Sensitivity Analysis of Risk Measures for Discrete Distributions

We consider the computation of quantiles and spectral risk measures for discrete distributions. This accounts for the empirical distributions of portfolio returns or outcomes of Monte Carlo simulations. We study the di¤erentiability of quantiles with respect to portfolio allocation. We show that quantiles and spectral risk measures are piecewise linear with respect to portfolio allocation. We also provide differentiability conditions for a given allocation and relate the gradient to conditional expectations. Eventually, we extend our results to spectral or distortion risk measures.