Risk measure satisfying Artzner's four axioms: monotonicity, translation invariance, sub-additivity and positive homogeneity.
A coherent risk measure satisfies Artzner's (1999) four axioms: (1) Monotonicity: if X ≤ Y almost surely, ρ(X) ≤ ρ(Y). (2) Translation invariance: ρ(X + c) = ρ(X) − c for any scalar c. (3) Sub-additivity: ρ(X + Y) ≤ ρ(X) + ρ(Y), meaning diversification can only improve the risk profile. (4) Positive homogeneity: ρ(λX) = λρ(X) for λ ≥ 0. VaR is generally not sub-additive (not coherent); TVaR (Expected Shortfall) is a coherent measure. Coherence matters for capital management: a non-coherent measure can suggest that combining risks increases required capital, which is counter-intuitive and incentivizes poor allocation.
VaR not coherent: two 95%-correlated bonds can each have VaR of 100 M EUR but a combined VaR of 210 M EUR > 200 M EUR (violates sub-additivity). TVaR does not have this problem.
coherent risk measure, mesure de risque cohérente, axiomes Artzner