4.5 Article

Entropic Value-at-Risk: A New Coherent Risk Measure

Journal

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
Volume 155, Issue 3, Pages 1105-1123

Publisher

SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10957-011-9968-2

Keywords

Chernoff inequality; Coherent risk measure; Conditional value-at-risk (CVaR); Convex optimization; Cumulant-generating function; Duality; Entropic value-at-risk (EVaR); g-entropic risk measure; Moment-generating function; Relative entropy; Stochastic optimization; Stochastic programming; Value-at-risk (VaR)

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This paper introduces the concept of entropic value-at-risk (EVaR), a new coherent risk measure that corresponds to the tightest possible upper bound obtained from the Chernoff inequality for the value-at-risk (VaR) as well as the conditional value-at-risk (CVaR). We show that a broad class of stochastic optimization problems that are computationally intractable with the CVaR is efficiently solvable when the EVaR is incorporated. We also prove that if two distributions have the same EVaR at all confidence levels, then they are identical at all points. The dual representation of the EVaR is closely related to the Kullback-Leibler divergence, also known as the relative entropy. Inspired by this dual representation, we define a large class of coherent risk measures, called g-entropic risk measures. The new class includes both the CVaR and the EVaR.

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