# Leptokurtosis

Leptokurtosis describes a probability function which is similar to the bell curve, but not quite.  The extreme cases, while still rare, are not as rare as expected.  This is often called “fat tails” because of the way this looks when you graph the probability function.

People studying the volatility of stocks usually find that the variance of a stock is leptokurtic.  This means that most of the time the stock moves around somewhat randomly.  But when it deviates from this random pattern, when for example it suddenly starts running in one direction or the other, it runs a lot further and a lot faster than you would expect.

Leptokurtosis is at the heart of what we do at Trade-Ideas.  We believe that the basic normal distribution of market moves is based on random noise.  Real trends exist, but they are hard to see through the noise.  Leptokurtosis shows that there is something in the stock price other than noise.  It can be small, and hard to see through the noise.  We use statistical analysis to identify these trends.

An example:  Let’s say you are examining a stock on a daily bar chart, and a 5 minute bar chart.  Let’s say that you see clear trends, patterns, or cycles on both charts.  First, notice that the patterns you see on each chart are not visible on the other chart.  The reason that you can see different patterns is that the 5 minute chart includes moves with a much sharper angle than the daily chart, but many of these moves cancel each other out, allowing the long term patterns to emerge.

Let’s model what we see on each graph as a normal bell curve.  The blue curve on the graph below represents what you see on the 5 minute chart, and the magenta curve represents what you see on the daily chart.  The blue curve is narrower than the magenta curve because the changes (measured in dollars) are smaller; you have to zoom into your chart to see these changes.  The blue curve is taller than the magenta curve, showing a higher probability, because the small changes shown on the 5 minute chart happen more frequently than the large changes shown on the daily chart.  Again, you have to zoom in to see all of the changes.

The yellow curve shows a combination of the blue and magenta curves.  As drawn above, the yellow curve shows price action driven 90% by the short term chart, and 10% by the long term chart.  This is a very reasonable model for a stock’s movement.  The yellow curve looks a lot like a normal bell curve.  But it’s different!

The yellow curve in the second picture was drawn by the same algorithm as the yellow curve in the first picture, but the second one is more leptokurtic, just so the example will be easy to see.  This curve represents a realistic model of a stock price.  The orange line is a normal bell curve which looks just like the yellow curve in the middle.  The yellow curve represents the naive model many people use for the stock price.  Notice that the orange curve is higher than the yellow curve near the middle.  Remember, the area near the middle represents smaller price changes.  So the normal bell curve is more likely than the leptokurtic one to have small changes in price.  As we move toward the ends, representing larger price moves, the yellow curve becomes higher than the orange one.  So, even though large price moves are rare for both curves, the leptokurtic curve is more likely to have a large price move than the normal curve.  (These are the “fat tails”.)  If we did not exaggerate the picture, the same problem would exist.  It would be harder to see, especially in the middle, but the surprises would still exist in the extreme cases.

How is this different from normal?  A normal bell curve is created by a large number of independent random events.  Adding more random events distorts the curve in a very specific way, and creates another normal bell curve.  The example above was caused by events which were not entirely random or independent.  Many institutional traders recieve large buy or sell orders to complete by the end of the day.  If this is the major cause of price fluxuations, the amount total amount a stock moves between one close and the next might be random.  Durring the day, however, the stock motion is anything but random.  Initially, the trader working the large order will do everything in his power to hide his intentions and defer any changes to the stock price.  He may even buy a little to hide the fact that he needs to sell a lot.  Then, when he gets close to his goal, he’ll use a large market order to finish his work for the day.  The stock price goes from artificially calm (or even moving in the wrong direction) to one large, sudden, unexpected move.

In the case above you could see some patterns on a short term graph which looked reasonable, right up until the moment where they failed big.  The short term trends and the long term trends are completely different.  It would be impossible to predict one just by knowing the other.  This is the nature of the problem of leptokurtosis.