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Risk Measures and Theories of Choice

Published online by Cambridge University Press:  10 June 2011

A. Tsanakas  and
E. Desli
A. Tsanakas
Affiliation:
Market Risk and Reserving Unit, Lloyd's of London, One Lime Street, London EC3M 7HA, U.K., Tel: +44(0)20-7327-5268, Fax: +44(0)20-7327-5658, Email: [email protected].
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Abstract

We discuss classes of risk measures in terms both of their axiomatic definitions and of the economic theories of choice that they can be derived from. More specifically, expected utility theory gives rise to the exponential premium principle, proposed by Gerber (1974), Dhaene et al. (2003), whereas Yaari's (1987) dual theory of choice under risk can be viewed as the source of the distortion premium principle (Denneberg, 1990; Wang, 1996). We argue that the properties of the exponential and distortion premium principles are complementary, without either of the two performing completely satisfactorily as a risk measure. Using generalised expected utility theory (Quiggin, 1993), we derive a new risk measure, which we call the distortion-exponential principle. This risk measure satisfies the axioms of convex measures of risk, proposed by Föllmer & Shied (2002a,b), and its properties lie between those of the exponential and distortion principles, which can be obtained as special cases.

Keywords

Risk MeasuresPremium Calculation PrinciplesCoherent Measures of RiskDistortion Premium PrincipleExponential Premium PrincipleExpected UtilityDual Theory of Choice Under RiskGeneralised Expected Utility
Type
Sessional meetings: papers and abstracts of discussions
Information
British Actuarial Journal , Volume 9 , Issue 4 , 01 October 2003 , pp. 959 - 991
Copyright
Copyright © Institute and Faculty of Actuaries 2003

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References

Allais, M. (1953). Le comportement de l'homme rationnel devant le risque: critique des postulates et axioms de l'ecole Americaine. Econometrica, 21, 503–546.Google Scholar
Artzner, P., Delbaen, F., Eber, J-M. & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9 (3), 203–228.CrossRefGoogle Scholar
Buhlmann, H. (1970). Mathematical methods in risk theory. Springer, Berlin.Google Scholar
Buhlmann, H. (1985). Premium calculation from top down. ASTIN Bulletin, 15 (2), 89–101.Google Scholar
Danielson, J., Embrechts, P., Goodhart, C., Keating, C., Muennich, F., Renault, O. & Shin, H.S. (2001). An academic response to Basel II. Financial Markets Group. Working paper, London School of Economics, http://risk.lse.ac.ukGoogle Scholar
Denneberg, D. (1990). Distorted probabilities and insurance premiums. Methods of Operations Research, 63, 3–5.Google Scholar
Denneberg, D. (1994). Non-additive measure and integral. Kluwer Academic Publishers, Dordrecht.Google Scholar
Dhaene, J., Denuit, M., Goovaerts, M.J., Kaas, R. & Vyncke, D. (2002). The concept of comonotonicity in actuarial science and finance: theory. Insurance: Mathematics and Economics, 31 (1), 3–33.Google Scholar
Dhaene, J., Goovaerts, M. & Kaas, R. (2003). Economic capital allocation derived from risk measures. North American Actuarial Journal, 7 (2).CrossRefGoogle Scholar
Fennema, H.P. & Wakker, P.P. (1996). A test of rank dependent utility in the context of ambiguity. Journal of Risk and Uncertainty, 13 (1), 19–36.Google Scholar
Föllmer, H. & Schied, A. (2002a). Convex measures of risk and trading constraints. Finance and Stochastics, 6 (4), 429–447.Google Scholar
Föllmer, H. & Schied, A. (2002b). Robust preferences and convex risk measures. Advances in Finance and Stochastics, Springer (to appear).Google Scholar
Georgantzis, N., Genius, M., Garca-Gallego, A. & Sabater-Grande, G. (2002). Do higher risk premia induce riskier options? An experimental test. Sixth International Congress on Insurance Mathematics and Economics, 15–17 July 2002.Google Scholar
Gerber, H.U. (1974). On additive premium calculation principles. ASTIN Bulletin, 7 (3), 215–222.Google Scholar
Gerber, H.U. & Pafumi, G. (1998). Utility functions: from risk theory to finance. North American Actuarial Journal, 2 (3), 74–100.Google Scholar
Goovaerts, M.J., de Vylder, F. & Haezendonck, J. (1984). Insurance premiums. Elsevier Science.Google Scholar
Joe, H. (1997). Multivariate models and dependence concepts. Chapman & Hall, London.Google Scholar
Luan, C. (2001). Insurance premium calculations with anticipated utility theory. ASTIN Bulletin, 31 (1), 23–35.Google Scholar
Machina, M. (1982). ‘Expected utility' analysis without the independence axiom. Econometrica, 50, 277–323.Google Scholar
Puppe, C. (1991). Distorted probabilities and choice under risk. Springer, Berlin.Google Scholar
Quiggin, J. (1982). A theory of anticipated utility. Journal of Economic Behavior and Organization, 3, 323–343.CrossRefGoogle Scholar
Quiggin, J. (1993). Generalized expected utility theory: the rank-dependent model. Kluwer Academic Publishers, Boston.CrossRefGoogle Scholar
Schmeidler, D. (1989). Subjective probability and expected utility without additivity. Econometrica, 57 (3), 571–587.Google Scholar
von Neumann, J.R. & Morgenstern, O. (1947). Theory of games and economic behaviour. Princeton University Press.Google Scholar
Wang, S. (1996). Premium calculation by transforming the premium layer density. ASTIN Bulletin, 26 (1), 71–92.Google Scholar
Wang, S.S., Young, V.R. & Panjer, H.H. (1997). Axiomatic characterization of insurance prices. Insurance: Mathematics and Economics, 21 (2), 173–183.Google Scholar
Yaari, M. (1987). The dual theory of choice under risk. Econometrica, 55 (1), 95–115.CrossRefGoogle Scholar