Portfolio Optimization under VaR Constraints Based on Dynamic Estimates of the Variance-Covariance Matrix

Abstract

The concept of Value at Risk (VaR) is eminently intuitive and easy to interpret. However, there are two major problems in its practical application: reliable estimations are difficult to perform and using the VaR as a constraint in portfolio optimization causes computational problems. Both problems are taken into account in the present application. First, the VaR based on estimates of the conditional covariance matrix with the ”Principal Components GARCH model” (PC GARCH model) is calculated. Second, recent advances in heuristic optimization (a modified version of Memetic Algorithms) are applied in order to deal with the computational problems. The computational study indicates that the proposed method for estimating the conditional covariance matrix, the PC GARCH model, results in much better approximations of the conditional volatility structure than a simple historical estimate. This advantage in modelling is exploited by the optimization algorithm to identify portfolios with higher expected return given a fixed VaR constraint. However, adjusting the portfolio to the dynamic approximations of the conditional volatility structure also results in some overconfidence with regard to the risk constraint.