Value-at-risk levels implied by risk estimators drawn from historical data

Frederik S. Herzberg

Abstract

ABSTRACT

Previous research on the accuracy of value-at-risk (VaR) estimators has mostly concentrated on statistical features of the estimators themselves, entailing considerable mathematical sophistication. We propose a more accessible approach to analyzing VaR or expected shortfall (ES) estimators, by studying the VaR level to which, under a given distribution of the losses, the estimate corresponds: under natural assumptions on the portfolioincrement distributions, every VaR (ES) estimator induces a [0, 1]-valued random variable, called the implied VaR level. For VaR (ES) estimators drawn from historical order statistics, we derive integral formulas for the moments of the implied VaR levels and evaluate them for practically relevant special cases. For analytic portfolio-increment densities and estimators that analytically depend on historical order statistics, the moments of the implied VaR level are multiple integrals of analytic functions, which in special cases can be simplified further.