Actuarial pricing, capital modelling and reserving

Pricing Squad

Issue 30 -- December 2019

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.

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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.


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.


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