In risk analysis, sensitivity measures quantify the extent to which the probability distribution of a model output is affected by changes (stresses) in individual random input factors. For input factors that are statistically dependent, we argue that a stress on one input should also precipitate stresses in other input factors. We introduce a novel sensitivity measure, termed cascade sensitivity, defined as a derivative of a risk measure applied on the output, in the direction of an input factor. The derivative is taken after suitably transforming the random vector of inputs, thus explicitly capturing the direct impact of the stressed input factor, as well as indirect effects via other inputs. Furthermore, alternative representations of the cascade sensitivity measure are derived, allowing us to address practical issues, such as incomplete specification of the model and high computational costs. The applicability of the methodology is illustrated through the analysis of a commercially used insurance risk model.