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arxiv: 2606.11855 · v1 · pith:GJFO7SUXnew · submitted 2026-06-10 · 🧮 math.OC

Distributionally Robust Reinsurance under Robust Optimized Certainty Equivalent Risk Measure

Xinqiao Xie , Taizhong Hu , Tiantian Mao This is my paper

Pith reviewed 2026-06-27 08:52 UTC · model grok-4.3

classification 🧮 math.OC
keywords distributionally robust optimizationoptimal reinsurancerobust risk measuresoptimized certainty equivalentmean-variance uncertainty setWasserstein uncertainty setfinite-dimensional reformulation
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The pith

DROR problems under ROCE risk measures reduce to finite-dimensional optimizations via three-point distributions for mean-variance uncertainty sets.

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

The paper defines a class of robust optimized certainty equivalents that generalize measures such as conditional value-at-risk and expectiles. It examines distributionally robust optimal reinsurance problems under these measures with two uncertainty sets on the loss distribution. For the mean-variance uncertainty set the optimization reduces to consideration of three-point distributions, which produces a single explicit formula covering the broad class. The same approach yields solvable finite-dimensional models when uncertainty is measured by Wasserstein distance, supporting direct numerical comparison of the two uncertainty types when choosing deductibles.

Core claim

We introduce robust optimized certainty equivalents (ROCE) as preference-robust risk measures and show that distributionally robust optimal reinsurance problems under ROCE with mean-variance uncertainty sets admit finite-dimensional reformulations because three-point distributions suffice; this produces a unified explicit formulation for a broad class of ROCE measures that recovers earlier results for conditional value-at-risk and expectiles, while the Wasserstein uncertainty set likewise yields a tractable finite-dimensional formulation.

What carries the argument

The restriction to three-point distributions inside the mean-variance uncertainty set, which converts the infinite-dimensional DROR problem under ROCE into an explicit finite-dimensional program.

If this is right

  • A unified explicit formulation applies across the broad class of ROCE risk measures in the reinsurance setting.
  • Earlier explicit results for conditional value-at-risk and expectiles are recovered as special cases.
  • Data-driven models become computationally tractable for both moment-based and Wasserstein uncertainty sets.
  • Systematic numerical comparison of the two uncertainty sets becomes feasible for optimal deductible design.

Where Pith is reading between the lines

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

  • The three-point reduction may extend to other convex risk measures whose worst-case expectations interact similarly with moment constraints.
  • The finite-dimensional structure could support extensions to multi-period or multi-line reinsurance contracts.
  • Testing the reformulated models on historical insurance portfolios would check whether they improve out-of-sample performance relative to non-robust alternatives.

Load-bearing premise

The convexity, monotonicity, and optimized-certainty-equivalent structure of ROCE are compatible with mean-variance and Wasserstein constraints in a way that permits the three-point and finite-dimensional reductions.

What would settle it

A concrete ROCE functional, mean-variance bounds, and reinsurance contract for which the optimal loss distribution under the infinite-dimensional problem has four or more support points would show the three-point restriction is insufficient.

Figures

Figures reproduced from arXiv: 2606.11855 by Taizhong Hu, Tiantian Mao, Xinqiao Xie.

Figure 1
Figure 1. Figure 1: Comparison of risk measurement values in the robust (solid) and non-robust cases (Log [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Worst-case risk as a function of the deductible for different Wasserstein radii, using [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Out-of-sample performance of two robust models and SAA model [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
read the original abstract

In this paper, we introduce a class of preference robust risk measures-\emph{robust optimized certainty equivalents} (ROCE)-which encompasses several widely used measures, including Conditional Value-at-Risk and expectiles, as special cases. Motivated by recent developments in distributionally robust optimal reinsurance (DROR), we investigate DROR problems under the ROCE risk measure and consider two prominent uncertainty sets: the mean-variance uncertainty set and the Wasserstein uncertainty set. For the mean-variance uncertainty set, we reformulate the infinite-dimensional optimization problem into a finite-dimensional one by showing that it suffices to consider three-point distributions. This leads to a unified and explicit formulation for a broad class of ROCE risk measures and offers a simplified framework that also recovers earlier results for Conditional Value-at-Risk and expectiles. For the Wasserstein uncertainty set, we also derive a tractable finite-dimensional formulation. The resulting data-driven models enable efficient computation and facilitate a systematic comparison between moment-based and Wasserstein-based uncertainty sets in the optimal deductible design. Numerical experiments are exhibited to illustrate the performance of our reformulated programs.

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.

Referee Report

2 major / 2 minor

Summary. The paper introduces robust optimized certainty equivalent (ROCE) risk measures, a class that includes CVaR and expectiles as special cases. It studies distributionally robust optimal reinsurance (DROR) problems under ROCE with mean-variance and Wasserstein uncertainty sets. For the mean-variance set, the infinite-dimensional DRO problem is claimed to reduce to a finite-dimensional program because three-point distributions suffice; this yields a unified explicit formulation across the ROCE class and recovers prior results for CVaR and expectiles. A tractable finite-dimensional reformulation is also derived for the Wasserstein set. Data-driven models are obtained and illustrated via numerical experiments comparing the two ambiguity sets in deductible design.

Significance. If the three-point sufficiency and finite-dimensional reductions are rigorously established, the work supplies a unified, computationally tractable framework for DROR under a broad family of risk measures. It recovers known special cases, enables direct comparison of moment-based versus Wasserstein ambiguity sets, and produces explicit data-driven programs that support efficient numerical solution.

major comments (2)
  1. [§3.2] §3.2 (mean-variance case): the argument that three-point distributions suffice must explicitly construct the worst-case distribution from the mean-variance constraints and the convex structure of ROCE; without this step the reduction from infinite- to finite-dimensional optimization remains unverified.
  2. [Theorem 4.1] Theorem 4.1 (unified formulation): the explicit finite-dimensional program for general ROCE must be shown to specialize exactly to the known CVaR and expectile formulations without extra parameters or hidden assumptions on the uncertainty set.
minor comments (2)
  1. Notation for the ROCE functional should be introduced with an explicit formula before the DROR problem is stated.
  2. [Numerical experiments] Numerical section: the tables comparing mean-variance and Wasserstein solutions should report both optimal deductibles and the attained ROCE values for direct comparison.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments that identify opportunities to strengthen the rigor of our arguments. We address each major comment below and will incorporate the requested clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [§3.2] §3.2 (mean-variance case): the argument that three-point distributions suffice must explicitly construct the worst-case distribution from the mean-variance constraints and the convex structure of ROCE; without this step the reduction from infinite- to finite-dimensional optimization remains unverified.

    Authors: We agree that an explicit construction of the worst-case distribution is required to rigorously establish the reduction. In the revision we will add a dedicated lemma in §3.2 that derives the three support points and associated probabilities directly from the mean-variance moment constraints together with the convexity of the ROCE functional, thereby verifying that the infinite-dimensional problem is attained at a three-point distribution. revision: yes

  2. Referee: [Theorem 4.1] Theorem 4.1 (unified formulation): the explicit finite-dimensional program for general ROCE must be shown to specialize exactly to the known CVaR and expectile formulations without extra parameters or hidden assumptions on the uncertainty set.

    Authors: We will include an additional corollary (or appendix subsection) that substitutes the specific ROCE representations for CVaR and expectiles into the general finite-dimensional program. This will demonstrate exact recovery of the known formulations under the identical mean-variance uncertainty set, with no additional parameters or hidden assumptions introduced. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rest on explicit properties of ROCE and uncertainty sets

full rationale

The paper defines ROCE as a new class encompassing CVaR and expectiles, then derives finite-dimensional reformulations for the mean-variance case by proving that three-point distributions suffice and for the Wasserstein case by direct tractability arguments. These steps are presented as consequences of convexity, monotonicity, and the optimized-certainty-equivalent structure interacting with the chosen ambiguity sets; no equation is shown to equal its own input by construction, no fitted parameter is relabeled as a prediction, and no load-bearing uniqueness result is imported solely via self-citation. The central claims therefore remain externally verifiable from the stated functional properties rather than reducing to the paper's own fitted values or prior self-references.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the introduction of the ROCE class and standard background assumptions from convex risk measure theory and distributionally robust optimization; no free parameters or invented entities with independent evidence are visible in the abstract.

axioms (1)
  • domain assumption ROCE satisfies convexity, monotonicity, and translation invariance as required for the reformulations to hold.
    These properties are invoked implicitly to encompass CVaR and expectiles and enable the three-point reduction.
invented entities (1)
  • Robust Optimized Certainty Equivalent (ROCE) no independent evidence
    purpose: New class of preference-robust risk measures
    Defined in the paper to generalize existing measures; no external falsifiable evidence provided in abstract.

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Works this paper leans on

51 extracted references · 1 canonical work pages

  1. [1]

    Acerbi, C. (2002). Spectral measures of risk: A coherent representation of subjective risk aversion. Journal of Banking and Finance , 26 (7), 1505--1518

    2002

  2. [2]

    Armbruster, B., and Delage, E. (2015). Decision making under uncertainty when preference information is incomplete. Management Science , 61 (1), 111--128

    2015

  3. [3]

    Arrow, K. J. (1963). Uncertainty and the welfare economics of medical care. American Economic Review , 53 (3), 941--973

    1963

  4. [4]

    Bartl, D., Drapeau, S., and Tangpi, L. (2020). Computational aspects of robust optimized certainty equivalents and option pricing. Mathematical Finance , 30 (1), 287--309

    2020

  5. [5]

    and Gianin, E

    Bellini, F., Klar, B., M\" u ller, A. and Gianin, E. R. (2014). Generalized quantiles as risk measures. Insurance: Mathematics and Economics , 54 , 41--48

    2014

  6. [6]

    Ben-Tal, A., and Teboulle, M. (1986). Expected utility, penalty functions, and duality in stochastic nonlinear programming. Management Science , 32 (11), 1445--1466

    1986

  7. [7]

    Ben-Tal, A., and Teboulle, M. (1987). Penalty functions and duality in stochastic programming via -divergence functionals. Mathematics of Operations Research , 12 (2), 224--240

    1987

  8. [8]

    Ben-Tal, A., and Teboulle, M. (2007). An old-new concept of convex risk measures: The optimized certainty equivalent. Mathematical Finance , 17 (3), 449--476

    2007

  9. [9]

    Boonen, T. J. and Jiang, W. (2024). Robust insurance design with distortion risk measures. European Journal of Operational Research, 316 (2), 694--706

    2024

  10. [10]

    J., and Jiang, W

    Boonen, T. J., and Jiang, W. (2025). Pareto-optimal insurance under robust distortion risk measures. European Journal of Operational Research, 324(2), 690--705

    2025

  11. [11]

    J., and Jiang, W

    Boonen, T. J., and Jiang, W. (2025). Distributionally robust insurance under the Wasserstein distance. Insurance: Mathematics and Economics, 120 , 61--78

    2025

  12. [12]

    Borch, K. (1960). An attempt to determine the optimum amount of stop loss reinsurance. Transactions of the 16th International Congress of Actuaries I , 597--610

    1960

  13. [13]

    and Chi, Y

    Cai, J. and Chi, Y. (2020). Optimal reinsurance designs based on risk measures: A review. Statistical Theory and Related Fields , 4 (1), 1--13

    2020

  14. [14]

    and Mao, T

    Cai, J., Jiao, Z. and Mao, T. (2025). Worst-case values of target semi-variances with applications to robust portfolio selection. European Journal of Operational Research, 327 (3), 905--921

    2025

  15. [15]

    Cai, J., Li, J. Y. M., and Mao, T. (2025). Distributionally robust optimization under distorted expectations. Operations Research, 73 (2), 969--985

    2025

  16. [16]

    Cai, J., Liu, F., and Yin, M. (2024). Worst-case risk measures of stop-loss and limited loss random variables under distribution uncertainty with applications to robust reinsurance. European Journal of Operational Research, 318 (1), 310--326

    2024

  17. [17]

    C ern\' y , A., Maccheroni, F., Marinacci, M., and Rustichini, A. (2012). On the computation of optimal monotone mean-variance portfolios via truncated quadratic utility. Journal of Mathematical Economics , 48 (6), 386--395

    2012

  18. [18]

    Chen, L., He, S., and Zhang, S. (2011). Tight bounds for some risk measures, with applications to robust portfolio selection. Operations Research , 59 (4), 847--865

    2011

  19. [19]

    C., Yam, S

    Cheung, K. C., Yam, S. C. P., Yuen, F. L., and Zhang, Y. (2020). Concave distortion risk minimizing reinsurance design under adverse selection. Insurance: Mathematics and Economics , 91 , 155--165

    2020

  20. [20]

    and Ye, Y

    Delage, E. and Ye, Y. (2010). Distributionally robust optimization under moment uncertainty with application to data-driven problems. Operations Research 58 (3): 595--612

    2010

  21. [21]

    F\" o llmer, H., and Schied, A. (2002). Convex measures of risk and trading constraints. Finance and stochastics , 6 (4), 429--447

    2002

  22. [22]

    Gao, R., Chen, X., and Kleywegt, A. J. (2024). Wasserstein distributionally robust optimization and variation regularization. Operations Research , 72 (3), 1177--1191

    2024

  23. [23]

    Gao, R., and Kleywegt, A. (2023). Distributionally robust stochastic optimization with Wasserstein distance. Mathematics of Operations Research , 48 (2), 603--655

    2023

  24. [24]

    Guo, S., Xu, H., and Zhang, S. (2024). Utility preference robust optimization with moment-type information structure. Operations Research , 72 (5), 2241--2261

    2024

  25. [25]

    F\" o llmer, H., and Schied, A. (2010). Convex and coherent risk measures. Encyclopedia of Quantitative Finance , 355--363

    2010

  26. [26]

    F\" o llmer, H., and Schied, A. (2016). Stochastic Finance: An Introduction in Discrete Time (Fourth Edition). Walter de Gruyter, Berlin

    2016

  27. [27]

    and Liu, F

    Cai, J., Lemieux, C. and Liu, F. (2016). Optimal reinsurance from the perspectives of both an insurer and a reinsurer. ASTIN Bulletin , 46 (3), 815--849

    2016

  28. [28]

    and Tan, K.S

    Cai J. and Tan, K.S. (2007). Optimal retention for a stop-loss reinsurance under the VaR and CTE risk measures. ASTIN Bulletin, 37(1), 93--112

    2007

  29. [29]

    and Weng, C

    Cai, J. and Weng, C. (2016). Optimal reinsurance with expectile. Scandinavian Actuarial Journal , 2016 (7), 624--645

    2016

  30. [30]

    C., Sung, K

    Cheung, K. C., Sung, K. C. J., Yam, S. C. P. and Yung, S. P. (2014). Optimal reinsurance under general law-invariant risk measures. Scandinavian Actuarial Journal , 2014 , 72--91

    2014

  31. [31]

    and Tan, K

    Chi, Y. and Tan, K. S. (2011). Optimal reinsurance under VaR and CVaR risk measures: a simplified approach. ASTIN Bulletin , 41 (2), 487--509

    2011

  32. [32]

    Kantorovich, L. (1958). On the translocation of masses. Journal of mathematical sciences , 133 (4)

    1958

  33. [33]

    M., Nguyen, V

    Kuhn, D., Esfahani, P. M., Nguyen, V. A., and Shafieezadeh-Abadeh, S. (2019). Wasserstein distributionally robust optimization: Theory and applications in machine learning. In Operations research and management science in the age of analytics , 130--166

    2019

  34. [34]

    Kusuoka, S. (2001). On law invariant coherent risk measures. In Advances in mathematical economics (pp. 83-95). Tokyo: Springer Japan

    2001

  35. [35]

    and Mao, T

    Liu, H. and Mao, T. (2022). Distributionally robust reinsurance with Value-at-Risk and Conditional Value-at-Risk. Insurance: Mathematics and Economics , 107 , 393--417

    2022

  36. [36]

    Maccheroni, F., Marinacci, M., and Rustichini, A. (2006). Ambiguity aversion, robustness, and the variational representation of preferences. Econometrica , 74 (6), 1447--1498

    2006

  37. [37]

    Mohajerin Esfahani, P., and Kuhn, D. (2018). Data-driven distributionally robust optimization using the Wasserstein metric: Performance guarantees and tractable reformulations. Mathematical Programming , 171 (1), 115--166

    2018

  38. [38]

    and Powell, J

    Newey, W. and Powell, J. (1987). Asymmetric least squares estimation and testing. Econometrica , 55 , 819---847

    1987

  39. [39]

    C., and Ruszczy\' n ski, A

    Pflug, G. C., and Ruszczy\' n ski, A. (2005). Measuring risk for income streams. Computational Optimization and Applications , 32 , 161--178

    2005

  40. [40]

    Rockafellar, R. T. and Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2, 21--42

    2000

  41. [41]

    Scarf, H. (1958). A min-max solution of an inventory problem. Arrow K, Karlin S, Scarf H, eds. Studies in the Mathematical Theory of Inventory and Production (Stanford University Press, Stanford, CA), 201--209

    1958

  42. [42]

    Shapiro, A., Dentcheva, D., and Ruszczynski, A. (2021). Lectures on Stochastic Programming: Modeling and Theory (Third Edition). Society for Industrial and Applied Mathematics, Philadelphia

    2021

  43. [43]

    Sion, M. (1958). On general minimax theorems. Pacific Journal of Mathematics, 8, 171--176

    1958

  44. [44]

    Villani, C. (2021). Topics in optimal transportation (Vol. 58). American Mathematical Soc

    2021

  45. [45]

    Villani, C. (2008). Optimal transport: old and new (Vol. 338, pp. 129-131). Berlin: springer

    2008

  46. [46]

    Wu, Q., Li, J. Y. M., and Mao, T. (2025). On generalization and regularization via wasserstein distributionally robust optimization. Management Science , 0, 0. https://doi.org/10.1287/mnsc.2023.03895

    work page doi:10.1287/mnsc.2023.03895 2025

  47. [47]

    and Hu, T

    Wu, Q., Mao, T. and Hu, T. (2024). Generalized optimized certainty equivalent with applications in the rank-dependent utility model. SIAM Journal on Financial Mathematics , 15(1), 255--294

    2024

  48. [48]

    Wu, Q., and Xu, H. (2022). Preference robust modified optimized certainty equivalent. SIAM Journal on Optimization , 32(4), 2662--2689

    2022

  49. [49]

    Wu, Q., and Xu, H. (2022). Preference Robust Modified Optimized Certainty Equivalent. SIAM Journal on Optimization , 32 (4), 2662--2689

    2022

  50. [50]

    Xie, X., Liu, H., Mao, T., and Zhu, X. (2023). Distributionally robust reinsurance with expectile. ASTIN Bulletin: The Journal of the IAA , 53 (1), 129--148

    2023

  51. [51]

    C., and Zhang, Y

    Yong, Y., Cheung, K. C., and Zhang, Y. (2024). Optimal reinsurance design under distortion risk measures and reinsurer’s default risk with partial recovery. ASTIN Bulletin: The Journal of the IAA , 54 (3), 738--766

    2024