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From the Equivalence Principle to Market Consistent Valuation

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Abstract

Insurance companies are exposed to many different types of risk, in particular actuarial as well as financial risks. As a consequence, the classical actuarial principle of pooling does not provide a sufficient basis for the valuation and risk management of the total portfolio of an insurance company. Instead, the methodology needs to be complemented by modern financial mathematics that enables a market consistent valuation. The current article provides an introduction to the fundamental principles of financial mathematics that were originally developed by Fischer Black, Robert Merton and Myron Scholes in the beginning of the 1970s. We will discuss the relevance of these concepts for insurance firms in the context of internal models and the computation of the market consistent embedded value (MCEV).

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References

  1. The Turner review—A regulatory response to the global banking crisis, published by the British Financial Services Authority, www.fsa.gov.uk/pubs/other/turner_review.pdf (2009)

  2. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9(3), 203–228 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bachelier, L.: Théorie de la spéculation. Les Grands Classiques Gauthier-Villars. [Gauthier-Villars Great Classics]. In: Sceaux, J.G. (ed.) Théorie mathématique du jeu. [Mathematical theory of games] (1995). Reprint of the 1900 original

    Google Scholar 

  4. Bielecki, T.R., Rutkowski, M.: Credit risk: modelling, valuation, and hedging. Springer Finance. Springer, Berlin (2002)

    MATH  Google Scholar 

  5. Black, F., Scholes, M.S.: The pricing of options and corporate liabilities. J. Polit. Econom. 81, 637–654 (1973)

    Article  Google Scholar 

  6. Box, G.E.P., Draper, N.R.: Empirical Model-building and Response Surfaces. Wiley Series in Probability and Mathematical Statistics: Applied Probability and Statistics. Wiley, New York (1987)

    MATH  Google Scholar 

  7. Brigo, D., Mercurio, F.: Interest Rate Models—Theory and Practice, 2nd edn. Springer Finance. Springer, Berlin (2006). With smile, inflation and credit

    MATH  Google Scholar 

  8. Bühlmann, H.: Multidimensional valuation. Finance 25, 15–29 (2004)

    Google Scholar 

  9. Cvitanić, J., Karatzas, I.: Generalized Neyman-Pearson lemma via convex duality. Bernoulli 7(1), 79–97 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Delbaen, F., Schachermayer, W.: The Mathematics of Arbitrage. Springer Finance. Springer, Berlin, (2006)

    MATH  Google Scholar 

  11. Donsker, M.D.: An invariance principle for certain probability limit theorems. Mem. Am. Math. Soc. 1951(6), 12 (1951)

    MathSciNet  Google Scholar 

  12. El Karoui, N., Quenez, M.-C.: Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control Optim. 33(1), 29–66 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  13. Favero, G.: Shortfall risk minimization under model uncertainty in the binomial case: adaptive and robust approaches. Math. Methods Oper. Res. 53(3), 493–503 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Favero, G., Runggaldier, W.J.: A robustness result for stochastic control. Syst. Control Lett. 46(2), 91–97 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  15. Filipović, D.: Term-Structure Models. Springer Finance. Springer, Berlin (2009). A graduate course

    Book  MATH  Google Scholar 

  16. Föllmer, H., Leukert, P.: Quantile hedging. Finance Stoch. 3(3), 251–273 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Föllmer, H., Leukert, P.: Efficient hedging: cost versus shortfall risk. Finance Stoch. 4(2), 117–146 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Föllmer, H., Schied, A.: Convex measures of risk and trading constraints. Finance Stoch. 6(4), 429–447 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  19. Föllmer, H., Schied, A.: Stochastic finance—an introduction in discrete time, 3rd edn. Graduate Textbook Series. Walter de Gruyter, Berlin (2011)

    Google Scholar 

  20. Föllmer, H., Schied, A., Weber, S.: Robust preferences and robust portfolio choice. In: Bensoussan, A., Zhang (eds.) Handbook of Numerical Analysis, Mathematical Modeling and Numerical Methods in Finance, pp. 29–89 (2009)

    Google Scholar 

  21. Frittelli, M., Rosazza Gianin, E.: Law invariant convex risk measures. Adv. Math. Econ. 7, 33–46 (2005)

    Article  Google Scholar 

  22. Gerber, H.U.: Life Insurance Mathematics, 3rd edn. Springer, Berlin (1997)

    MATH  Google Scholar 

  23. Giesecke, K., Schmidt, T., Weber, S.: Measuring the risk of large losses. J. Invest. Manag. 6(4), 1–15 (2008)

    Google Scholar 

  24. Heyde, C.C., Kou, S.G., Peng, X.H.: What is a good external risk measure: Bridging the gaps between robustness, subadditivity, and insurance risk measures. Preprint, Columbia University (2007)

  25. Itô, K.: Selected Papers. Springer, New York (1987). Edited and with an introduction by S.R.S. Varadhan and D.W. Stroock

    MATH  Google Scholar 

  26. Jaschke, S.: Valuation and risk management of life reinsurance. In: Talk, LUH-Kolloquium Versicherungs- und Finanzmathematik, Hannover (2010)

    Google Scholar 

  27. Kallenberg, O.: Foundations of modern probability. Probability and its Applications (New York), 2nd edn. Springer, New York (2002)

    MATH  Google Scholar 

  28. Kirch, M.: Efficient hedging in incomplete markets under model uncertainty. PhD thesis, Humboldt-Universität zu Berlin. http://edoc.hu-berlin.de/dissertationen/kirch-michael-2002-01-07/PDF/Kirch.pdf (2001)

  29. Kirch, M., Runggaldier, W.J.: Efficient hedging when asset prices follow a geometric Poisson process with unknown intensities. SIAM J. Control Optim. 43(4), 1174–1195 (2004/05) (electronic)

    Article  MathSciNet  Google Scholar 

  30. Klöppel, S., Schweizer, M.: Dynamic indifference valuation via convex risk measures. Math. Finance 17(4), 599–627 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Koller, M.: Stochastische Modelle in der Lebensversicherung, 2nd edn. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  32. Kramkov, D.O.: Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets. Probab. Theory Relat. Fields 105(4), 459–479 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kunze, M.: Verteilungsinvariante konvexe Risikomaße. Diploma thesis, Humboldt-Universität zu Berlin (2003)

  34. Kusuoka, S.: On law invariant coherent risk measures. Adv. Math. Econ. 3, 83–95 (2001)

    MathSciNet  Google Scholar 

  35. Merton, R.C.: Theory of rational option pricing. Bell J. Econom. Manag. Sci. 4, 141–183 (1973)

    Article  MathSciNet  Google Scholar 

  36. Milbrodt, H., Helbig, M.: Mathematische Methoden der Personenversicherung. Walter de Gruyter, Berlin (1999)

    Book  MATH  Google Scholar 

  37. Møller, T., Steffensen, M.: Market-valuation methods in life and pension insurance. International Series on Actuarial Science. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  38. Musiela, M., Rutkowski, M.: Martingale Methods in Financial Modelling, 2nd edn. Stochastic Modelling and Applied Probability, vol. 36. Springer, Berlin (2005)

    MATH  Google Scholar 

  39. Nakano, Y.: Efficient hedging with coherent risk measures. J. Math. Anal. Appl. 293(1), 345–354 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  40. Overhaus, M., Bemudez, A., Bühler, H., Ferraris, A., Jordinson, C., Lamnouar, A.: Credit Hybrid Derivatives. Wiley Finance. Wiley, New York (2007)

    Google Scholar 

  41. Petit, M.: Die verlorene Gleichung-Auf den Spuren von Wolfgang und Alfred Döblin. Eichborn Verlag (2005)

  42. Protter, P.E.: Stochastic integration and differential equations, 2nd edn. Stochastic Modelling and Applied Probability, vol. 21. Springer, Berlin, (2005). Version 2.1, Corrected third printing

    Google Scholar 

  43. Rudloff, B.: Convex hedging in incomplete markets. Appl. Math. Finance 14(5), 437–452 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  44. Rudloff, B.: Coherent hedging in incomplete markets. Quant. Finance 9(2), 197–206 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  45. Samuelson, P.A.: Rational theory of warrant pricing. Ind. Manage. Rev. 6(2), 13–39 (1951)

    Google Scholar 

  46. Schied, A.: On the Neyman-Pearson problem for law-invariant risk measures and robust utility functionals. Ann. Appl. Probab. 14(3), 1398–1423 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  47. Schied, A.: Risk measures and robust optimization problems. Stoch. Models 22(4), 753–831 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  48. Sekine, J.: Dynamic minimization of worst conditional expectation of shortfall. Math. Finance 14(4), 605–618 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  49. Wüthrich, M.V., Bühlmann, H., Furrer, H.: Market-consistent actuarial valuation. European Actuarial Academy (EAA) Series. Springer, Heidelberg (2010) (enlarged edition)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematische Stochastik, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Thomas Knispel & Stefan Weber

  2. Talanx AG, Quantitative Risk Management, Riethorst 2, 30659, Hannover, Germany

    Gerhard Stahl

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  1. Thomas Knispel
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  2. Gerhard Stahl
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  3. Stefan Weber
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Correspondence to Thomas Knispel.

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Knispel, T., Stahl, G. & Weber, S. From the Equivalence Principle to Market Consistent Valuation. Jahresber. Dtsch. Math. Ver. 113, 139–172 (2011). https://doi.org/10.1365/s13291-011-0022-y

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