Confidence Interval on Monte-Carlo-CVaR

Question

I use the Monte-Carlo Simulation for the computation of VaR and CVaR and wish to compute the 95% Confidence Interval of my result(not the confidence level of VaR). In the case of VaR this is simple the confidence interval on a quantile given by the formula $$\frac{r}{100}=\alpha+\sqrt\frac{\alpha(1-\alpha)}{m} N^{-1}(\frac{1-\beta}{2})$$ where $\alpha$ denotes the confidence level of VaR(the quantile) and $\beta$ the desired confidence Interval in percentage.

Question: How can I calculate the confidence Interval for CVaR? As the CVaR is a conditional expectation, I have two statistical errors, one on the quantile and one on the computation of expectation using Monte-Carlo. How can I find a formula for this.

Answer 1

One: Your VaR CI relies on normal approximation and might be (very) bad depending on the number of samples and the target function (P&L). Often it is better to use the exact approach based on the empirical distribution (see here: https://stats.stackexchange.com/a/284970/8298)

Two: To estimate CVaR confidence intervals you may use bootstrap confidence intervals (see here). The advantage of bootstrap is that it is simple to understand and implement and works for all kinds of cash-flows. This comes with a single disadvantage, it is computationally intensive.

In essence you calculate repeated independent estimates of your CVaR. This produces an empirical distribution of estimates, from which you can calculate standard deviation, confidence intervals and all kinds of statistics.

Be aware the "standard" CVaR estimator (average of everything beyond the quantile) is biased. I found the bias small/irrelevant in practice but then again that might depend. You can correct this bias with bootstrap as well (nicely explained here)