In keeping with the theme of recent weeks, I am doing a deeper dive into S&P 500 data. This week sees me updating an article originally published almost 8 years ago about the history of sigma events for the S&P 500. Or, as I like to call it the catalogue of, “People phreakin’ out.”

From 3 January 1950 through 30 June 2020, the average daily return of the S&P 500 was 0.03%, and the standard deviation was 0.99% (source: AIM Consulting, LLC; Yahoo! Finance). Since my original post the average daily return of 3 basis points has not changed, but the standard deviation has ticked up by 0.01%. That there is a positive skew of 3 basis points per day means that stock market returns are not a random walk. Why?

For those of you conversant with statistics, these results are remarkably similar to the mean and standard deviation of the normal distribution of 0 and 1, respectively. Many folks over the decades have pointed out, however, that stock market returns display extreme positive and negative daily returns — the proverbial “fat” tails. What do the data show when graphed?

As you can see, the overwhelming majority of daily returns fall within one standard deviation, or sigma, from the mean return of 0.03% per day. This is actually a characteristic not discussed as frequently as the stock market’s “fat tails.” Namely, that daily returns are leptokurtic until you reach the tails. Yet, the normal distribution holds that ~68% of returns should occur within one standard deviation of the mean, yet the actual number is a gigantic 93.7%.

Let’s layer in an image of a standard normal distribution curve above the history of the sigma events of the S&P 500 to see what it looks like…

As you can see the curves definitively do not look like one another. This is further evidence that stock market returns are not a random walk.

My colleague Michael Falk, CFA, CRC and I have our second episode of our new podcast, From the Research Chair on 9 July 2020 and our topic will be curiosity. Click on the embedded link to register for it.

Below is the numerical breakdown of the sigma graph topmost above and what it records are the number of daily returns that are within the number of standard deviations show above the data. For example, 7,349 is the number of negative daily returns that fell within one standard deviation for the whole data series. The line labeled 2012 displays the data as of 2012 when I first published this article. Last, the difference line shows the record of sigma events between the two dates.

I think it is fascinating to look at the data in the intervening 8 years from when I first published this article. Those results look much more like a normal distribution.

Market observers have noted that financial markets have become more volatile over time. A look at the number of sigma events by decade makes that clear.

For example, the number of normal trading days — as measured by the percentage of trading days that are a <1 sigma event — is sharply lower since the 1990s. After a significant dip in the number of “normal” trading days in the 2000s, they seemingly have made a rebound in the 2010s. All of that said, you can see that the number of 5+ sigma events begins to fill out after the 1970s.

The highest up sigma date in S&P 500 history came on 13 October 2008, when the S&P 500 surged upward registering as an 11.69 sigma event. Meanwhile, the largest negative sigma event was the famous 19 October 1987 crash, which was a whopping 20.65 sigma event. Wowza!

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