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Star-shaped Risk Measures

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Listed:
  • Erio Castagnoli
  • Giacomo Cattelan
  • Fabio Maccheroni
  • Claudio Tebaldi
  • Ruodu Wang

Abstract

In this paper monetary risk measures that are positively superhomogeneous, called star-shaped risk measures, are characterized and their properties studied. The measures in this class, which arise when the controversial subadditivity property of coherent risk measures is dispensed with and positive homogeneity is weakened, include all practically used risk measures, in particular, both convex risk measures and Value-at-Risk. From a financial viewpoint, our relaxation of convexity is necessary to quantify the capital requirements for risk exposure in the presence of liquidity risk, competitive delegation, or robust aggregation mechanisms. From a decision theoretical perspective, star-shaped risk measures emerge from variational preferences when risk mitigation strategies can be adopted by a rational decision maker.

Suggested Citation

  • Erio Castagnoli & Giacomo Cattelan & Fabio Maccheroni & Claudio Tebaldi & Ruodu Wang, 2021. "Star-shaped Risk Measures," Papers 2103.15790, arXiv.org, revised Apr 2022.
    • Handle: RePEc:arx:papers:2103.15790
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    Cited by:

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    3. Felix-Benedikt Liebrich, 2021. "Risk sharing under heterogeneous beliefs without convexity," Papers 2108.05791, arXiv.org, revised May 2022.
    4. Zou, Zhenfeng & Wu, Qinyu & Xia, Zichao & Hu, Taizhong, 2023. "Adjusted Rényi entropic Value-at-Risk," European Journal of Operational Research, Elsevier, vol. 306(1), pages 255-268.
    5. Felix-Benedikt Liebrich & Cosimo Munari, 2021. "Law-invariant functionals that collapse to the mean: Beyond convexity," Papers 2106.01281, arXiv.org, revised Jul 2021.
    6. Laeven, Roger J.A. & Rosazza Gianin, Emanuela & Zullino, Marco, 2024. "Law-invariant return and star-shaped risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 117(C), pages 140-153.
    7. Dejian Tian & Xunlian Wang, 2023. "Dynamic star-shaped risk measures and $g$-expectations," Papers 2305.02481, arXiv.org.
    8. Giulio Principi & Peter P. Wakker & Ruodu Wang, 2023. "Anticomonotonicity for Preference Axioms: The Natural Counterpart to Comonotonicity," Papers 2307.08542, arXiv.org, revised Dec 2024.
    9. Tolulope Fadina & Yang Liu & Ruodu Wang, 2021. "A Framework for Measures of Risk under Uncertainty," Papers 2110.10792, arXiv.org, revised Sep 2023.
    10. Marcelo Brutti Righi & Marlon Ruoso Moresco, 2022. "Star-Shaped deviations," Papers 2207.08613, arXiv.org.
    11. Martin Herdegen & Nazem Khan & Cosimo Munari, 2024. "Risk, utility and sensitivity to large losses," Papers 2405.12154, arXiv.org.
    12. Moresco, Marlon Ruoso & Righi, Marcelo Brutti, 2022. "On the link between monetary and star-shaped risk measures," Statistics & Probability Letters, Elsevier, vol. 184(C).
    13. Roger J. A. Laeven & Emanuela Rosazza Gianin & Marco Zullino, 2023. "Dynamic Return and Star-Shaped Risk Measures via BSDEs," Papers 2307.03447, arXiv.org, revised Jul 2023.
    14. Martin Herdegen & Nazem Khan, 2022. "$\rho$-arbitrage and $\rho$-consistent pricing for star-shaped risk measures," Papers 2202.07610, arXiv.org, revised May 2024.
    15. Zou, Zhenfeng & Hu, Taizhong, 2024. "Adjusted higher-order expected shortfall," Insurance: Mathematics and Economics, Elsevier, vol. 115(C), pages 1-12.
    16. Bingchu Nie & Dejian Tian & Long Jiang, 2024. "Set-valued Star-Shaped Risk Measures," Papers 2402.18014, arXiv.org.
    17. Francesco Fabbri & Giulio Principi & Lorenzo Stanca, 2024. "Absolute and Relative Ambiguity Attitudes," Papers 2406.01343, arXiv.org, revised Jun 2024.
    18. Max Nendel, 2025. "Lower semicontinuity of monotone functionals in the mixed topology on C b $C_{b}$," Finance and Stochastics, Springer, vol. 29(1), pages 261-287, January.
    19. Mario Ghossoub & Qinghua Ren & Ruodu Wang, 2024. "Counter-monotonic risk allocations and distortion risk measures," Papers 2407.16099, arXiv.org.
    20. 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.

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