artzner1999coherent

Coherent Measures of Risk

Authors: Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, David Heath · Year: 1999 · Venue: Mathematical Finance 9(3):203-228, DOI 10.1111/1467-9965.00068 Raw: md (full text, OCR/font-encoding caveat noted in file) · pdf not archived locally, source: ETH Zurich, Delbaen

Summary

Defines and axiomatizes coherent measures of risk: a risk measure $\rho :\mathcal{G}\to \mathbb{R}$ on the set $\mathcal{G}$ of real-valued random variables (future net worths) over a finite state space $\Omega$. The paper states four axioms a risk measure should satisfy to be usable for capital-requirement / margin-requirement regulation, proves value-at-risk (VaR) violates one of them (subadditivity), and gives a representation theorem: every coherent risk measure is the supremum of expected losses over a family of “generalized scenarios” (a set of probability measures / linear functionals), i.e. $\rho (X)={\sup }_{Q\in \mathcal{P}}{\mathbb{E}}_Q[-X/r]$ for a set $\mathcal{P}$ of measures. This is the foundational paper that established “coherence” as the standard adequacy criterion for a risk measure in mathematical finance.

Key Claims

• Definition 2.4 (Coherence). A risk measure satisfying all four of Axiom T (translation invariance), Axiom S (subadditivity), Axiom PH (positive homogeneity), and Axiom M (monotonicity) is called coherent.
• VaR is not subadditive (Section 3.3, worked counterexamples). Two explicit constructions (digital-option portfolio; independent identically distributed loss pair) show ${\mathrm{VaR}}_{\alpha }(X_1+X_2)>{\mathrm{VaR}}_{\alpha }(X_1)+{\mathrm{VaR}}_{\alpha }(X_2)$ is possible, and that the VaR acceptance set can fail to be convex — a strictly worse failure than non-subadditivity alone. VaR does satisfy T, PH, and M.
• Representation theorem (Section 4.1, Proposition 4.1). Every coherent risk measure on a finite $\Omega$ can be written as $\rho (X)={\sup }_{Q\in \mathcal{P}}{\mathbb{E}}_Q[-X/r]$ for some set $\mathcal{P}$ of probability measures (generalized scenarios); conversely any such supremum is coherent. This is the paper’s central structural result, obtained via a separating-hyperplane argument for convex sets.
• Tail conditional expectation (Section 5.1) is proposed as a concrete coherent measure, and shown to dominate VaR at the same confidence level under stated conditions while remaining less conservative than the extreme worst-case measure.

Method

Finite state space $\Omega$; risk = random variable $X\in \mathcal{G}={\mathbb{R}}^n$ (final net worth). Two parallel axiomatic tracks: (i) axioms on acceptance sets $\mathcal{A}\subset \mathcal{G}$ (Axioms 2.1-2.4: contains $L_{+}$, disjoint from strictly-negative cone, convex, positively homogeneous), and (ii) axioms on the risk measure ${\rho }_{\mathcal{A},r}(X)=\inf \{m\mid m\cdot r+X\in \mathcal{A}\}$ induced by an acceptance set and a reference instrument with return $r$. Section 2.4 proves the two axiom systems correspond bijectively. Section 3 audits VaR, SPAN, and SEC/NASD margin rules against the four axioms; Section 4 gives the scenario representation; Section 5 constructs tail conditional expectation and a credit-risk example.

Regime note. Pure mathematical-finance paper - no mechanical or spacecraft system, so the free-flying vs free-floating distinction does not arise. Relevance is purely as the axiomatic source of the coherence axioms themselves, imported into risk-aware planning/control as the adequacy criterion for a risk functional acting on a cost or constraint-violation random variable.

Relevance to thesis

Primary source for the four coherence axioms (translation invariance, subadditivity, positive homogeneity, monotonicity) cited by the risk-phase proof obligations. In particular, feeds proof obligation A2 (showing VaR fails subadditivity, motivating the use of a coherent alternative such as CVaR/tail conditional expectation for the risk-aware planning layer of the free-flying space manipulator). This page is the citation anchor: any proof invoking “the coherence axioms” or “VaR is not coherent” should cite artzner1999coherent with the specific axiom/section referenced above, not a paraphrase.

Connections

Topics: Coherent Risk Measures · Coherent Risk Metrics - both topic pages cited “Artzner et al. 1999” by name before this source page existed; now linked to artzner1999coherent directly. Sources: rockafellar2000optimization (CVaR, the canonical coherent measure) - relation not yet characterized in either page’s body.

Key Equations / Quotes

From the abstract:

“We present and justify a set of four desirable properties for measures of risk, and call the measures satisfying these properties ‘coherent’… We demonstrate the universality of scenario-based methods for providing coherent measures.”

The four axioms (Section 2.3-2.4, hand-verified against the raw OCR text - see caveat in the raw md file for why these are retyped here rather than quoted verbatim):

• Axiom T (Translation invariance). For all $X\in \mathcal{G}$ and all real $\alpha$: $\rho (X+\alpha \cdot r)=\rho (X)-\alpha$.
• Axiom S (Subadditivity). For all $X_1,X_2\in \mathcal{G}$: $\rho (X_1+X_2)\le \rho (X_1)+\rho (X_2)$.
• Axiom PH (Positive homogeneity). For all $\lambda \ge 0$ and all $X\in \mathcal{G}$: $\rho (\lambda X)=\lambda \rho (X)$.
• Axiom M (Monotonicity). For all $X,Y\in \mathcal{G}$ with $X\le Y$: $\rho (Y)\le \rho (X)$.

Definition 2.4 (verbatim modulo the raw file’s ligature corruption, decoded):

“Coherence: a risk measure satisfying the four axioms of translation invariance, subadditivity, positive homogeneity, and monotonicity, is called coherent.”

Open Questions

• Full text status: retrieved in full (24 pages, ETH Zurich author-hosted PDF) but the source PDF’s font encoding garbles ligatures and Greek letters on extraction - flagged in the raw md frontmatter/comment. Math above was hand-decoded against the labeled axiom names; if a downstream proof needs a different equation from the body (e.g. the exact representation-theorem statement in Section 4.1), re-verify against the raw file rather than trusting extracted symbols at face value.
• The rockafellar2000optimization (CVaR) link above is a bare co-reference, not yet characterized - state the precise relation (CVaR is the specific coherent measure obtained by… ) once that source page is reviewed against this one.