Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T14:20:22.923Z Has data issue: false hasContentIssue false
  • English
  • Français

Longevity trend risk over limited time horizons

Published online by Cambridge University Press:  21 May 2020

Stephen J. Richards Open the ORCID record for Stephen J. Richards [Opens in a new window] ,
Iain D. Currie Open the ORCID record for Iain D. Currie [Opens in a new window] ,
Torsten Kleinow Open the ORCID record for Torsten Kleinow [Opens in a new window]  and
Gavin P. Ritchie Open the ORCID record for Gavin P. Ritchie [Opens in a new window]
Stephen J. Richards*
Affiliation:
Longevitas Ltd, 24a Ainslie Place, EdinburghEH3 6AJ, UK
Iain D. Currie
Affiliation:
Department of Actuarial Mathematics and Statistics, Heriot-Watt University, EdinburghEH14 4AS, UK
Torsten Kleinow
Affiliation:
Department of Actuarial Mathematics and Statistics, Heriot-Watt University, EdinburghEH14 4AS, UK
Gavin P. Ritchie
Affiliation:
Longevitas Ltd, 24a Ainslie Place, EdinburghEH3 6AJ, UK
*
*Corresponding author. E-mail: stephen@longevitas.co.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider various aspects of longevity trend risk viewed through the prism of a finite time window. We show the broad equivalence of value-at-risk (VaR) capital requirements at a p-value of 99.5% to conditional tail expectations (CTEs) at 99%. We also show how deferred annuities have higher risk, which can require double the solvency capital of equivalently aged immediate anuities. However, results vary considerably with the choice of model and so longevity trend-risk capital can only be determined through consideration of multiple models to inform actuarial judgement. This model risk is even starker when trying to value longevity derivatives. We briefly discuss the importance of using smoothed models and describe two methods to considerably shorten VaR and CTE run times.

Keywords

Longevity trend riskValue-at-riskConditional tail expectationSolvency IIORSAImmediate annuityDeferred annuityIndex products
Type
Paper
Information
Annals of Actuarial Science , Volume 14 , Issue 2 , September 2020 , pp. 262 - 277
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.)

References

Akaike, H. (1987). Factor analysis and AIC. Psychometrica, 52, 317–333. ISSN 0033–3123. doi: https://doi.org/10.1007/BF02294359.CrossRefGoogle Scholar
Bank of England Prudential Regulatory Authority. Solvency II: internal model and matching adjustment update. Letter to UK-regulated insurers and reinsurers, March 2015. http://www.bankofengland.co.uk/pra/Pages/publications/ps/2016/ps3316.aspx.Google Scholar
Blake, D., Cairns, A.J.G., Dowd, K. & Kessler, A.R. (2018). Still living with mortality: the longevity risk transfer market after one decade. British Actuarial Journal, 24, e1. doi: https://doi.org/10.1017/S1357321718000314.Google Scholar
Boumezoued, A. (2020) Improving HMD mortality estimates with HFD fertility data. North American Actuarial Journal, 1–25.CrossRefGoogle Scholar
Cairns, A.J.G., Blake, D. & Dowd, K. (2006) A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration. Journal of Risk and Insurance, 73, 687–718. doi: https://doi.org/10.1111/j.1539-6975.2006.00195.x.CrossRefGoogle Scholar
Cairns, A.J.G. & El Boukfaoui, G. (2019) Basis risk in index based longevity hedges: a guide for longevity hedgers. North American Actuarial Journal, 1–22.CrossRefGoogle Scholar
Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D. & Khalaf-Allah, M. (2011). Mortality density forecasts: an analysis of six stochastic mortality models. Insurance: Mathematics and Economics, 48, 355–367.Google Scholar
Cairns, A.J.G., Blake, D., Dowd, K. & Kessler, A. (2015). Phantoms never die: living with unreliable mortality data. Journal of the Royal Statistical Society, Series A. doi: https://doi.org/10.1111/rssa.12159.Google Scholar
Currie, I.D. (2011). Modelling and forecasting the mortality of the very old. ASTIN Bulletin, 41, 419–427. doi: 10.2143/AST.41.2.2136983.Google Scholar
Currie, I.D. (2013). Smoothing constrained generalized linear models with an application to the Lee-Carter model. Statistical Modelling, 13(1), 69–93. doi: https://doi.org/10.1177%2F1471082X12471373CrossRefGoogle Scholar
Currie, I.D. (2020). Constraints, the identifiability problem and the forecasting of mortality. Annals of Actuarial Science, 1–30CrossRefGoogle Scholar
Currie, I.D., Durban, M. & Eilers, P.H. (2004). Smoothing and forecasting mortality rates. Statistical Modelling, 4, 279–298. doi: https://doi.org/10.1191%2F1471082X04st080oaCrossRefGoogle Scholar
Delwarde, A., Denuit, M. & Eilers, P.H.C. (2007). Smoothing the Lee-Carter and Poisson log-bilinear models for mortality forecasting: a penalized likelihood approach. Statistical Modelling, 7, 29–48. doi: https://doi.org/10.1177%2F1471082X0600700103CrossRefGoogle Scholar
Eilers, P.H.C. & Marx, B.D. (1996). Flexible smoothing with B-splines and penalties. Statistical Science, 11(2), 89–121. ISSN 08834237. http://www.jstor.org/stable/2246049.CrossRefGoogle Scholar
Hardy, M.R. (2006). An introduction to risk measures for actuarial applications. Casualty Actuarial Society.Google Scholar
Harrell, F.E. & Davis, C.E. (1982). A new distribution-free quantile estimator. Biometrika, 69, 635–640. ISSN 00063444. doi: https://doi.org/10.1093/biomet/69.3.635. http://www.jstor.org/stable/2335999.CrossRefGoogle Scholar
Hyndman, R. (2013). Bug 15396 – Initialization of regression coefficients in arima(). https://bugs.r-project.org/bugzilla/show_bug.cgi?id=15396 .Google Scholar
Kleinow, T. & Richards, S.J. (2016). Parameter risk in time-series mortality forecasts. Scandinavian Actuarial Journal, 2016(10), 1–25. doi: https://doi.org/10.1080/03461238.2016.1255655.Google Scholar
Langou, J. (2010). Lapack 3.2.2 release notes. http://www.netlib.org/lapack/lapack-3.2.2.html.Google Scholar
Lee, R.D. & Carter, L. (1992). Modeling and forecasting US mortality. Journal of the American Statistical Association, 87, 659–671. ISSN 01621459. http://www.jstor.org/stable/2290201.Google Scholar
Liu, Y. & Li, J.S.-H. (in press). An efficient method for mitigating longevity value-at-risk. North American Actuarial Journal, Special Issue for the 13th International Longevity Risk and Capital Markets Solutions Conference.Google Scholar
Macdonald, A.S., Richards, S.J. & Currie, I.D. (2018). Modelling Mortality with Actuarial Applications. Cambridge University Press. ISBN 978-1-107-04541-5.CrossRefGoogle Scholar
R Core, Team (2017). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/.Google Scholar
Renshaw, A.E. & Haberman, S. (2006). A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insurance: Mathematics and Economics, 38, 556–570.Google Scholar
Richards, S.J. & Currie, I.D. (2009). Longevity risk and annuity pricing with the Lee-Carter model. British Actuarial Journal, 15(II) No. 65, 317–365 (with discussion). doi: https://doi.org/10.1017/S1357321700005675.CrossRefGoogle Scholar
Richards, S.J., Currie, I.D., Kleinow, T. & Ritchie, G.P. (2019). A stochastic implementation of the APCI model for mortality projections. British Actuarial Journal, 24, e13. doi: 10.1017/S1357321718000260. https://www.longevitas.co.uk/apci.CrossRefGoogle Scholar
Richards, S.J., Currie, I.D. & Ritchie, G.P. (2014). A value-at-risk framework for longevity trend risk. British Actuarial Journal, 19(1), 116–167. doi: https://doi.org/10.1017/S1357321712000451. https://www.longevitas.co.uk/var.CrossRefGoogle Scholar
Richards, S.J., Kirkby, J.G. & Currie, I.D. (2006). The importance of year of birth in two-dimensional mortality data. British Actuarial Journal, 12(I), 5–61 (with discussion). doi: https://doi.org/10.1017/S1357321700004682.CrossRefGoogle Scholar
Wilmoth, J.R., Andreev, K., Jdanov, D., Glei, D.A. & Riffe, T. (2017). Methods Protocol for the Human Mortality Database. https://www.mortality.org/Public/Docs/MethodsProtocol.pdf.Google Scholar
Woods, S. (2016). Reflections on the 2015 Solvency II internal model approval process. Bank of England Prudential Regulation Authority. http://www.bankofengland.co.uk/pra/Pages/publications/ps/2016/ps3316.aspx.Google Scholar