Entropy Coherent and Entropy Convex Measures of Risk

Abstract

We introduce two subclasses of convex measures of risk, referred to as entropy coherent and entropy convex measures of risk. Entropy coherent and entropy convex measures of risk are special cases of -coherent and -convex measures of risk. Contrary to the classical use of coherent and convex measures of risk, which for a given probabilistic model entails evaluating a financial position by considering its expected loss, -coherent and -convex measures of risk evaluate a financial position under a given probabilistic model by considering its normalized expected -loss. We prove that (i) entropy coherent and entropy convex measures of risk are obtained by requiring -coherent and -convex measures of risk to be translation invariant; (ii) convex, entropy convex, and entropy coherent measures of risk emerge as certainty equivalents under variational, homothetic, and multiple priors preferences upon requiring the certainty equivalents to be translation invariant; and (iii) -convex measures of risk are certainty equivalents under variational and homothetic preferences if and only if they are convex and entropy convex measures of risk. In addition, we study the properties of entropy coherent and entropy convex measures of risk, derive their dual conjugate function, and characterize entropy coherent and entropy convex measures of risk in terms of properties of the corresponding acceptance sets.

Robust Portfolio Choice and Indifference Valuation

“…Multiple priors occurs when c(Q) = I M , the penalty function that is zero if Q ∈ M and ∞ otherwise, attaching the same plausibility to all probabilistic models in M . In the case that u is linear, multiple priors corresponds to coherent risk measures (Artzner et al [2]) and variational preferences corresponds to convex risk measures (Föllmer and Schied [26], Frittelli and Rosazza Gianin [29], Ruszczyński and Shapiro [67]); see Laeven and Stadje [47] for further results on these connections. We solve the portfolio choice and valuation problems, adopting certain versions of variational preferences (2.2).…”

“…Robust exponential utility functionals are also considered in Laeven and Stadje [47]. Then the driver function corresponding to the optimal portfolio choice is given by (cf.…”

A Celebration of the Ties That Bind Us: Connections between Actuarial Science and Mathematical Finance

“…• Technical and axiomatic characterization of risk measures: Furman and Zitikis 2008;Laeven and Stadje 2013).…”

Inf-convolution and optimal risk sharing with countable sets of risk measures

“…In Section 4, we also study in detail the framework of worst-case risk measures, an important class of set risk measures which appraise the risk of a collection of financial positions as the pointwise supremum of the individual risks (as measured by some traditional risk measure) of the positions in the set. This has become a prominent stream of studies such as Föllmer and Schied (2002), Laeven and Stadje (2013), Ang et al (2018), Wang and Ziegel (2018), Bartl et al (2019), Bellini et al (2018), Guo and Xu (2019), Qian et al (2019), Righi (2019), Bernard et al (2023), Fadina et al (2021), Righi (2024).…”