Finance Research Letters 1 (2004) 241–249 www.elsevier.com/locate/frl Bias of a Value-at-Risk estimator Yong Bao, Aman Ullah ∗ Department of Economics, University of California, Riverside, CA 92521, USA Received 17 March 2004; accepted 28 July 2004 Available online 15 September 2004 Abstract We develop the analytical second-order bias of a Value-at-Risk estimator based on an ARCH(1) volatility specification when the parameters are estimated by the method of quasi maximum like- lihood. We show that the bias results from two sources: assumption on the distribution of the standardized residuals and the parameter estimation error.  2004 Elsevier Inc. All rights reserved. Keywords: Value-at-Risk; Second-order bias 1. Introduction The Value-at-Risk (VaR) has received great attention from both regulatory and academic fronts as a measure of the risk exposure of the underlying asset or portfolio. In essence, it can be regarded as a quantile estimator. Loosely speaking, the method of calculating VaR can be classified into three types: parametric, nonparametric, and extreme-value-theory (EVT) based ones. The parametric one makes some explicit distributional assumption on the residuals and some functional form on the conditional second moment of the series, usually the GARCH-family models. The nonparametric approach includes the historical simulation, the Monte Carlo simulation, and the nonparametric quantile methods. The EVT method aims to model only the tail parts (extreme values) instead of the whole distribution. A practical question is how to choose an appropriate VaR estimator given the loss function * Corresponding author. E-mail addresses: yong.bao@email.ucr.edu (Y. Bao), aman.ullah@ucr.edu (A. Ullah). 1544-6123/$ – see front matter  2004 Elsevier Inc. All rights reserved. doi:10.1016/j.frl.2004.07.001