Value at Risk and Expected Shortfall: Improved Calculation Based on the Power Transformation Method

This article describes a new methodology to compute the value at risk and the expected shortfall using a power transformation technique. The methodology is an improvement of a recent method employing Johnson's system of distributions and is based on the idea of matching exactly the first four moments of the target portfolio distribution. The performance of this method is investigated in the context of jump-diffusion models with lognormal jumps. The authors show that it yields valid densities and quantile functions and that it is also superior to other techniques proposed in the literature, such as Cornish Fisher, Gram-Charlier, and Johnson distributions, in terms of relative error to the analytical benchmark. The power transformation is also applied to forecasting VaR for three equity indexes at different critical levels. The backtesting results support the validity of the newly proposed method.