# Monte-Carlo Quantile Confidence Intervals

I am looking at a Monte-Carlo engine with N simulations. This Monte-Carlo engine builds a distribution from which I would like to read the 99%-tile (called the VaR). The problem can be interpreted as Monte-Carlo VaR (value at risk). I use a confidence interval on this quantile obtained with 2 different methods:

1. Asymptotic CIs ( see Glasserman, Monte Carlo methods, p491) : $$\hat{x_p} \pm z_{\alpha/2} \frac{\sqrt{p(1-p)}}{f(x_p)\sqrt{N}}$$ where $f(x_p)$ is the density function and $x_p$ the quantile (VaR). To estimate the $f(x_p)$ one can use kernel density estimation or finite-differences.
2. Non-parametric CIs: Another approach is to use batching where the N samples are divided in non overlapping batches $b$ of size $\frac{N}{b}$ and we compute the empirical mean and standard deviation across batches and infer a confidence interval.

Having implemented both methods they give similar results, the batching providing slightly wider confidence intervals. Note that I am constructing 95% Confidence intervals. However, when I repeat the MC simulation $M$ times ($M$ MC runs) and look at the percentage of VaR values from this M runs that land into a given confidence interval (say the confidence interval obtained on the first run), this percentage is much higher than 5%.

Shouldn't I expect only around 5% of VaRs from the MC runs finishing outside any given 95% - confidence interval ? Or I am missing something.

Your help would be greatly appreciated.

• How large is $N$? These are asymptotic confidence intervals, so if $N$ is small, the intervals will not have the right coverage. Are the sample Monte Carlo or Markov chain Monte Carlo? – Greenparker Jul 2 '17 at 7:47