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Quantitative Risk Management / Loss Distributions, VaR, and Coherent Risk Measures

Coherent Risk Measures: Axioms and the Failure of VaR

VaR is translation invariant, monotone, and positive homogeneous — three properties that make good intuitive sense for a risk measure. But it can violate a fourth property, subadditivity, meaning the measured risk of a merged portfolio can exceed the sum of the component risks. This is not a minor technicality: it means that VaR-based risk limits can incentivise concentration rather than diversification, that regulatory capital can decrease when a bank legally splits itself into subsidiaries, and that VaR-based decentralised risk management — each desk's VaR bounded independently — cannot guarantee a bound on the firm's total VaR.
Artzner, Delbaen, Eber and Heath (1999) formalised what a good risk measure should do: they wrote down four axioms and called measures satisfying all four coherent. This lesson develops those axioms, shows concretely why VaR fails subadditivity, proves that ES satisfies it, and introduces the dual representation theorem — which reveals that every coherent risk measure is a worst-case expected loss over some set of probability measures, connecting the distribution-based and scenario-based approaches to risk measurement.
Learning objectives. By the end of this note you should be able to:
  • state and interpret the four coherence axioms (monotonicity, translation invariance, positive homogeneity, subadditivity);
  • prove that VaR satisfies three of the four axioms but fails subadditivity;
  • construct a concrete counterexample (defaultable bonds) demonstrating VaR superadditivity;
  • prove subadditivity of ES for continuous loss distributions;
  • state the dual representation theorem for coherent risk measures and explain its economic meaning;
  • distinguish coherent and convex risk measures and give the motivation for each.

In this lesson

    1. Risk measures as functions on loss distributions
    1. The four coherence axioms
    1. VaR: three axioms satisfied, one violated
    1. ES is coherent: subadditivity proof
    1. Convex risk measures
    1. The dual representation theorem

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