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

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.

[1]  Harry H. Panjer,et al.  MEASUREMENT OF RISK, SOLVENCY REQUIREMENTS AND ALLOCATION OF CAPITAL WITHIN FINANCIAL CONGLOMERATES , 2002 .

[2]  Stan Uryasev,et al.  Optimality conditions in portfolio analysis with general deviation measures , 2005, Math. Program..

[3]  Alejandro Balbás de la Corte,et al.  Mathematical methods in modern risk measurement: a survey , 2007 .

[4]  L. Montrucchio,et al.  RISK MEASURES: RATIONALITY AND DIVERSIFICATION , 2010 .

[5]  R. Rockafellar,et al.  Optimization of conditional value-at risk , 2000 .

[6]  Alejandro Balbás,et al.  Compatibility between pricing rules and risk measures: The CCVaR , 2009 .

[7]  Shabbir Ahmed,et al.  Convexity and decomposition of mean-risk stochastic programs , 2006, Math. Program..

[8]  R. Rockafellar,et al.  Generalized Deviations in Risk Analysis , 2004 .

[9]  Glyn A. Holton.  Value at Risk: Theory and Practice , 2003 .

[10]  Alexander Schied,et al.  Convex measures of risk and trading constraints , 2002, Finance Stochastics.

[11]  Gerhard Reinelt,et al.  A note on the Undirected Rural Postman Problem polytope , 2006, Math. Program..

[12]  M. Teboulle,et al.  AN OLD‐NEW CONCEPT OF CONVEX RISK MEASURES: THE OPTIMIZED CERTAINTY EQUIVALENT , 2007 .

[13]  Romain Deguest,et al.  Robustness and sensitivity analysis of risk measurement procedures , 2008 .

[14]  Alexander Shapiro,et al.  On Complexity of Stochastic Programming Problems , 2005 .

[15]  M. Frittelli,et al.  Putting order in risk measures , 2002 .

[16]  Naomi Miller,et al.  Risk-Averse Two-Stage Stochastic Linear Programming: Modeling and Decomposition , 2011, Oper. Res..

[17]  M. Pritsker.  Evaluating Value at Risk Methodologies: Accuracy versus Computational Time , 1996 .

[18]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[19]  Michel Denuit,et al.  Risk measurement with equivalent utility principles , 2006 .

[20]  Qihe Tang,et al.  A Comonotonic Image of Independence for Additive Risk Measures , 2004 .

[21]  F. Delbaen.  Coherent Risk Measures on General Probability Spaces , 2002 .

[22]  F. Delbaen.  Coherent risk measures , 2000 .

[23]  A. Schied.  Risk Measures and Robust Optimization Problems , 2006 .

[24]  Jan Dhaene,et al.  Modern Actuarial Risk Theory , 2001 .

[25]  N. El Karoui,et al.  CASH SUBADDITIVE RISK MEASURES AND INTEREST RATE AMBIGUITY , 2007, 0710.4106.

[26]  Bernt Øksendal,et al.  Advanced mathematical methods for finance , 2011 .

[27]  Shaun S. Wang,et al.  Axiomatic characterization of insurance prices , 1997 .

[28]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[29]  J. Lynch,et al.  A weak convergence approach to the theory of large deviations , 1997 .

[30]  C. Acerbi.  Spectral measures of risk: A coherent representation of subjective risk aversion , 2002 .

[31]  Hans U. Gerber,et al.  On Additive Premium Calculation Principles , 1974, ASTIN Bulletin.

[32]  Georg Ch. Pflug,et al.  Subdifferential representations of risk measures , 2006, Math. Program..

[33]  Alexander Shapiro,et al.  Optimization of Convex Risk Functions , 2006, Math. Oper. Res..

[34]  H. Föllmer,et al.  Stochastic Finance: An Introduction in Discrete Time , 2002 .

[35]  Stan Uryasev,et al.  Modeling and optimization of risk , 2011 .

[36]  R. A. Leibler,et al.  On Information and Sufficiency , 1951 .

[37]  Amir Ahmadi-Javid,et al.  An information-theoretic approach to constructing coherent risk measures , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[38]  Beatrice Acciaio,et al.  Dynamic risk measures , 2011 .

[39]  S. M. Ali,et al.  A General Class of Coefficients of Divergence of One Distribution from Another , 1966 .

[40]  Werner Römisch,et al.  Polyhedral Risk Measures in Stochastic Programming , 2005, SIAM J. Optim..

[41]  I. Vajda,et al.  Convex Statistical Distances , 2018, Statistical Inference for Engineers and Data Scientists.

[42]  Stanislav Uryasev,et al.  Conditional Value-at-Risk for General Loss Distributions , 2002 .

[43]  Ludger Rüschendorf,et al.  On convex risk measures on Lp-spaces , 2009, Math. Methods Oper. Res..

[44]  A. Ullah.  Entropy, divergence and distance measures with econometric applications , 1996 .