Risk Measures and Theories of Choice

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 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 Follmer and Shied (2002 a, b), and its properties lie between those of the exponential and distortion principles, which can be obtained as special cases.

[1]  M. Machina.  "Expected Utility" Analysis without the Independence Axiom , 1982 .

[2]  Jan Dhaene,et al.  Economic Capital Allocation Derived from Risk Measures , 2002 .

[3]  Hans U. Gerber,et al.  On Additive Premium Calculation Principles , 1974, ASTIN Bulletin.

[4]  Alexander Schied,et al.  Convex measures of risk and trading constraints , 2002, Finance Stochastics.

[5]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[6]  Cuncun Luan.  INSURANCE PREMIUM CALCULATIONS WITH ANTICIPATED UTILITY THEORY , 2001 .

[7]  Hans U. Gerber A.S.A.,et al.  Utility Functions: From Risk Theory to Finance , 1998 .

[8]  M. Yaari.  The Dual Theory of Choice under Risk , 1987 .

[9]  Clemens Puppe.  Distorted Probabilities and Choice under Risk , 1991 .

[10]  Philipp J. Schönbucher,et al.  Advances in Finance and Stochastics , 2002 .

[11]  D. Schmeidler.  Subjective Probability and Expected Utility without Additivity , 1989 .

[12]  J. Quiggin.  Generalized expected utility theory : the rank-dependent model , 1994 .

[13]  H. Fennema,et al.  A test of rank-dependent utility in the context of ambiguity , 1996 .

[14]  Shaun S. Wang,et al.  Axiomatic characterization of insurance prices , 1997 .

[15]  M. Allais.  Le comportement de l'homme rationnel devant le risque : critique des postulats et axiomes de l'ecole americaine , 1953 .

[16]  C. Goodhart,et al.  An academic response to Basel II , 2001 .

[17]  J. Quiggin.  A theory of anticipated utility , 1982 .

[18]  Satishs Iyengar,et al.  Multivariate Models and Dependence Concepts , 1998 .

[19]  Jan Dhaene,et al.  The Concept of Comonotonicity in Actuarial Science and Finance: Theory , 2002, Insurance: Mathematics and Economics.

[20]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[21]  Hans Bühlmann,et al.  Premium Calculation from Top Down , 1985, ASTIN Bulletin.

[22]  Shaun S. Wang.  Premium Calculation by Transforming the Layer Premium Density , 1996, ASTIN Bulletin.