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


Liebrich, Felix-Benedikt; Munari, Cosimo (2022). Law-Invariant Functionals that Collapse to the Mean: Beyond Convexity. Mathematics and Financial Economics, 16(3):447-480.

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.

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:03 Faculty of Economics > Department of Banking and Finance
Dewey Decimal Classification:330 Economics
Scopus Subject Areas:Physical Sciences > Statistics and Probability
Social Sciences & Humanities > Finance
Social Sciences & Humanities > Statistics, Probability and Uncertainty
Scope:Discipline-based scholarship (basic research)
Language:English
Date:28 March 2022
Deposited On:22 Aug 2022 07:00
Last Modified:27 Jun 2024 01:39
Publisher:Springer
ISSN:1862-9679
OA Status:Hybrid
Free access at:Official URL. An embargo period may apply.
Publisher DOI:https://doi.org/10.1007/s11579-022-00313-9
Official URL:https://link.springer.com/article/10.1007/s11579-022-00313-9
Other Identification Number:merlin-id:22591
  • Content: Published Version
  • Licence: Creative Commons: Attribution 4.0 International (CC BY 4.0)