IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1303.1690.html
   My bibliography  Save this paper

Coherence and elicitability

Author

Listed:
  • Johanna F. Ziegel

Abstract

The risk of a financial position is usually summarized by a risk measure. As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile based risk measures such as value at risk are elicitable. In this paper we show that law-invariant spectral risk measures such as expected shortfall are not elicitable unless they reduce to minus the expected value. Hence, it is unclear how to perform forecast verification or comparison. However, the class of elicitable law-invariant coherent risk measures does not reduce to minus the expected value. We show that it consists of certain expectiles.

Suggested Citation

  • Johanna F. Ziegel, 2013. "Coherence and elicitability," Papers 1303.1690, arXiv.org, revised Mar 2014.
    • Handle: RePEc:arx:papers:1303.1690
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1303.1690
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Kuan, Chung-Ming & Yeh, Jin-Huei & Hsu, Yu-Chin, 2009. "Assessing value at risk with CARE, the Conditional Autoregressive Expectile models," Journal of Econometrics, Elsevier, vol. 150(2), pages 261-270, June.
    2. Koenker,Roger, 2005. "Quantile Regression," Cambridge Books, Cambridge University Press, number 9780521845731, September.
    3. Acerbi, Carlo, 2002. "Spectral measures of risk: A coherent representation of subjective risk aversion," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1505-1518, July.
    4. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
    5. Tilmann Gneiting & Roopesh Ranjan, 2011. "Comparing Density Forecasts Using Threshold- and Quantile-Weighted Scoring Rules," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 29(3), pages 411-422, July.
    6. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Quantitative Finance, Taylor & Francis Journals, vol. 10(6), pages 593-606.
    7. Elyès Jouini & Walter Schachermayer & Nizar Touzi, 2006. "Law Invariant Risk Measures Have the Fatou Property," Post-Print halshs-00176522, HAL.
    8. So Yeon Chun & Alexander Shapiro & Stan Uryasev, 2012. "Conditional Value-at-Risk and Average Value-at-Risk: Estimation and Asymptotics," Operations Research, INFORMS, vol. 60(4), pages 739-756, August.
    9. Gneiting, Tilmann, 2011. "Making and Evaluating Point Forecasts," Journal of the American Statistical Association, American Statistical Association, vol. 106(494), pages 746-762.
    10. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Post-Print hal-00413729, HAL.
    11. Gneiting, Tilmann & Raftery, Adrian E., 2007. "Strictly Proper Scoring Rules, Prediction, and Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 359-378, March.
    12. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    13. Alois Pichler & Alexander Shapiro, 2012. "Uniqueness of Kusuoka Representations," Papers 1210.7257, arXiv.org, revised Feb 2013.
    14. repec:dau:papers:123456789/342 is not listed on IDEAS
    15. Gneiting, Tilmann & Ranjan, Roopesh, 2011. "Comparing Density Forecasts Using Threshold- and Quantile-Weighted Scoring Rules," Journal of Business & Economic Statistics, American Statistical Association, vol. 29(3), pages 411-422.
    16. Newey, Whitney K & Powell, James L, 1987. "Asymmetric Least Squares Estimation and Testing," Econometrica, Econometric Society, vol. 55(4), pages 819-847, July.
    Full references (including those not matched with items on IDEAS)

    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. Natalia Nolde & Johanna F. Ziegel, 2016. "Elicitability and backtesting: Perspectives for banking regulation," Papers 1608.05498, arXiv.org, revised Feb 2017.
    2. Ruodu Wang & Ričardas Zitikis, 2021. "An Axiomatic Foundation for the Expected Shortfall," Management Science, INFORMS, vol. 67(3), pages 1413-1429, March.
    3. Tobias Fissler & Hajo Holzmann, 2022. "Measurability of functionals and of ideal point forecasts," Papers 2203.08635, arXiv.org.
    4. Tobias Fissler & Fangda Liu & Ruodu Wang & Linxiao Wei, 2024. "Elicitability and identifiability of tail risk measures," Papers 2404.14136, arXiv.org, revised Jun 2024.
    5. Tobias Fissler & Johanna F. Ziegel, 2015. "Higher order elicitability and Osband's principle," Papers 1503.08123, arXiv.org, revised Sep 2015.
    6. Matthias Fischer & Thorsten Moser & Marius Pfeuffer, 2018. "A Discussion on Recent Risk Measures with Application to Credit Risk: Calculating Risk Contributions and Identifying Risk Concentrations," Risks, MDPI, vol. 6(4), pages 1-28, December.
    7. Del Brio, Esther B. & Mora-Valencia, Andrés & Perote, Javier, 2020. "Risk quantification for commodity ETFs: Backtesting value-at-risk and expected shortfall," International Review of Financial Analysis, Elsevier, vol. 70(C).
    8. Marcelo Brutti Righi, 2018. "A theory for combinations of risk measures," Papers 1807.01977, arXiv.org, revised May 2023.
    9. Ruodu Wang & Yunran Wei, 2020. "Risk functionals with convex level sets," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1337-1367, October.
    10. 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.
    11. 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.
    12. James Ming Chen, 2018. "On Exactitude in Financial Regulation: Value-at-Risk, Expected Shortfall, and Expectiles," Risks, MDPI, vol. 6(2), pages 1-28, June.
    13. Marcelo Brutti Righi & Paulo Sergio Ceretta, 2015. "Shortfall Deviation Risk: An alternative to risk measurement," Papers 1501.02007, arXiv.org, revised May 2016.
    14. 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.
    15. Mucahit Aygun & Fabio Bellini & Roger J. A. Laeven, 2023. "Elicitability of Return Risk Measures," Papers 2302.13070, arXiv.org, revised Mar 2023.
    16. Marcelo Brutti Righi & Fernanda Maria Muller & Marlon Ruoso Moresco, 2022. "A risk measurement approach from risk-averse stochastic optimization of score functions," Papers 2208.14809, arXiv.org, revised May 2023.
    17. Taylor, James W., 2022. "Forecasting Value at Risk and expected shortfall using a model with a dynamic omega ratio," Journal of Banking & Finance, Elsevier, vol. 140(C).
    18. Taylor, James W., 2021. "Evaluating quantile-bounded and expectile-bounded interval forecasts," International Journal of Forecasting, Elsevier, vol. 37(2), pages 800-811.
    19. Tobias Fissler & Silvana M. Pesenti, 2022. "Sensitivity Measures Based on Scoring Functions," Papers 2203.00460, arXiv.org, revised Jul 2022.
    20. Asimit, Alexandru V. & Bignozzi, Valeria & Cheung, Ka Chun & Hu, Junlei & Kim, Eun-Seok, 2017. "Robust and Pareto optimality of insurance contracts," European Journal of Operational Research, Elsevier, vol. 262(2), pages 720-732.

    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:1303.1690. 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.