the true mortality distribution is not the focus of the study (e.g., when the focus is on studying the effects of covariates on mortality progression rather than mortality itself). Finally, in the fourth section, longevity risk and longevity risk models are introduced. These models are different from the mathematical models in previous sections in that these longevity risk models describe the changes of life expectancy between generations of individuals taking into consideration the cohort effects of people born in different years. These models have a great potential in solving longevity risk problems such as those occur in individual financial planning, institutional portfolio management, social security and medical insurance at a country level.
This is a preview of subscription content, log in via an institution.
References
Aalen O (1978) Nonparametric inference for a family of counting processes. Ann Stat 6(4):701–726
Aglira B (2006) To freeze or not to freeze: observations on the U.S. pension landscape. Global retirement perspective. Mercer Human Resource Consulting, New York
Altshuler B (1970) Theory for the measurement of competing risks in animal experiments. Math Biosci 6:1–11
Amicable Society (1854) The charters, acts of parliament, and by-laws of the corporation of the amicable society for a perpetual assurance office. The Amicable Society for a Perpetual Assurance Office. South Yarra, Australia
Antolin P (2007) Longevity risk and private pensions. OECD working papers on insurance and private pensions, No. 3, OECD Publishing. https://doi.org/10.1787/261260613084
Austad SN (2006) Why women live longer than men: sex differences in longevity. Gend Med 3(2):79–92
Baker T, Simon J (2002) Embracing risk: the changing culture of insurance and responsibility. University of Chicago Press, Chicago
Baker DW, Wolf MS, Feinglass J (2007) Health literacy and mortality among elderly persons. Arch Intern Med 167(14):1503–1509. https://doi.org/10.1001/archinte.167.14.1503
Biffis E (2005) Affine processes for dynamic mortality and actuarial valuations. Insur Math Econ 37(3):443–468
Brockett PL (1984) General bivariate Makeham laws. Scand Actuar J 1984(3):150–156. https://doi.org/10.1080/03461238.1984.10413763
Brouhns N, Denuit M, Vermunt JK (2002) Measuring the longevity risk in mortality projections. Bull Swiss Assoc Actuar (2):105–130
Butrica B, Smith KE, Toder E (2009) How will the stock market collapse affect retirement incomes. The Retirement Policy Program at The Urban Institute, Brief No. 20. June
Cairns AJG, Blake D, Dowd K (2006) A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration. Insur Math Econ 73:687–718
Cairns AJG, Blake D, Dowd K, Coughlan GD, Epstein D, Ong A, Balevich I (2009) A quantitative comparison of stochastic mortality models using data from England and Wales and the United States. North Am Actuar J 13(1):1–35. https://doi.org/10.1080/10920277.2009.10597538
Carriere JF (2000) Bivariate survival models for coupled lives. Scand Actuar J 2000(1):17–32. https://doi.org/10.1080/034612300750066700
Carter LR, Lee R (1992) Modeling and forecasting US sex differentials in mortality. Int J Forecast 8(3):393–411
Cherubini U, Luciano E, Vecchiato W (2004) Copula methods in finance. Wiley, Hoboken
CIMA (2008) The pension liability. http://www.cimaglobal.com/Documents/ImportedDocuments/cid_execrep_pension_liability_Feb08.pdf
Clark G (1999) Betting on lives: the culture of life insurance in England, 1695–1775. Manchester University Press, Manchester
Currie ID, Durban M, Eilers PHC (2004) Smoothing and forecasting mortality rates. Stat Model 4:279–298
Dabrowska BM (1988) Kaplan-Meier estimate on the plane. Ann Stat 16(4):1475–1489
Dahl M (2004) Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts. Insur Math Econ 35:113–136
Dahl M (2005) On mortality and investment risk in life insurance. http://web.math.ku.dk/noter/filer/phd05md.pdf
De Moivre A (1725) Annuities upon lives. Annuities upon lives or the valuation of annuities on any number of lives as also of reversions. London, William Pearson publ. The second edition of annuities upon lives was published in 1743
De Moivre A (1752) Annuities on lives, with several tables, exhibiting at one view, the value of lives for different rates of interest, Fourth Edition. Printed for A. Millar, over against Catherine Street, in the Strand, London
Deng Y, Brockett PL, MacMinn RD (2012) Longevity/mortality risk modeling and securities pricing. J Risk Insur 79:697–721
El-Bar AMTA (2018) An extended Gompertz-Makeham distribution with application to lifetime data. Commun Stat Simul Comput 47(8):2454–2475
El-Gohary A, Alshamrani A, Al-Otaibi AN (2013) The generalized Gompertz distribution. Appl Math Model 37:13–24
Engle RF, Granger CWJ (1987) Co-integration and error correction: representation, estimation, and testing. Econometrica 55(2):251–276
Frees EW, Carriere J, Valdez E (1996) Annuity valuation with dependent mortality. J Risk Insur 63(2):229–261
Gavrilov LA, Gavrilova NS (1996) Mortality measurement at advanced ages: a study of the social security administration death master file. North Am Actuar J 63(2):229–261
Gavrilova NS, Gavrilov LA (2011) Mortality Measurement at Advanced Ages. North Am Actuar J 15(3):432–447
Gavrilova S, Lopez O, Philippe PS (2011) A simplified model for studying bivariate mortality under right-censoring. Demografie 53(2):109–128
GBD 2015 Mortality and Causes of Death Collaborators (2016) Global, regional, and national life expectancy, all-cause mortality, and cause-specific mortality for 249 causes of death, 1980–2015: a systematic analysis for the Global Burden of Disease Study 2015. Lancet 388:1459–1544
Gebhardtsbauer R (2006) The future of defined benefit (DB) plans – Keynote speech at the National Plan Sponsor Conference. The future of DB plans, Washington, DC
Girosi F, King G (2007) Understanding the Lee-Carter mortality forecasting method. https://gking.harvard.edu/files/gking/files/lc.pdf
Gompertz B (1825) On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philos Trans R Soc Lond 115(1825):513–583
Government Accountability Office (GAO) (2008) Fiscal year 2008 financial report of the United States Government (Report). https://www.gao.gov/financial_pdfs/fy2008/08frusg.pdf
Graunt J (1662) Natural and political observations mentioned in a following index, and made upon the bills of mortality. London, John Martin publ
Gribkova S, Lopez O, Saint-Pierre P (2013) A simplified model for studying bivariate mortality under right-censoring. J Multivar Analy 115(1):181–192. https://doi.org/10.1016/j.jmva.2012.10.005
Halley E (1693) An estimate of the degrees of the mortality of mankind, drawn from curious tables of the births and funerals at the city of Breslaw; with an attempt to ascertain the price of annuities upon lives. J Inst Actuar Assur Mag 18(4):251–262. (Reprinted in 1874)
Harlow M, Laurence R (2001) Growing up and growing old in ancient Rome: a life course approach. Routledge, Abingdon
Hougaard P, Harvald B, Holm NV (1992) Measuring the similarities between the lifetimes of adult Danish twins born between 1881–1930. J Am Stat Assoc 87(417):17–24
Hurd TR (2009) Credit risk modeling using time-changed Brownian motion. Int J Theor Appl Financ 12:1213–1230
Hustead EC (1988) The history of actuarial mortality tables in the United States. J Insur Med 20(4):12–16
Jagger C, Sutton CJ (1991) Death after marital bereavement – is the risk increased? Stat Med 10:395–404
James IR, Segal MR (1982) On a method of mortality analysis incorporating age-year interaction, with application to prostate cancer mortality. Biometrics 38(2):433–443
Kaplan EL, Meier P (1958) Nonparametric estimation from incomplete observations. J Am Stat Assoc 53(282):457–481
Keyfitz N, Cawell H (2005) Applied mathematical demography. 3rd edn. SpringerVerlag, New York
Kopf EW (1927) The early history of the annuity. L.W. Lawrence, New York
Kou SG, Wang H (2004) Option pricing under a double exponential jump diffusion model. Manag Sci 50(9):1178–1192
Lee R, Carter LR (1992) Modeling and forecasting U.S. Mortality. J Am Stat Assoc 87(419): 659–671
Leon AD (2011) Trends in European life expectancy: a salutary view. Int J Epidemiol 40:271–277. https://doi.org/10.1093/ije/dyr061
Lin DY, Ying Z (1993) A simple nonparametric estimator of the bivariate survival function under univariateCensoring. Biometrika 80(3):573–581
Luciano E, Schoutens W (2006) A multivariate jump-driven financial asset. Quant Finan 29. www.carloalberto.org
Luciano E, Semeraro P (2010) Multivariate time changes for Lvy asset models: characterization and calibration. J Comput Appl Math 233:1937–1953
Luciano E, Spreeuw J, Vigna E (2008) Modeling stochastic mortality for dependent lives. Insur Math Econ 43(2):234–244. https://doi.org/10.1016/j.insmatheco.2008.06.005
Makeham M (1860) On the law of mortality and the construction of annuity tables. Assur Mag J Inst Actuar 8(6):301–310
Manor O, Eisenbach Z (2003) Mortality after spousal loss: are there socio-demographic differences? Soc Sci Med 56:405–413
Marshall AW, Olkin I (2007) Life distributions. Structure of nonparametric, semiparametric and parametric families. Springer, New York
Mathers CD, Stevens GA, Boerma T, White RA, Tobias MI (2015) Causes of international increases in older age life expectancy Lancet 385(9967):540–548
Meara ER, Richards S, Cutler DM (2008) The gap gets bigger – changes in mortality and life expectancy, by education, 1981—2000. Health Aff (Millwood) 27(2):350–360. https://doi.org/10.1377/hlthaff.27.2.350
Milevsky MA, Promislow D (2001) Mortality derivatives and the option to annuitize. York university finance working paper, No. MM08-1
Mitchell D, Brockett PL, Mendoza-Arriaga R, Muthuraman K (2013) Credit risk modeling using time-changed Brownian motion. Insur Math Econ 52:275–285
Munnell AH, Golub-Sass F, Soto M, Vitagliano F (2006) Why are healthy employers freezing their pensions? Issue Brief 44, Chestnut Hill, MA Center for Retirement Research at Boston College (March)
Nelsen W (1969) Hazard plotting for incomplete failure data. J Qual Technol 1(1):27–52
Nelsen RB (2007) An introduction to copulas. Springer, New York
Nielsen JJ, Hulman A, Witte DR (2018) Spousal cardiometabolic risk factors and incidence of type 2 diabetes: a prospective analysis from the English Longitudinal Study of Ageing. Diabetologia 61(7):1572–1580. https://doi.org/10.1007/s00125-018-4587-1. Epub 2018 Mar 8
Ogborn M (2015) Equitable assurances. Routledge, Abingdon
Pearce J, Millett M, Struck M (2015) Burial, society and context in the Roman world. Oxbow Books, Oxford
Renshaw AE, Haberman S (2006) A cohort-based extension to the Lee–Carter model for mortality reduction factors. Insur Math Econ 38:556–570
Riffi M (2018) A generalized transmuted Gompertz-Makeham distribution. J Sci Eng Res 5(8):252–266
Samji H, Cescon A, Hogg RS, Modur SP, Althoff KN et al (2013) Closing the gap: increases in life expectancy among treated HIV-positive individuals in the United States and Canada. PLoS One 8(12):e81355. https://doi.org/10.1371/journal.pone.0081355
Seifter A, Singh S, McArdle PF, Ryan KA, Shuldiner AR, Mitchell BD, Schäffer AA (2014) Analysis of the bereavement effect after the death of a spouse in the Amish: a population-based retrospective cohort study. BMJ Open 4:e003670. https://doi.org/10.1136/bmjopen-2013-003670
Sklar A (1973) Random variables, joint distribution functions, and copulas. Kybernetika 9(6): 449–460
Spreeuw J (2006) Types of dependence and time-dependent association between two lifetimes in single parameter copula models. Scand Actuar J 2006(5):286–309. https://doi.org/10.1080/03461230600952880
U.S. Bureau of Labor Statistics (2008) Current labor statistics monthly labor review, June 2008. https://www.bls.gov/opub/mlr/2008/06/cls0806.pdf
Wilmoth JR (1993) Computational methods for fitting and extrapolating the Lee-Carter model of mortality change. Technical Report, University of California, Berkeley
Zhang Y, Brockett P (2019) Modeling stochastic mortality for joint lives through subordinators. Working paper
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this entry
Cite this entry
Brockett, P.L., Zhang, Y. (2019). Actuarial (Mathematical) Modeling of Mortality and Survival Curves. In: Sriraman, B. (eds) Handbook of the Mathematics of the Arts and Sciences. Springer, Cham. https://doi.org/10.1007/978-3-319-70658-0_69-1
Download citation
DOI: https://doi.org/10.1007/978-3-319-70658-0_69-1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-70658-0
Online ISBN: 978-3-319-70658-0
eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering