# Risk-minimization for life insurance liabilities with basis risk

- 379 Downloads
- 4 Citations

## Abstract

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.

## Keywords

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## Notes

### Acknowledgments

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

## References

- 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.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.Barbarin, J.: Valuation, Hedging and the Risk Management of Insurance Contracts. PhD thesis, Catholic University of Louvain, (2008b)Google Scholar
- 4.Barrieu, P., Albertini, L.: The Handbook of Insurance-Linked Securities. Wiley-Blackwell, London (2009)Google Scholar
- 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.Biagini, F., Cretarola, A.: Quadratic hedging methods for defaultable claims. Appl. Math. Optim.
**56**(3), 425–443 (2007)CrossRefMathSciNetzbMATHGoogle Scholar - 7.Biagini, F., Cretarola, A.: Local risk-minimization for defaultable markets. Math. Financ.
**19**(4), 669–689 (2009)CrossRefMathSciNetzbMATHGoogle Scholar - 8.Biagini, F., Cretarola, A.: Local risk-minimization with recovery process. Appl. Math. Optim.
**65**(3), 293–314 (2012)CrossRefMathSciNetzbMATHGoogle Scholar - 9.Biagini, F., Schreiber, I.: Risk-minimization for life insurance liabilities. SIAM J. Financ. Math.
**4**(1), 243–264 (2013)CrossRefMathSciNetzbMATHGoogle Scholar - 10.Biagini, F., Rheinländer, T., Widenmann, J.: Hedging mortality claims with longevity bonds. ASTIN Bull.
**43**(2), 123–157 (2013)CrossRefMathSciNetzbMATHGoogle Scholar - 11.Biagini, F., Botero, C., Schreiber, I.: Risk-minimization for life insurance liabilities with dependent mortality risk. Math. Financ. (2015) (accepted)Google Scholar
- 12.Bielecki, T.R., Rutkowski, M.: Credit Risk: Modeling, Valuation and Hedging. Springer Finance (2004)Google Scholar
- 13.Biffis, E.: Affine processes for dynamic mortality and actuarial valuations. Insur. Math. Econ.
**37**(3), 443–468 (2005)CrossRefMathSciNetzbMATHGoogle Scholar - 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.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.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.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.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.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.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.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.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.Dai, Q., Singleton, K.: Specification analysis of affine term structure models. J. Financ.
**55**(5), 1943–1978 (2000)CrossRefGoogle Scholar - 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.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.Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica
**68**, 1343–1376 (2000)CrossRefMathSciNetzbMATHGoogle Scholar - 27.Duffie, D., Filipović, D., Schachermayer, W.: Affine processes and applications in finance. Ann. Appl. Probab.
**13**(3), 984–1053 (2003)CrossRefMathSciNetzbMATHGoogle Scholar - 28.Filipović, D., Mayerhofer, E.: Affine diffusion processes: theory and applications. Radon Ser. Comput. Appl. Math.
**8**, 1–40 (2009)Google Scholar - 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.Friedman, A.: Stochastic Differential Equations and Applications. Springer, New York (1975)zbMATHGoogle Scholar
- 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.Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer, New York (2002)Google Scholar
- 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.Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1991)zbMATHGoogle Scholar
- 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.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.Møller, T.: Risk-minimizing hedging strategies for unit-linked life insurance contracts. ASTIN Bull.
**28**(1), 17–47 (1998)CrossRefGoogle Scholar - 38.Møller, T.: Risk-minimizing hedging strategies for insurance payment processes. Financ. Stoch.
**4**(5), 419–446 (2001)Google Scholar - 39.Norberg, R.: Optimal hedging of demographic risk in life insurance. Financ. Stoch.
**17**(1), 197–222 (2013)CrossRefMathSciNetzbMATHGoogle Scholar - 40.Øksendal, Bernt: Stochastic Differential Equations: an Introduction with Applications, 6th edn. Springer, New York (2010)Google Scholar
- 41.Protter, P.E.: Stochastic Integration and Differential Equations. Springer, New York (2003)Google Scholar
- 42.Riesner, M.: Hedging life insurance contracts in a lévy process financial market. Insur. Math. Econ.
**38**(3), 599–608 (2006)CrossRefMathSciNetzbMATHGoogle Scholar - 43.Schreiber, I.: Risk-minimization for life insurance liabilities. PhD thesis, University of Munich, (2012)Google Scholar
- 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.Schweizer, M.: Local risk-minimization for multidimensional assets and payment streams. Banach Cent. Publ.
**83**, 213–229 (2008)CrossRefMathSciNetGoogle Scholar - 46.Wüthrich, M.V., Merz, M.: Financial Modeling, Actuarial Valuation and Solvency in Insurance. Springer Finance (2013)Google Scholar