Advertisement

Annals of Operations Research

, Volume 254, Issue 1–2, pp 251–275 | Cite as

A quantitative comparison of risk measures

  • Alois PichlerEmail author
Original Paper

Abstract

The choice of a risk measure reflects a subjective preference of the decision maker in many managerial or real world economic problem formulations. To assess the impact of personal preferences it is thus of interest to have comparisons with other risk measures at hand. This paper develops a framework for comparing different risk measures. We establish a one-to-one relationship between norms and risk measures, that is, we associate a norm with a risk measure and conversely, we use norms to recover a genuine risk measure. The methods allow tight comparisons of risk measures and tight lower and upper bounds for risk measures are made available whenever possible. In this way we present a general framework for comparing risk measures with applications in numerous directions.

Keywords

Risk measures Dual representation Fenchel–Young inequality 

JEL Classification

90C15 60B05 62P05 

Notes

Acknowledgements

We would like to thank the editor of the journal and the referees for their commitment to assess and improve the paper.

References

  1. Acerbi, C. (2002). Spectral measures of risk: A coherent representation of subjective risk aversion. Journal of Banking & Finance, 26, 1505–1518. doi: 10.1016/S0378-4266(02)00281-9.CrossRefGoogle Scholar
  2. Ahmadi-Javid, A. (2012). Entropic Value-at-Risk: A new coherent risk measure. Journal of Optimization Theory and Applications, 155(3), 1105–1123. doi: 10.1007/s10957-011-9968-2.CrossRefGoogle Scholar
  3. Artzner, P., Delbaen, F., Eber, J.-M., Heath, D., & Ku, H. (2007). Coherent multiperiod risk adjusted values and Bellman’s principle. Annals of Operations Research, 152, 5–22. doi: 10.1007/s10479-006-0132-6.CrossRefGoogle Scholar
  4. Asamov, T., & Ruszczyński, A. (2014). Time-consistent approximations of risk-averse multistage stochastic optimization problems. Mathematical Programming, 153(2), 1–35. doi: 10.1007/s10107-014-0813-x.Google Scholar
  5. Bellini, F., & Caperdoni, C. (2007). Coherent distortion risk measures and higher-order stochastic dominances. North American Actuarial Journal, 11(2), 35–42. doi: 10.1080/10920277.2007.10597446.CrossRefGoogle Scholar
  6. Bellini, F., & Rosazza Gianin, E. (2008). On Haezendonck risk measures. Journal of Banking & Finance, 32(6), 986–994. doi: 10.1016/j.jbankfin.2007.07.007.CrossRefGoogle Scholar
  7. Bellini, F., & Rosazza Gianin, E. (2012). Haezendonck–Goovaerts risk measures and Orlicz quantiles. Insurance: Mathematics and Economics, 51(1), 107–114. doi: 10.1016/j.insmatheco.2012.03.005.Google Scholar
  8. Cheridito, P., & Kupper, M. (2011). Composition of time-consistent dynamic monetary risk measures in discrete time. International Journal of Theoretical and Applied Finance, 14(1), 137–162. doi: 10.1142/S0219024911006292.CrossRefGoogle Scholar
  9. Collado, R. A., Papp, D., & Ruszczyński, A. (2012). Scenario decomposition of risk-averse multistage stochastic programming problems. Annals of Operations Research, 200(1), 147–170. doi: 10.1007/s10479-011-0935-y.CrossRefGoogle Scholar
  10. De Lara, M., & Leclère, V. (2016). Building up time-consistency for risk measures and dynamic optimization. European Journal of Operational Research, 249, 177–187. doi: 10.1016/j.ejor.2015.03.046.CrossRefGoogle Scholar
  11. Delbaen, F. (2015). Remark on the paper ”Entropic Value-at-Risk: A new coherent risk measure” by Amir Ahmadi-Javid. In P. Barrieu (Ed.), Risk and stochastics. World Scientific, ISBN 978-1-78634-194-5.Google Scholar
  12. Denneberg, D. (1990). Distorted probabilities and insurance premiums. Methods of Operations Research, 63, 21–42.Google Scholar
  13. Densing, M. (2014). Stochastic progamming of time-consistent extensions of AVaR. SIAM Journal on Optimization, 24(3), 993–1010. doi: 10.1137/130905046.CrossRefGoogle Scholar
  14. Dentcheva, D., & Ruszczyński, A. (2003). Optimization with stochastic dominance constraints. SIAM Journal on Optimization, 14(2), 548–566. doi: 10.1137/S1052623402420528.CrossRefGoogle Scholar
  15. Dentcheva, D., Penev, S., & Ruszczyński, A. (2010). Kusuoka representation of higher order dual risk measures. Annals of Operations Research, 181, 325–335. doi: 10.1007/s10479-010-0747-5.CrossRefGoogle Scholar
  16. Dentcheva, D., Penev, S., & Ruszczyński, A. (2016). Statistical estimation of composite risk functionals and risk optimization problems. Annals of the Institute of Statistical Mathematics. doi: 10.1007/s10463-016-0559-8.
  17. Iancu, D. A., Petrik, M., & Subramanian, D. (2015). Tight approximations of dynamic risk measures. Mathematics of Operations Research, 40(3), 655–682. doi: 10.1287/moor.2014.0689.CrossRefGoogle Scholar
  18. Krokhmal, P. A. (2007). Higher moment coherent risk measures. Quantitative Finance, 7(4), 373–387. doi: 10.1080/14697680701458307.CrossRefGoogle Scholar
  19. Kusuoka, S. (2001). On law invariant coherent risk measures. In Advances in mathematical economics, Chapter 4 (Vol. 3, pp. 83–95). Springer. doi: 10.1007/978-4-431-67891-5.
  20. López-Díaz, M., Sordo, M. A., & Suárez-Llorens, A. (2012). On the \({L}_p\)-metric between a probability distribution and its distortion. Insurance: Mathematics and Economics, 51, 257–264. doi: 10.1016/j.insmatheco.2012.04.004.Google Scholar
  21. Luna, J. P., Sagastizábal, C., & Solodov, M. (2016). An approximatioin scheme for a class of risk-averse stochastic equilibrium problems. Mathematical Programming, 157(2), 451–481. doi: 10.1007/s10107-016-0988-4.CrossRefGoogle Scholar
  22. Miller, N., & Ruszczyński, A. (2011). Risk-averse two-stage stochastic linear programming: Modeling and decomposition. Operations Research, 59, 125–132. doi: 10.1287/opre.1100.0847.CrossRefGoogle Scholar
  23. Noyan, N., & Rudolf, G. (2014). Kusuoka representations of coherent risk measures in general probability spaces. Annals of Operations Research, 229, 591–605. doi: 10.1007/s10479-014-1748-6. (ISSN 0254-5330).CrossRefGoogle Scholar
  24. Pflug, G. C. (2000). Some remarks on the Value-at-Risk and the Conditional Value-at-Risk, Chapter 15. In S. Uryasev (Ed.), Probabilistic constrained optimization (Vol. 49, pp. 272–281). New York: Springer.CrossRefGoogle Scholar
  25. Pflug, G. C., & Pichler, A. (2014). Multistage stochastic optimization. Springer Series in Operations Research and Financial Engineering: Springer. ISBN 978-3-319-08842-6. doi: 10.1007/978-3-319-08843-3.
  26. Pflug, G. C., & Pichler, A. (2016). Time-consistent decisions and temporal decomposition of coherent risk functionals. Mathematics of Operations Research, 41(2), 682–699. doi: 10.1287/moor.2015.0747.CrossRefGoogle Scholar
  27. Pflug, G. C., & Römisch, W. (2007). Modeling, measuring and managing risk. River Edge, NJ: World Scientific. doi: 10.1142/9789812708724.CrossRefGoogle Scholar
  28. Pflug, G. C., & Ruszczyński, A. (2005). Measuring risk for income streams. Computational Optimization and Applications, 32(1–2), 161–178, ISSN 0926-6003. doi: 10.1007/s10589-005-2058-3.
  29. Philpott, A. B., & de Matos, V. L. (2012). Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion. European Journal of Operational Research, 218(2), 470–483. doi: 10.1016/j.ejor.2011.10.056.CrossRefGoogle Scholar
  30. Philpott, A. B., de Matos, V. L., & Finardi, E. (2013). On solving multistage stochastic programs with coherent risk measures. Operations Research, 61(4), 957–970. doi: 10.1287/opre.2013.1175.CrossRefGoogle Scholar
  31. Pichler, A. (2013). The natural Banach space for version independent risk measures. Insurance: Mathematics and Economics, 53(2), 405–415. doi: 10.1016/j.insmatheco.2013.07.005.Google Scholar
  32. Pichler, A. (2013). Premiums and reserves, adjusted by distortions. Scandinavian Actuarial Journal, 2015(4), 332–351. doi: 10.1080/03461238.2013.830228.CrossRefGoogle Scholar
  33. Pichler, A., & Shapiro, A. (2015). Minimal representations of insurance prices. Insurance: Mathematics and Economics, 62, 184–193. doi: 10.1016/j.insmatheco.2015.03.011.Google Scholar
  34. Rockafellar, R. T., & Uryasev, S. (2000). Optimization of Conditional Value-at-Risk. Journal of Risk, 2(3), 21–41. doi: 10.21314/JOR.2000.038.CrossRefGoogle Scholar
  35. Rockafellar, R.T., Uryasev, S., & Zabarankin, M. (2006). Generalized deviations in risk analysis. Finance and Stochastics, 10, 51–74, ISSN 0949-2984. doi: 10.1007/s00780-005-0165-8.
  36. Ruszczyński, A. (2010). Risk-averse dynamic programming for Markov decision processes. Mathematical Programming Series B, 125, 235–261.CrossRefGoogle Scholar
  37. Ruszczyński, A., & Shapiro, A. (2006). Conditional risk mappings. Mathematics of Operations Research, 31(3), 544–561. doi: 10.1287/moor.1060.0204.CrossRefGoogle Scholar
  38. Ruszczyński, A., & Yao, J. (2015). A risk-averse analog of the Hamilton–Jacobi–Bellman equation. In Proceedings of the Conference on Control and its Applications, Chapter 62 (pp. 462–468). Society for Industrial & Applied Mathematics (SIAM). doi: 10.1137/1.9781611974072.63.
  39. Shapiro, A. (2010). Analysis of stochastic dual dynamic programming method. European Journal of Operational Research, 209, 63–72.CrossRefGoogle Scholar
  40. Shapiro, A. (2013). On Kusuoka representation of law invariant risk measures. Mathematics of Operations Research, 38(1), 142–152. doi: 10.1287/moor.1120.0563.CrossRefGoogle Scholar
  41. Shapiro, A. (2016). Rectangular sets of probability measures. Operations Research, 64(2), 528–541. doi: 10.1287/opre.2015.1466.CrossRefGoogle Scholar
  42. Shapiro, A., Dentcheva, D., & Ruszczyński, A. (2009). Lectures on stochastic programming. In MOS-SIAM series on optimization. SIAM. doi: 10.1137/1.9780898718751.
  43. Sundaresan, K. (1969). Extreme points of the unit cell in Lebesgue–Bochner function spaces. Proceedings of the American Mathematical Society, 23(1), 179–184. doi: 10.2307/2037513.Google Scholar
  44. van Heerwaarden, A. E., & Kaas, R. (1992). The Dutch premium principle. Insurance: Mathematics and Economics, 11, 223–230. doi: 10.1016/0167-6687(92)90049-H.Google Scholar
  45. Wang, S. S. (1995). Insurance pricing and increased limits ratemaking by proportional hazards transforms. Insurance: Mathematics and Economics, 17, 43–54. doi: 10.1016/0167-6687(95)00010-P.Google Scholar
  46. Wojtaszczyk, P. (1991). Banach spaces for analysts. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  47. Wozabal, D. (2010). A framework for optimization under ambiguity. Annals of Operations Research, 193(1), 21–47. doi: 10.1007/s10479-010-0812-0.CrossRefGoogle Scholar
  48. Wozabal, D. (2014). Robustifying convex risk measures for linear portfolios: A nonparametric approach. Operations Research, 62(6), 1302–1315. doi: 10.1287/opre.2014.1323.CrossRefGoogle Scholar
  49. Xin, L., & Shapiro, A. (2012). Bounds for nested law invariant coherent risk measures. Operations Research Letters, 40, 431–435. doi: 10.1016/j.orl.2012.09.002.CrossRefGoogle Scholar
  50. Young, V. R. (2006). Premium Principles. Encyclopedia of Actuarial Science. Wiley Pennsylvania State University. ISBN 9780470012505. doi: 10.1002/9780470012505.tap027.

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Technische Universität Chemnitz, Fakultät für MathematikChemnitzGermany

Personalised recommendations