Excel
02 Dec 2008
Apr
30
Extreme Value Theory, The Idea: Part 1
by David Harper, CFA, FRM, CIPM
Extremes are not normal
The traditional (parametric) way to get at value at risk (VaR) is to fit a normal distribution to a set of loss observations. It is a misnomer to say we "fit" the distribution; it never fits. It's more like we artificially "impose" a distribution; we almost always force a curve. The traits of the distribution inform a VaR estimate. For example, based on a sample mean and standard deviation of x and y, we estimate with 95% confidence that losses will not exceed 1.645 standard deviations. Extreme value theory (EVT) attempts to remedy a glaring problem here, with the traditional approach: the normal distribution is designed for, it really cares about, observations near the middle (it is anchored in a "central tendency"). Further, as a key theme in the FRM curriculum, we know that actual returns are fat-tails (they are leptokurtotic, that is, they show excess kurtosis) . Given this, you can't blame folks for critiquing the traditional parametric VaR approach.
How this works
Let's just assume we have a given set of losses over some time sequence. For argument's sake, as illustrated below the losses tend to run from one dollar to eight dollars:
The key insight of EVT is that, instead of fitting a distribution to the entire dataset, we fit a distribution directly to the tail. That's focusing the solution on the problem. From a risk perspective, we care about the shape of the tail, where losses occur. We then broadly have two ways to approach this.
Block Maxima
The classical way is to divide the time series into "blocks" of equal length. In our illustration, the total series in 100 units, so we chop it up into 10-unit intervals. Within each interval, there will be a single largest loss: a block maxima. In our case, we will have a set of ten local block maxima (i.e., one for each interval). This is a dataset within the dataset; we can then directly characterize this data subset with a special distribution!
Peaks over threshold
A more modern approach is to set a threshold. In our case, let's just say the threshold is $5.10. Then we can collect the losses that are greater than our threshold: the "peaks over threshold."
Whereas block maxima divides the series into time-based intervals, peaks over threshold (POTS) is not concerned with the timing of the extremes, only if they exceed the threshold. But both approaches generate a dataset; and like we use the normal distribution to describe the whole dataset, we can use a special distribution to describe the subset of extreme tail losses.
Comments
David--I have just initated work upon a new computer. Even at my age, life is one new thrill after another!!! However, I have not accessed your site for a few weeks and was delighted to see the results of your on going effort. The best program in the country.
John
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