Modeling with Power Law distributions

Power Law distributions can explain data sets with extreme values, which includes a whole lot of things in nature, and pretty much everything in financial economics; Normal/Gaussian distributions can not. At the same time, Power Law models are just as, if not more flexible and mathematically elegant.

So here's the walk through,

Definition source: Forecasting Extreme Financial Risk

The Tail Exponent alpha "a" is the key to its tail fatness. A number of existing research papers suggest that actual financial returns resemble a Power Law distribution with the tail exponent between 2 and 3.

An example:

Given a probability P= .2, i.e. 20% for a security to return a minimum of .01/day, so P(.01)= .2

Let the Tail Exponent= 2, we can now work out the probability of it returning .05 (5 times .01/day) in a day. So we want to find P(.05)

P(.05)= P(.01)(5)^(-2)
P(.05)= .2(.04)
P(.05)= .008 = .8%

That was pretty easy right?

Obviously, better probability estimations is critical for anyone planning business models where it's all about "calculated risks".