The name of the fact (Non-Gaussian distribution of returns) is pretty much self-explanatory. It was observed in the literature that (daily) asset returns exhibit the following:

• Negative skewness (third moment): Large negative returns occur more frequently than large positive ones.
• Excess kurtosis (fourth moment) : Large (and small) returns occur more often than expected.

We break down investigating this fact into three parts.

Histogram of returns

The first step of investigating this fact was to plot a histogram visualizing the distribution of returns. To do so, we used sns.distplot while setting kde=False (which does not use the Gaussian kernel density estimate) and norm_hist=True (this plot shows density instead of the count).

To see the difference between our histogram and Gaussian distribution, we superimposed a line representing the PDF of the Gaussian distribution with the mean and standard deviation coming from the considered return series.

First, we specified the range over which we calculated the PDF by using np.linspace (we set the number of points to 1,000, generally the more points the smoother the line) and then calculated the PDF using scs.norm.pdf. The default arguments correspond to the standard normal distribution, that is, with zero mean and unit variance. That is why we specified the loc and scale arguments as the sample mean and standard deviation, respectively.

To verify the existence of the previously mentioned patterns, we should look at the following:

• Negative skewness: The left tail of the distribution is longer, while the mass of the distribution is concentrated on the right side of the distribution.
• Excess kurtosis: Fat-tailed and peaked distribution.

The second point is easier to observe on our plot, as there is a clear peak over the PDF and we see more mass in the tails.

Q-Q plot

 After inspecting the histogram, we looked at the Q-Q (quantile-quantile) plot, on which we compared two distributions (theoretical and observed) by plotting their quantiles against each other. In our case, the theoretical distribution is Gaussian (Normal) and the observed one comes from the S&P 500 returns.

To obtain the plot, we used the sm.qqplot function. If the empirical distribution is Normal, then the vast majority of the points will lie on the red line. However, we see that this is not the case, as points on the left side of the plot are more negative (that is, lower empirical quantiles are smaller) than expected in the case of the Gaussian distribution, as indicated by the line. This means that the left tail of the returns distribution is heavier than that of the Gaussian distribution. Analogical conclusions can be drawn about the right tail, which is heavier than under normality.

Descriptive statistics

The last part involves looking at some statistics. We calculated them using the appropriate methods of pandas Series/DataFrames. We immediately see that the returns exhibit negative skewness and excess kurtosis. We also ran the Jarque-Bera test (scs.jarque_bera) to verify that returns do not follow a Gaussian distribution. With a p-value of zero, we reject the null hypothesis that sample data has skewness and kurtosis matching those of a Gaussian distribution.