Extreme Value Theory (EVT) Distribution
Normal is good for central
Value at risk for risk management which is the same and normal distribution or bell curve. First moment
is at 0 or the mean beeing 0 (peak of curve). The second moment is volatility and covariance are 1. The
third moment is the measuremean of tilt or scew of the curve. Normal is always symmetrical (mirror ob
both sides) which means the skew is 0. The fourth moment is the tail weight or density where you a
problem. It is understood that asset returns are fat tails. The normal will not capture properly the
probability where losses can be severe. The normal is a good measure of Central Tendency where the
probability will end up at the middle (mean of 0). It is not very good where losses are in the tail. There
are operational risk terms of losses ofLFHS (Less Frequency with High Severity). So for the fourth
momem where tail is skinny, you need to be aware of the extreme tail losses.
This is where Extreme Value Theory comes in to play. You draw a line somewhere along the tail. You
would describe a distribution where you are characterized by the losses in the tail. You just zoom in the
tail and you have a special distribution for the tail. You could have a parent distribution with measure of
central tendency but you care about the EVT.
EVT: A distribution specifically for the 'extreme' tail loss.
This becomes a child distribution focusing on the losses in the tail.
You could have losses over time in a chart over 100 days. Losses in dollars could be Y axix while Time in
days is X. What ways are there do describe extreme losses?
First approach is to use Block Maxima tends to fit GEV. It divides time into chunks. If you have 100 days,
you could divide into ten separate 10 day blocks. For each block, you take the maximum loss. You will
then get 10 local block maximums losses (maxima). This data could be characterized.
The second approach is to use Peaks over Threashold (GPD) which is more modern. You select a
threshold as in a bar. Anything over that threshold is considered an extreme loss. You may not get
enough data to work with.
Generalized extreme value (GEV) distribution
Traditional probability could be described as
F(x)=P(x<=x)
F(x) is cumulative function which is normal distribution or bell curve.
The GEV is
Select a threshold called U (green dash line). Random varliable is X.
The function becomes the probability that X-u (X over threshold) <= y. This will be in the child
distribution. The conditional pipe (|) only if X>u threshold.
http://www.youtube.com/watch?v=o-cpu1IH3tM