We extend the duality between exponential integrals and relative entropy to a variational formula for exponential integrals involving the Renyi divergence. This formula characterizes the dependence of risk-sensitive functionals and related quantities determined by tail behavior to perturbations in the underlying distributions, in terms of the Renyi divergence. The characterization gives rise to upper and lower bounds that are meaningful for all values of a large deviation scaling parameter, allowing one to quantify in explicit terms the robustness of risk-sensitive costs. As applications we consider problems of uncertainty quantification when aspects of the model are not fully known, as well their use in bounding tail properties of an intractable model in terms of a tractable one.