E - Companion of “ On the Measurement of Economic Tail Risk

Proof. Without loss of generality, we only need to prove for the case s= 1, as ρ satisfies Axioms A1-A5 if and only if 1 s ρ satisfies Axioms A1-A5 (with s= 1 in Axiom A3). The “only if” part. First, we show that (2) holds for any X ∈L∞(Ω,F , P ). Define the set function ν(E) := ρ(1E),E ∈F . Then, it follows from Axiom A2 and A3 that ν is monotonic, ν(∅) = 0, and ν(Ω) = 1. ForM ≥ 1, define LM := {X | |X| ≤M}. For any X ∈L∞(Ω,F , P ), let M0 be the essential supremum of |X| and denote X0 := min(M0,max(X,−M0)). Then X0 ∈ LM0 and X = X0 a.s., which implies that ρ(X) = ρ(X0) (by Axiom A4) and ν(X > x) = ν(X0 > x), ∀x. Since ρ satisfies Axioms A1-A3 on L∞(Ω,F , P ), it follows that ρ satisfies the conditions (i)-(iii) of the Corollary in Section 3 of Schmeidler (1986) (with B(K) in the corollary defined to be L1+M0). Hence, it follows from the Corollary that