Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T15:07:50.706Z Has data issue: false hasContentIssue false
  • English
  • Français

Key Q-Duration: A Framework for Hedging Longevity Risk

Published online by Cambridge University Press:  09 August 2013

Johnny Siu-Hang Li  and
Ancheng Luo
Ancheng Luo
Affiliation:
Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada, N2L3G1, Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

When hedging longevity risk with standardized contracts, the hedger needs to calibrate the hedge carefully so that it can effectively reduce the risk. In this article, we present a calibration method that is based on matching mortality rate sensitivities. Specifically, we introduce a measure called key q-duration, which allows us to estimate the price sensitivity of a life-contingent liability to each portion of the underlying mortality curve. Given this measure, one can easily construct a longevity hedge with a small number of q-forward contracts. We further propose an extension for hedging the longevity risk associated with multiple birth cohorts, and another extension for accommodating population basis risk.

Keywords

Cairns-Blake-Dowd modelq-forwardsSecuritization
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 42 , Issue 2 , November 2012 , pp. 413 - 452
Copyright
Copyright © International Actuarial Association 2012

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

Barbarin, J. (2008) Heath-Jarrow-Morton Modelling of Longevity Bonds and the Risk Minimization of Life Insurance Portfolios. Insurance: Mathematics and Economics, 43, 41–55.Google Scholar
Blake, D., Cairns, A.J.G. and Dowd, K. (2006) Living with Mortality: Longevity Bonds and Other Mortality-Linked Securities. British Actuarial Journal, 12, 153–197.Google Scholar
Brouhns, N., Denuit, M., and Keilegom, I.V. (2005) Bootstrapping the Poisson Log-bilinear Model for Mortality Forecasting. Scandinavian Actuarial Journal, 31, 212–224.Google Scholar
Cairns, A.J.G., Blake, D., and Dowd, K. (2006) A Two-Factor Model for Stochastic Mortality with Parameter Uncertainty: Theory and Calibration. Journal of Risk and Insurance, 73, 687–718.CrossRefGoogle Scholar
Cairns, A.J.G., Blake, D., and Dowd, K. (2008) Modelling and Management of Mortality Risk: a Review. Scandinavian Actuarial Journal, 108, 79–113.Google Scholar
Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D., Ong, A. and Balevich, I. (2009) A Quantitative Comparison of Stochastic Mortality Models Using Data from England and Wales and the United States. North American Actuarial Journal, 13, 1–35.Google Scholar
Cairns, A.J.G. (2011) Modelling and Management of Longevity Risk: Approximations to Survivor Functions and Dynamic Hedging. Insurance: Mathematics and Economics, 49, 438–453.Google Scholar
Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D. and Khalaf-Allah, M. (2011a) Bayesian Stochastic Mortality Modelling for Two Populations. ASTIN Bulletin, 41, 29–55.Google Scholar
Cairns, A.J.G., Dowd, K., Blake, D. and Coughlan, G.D. (2011b) Longevity Hedge Effectiveness: A Decomposition. Working paper, Heriot-Watt University.Google Scholar
Coughlan, G. (2009) Longevity Risk Transfer: Indices and Capital Market Solutions. In Barrieu, P.M. and Albertini, L. (eds.) The Handbook of Insurance Linked Securities. London: Wiley.Google Scholar
Coughlan, G., Epstein, D., Ong, A., Sinha, A., Hevia-Portocarrero, J., Gingrich, E., Khalaf-Allah, M. and Joseph, P. (2007) LifeMetrics: A Toolkit for Measuring and Managing Longevity and Mortality Risks. Available at http://www.jpmorgan.com/pages/jpmorgan/investbk/solutions/lifemetrics/library.Google Scholar
Coughlan, G.D., Khalaf-Allah, M., Ye, Y., Kumar, S., Cairns, A.J.G., Blake, D. and Dowd, K. (2010) Longevity Hedging: A Framework for Longevity Basis Risk Analysis and Hedge Effectiveness. North American Actuarial Journal, 15, 150–176.Google Scholar
Dahl, M., Melchior, M. and Moller, T. (2008) On Systematic Mortality Risk and Risk-Minimization with Survivor Swaps. Scandinavian Actuarial Journal, 2, 114–146.CrossRefGoogle Scholar
Denuit, M., Devolder, P., and Goderniaux, A.-C. (2007) Securitization of Longevity Risk: Pricing Survivor Bonds with Wang Transform in the Lee-Carter Framework. Journal of Risk and Insurance, 74, 87–113.Google Scholar
Dowd, K., Blake, D., Cairns, A.J.G. and Dawson, P. (2006) Survivor swaps. Journal of Risk and Insurance, 73, 1–17.Google Scholar
Dowd, K., Cairns, A.J.G., Blake, D., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2011) A Gravity Model of Mortality Rates for Two Related Populations. North American Actuarial Journal, 15, 334–356.Google Scholar
Dowd, K., Cairns, A.J.G., Blake, D., Coughlan, G.D., Epstein, D. and Khalaf-Allah, M. (2010) Backtesting Stochastic Mortality Models: An Ex-Post Evaluation of Multi-Period-Ahead Density Forecasts. North American Actuarial Journal, 14, 281–298.CrossRefGoogle Scholar
Hair, J.F., Black, W.C., Babin, B.J., Anderson, R.E. and Tatham, R.L. (2006) Multivariate Data Analysis, Sixth Edition. Upper Saddle River, RJ: Prentice Hall.Google Scholar
Ho, T.S.Y. (1992) Key Rate Durations: Measures of Interest Rate Risks. Journal of Fixed Income, 2, 29–44.CrossRefGoogle Scholar
Human Mortality Database. University of California, Berkeley (USA), and Max Planck Institute of Demographic Research (Germany). Available at www.mortality.org or www.humanmortality.de (data downloaded on 1 April 2011).Google Scholar
Jarner, S.F. and Kryger, E.M. (2011) Modelling Adult Mortality in Small Populations: The SAINT Model. ASTIN Bulletin, 41, 377–418.Google Scholar
Johnson, R. and Wichern, D. Applied Multivariate Statistical Analysis, 6th Edition. Upper Saddle River, NJ: Prentice Hall.Google Scholar
Lee, R. and Carter, L. (1992) Modeling and Forecasting U.S. Mortality. Journal of the American Statistical Association, 87, 659–671.Google Scholar
Li, N. and Lee, R. (2005) Coherent Mortality Forecasts for a Group of Population: An Extension of the Lee-Carter Method. Demography, 42, 575–594.Google Scholar
Li, J.S.-H. and Hardy, M.R. (2011) Measuring Basis Risk in Longevity Hedges. North American Actuarial Journal, 15, 177–200.CrossRefGoogle Scholar
Li, J.S.-H., Ng, A.C.Y. and Chan, W.S. (2011) On the Calibration of Mortality Forward Curves. Journal of Futures Markets, 31, 941–970.Google Scholar
Plat, R. (2010) One-Year Value-at-Risk for Longevity and Mortality. Insurance: Mathematics and Economics, 49, 462–470.Google Scholar
Saunders, D.R. (1962) Trans-varimax: Some Properties of the Ratiomax and Equamax Criteria for Blind Orthogonal Rotation. Paper presented at the meeting of the American Psychological Association, St. Louis.Google Scholar
Stevens, R., De Waegenaere, A. and Melenberg, B. (2011) Longevity Risk and Natural Hedge Potential in Portfolios of Life Insurance Products: the Effect of Investment Risk. CentER Discussion Paper No. 2011-036.Google Scholar
Wills, S. and Sherris, M. (2008) Integrating Financial and Demographic Longevity Risk Models: An Australian Model for Financial Applications. UNSW Australian School of Business Research Paper No. 2008ACTL05.Google Scholar
Wills, S. and Sherris, M. (2010) Securitization, Structuring and Pricing of Longevity Risk. Insurance: Mathematics and Economics, 46, 173–185.Google Scholar
Zhou, R. and Li, J.S.-H. (2010) A Cautionary Note on Pricing Longevity Index Swaps. Scandinavian Actuarial Journal, first published on: 04 September 2010 (iFirst).Google Scholar
Zhou, R., Li, J.S.-H. and Tan, K.S. (2011) Pricing Standardized Mortality Securitizations: A Two-Population Model with Transitory Jump Effects. Paper presented in the Seventh International Longevity Risk and Capital Markets Solutions Conference, Frankfurt, Germany.Google Scholar