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Entropy Coherent and Entropy Convex Measures of Risk

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  • Laeven, R.J.A.

    (Tilburg University, Center For Economic Research)

  • Stadje, M.A.

    (Tilburg University, Center For Economic Research)

Abstract

We introduce two subclasses of convex measures of risk, referred to as entropy coherent and entropy convex measures of risk. Entropy coherent and entropy convex measures of risk are special cases of (phi)-coherent and (phi)-convex measures of risk. Contrary to the classical use of coherent and convex measures of risk, which for a given probabilistic model entails evaluating a financial position by considering its expected loss, (phi)-coherent and (phi)-convex measures of risk evaluate a financial position under a given probabilistic model by considering its normalized expected (phi)-loss. We prove that (i) entropy coherent and entropy convex measures of risk are obtained by requiring (phi)-coherent and (phi)-convex measures of risk to be translation invariant; (ii) convex, entropy convex, and entropy coherent measures of risk emerge as certainty equivalents under variational, homothetic, and multiple priors preferences upon requiring the certainty equivalents to be translation invariant; and (iii) (phi)-convex measures of risk are certainty equivalents under variational and homothetic preferences if and only if they are convex and entropy convex measures of risk. In addition, we study the properties of entropy coherent and entropy convex measures of risk, derive their dual conjugate function, and characterize entropy coherent and entropy convex measures of risk in terms of properties of the corresponding acceptance sets.
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Suggested Citation

  • Laeven, R.J.A. & Stadje, M.A., 2011. "Entropy Coherent and Entropy Convex Measures of Risk," Discussion Paper 2011-031, Tilburg University, Center for Economic Research.
    • Handle: RePEc:tiu:tiucen:08f59c7c-7302-47f9-9a9b-b606762fd2f7
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    Citations

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    Cited by:

    1. Dhaene, Jan & Laeven, Roger J.A. & Zhang, Yiying, 2022. "Systemic risk: Conditional distortion risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 102(C), pages 126-145.
    2. Arai, Takuji & Asano, Takao & Nishide, Katsumasa, 2019. "Optimal initial capital induced by the optimized certainty equivalent," Insurance: Mathematics and Economics, Elsevier, vol. 85(C), pages 115-125.
    3. Roger J. A. Laeven & John G. M. Schoenmakers & Nikolaus F. F. Schweizer & Mitja Stadje, 2020. "Robust Multiple Stopping -- A Pathwise Duality Approach," Papers 2006.01802, arXiv.org, revised Sep 2021.
    4. Thai Nguyen & Mitja Stadje, 2020. "Utility maximization under endogenous pricing," Papers 2005.04312, arXiv.org, revised Mar 2024.
    5. Knispel, Thomas & Laeven, Roger J.A. & Svindland, Gregor, 2016. "Robust optimal risk sharing and risk premia in expanding pools," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 182-195.
    6. Goovaerts, Marc J. & Kaas, Rob & Laeven, Roger J.A., 2011. "Worst case risk measurement: Back to the future?," Insurance: Mathematics and Economics, Elsevier, vol. 49(3), pages 380-392.
    7. Kaluszka, M. & Laeven, R.J.A. & Okolewski, A., 2012. "A note on weighted premium calculation principles," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 379-381.
    8. Marcelo Brutti Righi, 2018. "A theory for combinations of risk measures," Papers 1807.01977, arXiv.org, revised May 2023.
    9. Bellini, Fabio & Bignozzi, Valeria & Puccetti, Giovanni, 2018. "Conditional expectiles, time consistency and mixture convexity properties," Insurance: Mathematics and Economics, Elsevier, vol. 82(C), pages 117-123.
    10. Thomas Knispel & Roger J. A. Laeven & Gregor Svindland, 2021. "Asymptotic Analysis of Risk Premia Induced by Law-Invariant Risk Measures," Papers 2107.01730, arXiv.org.
    11. Stephen J. Mildenhall, 2017. "Actuarial Geometry," Risks, MDPI, vol. 5(2), pages 1-44, June.
    12. Roger J. A. Laeven & Mitja Stadje, 2014. "Robust Portfolio Choice and Indifference Valuation," Mathematics of Operations Research, INFORMS, vol. 39(4), pages 1109-1141, November.
    13. Shushi, Tomer & Yao, Jing, 2020. "Multivariate risk measures based on conditional expectation and systemic risk for Exponential Dispersion Models," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 178-186.

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    More about this item

    Keywords

    Multiple priors; Variational and homothetic preferences; Robustness; Convex risk measures; Exponential utility; Relative entropy; Translation invariance; Convexity; Indifference valuation;
    All these keywords.

    JEL classification:

    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G20 - Financial Economics - - Financial Institutions and Services - - - General

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