Meyer Risk Measures in Stochastic Orders

• Meyer risk measures are monetary risk measures defined via threshold utility functions that generalize second-order stochastic dominance (SSD).
• They are represented as lower envelopes of base risk measures, illustrating trade-offs between cash-additivity, convexity, and positive homogeneity.
• Their applications span portfolio optimization and empirical risk assessment, with nontrivial cases emerging mainly within the exponential (CARA) utility framework.

Meyer risk measures are monetary risk measures consistent with stochastic orders defined via threshold utility functions, specifically those generalizing second-order stochastic dominance (SSD) through what are termed $v$-SD orders. These represent a family of risk measures that interpolate between classical SSD and more restrictive or refined utility-based orders. The $v$-SD order employs utility functions at least as risk averse as a selected benchmark $v$, leading to risk measures that respect the resulting stochastic order. Meyer risk measures exhibit intricate links to convexity, positive homogeneity, and duality representations, and their structural restrictions depend crucially on the choice of the threshold utility. The framework not only extends SSD-consistent risk measures but also provides impossibility theorems and applications in portfolio optimization and empirical risk comparisons of financial data (Laudagé et al., 29 Sep 2025).

1. $v$-SD Orders and Definition of Meyer Risk Measures

A core concept underpinning Meyer risk measures is the $v$-SD order, constructed via a threshold utility function $v:\mathbb{R} \to \mathbb{R}$ that is twice differentiable and strictly increasing. The set of admissible test utilities is

$$\mathcal{U}_v(\mathbb{R}) = \left\{ u \in \mathcal{U}(\mathbb{R}) : -\frac{u''(x)}{u'(x)} \geq -\frac{v''(x)}{v'(x)} \;\forall x \right\},$$

where the Arrow–Pratt measure $-u''(x)/u'(x)$ quantifies absolute risk aversion. For two random variables $X, Y$, the $v$-SD order is defined by: $$X \leq_v Y \quad \text{iff} \quad \mathbb{E}[u(X)] \leq \mathbb{E}[u(Y)] \quad \forall u \in \mathcal{U}_v(\mathbb{R}).$$ A monetary risk measure $\rho$ is $v$-SD-consistent ("v-Meyer") if

$X \leq_v Y \implies \rho(X) \geq \rho(Y).$

This definition generalizes SSD ($v(x) = x$), allowing the representation of fractional stochastic dominance and interpolations excluding low-risk-aversion utility functions.

2. Structural Representations and Restrictions

Any $v$-Meyer risk measure $\rho$ can be represented as the lower envelope of a family of base risk measures. For a benchmark random variable $Z$, the base measure is

$$\rho_{Z,v}(X) = \inf \{ m \in \mathbb{R} : Z \;v\; X + m \},$$

where "$Z \;v\; X + m$" signifies acceptability in the $v$-SD sense after a capital injection $m$. This construction adapts classical monetary risk measure representations using acceptance sets and capital requirements.

Key impossibility theorems, such as those restricting nontrivial positively homogeneous or convex $v$-Meyer risk measures, arise from the structure of $v$. If $v$ is chosen outside the exponential (CARA) class (i.e., non-constant absolute risk aversion), admissible measures either collapse to the worst-case risk measure $M(X) = \inf\{x : X \leq x\}$ or do not exist. The exponential utility subclass (CARA) allows for tractable representations and nontrivial $v$-Meyer risk measures. These results illuminate the deep link between monetary axioms, SSD, and stronger stochastic orders.

3. Relationship to Second-Order Stochastic Dominance (SSD)

Meyer risk measures generalize SSD by permitting fractional stochastic dominance. When $v(x) = x$, $v$-SD coincides with SSD, which compares prospects via all increasing concave utility functions. For nonconstant $v$ (e.g., logistic or SAHARA utilities), the $v$-SD order excludes test utilities with insufficient risk aversion and yields a continuum of dominance concepts between SSD and more restrictive orders.

The structural consequences of $v$ are:

• If the Arrow–Pratt risk aversion measure of $v$ diverges (e.g., $\liminf_{x\to\infty} R_v^A(x) = \infty$), only the worst-case risk measure is cash-additive and $v$-SD-consistent.
• Positive homogeneity or convexity combined with $v$-SD consistency typically restricts admissible risk measures to trivial cases, especially outside the CARA utility class.

4. Applications: Portfolio Optimization and Empirical Risk Assessment

Two applications are developed:

(a) Portfolio Optimization

A risk minimization problem is formulated using loss-based return risk measures (RRMs) respecting SSD or $v$-SD consistency. For losses $L$ and a set of benchmark profiles $g$,

$$\kappa(L) = \inf_{g \in \mathcal{G}} \sup_{p \in [0,1]} \left( \text{ES}_p(-L) / g(p) \right),$$

where ES$_p(-L)$ denotes Expected Shortfall. The optimization

$$\min \kappa(L) \quad \text{subject to} \quad \mathbb{E}_\mathcal{Q}[L] \geq x_0$$

reveals solutions with multiplicative structures (contrasting with additive decompositions in classic SSD).

(b) Empirical Comparison

Risk measures derived from Meyer risk measures, especially those with exponential utilities $v_c(x) = -e^{c x}$, are applied to real log-return data (S&P 500, FTSE, DAX, etc.). By plotting

$$\rho_c(X) = \begin{cases} \frac{1}{|c|} \log \left( \frac{\text{ES}_p(-e^{c X})}{\text{ES}_p(-e^{c Y})} \right), & c < 0 \ \text{ES}_p(X) - \text{ES}_p(Y), & c = 0 \ \frac{1}{c} \log \left( \frac{\text{ES}_p(e^{c Y})}{\text{ES}_p(e^{c X})} \right), & c > 0 \end{cases}$$

benchmarking risk aversion, one can compare risk across indices and regimes, illustrating the sensitivity of Meyer risk measures to the chosen utility parameter.

5. Implications, Limitations, and Further Research

The investigation reveals strong constraints imposed by cash-additivity and $v$-SD consistency on the admissible risk measures. Additional axiomatic requirements—convexity, positive homogeneity, star-shapedness—tighten these restrictions further, often precluding nontrivial examples outside the exponential utility family.

The representation of Meyer risk measures as lower envelopes of base risk measures provides a unifying perspective. However, outside the constant risk aversion case, these base measures may fail to be $v$-SD-consistent themselves, indicating nontrivial interplay between acceptance set constructions and stochastic orders.

Key areas for further investigation include:

• Classification of all $v$ for which a given risk measure $\rho$ is $v$-SD-consistent.
• Extensions to risk minimization and risk sharing beyond SSD to general $v$-SD consistent measures.
• Analysis of stochastic orders that exclude agents with excessive risk aversion, further refining the consensus on acceptability.
• Systematic paper of return risk sharing problems within these frameworks for applications in insurance and portfolio management.

6. Comparative Table: Meyer Risk Measures vs. Classical SSD Risk Measures

Property	Classical SSD Risk Measure	Meyer Risk Measure (General $v$)	Meyer Risk Measure (Exponential $v$)
Underlying order	All concave utility	Utilities $\geq$ risk aversion of $v$	CARA utilities only
Nontrivial examples	Yes	Rare/mostly worst-case	Yes
Positive homogeneity	Often possible	Rare, only for CARA $v$	Possible
Convexity	Achievable	Heavily restricted	Admissible
Envelope representation	Yes	Yes	Yes
Sensitivity to $v$	N/A	High	Parameter $c$ tracks risk aversion

7. Summary

Meyer risk measures formalize the idea of monetary risk measures that are compatible with fractional stochastic orders determined by a threshold utility. This framework generalizes SSD and enables nuanced risk quantification sensitive to investor risk aversion but introduces severe mathematical restrictions. Nontrivial Meyer risk measures largely exist only within the exponential utility family. Applications include portfolio optimization and empirical risk rankings sensitive to the risk aversion profile. Outstanding questions remain in the axiomatic classification and practical deployment of these measures in modern financial contexts (Laudagé et al., 29 Sep 2025).