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Delta–Gamma hedging of mortality and interest rate risk

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  • Luciano, Elisa
  • Regis, Luca
  • Vigna, Elena

Abstract

One of the major concerns of life insurers and pension funds is the increasing longevity of their beneficiaries. This paper studies the hedging problem of annuity cash flows when mortality and interest rates are stochastic. We first propose a Delta–Gamma hedging technique for mortality risk. The risk factor against which to hedge is the difference between the actual mortality intensity in the future and its “forecast” today, the forward intensity. We specialize the hedging technique first to the case in which mortality intensities are affine, then to Ornstein–Uhlenbeck and Feller processes, providing actuarial justifications for this selection. We show that, without imposing no arbitrage, we can get equivalent probability measures under which the HJM condition for no arbitrage is satisfied. Last, we extend our results to the presence of both interest rate and mortality risk. We provide a UK calibrated example of Delta–Gamma hedging of both mortality and interest rate risk.

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Bibliographic Info

Article provided by Elsevier in its journal Insurance: Mathematics and Economics.

Volume (Year): 50 (2012)
Issue (Month): 3 ()
Pages: 402-412

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Handle: RePEc:eee:insuma:v:50:y:2012:i:3:p:402-412

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Web page: http://www.elsevier.com/locate/inca/505554

Related research

Keywords: Longevity risk; Insurance pricing and hedging; Delta–Gamma coverage; No-arbitrage in insurance;

References

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  1. Elisa Luciano & Jaap Spreeuw & Elena Vigna, 2007. "Modelling Stochastic Mortality for Dependent Lives," CeRP Working Papers 58, Center for Research on Pensions and Welfare Policies, Turin (Italy).
  2. Blake, David & De Waegenaere, Anja & MacMinn, Richard & Nijman, Theo, 2010. "Longevity risk and capital markets: The 2008-2009 update," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 135-138, February.
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  6. David Blake & Andrew Cairns & Kevin Dowd & Richard MacMinn, 2006. "Longevity Bonds: Financial Engineering, Valuation, and Hedging," Journal of Risk & Insurance, The American Risk and Insurance Association, The American Risk and Insurance Association, vol. 73(4), pages 647-672.
  7. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "A Theory of the Term Structure of Interest Rates," Econometrica, Econometric Society, Econometric Society, vol. 53(2), pages 385-407, March.
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  9. Milevsky, Moshe A. & David Promislow, S., 2001. "Mortality derivatives and the option to annuitise," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 299-318, December.
  10. Darrell Duffie & Jun Pan & Kenneth Singleton, 1999. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," NBER Working Papers 7105, National Bureau of Economic Research, Inc.
  11. Dahl, Mikkel & Moller, Thomas, 2006. "Valuation and hedging of life insurance liabilities with systematic mortality risk," Insurance: Mathematics and Economics, Elsevier, vol. 39(2), pages 193-217, October.
  12. Robert Jarrow & Stuart Turnbull, 1994. "Delta, gamma and bucket hedging of interest rate derivatives," Applied Mathematical Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 1(1), pages 21-48.
  13. Alex Cowley & J. David Cummins, 2005. "Securitization of Life Insurance Assets and Liabilities," Journal of Risk & Insurance, The American Risk and Insurance Association, The American Risk and Insurance Association, vol. 72(2), pages 193-226.
  14. Biffis, Enrico, 2005. "Affine processes for dynamic mortality and actuarial valuations," Insurance: Mathematics and Economics, Elsevier, vol. 37(3), pages 443-468, December.
  15. Heath, David & Jarrow, Robert & Morton, Andrew, 1992. "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation," Econometrica, Econometric Society, Econometric Society, vol. 60(1), pages 77-105, January.
  16. Dahl, Mikkel, 2004. "Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts," Insurance: Mathematics and Economics, Elsevier, vol. 35(1), pages 113-136, August.
  17. Duffie, Darrell & Singleton, Kenneth J, 1999. "Modeling Term Structures of Defaultable Bonds," Review of Financial Studies, Society for Financial Studies, Society for Financial Studies, vol. 12(4), pages 687-720.
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Citations

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Cited by:
  1. Elisa Luciano & Luca Regis & Elena Vigna, 2012. "Single and cross-generation natural hedging of longevity and financial risk," ICER Working Papers, ICER - International Centre for Economic Research 04-2012, ICER - International Centre for Economic Research.
  2. Elisa Luciano & Luca Regis, 2012. "Demographic risk transfer: is it worth for annuity providers?," ICER Working Papers, ICER - International Centre for Economic Research 11-2012, ICER - International Centre for Economic Research.
  3. Elisa Luciano & Luca Regis, 2013. "Efficient versus inefficient hedging strategies in the presence of financial and longevity (value at) risk," Carlo Alberto Notebooks, Collegio Carlo Alberto 308, Collegio Carlo Alberto.

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