Equivalence of Robust VaR and CVaR Optimization

We show that robust optimization of the VaR and CVaR risk measures with a minimum return constraint under distribution ambiguity reduce to the same second order cone program. We use this result to formulate models for robust risk optimization under joint ambiguity in distribution, mean returns and covariance matrices, under ellipsoidal ambiguity sets. We also obtain models for robust VaR and CVaR optimization for polytopic and interval ambiguity sets of the means and covariance. The models unify and/or extend several existing models. We also propose an algorithm and a heuristic for constructing an ellipsoidal ambiguity set from point estimates given by multiple securities analysts, and show how to overcome the well-known conservatism of robust optimization models. Using CDS spread return data from eurozone crisis countries we illustrate that investment strategies using robust optimization models perform well even out-of-sample. Finally, using a controlled experiment we show how the well-known sensitivity of CVaR to mis-specifications of the first four moments of the distribution is alleviated with the robust models.