Modeling extremes | Vose Software

# Modeling extremes

Imagine that we have a reasonably large dataset of the impacts of natural disasters that an insurance or reinsurance company covers. It is quite common to fit such a dataset, or at least the high-end tail values to a Pareto distribution because this has a longer tail than any other distribution (excepting curiosities like the Cauchy, Slash and Lévy which have infinite tails but are symmetric). An insurance company will often run a stress test of a 'worst case' scenario where several really high impacts hit the company within a certain period. So, for example, we might ask what could be the size of the largest of 10 000 impacts drawn from a fitted Pareto(5,2) distribution modelling the impact of a risk in \$billion.

Order statistics tells us that the cumulative probability U of the largest of n samples drawn from a continuous distribution will follow a Beta(n,1) distribution. We can use this U value to invert the cdf of the Pareto distribution. A Pareto(q,a) distribution has cdf:

giving

Thus, we can directly generate what the value of the largest of 10 000 values drawn from this distribution might be in \$billion as follows:

=2/(1-VoseBeta(10000,1))^(1/5)

One can also determine, for example, the value for which we are 95% sure that the largest of 10 000 risk impacts will not exceed by simply finding the 95th percentile of the Beta distribution and use that in the above equation instead:

=2/(1-BETAINV(0.95,10000,1))^(1/5)

The same method can be applied to any distribution for which we have the inverse to the cdf. The principle can also be extended to model the largest set of values, or smallest, as shown in the example model below which allows you to simulate the sum of the largest several possible values (in this case six since the array function has been input to cover six cells). ModelRisk can perform these extreme calculations for all of its univariate distributions: use the Extreme Values window for this.

Example model  Pareto_extremes - determination of extreme values for 10 000 independent random variables drawn from a Pareto(5,2) distribution.

in ModelRisk