Quantiles play an important role in modelling quality of service in the service industry and in modelling risk in the financial industry. The recent discovery that efficient simulation-based estimators can be obtained for quantile sensitivities has led to an intensive search for sample-path differentiation-based estimators for quantile sensitivities. In this paper, we present a novel approach to quantile sensitivity estimation. Our approach elaborates on the concept of measure-valued differentiation. Thereby, we overcome the main obstacle of the sample-path approach, which is the requirement that the sample cost have to be Lipschitz continuous with respect to the parameter of interest. Specifically, we perform a sensitivity analysis of the value at risk in financial models. In addition, we discuss an application of our sensitivity estimator to queueing networks.