Risk-minimization for life insurance liabilities with basis risk
Tóm tắt
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.
Tài liệu tham khảo
Ansel, J.P., Stricker, C.: Décomposition de Kunita-Watanabe. In: Séminaire de Probabilités XXVII, Lecture Notes in Mathematics Volume 1557. Springer, New York (1993)
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)
Barbarin, J.: Valuation, Hedging and the Risk Management of Insurance Contracts. PhD thesis, Catholic University of Louvain, (2008b)
Barrieu, P., Albertini, L.: The Handbook of Insurance-Linked Securities. Wiley-Blackwell, London (2009)
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)
Biagini, F., Cretarola, A.: Quadratic hedging methods for defaultable claims. Appl. Math. Optim. 56(3), 425–443 (2007)
Biagini, F., Cretarola, A.: Local risk-minimization for defaultable markets. Math. Financ. 19(4), 669–689 (2009)
Biagini, F., Cretarola, A.: Local risk-minimization with recovery process. Appl. Math. Optim. 65(3), 293–314 (2012)
Biagini, F., Schreiber, I.: Risk-minimization for life insurance liabilities. SIAM J. Financ. Math. 4(1), 243–264 (2013)
Biagini, F., Rheinländer, T., Widenmann, J.: Hedging mortality claims with longevity bonds. ASTIN Bull. 43(2), 123–157 (2013)
Biagini, F., Botero, C., Schreiber, I.: Risk-minimization for life insurance liabilities with dependent mortality risk. Math. Financ. (2015) (accepted)
Bielecki, T.R., Rutkowski, M.: Credit Risk: Modeling, Valuation and Hedging. Springer Finance (2004)
Biffis, E.: Affine processes for dynamic mortality and actuarial valuations. Insur. Math. Econ. 37(3), 443–468 (2005)
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
Blake, D., Cairns, A.J.G., Dowd, K.: The birth of the life market. Asia-Pac. J. Risk Insur. 3(1), 6–36 (2008)
Blanchet-Scalliet, C., Jeanblanc, M.: Hazard rate for credit risk and hedging defaultable contingent claims. Financ. Stoch. 8(1), 145–159 (2004)
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)
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)
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)
Dahl, M., Møller, T.: Valuation and hedging of life insurance liabilities with systematic mortality risk. Insur. Math. Econ. 39(2), 193–217 (2006)
Dahl, M., Melchior, M., Møller, T.: On systematic mortality risk and risk-minimization with survivor swaps. Scand. Actuar. J. 2–3, 114–146 (2008)
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)
Dai, Q., Singleton, K.: Specification analysis of affine term structure models. J. Financ. 55(5), 1943–1978 (2000)
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)
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)
Duffie, D., Pan, J., Singleton, K.: Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 1343–1376 (2000)
Duffie, D., Filipović, D., Schachermayer, W.: Affine processes and applications in finance. Ann. Appl. Probab. 13(3), 984–1053 (2003)
Filipović, D., Mayerhofer, E.: Affine diffusion processes: theory and applications. Radon Ser. Comput. Appl. Math. 8, 1–40 (2009)
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)
Friedman, A.: Stochastic Differential Equations and Applications. Springer, New York (1975)
Henriksen, L.F.B., Møller, T.: Local risk-minimization with longevity bonds. Appl. Stoch. Models Bus. Ind. 31(2), 241–263 (2015)
Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer, New York (2002)
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)
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1991)
Li, J.S.-H., Hardy, M.R.: Measuring basis risk in longevity hedges. N. Am. Actuar. J. 15(2), 177–200 (2011)
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)
Møller, T.: Risk-minimizing hedging strategies for unit-linked life insurance contracts. ASTIN Bull. 28(1), 17–47 (1998)
Møller, T.: Risk-minimizing hedging strategies for insurance payment processes. Financ. Stoch. 4(5), 419–446 (2001)
Norberg, R.: Optimal hedging of demographic risk in life insurance. Financ. Stoch. 17(1), 197–222 (2013)
Øksendal, Bernt: Stochastic Differential Equations: an Introduction with Applications, 6th edn. Springer, New York (2010)
Protter, P.E.: Stochastic Integration and Differential Equations. Springer, New York (2003)
Riesner, M.: Hedging life insurance contracts in a lévy process financial market. Insur. Math. Econ. 38(3), 599–608 (2006)
Schreiber, I.: Risk-minimization for life insurance liabilities. PhD thesis, University of Munich, (2012)
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)
Schweizer, M.: Local risk-minimization for multidimensional assets and payment streams. Banach Cent. Publ. 83, 213–229 (2008)
Wüthrich, M.V., Merz, M.: Financial Modeling, Actuarial Valuation and Solvency in Insurance. Springer Finance (2013)