Entropy Coherent and Entropy Convex Measures of Risk

We introduce two subclasses of convex measures of risk, referred to as entropy coherent and entropy convex measures of risk. We prove that convex, entropy convex and entropy coherent measures of risk emerge as certainty equivalents under variational, homothetic and multiple priors preferences, respectively, upon requiring the certainty equivalents to be translation invariant. In addition, we study the properties of entropy coherent and entropy convex measures of risk, derive their dual conjugate function, and prove their distribution invariant representation. Some financial applications and examples of entropy coherent and entropy convex measures of risk are also investigated.Multiple priors;Variational and homothetic preferences;Robustness;Convex risk measures;Exponential utility;Relative entropy;Translation invariance;Convexity;Indifference valuation

Entropy Coherent and Entropy Convex Measures of Risk

We introduce entropy coherent and entropy convex measures of risk and prove a collection of axiomatic characterization and duality results. We show in particular that entropy coherent and entropy convex measures of risk emerge as negative certainty equivalents in (the regular and a generalized version, respectively, of) the popular maxmin expected utility theory of Gilboa and Schmeidler [12] whenever the negative certainty equivalents are translation invariant. In addition, we derive the dual conjugate function for entropy coherent and entropy convex measures of risk, and prove their distribution invariant representation. Keywords: Robust preferences; Convex risk measures; Exponential utility; Relative entropy; Translation invariance; Convexity

Entropy coherent and entropy convex measures of risk

We introduce entropy coherent and entropy convex measures of risk and prove a collection of axiomatic characterization and duality results. We show in particular that entropy coherent and entropy convex measures of risk emerge as negative certainty equivalents in (the regular and a generalized version, respectively, of) the popular maxmin expected utility theory of Gilboa and Schmeidler [12] whenever the negative certainty equivalents are translation invariant. In addition, we derive the dual conjugate function for entropy coherent and entropy convex measures of risk, and prove their distribution invariant representation. Keywords: Robust preferences; Convex risk measures; Exponential utility; Relative entropy; Translation invariance; Convexity

Determination of Risk Pricing Measures from Market Prices of Risk

A new insurance provider or a regulatory agency may be interested in determining a risk measure consistent with observed market prices of a collection of risks. Using a relationship between distorted coherent risk measures and spectral risk measures, we provide a method for reconstruction distortion functions from the observed prices of risk. The technique is based on an appropriate application of the method on maximum entropy in the mean.

Robust Optimal Risk Sharing and Risk Premia in Expanding Pools

We consider the problem of optimal risk sharing in a pool of cooperative agents. We analyze the asymptotic behavior of the certainty equivalents and risk premia associated with the Pareto optimal risk sharing contract as the pool expands. We first study this problem under expected utility preferences with an objectively or subjectively given probabilistic model. Next, we develop a robust approach by explicitly taking uncertainty about the probabilistic model (ambiguity) into account. The resulting robust certainty equivalents and risk premia compound risk and ambiguity aversion. We provide explicit results on their limits and rates of convergence, induced by Pareto optimal risk sharing in expanding pools

Merging of Opinions under Uncertainty

We consider long-run behavior of agents assessing risk in terms of dynamic convex risk measures or, equivalently, utility in terms of dynamic variational preferences in an uncertain setting. By virtue of a robust representation, we show that all uncertainty is revealed in the limit and agents behave as expected utility maximizer under the true underlying distribution regardless of their initial risk anticipation. In particular, risk assessments of distinct agents converge. This result is a generalization of the fundamental Blackwell-Dubins Theorem, cp. [Blackwell & Dubins, 62], to convex risk. We furthermore show the result to hold in a non-time-consistent environment.Dynamic Convex Risk Measures, Multiple Priors, Uncertainty, Robust Representation, Time-Consistency, Blackwell-Dubins

Merging of Opinions under Uncertainty

We consider long-run behavior of agents assessing risk in terms of dynamic convex risk measures or, equivalently, utility in terms of dynamic variational preferences in an uncertain setting. By virtue of a robust representation, we show that all uncertainty is revealed in the limit and agents behave as expected utility maximizer under the true underlying distribution regardless of their initial risk anticipation. In particular, risk assessments of distinct agents converge. This result is a generalization of the fundamental Blackwell-Dubins Theorem, cp. [Blackwell & Dubins, 62], to convex risk. We furthermore show the result to hold in a non -time-consistent environment.Dynamic Convex Risk Measures, Multiple Priors, Uncertainty, Robust Representation, Time-Consistency, Blackwell-Dubins.

Conditional and Dynamic Convex Risk Measures

We extend the definition of a convex risk measure to a conditional framework where additional information is available. We characterize these risk measures through the associated acceptance sets and prove a representation result in terms of conditional expectations. As an example we consider the class of conditional entropic risk measures. A new regularity property of conditional risk measures is defined and discussed. Finally we introduce the concept of a dynamic convex risk measure as a family of successive conditional convex risk measures and characterize those satisfying some natural time consistency properties.Conditional convex risk measure, robust representation, regularity, entropic risk measure, dynamic convex risk measure, time consistency