1. Risk sharing in a rapidly changing world

Risk sharing is one of the foundational principles of actuarial science: by pooling exposures and spreading losses, societies transform economic risks – as well as their associated uncertainty – into quantitatively manageable events. Yet the 2020s have reminded us that the landscape of risks – and of the mechanisms that absorb them – is evolving at an unprecedented pace. Climate-induced catastrophes occur with rising frequency and severity; demographic shifts compound longevity and social security risks; cyber threats materialise across interconnected digital infrastructures; and financial markets propagate shocks globally within seconds.

Against this backdrop, the design of efficient, equitable and resilient risk-sharing arrangements has become more urgent than ever. In this environment, classical actuarial models are being supplemented – and at times challenged – by new methodologies and technologies. Analytical tools such as game theory, robust optimisation, machine learning and distributed ledger technologies are integrated into frameworks of insurance, pensions and financial regulation at an increasingly speed. This blending of traditional theory with modern computation reflects a broader trend: risk sharing, both as theoretical concepts and practical mechanisms, must evolve to stay relevant.

This special issue responds to this momentum. Our aim is two-fold:

• • To showcase conceptual advances that extend the theoretical foundations of risk sharing under uncertainty,

• • To highlight innovations in practice that translate theory into mechanisms capable of protecting households, firms and societies under real-world constraints.

2. Overview of accepted contributions

The 12 articles collected in this special issue reflect the diversity and evolution of contemporary risk-sharing research. They span topics such as economic models under uncertainty, fairness and heterogeneity in insurance and retirement systems, technological innovation, and applied actuarial frameworks. Together, these contributions enrich several strands of literature and point to new directions for both theoretical and practical developments in modern financial and insurance contexts.

This editorial introduces these papers, highlighting their key insights, how they relate to one another and how they contribute to the broader, developing discourse on risk sharing.

3. Equity and fairness in longevity and retirement risk sharing

Longevity heterogeneity across socioeconomic groups raises long-standing questions about fairness in pooled retirement products. Early work such as Yaari (Reference Yaari1965) formalised the welfare gains from annuitisation in frictionless markets, while more recent studies – for example Piggott et al. (Reference Piggott, Valdez and Detzel2005), Sabin (Reference Sabin2010) and Milevsky and Salisbury (Reference Milevsky and Salisbury2015) – demonstrate how product design can mitigate inequities caused by mortality gradients. Against this backdrop, the four papers in this category push the frontier by uniting modern mortality modelling with redistributive or Pareto-efficient mechanisms.

Aragona et al. (2025) advance the concept of Group Self-Annuitisation (GSA) by incorporating explicit redistribution rules to offset socioeconomic mortality gaps. Building on the two-cohort design of Piggott et al. (Reference Piggott, Valdez and Detzel2005), they generalise the framework to an arbitrary number of groups and derive closed-form transfer rates that preserve incentive compatibility while satisfying actuarial fairness principles similar to those discussed in the tontine literature (Chen and Rach, Reference Chen and Rach2023).

Kang et al. (2025) explore fairness in tontine schemes using mortality models with long-range dependence. Integrating the Volterra framework with an innovative fairness measure (the f-value), they propose a hybrid optimisation model that accounts for age and wealth heterogeneity. Their analysis complements the constant relative risk aversion results of Sabin (Reference Sabin2010) and the load-adjusted tontine pricing in Chen et al. (Reference Chen, Guillen and Rach2021), showing that path-dependent survival dynamics can either amplify or dampen perceived inequities in payout profiles.

Hanbali et al. (2025) introduce time-varying Pareto-optimal designs for annuities. Their heuristic tackles a challenging optimisation problem and addresses the dynamic nature of annuity markets. By allowing benefit adjustments reminiscent of the ‘money-back-in-expectation’ constraint analysed by Milevsky and Salisbury (Reference Milevsky and Salisbury2025), the authors provide a tractable bridge between the static optimal-retirement-income model of Yaari (Reference Yaari1965) and the adaptive ‘tonuity’ concept of Chen et al. (Reference Chen, Hieber and Klein2019).

Tao et al. (2025) formulate a continuous-time stochastic control model for dynamic risk exchange among heterogeneous agents and derive explicit Pareto-optimal rules for investments, consumption and enter-entity transfers. Their framework generalises the classical sharing results of Borch (Reference Borch1960) and extends the continuous-time risk-exchange prototype of Iwaki (Reference Iwaki2002) by allowing risk and wealth to co-evolve over an entire planning horizon. When applied to a Target Benefit Pension (TBP) plan, the mechanism demonstrates how calibrated transfer schedules can absorb longevity-driven funding shocks without breaching solvency bounds, thereby complementing recent TBP design studies by Wang and Lu (Reference Wang and Lu2019) and Zhu et al. (Reference Zhu, Hardy and Saunders2021).

Zhou et al. (2025) extend the theory of risk sharing in mortality pooling products by incorporating stochastic and cohort-correlated mortality dynamics. Motivated by the empirical evidence that mortality rates are both random and correlated across groups, the authors propose a novel ‘joint expectation rule’ that preserves actuarial fairness while accounting for these complexities. Their model generalises and analytically compares established risk-sharing rules – including the proportional, regression and alive-only rules – under systematic mortality shocks. Through extensive simulation and analytical characterisation, they demonstrate how factors such as account balances, pool size and inter-cohort correlation shape the distribution of mortality credits. Building on foundations laid by Denuit (Reference Denuit2019) and Fullmer and Sabin (Reference Fullmer and Sabin2018), this paper provides a unifying extension to settings with stochastic and dependent mortality. The results offer design insights into dynamic pooling structures with heterogeneous and evolving membership and show that the choice of sharing rule crucially affects the equity and stability of retirement income under longevity risk.

4. Peer-to-peer and decentralised risk sharing

Recent years have witnessed a surge of interest in decentralised or peer-to-peer (P2P) insurance. Foundational work on comonotone conditional-mean risk-sharing (Denuit and Dhaene, Reference Denuit and Dhaene2012; Denuit et al., Reference Denuit, Dhaene and Robert2022) shows how pooling can reduce losses in convex order, yet those rules are computationally demanding. To retain tractability, later studies adopted linear allocation rules – variance minimisation in Feng (Reference Feng2023), mutual-aid efficiency in Abdikerimova and Feng (Reference Abdikerimova and Feng2022), while still pursuing fairness and participation incentives.

Yang and Wei (2025) revisit this variance-based strand and deliver the first theoretical proof that within a very broad admissible class, total variance is minimised when linear sharing is applied to residual (mean-centred) risks rather than the raw losses. Their result unifies earlier heuristic insights in Feng (Reference Feng2023), clarifies why residualisation preserves actuarial fairness and echoes the ‘risk-anonymity’ principle of Jiao et al. (Reference Jiao, Kou, Liu and Wang2022). By adding market-oriented constraints – variance reduction and $\gamma$-retention consistency – the paper extends the optimisation framework of Abdikerimova and Feng (Reference Abdikerimova and Feng2022) and proves robustness against model estimation error.

Boonen et al. (2025) revisit the optimal design of P2P insurance by synthesising ideas from both reinsurance and mutual-aid literatures. While prior models often treated P2P pooling in isolation or assumed homogeneous agents (e.g., Abdikerimova and Feng, Reference Abdikerimova and Feng2022), their study introduces a hybrid framework that integrates traditional insurance for catastrophic losses with decentralised cost-sharing for moderate claims via a common deposit fund. Within a mean-variance setting, they derive closed-form solutions for the optimal allocation of both fixed (premium) and variable (refundable deposit) costs under heterogeneity. The analysis highlights a trade-off between fixed and variable contributions driven by risk aversion and expected loss, offering a unified characterisation of ‘buyer’ and ‘seller’ types within the pool. Their fairness condition extends classical actuarial principles to dual-cost settings, while their individual-rationality constraints ensure robust participation incentives, generalising earlier contributions in altruistic and mutual-aid risk sharing.

5. Risk sharing and reinsurance design under ambiguity

Ambiguity, sometimes also called model uncertainty in actuarial science and risk management, has been a focal point in modern economic theory (Wakker, Reference Wakker2010). Ambiguity concerns how decisions can be made without full probabilistic or statistical modelling of the underlying risks. In the recent decade, developments are made on how ambiguity affects the outcome of risk sharing and reinsurance design.

Landriault et al. (2025) address model uncertainty in reinsurance design by considering worst-case reinsurance strategies under likelihood-ratio ambiguity. Their analysis bridges robust optimisation and insurance design, offering concrete guidance for insurers in adversarial environments. Builiding on the likelihood-robust framework of Wang et al. (Reference Wang, Glynn and Ye2016) and its extension in Liu et al. (Reference Liu, Mao, Wang and Wei2022), the paper generalises the classic min-max approach of Scarf et al. (Reference Scarf, Arrow and Karlin1957) to the reinsurance setting. It further connects to the growing literature on distributionally robust reinsurance – for example, Hu et al. (Reference Hu, Yang and Zhang2015). By characterising distortion risk measures for which the insurer’s optimal ceded-loss function remains unchanged under worst-case scenarios, the authors deliver ready-to-implement rules that close an important gap between theory and practice.

Ghossoub et al. (2025) move beyond the quadratic-loss paradigm. Extending the settings of distortion risk measures in Liu (Reference Liu2020) and quantile-based preferences in Embrechts et al. (Reference Embrechts, Liu and Wang2018), they characterise Pareto-optimal P2P allocations when each agent evaluates risk through a robust (set-valued) distortion functional. Their comonotone representation theorem generalises classical Borch rules to settings with heterogeneous beliefs and tail-sensitive preferences, linking to ambiguity-averse models such as Chateauneuf et al. (Reference Chateauneuf, Dana and Tallon2000) and Dana (Reference Dana2004).

6. Innovations in risk models and risk measures

Several papers develop new models or measures that can be used in risk sharing problems. They either extend classic models to new types of risks or introduce new models inspired by technological innovations in the insurance market.

Müller (2025) revisits the classical insurance paradigm by considering infinite-mean risk models. Building on seminar insights by Fama (Reference Fama1965) and Samuelson (Reference Samuelson1967) on stable distributions, subsequent evident of the non-diversifiable trap in Ibragimov et al. (Reference Ibragimov, Jaffee and Walden2009) and Hofert and Wüthrich (Reference Hofert and Wüthrich2012), and the more recent development in Chen et al. (Reference Chen, Embrechts and Wang2025b) and Chen and Shneer (Reference Chen and Shneer2025), Müller unifies these strands through a general skewness-based criterion, showing that any distribution more skewed than a Cauchy may thwart the benefits of pooling. In contrast to the conventional belief that pooling benefits the risk risk-averse, his work illustrates how heavy-tailed laws such that Pareto or Fréchet can trigger adverse selection into risky pools, extending the observations made by Chen et al. (Reference Chen, Embrechts and Wang2025a). This broad generalisation deepens our understanding of risk pooling in catastrophic scenarios and highlights when diversification may backfire.

Laudagé et al. (2025) propose multi-asset return risk measures (MARRMs), a natural generalisation of the return risk measures of Bellini et al. (Reference Bellini, Laeven and Rosazza Gianin2018). Their framework marries the relative-performance perspective of RRMs with the capital-adequacy logic of multi-asset risk measures (MARMs) developed by Frittelli and Scandolo (Reference Frittelli and Scandolo2006) and rigorously analysed by Farkas et al. (Reference Farkas, Koch-Medina and Munari2015). In doing so, they offer a unified, multiplicative risk-sharing interpretation that complements the additive inf-convoluation paradigm of classical MARMs.

Lin and Zeng (2025) study how unobservable access to advanced digital technology shapes competition in insurance markets. Their model combines search-cost frictions–à la Ellison and Wolitzky (Reference Ellison and Wolitzky2012) – with asymmetric information, extending the technology heterogeneity arguments of Eling and Lehmann (Reference Eling and Lehmann2018) and the cost-based competition framework of Eling et al. (Reference Eling, Jia, Lin and Rothschild2022). By showing that technology adoption can lower an insurer’s market share when accessibility is hidden from customers, they reconcile mixed empirical findings on InsurTech penetration and echo classic insights on information asymmetry in insurance (e.g., Rothschild and Stiglitz, Reference Rothschild and Stiglitz1978). The paper thus links the digital-divide narrative in industrial economics to actuarial questions of market structure and welfare under imperfect information.

7. Conclusion

This special issue highlights the multifaceted nature of risk sharing today. From theoretical refinements under extreme risks and ambiguity to applied designs of equitable retirement systems, and from decentralised insurance markets to innovations in risk measurement and technology, the contributions herein showcase the vitality and relevance of this field.