Finiteness of the folding score for law-invariant coherent risk measures

Determine whether, for every law-invariant coherent risk measure p defined on L-infinity that is not equal to the expectation, the folding score Sp—defined as the supremum over all X in L-infinity of p(|X|) divided by the maximum of p(X) and p(-X)—is finite. Equivalently, establish whether the folding score remains bounded for all non-mean law-invariant coherent risk measures on L-infinity.

Background

The paper introduces the folding score Sp of a risk measure p to quantify how large p(|X|) can be relative to p(X) and p(-X). Theorem 3.1 shows that for coherent distortion risk measures on L1, the folding score is finite unless p equals the expectation. Appendix A explores whether this finiteness extends beyond distortion risk measures.

Counterexamples in Appendix A demonstrate that the finiteness property fails for several broader classes: law-invariant convex risk measures, general coherent risk measures, and coherent Choquet risk measures. However, for the intermediate class of law-invariant coherent risk measures, neither a counterexample nor a general proof is known. Using Kusuoka representation, the authors note that if the representing set of distortions is finite, the finiteness follows, but for infinite sets the situation remains unclear.