# Pricing Squad

## Welcome back to Pricing Squad!

Pricing Squad is the newsletter for fellow pricing practitioners and actuaries in general insurance. Today's issue is about Generalized Pareto Distribution.

## Get in touch

If you are looking for a contractor or need actuarial help in 2020, get in touch. Email@iwanik.co.uk.

The last Pricing Squad, in early 2019, explained how to use Pareto distribution which is a heavy-tailed severity distribution.

Nine out of ten times a simple Pareto distribution is sufficient to model heavy-loss tails. However, there are lines of business for which a Pareto tail is still not heavy enough, such as major environmental damage, US M&A lawyers, or catastrophic accidents involving fleets of trucks and trains.

Below is a typical tail of such a distribution function. You can see how the tail curves upward on the log-log scale. The more curved the log-log chart the heavier the distribution.

## GPD

For the most extreme cases you can rely on the Generalized Pareto Distribution with its extra tail parameter xi. CDF(x) = 1 - [1 + xi (x-mu) / sigma)] ^ (-1/xi).

If you select xi > 1, the distribution tail becomes so heavy it has no expected value. In this case we can only talk about limited expected value = mu + sigma / (1-xi) - sigma / (1-xi) * [1+xi (cap-mu) / sigma)] ^ (1-1/xi).

Let me know if you encounter a dataset for which a GPD tail is not heavy enough!

## How to estimate GPD

You can estimate parameters of the Generalized Pareto Distribution can be estimated using the maximum likelihood method. Use PDF(x) = 1/sigma * (1+xi (x-mu) / sigma) ^ (-1-1/xi) and Excel Solver to search for parameters mu, sigma and xi which maximise the likelihood.

Alternatively, you can use the maximum likelihood method introduced by Smith and extended by Grimshaw.

## How to simulate GPD

Once you have selected parameters mu, sigma and xi then you can use the inverted CDF to simulate from the Generalized Pareto Distribution. First, randomise a number p between 0 and 1. Then calculate invCDF(p) = mu + sigma [(1-p) ^ -xi - 1] / xi.

Done.

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