The historical VaR is a non-paramet

The historical VaR is a non-parametric approach to estimate the VaR and ES from historical data over an estimation window. The VaR is a percentile, and there are alternative ways to estimate the percentile of a distribution based on a finite sample. One common approach is to use the prctile function. An alternative approach is to sort the data and determine a cut point based on the sample size and VaR confidence level. Similarly, there are alternative approaches to estimate the ES based on a finite sample.The hHistoricalVaRES local function on the bottom of this example uses a finite-sample approach for the estimation of VaR and ES following the methodology described in [5]. In a finite sample, the number of observations below the VaR may not match the total tail probability corresponding to the VaR level. For example, for 100 observations and a VaR level of 97.5%, the tail observations are 2, which is 2% of the sample, however the desired tail probability is 2.5%. It could be even worse for samples with repeated observed values, for example, if the second and third sorted values were the same, both equal to the VaR, then only the smallest observed value in the sample would have a value less than the VaR, and that is 1% of the sample, not the desired 2.5%. The method implemented in hHistoricalVaRES makes a correction so that the tail probability is always consistent with the VaR level; see [5] for details.