Improving Value-at-risk Estimates by Combining Kernel Estimation with Historical Simulation

We thank seminar participants at the OCC and the 1996 Chicago Fed Bank Structure Conference for their comments. We also thank Rene Stulz for his suggestions. The views expressed herein are those of the authors and do not necessarily represent the views of the Office of the Comptroller of the Currency or members of its staff. Address correspondence to Barry Schachter, mail stop 6-8, In this paper we develop an improvement on one of the more popular methods for Value-at-Risk measurement, the historical simulation approach. The procedure we employ is the following: First, the density of the return on a portfolio is estimated using a non-parametric method, called a Gaussian kernel. Second, we derive an expression for the density of any order statistic of the return distribution. Finally, because the density is not analytic, we employ Gauss-Legendre integration to obtain the moments of the density of the order statistic, the mean being our Value-at-Risk estimate, and the standard deviation providing us with the ability to construct a confidence interval around the estimate. We apply this method to trading portfolios provided by a financial institution.