Paper: arXiv 2507.09181

Authors: Mücahit Aygün, Fabio Bellini, Roger J. A. Laeven

Abstract

We introduce a generalized version of Orlicz premia, based on possibly non-convex loss functions. We show that this generalized definition covers a variety of relevant examples, such as the geometric mean and the expectiles, while at the same time retaining a number of relevant properties. We establish that cash-additivity leads to $L^p$-quantiles, extending a classical result on ‘collapse to the mean’ for convex Orlicz premia. We then focus on the geometrically convex case, discussing the dual representation of generalized Orlicz premia and comparing it with a multiplicative form of the standard dual representation for the convex case. Finally, we show that generalized Orlicz premia arise naturally as the only elicitable, positively homogeneous, monotone and normalized functionals.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical generalization of Orlicz premia, providing elegant mathematical characterizations and extending classical results like ‘collapse to the mean’ to expectiles and geometric convexity. However, as a purely theoretical quantitative finance/actuarial mathematics paper, it lacks empirical data, backtests, or numerical applications.

Research Flowchart

  flowchart TD
    A[Research Goal: Generalize Orlicz premia using non-convex loss functions] --> B[Input: Loss functions, Geometric convexity, & Axiomatic properties]
    B --> C[Methodology: Axiomatic characterization & Dual representation analysis]
    C --> D[Computational & Theoretical Derivations]
    D --> E[Finding 1: Covers geometric mean, expectiles, & cash-additivity yields Lp-quantiles]
    D --> F[Finding 2: Arise as the unique elicitable, positively homogeneous, monotone, & normalized functionals]