Demographic and financial factors are key risk-drivers for insurance companies and pension funds. This paper proposes a systematic investigation for deepening our understanding how these risk drivers affect the annuity cost. We employ local and global sensitivity methods. For local sensitivity, we derive closed form expressions for the differential importance measures of perturbed annuities and connect them to the entropy of the annuity cost. For global sensitivity, we compare variance-based, moment-independent sensitivity measures and Shapley effects. In particular, moment-independent sensitivity measures and Shapley effects are compared for the first time in the case of dependent risk factors. Our framework encompasses and extends several previous results on the sensitivity analysis of annuity models. From a methodological viewpoint, the techniques compared in this paper can support analysts in building annuity models and in verifying the impact of risk drivers in their models. Numerical results using the U.S. 1990 and the U.K. 1990–1994 mortality tables show that the demographic factor is the most important risk source in low-interest rate contexts. However, when uncertainty on the two risk sources is taken into account, the financial factor becomes the global key-driver of risk. Also, interactions among the two factors appear quantitatively significant.