Estimating Value-at-risk with a Precision Measure 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 comments and suggestions and Amy Crews for guidance concerning kernel estimation. The views expressed herein are those of the authors and do not necessarily represent the views of the Chase Manhattan Bank or any of its staff or of the Office of the Comptroller of the Currency or members of its staff. In this paper we propose an alternative way to implement the historical simulation approach to Value-at-Risk (VaR) measurement, employing a non-parametric kernel quantile estimator (Sheather and Marron (1990)) of the probability density function (pdf) of the returns on a portfolio. Then we derive an expression for the pdf of any order statistic of the return distribution. Finally, because that pdf is not analytic, we employ numerical integration to obtain the moments of the order statistic, the mean being the VaR estimate, and the standard deviation allowing the construction of a confidence interval around the estimate. We apply this method to trading portfolios provided by a financial institution.