Mathematics and Financial Economics

, Volume 10, Issue 2, pp 151–178 | Cite as

Risk-minimization for life insurance liabilities with basis risk

  • Francesca BiaginiEmail author
  • Thorsten Rheinländer
  • Irene Schreiber


In this paper we study the hedging of typical life insurance payment processes in a general setting by means of the well-known risk-minimization approach. We find the optimal risk-minimizing strategy in a financial market where we allow for investments in a hedging instrument based on a longevity index, representing the systematic mortality risk. Thereby we take into account and model the basis risk that arises due to the fact that the insurance company cannot perfectly hedge its exposure by investing in a hedging instrument that is based on the longevity index, not on the insurance portfolio itself. We also provide a detailed example within the context of unit-linked life insurance products where the dependency between the index and the insurance portfolio is described by means of an affine mean-reverting diffusion process with stochastic drift.


Life insurance payment processes Risk-minimization  Martingale representation Basis risk Affine mortality structure 

Mathematics Subject Classification

62P05 91G80 91G20 62P20 

JEL Classification

C02  G19  G10 



We wish to thank an anonymous referee for advices and remarks, which contributed to improve the paper.


  1. 1.
    Ansel, J.P., Stricker, C.: Décomposition de Kunita-Watanabe. In: Séminaire de Probabilités XXVII, Lecture Notes in Mathematics Volume 1557. Springer, New York (1993)Google Scholar
  2. 2.
    Barbarin, J.: Heath–Jarrow–Morton modelling of longevity bonds and the risk minimization of life insurance portfolios. Insur. Math. Econ. 43(1), 41–55 (2008a)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Barbarin, J.: Valuation, Hedging and the Risk Management of Insurance Contracts. PhD thesis, Catholic University of Louvain, (2008b)Google Scholar
  4. 4.
    Barrieu, P., Albertini, L.: The Handbook of Insurance-Linked Securities. Wiley-Blackwell, London (2009)Google Scholar
  5. 5.
    Barrieu, P., Bensusan, H., El Karoui, N., Hillairet, C., Loisel, S., Ravanelli, S.C., Salhi, Y.: Understanding, modelling and managing longevity risk: key issues and main challenges. Scand. Actuar. J. 3, 203–231 (2012)CrossRefGoogle Scholar
  6. 6.
    Biagini, F., Cretarola, A.: Quadratic hedging methods for defaultable claims. Appl. Math. Optim. 56(3), 425–443 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Biagini, F., Cretarola, A.: Local risk-minimization for defaultable markets. Math. Financ. 19(4), 669–689 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Biagini, F., Cretarola, A.: Local risk-minimization with recovery process. Appl. Math. Optim. 65(3), 293–314 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Biagini, F., Schreiber, I.: Risk-minimization for life insurance liabilities. SIAM J. Financ. Math. 4(1), 243–264 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Biagini, F., Rheinländer, T., Widenmann, J.: Hedging mortality claims with longevity bonds. ASTIN Bull. 43(2), 123–157 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Biagini, F., Botero, C., Schreiber, I.: Risk-minimization for life insurance liabilities with dependent mortality risk. Math. Financ. (2015) (accepted)Google Scholar
  12. 12.
    Bielecki, T.R., Rutkowski, M.: Credit Risk: Modeling, Valuation and Hedging. Springer Finance (2004)Google Scholar
  13. 13.
    Biffis, E.: Affine processes for dynamic mortality and actuarial valuations. Insur. Math. Econ. 37(3), 443–468 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Biffis, E., Blake, D., Pitotti, L., Sun, A.: The cost of counterparty risk and collateralization in longevity swaps. J. Risk Insur. (2014). doi: 10.1111/jori.12055
  15. 15.
    Blake, D., Cairns, A.J.G., Dowd, K.: The birth of the life market. Asia-Pac. J. Risk Insur. 3(1), 6–36 (2008)Google Scholar
  16. 16.
    Blanchet-Scalliet, C., Jeanblanc, M.: Hazard rate for credit risk and hedging defaultable contingent claims. Financ. Stoch. 8(1), 145–159 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Cairns, A.J.G., Blake, D., Dowd, K.: Pricing death: frameworks for the valuation and securitization of mortality risk. ASTIN Bull. 36(1), 79–120 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Khalaf-Allah, M.: Bayesian stochastic mortality modelling for two populations. ASTIN Bull. 41(1), 29–59 (2011)MathSciNetGoogle Scholar
  19. 19.
    Coughlan, G.D., Khalaf-Allah, M., Ye, Y., Kumar, S., Cairns, A.J.G., Blake, D., Dowd, K.: Longevity hedging 101: a framework for longevity basis risk analysis and hedge effectiveness. N. Am. Actuar. J. 15(2), 150–176 (2011)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Dahl, M., Møller, T.: Valuation and hedging of life insurance liabilities with systematic mortality risk. Insur. Math. Econ. 39(2), 193–217 (2006)CrossRefzbMATHGoogle Scholar
  21. 21.
    Dahl, M., Melchior, M., Møller, T.: On systematic mortality risk and risk-minimization with survivor swaps. Scand. Actuar. J. 2–3, 114–146 (2008)CrossRefGoogle Scholar
  22. 22.
    Dahl, M., Glar, S., Møller, T.: Mixed dynamic and static risk-minimization with an application to survivor swaps. Eur. Actuar. J. 1, 233–260 (2011)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Dai, Q., Singleton, K.: Specification analysis of affine term structure models. J. Financ. 55(5), 1943–1978 (2000)CrossRefGoogle Scholar
  24. 24.
    Deelstra, G., Delbaen, F.: Convergence of discretized stochastic (interest rate) processes with stochastic drift term. Appl. Stoch. Models Data Anal. 14(1), 77–84 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Dowd, K., Cairns, A.J.G., Blake, D., Coughlan, G.D., Khalaf-Allah, M.: A gravity model of mortality rates for two related populations. N. Am. Actuar. J. 15(2), 334–356 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Duffie, D., Filipović, D., Schachermayer, W.: Affine processes and applications in finance. Ann. Appl. Probab. 13(3), 984–1053 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Filipović, D., Mayerhofer, E.: Affine diffusion processes: theory and applications. Radon Ser. Comput. Appl. Math. 8, 1–40 (2009)Google Scholar
  29. 29.
    Föllmer, H., Sondermann, D.: Hedging of non-redundant contingent claims. In: Mas-Colell, A., Hildenbrand, W., (eds.), Contributions to Mathematical Economics, pp. 205–223. North Holland, Amsterdam (1986)Google Scholar
  30. 30.
    Friedman, A.: Stochastic Differential Equations and Applications. Springer, New York (1975)zbMATHGoogle Scholar
  31. 31.
    Henriksen, L.F.B., Møller, T.: Local risk-minimization with longevity bonds. Appl. Stoch. Models Bus. Ind. 31(2), 241–263 (2015)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer, New York (2002)Google Scholar
  33. 33.
    Jarner, S.F., Kryger, E.M.: Modelling adult mortality in small populations: The SAINT model. Pensions Institute Working Paper PI-0902, Cass Business School, London (2009)Google Scholar
  34. 34.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1991)zbMATHGoogle Scholar
  35. 35.
    Li, J.S.-H., Hardy, M.R.: Measuring basis risk in longevity hedges. N. Am. Actuar. J. 15(2), 177–200 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Li, N., Lee, R.: Coherent mortality forecasts for a group of populations: an extension to the Lee–Carter method. Demography 42(3), 575–594 (2005)CrossRefGoogle Scholar
  37. 37.
    Møller, T.: Risk-minimizing hedging strategies for unit-linked life insurance contracts. ASTIN Bull. 28(1), 17–47 (1998)CrossRefGoogle Scholar
  38. 38.
    Møller, T.: Risk-minimizing hedging strategies for insurance payment processes. Financ. Stoch. 4(5), 419–446 (2001)Google Scholar
  39. 39.
    Norberg, R.: Optimal hedging of demographic risk in life insurance. Financ. Stoch. 17(1), 197–222 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  40. 40.
    Øksendal, Bernt: Stochastic Differential Equations: an Introduction with Applications, 6th edn. Springer, New York (2010)Google Scholar
  41. 41.
    Protter, P.E.: Stochastic Integration and Differential Equations. Springer, New York (2003)Google Scholar
  42. 42.
    Riesner, M.: Hedging life insurance contracts in a lévy process financial market. Insur. Math. Econ. 38(3), 599–608 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  43. 43.
    Schreiber, I.: Risk-minimization for life insurance liabilities. PhD thesis, University of Munich, (2012)Google Scholar
  44. 44.
    Schweizer, M.: A guided tour through quadratic hedging approaches. In: Jouini, E., Cvitanic, J., Musiela, M. (eds.) Option Pricing, Interest Rates and Risk Management, pp. 538–574. Cambridge University Press, Cambridge (2001)CrossRefGoogle Scholar
  45. 45.
    Schweizer, M.: Local risk-minimization for multidimensional assets and payment streams. Banach Cent. Publ. 83, 213–229 (2008)CrossRefMathSciNetGoogle Scholar
  46. 46.
    Wüthrich, M.V., Merz, M.: Financial Modeling, Actuarial Valuation and Solvency in Insurance. Springer Finance (2013)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Francesca Biagini
    • 1
    Email author
  • Thorsten Rheinländer
    • 2
  • Irene Schreiber
    • 3
  1. 1.Department of MathematicsUniversity of MunichMunichGermany
  2. 2.Financial and Actuarial Mathematics GroupVienna University of TechnologyViennaAustria
  3. 3.MunichGermany

Personalised recommendations