Generalized Pareto distribution (GPD)

$gpd(x) = 1 - (1+\frac{\xi X}{\beta})^{\frac{-1}{\xi}}$

note its similarity to the GEV(generalized extreme value distribution).

with $\xi > 0$ this distribution also shows significant fat tails. $\beta$ is a scaling factor.

This distribution was derived using the peaks-over-threshold method: the conditional probability of a loss above a defined threshold.