Equivalence of [$S$-PE] and [$S$-PE*] for $S$-distortion risk measures

Determine whether scenario-based pooling effect consistency [$S$-PE] is equivalent to its zero-attachment variant [$S$-PE*] for $S$-distortion risk measures. Specifically, ascertain whether, for Choquet risk measures of the form ρ(X) = ∫ X d(g ∘ P^S) with an $S$-distortion function g:[0,1]^s → ℝ, requiring non-increasing risk under asset pooling for all attachment points K ∈ [0,1) is equivalent to requiring it only for K = 0.

Background

Under $S$-law invariance, Choquet risk measures admit the representation ρ(X)=∫ X d(g∘P^S), where g:[0,1]^s→ℝ is an $S$-distortion function and P^S denotes the vector of conditional probabilities across s scenarios. These $S$-distortion risk measures are central to analyzing scenario-based rating criteria.

The pooling effect consistency properties [$S$-PE] and [$S$-PE*] formalize desirable behavior under asset pooling: for sequences of losses that are conditionally iid under each scenario, the normalized senior-tranche loss ([(L^{(ℓ)}−K)_+]/[1−K]) should be non-increasing in the pool size ℓ for all K∈[0,1) in [$S$-PE], while [$S$-PE*] imposes the same requirement only for K=0.

The paper establishes strong characterization results linking [$S$-PE], [$S$-RA], and [QC] under a continuity assumption on g, but without continuity only implication chains are available. Whether [$S$-PE] and [$S$-PE*] coincide for $S$-distortion risk measures remains unresolved, and resolving this would clarify if checking pooling consistency at K=0 suffices to guarantee it for all K∈[0,1).