IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1411.0426.html
   My bibliography  Save this paper

Risk measures with the CxLS property

Author

Listed:
  • Freddy Delbaen
  • Fabio Bellini
  • Valeria Bignozzi
  • Johanna F. Ziegel

Abstract

In the present contribution we characterize law determined convex risk measures that have convex level sets at the level of distributions. By relaxing the assumptions in Weber (2006), we show that these risk measures can be identified with a class of generalized shortfall risk measures. As a direct consequence, we are able to extend the results in Ziegel (2014) and Bellini and Bignozzi (2014) on convex elicitable risk measures and confirm that expectiles are the only elicitable coherent risk measures. Further, we provide a simple characterization of robustness for convex risk measures in terms of a weak notion of mixture continuity.

Suggested Citation

  • Freddy Delbaen & Fabio Bellini & Valeria Bignozzi & Johanna F. Ziegel, 2014. "Risk measures with the CxLS property," Papers 1411.0426, arXiv.org.
    • Handle: RePEc:arx:papers:1411.0426
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1411.0426
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Freddy Delbaen, 2013. "A Remark on the Structure of Expectiles," Papers 1307.5881, arXiv.org.
    2. Paul Embrechts & Giovanni Puccetti & Ludger Rüschendorf & Ruodu Wang & Antonela Beleraj, 2014. "An Academic Response to Basel 3.5," Risks, MDPI, vol. 2(1), pages 1-24, February.
    3. Stefan Weber, 2006. "Distribution‐Invariant Risk Measures, Information, And Dynamic Consistency," Mathematical Finance, Wiley Blackwell, vol. 16(2), pages 419-441, April.
    4. Gneiting, Tilmann, 2011. "Making and Evaluating Point Forecasts," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 746-762.
    5. Volker Krätschmer & Alexander Schied & Henryk Zähle, 2014. "Comparative and qualitative robustness for law-invariant risk measures," Finance and Stochastics, Springer, vol. 18(2), pages 271-295, April.
    6. Volker Kratschmer & Alexander Schied & Henryk Zahle, 2012. "Comparative and qualitative robustness for law-invariant risk measures," Papers 1204.2458, arXiv.org, revised Jan 2014.
    7. Bellini, Fabio & Klar, Bernhard & Müller, Alfred & Rosazza Gianin, Emanuela, 2014. "Generalized quantiles as risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 54(C), pages 41-48.
    8. Johanna F. Ziegel, 2013. "Coherence and elicitability," Papers 1303.1690, arXiv.org, revised Mar 2014.
    9. Dekel, Eddie, 1986. "An axiomatic characterization of preferences under uncertainty: Weakening the independence axiom," Journal of Economic Theory, Elsevier, vol. 40(2), pages 304-318, December.
    10. Patrick Cheridito & Tianhui Li, 2009. "Risk Measures On Orlicz Hearts," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 189-214, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Bellini, Fabio & Laeven, Roger J.A. & Rosazza Gianin, Emanuela, 2021. "Dynamic robust Orlicz premia and Haezendonck–Goovaerts risk measures," European Journal of Operational Research, Elsevier, vol. 291(2), pages 438-446.
    2. Freddy Delbaen, 2021. "Commonotonicity and time-consistency for Lebesgue-continuous monetary utility functions," Finance and Stochastics, Springer, vol. 25(3), pages 597-614, July.
    3. Wang, Ruodu & Ziegel, Johanna F., 2015. "Elicitable distortion risk measures: A concise proof," Statistics & Probability Letters, Elsevier, vol. 100(C), pages 172-175.
    4. Tobias Fissler & Johanna F. Ziegel, 2015. "Higher order elicitability and Osband's principle," Papers 1503.08123, arXiv.org, revised Sep 2015.
    5. Mucahit Aygun & Fabio Bellini & Roger J. A. Laeven, 2023. "Elicitability of Return Risk Measures," Papers 2302.13070, arXiv.org, revised Mar 2023.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Krätschmer Volker & Schied Alexander & Zähle Henryk, 2015. "Quasi-Hadamard differentiability of general risk functionals and its application," Statistics & Risk Modeling, De Gruyter, vol. 32(1), pages 25-47, April.
    2. Ruodu Wang & Johanna F. Ziegel, 2014. "Distortion Risk Measures and Elicitability," Papers 1405.3769, arXiv.org, revised May 2014.
    3. Fissler Tobias & Ziegel Johanna F., 2021. "On the elicitability of range value at risk," Statistics & Risk Modeling, De Gruyter, vol. 38(1-2), pages 25-46, January.
    4. Tobias Fissler & Jana Hlavinová & Birgit Rudloff, 2021. "Elicitability and identifiability of set-valued measures of systemic risk," Finance and Stochastics, Springer, vol. 25(1), pages 133-165, January.
    5. Yannick Armenti & Stéphane Crépey & Samuel Drapeau & Antonis Papapantoleon, 2018. "Multivariate Shortfall Risk Allocation and Systemic Risk," Working Papers hal-01764398, HAL.
    6. Samuel Drapeau & Mekonnen Tadese, 2019. "Dual Representation of Expectile based Expected Shortfall and Its Properties," Papers 1911.03245, arXiv.org.
    7. Damiano Rossello, 2022. "Performance measurement with expectiles," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 45(1), pages 343-374, June.
    8. Samuel Drapeau & Mekonnen Tadese, 2019. "Relative Bound and Asymptotic Comparison of Expectile with Respect to Expected Shortfall," Papers 1906.09729, arXiv.org, revised Jun 2020.
    9. Matteo Burzoni & Ilaria Peri & Chiara Maria Ruffo, 2016. "On the properties of the Lambda value at risk: robustness, elicitability and consistency," Papers 1603.09491, arXiv.org, revised Feb 2017.
    10. Carole Bernard & Ludger Rüschendorf & Steven Vanduffel & Ruodu Wang, 2017. "Risk bounds for factor models," Finance and Stochastics, Springer, vol. 21(3), pages 631-659, July.
    11. Henryk Zähle, 2022. "A concept of copula robustness and its applications in quantitative risk management," Finance and Stochastics, Springer, vol. 26(4), pages 825-875, October.
    12. Righi, Marcelo Brutti & Müller, Fernanda Maria & Moresco, Marlon Ruoso, 2020. "On a robust risk measurement approach for capital determination errors minimization," Insurance: Mathematics and Economics, Elsevier, vol. 95(C), pages 199-211.
    13. Edgars Jakobsons & Steven Vanduffel, 2015. "Dependence Uncertainty Bounds for the Expectile of a Portfolio," Risks, MDPI, vol. 3(4), pages 1-25, December.
    14. Tadese, Mekonnen & Drapeau, Samuel, 2020. "Relative bound and asymptotic comparison of expectile with respect to expected shortfall," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 387-399.
    15. Volker Krätschmer & Henryk Zähle, 2017. "Statistical Inference for Expectile-based Risk Measures," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(2), pages 425-454, June.
    16. Werner Ehm & Tilmann Gneiting & Alexander Jordan & Fabian Krüger, 2016. "Of quantiles and expectiles: consistent scoring functions, Choquet representations and forecast rankings," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 78(3), pages 505-562, June.
    17. Lauer Alexandra & Zähle Henryk, 2016. "Nonparametric estimation of risk measures of collective risks," Statistics & Risk Modeling, De Gruyter, vol. 32(2), pages 89-102, March.
    18. Jakobsons Edgars, 2016. "Scenario aggregation method for portfolio expectile optimization," Statistics & Risk Modeling, De Gruyter, vol. 33(1-2), pages 51-65, September.
    19. Paul Embrechts & Giovanni Puccetti & Ludger Rüschendorf & Ruodu Wang & Antonela Beleraj, 2014. "An Academic Response to Basel 3.5," Risks, MDPI, vol. 2(1), pages 1-24, February.
    20. Lauer, Alexandra & Zähle, Henryk, 2017. "Bootstrap consistency and bias correction in the nonparametric estimation of risk measures of collective risks," Insurance: Mathematics and Economics, Elsevier, vol. 74(C), pages 99-108.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1411.0426. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.