Skip to main content
SpringerLink
Log in

A quantitative comparison of risk measures

  • Original Paper
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

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.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Also law invariant, or distribution based.

  2. (NP) is mnemonic for positive.

  3. \(x_{+}:=\max \left\{ 0,x\right\} . \)

  4. U is uniformly distributed, iff \(P(U\le u)=u\).

References

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

  • Denneberg, D. (1990). Distorted probabilities and insurance premiums. Methods of Operations Research, 63, 21–42.

    Google Scholar 

  • Densing, M. (2014). Stochastic progamming of time-consistent extensions of AVaR. SIAM Journal on Optimization, 24(3), 993–1010. doi:10.1137/130905046.

    Article  Google Scholar 

  • Dentcheva, D., & Ruszczyński, A. (2003). Optimization with stochastic dominance constraints. SIAM Journal on Optimization, 14(2), 548–566. doi:10.1137/S1052623402420528.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

  • 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.

    Article  Google Scholar 

  • Krokhmal, P. A. (2007). Higher moment coherent risk measures. Quantitative Finance, 7(4), 373–387. doi:10.1080/14697680701458307.

    Article  Google Scholar 

  • 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.

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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).

    Article  Google Scholar 

  • 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.

    Chapter  Google Scholar 

  • 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.

  • 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.

    Article  Google Scholar 

  • Pflug, G. C., & Römisch, W. (2007). Modeling, measuring and managing risk. River Edge, NJ: World Scientific. doi:10.1142/9789812708724.

    Book  Google Scholar 

  • 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.

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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 

  • Pichler, A. (2013). Premiums and reserves, adjusted by distortions. Scandinavian Actuarial Journal, 2015(4), 332–351. doi:10.1080/03461238.2013.830228.

    Article  Google Scholar 

  • 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 

  • 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.

    Article  Google Scholar 

  • 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.

  • Ruszczyński, A. (2010). Risk-averse dynamic programming for Markov decision processes. Mathematical Programming Series B, 125, 235–261.

    Article  Google Scholar 

  • Ruszczyński, A., & Shapiro, A. (2006). Conditional risk mappings. Mathematics of Operations Research, 31(3), 544–561. doi:10.1287/moor.1060.0204.

    Article  Google Scholar 

  • 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.

  • Shapiro, A. (2010). Analysis of stochastic dual dynamic programming method. European Journal of Operational Research, 209, 63–72.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Shapiro, A. (2016). Rectangular sets of probability measures. Operations Research, 64(2), 528–541. doi:10.1287/opre.2015.1466.

    Article  Google Scholar 

  • Shapiro, A., Dentcheva, D., & Ruszczyński, A. (2009). Lectures on stochastic programming. In MOS-SIAM series on optimization. SIAM. doi:10.1137/1.9780898718751.

  • 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 

  • 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 

  • 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 

  • Wojtaszczyk, P. (1991). Banach spaces for analysts. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Wozabal, D. (2010). A framework for optimization under ambiguity. Annals of Operations Research, 193(1), 21–47. doi:10.1007/s10479-010-0812-0.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Young, V. R. (2006). Premium Principles. Encyclopedia of Actuarial Science. Wiley Pennsylvania State University. ISBN 9780470012505. doi:10.1002/9780470012505.tap027.

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Technische Universität Chemnitz, Fakultät für Mathematik, Chemnitz, Germany

    Alois Pichler

Authors
  1. Alois Pichler
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alois Pichler.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pichler, A. A quantitative comparison of risk measures. Ann Oper Res 254, 251–275 (2017). https://doi.org/10.1007/s10479-017-2397-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-017-2397-3

Keywords

JEL Classification

Search

Navigation