Stay with me this is complicated but there is a message in the letter that anyone can take away so don’t stress trying to understand the mathematical formulae as my point is hopefully intuitive, amidst some non-linearity for the geeks.
If human heights were distributed by a power law similar to that of city populations, and if we calibrate the mean height at 5 feet 9 inches, then the United States would include one person the height of the Empire State Building, over 10,000 people taller than giraffes, and 180 million people less than 7 inches tall (Clauset, Young, and Gleditsch 2007). Hmmmm that doesn’t sound very normal.
The probability of an event with a power law distribution is proportional to its size raised to a negative exponent. The larger the event, the less likely it is to occur. The simplest example of a power law is a square. If you square (double) the length of a side from 2 to 4 inches then the area will quadruple (area will increase from 4 to 16). Here the exponent = 2.
Ok I am now getting to the good stuff. Even when the probability of an event is said to happen very rarely like one in a million it is super important to understand if the nature of the event being considered is part of the power law distribution family or as some describe a fat tail distribution. This is much more important than we are naively lead to believe, as power law distributions cause unlikely events to happen much more frequently than a typical normal distribution probability analysis would lead us to believe.
For instance an earthquake larger than 9 on the Richter scale occurs each day with a probability of one in a million. Within a century an earthquake of that size would occur with a probability of 3.5%, suddenly not as unlikely as you thought before.
Using a power law distribution with an exponent of 2 would make a one in a million event terrorist attack result in 800 deaths. However, if you followed a normal distribution it would involve fewer than 50 deaths (I am skipping the calcs – just trust me).
There is a feature required in power law fat tail distribution’s, that of non-independence, things like feedback loops or interconnectedness create the power.
Nature is filled with power laws, don’t ask me why. It seems this is the nature of complex systems and how they self regulate themselves. Power laws are perfect examples of natures unfairness where the bulk of wealth, fame, population density is concentrated with the few. Power laws express themselves more frequently and more dramatically than normal distributions.
When it comes to trading, the bulk of the profits will be made by a few trades / small group of traders compared to the trading account / universe of trader. When it comes to losing much more in your trading accounts than the backtests you ran predicted you are coming face to face with power laws that normal models underestimate the frequency of large events.
When you think a one in a 100 year pandemic or flood will not likely occur anytime soon, think again the probabilities are much higher than you think, just look at the recent Australian floods. If you think a world war or a nuclear war is a very low probability don’t be naive. Wars just like natural events are subject to power laws.
My message is bleak but it is also a wake up call to get real. When dealing with probabilities you have to use the right models to do your calculations otherwise you will always be unprepared. That is why we run for shelter every time an earthquake warning siren sounds. I write this while my colleagues are literally sitting in bomb shelters through long nights trying to avoid becoming a statistic running for cover with every siren (this is the truth Vlad shared a picture of him in a bunker at 5am trying to stay safe).
I could go on at length describing how financial time series are not normally distributed, yet most models used in modern finance assume a bell curve shape. I could suggest that power law distributions are easily fit to a suitable model, however it is not always easy to fit a suitable model to a particular power law distribution. This is complex stuff made a little easier with the brute force computers bring to the table. I pay my respects once again to my colleague Vlad who is an expert in the field of non normal distributions, power laws being one of them.
I am way to weak a student of mathematics to understand this subject at any great depth. What I can contribute to the subject is awareness. Be aware. Be curious. Be suspicious.
But most importantly don’t be the fool who dismisses the power of the power law.