Hostname: page-component-cb9f654ff-65tv2 Total loading time: 0 Render date: 2025-08-25T13:31:06.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Analytic expressions for annuities based on Makeham–Beard mortality laws

Published online by Cambridge University Press:  28 April 2020

David C. Bowie
David C. Bowie*
Affiliation:
Hymans Robertson LLP, Exchange Place One, 1 Semple Street, Edinburgh, EH3 8BL, UK
*
*Corresponding author. E-mail: david.bowie@hymans.co.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

This note derives analytic expressions for annuities based on a class of parametric mortality “laws” (the so-called Makeham–Beard family) that includes a logistic form that models a decelerating increase in mortality rates at the higher ages. Such models have been shown to provide a better fit to pensioner and annuitant mortality data than those that include an exponential increase. The expressions derived for evaluating single life and joint life annuities for the Makeham–Beard family of mortality laws use the Gauss hypergeometric function and Appell function of the first kind, respectively.

Keywords

AnnuitiesMakeham–Beard mortality lawGauss hypergeometric functionAppell function

Information

Type
Paper
Information
Annals of Actuarial Science , Volume 15 , Issue 1 , March 2021 , pp. 1 - 13
Copyright
© Institute and Faculty of Actuaries 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abramowitz, M. & Stegun, I.A. (eds) (1972). Hypergeometric functions, Ch.15 in Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, 9th Printing (pp. 555–566). Dover, New York.Google Scholar
Appell, P. (1925). Sur les fonctions hypergéométriques de plusieurs variables. In Mémoir. Sci. Math. Gauthier-Villars, Paris. Google Scholar
Beard, R.E. (1959). Note on some mathematical mortality models. In G.E.W. Wolstenholme & M. O’Connor (Eds.), The Lifespan of Animals (pp. 302–311). Little, Brown, Boston. Google Scholar
Borchers, H. W. (2019). Pracma: Practical Numerical Math Functions. R package, version 2.2.5. https://CRAN.R-project.org/package=pracma Google Scholar
Bove, D.S. with contributions by Colavecchia, F.D., Forrey, R.C., Gasaneo, G., Michel, N.L.J., Shampine, L.F., Stoitsov, M.V. & Watts, H.A. (2013). Appell: Compute Appell’s F1 Hypergeometric Function. R package version 0.0–4. https://CRAN.R-project.org/package=appell Google Scholar
CMI Working Paper 100. (2017). A second report on high age mortality. Available online at the address https://www.actuaries.org.uk/learn-and-develop/continuous-mortality-investigation/cmi-working-papers/other/cmi-working-paper-100 [accessed February 2019].Google Scholar
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. Philosophical Transactions of the Royal Society, 115, 513–585.Google Scholar
Makeham, W.M. (1860). On the law of mortality and the construction of annuity tables. The Assurance Magazine, and Journal of the Institute of Actuaries, 8(6), 301–310.CrossRefGoogle Scholar
Milevsky, M.A., Salisbury, T.S. & Chigodaev, A. (2016). The implied longevity curve: How long does the market think you are going to live? Journal of Investment Consulting, 17, 11–21.Google Scholar
Missov, T.I. (2013). Gamma-Gompertz life expectancy at birth. Demographic Research, 28, 259–270.10.4054/DemRes.2013.28.9CrossRefGoogle Scholar
Perks, W. (1932). On some experiments in the graduation of mortality statistics. Journal of the Institute of Actuaries (1886-1994), 63(1), 12–57.10.1017/S0020268100046680CrossRefGoogle Scholar
R Core Team (2018). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/ Google Scholar
Richards, S. (2008). Applying survival models to pensioner mortality data. British Actuarial Journal, 14(II), 257–326.10.1017/S1357321700001720CrossRefGoogle Scholar
Richards, S. (2012) A handbook of parametric survival models for actuarial use. Scandinavian Actuarial Journal, (4), 233–257.10.1080/03461238.2010.506688CrossRefGoogle Scholar
Richards, S. (2019) The cascade model of mortality, Longevitas Information Matrix, Available online at the address https://www.longevitas.co.uk/site/informationmatrix/thecascademodelofmortality.html [accessed February 2019].Google Scholar
Schlosser, M.J. (2013) Multiple hypergeometric series – Appell Series and Beyond, arXiv:1305.1966v1 [math.CA] [accessed February 2019].CrossRefGoogle Scholar
Seal, H. (1962). Actuarial note on the calculation of isolated (Makeham) joint annuity values. Transactions of the Faculty of Actuaries, 28, 91–98, doi:10.1017/S0071368600007394 CrossRefGoogle Scholar
Whelan, S. (2009). Mortality in Ireland at advanced ages, 1950-2006: Part 2: graduated rates. Annals of Actuarial Science, 4(1), 67–104. doi:10.1017/S1748499500000609 CrossRefGoogle Scholar
Wolfram Research, Inc., (2019). Mathematica, Version 12.0, Champaign, IL.Google Scholar