Contextual Conditional Value-at-Risk: Estimation and Optimization

Abstract

Conditional value-at-risk (CVaR) is a widely used measure for quantifying and controlling tail risk. In many applications, however, the loss distribution varies with observable contextual information, giving rise to contextual CVaR, which evaluates tail risk under a context-specific conditional distribution. Rather than estimating the conditional distribution from historical data, we exploit the Rockafellar–Uryasev representation to formulate both contextual CVaR estimation and optimization as optimization problems involving conditional expectations. Based on this formulation, we develop a unified data-driven predict-then-optimize framework. For a broad class of smoothing methods, the predicted objective admits a context-dependent weighted empirical formulation. When the weights are nonnegative and normalized, they induce a context-specific empirical probability distribution and preserve the convexity of the optimization problems. We study three representative methods, k-nearest neighbors, kernel smoothing, and kernel ridge regression, and establish consistency and explicit convergence rates for both estimation and optimization, thereby providing theoretical guarantees for the proposed framework. Numerical experiments in portfolio and inventory applications demonstrate the effectiveness of the proposed framework and support the theoretical findings.