Are market returns log normal? A quantitative answer.

In "Comparison of Historical Volatilities of Market Returns and of Normal Returns" I have compared the statistics of GE daily log returns with the statistics of computer generated data that are normally distributed with the same mean and standard deviation of the GE market data. However, the comparison was purely qualitative, that is, I just calculated the histograms of quantities such as mean, standard deviation, skewness and kurtosis for samples obtained by sliding a 21 day window over the entire data set available and compare them with the same quantities for the normally distributed data.

Now I would like to make a quantitative comparison, using the Kolmogorov-Smirnov test.

First, I calculate the empirical CDF's (Cumulative Distribution Functions) from both the GE market data (red) and the normally generated data (blue):


How to get the empirical CDF from the data is explained, for example, at the College of Saint Benedict and Saint John's University web site and at the National Institute of Standards and Technology web site.

Second, I calculate the absolute value of the difference between the two CDF's, for each statistics:


The quantity that I need is the maximum of this difference, multiplied by the square root of the total sample size. This is a measure of the "distance" between the empirical distributions that I am comparing.

Now I need to know what is the probability that two empirical distributions come from the same distribution, or that they are just two different statistical realizations of the same distribution, given the value of their Kolmogorov-Smirnov distance. In other words, I want to know what is the chance that the GE market data and the computer generated data, have the same probability distribution. Since I already know that the computer generated data are normally distributed, this is the same chance that the GE daily log returns are normally distributed as well.

This probability is given by the Kolmogorov-Smirnov distribution:


It is clear that once the distance gets larger than, say, 1.5, the chance of the two distributions being the same is basically negligible.

Now, this is what I get.

In the case of the 21 day mean, the maximum of the difference between CDF's is 0.042, which, multiplied by the square root of the number of data points, becomes 4.42.

The maximum difference between the CDF's of the 21 day historical volatility is 0.349, which becomes 36.76.

In the case of the 21 day skewness I get 0.085 and 8.95, and in the case of the 21 day kurtosis, 0.139 and 14.64.

With such large values of the distance between the CDF's, the probability of them being the same, as given by the Kolmogorov-Smirnov distribution, is practically zero.

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