Abstract

In risk management, an internal model consists of three elements: (i) a random vector of input risk factors, (ii) a real valued aggregation function, and (iii) the output, which is a random variable obtained by applying the aggregation function to the vector of risk factors. Sensitivity analysis often requires evaluation of changes in the distribution of the output, when the distribution of risk factors is varied with reference to a baseline input distribution. We introduce a novel sensitivity measure termed cascade sensitivity, which quantifies the extent to which small changes (stresses) in an individual risk factor affect the output risk measure. The proposed method captures the direct impact on the output, as well as indirect effects via other risk factors that are dependent with the one being stressed. In this way, the dependence between risk factors is explicitly taken into account. Alternative representations of the cascade sensitivity measure are provided, requiring either knowledge of the aggregation function's gradient or additional evaluations of the model under alternative distributional assumptions. This is of particular interest in applications, where the distribution of the output is typically determined via simulation methods.  Furthermore we give a representation of the proposed sensitivity measure that can be calculated from a single simulated sample starting from the baseline model, without the need to simulate under a different model specification or to explicitly study the properties of the aggregation function. This makes the proposed method attractive for practical applications, as is illustrated through numerical examples.