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Quasi Risk-Neutral Pricing in Insurance

  • Harry Niederau (a1) and Peter Zweifel (a2)

Abstract

This contribution shows that for certain classes of insurance risks, pricing can be based on expected values under a probability measure ℙ* amounting to quasi risk-neutral pricing. This probability measure is unique and optimal in the sense of minimizing the relative entropy with respect to the actuarial probability measure ℙ, which is a common approach in the case of incomplete markets. After expounding the key elements of this theory, an application to a set of industrial property risks is developed, assuming that the severity of losses can be modeled by “Swiss Re Exposure Curves”, as discussed by Bernegger (1997). These curves belong to a parametric family of distribution functions commonly used by pricing actuaries. The quasi risk-neutral pricing approach not only yields risk exposure specific premiums but also Risk Adjusted Capital (RAC) values on the very same level of granularity. By way of contrast, the conventional determination of RAC is typically considered on a portfolio level only.

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Antal, P. (1997) Mathematische Methoden der Rückversicherung, lecture notes ETHZ, Zurich.
Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999) Coherent Measures of Risk, Mathematical Finance, 9(3), 203228.
Bernegger, S. (1997) The Swiss Re Exposure Curves and MBBEFD Distribution Class, Astin Bulletin, 27(1), 99111.
Bühlmann, H. (1980) An Economic Premium Principle, Astin Bulletin, 11, 5260.
Daykin, C.D., Pentikäinen, T. and Pesonen, M. (1994) Practical Risk Theory for Actuaries, Monographs on Statistics and Applied Probability, 53, Chapman and Hall.
Delbaen, F. and Haezendonck, J. (1989) A Martingale Approach to Premium Calculation Principles in an Arbitrage-Free Market, Insurance: Mathematics and Economics, 8(4), 269277.
Denneberg, D. (1994) Non-Additive Measure and Integral, Boston: Kluver Academic Publishers.
Denuit, M., Dhaene, J.S., Goovaerts, M., Kaas, R. and Vyncke, D. (2002a) The Concept of Comonotonicity in Actuarial Science and Finance: Theory, Insurance: Mathematics & Economics, 31(1), 333.
Denuit, M., Dhaene, J.S., Goovaerts, M., Kaas, R. and Vyncke, D. (2002b) The Concept of Comonotonicity in Actuarial Science and Finance: Applications, Insurance: Mathematics & Economics, 31(2), 133161.
Denuit, M., Dhaene, J.S., Goovaerts, M. and Kaas, R. (2005) Actuarial Theory for Dependent Risks – Measures, Orders and Models, New York: John Wiley.
Dhaene, J.S., Vanduffel, Q., Tang, M.J., Goovaerts, R., Kaas, R. and Vyncke, D. (2006) Risk Measures and Comonotonicity: A Review, Stochastic Models, 22, 573606.
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance, Heidelberg: Springer.
Föllmer, H., Schweizer, M. (1991) Hedging of Contingent Claims Under Incomplete Information, Applied Stochastic Analysis (Davis, M. H. and Elliott, R.J., eds.), 389414, Gordon and Breach.
Gerber, H.U. and Shiu, E.S. (1994) Option Pricing by Esscher Transforms (with discussion), Transactions of the Society of Actuaries, 46, 99191.
Goovaerts, M.J. and Dhaene, J. (1998) On the Characterization on Wang's Class of Premium Principles, Transactions of the 26th International Congress of Actuaries, 4, 121134.
Hardy, G., Littlewood, J. and Polya, G. (1929) Some Simple Inequalities Satisfied by Convex Functions, Messenger of Mathematics, 58, 48152.
Hürlimann, W. (1998) On Stop-Loss Order and the Distortion Pricing Principle, Astin Bulletin, 28(1), 119134.
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, New York: John Wiley.
Niederau, H. (2000) Pricing Risk in Incomplete Markets: An Application to Industrial Reinsurance, Doctoral Thesis at the Socioeconomic Institute at the University of Zurich.
Pelsser, A. (2008) On the Applicability of the Wang Transform for Pricing Financial Risks, Astin Bulletin, 38(1), 171181.
Reesor, R.M. and McLeish, D.L. (2003) Risk, Entropy, and the Transformation of Distributions, North American Actuarial Journal, 7(2), 128144.
Shaked, M. and Shantikumar, J.G. (1997) Stochastic Orders and their Applications, Boston: Academic Press.
Varian, H.R. (1992) Micro-economic Analysis, New York: W.W. Norton & Company.
Wang, S., (2003) Cat Bond Pricing using Probability Transforms, Geneva Papers, 278, 1929.
Wang, S. (2000) A Class of Distortion Operators for Pricing Financial and Insurance Risks, Journal of Risk and Insurance, 67(1), 1536.
Wang, S. (1995) Insurance Pricing and Increased Limits Ratemaking by Proportional Hazard Transforms, Insurance Mathematics and Economics, 17, 4354.
Zweifel, P. and Auckenthaler, Ch. (2008) On the Feasibility of Insurer's Investment Policies, Journal of Risk and Insurance, 75(1), 193206.

Keywords

Quasi Risk-Neutral Pricing in Insurance

  • Harry Niederau (a1) and Peter Zweifel (a2)

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