Continuity of VaR and Continuous Differentiability of CVaR under Decision-Dependent Losses

Abstract

Value-at-risk (VaR) and conditional value-at-risk (CVaR) are widely used in risk-aware optimization and equilibrium models. When the loss depends on a decision variable, the induced distribution, the VaR threshold, and the CVaR tail set all change with the decision. This makes the regularity of the VaR and CVaR maps nontrivial. We give simple sufficient conditions under which the VaR map is continuous and the corresponding CVaR map is continuously differentiable. The assumptions are local around the VaR level and rely on dominated pathwise differentiability of the scenario-wise loss. We also derive the CVaR gradient formula, thereby justifying first-order analysis for decision-dependent tail-risk models.