Journal of Optimization Theory and Applications

, Volume 155, Issue 3, pp 1105–1123

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


DOI: 10.1007/s10957-011-9968-2

Cite this article as:
Ahmadi-Javid, A. J Optim Theory Appl (2012) 155: 1105. doi:10.1007/s10957-011-9968-2


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.


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) 

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Industrial EngineeringAmirkabir University of Technology (Tehran Polytechnic)TehranIran

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