Definition
A coherent risk measure (Artzner-Delbaen-Eber-Heath 1999) is a functional ρ mapping a future net worth (a random variable X) to the amount of capital required to make the position acceptable, that satisfies four axioms: translation invariance, subadditivity, positive homogeneity, and monotonicity. The definition is dual to an acceptance set — the set of positions requiring no extra capital — and ρ(X) is the minimum amount that, added and invested in a reference "prudent" asset, moves X into that set.
Key Ideas
- The four axioms (with r the return on a reference riskless asset):
- Translation invariance: ρ(X+αr)=ρ(X)−α — adding cash lowers required capital one-for-one.
- Subadditivity: ρ(X1+X2)≤ρ(X1)+ρ(X2) — merging positions cannot increase total risk; this is the axiom that encodes diversification.
- Positive homogeneity: ρ(λX)=λρ(X), λ≥0 — risk scales with size (together with subadditivity, ρ is convex).
- Monotonicity: X≤Y⇒ρ(Y)≤ρ(X).
- VaR is not coherent. Value at risk is a quantile of the loss distribution and can violate subadditivity: the VaR of a combined portfolio can exceed the sum of the standalone VaRs, so VaR can penalize diversification and be reduced by splitting a book across accounts. It also ignores the size of losses in the tail beyond the quantile.
- Expected shortfall is coherent. Tail conditional expectation / expected shortfall (average loss beyond the VaR quantile) satisfies all four axioms and is the standard coherent "repair" of VaR — the basis for the Basel shift from VaR to ES.
- FRTB makes it regulation (Basel Committee 2016): the Fundamental Review of the Trading Book replaces the internal-models 99% VaR charge with 97.5% Expected Shortfall, calibrated to a stress period and scaled by risk-factor-specific liquidity horizons (10–120 days), with desk-level model approval. Tellingly, because ES is not elicitable (hard to backtest in isolation), the standard still backtests VaR at the 97.5th and 99th percentiles even though it capitalises on ES — a concrete instance of the estimation/validation tension below.
- Representation theorem. ρ is coherent iff it is a worst case of expected loss over a set P of probability measures ("generalized scenarios"): ρ(X)=supP∈PEP[−X/r]. Coherence is exactly worst-case-over-scenarios, so scenario/stress-test methods are the universal construction.
- A class, not a unique measure. The axioms characterize a family; choosing a specific coherent measure requires additional economic input (e.g., the confidence level for expected shortfall).
Why It Matters
- Reframed capital regulation. By showing the industry-standard VaR fails a diversification axiom, the coherence framework motivated the regulatory move toward expected shortfall (Basel 2.5/III) and gave a principled language for evaluating any proposed risk measure.
- Diversification consistency. Subadditivity guarantees that portfolio aggregation and decentralization of risk limits behave sensibly — a property risk managers rely on when allocating capital across desks.
- Bridges estimation and axiomatics. Much of the VaR literature perfects the estimation of the conditional quantile (GARCH filters, CAViaR, filtered historical simulation); coherence addresses the prior question of whether the quantile is the right functional to estimate at all.
Open Questions
- Estimation of expected shortfall. ES is harder to estimate and to backtest than VaR (it is not elicitable in isolation), which complicated the regulatory transition.
- Convex vs. coherent. Dropping positive homogeneity (to reflect liquidity/size effects) gives the broader class of convex risk measures; when the extra generality is needed in practice is unsettled.
- Choice within the class. The axioms leave the confidence level and scenario set unspecified, so model and parameter risk re-enter through those choices.
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