Autoregressive conditional moments in VaR estimate with Gram‐Charlier and Cornish‐Fisher expansions

This paper proposes a model to compute value at risk by means of Gram‐Charlier and Cornish‐Fisher expansions. In this model VaR is a function of conditional mean, volatility, skewness and kurtosis where all the moments are computed via a time‐varying dynamic. An appropriate form of Gram‐Charlier density is used to estimate the parameters of the equations proposed for the first four autoregressive conditional moments. Since skewness and kurtosis appear directly as parameters in the functional form of the density, it is possible to estimate simply the third and fourth moments with the maximum likelihood method. By interpreting the VaR as the quantile of future asset values conditional on current information, a Cornish‐Fisher expansion is used to compute VaR as a function of the first four conditional moments that appear directly in the VaR formula. The main goal of the present analysis is to confirm some stylised facts of financial data such as volatility clustering, asymmetry and fat‐tails. An evaluation of the predictive performance of four conditional moments in VaR computation context is provided in the last part of the paper.