Bootstrap Confidence Intervals for Monte Carlo CVaR
Summary
The document addresses uncertainty in a Monte Carlo estimate of conditional value at risk (CVaR), which depends both on estimating a tail quantile and averaging outcomes beyond that threshold. It recommends bootstrap confidence intervals: repeatedly resample the simulation outcomes, recompute CVaR for each resample, and use the resulting distribution of estimates to assess uncertainty and construct intervals.
This approach is presented as straightforward and applicable across different cash-flow settings, but computationally intensive. The answer also cautions that a normal-approximation interval for VaR can perform poorly depending on sample size and the P&L distribution, and points to empirical-distribution methods as an alternative. Averaging observations beyond an estimated quantile can introduce bias; the answer says this may be small in practice but depends on the case, and that bootstrap techniques can also help address it. No numerical example or comparative performance evidence is provided.
Key ideas
- Use repeated bootstrap resamples to estimate the sampling distribution of CVaR.
- Construct confidence intervals from the resulting distribution of CVaR estimates.
- Bootstrap methods are flexible but require substantial computation.
- Normal-approximation intervals for VaR may be unreliable for some samples or P&L distributions.
- The usual tail-average CVaR estimator can be biased, with the size of bias depending on the setting.
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# Confidence Interval on Monte-Carlo-CVaR
# Confidence Interval on Monte-Carlo-CVaR
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 by g g (score 3, accepted)
https://quant.stackexchange.com/a/34970
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)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.