How valuable is your VaR? Large sample confidence intervals for normal VaR

Abstract

Little is known about the distribution of the ‘value-at-risk’ (VaR) estimate and the associated estimation risk. In the case of the normal VaR, the key problem comes from the fact that it is estimated using a couple of parameters whose estimates are distributed differently. Previous research has either neglected uncertainty around the mean parameter, or resorted to simulations. By contrast, this paper derives analytical results for the normal VaR with the help of asymptotic theory and the so-called ‘delta method’. Properties of the estimation errors are then explored in detail and the VaR estimation risk is broken down into its various components. It is then shown, among other things, that the fraction of error owing to mean uncertainty is limited in a prudential context. In other words, the approximate approach defended by Jorion and Chappell and Dowd is shown to still be relevant.