### Update on the Distribution of Returns

When I first calculated the distribution of returns, I used data from SPY, which only goes back to 1993. Today I finally got around to checking with data for SPX, which goes back to 1950. I didn't do the full analysis. I just used the 16 day variance normalization. The normalized returns are very close to a normal distribution. The lowest z-score is -3.84 and the highest is 3.53. The '87 crash has a z-score of -3.75 (third lowest). I have two graphs showing the distribution.

The first one would be a straight line if the distribution were normal.

The second one shows the actual cdf in blue with the normal cdf in green. You can see the slight negative fat tail in the second graph where the blue line dips below the green one. But then the blue line goes above the green line, showing that slightly negative returns are less likely than expected.

At any rate, this shows that the normalized returns are very close to a normal distribution.

The first one would be a straight line if the distribution were normal.

The second one shows the actual cdf in blue with the normal cdf in green. You can see the slight negative fat tail in the second graph where the blue line dips below the green one. But then the blue line goes above the green line, showing that slightly negative returns are less likely than expected.

At any rate, this shows that the normalized returns are very close to a normal distribution.

Labels: market statistics