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

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 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.

This is a preview of subscription content, log in to check access.

References

  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)

    Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Frittelli, M., Rosazza Gianin, E.: Putting order in risk measures. J. Bank. Finance 26, 1473–1486 (2002)

    Article  Google Scholar 

  8. 8.

    Ruszczynski, A., Shapiro, A.: Optimization of convex risk functions. Math. Oper. Res. 31, 433–452 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Kaina, M., Ruschendorf, L.: On convex risk measures on L p-spaces. Math. Methods Oper. Res. 69, 475–495 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Pflug, G.Ch.: Subdifferential representations of risk measures. Math. Program. 108, 339–354 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Schied, A.: Risk measures and robust optimization problems. Stoch. Models 22, 753–831 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Balbas, A.: Mathematical methods in modern risk measurement: A survey. RACSAM Ser. Appl. Math. 101, 205–219 (2007)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    El Karoui, N., Ravanelli, C.: Cash sub-additive risk measures under interest rate ambiguity. Math. Finance 19, 561–590 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Risk measures: rationality and diversification. Math. Finance 21, 743–774 (2011)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Balbas, A., Balbas, R.: Compatibility between pricing rules and risk measures: The CCVaR. RACSAM Ser. Appl. Math. 103, 251–264 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Cont, R., Deguest, R., Scandolo, G.: Robustness and sensitivity analysis of risk measurement procedures. Quant. Finance 10, 593–606 (2010)

    MathSciNet  MATH  Article  Google 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)

    Google 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)

    MathSciNet  MATH  Article  Google 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)

  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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Ahmed, S.: Convexity and decomposition of mean-risk stochastic programs. Math. Program. 106, 447–452 (2006)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Eichhorn, A., Romisch, W.: Polyhedral risk measures in stochastic programming. SIAM J. Optim. 16, 69–95 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Miller, N., Ruszczyński, A.: Risk-averse two-stage stochastic linear programming: Modeling and decomposition. Oper. Res. 59, 125–132 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Krokhmal, P., Zabarankin, M., Uryasev, S.: Modeling and optimization of risk. Surv. Oper. Res. Manag. Sci. 16, 49–66 (2011)

    Article  Google 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)

    Article  Google 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)

    Article  Google Scholar 

  36. 36.

    Acerbi, C.: Spectral risk measures: A coherent representation of subjective risk aversion. J. Bank. Finance 26, 1505–1518 (2002)

    Article  Google 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)

    MathSciNet  MATH  Article  Google 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)

    Google Scholar 

  40. 40.

    Dupuis, P., Ellis, R.S.: A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York (1997)

    Google 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)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Csiszar, I.: Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hung. 2, 299–318 (1967)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Liese, F., Vajda, I.: Convex Statistical Distances. Teubner, Leipzig (1987)

    Google Scholar 

  45. 45.

    Ullah, A.: Entropy, divergence and distance measures with econometric applications. J. Stat. Plan. Inference 49, 137–162 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Kullback, S., Leibler, R.A.: On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)

    MathSciNet  MATH  Article  Google 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)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to A. Ahmadi-Javid.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ahmadi-Javid, A. Entropic Value-at-Risk: A New Coherent Risk Measure. J Optim Theory Appl 155, 1105–1123 (2012). https://doi.org/10.1007/s10957-011-9968-2

Download citation

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)