Estimating VaR with Order Statistics

Abstract

“Value at Risk (VaR), despite its known shortcomings, has probably become the most widely used standard for assessing market risk exposure. One difficulty in using VaR that is not frequently considered explicitly, is that it must be estimated from observed data and is, therefore, subject to estimation error. A given VaR estimate should actually be thought of as a draw from a probability distribution around the true value. How accurate an estimate of the 1% tail of the returns distribution will be depends on the number of data points in the sample and the form of the returns distribution itself. In this article, Dowd shows how the theory of order statistics can be used to examine the sampling distribution of a VaR estimate. He finds that confidence intervals can be uncomfortably wide even for rather large sample sizes, and the problem is a lot worse for fat-tailed distributions than for the normal.”