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On the link between monetary and star-shaped risk measures

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  • Moresco, Marlon Ruoso
  • Righi, Marcelo Brutti

Abstract

Recently, Castagnoli et al. (2021) introduced the class of star-shaped risk measures as a generalization of convex and coherent ones, proving that there is a representation as the pointwise minimum of some family composed by convex risk measures. Concomitantly, Jia et al. (2020) prove a similar representation result for monetary risk measures, which are more general than star-shaped ones. Then, there is a question on how both classes are connected. In this letter, we provide an answer by casting light on the importance of the acceptability of 0, which is linked to the property of normalization. We then show that under mild conditions, a monetary risk measure is only a translation away from star-shapedness.

Suggested Citation

  • Moresco, Marlon Ruoso & Righi, Marcelo Brutti, 2022. "On the link between monetary and star-shaped risk measures," Statistics & Probability Letters, Elsevier, vol. 184(C).
    • Handle: RePEc:eee:stapro:v:184:y:2022:i:c:s016771522100290x
    DOI: 10.1016/j.spl.2021.109345
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    References listed on IDEAS

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    1. Guangyan Jia & Jianming Xia & Rongjie Zhao, 2020. "Monetary Risk Measures," Papers 2012.06751, arXiv.org.
    2. Righi, Marcelo Brutti, 2024. "Star-shaped acceptability indexes," Insurance: Mathematics and Economics, Elsevier, vol. 117(C), pages 170-181.
    3. Felix-Benedikt Liebrich, 2021. "Risk sharing under heterogeneous beliefs without convexity," Papers 2108.05791, arXiv.org, revised May 2022.
    4. Elyès Jouini & Walter Schachermayer & Nizar Touzi, 2006. "Law Invariant Risk Measures Have the Fatou Property," Post-Print halshs-00176522, HAL.
    5. Hans Föllmer & Alexander Schied, 2002. "Convex measures of risk and trading constraints," Finance and Stochastics, Springer, vol. 6(4), pages 429-447.
    6. Song, Yongsheng & Yan, Jia-An, 2009. "Risk measures with comonotonic subadditivity or convexity and respecting stochastic orders," Insurance: Mathematics and Economics, Elsevier, vol. 45(3), pages 459-465, December.
    7. Frittelli, Marco & Rosazza Gianin, Emanuela, 2002. "Putting order in risk measures," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1473-1486, July.
    8. Erio Castagnoli & Giacomo Cattelan & Fabio Maccheroni & Claudio Tebaldi & Ruodu Wang, 2021. "Star-shaped Risk Measures," Papers 2103.15790, arXiv.org, revised Apr 2022.
    9. 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.
    10. repec:dau:papers:123456789/342 is not listed on IDEAS
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    1. Righi, Marcelo Brutti, 2024. "Star-shaped acceptability indexes," Insurance: Mathematics and Economics, Elsevier, vol. 117(C), pages 170-181.
    2. 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.
    3. 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).
    4. Marcelo Brutti Righi & Marlon Ruoso Moresco, 2022. "Star-Shaped deviations," Papers 2207.08613, arXiv.org.
    5. 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.
    6. Dejian Tian & Xunlian Wang, 2023. "Dynamic star-shaped risk measures and $g$-expectations," Papers 2305.02481, arXiv.org.
    7. 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.
    8. Bingchu Nie & Dejian Tian & Long Jiang, 2024. "Set-valued Star-Shaped Risk Measures," Papers 2402.18014, arXiv.org.

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