A Functional-Analytic Sensitivity Methods for Actuarial Valuation

Actuarial valuation depends heavily on assumptions involving mortality, lapse behavior, and dis count rates. Small changes in these assumptions may produce material changes in liability values, leading to significant model risk for insurers and affecting long-term reserve adequacy and risk management decisions. This paper develops a functional-analytic framework for studying stability and sensitivity in de terministic actuarial valuation. The valuation process is formulated as a nonlinear operator acting on an infinite-dimensional space of model assumptions. Within this framework, we establish Lip schitz continuity, Fr´echet differentiability, and second-order sensitivity properties of the valuation operator under perturbations in mortality, lapse, and discount assumptions. Explicit first- and second-order sensitivity formulas are derived and interpreted in an actuarial setting. The theoretical results are illustrated through a numerical example using mortality data from the Social Security Administration actuarial life table. The numerical results show close agreement between exact perturbed valuations and the corresponding Fr´echet approximations. The framework developed in this paper provides a mathematically rigorous approach for studying valuation sensitivity, assumption uncertainty, and model risk in deterministic actuarial models.