Artzner-Delbaen-Eber-Heath (1999) Coherent Measures of Risk

Summary

This paper puts risk measurement on an axiomatic footing. It proposes four properties that any sensible measure of financial risk — one used to set capital or margin requirements — ought to satisfy, and calls a measure coherent if it satisfies all four. The central negative result is that value at risk (VaR) is not coherent: it can violate subadditivity, so it may penalize diversification and reward the splitting of a portfolio across accounts. The central positive result is a representation theorem: every coherent risk measure is the worst-case expected loss over a family of probability measures ("generalized scenarios"). The authors propose tail conditional expectation (expected shortfall) as a canonical coherent measure and show how to repair the subadditivity failure of quantile-based methods.

Key Claims

• The four coherence axioms for a risk measure $\rho$ on future net worths (with $r$ the return of a reference "prudent" asset):
  • Translation invariance: $\rho(X+\alpha r)=\rho(X)-\alpha$ — adding cash reduces required capital one-for-one, so $\rho(X+\rho(X)r)=0$.
  • Subadditivity: $\rho(X_1+X_2)\le\rho(X_1)+\rho(X_2)$ — "a merger does not create extra risk"; encodes diversification.
  • Positive homogeneity: $\rho(\lambda X)=\lambda\rho(X)$ for $\lambda\ge0$ — risk scales with position size.
  • Monotonicity: $X\le Y\Rightarrow\rho(Y)\le\rho(X)$ — a dominated position is riskier.
  • (An auxiliary relevance axiom rules out assigning zero risk to a position that can only lose.)
• Risk measures ↔ acceptance sets. $\rho(X)$ is defined as the minimum extra capital that, invested in the reference instrument, makes the position acceptable; the axioms on $\rho$ correspond one-to-one to convexity/closedness properties of the acceptance set.
• VaR is not coherent. Because it is a quantile, VaR can fail subadditivity — the VaR of a merged portfolio can exceed the sum of the parts — so it discourages diversification and can be gamed by splitting positions. The SEC/NASD and SPAN margin rules are analyzed in the same framework (and shown to be essentially dual).
• Representation theorem. A measure is coherent iff it can be written as a worst case over a set $\mathcal P$ of probability measures ("generalized scenarios"), $\rho(X)=\sup_{P\in\mathcal P}\mathbb E_P[-X/r]$ — a consequence of the separation theorem for convex sets. Scenario-based methods are thus universal for constructing coherent measures.
• Tail conditional expectation. The paper singles out expected loss beyond the quantile — tail conditional expectation / expected shortfall — as a coherent, regulator-acceptable measure and the natural coherent "repair" of VaR.

Concepts Introduced or Extended

• Coherent Risk Measure
• Value at Risk

Entities Mentioned

• Philippe Artzner
• Freddy Delbaen
• Jean-Marc Eber
• David Heath

Quotes

"We present and justify a set of four desirable properties for measures of risk, and call the measures satisfying these properties 'coherent'."

"A merger does not create extra risk." (motivating subadditivity)

My Take

This is one of those rare papers that changed practice by changing definitions: by writing down what a risk measure should do and showing VaR fails the diversification axiom, it reframed the entire capital-requirements debate and launched expected shortfall toward its eventual adoption in Basel. The representation theorem is the mathematical core — coherence is exactly worst-case-over-scenarios — and it is what ties the abstract axioms to the concrete scenario methods practitioners already used. In the wiki's terms it is the theoretical counterweight to the GARCH-based conditional-quantile machinery on the VaR page: those methods estimate the quantile well, but Artzner et al. show the quantile itself is the wrong functional when portfolios are aggregated. The main caveat the authors stress themselves — the axioms pin down a class, not a unique measure, so the final choice still rests on economic judgment.