IDEAS home Printed from https://ideas.repec.org/b/spr/mathfi/v16y2022i3d10.1007_s11579-022-00313-9.html
   My bibliography  Save this book

Law-Invariant Functionals that Collapse to the Mean: Beyond Convexity

Author

Listed:
  • Felix-Benedikt Liebrich

    (Leibniz University Hannover)

  • Cosimo Munari

    (University of Zurich)

Abstract

We establish general “collapse to the mean” principles that provide conditions under which a law-invariant functional reduces to an expectation. In the convex setting, we retrieve and sharpen known results from the literature. However, our results also apply beyond the convex setting. We illustrate this by providing a complete account of the “collapse to the mean” for quasiconvex functionals. In the special cases of consistent risk measures and Choquet integrals, we can even dispense with quasiconvexity. In addition, we relate the “collapse to the mean” to the study of solutions of a broad class of optimisation problems with law-invariant objectives that appear in mathematical finance, insurance, and economics. We show that the corresponding quantile formulations studied in the literature are sometimes illegitimate and require further analysis.

Suggested Citation

  • Felix-Benedikt Liebrich & Cosimo Munari, 2022. "Law-Invariant Functionals that Collapse to the Mean: Beyond Convexity," Mathematics and Financial Economics, Springer, volume 16, number 2, December.
    • Handle: RePEc:spr:mathfi:v:16:y:2022:i:3:d:10.1007_s11579-022-00313-9
    DOI: 10.1007/s11579-022-00313-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11579-022-00313-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s11579-022-00313-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Zuo Quan Xu, 2016. "A Note On The Quantile Formulation," Mathematical Finance, Wiley Blackwell, vol. 26(3), pages 589-601, July.
    2. R. Rockafellar & Stan Uryasev & Michael Zabarankin, 2006. "Generalized deviations in risk analysis," Finance and Stochastics, Springer, vol. 10(1), pages 51-74, January.
    3. Bellini, Fabio & Koch-Medina, Pablo & Munari, Cosimo & Svindland, Gregor, 2021. "Law-invariant functionals that collapse to the mean," Insurance: Mathematics and Economics, Elsevier, vol. 98(C), pages 83-91.
    4. De Giorgi, Enrico, 2005. "Reward-risk portfolio selection and stochastic dominance," Journal of Banking & Finance, Elsevier, vol. 29(4), pages 895-926, April.
    5. Amarante Massimiliano, 2021. "Bipolar behavior of submodular, law-invariant capacities," Statistics & Risk Modeling, De Gruyter, vol. 38(3-4), pages 65-70, July.
    6. Niushan Gao & Denny Leung & Cosimo Munari & Foivos Xanthos, 2018. "Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces," Finance and Stochastics, Springer, vol. 22(2), pages 395-415, April.
    7. Wang, Shaun S., 2002. "A Universal Framework for Pricing Financial and Insurance Risks," ASTIN Bulletin, Cambridge University Press, vol. 32(2), pages 213-234, November.
    8. Chateauneuf, Alain & Eichberger, Jurgen & Grant, Simon, 2007. "Choice under uncertainty with the best and worst in mind: Neo-additive capacities," Journal of Economic Theory, Elsevier, vol. 137(1), pages 538-567, November.
    9. Wakker, Peter P. & Yang, Jingni, 2021. "Concave/convex weighting and utility functions for risk: A new light on classical theorems," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 429-435.
    10. repec:dau:papers:123456789/5392 is not listed on IDEAS
    11. Niushan Gao & Cosimo Munari, 2020. "Surplus-Invariant Risk Measures," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1342-1370, November.
    12. L. Rüschendorf & Steven Vanduffel, 2020. "On the construction of optimal payoffs," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 43(1), pages 129-153, June.
    13. Johannes Leitner, 2005. "A Short Note On Second‐Order Stochastic Dominance Preserving Coherent Risk Measures," Mathematical Finance, Wiley Blackwell, vol. 15(4), pages 649-651, October.
    14. Alexander Schied, 2004. "On the Neyman-Pearson problem for law-invariant risk measures and robust utility functionals," Papers math/0407127, arXiv.org.
    15. Erio Castagnoli & Fabio Maccheroni & Massimo Marinacci, 2004. "Choquet Insurance Pricing: A Caveat," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 481-485, July.
    16. 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.
    17. Fabio Bellini & Pablo Koch-Medina & Cosimo Munari & Gregor Svindland, 2020. "Law-invariant functionals that collapse to the mean," Papers 2009.04144, arXiv.org, revised Jan 2021.
    18. Elyès Jouini & Walter Schachermayer & Nizar Touzi, 2006. "Law Invariant Risk Measures Have the Fatou Property," Post-Print halshs-00176522, HAL.
    19. Jean-Yves Jaffray & Fabrice Philippe, 1997. "On the Existence of Subjective Upper and Lower Probabilities," Mathematics of Operations Research, INFORMS, vol. 22(1), pages 165-185, February.
    20. James P. Quirk & Rubin Saposnik, 1962. "Admissibility and Measurable Utility Functions," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 29(2), pages 140-146.
    21. Nendel, Max & Riedel, Frank & Schmeck, Maren Diane, 2021. "A decomposition of general premium principles into risk and deviation," Insurance: Mathematics and Economics, Elsevier, vol. 100(C), pages 193-209.
    22. Simone Cerreia-Vioglio & Fabio Maccheroni & Massimo Marinacci, 2017. "Stochastic Dominance Analysis Without the Independence Axiom," Management Science, INFORMS, vol. 63(4), pages 1097-1109, April.
    23. Marcelo Brutti Righi, 2019. "A composition between risk and deviation measures," Annals of Operations Research, Springer, vol. 282(1), pages 299-313, November.
    24. Aouani, Zaier & Chateauneuf, Alain, 2008. "Exact capacities and star-shaped distorted probabilities," Mathematical Social Sciences, Elsevier, vol. 56(2), pages 185-194, September.
    25. Massoomeh Rahsepar & Foivos Xanthos, 2020. "On the extension property of dilatation monotone risk measures," Papers 2002.11865, arXiv.org.
    26. Massimo Marinacci, 2000. "A uniqueness theorem for convex-ranged probabilities," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 23(2), pages 121-132.
    27. repec:dau:papers:123456789/342 is not listed on IDEAS
    28. 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.
    29. Felix-Benedikt Liebrich & Gregor Svindland, 2019. "Risk sharing for capital requirements with multidimensional security markets," Finance and Stochastics, Springer, vol. 23(4), pages 925-973, October.
    30. Liu, Fangda & Cai, Jun & Lemieux, Christiane & Wang, Ruodu, 2020. "Convex risk functionals: Representation and applications," Insurance: Mathematics and Economics, Elsevier, vol. 90(C), pages 66-79.
    31. Freddy Delbaen, 2021. "Commonotonicity and time-consistency for Lebesgue-continuous monetary utility functions," Finance and Stochastics, Springer, vol. 25(3), pages 597-614, July.
    32. Samuel Drapeau & Michael Kupper, 2013. "Risk Preferences and Their Robust Representation," Mathematics of Operations Research, INFORMS, vol. 38(1), pages 28-62, February.
    33. Massimo Marinacci, 2002. "Probabilistic Sophistication and Multiple Priors," Econometrica, Econometric Society, vol. 70(2), pages 755-764, March.
    34. Bogdan Grechuk & Anton Molyboha & Michael Zabarankin, 2009. "Maximum Entropy Principle with General Deviation Measures," Mathematics of Operations Research, INFORMS, vol. 34(2), pages 445-467, May.
    35. Ghirardato, Paolo & Maccheroni, Fabio & Marinacci, Massimo, 2004. "Differentiating ambiguity and ambiguity attitude," Journal of Economic Theory, Elsevier, vol. 118(2), pages 133-173, October.
    36. Machina, Mark J & Schmeidler, David, 1992. "A More Robust Definition of Subjective Probability," Econometrica, Econometric Society, vol. 60(4), pages 745-780, July.
    37. Fabio Bellini & Pablo Koch-Medina & Cosimo Munari & Gregor Svindland, 2018. "Law-invariant functionals on general spaces of random variables," Papers 1808.00821, arXiv.org, revised Jan 2021.
    38. Volker Kratschmer & Alexander Schied & Henryk Zahle, 2012. "Comparative and qualitative robustness for law-invariant risk measures," Papers 1204.2458, arXiv.org, revised Jan 2014.
    39. Elisa Mastrogiacomo & Emanuela Rosazza Gianin, 2015. "Portfolio Optimization with Quasiconvex Risk Measures," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 1042-1059, October.
    40. Wang, Shaun S. & Young, Virginia R. & Panjer, Harry H., 1997. "Axiomatic characterization of insurance prices," Insurance: Mathematics and Economics, Elsevier, vol. 21(2), pages 173-183, November.
    41. Liebrich, Felix-Benedikt & Svindland, Gregor, 2019. "Efficient allocations under law-invariance: A unifying approach," Journal of Mathematical Economics, Elsevier, vol. 84(C), pages 28-45.
    42. Zuo Quan Xu, 2013. "A New Characterization of Comonotonicity and its Application in Behavioral Finance," Papers 1311.6080, arXiv.org, revised Jun 2014.
    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. Yi Shen & Zachary Van Oosten & Ruodu Wang, 2024. "Partial Law Invariance and Risk Measures," Papers 2401.17265, arXiv.org, revised Jun 2024.
    2. Max Nendel & Jan Streicher, 2023. "An axiomatic approach to default risk and model uncertainty in rating systems," Papers 2303.08217, arXiv.org, revised Sep 2023.
    3. Martin Herdegen & Nazem Khan & Cosimo Munari, 2024. "Risk, utility and sensitivity to large losses," Papers 2405.12154, arXiv.org.
    4. Felix-Benedikt Liebrich, 2024. "Risk sharing under heterogeneous beliefs without convexity," Finance and Stochastics, Springer, vol. 28(4), pages 999-1033, October.
    5. Nendel, Max & Streicher, Jan, 2023. "An axiomatic approach to default risk and model uncertainty in rating systems," Journal of Mathematical Economics, Elsevier, vol. 109(C).
    6. Muqiao Huang & Ruodu Wang, 2024. "Coherent risk measures and uniform integrability," Papers 2404.03783, arXiv.org, revised Oct 2024.

    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. Felix-Benedikt Liebrich & Cosimo Munari, 2021. "Law-invariant functionals that collapse to the mean: Beyond convexity," Papers 2106.01281, arXiv.org, revised Jul 2021.
    2. Shengzhong Chen & Niushan Gao & Denny Leung & Lei Li, 2021. "Automatic Fatou Property of Law-invariant Risk Measures," Papers 2107.08109, arXiv.org, revised Jan 2022.
    3. Nendel, Max & Streicher, Jan, 2023. "An axiomatic approach to default risk and model uncertainty in rating systems," Journal of Mathematical Economics, Elsevier, vol. 109(C).
    4. Marcelo Brutti Righi & Marlon Ruoso Moresco, 2024. "Inf-convolution and optimal risk sharing with countable sets of risk measures," Annals of Operations Research, Springer, vol. 336(1), pages 829-860, May.
    5. Ruodu Wang & Yunran Wei & Gordon E. Willmot, 2020. "Characterization, Robustness, and Aggregation of Signed Choquet Integrals," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 993-1015, August.
    6. Max Nendel & Jan Streicher, 2023. "An axiomatic approach to default risk and model uncertainty in rating systems," Papers 2303.08217, arXiv.org, revised Sep 2023.
    7. Marcelo Brutti Righi, 2018. "A theory for combinations of risk measures," Papers 1807.01977, arXiv.org, revised May 2023.
    8. 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.
    9. Bellini, Fabio & Koch-Medina, Pablo & Munari, Cosimo & Svindland, Gregor, 2021. "Law-invariant functionals that collapse to the mean," Insurance: Mathematics and Economics, Elsevier, vol. 98(C), pages 83-91.
    10. Fabio Bellini & Pablo Koch-Medina & Cosimo Munari & Gregor Svindland, 2020. "Law-invariant functionals that collapse to the mean," Papers 2009.04144, arXiv.org, revised Jan 2021.
    11. Samuel Solgon Santos & Marlon Ruoso Moresco & Marcelo Brutti Righi & Eduardo de Oliveira Horta, 2023. "A note on the induction of comonotonic additive risk measures from acceptance sets," Papers 2307.04647, arXiv.org, revised Jul 2023.
    12. Xia Han & Ruodu Wang & Qinyu Wu, 2023. "Monotonic mean-deviation risk measures," Papers 2312.01034, arXiv.org, revised Aug 2024.
    13. Enrico G. De Giorgi & David B. Brown & Melvyn Sim, 2010. "Dual representation of choice and aspirational preferences," University of St. Gallen Department of Economics working paper series 2010 2010-07, Department of Economics, University of St. Gallen.
    14. Albrecht, Peter & Huggenberger, Markus, 2017. "The fundamental theorem of mutual insurance," Insurance: Mathematics and Economics, Elsevier, vol. 75(C), pages 180-188.
    15. Liu, Peng & Wang, Ruodu & Wei, Linxiao, 2020. "Is the inf-convolution of law-invariant preferences law-invariant?," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 144-154.
    16. Marcelo Brutti Righi & Paulo Sergio Ceretta, 2015. "Shortfall Deviation Risk: An alternative to risk measurement," Papers 1501.02007, arXiv.org, revised May 2016.
    17. Maria Arduca & Cosimo Munari, 2021. "Risk measures beyond frictionless markets," Papers 2111.08294, arXiv.org.
    18. Marcelo Brutti Righi & Marlon Ruoso Moresco, 2022. "Star-Shaped deviations," Papers 2207.08613, arXiv.org.
    19. Xia Han & Ruodu Wang & Xun Yu Zhou, 2022. "Choquet regularization for reinforcement learning," Papers 2208.08497, arXiv.org.
    20. Santos, Samuel S. & Moresco, Marlon R. & Righi, Marcelo B. & Horta, Eduardo, 2024. "A note on the induction of comonotonic additive risk measures from acceptance sets," Statistics & Probability Letters, Elsevier, vol. 208(C).

    More about this item

    Keywords

    Law invariance; Quasiconvex functionals; Consistent risk measures; Nonconvex Choquet integrals; Optimisation problems;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty

    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:spr:mathfi:v:16:y:2022:i:3:d:10.1007_s11579-022-00313-9. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    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.