Why Risk is so Hard to Measure
Tables and plots
The results in the paper are a subset of the tables and figures here. The tables here contain results from the Pareto distribution and cases were both VaR and ES have the same 99% probability.
We simulated from the student and the Pareto, calculating VaR and ES of both 97 5% probability and 99% probability. We repeat this $$10^7$$ times, so can use those $$10^7$$ realizations of the risk measures to formulate some interesting statistics. In particular, we report the mean, standard error and the 99% empirical confidence interval. The last is calculated by taking the 0.5% and 99.5% quartiles across the realized risk measurements.
The results in the paper are reported slightly differently, there we adjusted the numbers by the true values, but not below.