Comonotonic Risk Sharing

Updated 1 December 2025

Comonotonic risk sharing is a method that allocates aggregate risk by aligning agents’ exposures through common shocks and quantile representations, ensuring Pareto optimality.
It employs distortion risk measures—such as Value-at-Risk and Expected Shortfall—to facilitate additivity and layer-wise risk allocation in risk management.
Extensions like partial and weak comonotonicity address practical pooling limitations and model ambiguity in nonconvex and heterogeneous preference settings.

Comonotonic risk sharing is a foundational concept in quantitative risk management, insurance mathematics, and mathematical economics. It formalizes the optimal allocation of aggregate risk among multiple agents under conditions of law-invariance, monotonicity, and—in the classical setting—concavity or subadditivity of risk preferences. The structure and uniqueness of comonotonic allocations have far-reaching implications, as do generalizations encompassing nonconvex measures, model ambiguity, “partial” or “weak” comonotonicity, and heterogeneous beliefs.

1. Definition and Foundational Principles

A collection of random variables $(X_1, \dots, X_n)$ on a probability space is comonotonic if there exists a random variable $Z$ and nondecreasing functions $f_i$ such that $X_i = f_i(Z)$ for each agent $i$ (Xu, 2013, Lauzier et al., 2023, Ghossoub et al., 4 Jun 2024). Equivalently,

$(X_i(\omega) - X_i(\omega'))(X_j(\omega) - X_j(\omega')) \ge 0 \;\; \forall\,\omega,\omega',\;\forall\,i,j.$

This joint monotonicity ensures that the components cannot decrease or increase independently—they are inextricably ordered by common shocks.

A key result is the quantile (copula) representation: for marginal quantile functions $Q_i$ and a Uniform(0,1) random variable $U$ , $(Q_1(U), ..., Q_n(U))$ is comonotonic, and every comonotonic vector can be represented in this way (Xu, 2013). Comonotonicity corresponds to the Fréchet–Hoeffding upper bound copula.

Agents equipped with law-invariant, concave, and translation-invariant preferences always admit Pareto-optimal risk allocations that are comonotonic. This is formalized through “comonotone improvement” theorems, which state any non-comonotonic allocation can be strictly improved via Schur-concave preferences (Ghossoub et al., 4 Jun 2024, Ghossoub et al., 21 Oct 2025).

Distortion risk measures, $p_h(X) = \int_{-\infty}^{0}\bigl(h(P(X>x))-h(1)\bigr)dx + \int_{0}^{\infty} h(P(X>x))dx$ , encompass prominent capital requirement functionals, including Value-at-Risk, Expected Shortfall, and various deviation-type penalties (Lauzier et al., 2023, Assa, 2015).

For agents $i=1,\dots,n$ with distortion functions $h_i$ , comonotonicity induces a powerful additivity: for comonotonic $(X_1, ..., X_n)$ , $p_h(X_1+\cdots+X_n) = p_h(X_1) + ... + p_h(X_n)$ for all distortion riskmetrics $p_h$ (Huang, 9 Jun 2025). When all agents use concave $h_i$ , Pareto-optimal allocations are comonotonic and may be constructed via infimal convolution and Lagrange multiplier strategies (Lauzier et al., 2023, Ghossoub et al., 21 Oct 2025, Ghossoub et al., 4 Jun 2024).

Layer-wise Allocation Rule

In the "Yaari-Dual" case (comonotone-additive concave functionals), the optimal rule is “tranche-by-tranche” (Ghossoub et al., 4 Jun 2024, Assa, 2015):

At each quantile (layer) $t$ , assign the loss to the agent $i$ minimizing $h_i(P(X>x))$ or, in the spectral representation, the "most optimistic" convex distortion $T_i$ .
Integrate the corresponding indicator to obtain the retention function.

This construction produces the unique (up to sets of measure zero or constant shifts) comonotone Pareto-optimal risk allocation.

3. Comonotonicity, Additivity, and Extensions

Partial and Weak Comonotonicity

Classical comonotonicity is too restrictive for many practical pooling problems. Several generalizations have been developed:

Partial comonotonicity: Defined via $K$ -concentration, where comonotonic order is imposed only on events at probability levels $K\subset [0,1]$ ; it nests both strong comonotonicity ( $K=[0,1]$ ) and $p$ -concentration ( $K=\{p\}$ , which characterizes Expected Shortfall) (Huang, 9 Jun 2025).
Weak comonotonicity: Relaxation based on averaging comonotonicity over product measures or restricting to tail events, interpolating between independence and strong comonotonicity. The optimal allocation is calibrated by the level of tail alignment—parameterized by $\beta$ —with resulting closed-form solutions for quantile-based risk sharing that interpolate between pooled and fully comonotonic extremes (Wang et al., 2018).

Additivity and Characterization of Riskmetrics

For a riskmetric $\rho_h$ , additivity on partial comonotonic dependence structures is equivalent to piecewise linearity of $h$ on specific domains (determined by the structure), tying dependence directly to the “shape” of the riskmetric (Huang, 9 Jun 2025). In the case of full comonotonicity, any Choquet integral-based riskmetric is additive; for $p$ -concentration, only those $h$ corresponding to Expected Shortfall at level $p$ are (Huang, 9 Jun 2025, Lauzier et al., 2023).

Lambda Value-at-Risk under Ambiguity

When ambiguity is modeled by sets of probability measures (e.g., $\phi$ -divergence or likelihood-ratio constraints), one replaces probability measures with capacities, leading to robustified risk measures (Liu et al., 1 Nov 2025). For agents with increasing Lambda ( $\Lambda$ ) functions, the comonotonic infimal convolution problem admits a trivial solution: optimality is achieved by concentrating the entire loss on the single agent with the lowest robust $\Lambda$ VaR, with all others getting zero (Liu et al., 1 Nov 2025). This “winner-take-all” allocation collapses the block downset to a singleton, in contrast to the layer-wise tranching of convex risk-sharing.

Nonconvex and Heterogeneous Preference Models

In nonconvex or even star-shaped law-invariant settings, existence and structure of Pareto-optimal allocations persist. Locally comonotone improvement, under scenario partitions and minimal admissibility or consistency assumptions, produces compact solution sets even when full convexity is absent (Liebrich, 2021). Optimal allocations exhibit blockwise (scenario-by-scenario) comonotonicity, possibly mixed with deterministic cash transfers for further fairness selection.

5. Mixed and Counter-monotonic Structures

Optimal sharing sometimes deviates from pure comonotonicity:

Counter-monotonicity: For convex (risk-seeking) distortion functions, the optimum may be counter-monotonic—agents split risk in perfectly anti-aligned layers (Ghossoub et al., 21 Oct 2025, Lauzier et al., 2023).
Mixture and Non-convex Measures: For certain variability measures (inter-quantile difference, Range-Value-at-Risk), the optimum is a mixture: comonotonic sharing in one regime (e.g., upper tail) and counter-monotonic in another (e.g., lower tail or gains). Explicit allocations distribute increments among agents using event partitions, yielding sharp bounds tied to dependence uncertainty (Liu et al., 26 Nov 2025, Lauzier et al., 2023).

Examples from Extended Convolution Bounds

In the case of averaged-quantile risk metrics, the region $\{U_X>\beta\}$ (upper tail) features comonotonic allocations (all agents bear equal fractional loss), whereas in $\{U_X\le\beta\}$ (lower tail), counter-monotonic permutations ensure only one agent incurs the maximal residual at a time (Liu et al., 26 Nov 2025).

Setting	Distortion Shape	Structure of Optimum Allocation
Concave distortion (risk-averse)	Concave	Comonotonic, layer/tranche allocation
Convex distortion (risk-seeking)	Convex	Counter-monotonic, anti-aligned shares
Non-convex/inter-quantile	Non-convex, jump	Mixed: comonotonic in core, counter-monotonic in tails
Lambda-VaR under ambiguity	Increasing $\Lambda$	Fully concentrated (single-agent)
Heterogeneous beliefs	Any	Locally comonotonic (scenario-based)

The theoretical underpinnings across these cases rely on infimal convolution, Schur-concavity, Choquet integrals, scenario/local improvements, and copula/quantile representations, as formally demonstrated in a series of recent arXiv works (Xu, 2013, Lauzier et al., 2023, Liebrich, 2021, Huang, 9 Jun 2025, Ghossoub et al., 4 Jun 2024, Ghossoub et al., 21 Oct 2025, Liu et al., 1 Nov 2025, Liu et al., 26 Nov 2025, Assa, 2015, Wang et al., 2018).

7. Economic and Regulatory Implications

Comonotonic risk sharing reveals regimes in which all optimal allocations are fully risk-aligned, eliminating cross-state hedging and layering risk according to agents’ marginal costs or risk distortions. These solutions underpin the logic of layered (tranche-based) reinsurance, capital allocation, and margin design in financial institutions (Lauzier et al., 2023, Ghossoub et al., 4 Jun 2024). The shape of the distortion function precisely determines whether comonotonic, counter-monotonic, or mixed structures are optimal—including when risk preferences are non-convex or model beliefs are heterogeneous.

Under ambiguity, the optimal structure may collapse to concentrated allocations, emphasizing the qualitative sensitivity of collective risk management to agents’ specification of uncertainty. Partial and weak comonotonicity frameworks provide calibrated, intermediate dependence structures accommodating regulatory objectives and real-world pooling limitations (Huang, 9 Jun 2025, Wang et al., 2018).

Comonotonic risk sharing remains central to the structure, existence, and uniqueness of Pareto-optimal allocations in risk pooling, with explicit construction principles now elucidated for a broad scope of risk measures and ambiguity models.