Robust VaR and CVaR optimization under joint ambiguity in distributions, means, and covariances

We develop robust models for optimization of the VaR (value at risk) and CVaR (conditional value at risk) risk measures with a minimum expected return constraint under joint ambiguity in distribution, mean returns, and covariance matrix. We formulate models for ellipsoidal, polytopic, and interval ambiguity sets of the means and covariances. The models unify and/or extend several existing models. We also show how to overcome the well-known conservativeness of robust optimization models by proposing an algorithm and a heuristic for constructing joint ellipsoidal ambiguity sets from point estimates given by multiple securities analysts. 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. Finally, applying the model to the active management of portfolios of sovereign credit default swaps (CDS) from Eurozone core and periphery, and Central, Eastern and South-Eastern Europe countries, we illustrate that investment strategies using robust optimization models perform well out-of-sample, even during the eurozone crisis. We consider both buy-and-hold and active management strategies.