General Bounds on Functionals of the Lifetime under Life Table Constraints

2026-03-06

AI Summary

Life tables provide survival probabilities only at integer ages and lack information on the distribution of deaths between ages, forcing the pricing of annuities and other insurance products to rely on subjective fractional-age assumptions that compromise robustness. This work addresses this limitation by constructing, for the first time, a set of mortality trajectories consistent with a given life table under the sole constraint of known integer-age survival probabilities. Two complementary frameworks are proposed: one requiring paths to match the life table almost surely, and another permitting deviations while preserving agreement in expectation. Using probabilistic and optimization techniques, sharp upper and lower bounds on life-contingent functionals are derived across all compatible models—without invoking any fractional-age assumptions. This approach quantifies the admissible range of insurance contract values, offering a model-free, robust foundation for mortality risk assessment and product pricing.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

In life insurance, life tables are used to estimate the survival distribution of individuals from a given population. However, these tables only provide survival probabilities at integer ages but no information about the distribution of deaths between two consecutive integer values. Many actuarial quantities, such as variable annuities, are functionals of the lifetime and computing them requires full information about mortality rates. One frequent solution is to postulate fractional age assumptions or mortality rate models, but it turns out that the results of the computations strongly depend on these assumptions, which makes it difficult to generalize them. We hence derive upper and lower bounds of functionals of the lifetime with respect to mortality rates, which are compatible with the observed life table at integer ages. We derive two sets of results under distinct assumptions. In the first, we assume that each mortality trajectory is almost surely consistent with all the given one-year survival probabilities from the table. In the second, we consider a relaxed formulation that allows for deviations of the mortality rates while still being consistent in expectation with the given one-year reference survival probabilities. These distinct yet complementary approaches provide a new robust framework for managing mortality risk in life insurance. They characterize the worst- and best-case contract values over all mortality processes that remain compatible with the observed life-table information, thereby enabling insurers to quantify the impact on prices of deviations of the observed mortality rates from their mortality assumptions/models.

Problem

• life tables
• mortality rates
• lifetime functionals
• actuarial valuation
• fractional age assumptions

Innovation

• mortality bounds
• life table constraints
• lifetime functionals
• robust actuarial modeling
• fractional age assumptions