4.1.3 Coherent Risk Measures

Basic Prerequisites: 02 · VaR Definition & Flaws and Probability & Measure Theory.

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1. Intuition & Practical Objective

Before Artzner–Delbaen–Eber–Heath (1999), “risk measure” meant whatever a bank computed. Their contribution was to ask: what properties must any number claiming to be “risk” satisfy to be usable for regulating capital? Four axioms. A measure satisfying all four is coherent. VaR fails one of them; ES passes all.

This page states the axioms, connects them to a set of acceptable positions (the deeper primitive), gives the scenario representation (every coherent measure is a worst-case expectation over a set of probability scenarios), and shows - numerically - which axiom VaR breaks.

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2. Mathematical Ground Truth & Derivations

2.1 The four axioms (Artzner Def. 2.4)

Fix the future net worth $X$ and a reference return $r$ (typically $r=1$; think of $X$ as P&L). A measure $\rho :\mathcal{G}\to \mathbb{R}$ is coherent iff:

Axiom	Statement	Meaning
T Translation invariance	$\rho (X+\alpha r)=\rho (X)-\alpha$	adding sure cash $\alpha$ reduces risk by $\alpha$ (risk is a capital number)
S Subadditivity	$\rho (X_1+X_2)\le \rho (X_1)+\rho (X_2)$	”a merger does not create extra risk”
PH Positive homogeneity	$\rho (\lambda X)=\lambda \rho (X),\lambda \ge 0$	risk scales with size (no liquidity/concavity)
M Monotonicity	$X\le Y\Rightarrow \rho (Y)\le \rho (X)$	more net worth is never riskier

Consequences useful in practice: T gives $\rho (X+\rho (X)r)=0$ (the risk measure is the capital that makes the position acceptable); T + PH give $\rho (\alpha (-r))=\alpha$. Artzner notes M rules out mean–standard-deviation measures $\rho (X)=-{\mathbb{E}}_P[X]+\alpha {\sigma }_P(X)$, and S rules out semi-variance-type measures - i.e. the axioms are restrictive.

2.2 The deeper primitive: acceptance sets (Artzner §2.2–2.4)

Define the acceptance set $\mathcal{A}=\{X:\rho (X)\le 0\}$ - positions that need no extra capital. The measure is recovered as

$${\rho }_{\mathcal{A},r}(X)=\inf \{m:mr+X\in \mathcal{A}\}.$$

Axioms on $\mathcal{A}$ (contains the positive orthant $L^{+}$; avoids the strictly-negative orthant $L^{--}$; is convex; is a positively homogeneous cone) are equivalent to coherence of $\rho$ (Props. 2.1–2.2). This is why VaR’s failure bites: its acceptance set is not convex (02 · §2.2).

2.3 Representation: coherence ⇔ worst-case scenarios (Artzner §4.1)

Proposition 4.1 (Artzner et al.). $\rho$ is coherent iff there is a family $\mathcal{P}$ of probability measures on the states of the world such that

$$\boxed{\rho (X)=\sup \{{\mathbb{E}}_P[-X/r]\mid P\in \mathcal{P}\}}$$

i.e. every coherent risk measure is a supremum of expected losses over a set of “generalized scenarios.” Conversely (Prop. 3.1) any ${\rho }_{\mathcal{P}}(X)={\sup }_{P\in \mathcal{P}}{\mathbb{E}}_P[-X/r]$ is coherent (satisfying relevance iff ${\bigcup }_P\mathrm{supp}P=\Omega$). Adding more scenarios makes the measure more conservative. This is the theoretical roof over all coherent measures - including ES, which is the special case where $\mathcal{P}$ is the set of all measures agreeing with the base measure on the tail (equivalently the spectral representation in 06 · Advanced Extensions).

2.4 ES as coherence repair (Artzner §5.1)

Artzner’s concrete repair uses the tail conditional expectation (TailVaR):

$${\mathrm{TCE}}_{\alpha }(X)=-{\mathbb{E}}_P[X/r\mid X/r\le -{\mathrm{VaR}}_{\alpha }(X)],{\mathrm{WCE}}_{\alpha }(X)=-\inf \{{\mathbb{E}}_P[X/r\mid A]:\mathbb{P}(A)>\alpha \}.$$

He proves ${\mathrm{TCE}}_{\alpha }\le {\mathrm{WCE}}_{\alpha }$ (Prop. 5.1), with equality when $\mathbb{P}$ is uniform and the discounted outcomes are distinct (Prop. 5.3). And the sharpest statement of VaR’s status (Prop. 5.2):

$${\mathrm{VaR}}_{\alpha }(X)=\inf \{\rho (X):\rho \text{coherent},\rho \ge {\mathrm{VaR}}_{\alpha }\},$$

i.e. VaR is the least coherent measure that dominates it. Any coherent measure you pick (ES, WCE) must be at least as large as VaR - coherence costs conservatism, and this identity quantifies the gap.

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3. Computational Implementation - the axioms, checked numerically

We encode discrete loss distributions as pmfs and test all four axioms for VaR and ES on the defaultable-bond example. Stdlib only.

Both measures satisfy T, PH, M. VaR fails S; ES passes. That single False is the entire twentieth-century risk-management debate.

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4. Failure Modes & First-Principles Breakdowns

1. Scenario-set mis-specification. The representation $\rho (X)={\sup }_{P\in \mathcal{P}}{\mathbb{E}}_P[-X/r]$ is only as good as $\mathcal{P}$. Too small and the measure is optimistic (“model risk” - Artzner’s Remark: add other models’ distributions to $\mathcal{P}$); too large and every position looks maximally risky. ES corresponds to a specific $\mathcal{P}$; choosing “coherent” does not remove the modelling choice.
2. Homogeneity is an idealisation. Axiom PH ignores liquidity: if liquidating a position moves the price, then $\rho (\lambda X)<\lambda \rho (X)$, and the axiom over-states large-position risk (Artzner’s own caveat on Axiom PH). This is the formal reason liquidity-adjusted risk measures are needed (Liquidity Risk & Margin Spirals).
3. Coherence ≠ correctness of the loss distribution. The axioms constrain the functional, not the distribution. A coherent measure on a wrong $\mathbb{P}$ is still wrong. Coherence buys aggregation safety, not predictive accuracy (Estimation & Backtesting).
4. Dominance cost of repair (Prop. 5.2). Any coherent measure dominating VaR is at least VaR. Replacing VaR with ES raises the reported capital number - a governance decision as much as a mathematical one, which is exactly why the Basel switch needed a calibration ($97.5\%$ ES $\approx$ $99\%$ VaR; 06 · §Basel).

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5. References

• Artzner, Delbaen, Eber & Heath, Coherent Measures of Risk (1999)
• Föllmer & Schied, Stochastic Finance: An Introduction in Discrete Time
• McNeil, Frey & Embrechts, Quantitative Risk Management (2015)
• Acerbi & Tasche, On the Coherence of Expected Shortfall, J. Banking & Finance 26(7) (2002)

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6. Connected Graph Bridges

• Back: 02 · VaR Definition & Flaws · Index Hub
• Forward: 04 · Expected Shortfall · 06 · Spectral & Euler
• Base: Probability & Measure Theory