# On the optimal hedge ratio in index-based longevity risk hedging

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## Abstract

In an index-based longevity hedge, the so-called longevity basis risk arises from the potential mismatch between the hedging instrument and the annuity portfolio being hedged, due to the differences in the underlying populations and payoff structures. To reduce the impact of this longevity basis risk and increase the hedge effectiveness, an optimal position in the hedging instrument should be determined appropriately with regard to the nature of the two populations and also the timing of the payments. In this paper, we examine some analytical results on the optimal hedge ratio in hedging the longevity exposure of an annuity portfolio with index-based longevity- or mortality-linked securities. This optimal hedge ratio may serve as a convenient starting point for constructing an index-based hedge. We also conduct a cost–benefit analysis using different financial objectives. Our results are based on data of Australian public sector pensioners.

## Keywords

Index-based longevity hedge Longevity swap S-forward*q*-forward Longevity basis risk

## Notes

### Acknowledgements

The authors thank Mercer Australia for providing the mortality datasets to the Longevity Basis Risk research project (Phase 2), which contributes to some parts of this paper. The Phase 2 project was sponsored jointly by the Institute and Faculty of Actuaries (IFoA) and the Life and Longevity Markets Association (LLMA).

## References

- 1.Blake D, Burrows W (2001) Survivor bonds: helping to hedge mortality risk. J Risk Insur 68(2):339–348CrossRefGoogle Scholar
- 2.Cairns AJG (2013) Robust hedging of longevity risk. J Risk Insur 80(3):621–648CrossRefGoogle Scholar
- 3.Cairns AJG, Blake D, Dowd K (2006) A two-factor model for stochastic mortality with parameter uncertainty: theory and calibration. J Risk Insur 73(4):687–718CrossRefGoogle Scholar
- 4.Cairns AJG, Blake D, Dowd K (2008) Modelling and management of mortality risk: a review. Scand Actuar J 2008(2–3):79–113MathSciNetCrossRefGoogle Scholar
- 5.Chan WS, Li JSH, Li J (2014) The CBD mortality indexes: modeling and applications. N Am Actuar J 18(1):38–58MathSciNetCrossRefGoogle Scholar
- 6.Coughlan GD, Epstein D, Sinha A, Honig P (2007) q-forwards: derivatives for transferring longevity and mortality risk, JPMorgan. http://www.lifemetrics.com. Accessed 1 Feb 2018
- 7.Coughlan GD, Khalaf-Allah M, Ye Y, Kumar S, Cairns AJG, Blake D, Dowd K (2011) Longevity hedging 101: a framework for longevity basis risk analysis and hedge effectiveness. N Am Actuar J 15(2):150–176MathSciNetCrossRefGoogle Scholar
- 8.Dowd K (2003) Survivor bonds: a comment on Blake and Burrows. J Risk Insur 70(2):339–348CrossRefGoogle Scholar
- 9.Haberman S, Kaishev V, Millossovich P, Villegas A, Baxter S, Gaches A, Gunnlaugsson S, Sison M (2014) Longevity basis risk: a methodology for assessing basis risk. CASS business school, Hymans Robertson LLP, Institute and Faculty of Actuaries, Life and Longevity Markets Association. https://www.actuaries.org.uk/learn-and-develop/research-and-knowledge/actuarial-research-centre-arc/commissioned-projects/longevity-basis-risk. Accessed 8 Nov 2015
- 10.Human Mortality Database (HMD) (2017) University of California, Berkeley (USA) and Max Planck Institute for Demographic Research (Germany). http://www.mortality.org. Accessed 6 Apr 2017
- 11.James E, Song X (2001) Annuities markets around the world: money’s worth and risk intermediation. American Economic Association Meetings 2001. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.492.8023&rep=rep1&type=pdf. Accessed 13 Oct 2018
- 12.Keating C, Shadwick WF (2002) A universal performance measure. The Finance Development Centre, London. https://www.actuaries.org.uk/documents/universal-performance-measure. Accessed 13 Oct 2018
- 13.Kleinow T (2015) A common age effect model for the mortality of multiple populations. Insur Math Econ 63:147–152MathSciNetCrossRefGoogle Scholar
- 14.Koissi MC, Shapiro AF, Högnäs G (2006) Evaluating and extending the Lee-Carter model for mortality forecasting: bootstrap confidence interval. Insur Math Econ 38:1–20MathSciNetCrossRefGoogle Scholar
- 15.Lee RD, Carter LR (1992) Modeling and forecasting U.S. mortality. J Am Stat Assoc 87(419):659–671zbMATHGoogle Scholar
- 16.Li J (2014) A quantitative comparison of simulation strategies for mortality projection. Ann Actuar Sci 8(2):281–297CrossRefGoogle Scholar
- 17.Li J, Dacorogna M, Tan CI (2014) The impact of joint mortality modelling on hedging effectiveness of mortality derivatives. Longevity 10: Tenth International Longevity Risk and Capital Markets Solutions Conference, Cass Business School, City University of London. https://www.cass.city.ac.uk/faculties-and-research/centres/pensions-institute/events/longevity-10/programme. Accessed 4 Sept 2014
- 18.Li J, Li JSH, Tan CI, Tickle L (2018) Assessing basis risk in index-based longevity swap transactions. Ann Actuar Sci 2:22. https://doi.org/10.1017/s1748499518000179 CrossRefGoogle Scholar
- 19.Li JSH, Luo A (2012) Key q-duration: a framework for hedging longevity risk. ASTIN Bull 42(2):413–452MathSciNetzbMATHGoogle Scholar
- 20.Life and Longevity Markets Association (LLMA) (2010) Technical note: the S-forward. https://llma.org/. Accessed 29 Oct 2010
- 21.Sortino FA, Price LN (1994) Performance measurement in a downside risk framework. J Invest 3(3):59–64CrossRefGoogle Scholar
- 22.Villegas AM, Haberman S, Kaishev VK, Millossovich P (2017) A comparative study of two-population models for the assessment of basis risk in longevity hedges. ASTIN Bull 47(3):631–679MathSciNetCrossRefGoogle Scholar
- 23.Zhou R, Li JSH, Tan KS (2013) Pricing standardized mortality securitizations: a two-population model with transitory jump effects. J Risk Insur 80(3):733–774CrossRefGoogle Scholar