Journal of Optimization Theory and Applications

, Volume 155, Issue 3, pp 1105–1123 | Cite as

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

  • A. Ahmadi-JavidEmail author


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) 


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  1. 1.
    Ahmadi-Javid, A.: Stochastic optimization via entropic value-at-risk: A new coherent risk measure. In: International Conference on Operations Research and Optimization, January 2011, Tehran, Iran (2011) Google Scholar
  2. 2.
    Ahmadi-Javid, A.: An information–theoretic approach to constructing coherent risk measures. In: Proceedings of IEEE International Symposium on Information Theory, August 2011, St. Petersburg, Russia, pp. 2125–2127 (2011) CrossRefGoogle Scholar
  3. 3.
    Artzner, Ph., Delbaen, F., Eber, J.M., Heath, D.: Thinking coherently. Risk 10, 68–71 (1997) Google Scholar
  4. 4.
    Artzner, Ph., Delbaen, F., Eber, J.M., Heath, D.: Coherent risk measures. Math. Finance 9, 203–228 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Delbaen, F.: Coherent risk measures on general probability spaces. In: Sandmann, K., Schonbucher, P.J. (eds.) Advances in Finance and Stochastic, Essays in Honor of Dieter Sondermann, pp. 1–38. Springer, Berlin (2002) Google Scholar
  6. 6.
    Föllmer, H., Schied, A.: Convex risk measures and trading constraints. Finance Stoch. 6, 429–447 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Frittelli, M., Rosazza Gianin, E.: Putting order in risk measures. J. Bank. Finance 26, 1473–1486 (2002) CrossRefGoogle Scholar
  8. 8.
    Ruszczynski, A., Shapiro, A.: Optimization of convex risk functions. Math. Oper. Res. 31, 433–452 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Kaina, M., Ruschendorf, L.: On convex risk measures on L p-spaces. Math. Methods Oper. Res. 69, 475–495 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Pflug, G.Ch.: Subdifferential representations of risk measures. Math. Program. 108, 339–354 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Schied, A.: Risk measures and robust optimization problems. Stoch. Models 22, 753–831 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Balbas, A.: Mathematical methods in modern risk measurement: A survey. RACSAM Ser. Appl. Math. 101, 205–219 (2007) MathSciNetzbMATHGoogle Scholar
  13. 13.
    El Karoui, N., Ravanelli, C.: Cash sub-additive risk measures under interest rate ambiguity. Math. Finance 19, 561–590 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Risk measures: rationality and diversification. Math. Finance 21, 743–774 (2011) MathSciNetzbMATHGoogle Scholar
  15. 15.
    Balbas, A., Balbas, R.: Compatibility between pricing rules and risk measures: The CCVaR. RACSAM Ser. Appl. Math. 103, 251–264 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Cont, R., Deguest, R., Scandolo, G.: Robustness and sensitivity analysis of risk measurement procedures. Quant. Finance 10, 593–606 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Acciaio, B., Penner, I.: Dynamic risk measures. In: Di Nunno, G., Oksendal, B. (eds.) Advanced Mathematical Methods for Finance, pp. 1–34. Springer, Berlin (2011) CrossRefGoogle Scholar
  18. 18.
    Goovaerts, M.J., De Vijlder, F., Haezendonck, J.: Insurance Premiums. North-Holland, Amsterdam (1984) Google Scholar
  19. 19.
    Kaas, R., Goovaerts, M.J., Dhaene, J., Denuit, M.: Modern Actuarial Risk Theory. Kluwer Academic, Dordrecht (2001) Google Scholar
  20. 20.
    Wang, S.S., Young, V.R., Panjer, H.H.: Axiomatic characterization of insurance prices. Insur. Math. Econ. 21, 173–183 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Panjer, H.H.: Measurement of risk, solvency requirements and allocation of capital within financial conglomerates. Institute of Insurance and Pension Research, Research Report 1-15, University of Waterloo (2002) Google Scholar
  22. 22.
    Goovaerts, M.J., Kaas, R., Laeven, R.J.A., Tang, Q.: A comonotonic image of independence for additive risk measures. Insur. Math. Econ. 35, 581–594 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Denuit, M., Dhaene, J., Goovaerts, M.J., Kaas, R., Laeven, R.J.A.: Risk measurement with equivalent utility principles. In: Ruschendorf, L. (ed.) Risk Measures: General Aspects and Applications (special issue). Statistics and Decisions, vol. 24, pp. 1–26 (2006) Google Scholar
  24. 24.
    Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Generalized deviations in risk analysis. Finance Stoch. 10, 51–74 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Rockafellar, R.T., Uryasev, S., Zabarankin, M.: Optimality conditions in portfolio analysis with generalized deviation measures. Math. Program. 108, 515–540 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Ahmed, S.: Convexity and decomposition of mean-risk stochastic programs. Math. Program. 106, 447–452 (2006) MathSciNetCrossRefGoogle Scholar
  27. 27.
    Eichhorn, A., Romisch, W.: Polyhedral risk measures in stochastic programming. SIAM J. Optim. 16, 69–95 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Miller, N., Ruszczyński, A.: Risk-averse two-stage stochastic linear programming: Modeling and decomposition. Oper. Res. 59, 125–132 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Krokhmal, P., Zabarankin, M., Uryasev, S.: Modeling and optimization of risk. Surv. Oper. Res. Manag. Sci. 16, 49–66 (2011) CrossRefGoogle Scholar
  30. 30.
    Markowitz, H.: Portfolio Selection. J. Finance 7, 77–91 (1952) Google Scholar
  31. 31.
    Pritsker, M.: Evaluating value at risk methodologies. J. Financ. Serv. Res. 12, 201–242 (1997) CrossRefGoogle Scholar
  32. 32.
    Guldimann, T.: The story of risk metrics. Risk 13, 56–58 (2000) Google Scholar
  33. 33.
    Holton, G.: Value-at-Risk: Theory and Practice. Academic Press, San Diego (2002) Google Scholar
  34. 34.
    Rockafellar, R.T., Uryasev, S.: Optimization of conditional value-at-risk. J. Risk 2, 21–41 (2000) Google Scholar
  35. 35.
    Rockafellar, R.T., Uryasev, S.: Conditional value-at-risk for general loss distributions. J. Bank. Finance 26, 1443–1471 (2002) CrossRefGoogle Scholar
  36. 36.
    Acerbi, C.: Spectral risk measures: A coherent representation of subjective risk aversion. J. Bank. Finance 26, 1505–1518 (2002) CrossRefGoogle Scholar
  37. 37.
    Chernoff, H.: A measure of asymptotic efficiency for tests of a hypothesis based on a sum of observations. Ann. Stat. 23, 493–507 (1952) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Gerber, H.U.: On additive premium calculation principles. ASTIN Bull. 7, 215–222 (1974) Google Scholar
  39. 39.
    Follmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time. de Gruyter, Berlin (2004) CrossRefGoogle Scholar
  40. 40.
    Dupuis, P., Ellis, R.S.: A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York (1997) zbMATHCrossRefGoogle Scholar
  41. 41.
    Shapiro, A., Nemirovski, A.: On complexity of stochastic programming problems. In: Jeyakumar, V., Rubinov, A.M. (eds.) Continuous Optimization: Current Trends and Applications, pp. 111–144. Springer, New York (2005) Google Scholar
  42. 42.
    Ali, S.M., Silvey, S.D.: A general class of coefficients of divergence of one distribution from another. J. R. Stat. Soc. B 28, 131–142 (1966) MathSciNetzbMATHGoogle Scholar
  43. 43.
    Csiszar, I.: Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hung. 2, 299–318 (1967) MathSciNetzbMATHGoogle Scholar
  44. 44.
    Liese, F., Vajda, I.: Convex Statistical Distances. Teubner, Leipzig (1987) zbMATHGoogle Scholar
  45. 45.
    Ullah, A.: Entropy, divergence and distance measures with econometric applications. J. Stat. Plan. Inference 49, 137–162 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951) MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Ben-Tal, A., Teboulle, M.: An old-new concept of convex risk measures: the optimized certainty equivalent. Math. Finance 17, 449–476 (2007) MathSciNetzbMATHCrossRefGoogle Scholar

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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