May 01

Extreme Value Theory (EVT) - Practice Question (Par 3+)

by David Harper, CFA, FRM, CIPM

We want to fit a distribution to extreme operational losses (operational losses with "low frequency but high severity"). However, we do not want to set the threshold (i.e., the loss level beyond which is defined "extreme") and, conveniently, our loss data-set is rather well distributed over time. Each historical interval contains a representative maxima.

Questions:

Which distribution might we use?
What is the key parameter?
How do we change the parameter to simulate fatter tails?
What is the impact of the this parameter change on the second moment of the EVT distribution?

(don't peek until you try)

Answers:

The generalized extreme value (GEV) distribution fits the block-maxima approach. The block-maxima approach is deemed "less modern" than peaks over threshold (POTS) but does offer the advantage that we do not need to choose a threshold.
The key parameter is the shape parameter (Greek beta). When the parent distribution (the overall operational loss distribution) is fat-tailed, shape > 0 which indicates a Frechet distribution. Financial returns, therefore are GEV-Frechet where higher shape implies fatter tail.
See above
This is tricky at first because the GEV is the "child" distribution: it characterizes the tail. Say the extreme losses occur with 1% probability; i.e., the CDF characterizes non-extreme losses as CDF P [X < extreme] = 99%. This is where the "child" GEV starts. The 50th percentile of the GEV is not the 50th percentile of the original "parent" operational loss distribution, it's the 50th percentile of the child distribution that starts over in the tail (at 99% percentile of the parent). As such, a higher shape param implies fatter tails which is reflected by a less disperse GEV (child distribution). Fatter parent tails are achieved with a less disperse child distribution.