Equal Risk Contribution portfolios for CVaR and CVaR-deviation risk measures

Equal Risk Contribution (ERC), also called Risk Parity (RP), is a strategy for asset allocation that aims at equally sharing the risk among all the assets of the selected portfolio. In this paper we propose new developments of the ERC approach using Conditional Value-at-Risk (CVaR) and CVaR-Deviation as risk measures. Under appropriate conditions, we provide a way to ?nd CVaR and CVaR-Deviation ERC portfolios as solutions of a convex optimization problem. For asset allocation models, an important issue concerns the stability of the selected portfolios w.r.t. errors in the estimates of the input model parameters. This is a critical point, since the optimization phase in portfolio selection models tends to amplify these errors and thus cause instability of the optimal portfolio weights. We perform an analysis to examine the sensitivity to estimation errors of Minimum-Risk and ERC portfolios with CVaR and CVaR-Deviation, both on arti?cial and real investment universes. As expected, the estimation errors for the parameters needed to implement portfolio selection models, particularly those with CVaR, tend to rapidly increase with the number of assets in the market when the length of data is limited. Therefore, we also propose a way to tackle this issue representing the future assets returns by a simulation model called Historical Filtered Bootstrap. We then perform an empirical analysis to determine the best range for the number of simulated scenarios, taking into account the performances of the portfolios provided by the models, the sensitivity to input errors, and the computational burden.