Why Monte Carlo VaR Estimates Need Large Samples for Rare Mixtures
Article Quant Q&A · Author: Mh Aztec
Summary
The document explains why simulated estimates of a 99% value at risk can vary when the loss distribution combines a standard normal variable with a rare negative jump. The jump occurs with probability 0.009, so a modest simulation may contain few observations from that component. Since VaR is itself a tail quantile, the estimate depends on adequately sampling the observations around the relevant cutoff. The responses attribute the instability to finite sample size, the tail probability, and the distribution’s skew, rather than to mixing discrete and continuous variables. One response gives a rough event-count calculation: at 500,000 simulations, about 45 jump events are expected in the 1% tail under its simplifying assumption. Another illustrates how quantile estimates change with sample size and jump magnitude.
The discussion is explanatory rather than a formal convergence analysis. Its event-count argument assumes the jump component drives the quantile, and the example does not establish that 500,000 draws are universally needed. A response also suggests quasi-Monte Carlo as an alternative, but provides no comparative results supporting that suggestion.
Key ideas
- Tail quantiles can be unstable when a simulation produces very few observations near the target tail.
- A rare jump can make a mixture distribution’s quantile estimate sensitive to sample composition.
- Increasing the number of simulations generally improves empirical estimation, though the required count depends on the distribution and target quantile.
- The document’s event-count calculation relies on the simplifying assumption that the rare jump drives the VaR tail.
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Full text
# VaR estimate with Monte Carlo simlation
# VaR estimate with Monte Carlo simlation
i want to verify the theoretical VaR 99% for the following Random Variable:
\begin{align*} X=\epsilon + \nu, \end{align*} $\epsilon \sim \mathcal{N}(0,1)$, \begin{align*} \nu= \begin{cases}\begin{array}{ll}0 & \text{with prob.}\enspace 0.991 \\ -10& \text{with prob.}\enspace 0.009 \end{array} \end{cases}. \end{align*} So i poved that the theoretical 99% VaR is 3.088. Then i ran some monte carlo simulations in excel( $\epsilon$ as a standard normal and $\nu$ as a bernoulli and so on and used the quantile function). Now i went up to 20000 simulations in excel, but the VaR estimate was very sensible to different simulations , varying from 2-8, but i already saw that most of the estimates were close to the theoretical one. I coded the same monte carlo simulation in MATLAB and had to increase the simulation amount to 500000 in order to get a very stable and precise estimate.
Now i am interested in the reason, 500000 seems very high to me, i have never experienced a situation were it was necessary to run more than 10000 simlations. Is it because X is a mixture of a continous and discrete Random Variable? Thanks.
## Answer by rbm (score 4)
https://quant.stackexchange.com/a/33519
To answer you question "is it because X is a mixture of a continous and discrete Random Variable": the answer is no. The mean reasons are (1) the sample size (which is limited / countable) (2) the fact that you're trying to get the tail value and (3) the shape of the KDE (distribution).The theoretical value of 3.088 will be emprirically calculated if and only if you have sufficiently large sample size.
(@Jack JackGuRae says It is about simulations are not actually random which isn't exactly true, as the pseudo-random number generators use algorithms that do ensure that big enough samples are properly distributed, i.e. simulating 1000 dice rolls will give you p=1/6 for each side to let's say 3 d.p.)
(@msitt is spot on with the tail remark)
As you've mentioned before, it is true that for some of the simulations, you needed considerably smaller sample size, but in this case you are trying to sample from a highly left-skewed distribution. Consider following code to plot the KDE:
```
sims <- 10e6
errors <- rnorm(sims)
v <- ifelse(runif(sims) <= 0.009, -10.0, 0.00)
x <- v + errors
plot(density(x), col = 2, lwd = 2)
abline(v=unname(quantile(x,probs=c(0.01))))
```
where the vertical line is the 1% quantile. Chances of having a small sample with this particular value for 1% quantile are very small indeed.
Further example where we parametrize the quantile calculation based on the number if simulations and the Bernoulli mode and plot the quantiles based on the number of sims:
```
set.seed (3134)
quant <- function(sims, tailmode)
{
errors <- rnorm(sims)
v <- ifelse(runif(sims) <= 0.009, tailmode, 0.00)
x <- v + errors
return(unname(quantile(x,probs=c(0.01))))
}
sims <- seq(100, 20000, len=20)
par(mfrow=c(2,2))
plot(sims, Vectorize(quant, c("sims","tailmode"))(sims,-10), col = 2, pch=16, ylim = c(-9, -1), ylab = "", xlab="", main="-10")
plot(sims, Vectorize(quant, c("sims","tailmode"))(sims,-5 ), col = 2, pch=16, ylim = c(-9, -1), ylab = "", xlab="", main="-5")
plot(sims, Vectorize(quant, c("sims","tailmode"))(sims,-1 ), col = 2, pch=16, ylim = c(-9, -1), ylab = "", xlab="", main="-1")
plot(sims, Vectorize(quant, c("sims","tailmode"))(sims, 0 ), col = 2, pch=16, ylim = c(-9, -1), ylab = "", xlab="", main="0")
```
gives
which clearly shows that the less skewed your distribution is (i.e. the Bernoulli mode increasing), the faster the calculated 1% quantile converges to its model value.
## Answer by msitt (score 1)
https://quant.stackexchange.com/a/33515
The reason you need to run so many simulations is because you are trying to estimate a tail property which has a very small chance of happening.
Let's do a quick back of the envelope calculation: The 99% VaR will occur 1% of the time. With the way your binomial variable is set up let's assume only $\nu=-10$ contributes to the VaR. This happens with probability 0.009. If you run 1% * 0.009 = 11,111 simulations, you will only expect to get 1 tail event. With 500,000 simulations, you get 45 tail events, which is pretty much exactly what I would expect to get convergence.
## Answer by Woraphon T (score -1)
https://quant.stackexchange.com/a/33512
It is not about a mixture of a continuous and discrete Random Variable.
It is about simulations are not actually random, even using 50,000 trials. That's why you get varying results. (It exhibits high discrepancy)
Simulating random variable from pseudo-random function generator, very large simulations is needed to ensure convergence.
Try using quasi-Monte Carlo Method. It uses quasi-random sequences. Plus, it is readily available in MATLAB. look here
Actually, with QMC, you may only need a few thousands simulations.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.