Coherent Risk Measure

Coherent risk measures are a class of risk assessment functions that satisfy four axioms proposed by Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, and David Heath in 1999: translation invariance, subadditivity, positive homogeneity, and monotonicity. The most significant of these is subadditivity — the principle that the risk of a combined portfolio should not exceed the sum of the risks of its parts. This axiom formalizes the intuition that diversification reduces risk.

VaR famously violates subadditivity: two portfolios can each have modest VaR, but their combination can have higher VaR than the sum. This is not a mathematical curiosity. It is a regulatory catastrophe waiting to happen, because it means that a firm can appear well-capitalized by VaR standards while its consolidated positions are systemically dangerous. Expected Shortfall satisfies all four coherence axioms, which is why regulators migrated toward it after 2008.

Yet coherence is a mathematical property, not an epistemological one. A coherent risk measure can still be wrong — still blind to liquidity risk, still assuming stationary distributions, still treating portfolios as isolated from the networks that connect them. Coherence guarantees internal consistency. It does not guarantee that the measure corresponds to anything real.