TY - JOUR

T1 - Law-Invariant Functionals that Collapse to the Mean

T2 - Beyond Convexity

AU - Liebrich, Felix-Benedikt

AU - Munari, Cosimo

N1 - Funding Information: We would like to thank two anonymous referees, the Editor, the participants of the Weekly Seminars on Risk Management and Actuarial Science at the University of Waterloo and of the Mathematical Finance Seminar at the University of Bielefeld, and Ruodu Wang for their valuable input which helped improving earlier versions of this manuscript. Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

PY - 2022/7

Y1 - 2022/7

N2 - 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.

AB - 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.

KW - q-fin.MF

KW - math.PR

KW - q-fin.RM

KW - Consistent risk measures

KW - Nonconvex Choquet integrals

KW - Law invariance

KW - Optimisation problems

KW - Quasiconvex functionals

UR - http://www.scopus.com/inward/record.url?scp=85127302611&partnerID=8YFLogxK

U2 - 10.1007/s11579-022-00313-9

DO - 10.1007/s11579-022-00313-9

M3 - Article

VL - 16

SP - 447

EP - 480

JO - Mathematics and Financial Economics

JF - Mathematics and Financial Economics

SN - 1862-9679

IS - 3

ER -