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Law-invariant return and star-shaped risk measures

Author

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  • Laeven, Roger J.A.
  • Rosazza Gianin, Emanuela
  • Zullino, Marco

Abstract

This paper presents novel characterization results for classes of law-invariant star-shaped functionals. We begin by establishing characterizations for positively homogeneous and star-shaped functionals that exhibit second- or convex-order stochastic dominance consistency. Building on these characterizations, we proceed to derive Kusuoka-type representations for these functionals, shedding light on their mathematical structure and intimate connections to Value-at-Risk and Expected Shortfall. Furthermore, we offer representations of general law-invariant star-shaped functionals as robustifications of Value-at-Risk. Notably, our results are versatile, accommodating settings that may, or may not, involve monotonicity and/or cash-additivity. All of these characterizations are developed within a general locally convex topological space of random variables, ensuring the broad applicability of our results in various financial, insurance and probabilistic contexts.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:insuma:v:117:y:2024:i:c:p:140-153
    DOI: 10.1016/j.insmatheco.2024.04.006
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    1. Yi Shen & Zachary Van Oosten & Ruodu Wang, 2024. "Partial Law Invariance and Risk Measures," Papers 2401.17265, arXiv.org, revised Jun 2024.
    2. Bingchu Nie & Dejian Tian & Long Jiang, 2024. "Set-valued Star-Shaped Risk Measures," Papers 2402.18014, arXiv.org.

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    More about this item

    Keywords

    Return risk measures; Star-shapedness; Law-invariance; SSD- and CSD-consistency; Value-at-risk; Expected shortfall;
    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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