May 01

Extreme Value Theory (EVT), The Idea: Part 2

by David Harper, CFA, FRM, CIPM

FRM | Risk |

We previously looked at two approaches to estimating losses under an extreme value theory (EVT) approach. Peaks over threshold (POT) collects losses greater than some threshold...

 

while block maxima divides the dataset into equally-spaced time intervals and collects the largest loss ("maxima") within each interval:

 

Either way, the dataset of extreme losses can be characterized by a distribution...

But instead of the normal distribution, models under the POT approach tend to be fitted with the generalized Pareto distribution (GPD). Models under the block maxima tend to be fitted with the generalized extreme value distribution (GEV). The Kalyvas reading gives the GEV distribution as a cumulative distribution function (CDF) rather than a probability density function (pdf); by convention, a CDF is capitalized: 

 

Okay, so we have a distribution that potentially fits the extreme tail of losses. The distribution has three parameters: location, scale and shape. Where are location and scale? They are "inside" the y. The (y) is normalized variable. The (y) is equal to (datapoint minus location) divided by (scale). Or: y = (datapoint - mean) divided by (standard deviation). So, (y) is the random variable characterized by location and scale.

The "shape" parameter, given above by the Greek xi is important: the shape parameter measures the speed at which the tail disappears (is it fat or skinny?). And we know that financial data (e.g., stock returns) exhibit fat tails. So, we are generally only concerned with the GEV distribution where the shape parameter is greater than zero (shape > 0). You may note this is called the Frechet distribution. Next post we will clarify some of the potential confusion around shape, scale, and tail index...

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