Value-at-Risk in Portfolio Optimization: Properties and Computational Approach ⁄

Value-at-Risk (VAR) is an important and widely used measure of the extent to which a given portfolio is subject to risk present in financial markets. In this paper, we present a method of calculating a portfolio that gives the optimal VAR among those which yield at least some specified expected return. This method allows us to calculate the mean-VAR-efficient frontier. The method is based on the approximation of historical VAR by smoothed VAR (SVAR), which filters out local irregular behavior of the historical VAR function. Moreover, we compare VAR as a risk measure to other well-known measures of risk, such as conditional value-at risk (CVAR) and the standard deviation. We show that the resulting efficient frontiers are quite different. An investor who wants to controls his or her VAR should not look at portfolios lying on other than the VAR efficient frontier, although the calculation of this frontier is algorithmically more complex than other frontiers. We support this conjecture by presenting the results of a large-scale experiment with a representative selection of stock and bond indices from developed and emerging markets that involved the computation of many thousand VAR-optimal portfolios.