Coherent risk measure

A coherent risk measure is a function ρ that assigns capital to a random outcome X and satisfies four axioms, set by Artzner, Delbaen, Eber and Heath, "Coherent Measures of Risk," Mathematical Finance, 1999; VaR fails one of them. A measure ρ is coherent if it is monotone (an outcome that is always worse needs at least as much capital), translation invariant (ρ(X + c) = ρ(X) − c, so adding a sure amount cuts required capital by the same amount), sub-additive (ρ(X + Y) ≤ ρ(X) + ρ(Y), so pooling never costs more than the sum) and positively homogeneous (ρ(λX) = λρ(X) for λ ≥ 0). VaR is not sub-additive in general. On heavy-tailed or lumpy losses, such as credit exposures subject to default, it can show combined capital above the sum of the standalone figures, which rewards splitting risks instead of pooling them. Under elliptical distributions, the multivariate normal first among them, VaR becomes sub-additive again, which is part of why it lasted so long in models. TVaR, or Expected Shortfall, meets all four axioms. For a continuous distribution it is the average loss given that the VaR at the same level has been breached. Coherence often gets confused with convexity. Convex risk measures replace sub-additivity and homogeneity with convexity alone, to allow a doubled position to need more than twice the capital when liquidity is thin; every coherent measure is convex, and the converse fails. Spectral risk measures, which weight quantiles more heavily toward the tail, form a broad family of coherent measures with Expected Shortfall as the simplest case. In practice, sub-additivity and homogeneity are what let a group allocate capital to its lines by Euler contributions without any line getting more than its standalone capital. The Swiss Solvency Test adopted Expected Shortfall at 99%; the Basel Committee adopted it at 97.5% for market risk in the Fundamental Review of the Trading Book (FRTB); Solvency II stayed with 99.5% VaR.