Approximations for the Value-at-Risk Approach to Risk-Return Analysis

An evergreen debate in Finance concerns the rules for making portfolio hedge decisions. A traditional tool proposed in the literature is the well-known standard deviation based Sharpe Ratio, which has been recently generalized in order to involve also other popular risk measures p, such as VaR (Value-at-Risk) or CVaR (Conditional Value at Risk). This approach gives the correct choice of portfolio selection in a mean-p world as long as p is homogeneous of order 1. But, unfortunately, in important cases calculating the exact incremental Sharpe Ratio for ranking profitable portfolios turns out to be computationally too costly. Therefore, more easy-to-use rules for a rapid portfolio selection are needed. The research in this direction for VaR is just the aim of the paper. Approximation formulae are carried out which are based on certain derivatives of VaR and involve quantities similar to the skewness and kurtosis of the random variables under consid-eration. Starting point for the approximations is the observation that the partial derivatives of portfolio VaR with respect to the portfolio weights are just the conditional expectations of the asset returns given that the portfolio return equals VaR. Since the conditional expec-tation of a random variable Y given another random variable X can be considered the best possible regression of Y versus X in least squares sense, the idea is to replace the conditional expectation by polynomial regression or, more generally, by finite-dimensional regression of Y versus X. In case of the variables obeying an elliptical joint distribution, the resulting approximation formulae coincide with the exact formula for the standard deviation taken as risk measure. By means of a number of numerical examples and counter-examples the properties of the formulae are discussed.