Coherent Risk Measures (一致风险测度)

Definition. A coherent risk measure (Artzner et al. 1999) is a functional \rho: \mathcal Z\to\mathbb R on a space of real-valued random payoffs / losses that satisfies four axioms: monotonicity, translation invariance, sub-additivity, and positive homogeneity. It quantifies the capital required to make a position acceptable to a supervisory body with a particular risk attitude.

Axioms (Artzner–Eber–Heath 1999)

Let \mathcal Z be a linear space of measurable functions on (\Omega,\mathcal F), with Z\in\mathcal Z interpreted as the loss of a position (positive = loss, negative = gain). \rho:\mathcal Z\to\mathbb R is coherent if for all Z,W\in\mathcal Z:

Monotonicity. Z\le W \Rightarrow \rho(Z)\le\rho(W). If a position is pointwise worse, the risk is at least as large.
Translation invariance. \rho(Z+c) = \rho(Z) + c for c\in\mathbb R. Adding a sure cash inflow c reduces the capital requirement by c.
Sub-additivity. \rho(Z+W)\le\rho(Z)+\rho(W). The risk of a merged book is at most the sum of the individual risks — diversification never hurts.
Positive homogeneity. \rho(\lambda Z) = \lambda\,\rho(Z) for \lambda\ge 0. Scaling exposure scales the risk linearly.

A risk measure satisfying all four is coherent; one satisfying all but positive homogeneity is convex; one satisfying only 1 and 3 is a monetary risk measure (Föllmer & Schied 2002).

Why coherent, why now

The axioms pin down which functionals are acceptable as a risk measure. The breakthrough of Artzner et al. was showing that Value-at-Risk (VaR) fails sub-additivity — two separately VaR-bounded positions can produce a merged position that violates the VaR limit — and proposing a replacement theory in which Expected Shortfall / CVaR is the minimal coherent upper bound on VaR. This is now the foundation of Basel III/IV market-risk capital rules.

The DRO = coherent risk theorem

The bridge that makes DRO an axiomatic risk theory (Rahimian & Mehrotra §3.3.1):

Theorem (Hoffman 1979 / Shapiro 2017). For a convex and compact ambiguity set \mathcal P\subseteq\mathcal M(\Omega,\mathcal F) and R_P[Z]=\mathbb E_P[Z], the worst-case expectation operator \rho_{\mathcal P}(Z) := \sup_{P\in\mathcal P}\ \mathbb E_P[Z] is a coherent risk measure. Conversely, every law-invariant coherent risk measure on a suitable space of bounded measurable losses admits a representation of this form.

Equivalently: DRO over a convex compact ambiguity set IS a coherent risk measure, and the worst-case distribution P^\ast is the "adversarial" scenario that attains the supremum. DRO with \mathcal P = \{P : \mathbb E_P[\tilde\xi]=\mu_0,\ \mathrm{Cov}_P(\tilde\xi)=\Sigma_0\} recovers the worst-case expected loss over all distributions with given first two moments — the classical portfolio DRO of Scarf, El Ghaoui et al., Delage & Ye.

Representations

Optimised certainty equivalent (Ben-Tal & Teboulle 2007)

For a convex, lower-semicontinuous utility u and a strictly convex dual function,

\rho_u(Z) = \inf_{t\in\mathbb R}\ \{t + \mathbb E[u^{-1}(Z-t)]\}. This is the law-invariant coherent risk measure representation. For u(t) = -e^{-\alpha t} we recover entropic risk; for u(t)=\min(t,0) the infimum is -\infty, but the related CVaR is a coherent special case. ### Convex dual For a coherent \rho on L^\infty, \rho(Z) = \sup_{Q\in\mathcal Q}\ \mathbb E_Q[Z]

for some convex set \mathcal Q of "risk-adjusted probabilities" (the risk envelope). The set \mathcal Q is exactly the ambiguity set \mathcal P in the DRO representation. The duality between \mathcal Q and the choice of \rho is the same duality as between \mathcal P and the resulting DRO objective.

Common coherent risk measures

Risk measure	\rho(Z)	Coherent?	Notes
Expected loss	\mathbb E[Z]	yes (degenerate)	Risk-neutral; no tail sensitivity
CVaR_\alpha (Expected Shortfall)	\mathbb E[Z\mid Z\ge \mathrm{VaR}_\alpha(Z)]	yes	Basel III/IV standard; sub-additive; smallest coherent upper bound on VaR
Entropic risk	\frac{1}{\alpha}\log\mathbb E[e^{\alpha Z}]	yes	Smooth; dual to exponential tilting
Worst-case	\mathrm{ess\,sup}(Z)	yes	Trivially coherent; DRO with \mathcal P = all distributions on the support
VaR_\alpha	\inf\{z: P(Z\le z)\ge\alpha\}	no (fails sub-additivity)	The motivating example of why coherence matters
Mean-variance	\mathbb E[Z]+\lambda\,\sqrt{\mathrm{Var}(Z)}	yes (for \lambda\ge 0)	Markowitz-style risk adjustment

DRO recipes by risk measure

For each coherent risk measure, the ambiguity set is the risk envelope — but in DRO the inverse construction is more useful: pick a tractable \mathcal P and get a coherent \rho_\mathcal P for free. Standard recipes (Rahimian & Mehrotra Table 1):

\mathcal P choice	Resulting worst-case expectation	Coherent?
Wasserstein ball (§5.1.1)	\sup_{P: W_p(P,\hat P_N)\le\epsilon}\mathbb E_P[Z]	yes for convex compact Wasserstein ball
\phi-divergence ball	\sup_{P: d_\phi(P,P_0)\le\epsilon}\mathbb E_P[Z]	yes (Ben-Tal et al. 2013 — soft robust)
Mean-covariance (Delage-Ye)	\sup_{P: (\mu,\Sigma)\in\mathcal U}\mathbb E_P[Z] for an ellipsoidal \mathcal U	yes
Chebyshev (mean + variance)	\sup_{P: \mathbb E_P[\tilde\xi]=\mu_0,\ \mathrm{Var}_P(\tilde\xi)\le\sigma_0^2}\mathbb E_P[Z]	yes (Scarf 1958)
Full-support (no constraints)	\mathrm{ess\,sup}(Z)	yes (trivially — this is RO)

Quant-finance implications

Risk capital can be read off as \rho(Z) for a portfolio loss Z.
Portfolio constraints can be formulated as \rho(\text{return})\ge \text{floor} — this is the "DRO mean-CVaR" portfolio of Iyengar & Ma, Krokhmal, Palmquist, Uryasev.
Model risk is naturally expressed: if we are uncertain about the distribution, the worst-case expectation is the risk capital; the choice of \mathcal P is the prior on the prior.

与 CVaR 优化的关系

一致风险测度是一套公理，CVaR 优化是把其中一个具体测度装进求解器。Artzner 等 1999 给出了公理与表示定理（情景法：\rho 相干 \iff 存在一族概率测度 P 使 \rho(X)=\sup_{P}\mathbb E_P[-X/r]）；Rockafellar 与 Uryasev 则回答了下一个问题：给定一个相干测度，怎么把它写成能算的形式。

对 CVaR 而言答案是分段线性化：引入分位点 \zeta 与辅助变量 z_j \ge f(x,y^{(j)})-\zeta，把 \mathrm{CVaR}_\alpha(x)=\min_\zeta F(x,\zeta) 化成线性规划，于是"最小化 CVaR"可以直接送进 LP 求解器——讲义称可用于超过 1,000,000 个工具与情景的问题。详见CVaR 优化。

两者的分工可以这样记：公理回答"这样的风险度量可不可接受"，LP 刻画回答"这个可接受的度量怎么算得动"。Artzner 等的论文还给出了尾部条件期望 TCE 与最坏条件期望 WCE 这对构造，说明"相干"这一条件本身留下了很大的自由度——最终选哪个测度仍要靠额外经济考虑。

Cross-references

concepts/Distributionally Robust Optimization — DRO = coherent risk (for convex compact \mathcal P)
concepts/Wasserstein Ambiguity Set — most-used data-driven \mathcal P giving a coherent \rho_\mathcal P
concepts/Robust Optimization — degenerate case \mathcal P = all on support → \rho = \mathrm{ess\,sup}
concepts/Volatility Targeting — a practical risk-budget layer; DRO formalises what it really means to "target vol"
concepts/Risk Parity — ERC uses CVaR as the risk measure in its risk-budgeting formulation