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J Optim Theory Appl (2012) 155:11051123
DOI 10.1007/s10957-011-9968-2
Received: 24 February 2011 / Accepted: 12 November 2011 / Published online: 17 December 2011 Springer Science+Business Media, LLC 2011
Abstract 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 efciently solvable when the EVaR is incorporated. We also prove that if two distributions have the same EVaR at all condence 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 dene a large class of coherent risk measures, called g-entropic risk measures. The new class includes both the CVaR and the EVaR.
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)
1 Introduction
In the last twenty years, a great deal of effort has gone into achieving suitable methods of measuring risk. A risk measure assigns to a random outcome or risk position a real number that scalarizes the degree of risk involved in that random outcome. This concept has found many applications in different areas such as nance, actuary science, operations research and management science.
A. Ahmadi-Javid ( )
Department of Industrial Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Irane-mail: mailto:[email protected]
Web End [email protected]
Entropic Value-at-Risk: A New Coherent Risk Measure
A. Ahmadi-Javid
1106 J Optim Theory Appl (2012) 155:11051123
Most theoretical research papers studying risk measures take two main approaches. In the rst approach, after specifying a set of desirable properties for a risk measure, the class of risk measures satisfying those properties is studied. In the second approach, a particular risk measure is investigated from different points of view, including interpretation, axiomatic properties, alternative representations, computational issues, and optimization of models incorporating that type of risk measure. This paper introduces a new risk measure and thus ts in the category of...