A relative robust approach on expected returns with bounded CVaR for portfolio selection

A robust optimization model to find a stable investment portfolio is proposed under twofold uncertainty sources: the random nature of returns for a given economic scenario which is in itself unknown. Our model combines expected returns together with risk and regret measures in order to find a solution ensuring acceptable returns while the investor is protected from the market volatility. More formally, we formulate a model that minimizes the maximum regret on the expected returns while the conditional value-at-risk is upper bounded under different scenario settings. Several mathematical formulations are analyzed. Duality relations drive us to obtaining bounds on the optimal objective value of the problem in order to develop a cutting plane approach. We show experimentally that, despite the large number (hundreds of thousands) of constraints and variables of the resulting problem, an optimal portfolio can be found in a few seconds. Finally, our model is tested in a financial decision making environment by simulating its application in different markets indexes and under different underlying economic conditions. It will be seen that using scenarios usually improves the realized portfolio returns.