Axiomatic characterization of insurance prices

Abstract

In this paper, we take an axiomatic approach to characterize insurance prices in a competitive market setting. We present four axioms to describe the behavior of market insurance prices. From these axioms it follows that the price of an insurance risk has a Choquet integral representation with respect to a distorted probability, (Yanri, 1987). We propose an additional axiom for reducing compound risks. This axiom determines that the distortion function is a power function.

Optimal risk sharing under distorted probabilities

“…For more background on such risk measures H g , see Yaari [33] who discusses evaluating random variables in a theory of risk that is dual to expected utility. See also Wang et al [31] for an axiomatic definition of distortion risk measures as law-invariant, coherent and comonotone-additive functionals on P. Two noteworthy examples of distortion risk measures are (1) the Average Value-at-Risk at level 1 − α −1 (AVaR) obtained by taking g( p) = min(αp, 1) for some α > 1 and (2) the proportional hazards transform g( p) = p c for some 0 < c < 1.…”

Risk capital allocation and cooperative pricing of insurance liabilities

“…This type of risk measures have been axiomatically defined in the context of insurance pricing by Denneberg (1990) and Wang et al (1997). They are termed distortion principles, since the non-linear function g 'distorts' the physical probability measure P 0 .…”

“…In this paper we calculate an analytic formula for the Aumann-Shapley value using quantile derivatives (Tasche 2000b), for the case that the risk measure belongs to the class of distortion principles (Denneberg (1990), Wang et al (1997)). We obtain a representation of the Aumann-Shapley value, i.e.…”

Optimal insurance in the presence of reinsurance

“…A risk measure is called a distortion risk measure with distortion function g ∈ G, denoted by ρ g , if and only if it admits the following representation Note that the second term in (1) vanishes for non-negative risk. Distortion risk measures are based on dual utilities (Yaari 1987), and are introduced by Wang et al (1997) as a premium principle. If the distortion function g is concave, the distortion risk measure is coherent, as defined by Artzner et al (1999).…”