Differential quantile-based sensitivity in discontinuous models

Differential sensitivity measures provide valuable tools for interpreting complex computational models, as used in applications ranging from simulation to algorithmic prediction. Taking the derivative of the model output in direction of a model parameter can reveal input–output relations and the relative importance of model parameters and input variables. Nonetheless, it is unclear how such derivatives should be taken when the model function has discontinuities and/or input variables are discrete. We present a general framework for addressing such problems, considering derivatives of quantile-based output risk measures, with respect to distortions to random input variables (risk factors), which impact the model output through step-functions. We prove that, subject to weak technical conditions, the derivatives are well-defined and we derive the corresponding formulas. We apply our results to the sensitivity analysis of compound risk models and to a numerical study of reinsurance credit risk in a multi-line insurance portfolio. • Local sensitivities calculated when input–output relation is discontinuous. • Quantile-based risk measures: Value-at-Risk, Expected Shortfall applied to output. • We introduce differential sensitivities to discrete inputs factors. • Sensitivity measures introduced are suitable for credit risk models. • Numerical case study on reinsurance credit default.