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A Review and Some Complements on Quantile Risk Measures and Their Domain

Author

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  • Sebastian Fuchs

    (Faculty of Economics and Management, Free University of Bozen-Bolzano, 39100 Bolzano, Italy)

  • Ruben Schlotter

    (Fakultät für Mathematik, Technische Universität Chemnitz, 09126 Chemnitz, Germany)

  • Klaus D. Schmidt

    (Fachrichtung Mathematik, Technische Universität Dresden, 01062 Dresden, Germany)

Abstract

In the present paper, we study quantile risk measures and their domain. Our starting point is that, for a probability measure Q on the open unit interval and a wide class L Q of random variables, we define the quantile risk measure ϱ Q as the map that integrates the quantile function of a random variable in L Q with respect to Q . The definition of L Q ensures that ϱ Q cannot attain the value + ∞ and cannot be extended beyond L Q without losing this property. The notion of a quantile risk measure is a natural generalization of that of a spectral risk measure and provides another view of the distortion risk measures generated by a distribution function on the unit interval. In this general setting, we prove several results on quantile or spectral risk measures and their domain with special consideration of the expected shortfall. We also present a particularly short proof of the subadditivity of expected shortfall.

Suggested Citation

  • Sebastian Fuchs & Ruben Schlotter & Klaus D. Schmidt, 2017. "A Review and Some Complements on Quantile Risk Measures and Their Domain," Risks, MDPI, vol. 5(4), pages 1-16, November.
    • Handle: RePEc:gam:jrisks:v:5:y:2017:i:4:p:59-:d:117902
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    References listed on IDEAS

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    1. Georg Ch Pflug & Werner Römisch, 2007. "Modeling, Measuring and Managing Risk," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 6478, September.
    2. 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.
    3. 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.
    4. Pichler, Alois, 2013. "The natural Banach space for version independent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 53(2), pages 405-415.
    5. Embrechts Paul & Wang Ruodu, 2015. "Seven Proofs for the Subadditivity of Expected Shortfall," Dependence Modeling, De Gruyter, vol. 3(1), pages 1-15, October.
    6. Wang, Shaun & Dhaene, Jan, 1998. "Comonotonicity, correlation order and premium principles," Insurance: Mathematics and Economics, Elsevier, vol. 22(3), pages 235-242, July.
    7. Rama Cont & Romain Deguest & Giacomo Scandolo, 2010. "Robustness and sensitivity analysis of risk measurement procedures," Post-Print hal-00413729, HAL.
    8. Alexander J. McNeil & Rüdiger Frey & Paul Embrechts, 2015. "Quantitative Risk Management: Concepts, Techniques and Tools Revised edition," Economics Books, Princeton University Press, edition 2, number 10496.
    9. Greselin, Francesca & Zitikis, Ricardas, 2015. "Measuring economic inequality and risk: a unifying approach based on personal gambles, societal preferences and references," MPRA Paper 65892, University Library of Munich, Germany.
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    Cited by:

    1. Barczy, Mátyás & K. Nedényi, Fanni & Sütő, László, 2023. "Probability equivalent level of Value at Risk and higher-order Expected Shortfalls," Insurance: Mathematics and Economics, Elsevier, vol. 108(C), pages 107-128.
    2. Fuchs Sebastian & Trutschnig Wolfgang, 2020. "On quantile based co-risk measures and their estimation," Dependence Modeling, De Gruyter, vol. 8(1), pages 396-416, January.
    3. Silvia Faroni & Olivier Le Courtois & Krzysztof Ostaszewski, 2022. "Equivalent Risk Indicators: VaR, TCE, and Beyond," Risks, MDPI, vol. 10(8), pages 1-19, July.
    4. Fuchs Sebastian & Trutschnig Wolfgang, 2020. "On quantile based co-risk measures and their estimation," Dependence Modeling, De Gruyter, vol. 8(1), pages 396-416, January.
    5. Zou, Zhenfeng & Hu, Taizhong, 2024. "Adjusted higher-order expected shortfall," Insurance: Mathematics and Economics, Elsevier, vol. 115(C), pages 1-12.
    6. 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.
    7. Matyas Barczy & Fanni K. Ned'enyi & L'aszl'o SutH{o}, 2022. "Probability equivalent level of Value at Risk and higher-order Expected Shortfalls," Papers 2202.09770, arXiv.org, revised Nov 2022.

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