Estimating Monte Carlo Value-at-Risk Sampling Error
Summary
The document presents ways to assess how the number of Monte Carlo scenarios affects an estimated value at risk. One approach repeats the VaR calculation many times at a fixed scenario count, then uses the sample standard deviation of those estimates as an empirical measure of estimation variability. Repeating that process across scenario counts produces a plot of scenario count against variability. A second suggestion is to add simulated paths progressively and monitor changes in VaR, stopping when recent changes fall below a chosen tolerance.
The example uses simulated normally distributed returns to illustrate the repeated-estimation approach, while another answer describes returns as a linear combination of factor realizations. These are practical diagnostics, not a universal closed-form error formula. The repeated estimates must be comparable and independently generated for the standard deviation to be informative; the stopping rule depends on a chosen tolerance and can be unstable. The document does not establish a guaranteed confidence interval or convergence rate for arbitrary distributions or risk models.
Key ideas
- Repeated VaR estimates at a fixed scenario count can be summarized by their sample standard deviation.
- Repeating the experiment at different scenario counts shows how estimated variability changes with simulation size.
- An incremental approach can monitor recent VaR changes and stop when they fall below a chosen tolerance.
- The proposed diagnostics do not provide a universal error formula or guarantee convergence.
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# How to Calculate a Monte Calo VaR estimation error
# How to Calculate a Monte Calo VaR estimation error
I'm performing a Monte Carlo to calculate value at risk (with a 3 dimension risk factor) Now, I would like to calculate the error of the estimation of the VaR with respect to the number of simulations (drawing a graph of estimation error with respect the number of simulations)
What is the formula for the error on the VaR?
## Answer by user25064 (score 4)
https://quant.stackexchange.com/a/10351
- Do $N$ MC simulations of $M$ samples, calculating your estimate of VaR for each one $\{\widehat{VaR}_i\}_{i=1}^N$ and you now have an IID sample!
- Take the sample (or unbiased) standard deviation for your estimate of VaR (this is probably what you mean by error) $SD(\widehat{VaR})=\sqrt{\frac{1}{N-1} \sum_{i=1}^N (\widehat{VaR}_i - \overline{VaR})^2}$ and of course $\overline{VaR}=\frac{1}{N}\sum_{i=1}^N\widehat{VaR}_i$
- Increase $M$ to get your plot, plot $M$ against $SD(\widehat{VaR})$ for each value $M \in [\underline{M}, \overline{M}]$ you might want to use something like $\underline{M}=50$ and $\overline{M}=1000$ depending on the application.
Edit There probably are more tractable things to do but by the fact that OP is already in Monte-Carlo world, this is the Monte-Carlo answer.
Edit 2
```
N = 1000
M = seq(50, 1000, by=10)
VaRstdevs = rep(0, length(M))
i=1
for(nscenarios in M) {
varsample = rep(0, N)
for(sim in 1:N) {
samp = rnorm(nscenarios, 0, 0.3/sqrt(252)) # 30% annualized sd MC sim
varsample[sim] = -1.0*quantile(samp, 0.05) # VaR 95%
}
VaRstdevs[i] = sd(varsample)
i=i+1
}
plot(M, VaRstdevs)
```
## Answer by Aksakal almost surely binary (score 2)
https://quant.stackexchange.com/a/10382
Let's say your return realization for path $i$ is $r_i = \beta\cdot f_i$, where $f_i=(f_{1i}, f_{2i}, f_{3i})$ - factors realizations, and $\beta$ - factor coefficients. So, your VaR is $VaR=percentile(r_i,\alpha)$, where $\alpha$ - confidence.
The simplest Monte Carlo stopping criterion is to keep adding paths $i$ and computing VaR on the growing sample until VaR "stops changing". For instance you can keep track of the MAX change in VaR during the last N paths, and wait until it becomes smaller than the required tolerance.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.