General Bounds on Functionals of the Lifetime under Life Table Constraints

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.