Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T13:58:17.287Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi Risk-Neutral Pricing in Insurance

Published online by Cambridge University Press:  09 August 2013

Harry Niederau  and
Peter Zweifel
Harry Niederau
Affiliation:
Dr. Niederau Consulting & Research, Hochstrasse 26, 8044 Zurich, Switzerland, E-mail: info@niederau-research.com, Tel.: +41 43 537 03 51, Fax: +41 43 537 07 49
Peter Zweifel
Affiliation:
Socioeconomic Institute of the University of Zurich, Hottingerstrasse 10, 8032 Zurich, Switzerland, E-mail: pzweifel@soi.uzh.ch, Tel.: +41 44 634 22 05, Fax: +41 44 634 49 07
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

Incomplete marketsinsurance pricesLorenz orderproperty risksquasi risk-neutral pricingrisk aversionrisk neutralityrisk managementRisk Adjusted Capital (RAC)
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 39 , Issue 1 , May 2009 , pp. 317 - 337
Copyright
Copyright © International Actuarial Association 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Antal, P. (1997) Mathematische Methoden der Rückversicherung, lecture notes ETHZ, Zurich.Google Scholar
Artzner, P., Delbaen, F., Eber, J.M. and Heath, D. (1999) Coherent Measures of Risk, Mathematical Finance, 9(3), 203–228.Google Scholar
Bernegger, S. (1997) The Swiss Re Exposure Curves and MBBEFD Distribution Class, Astin Bulletin, 27(1), 99–111.Google Scholar
Bühlmann, H. (1980) An Economic Premium Principle, Astin Bulletin, 11, 52–60.CrossRefGoogle Scholar
Daykin, C.D., Pentikäinen, T. and Pesonen, M. (1994) Practical Risk Theory for Actuaries, Monographs on Statistics and Applied Probability, 53, Chapman and Hall.Google Scholar
Delbaen, F. and Haezendonck, J. (1989) A Martingale Approach to Premium Calculation Principles in an Arbitrage-Free Market, Insurance: Mathematics and Economics, 8(4), 269–277.Google Scholar
Denneberg, D. (1994) Non-Additive Measure and Integral, Boston: Kluver Academic Publishers.Google Scholar
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), 3–33.Google Scholar
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), 133–161.Google Scholar
Denuit, M., Dhaene, J.S., Goovaerts, M. and Kaas, R. (2005) Actuarial Theory for Dependent Risks – Measures, Orders and Models, New York: John Wiley.Google Scholar
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, 573–606.Google Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance, Heidelberg: Springer.CrossRefGoogle Scholar
Föllmer, H., Schweizer, M. (1991) Hedging of Contingent Claims Under Incomplete Information, Applied Stochastic Analysis (Davis, M. H. and Elliott, R.J., eds.), 389–414, Gordon and Breach.Google Scholar
Gerber, H.U. and Shiu, E.S. (1994) Option Pricing by Esscher Transforms (with discussion), Transactions of the Society of Actuaries, 46, 99–191.Google Scholar
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, 121–134.Google Scholar
Hardy, G., Littlewood, J. and Polya, G. (1929) Some Simple Inequalities Satisfied by Convex Functions, Messenger of Mathematics, 58, 48–152.Google Scholar
Hürlimann, W. (1998) On Stop-Loss Order and the Distortion Pricing Principle, Astin Bulletin, 28(1), 119–134.CrossRefGoogle Scholar
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, New York: John Wiley.Google Scholar
Niederau, H. (2000) Pricing Risk in Incomplete Markets: An Application to Industrial Reinsurance, Doctoral Thesis at the Socioeconomic Institute at the University of Zurich.Google Scholar
Pelsser, A. (2008) On the Applicability of the Wang Transform for Pricing Financial Risks, Astin Bulletin, 38(1), 171–181.Google Scholar
Reesor, R.M. and McLeish, D.L. (2003) Risk, Entropy, and the Transformation of Distributions, North American Actuarial Journal, 7(2), 128–144.Google Scholar
Shaked, M. and Shantikumar, J.G. (1997) Stochastic Orders and their Applications, Boston: Academic Press.Google Scholar
Varian, H.R. (1992) Micro-economic Analysis, New York: W.W. Norton & Company.Google Scholar
Wang, S., (2003) Cat Bond Pricing using Probability Transforms, Geneva Papers, 278, 19–29.Google Scholar
Wang, S. (2000) A Class of Distortion Operators for Pricing Financial and Insurance Risks, Journal of Risk and Insurance, 67(1), 15–36.CrossRefGoogle Scholar
Wang, S. (1995) Insurance Pricing and Increased Limits Ratemaking by Proportional Hazard Transforms, Insurance Mathematics and Economics, 17, 43–54.Google Scholar
Zweifel, P. and Auckenthaler, Ch. (2008) On the Feasibility of Insurer's Investment Policies, Journal of Risk and Insurance, 75(1), 193–206.CrossRefGoogle Scholar