Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-23T23:10:16.064Z Has data issue: false hasContentIssue false
  • English
  • Français

Pricing Death: Frameworks for the Valuation and Securitization of Mortality Risk*

Published online by Cambridge University Press:  17 April 2015

Andrew J.G. Cairns ,
David Blake  and
Kevin Dowd
Andrew J.G. Cairns
Affiliation:
Department of Actuarial Mathematics and Statistics, School of Mathematical and Computer Sciences, Heriot-Watt University Edinburgh, EH14 4AS, United Kingdom, E-mail: A.Cairns@ma.hw.ac.uk
David Blake
Affiliation:
Pensions Institute Cass Business School, 106 Bunhill Row, London, EC1Y 8TZ, United Kingdom
Kevin Dowd
Affiliation:
Centre for Risk & Insurance Studies, Nottingham University Business School, Jubilee Campus, Nottingham, NG8 1BB, United Kingdom
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is now widely accepted that stochastic mortality – the risk that aggregate mortality might differ from that anticipated – is an important risk factor in both life insurance and pensions. As such it affects how fair values, premium rates, and risk reserves are calculated.

This paper makes use of the similarities between the force of mortality and interest rates to examine how we might model mortality risks and price mortality-related instruments using adaptations of the arbitrage-free pricing frameworks that have been developed for interest-rate derivatives. In so doing, the paper pulls together a range of arbitrage-free (or risk-neutral) frameworks for pricing and hedging mortality risk that allow for both interest and mortality factors to be stochastic. The different frameworks that we describe – short-rate models, forward-mortality models, positive-mortality models and mortality market models – are all based on positive-interest-rate modelling frameworks since the force of mortality can be treated in a similar way to the short-term risk-free rate of interest. While much of this paper is a review of the possible frameworks, the key new development is the introduction of mortality market models equivalent to the LIBOR and swap market models in the interest-rate literature.

These frameworks can be applied to a great variety of mortality-related instruments, from vanilla longevity bonds to exotic mortality derivatives.

Keywords

Stochastic mortalityterm structure of mortalitysurvivor indexspot survival probabilitiesspot force of mortalityforward mortality surfaceshort-rate modelsforward mortality modelspositive mortality frameworkmortality market modelsannuity market modelSCOR market model
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 36 , Issue 1 , May 2006 , pp. 79 - 120
Copyright
Copyright © ASTIN Bulletin 2006

Footnotes

*

First version presented at the 14th International AFIR Colloquium, Boston, 2004, under the title Pricing Frameworks for Securitization of Mortality Risk, available online at http://afir2004.soa.org/afir_papers.htm.

References

Ahn, A.-H., Dittmar, R.F. and Gallant, A.R. (2002) Quadratic term structure models: theory and evidence. Review of Financial Studies, 15, 243–288.CrossRefGoogle Scholar
Artzner, P. and Delbaen, F. (1995) Default risk insurance and incomplete markets. Mathematical Finance, 5, 187–195.CrossRefGoogle Scholar
Beelders, O. and Colarossi, D. (2004) Modelling mortality risk with extreme value theory: The case of Swiss Re’s mortality-indexed bond. Global Association of Risk Professionals, 4 (July/August), 26–30.Google Scholar
Biffis, E. (2005) Affine processes for dynamic mortality and actuarial valuations. Insurance: Mathematics and Economics, 37, 443–468.Google Scholar
Biffis, E. and Millossovich, P. (2004) The fair value of guaranteed annuity options. Working paper, Bocconi University, Milan.Google Scholar
Biffis, E. and Millossovich, P. (2005) A bidimensional approach to mortality risk. Working paper, Bocconi University, Milan.Google Scholar
Black, F. and Karasinski, P. (1991) Bond and option pricing when short rates are log-normal, Financial Analysts Journal, July-Aug, 52–59.CrossRefGoogle Scholar
Blake, D. and Burrows, W. (2001) Survivor bonds: Helping to hedge mortality risk. Journal of Risk and Insurance, 68, 339–348.CrossRefGoogle Scholar
Blake, D., Cairns, A.J.G. and Dowd, K. (2003) Pensionmetrics II: Stochastic pension plan design during the distribution phase. Insurance: Mathematics and Economics, 33, 29–47.Google Scholar
Blake, D., Cairns, A.J.G. and Dowd, K. (2006) Living with mortality: longevity bonds and other mortality-linked securities. To appear in British Actuarial Journal.CrossRefGoogle Scholar
Brace, A., Gatarek, D. and Musiela, M. (1997) The market model of interest-rate dynamics. Mathematical Finance, 7, 127–155.CrossRefGoogle Scholar
Brigo, D. and Mercurio, F. (2001) Interest Rate Models: Theory and Practice. Springer, Berlin.CrossRefGoogle Scholar
Brouhns, N., Denuit, M. and Vermunt, J.K. (2002) A Poisson log-bilinear regression approach to the construction of projected life tables. Insurance: Mathematics and Economics, 31, 373–393.Google Scholar
Cairns, A.J.G. (2004a) A family of term-structure models for long-term risk management and derivative pricing. Mathematical Finance, 14, 415–444.CrossRefGoogle Scholar
Cairns, A.J.G. (2004b) Interest-Rate Models: An Introduction. Princeton University Press, Princeton.CrossRefGoogle Scholar
Cairns, A.J.G., Blake, D. and Dowd, K. (2006) A two-factor model for stochastic mortality with parameter uncertainty. To appear in the Journal of Risk and Insurance.CrossRefGoogle Scholar
Cairns, A.J.G., Blake, D., Dawson, P. and Dowd, K. (2005) Pricing the risk on longevity bonds. Life and Pensions, October, 41–44.Google Scholar
Cairns, A.J.G. and Garcia Rosas, S.A. (2004) A family of term-structure models with stochastic volatility. In Proceedings of the 3rd World Congress of the Bachelier Society, Chicago, and Proceedings of the 35th International ASTIN Colloquium, Bergen.Google Scholar
CMI (1978) Proposed standard tables for life office pensioners and annuitants. Continuous Mortality Investigation Reports, 3, 1–30.Google Scholar
CMI (1999) Projection factors for mortality improvement. Continuous Mortality Investigation Reports, 17, 89–108.Google Scholar
Cox, S.H., Fairchild, J.R. and Pedersen, H.W. (2000) Economic aspects of securitization of risk. ASTIN Bulletin, 30, 157–193.CrossRefGoogle Scholar
Cox, J., Ingersoll, J. and Ross, S. (1985) A theory of the term-structure of interest rates. Econometrica, 53, 385–408.CrossRefGoogle Scholar
Cowley, A. and Cummins, J.D. (2005) Securitization of life insurance assets and liabilities. Journal of Risk and Insurance, 72, 193–226.CrossRefGoogle Scholar
Currie, I.D., Durban, M. and Eilers, P.H.C. (2004) Smoothing and forecasting mortality rates. Statistical Modelling, 4, 279–298.CrossRefGoogle Scholar
Dahl, M. (2004) Stochastic mortality in life insurance: Market reserves and mortality-linked insurance contracts. Insurance: Mathematics and Economics, 35, 113–136.Google Scholar
Dahl, M. and Møller, T. (2005) Valuation and hedging of life insurance liabilities with systematic mortality risk. In Proceedings of the 15th International AFIR Colloquium, Zurich. (Available online at http://www.afir2005.ch)Google Scholar
Doherty, N.A. (1997) Innovations in managing catastrophe risk. Journal of Risk and Insurance, 64, 713–718.CrossRefGoogle Scholar
Doherty, N.A. and Richter, A. (2002) Moral hazard, basis risk, and gap insurance. Journal of Risk and Insurance, 69, 9–24.CrossRefGoogle Scholar
Dowd, K., Blake, D., Cairns, A.J.G. and Dawson, P. (2006) Survivor swaps. Journal of Risk and Insurance, 73, 1–17.CrossRefGoogle Scholar
Duffie, D. and Kan, R. (1996) A yield-factor model of interest rates. Mathematical Finance, 6, 379–406.CrossRefGoogle Scholar
Flesaker, B. and Hughston, L.P. (1996) Positive interest. Risk, 9(1), 46–49.Google Scholar
Forfar, D.O. and Smith, D.M. (1987) The changing shape of English life Tables. Transactions of the Faculty of Actuaries, 40, 98–134.CrossRefGoogle Scholar
Froot, K.A. and Stein, J.C. (1998) Risk management, capital budgeting, and capital structure policy for financial institutions: an integrated approach. Journal of Financial Economics, 47, 55–82.CrossRefGoogle Scholar
Heath, D., Jarrow, R. and Morton, A. (1992) Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation. Econometrica, 60, 77–105.CrossRefGoogle Scholar
James, J. and Webber, N. (2000) Interest Rate Modelling. Wiley: Chichester.Google Scholar
Jamshidian, F. (1997) LIBOR and swap market models and measures. Finance and Stochastics, 1, 293–330.CrossRefGoogle Scholar
Joint Mortality Investigation Committee (JMIC) (1974) Considerations affecting the preparation of standard tables of mortality. Journal of the Institute of Actuaries, 101, 135–201.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S.E. (1998) Methods of Mathematical Finance. Springer-Verlag: New York.Google Scholar
Lando, D. (2004) Credit Risk Modeling: Theory and Applications. Princeton University Press: Princeton.CrossRefGoogle Scholar
Lane, M.N. (2000) Pricing risk transfer transactions. ASTIN Bulletin, 30, 259–293.CrossRefGoogle Scholar
Lee, P.J. (2000a) A general framework for stochastic investigations of mortality and investment risks. Presented at the Wilkiefest, Heriot-Watt University, March 2000.Google Scholar
Lee, R. (2000b) The Lee-Carter method for forecasting mortality, with various extensions and applications. North American Actuarial Journal, 4, 80–93.CrossRefGoogle Scholar
Lee, R.D. and Carter, L.R. (1992) Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87, 659–675.Google Scholar
Lin, Y. and Cox, S.H. (2005a) Securitization of mortality risks in life annuities. Journal of Risk and Insurance, 72, 227–252.CrossRefGoogle Scholar
Lin, Y. and Cox, S.H. (2005b) A mortality securitization model. Working paper, Georgia State University.Google Scholar
Luciano, E. and Vigna, E. (2005) Non mean reverting affine processes for stochastic mortality. In Proceedings of the 15th International AFIR Colloquium, Zurich. (Available online at http://www.afir2005.ch)CrossRefGoogle Scholar
Macdonald, A.S., Cairns, A.J.G., Gwilt, P.L. and Miller, K.A. (1998) An international comparison of recent trends in population mortality. British Actuarial Journal, 4, 3–141.CrossRefGoogle Scholar
Macdonald, A.S., Bartlett, D., Berman, C., Daykin, C., Grimshaw, D. and Savill, P. (2003) Mortality improvements and the cohort effect. CMI Working Papers 1 and 2; Presented to the Staple Inn Actuarial Society on 11 March 2003; 46 pages. (Available online at http://www.sias.org.uk)Google Scholar
Marocco, P. and Pitacco, E. (1998) Longevity risk and life annuity reinsurance. In Transactions of the 26th International Congress of Actuaries, Birmingham, 6, 453–479.Google Scholar
Milevsky, M.A. and Promislow, S.D. (2001) Mortality derivatives and the option to annuitise. Insurance: Mathematics and Economics, 29, 299–318.Google Scholar
Miltersen, K.R. and Persson, S.-A. (2005) Is mortality dead? Stochastic forward force of mortality determined by no arbitrage. Working paper, University of Bergen.Google Scholar
Miltersen, K., Sandmann, K. and Sondermann, D. (1997) Closed-form solutions for term structure derivatives with log-normal interest rates. Journal of Finance, 52, 409–430.CrossRefGoogle Scholar
Muermann, A. (2004) Catastrophe derivatives. In The Encyclopaedia of Actuarial Science, Volume 1, pages 231–236, Wiley, Chichester.Google Scholar
Olivier, P. and Jeffery, T. (2004) Stochastic mortality models. Presentation to the Society of Actuaries of Ireland. (Available online at http://www.actuaries.ie/Resources/events_papers/PastCalendarListing.htm)Google Scholar
Pelsser, A. (2003) Pricing and hedging guaranteed annuity options via static option replication. Insurance: Mathematics and Economics, 33, 283–296.Google Scholar
Perks, W. (1932) On some experiments in the graduation of mortality statistics. Journal of the Institute of Actuaries, 63, 12–57.CrossRefGoogle Scholar
Pitacco, E. (2002) Longevity risk in living benefits. Presented at the third annual CeRP conference “Developing an Annuity Market in Europe”, Turin.Google Scholar
Rebonato, R. (2002) Modern Pricing of Interest-Rate Derivatives. Princeton University Press: Princeton.Google Scholar
Renshaw, A.E. and Haberman, S. (2003) Lee-Carter mortality forecasting with age-specific enhancement. Insurance: Mathematics and Economics, 33, 255–272.Google Scholar
Rogers, L.C.G. (1997) The potential approach to the term-structure of interest rates and foreign exchange rates. Mathematical Finance, 7, 157–164.CrossRefGoogle Scholar
Rutkowski, M. (1997) A note on the Flesaker & Hughston model of the term structure of interest rates. Applied Mathematical Finance, 4, 151–163.CrossRefGoogle Scholar
Schmock, U. (1999) Estimating the value of the WINCAT coupons of the Wintherthur Insurance Convertible Bond: A study of the model risk. ASTIN Bulletin, 29, 101–163.CrossRefGoogle Scholar
Schönbucher, P. (2003) Credit Derivatives Pricing Models: Models, Pricing and Implementation. Wiley: Chichester.Google Scholar
Smith, A.D. (2005) Stochastic mortality modelling. Presentation at the Workshop on the Interface between Quantitative Finance and Insurance, Edinburgh, April 2005. (Available online at www.icms.org.uk/meetings/2005/quantfinance/sci_prog.html)Google Scholar
Vasicek, O. (1977) An equilibrium characterisation of the term structure. Journal of Financial Economics, 5, 177–188.CrossRefGoogle Scholar
Wang, S.S. (1996) Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26, 71–92.CrossRefGoogle Scholar
Wang, S.S. (2000) A class of distortion operations for pricing financial and insurance risks. Journal of Risk and Insurance, 67, 15–36.CrossRefGoogle Scholar
Wang, S.S. (2002) A universal framework for pricing financial and insurance risks. ASTIN Bulletin, 32, 213–234.CrossRefGoogle Scholar
Wang, S.S. (2003) Equilibrium pricing transforms: new results using Buhlmann’s 1980 economic model. ASTIN Bulletin, 33, 57–73.CrossRefGoogle Scholar
Willets, R.C. (1999) Mortality in the next millenium. Presented to the Staple Inn Actuarial Society on 7 December 1999. (Available online at www.sias.org.uk.)Google Scholar
Yang, S. (2001) Reserving, pricing and hedging for guaranteed annuity options. PhD Thesis, Heriot-Watt University, Edinburgh.Google Scholar
Yang, S., and Huang, H.C. (2005) Optimal contribution rate and asset allocation for defined contribution pension plan incorporating longevity risk. Working paper, Soochow University, Taiwan.Google Scholar