Abstract The limited stop-loss transform, along with the stop-loss and limited loss transforms – which are special or limiting cases of the limited stop-loss transform – is one of the most important transforms used in insurance, and it also appears extensively in many other fields including finance, economics, and operations research. When the distribution of the underlying loss is uncertain, the worst-case risk measure for the limited stop-loss transform plays a key role in many quantitative risk management problems in insurance and finance. In this paper, we derive expressions for the worst-case distortion risk measure of the limited stop-loss transform, as well as for the stop-loss and limited loss transforms, when the distribution of the underlying loss is uncertain and lies in a general $k$ -order Wasserstein ball that contains a reference distribution. We also identify the worst-case distributions under which the worst-case distortion risk measures are attained. Additionally, our results also recover the findings of Guan et al. ((2023) North American Actuarial Journal, 28(3), 611–625), regarding the worst-case stop-loss premium over a $k$ -order Wasserstein ball. Furthermore, we use numerical examples to illustrate the worst-case distributions and the worst-case risk measures derived in this paper. We also examine the effects of the reference distribution, the radius of the Wasserstein ball, and the retention levels of limited stop-loss reinsurance on the premium for this type of reinsurance.

Abstract The bonus-malus system (BMS) is a widely recognized and commonly employed risk management tool. A well-designed BMS can match expected insurance payments with estimated claims even in a diverse group of risks. Although there has been abundant research on improving bonus-malus (BM) systems, one important aspect has been overlooked: the stationary probability of a BMS satisfies the monotone likelihood ratio property. The monotone likelihood ratio for stationary probabilities allows us to better understand how riskier policyholders are more likely to remain in higher premium categories, while less risky policyholders are more likely to move toward lower premiums. This study establishes this property for BMSs that are described by an ergodic Markov chain with one possible claim and a transition rule +1/-d. We derive this result from the linear recurrences that characterize the stationary distribution; this represents a novel analytical approach in this domain. We also illustrate the practical implications of our findings: in the BM design problem, the premium scale is automatically monotonic.

Abstract Introduced over a century ago, Whittaker–Henderson smoothing remains widely used by actuaries in constructing one-dimensional and two-dimensional experience tables for mortality, disability, and other life insurance risks. In this paper, we reinterpret this smoothing technique within a modern statistical framework and address six practically relevant questions about its use. First, we adopt a Bayesian perspective on this method to construct credible intervals. Second, in the context of survival analysis, we clarify how to choose the observation and weight vectors by linking the smoothing technique to a maximum likelihood estimator. Third, we improve accuracy by relaxing the method’s reliance on an implicit normal approximation. Fourth, we select the smoothing parameters by maximizing a marginal likelihood function. Fifth, we improve computational efficiency when dealing with numerous observation points and consequently parameters. Finally, we develop an extrapolation procedure that ensures consistency between estimated and predicted values through constraints.

Abstract Risk-sharing rules have been applied to mortality pooling products to ensure these products are actuarially fair and self-sustaining. However, most of the existing studies on the risk-sharing rules of mortality pooling products assume deterministic mortality rates, whereas the literature on mortality models provides empirical evidence suggesting that mortality rates are stochastic and correlated between cohorts. In this paper, we extend existing risk-sharing rules and introduce a new risk-sharing rule, named the joint expectation (JE) rule, to ensure the actuarial fairness of mortality pooling products while accounting for stochastic and correlated mortality rates. Moreover, we perform a systematic study of how the choice of risk-sharing rule, the volatility and correlation of mortality rates, pool size, account balance, and age affect the distribution of mortality credits. Then, we explore a dynamic pool that accommodates heterogeneous members and allows new entrants, and we track the income payments for different members over time. Furthermore, we compare different risk-sharing rules under the scenario of a systematic shock in mortality rates. We find that the account balance affects the distribution of mortality credits for the regression rule, while it has no effect under the proportional, JE, and alive-only rules. We also find that a larger pool size increases the sensitivity to the deviation in total mortality credits for cohorts with mortality rates that are volatile and highly correlated with those of other cohorts, under the stochastic regression rule. Finally, we find that risk-sharing rules significantly influence the effect of longevity shocks on fund balances since, under different risk-sharing rules, fund balances have different sensitivities to deviations in mortality credits.

Abstract This paper examines the optimal design of peer-to-peer (P2P) insurance models, which combines outside insurance purchases with P2P risk sharing and heterogeneous risk. Participants contribute deposits to collectively cover the premium for group-based insurance against tail risks and to share uncovered losses. We analyze the cost structure by decomposing it into a fixed premium for outside coverage and a variable component for shared losses, the latter of which may be partially refunded if aggregate losses are sufficiently low. We derive closed-form solutions to the optimal sharing rule that maximizes a mean-variance objective from the perspective of a central or social planner, and we characterize its theoretical properties. Building on this foundation, we further investigate the choice of deposit for the common fund. Finally, we also provide numerical illustrations.

Abstract The special issue on risk sharing contains 12 papers. This editorial introduces these papers by briefly discussing their contents and contributions.

Abstract This paper studies a long-standing problem of risk exchange and optimal resource allocation among multiple entities in a continuous-time pure risk-exchange economy. We establish a novel risk exchange mechanism that allows entities to share and transfer risks dynamically over time. To achieve Pareto optimality, we formulate the problem as a stochastic control problem and derive explicit solutions for the optimal investment, consumption, and risk exchange strategies using a martingale method. To highlight practical applications of the solution to the proposed problem, we apply our results to a target benefit pension plan, featuring the potential benefits of risk sharing within this pension system. Numerical examples show the sensitivity of investment portfolios, the adjustment item, and allocation ratios to specific parameters. It is observed that an increase in the aggregate endowment process results in a rise in the adjustment item. Furthermore, the allocation ratios exhibit a positive correlation with the weights of the agents.

Abstract This paper investigates time-varying risk sharing between annuity buyer and provider. It explores Pareto optimal (PO) and viable Pareto optimal (VPO) risk-sharing designs, in which the share of the reserve deviation transferred to the policyholder varies over time. The optimization problem, based on a weighted average of mean-variance preferences, results in a complex quartic objective function. Such optimization problems are difficult to solve, and checking their convexity is known to be NP-hard. A heuristic method is introduced to simplify the problem, providing a closed-form solution that closely approximates the numerical results. The paper also highlights factors influencing the existence of VPO designs, with age playing a critical role, thereby suggesting the suitability of these designs as retirement products.