Why Risk is so Hard to Measure

ES and VaR precision plots

These are the same results as in the table but presented graphically. All for 99%.

The notable result is that the 99% confidence bound is quite asymmetric, and that on the upside it is much larger than a asymmetric bound coming from the standard errors while on the downside the opposite result obtains


The plot shows the true risk, along with the mean value across the simulations, and a range of tail thicknesses (), as well as the two standard error bounds and the 99% empirical confidence bounds (Q99). We show two sample sizes, 300 days and 50 years (12500 days) to show the small sample and asymptotic properties.


VaR

Student-t

Pareto


ES

Student-t

Pareto