Differential Quantile-Based Sensitivity in Discontinuous Models

TL;DR

This paper develops a framework for calculating derivatives of quantile-based risk measures in models with discontinuities or discrete inputs, enabling better sensitivity analysis of complex, real-world models.

Contribution

It introduces a general method for defining and computing derivatives of quantile-based measures in discontinuous models, filling a gap in sensitivity analysis techniques.

Findings

Derivatives of quantile-based risk measures are well-defined under weak conditions.

Formulas for derivatives are derived for models with step-function impacts.

Application to insurance risk models demonstrates practical utility.

Abstract

Differential sensitivity measures provide valuable tools for interpreting complex computational models 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 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.