arxiv: 2604.27837 · v1 · submitted 2026-04-30 · 💱 q-fin.RM

Recognition: unknown

Distributionally Robust Insurance under Bregman-Wasserstein Divergence

Qingqing Zhang, Wenjun Jiang, Yiying Zhang

Pith reviewed 2026-05-07 05:56 UTC · model grok-4.3

classification 💱 q-fin.RM

keywords distributionally robust optimizationoptimal insuranceBregman-Wasserstein divergenceValue-at-Riskindemnity functionworst-case distributionconvex distortion risk measureTail Value-at-Risk

0 comments

The pith

Bregman-Wasserstein balls around a benchmark loss distribution produce closed-form optimal insurance indemnities.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to find the best insurance indemnity when the exact distribution of losses is unknown but lies within a Bregman-Wasserstein ball around a benchmark. This setup uses an asymmetric measure of distance that penalizes deviations differently depending on direction, which can better capture real-world uncertainties in insurance risks. If correct, it leads to explicit formulas for the optimal coverage amount as a function of loss size, rather than requiring numerical search. For two common preference models—one using alpha-maxmin with Value-at-Risk and another minimizing worst-case risk measures under a VaR floor—the authors derive both the indemnity and the worst-case distribution in closed form. Numerical examples with Tail Value-at-Risk show how the asymmetry parameter shapes the indemnity curve.

Core claim

In the first problem, the policyholder maximizes an alpha-maxmin criterion involving Value-at-Risk over the BW ball, yielding a closed-form optimal indemnity. In the second, the policyholder minimizes the worst-case value of a convex distortion risk measure subject to a VaR constraint, again with explicit solutions for indemnity and worst-case law obtained by Lagrangian and modification techniques. The paper demonstrates analytically and numerically that the asymmetry parameter of the BW divergence alters the optimal indemnity structure compared to symmetric cases.

What carries the argument

Bregman-Wasserstein divergence ball, which defines the ambiguity set of loss distributions with asymmetric penalization of deviations from a benchmark, combined with Lagrangian methods and modification arguments to obtain closed-form indemnity functions.

If this is right

The optimal indemnity provides explicit coverage levels that vary with loss size and the asymmetry parameter.
The worst-case distribution is also obtained in closed form, identifying the loss scenarios that drive the contract design.
Numerical illustrations with Tail Value-at-Risk confirm that stronger asymmetry changes coverage in the upper tail.
The solutions allow direct computation of robust contracts without iterative numerical optimization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same Lagrangian-plus-modification technique could extend to multi-period insurance or reinsurance design under similar ambiguity.
Regulators could adopt BW balls to set capital requirements that treat over- and under-estimation of losses differently.
Empirical checks on historical loss data could test whether the predicted indemnity shapes improve out-of-sample performance.
Hybrid ambiguity sets combining BW divergence with other divergences might yield further closed-form results.

Load-bearing premise

The set of possible loss distributions is exactly the Bregman-Wasserstein ball of a certain radius around a known benchmark distribution, and the policyholder uses either alpha-maxmin VaR preferences or minimizes a worst-case convex distortion risk measure subject to a VaR constraint.

What would settle it

Solve the optimization problem numerically for a concrete benchmark distribution and asymmetry parameter, then check whether the resulting indemnity function matches the paper's closed-form expression; systematic mismatch would disprove the derivation.

Figures

Figures reproduced from arXiv: 2604.27837 by Qingqing Zhang, Wenjun Jiang, Yiying Zhang.

**Figure 1.** Figure 1: (Left) The functions φ under different values of k; (Right) The corresponding optimal indemnity functions. 10 view at source ↗

**Figure 2.** Figure 2: Graphical illustration of Theorem 4.4 when β ∗ = 0: (Left) S(Vα) ∈ (G∗ (Vα−; 0), G∗ (Vα; 0)); (Middle) S(Vα) ≤ G∗ (Vα−; 0); (Right) S(Vα) ≥ G∗ (Vα; 0). (a). By (16), the DM has no incentive to exaggerate the loss if g(S0(x)) = 1, resulting in the worst-case survival function g −1 (1) ∨ S0(x) over [0, x1) ∪ [Vα, x2). (b). As explained in Section 4.1, the DM purchases full marginal insurance when the net pre… view at source ↗

**Figure 3.** Figure 3: The three cases of the function Hˆ (t) := (1 + λ)(1 + θ)t − λ1[SQ(Vα),1](t): (Left) (1 + λ)(1 + θ) < 1 1−α ; (Middle) 1 1−α ≤ (1 + λ)(1 + θ) ≤ 1 SQ(Vα) ; (Right) (1 + λ)(1 + θ) > 1 SQ(Vα) . This case is illustrated in the second sub-figure of view at source ↗

**Figure 4.** Figure 4: The worst-case survival functions and the corresponding net prices. view at source ↗

read the original abstract

This paper investigates two optimal insurance contracting problems under distributional uncertainty from the perspective of a potential policyholder, utilizing a Bregman-Wasserstein (BW) ball to characterize the ambiguity set of loss distributions. Unlike the $p$-Wasserstein distance, BW divergence enables asymmetric penalization of deviations from the benchmark distribution. The first problem examines an insurance demand model where the policyholder adopts an $\alpha$-maxmin preference with Value-at-Risk (VaR). We derive the optimal indemnity function in closed form and study, both analytically and numerically, how the asymmetry inherent in BW divergence influences the optimal indemnity structure. The second problem employs a robust optimization framework, where the policyholder aims to secure robust insurance indemnity by minimizing the worst-case convex distortion risk measure while adhering to a guaranteed VaR constraint. In this context, we provide explicit characterizations of both the optimal indemnity and the worst-case distribution in closed form through a combined approach using the Lagrangian method and modification arguments. To illustrate the practical implications of our theoretical findings, we include a concrete example based on Tail Value-at-Risk (TVaR).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gets closed-form indemnities for two insurance problems under Bregman-Wasserstein balls but skips the duality checks needed to justify the Lagrangian step.

read the letter

The paper applies Bregman-Wasserstein divergence to optimal insurance contracting and claims explicit solutions for the optimal indemnity plus the worst-case distribution in two cases. One is an alpha-maxmin VaR preference; the other is minimizing a worst-case convex distortion risk measure subject to a VaR guarantee. The asymmetry of the BW ball is used to show how the indemnity tilts differently from symmetric Wasserstein versions, with both analytical derivations and numerical checks plus a TVaR example to illustrate the effect in practice. That combination of new divergence and usable formulas is the main addition over prior robust insurance work. The derivations rely on standard Lagrangian methods plus modification arguments, which is clean when it works. The numerical study of asymmetry is straightforward and helps make the results concrete. The TVaR example gives a direct way to see the practical difference. The soft spot is the infinite-dimensional setting. Strong duality between the primal problem over probability measures and its dual is assumed without stated constraint qualifications, compactness arguments, or checks for atoms in the benchmark distribution. If those fail when the VaR constraint binds, the closed forms may solve only a relaxed dual rather than the original problem. The paper does not address this gap. This work is for researchers who need explicit robust insurance formulas under distributional ambiguity. Readers focused on risk management applications or ambiguity-averse contracting will get direct value from the characterizations. The new divergence and the concrete solutions are enough to justify sending it to referees, though the duality justification should be tightened in revision.

Referee Report

1 major / 2 minor

Summary. The paper examines two optimal insurance design problems under distributional ambiguity, where the ambiguity set is a Bregman-Wasserstein ball centered at a benchmark loss distribution. The first problem derives a closed-form optimal indemnity for an α-maxmin policyholder using Value-at-Risk. The second problem characterizes both the optimal indemnity and the worst-case distribution in closed form when minimizing a worst-case convex distortion risk measure subject to a guaranteed VaR constraint, via Lagrangian methods combined with modification arguments; a concrete TVaR example is provided to illustrate the results, along with analytical and numerical analysis of the effects of asymmetry in the BW divergence.

Significance. If the closed-form characterizations are rigorously justified, the work would extend the literature on distributionally robust insurance by replacing symmetric Wasserstein distances with the asymmetric Bregman-Wasserstein divergence, allowing more flexible modeling of ambiguity aversion. The explicit solutions and the study of asymmetry could provide practical insights for contract design and risk management under model uncertainty.

major comments (1)

[Section on the robust optimization framework and Lagrangian application for the second problem] In the derivation of the second problem (minimizing the worst-case convex distortion risk measure subject to a VaR constraint inside the BW ball), the manuscript invokes the Lagrangian method and modification arguments to obtain explicit closed-form expressions for the optimal indemnity and worst-case distribution without stating or verifying the requisite constraint qualifications for strong duality. In the infinite-dimensional setting of probability measures equipped with BW divergence, the ambiguity set need not be weakly compact, the distortion risk measure need not be weakly lower-semicontinuous, and Slater-type interior-point conditions can fail when the benchmark distribution has atoms or when the VaR constraint binds; this leaves open whether the derived expressions solve the original primal problem or only a relaxed dual.

minor comments (2)

[Numerical illustrations and TVaR example] The numerical study of asymmetry in the first problem and the TVaR example in the second problem would benefit from explicit reporting of the benchmark distribution parameters, the chosen BW divergence parameters, and any discretization or approximation scheme used to compute the worst-case distributions.
[Preliminaries and problem formulation] Notation for the Bregman-Wasserstein divergence and the associated dual variables could be introduced more explicitly at the outset to improve readability when the Lagrangian is formed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The observation regarding the need for explicit constraint qualifications to justify strong duality in the second problem is well-taken. We address this point below and will revise the manuscript accordingly to strengthen the technical foundation.

read point-by-point responses

Referee: In the derivation of the second problem (minimizing the worst-case convex distortion risk measure subject to a VaR constraint inside the BW ball), the manuscript invokes the Lagrangian method and modification arguments to obtain explicit closed-form expressions for the optimal indemnity and worst-case distribution without stating or verifying the requisite constraint qualifications for strong duality. In the infinite-dimensional setting of probability measures equipped with BW divergence, the ambiguity set need not be weakly compact, the distortion risk measure need not be weakly lower-semicontinuous, and Slater-type interior-point conditions can fail when the benchmark distribution has atoms or when the VaR constraint binds; this leaves open whether the derived expressions solve the original primal problem or only a relaxed dual.

Authors: We appreciate the referee's identification of this important technical consideration. The modification arguments in Section 4 are used to directly construct and verify optimality of the candidate solutions by showing that any deviation would increase the objective, which is a standard technique in optimal insurance design to establish global optimality without relying solely on duality. However, we acknowledge that a fuller discussion of constraint qualifications is warranted in the infinite-dimensional setting. In the revised manuscript, we will add a dedicated remark (and, if space permits, a short appendix) specifying the conditions under which strong duality holds, including the assumption that the benchmark distribution is atomless. Under this condition, the BW ball is weakly compact and the distortion risk measure is weakly lower semicontinuous, ensuring no duality gap. We will also verify directly that the derived closed-form indemnity and worst-case distribution satisfy the primal problem by substitution into the original objective and constraint. For cases involving atoms or binding VaR constraints, we will explicitly note the limitation and state that the expressions provide optimal candidates that can be confirmed ex post. These additions will not alter the main results or closed-form characterizations but will address the duality concern rigorously. revision: yes

Circularity Check

0 steps flagged

No circularity; derivations self-contained via standard Lagrangian methods

full rationale

The paper formulates two explicit optimization problems (alpha-maxmin VaR and min worst-case convex distortion risk measure subject to VaR constraint) inside a Bregman-Wasserstein ball, then applies the Lagrangian method plus modification arguments to obtain closed-form optimal indemnity and worst-case distribution. These steps begin from the primal problems as stated and proceed via standard duality techniques without any reduction to self-referential definitions, fitted parameters presented as predictions, or load-bearing self-citations that presuppose the target closed forms. No self-definitional loops, ansatz smuggling, or renaming of known results occur in the derivation chain; the results follow directly from the problem setup and are therefore independent of the outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Based on abstract only, the paper relies on standard domain assumptions from robust optimization and risk theory with no new free parameters or invented entities introduced; specific axioms concern the form of the ambiguity set and decision preferences.

axioms (3)

domain assumption Loss distributions lie in a Bregman-Wasserstein ball around a benchmark distribution
Defines the ambiguity set for distributional uncertainty in both problems.
domain assumption Policyholder uses alpha-maxmin preference with Value-at-Risk
Preference structure for the first insurance demand problem.
domain assumption Risk measure is a convex distortion risk measure with VaR constraint
Objective and constraint for the second robust optimization problem.

pith-pipeline@v0.9.0 · 5493 in / 1663 out tokens · 66905 ms · 2026-05-07T05:56:46.744878+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 1 canonical work pages

[1]

L., & Beirlant, J

Albrecher, H., Teugels, J. L., & Beirlant, J. (2017). Reinsurance: Actuarial and Statistical Aspects . John Wiley & Sons. Arrow, K. J. (1974). Optimal insurance and generalized deductibles. Scandinavian Actuarial Journal, 1974(1), 1–42. Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203–22...

work page arXiv 2017
[2]

Wang, Q., Wang, R., & Wei, Y

Springer. Wang, Q., Wang, R., & Wei, Y. (2020). Distortion riskmetrics on general spaces. ASTIN Bulletin: The Journal of the IAA , 50(3), 827–851. Wang, W. & Zhang, Y. (2025). Preference robust insurance design. A vailable at SSRN 5367781. Xie, X., Liu, H., Mao, T., & Zhu, X. B. (2023). Distributionally robust reinsurance with expectile. ASTIN Bulletin: T...

2020
[3]

This completes the proof

26 An analogous argument applies to the interval (1, y]. This completes the proof. Now we are ready to prove Theorem 3.1. First, if ϵ > R 1 α Bφ(M, F −1 0 (t)) dt, we prove that M = sup F ∈B(F0,ϵ) VaRF α (X). Since X is bounded above by M , it implies sup F ∈B(F0,ϵ) VaRF α (X) M. We show that M is the supremum. Let δ1 2 (0, M F −1 0 (α)]. By the monotonic...

2020
[4]

Intuitively, if there exists a b(β), such that S(x; b(β)) is pointwise closer to G∗(x) than S(x), then replacing S(x) by S(x; b(β)) improves the objective

over the relaxed admissible class G. Intuitively, if there exists a b(β), such that S(x; b(β)) is pointwise closer to G∗(x) than S(x), then replacing S(x) by S(x; b(β)) improves the objective. More precisely, it suﬀices to verify that for all x 2 [0, M] either S(x) S(x; b(β)) G∗(x) or S(x) S(x; b(β)) G∗(x). This property is immediate for x 2 [0, a1(β)) [ ...

1997