A note on the induction of comonotonic additive risk measures from acceptance sets

Paper: arXiv 2307.04647

Authors: Samuel Solgon Santos, Marlon Ruoso Moresco, Marcelo Brutti Righi, Eduardo de Oliveira Horta

Abstract

We present simple general conditions on the acceptance sets under which their induced monetary risk and deviation measures are comonotonic additive. We show that acceptance sets induce comonotonic additive risk measures if and only if the acceptance sets and their complements are stable under convex combinations of comonotonic random variables. A generalization of this result applies to risk measures that are additive for random variables with a priori specified dependence structures, e.g., perfectly correlated, uncorrelated, or independent random variables.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical characterization of comonotonic additive risk measures derived from acceptance sets using functional analysis and probability theory. However, it is a purely theoretical note lacking any empirical validation, backtesting, or real-world market data application.

Research Flowchart

  flowchart TD
    A[Research Goal: Conditions for acceptance sets to<br>induce comonotonic additive risk measures] --> B[Inputs: Acceptance sets, Comonotonic RVs,<br>Specified dependence structures]
    B --> C[Methodology: Analyze stability under convex<br>combinations of comonotonic random variables]
    C --> D[Generalization Step: Extend analysis to other<br>dependence structures e.g., independent, uncorrelated]
    D --> E[Key Finding: Risk/deviation measures are comonotonic additive<br>iff acceptance sets AND complements are comonotonically convex-stable]