# How many data points are required to perform a fitting of GPD?

A friend of mine told me that their firm is using Extreme Value Theory (EVT) to compute value of the Expected Shortfall 99% of a portfolio for their asset allocation process. To do so, they try to fit the parameters of the Generalized Pareto Distribution using the historical sample of their portfolio, and they use a formula to get the value of $ES_{0.99}$.

However, I they are using monthly data points, which means that the size of the sample is pretty (very) limited. When I asked what the minimum number of point required to performed the fitting was, I was told that their algorithm was asking for a minimum of 10 points.

I am wondering

1. if that's enough?
2. how could we compute some kind of quantitative value indicating "how wrong" the fitting might be given the give size of the sample $N$? (this method could be specific to GPD, but of course a generic approach would be great).
• They are definitely going to get poor stability in their estimates of tail risk. Of course, the result only has to be better than what they had before which is almost certainly a low bar. – Brian B Mar 26 '12 at 20:14

Playing around with some random numbers from the GPD, I'm not convinced that a sample of 10 will give anything like useful results. The code for generation was

require("gPdtest")
reps = 10000
ValuesAct = rgp(reps, 0.5,0.5)
plot(density(ValuesAct))
quantile(ValuesAct, probs=0.999)

loops = 10000
results = matrix(rep(NA,2*loops), ncol = 2)
colnames(results) = c("shape", "scale")

for(i in 1:loops)
{
ValuesSample = sample(ValuesAct,10,replace=F)
results[i,] = gpd.fit(ValuesSample, "amle")
}

plot(density(results[,2]))
lines(density(results[,1]), col = "green")
abline(v=0.5)
legend("topright", c("Shape", "Scale"), col = c("green", "black"), lwd=1)


Given a sample you can estimate the parameters and calculate the standard deviation of these estimations. You could use this as a measure of quality.

Creating a measure given only the sample size $N$ seems very hard to me since the quality of the estimation may depend on the values of the parameters.