### Kurtosis and fat tails in IBM's price changes

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**Sat Oct 06, 2012 11:33 am**A friend loaned me the book Black Swan by Taleb. It reinforces that the risk in markets is much greater than a normal distribution would suggest. I did some research to verify this myself. Using IBM daily close prices from 1962 - 2012 from Yahoo! Finance, I first plotted percentage price changes in a histogram (see attached). Looks pretty similar to a normal distribution, i.e., to a bell curve, although the upward bias of the stock market is evident in the much greater instances of 1 - 2% moves as opposed to -1 - -2% moves. Curiously, the average of the price changes is slightly negative, which indicates there must be a large number of -1% - 0% price changes. I then calculated the standard deviation (sigma) of the price changes, which equals almost 2%. In a normal distribution 68.2% of observations occur within 1 sigma of the mean, so if IBM price changes are normally distributed, then 68.2% of price changes should lie between -2% and 2%. In fact, 83.7% of IBM's price changes are within 1 sigma of the mean. This indicates that the distribution of IBM's price changes exhibits positive kurtosis, which appears as a taller, narrower hump than the hump of the normal distribution. It also seems to indicate that IBM is in fact LESS risky than a hypothetical investment that exhibits a normal distribution. However, in looking at much larger moves, you find that they are much more probable in IBM's price distribution than in a normal one. For example, the probability of a greater than 5-sigma move on any given day for a normally-distributed hypothetical investment is 1 in 1.74 million. The historical probability of a similar move, that is, up or down more than 10% in one day, by IBM? About 1 in 500. For a greater than 6-sigma move? Normal: 1 in 506 million. IBM: 1 in 900. Historically, IBM was 562,000 times more likely to exhibit a greater than 6-sigma move on any given day than a normal distribution would suggest. This greater frequency of large moves in IBM's price distribution is likened as "fat tails" to the left and right of the hump. For example, you see that IBM closed up greater than 15% 9 times throughout the data period, which is a far, FAR greater frequency than expected if IBM prices were normally distributed. You could say that these bigger moves pose more risk for the trader and that the fat tails in IBM's price distribution make IBM much more risky than a hypothetical, normally-distributed investment.

So how does this impact system design? For one, any statistical measurements that assume that price changes are normally distributed and that assign a probability to a particular degree of move grossly underestimate the actual risk if that move is more than 3-sigma. More broadly, the lesson here is that before utilizing tools that assume a normal distribution, make sure what you are measuring is ACTUALLY normally distributed, otherwise you have to take your probability estimates with a grain of salt.

Still learning. Need to learn more about what tools you can use to measure probability and determine statistical significance for non-normally distributed data series.

So how does this impact system design? For one, any statistical measurements that assume that price changes are normally distributed and that assign a probability to a particular degree of move grossly underestimate the actual risk if that move is more than 3-sigma. More broadly, the lesson here is that before utilizing tools that assume a normal distribution, make sure what you are measuring is ACTUALLY normally distributed, otherwise you have to take your probability estimates with a grain of salt.

Still learning. Need to learn more about what tools you can use to measure probability and determine statistical significance for non-normally distributed data series.