General bounds on functionals of the lifetime under life table constraints in a joint actuarial-financial framework

Abstract

In life insurance, life tables provide survival probabilities at integer ages but leave the distribution of deaths within each year unspecified. This incompleteness is particularly relevant for modern insurance products such as variable annuities, whose payoffs depend jointly on lifetime uncertainty and financial market performance. The valuation of such contracts must therefore be carried out in a joint actuarial-financial framework, as their values depend not only on the full information about mortality rates but also on the interaction between mortality risk, asset dynamics and embedded guarantees. Standard approaches impose fractional-age assumptions or mortality rate models, but the resulting valuations strongly depend on these assumptions, which makes it difficult to generalize them. We hence derive upper and lower bounds of hybrid functionals of the lifetime with respect to mortality rates, which are compatible with the observed life table at integer ages and the given financial market. 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 tabulated one-year survival probabilities. 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 reference life table. These distinct yet complementary formulations characterize the worst- and best-case contract values over all mortality processes that remain compatible with the observed life-table information and the financial market. The resulting bounds provide insurers with a practical and robust risk-management tool for assessing the impact of adverse mortality dynamics on variable annuities, quantifying the uncertainty arising from incomplete life-table information, and incorporating mortality model risk into pricing decisions.