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 speci ed expected return. This method allows to calculate the mean-VaR ef cient frontier. The method is based on approximation of historic VaR by smoothed VaR (SVaR) which lters 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 ef cient frontiers are quite different. An investor, who wants to control his VaR should not look at portfolios lying on other than the VaR ef cient frontier, although the calculation of this frontier is algorithmically more complex compared to other frontiers.