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Home > Financial Risk Management Best Practice > A Value-At-Risk Framework for Longevity Trend Risk

Financial Risk Management Best Practice

A Value-At-Risk Framework for Longevity Trend Risk

by Stephen J. Richards

This Chapter Covers

  • Longevity risk faced by annuity portfolios and defined-benefit pension schemes is typically long-term, i.e. the risk is of an adverse trend which unfolds over a long period of time.

  • However, there are circumstances in which it is useful to know by how much expectations of future mortality rates might change over a single year.

  • Such an approach lies at the heart of the one-year, value-at-risk view of reserves, and also for the pending Solvency II regime for insurers in the European Union.

  • This chapter describes a framework for determining how much a longevity liability might change based on new information over the course of one year.

  • It is a general framework and can accommodate a wide choice of stochastic projection models, thus allowing the user to explore the importance of model risk.


“Whereas a catastrophe can occur in an instant, longevity risk takes decades to unfold.” The Economist (2012)

Longevity risk is different from many others faced by insurers and pension schemes because the risk lies in the long-term trend taken by mortality rates. However, although longevity is typically a long-term risk, it is often necessary to pose questions over a short-term horizon, such as a year.

Two useful questions in risk management and reserving are “what could happen over the coming year to change the best-estimate projection?” and “by how much could a reserve change if new information became available?” The pending Solvency II regulations for insurers and reinsurers in the European Union are concerned with reserves being adequate in 99.5% of situations that might arise over the coming year. Insurers already have to do this as part of the International Congress of Actuaries (ICA) regime in the United Kingdom.

This chapter describes a framework for answering such questions, and for setting reserve requirements for longevity risk based on a one-year horizon instead of the more natural long-term approach. The chapter draws heavily on the paper by Richards et al. (forthcoming). The framework presented here is general, and can work with any stochastic projection model which can be fitted to data and is capable of generating sample paths. As with previous work in this area, such as Börger (2010), Plat (2011), and Cairns (2011), we will work with all-cause mortality rates rather than with rates disaggregated by cause of death.

In considering the insurer solvency capital requirement (SCR) for longevity risk, Börger (2010) concluded that “the computation of the SCR for longevity risk via the VaR approach obviously requires stochastic modelling of mortality.” Similarly, Plat (2011) stated that “naturally this requires stochastic mortality rates.” This chapter therefore only considers stochastic mortality as a solution to the value-at-risk (VaR) question of longevity risk. The VaR framework presented here requires stochastic projection models.

Cairns (2011) warns of the risks in relying on a single model by posing the oft-overlooked questions “what if the parameters … have been miscalibrated?” and “what if the model itself is wrong?” Cairns further writes that any solution “should be applicable to a wide range of stochastic mortality models.” The framework described below works with a wide variety of models, enabling practitioners to explore the impact of model risk on capital requirements.


The data used in this chapter are the all-cause number of deaths at age x last birthday during each calendar year y, split by gender. Corresponding midyear population estimates are also given.

The data therefore lend themselves to modeling the force of mortality, µx + ½, y + ½, without further adjustment. We use data provided by the Office for National Statistics (ONS) for England and Wales for the calendar years 1961–2010 inclusive. This particular data set has death counts and estimated exposures at individual ages up to age 104. We will work here with the subset of ages 50–104, which is most relevant for the insurance products sold at around retirement ages. The deaths and exposures in the age group labeled “105+” were not used. More detailed discussion of this data set, particularly regarding the estimated exposures, can be found in Richards (2008). Note that the ONS has subsequently revised the population estimates for 2002–10.

One consequence of only having data to age 104 is that one has to decide how to calculate annuity factors for comparison. One option would be to create an arbitrary extension of the projected mortality rates up to (say) age 120. Another alternative is simply to look at temporary annuities to avoid artefacts arising from the arbitrary extrapolation. We use the latter approach here, and we therefore calculate continuously paid temporary annuity factors. Restricting our calculations to temporary annuities has no meaningful consequences at the main ages of interest, as shown by the examples in Richards et al. (forthcoming). Although we have opted for the temporary annuity solution, it is worth noting that the models of Currie et al. (2004) and Cairns et al. (2006) are capable of simultaneously extrapolating mortality rates to higher (and lower) ages at the same time as projecting forward in time. These models therefore deserve a special place in the actuarial toolkit, and the subject is discussed in more detail by Richards and Currie (2011) and Currie (2011).

In this chapter we are concerned only with longevity trend risk. However, there are other aspects of longevity risk that an insurer or pension scheme needs to take into account, and an overview of these is given in Richards et al. (forthcoming).

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Further reading


  • Börger, Matthias. “Deterministic shock vs. stochastic value-at-risk: An analysis of the Solvency II standard model approach to longevity risk.” Blätter der DGVFM 31:2 (October 2010): 225–259. Online at:
  • Cairns, Andrew J. G. “Modelling and management of longevity risk: Approximations to survival functions and dynamic hedging.” Insurance: Mathematics and Economics 49:3 (November 2011): 438–453. Online at:
  • Cairns, Andrew J. G., David Blake, and Kevin Dowd. “A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration.” Journal of Risk and Insurance 73:4 (December 2006): 687–718. Online at:
  • Currie, Iain D. “Modelling and forecasting the mortality of the very old.” ASTIN Bulletin 41:2 (2011): 419–427. Online at:
  • Currie, Iain D. “Forecasting with the age-period-cohort model?” Proceedings of 27th International Workshop on Statistical Modelling, Prague, July 16–22, 2012; pp. 87–92. Online at:
  • Currie, Iain D., Maria Durban, and Paul H. C. Eilers. “Smoothing and forecasting mortality rates.” Statistical Modelling 4:4 (December 2004): 279–298. Online at:
  • Harrell, Frank E., and C. E. Davis. “A new distribution-free quantile estimator.” Biometrika 69:3 (December 1982): 635–640. Online at:
  • Lee, Ronald D., and Lawrence Carter. “Modeling and forecasting U.S. mortality.” Journal of the American Statistical Association 87:419 (September 1992): 659–671. Online at: [PDF].
  • Plat, Richard. “One-year value-at-risk for longevity and mortality.” Insurance: Mathematics and Economics 49:3 (November 2011): 462–470. Online at:
  • Richards, S. J. “Detecting year-of-birth mortality patterns with limited data.” Journal of the Royal Statistical Society, Series A 171:1 (January 2008): 279–298. Online at:
  • Richards, S. J., and I. D. Currie. “Extrapolating mortality projections by age.” Life and Pension Risk (June 2011): 34–38. Online at:
  • Richards, S. J., I. D. Currie, and G. P. Ritchie. “A value-at-risk framework for longevity trend risk.” British Actuarial Journal (forthcoming). Online at:
  • The Economist. “How innovation happens: The ferment of finance.” In “Playing with fire,” special report on financial innovation. February 25, 2012; p. 8. Online at:
  • Willets, Richard C. “The cohort effect: Insights and explanations.” British Actuarial Journal 10:4 (October 2004): 833–877. Online at:


  • Continuous Mortality Investigation (CMI) Bureau. “Continuous mortality investigation reports.” No. 17. Institute of Actuaries and Faculty of Actuaries, July 1999. Online at:
  • Continuous Mortality Investigation (CMI) Mortality Sub-Committee. “An interim basis for adjusting the ‘92’ series mortality projections for cohort effects.” Working paper no. 1. December 2002. Online at:
  • European Commission, Internal Market and Services Directorate General. “QIS5 technical specifications: Annex to call for advice from CEIOPS on QIS5.” July 5, 2010; p. 152.
  • Willets, Richard C. “Mortality in the next millennium.” Staple Inn Actuarial Society (SIAS), December 7, 1999. Online at:

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