Risk measurement with equivalent utility principles

Abstract

Summary: Risk measures have been studied for several decades in the actuarial literature, where they appeared under the guise of premium calculation principles. Risk measures and properties that risk measures should satisfy have recently received considerable attention in the financial mathematics literature. Mathematically, a risk measure is a mapping from a class of random variables to the real line. Economically, a risk measure should capture the preferences of the decision-maker. This paper complements the study initiated in Denuit, Dhaene & Van Wouwe (1999) and considers several theories for decision under uncertainty: the classical expected utility paradigm, Yaari's dual approach, maximin expected utility theory, Choquet expected utility theory and Quiggin's rank-dependent utility theory. Building on the actuarial equivalent utility pricing principle, broad classes of risk measures are generated, of which most classical risk measures appear to be particular cases. This approach shows that most risk measures studied recently in the financial mathematics literature disregard the utility concept (i.e., correspond to linear utilities), restricting their applicability. Some alternatives proposed in the literature are discussed.

Expected Utility and Catastrophic Risk

“…Furthermore, applications to insurance premium calculation and risk measurement using the equivalent (or zero) utility principle or the certainty equivalent (or mean value) principle (see Goovaerts, De Vylder andHaezendonck, 1984, andDenuit et al, 2006) are straightforward.…”

Risk Measures and Economic Capital for (Re)Insurers

“…More detailed discussions of risk measures resulting from alternative theories of choice under risk and references to the associated economics literature are given in [24], [32].…”

A note on the Swiss Solvency Test risk measure

“…At the same time, there has been a well-established axiomatic theory of risk measures in the financial and insurance mathematics literature, see e.g. Artzner et al (1999), Denuit et al (2006), Föllmer and Schied (2004), Kaas et al (2001), Artzner et al (2004) and Cheridito et al (2006) for the multiperiod case. In this context, it is worth mentioning that "optimal", not necessarily coherent, assessment of risk capital is still the subject of on-going research, see for instance Danielsson et al (2007), Dhaene et al (in press), Föllmer and Schied (2002) and Rootzen and Klüppelberg (1999).…”