Estimation of Conditional Value-at-Risk in Linear Model

The conditional value-at-risk ( \(\textsf{CVaR}\) ) represents a popular risk measure often exploited e.g. within portfolio optimization. The situation with a nuisance linear regression is considered here; in other words, we do not observe directly the loss Z of interest, but only \(Y=\beta _0+\textbf{X}{\boldsymbol{\beta }}+Z\) , where the covariates are not under our control. We propose a novel estimator of \(\mathsf CVaR(Z)\) based on the averaged two-step regression quantile combined with an R-estimate of regression parameters.