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 _nancial markets. Considerable amount of research was dedicated during recent years to development of acceptable methods for evaluation of this risk measure. In this paper, we present a method of calculating the portfolio which gives the optimal VaR among those, which yield at least some specified expected return. This method allows to calculate the mean-VaR efficient frontier. The method is based on approximation of historic VaR by smoothed VaR (SVaR) which filters out local irregular behavior of historic VaR function. Moreover, we compare the VaR as a risk measure to other well known measures of risk such as the conditional value at risk (CVaR) and the standard deviation. It turns out, that the corresponding efficient frontiers are quite different. An investor, who wants to control his VaR should not look at portfolios lying on other than the VaR efficient frontier, although the calculation of this frontier is algorithmically more complex compared to other frontiers.