In Notional Defined Contribution (NDC) pension schemes, the annuity divisor plays a pivotal role in ensuring actuarial fairness and financial sustainability. However, depending on how it incorporates factors such as longevity improvements, gender, frontloading, indexation, marital status, and survivor benefits, the annuity divisor can generate significant redistributive effects both across individuals and between generations. This paper examines the actuarial fairness and redistributive properties of the uniquely designed Italian NDC annuity divisor. Specifically, it analyses the implications of: using period rather than cohort annuity divisors; the sex gradient in longevity; the inclusion of survivor pensions through one-head, gender-specific reversionary annuity factors undifferentiated by marital status; the frontloading of pension payments; and other demographic and economic institutional parameters. The study adopts a prospective analytical framework to generate forecasts of period and cohort unisex and gender-specific annuity divisors and annuity rates up to the year 2100, employing a Bayesian model combination of stochastic mortality models. The findings confirm that the Italian annuity divisor design entails non-negligible distributional and financial sustainability effects, leading to systematic redistribution both within and between generations. The direction and magnitude of the resulting implicit tax/subsidy effects vary by sex, retirement age, and time. These outcomes are critically influenced by the use of a period approach, the frontloading of pension benefits while maintaining price indexation, and the inclusion of survivor pensions in the divisor formula. Ultimately, the Italian annuity divisor design results in implicit cross-subsidisation from single individuals to married couples and may introduce incentive distortions that affect both labour supply and retirement decisions.
Across fields including insurance and finance, risk assessment is indispensable, involving two key dimensions: the magnitude of potential losses and their frequency of occurrence. While the size of losses is often emphasized, overlooking frequency may result in incomplete risk evaluations. Drawdowns and ranges are two fundamental path-dependent measures for analyzing underlying risk processes, with drawdowns quantifying declines relative to prior peak values, and ranges capturing overall variability. This paper studies the frequency of deep drawdowns and ranges for L & eacute;vy risk processes through the characterization of their n-th occurrence times and associated quantities of interest. Numerical studies are also conducted to analyze inherent properties and explore potential applications.
This paper investigates an optimal reinsurance-investment problem with model uncertainty, in which future claims are correlated with historical ones. The insurer is allowed to purchase proportional reinsurance and invest in a risk-free bond and a stock. The insurance and reinsurance premiums are all calculated according to a principle depending on historical claims. Furthermore, we assume that the insurer and the reinsurer have different extrapolation intensities. Besides, the insurer is ambiguous about the claim process and the stock’s price process. The insurer aims to maximize the expected exponential utility of terminal wealth under model uncertainty. By applying the dynamic programming approach, we establish the corresponding Hamilton-Jacobi-Bellman (HJB) equation and derive the optimal reinsurance-investment strategy and value function by using the eigenvalue method. Finally, the numerical analysis is provided to illustrate our results. We find that the expectation of the insurer’s optimal reinsurance strategy is no longer a monotonically increasing function, which is different from the results for the case where the insurer and the reinsurer have the same extrapolation intensity.
Group self-annuitization schemes efficiently pool longevity risk but expose participants to significant benefit volatility. We propose a benefit-stabilization mechanism based on a zero-cost collar (ZCC) implemented through option-based hedging, relaxing the perfect replication assumption commonly adopted in the literature by explicitly incorporating basis risk. Optimal cross-hedging strategies are derived within a quadratic risk-minimization framework under stochastic volatility. We show that the ZCC strategy stabilizes income and extends the period over which benefit payouts remain steady. Put option-based hedging strategies provide stronger downside protection and improve the sustainability of long-term benefits by aligning with the convexity and volatility sensitivity of the downside liability. We further show that optimal rebalancing frequency depends critically on the hedge’s gamma exposure: long-gamma strategies benefit from more frequent rebalancing to reduce hedging errors, whereas short-gamma strategies require lower rebalancing frequencies to limit trading losses. Sensitivity analyses confirm the robustness of the stabilization mechanism across alternative market environments and hedging instruments.
The estimation of mortality and disease incidence rates for small populations poses significant challenges for insurers in meeting regulatory compliance requirements, such as those imposed by the European prudential framework Solvency II. These difficulties primarily arise from data scarcity and the substantial volatility inherent in estimates derived from limited observations. To alleviate these issues, several methodologies have been proposed that effectively enlarge the underlying data set—for instance, through multi-population mortality models or the incorporation of reference populations—to reduce the stochastic variation in small-sample estimates. Such approaches, however, rest critically on the assumption that the reference population exhibits comparable demographic and epidemiological characteristics to those of the target population. A frequently adopted strategy involves pooling data from geographically adjacent districts; nevertheless, neighboring areas often differ markedly in their socioeconomic and demographic compositions, which may induce pronounced heterogeneity in age-specific mortality and incidence patterns.In this study, we propose a data-driven framework for identifying suitable reference populations to enhance the goodness-of-fit of mortality estimation in small populations. Specifically, cluster analysis is employed to group small populations together with their neighboring districts, thereby constructing reference populations using techniques grounded in the heterogeneity measures proposed by Lee (2003). The study areas are partitioned into clusters exhibiting intra-group homogeneity and substantial inter-group heterogeneity in mortality rates. The proposed methodology is empirically validated using mortality data at the county and township levels in Taiwan. Based on the constructed reference populations, life tables are established, and life expectancy estimates for small populations are subsequently derived.
