The S&P 500 (Index: SPX) is sometimes very volatile. But much more often, it is not.

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That characteristic is driven home in our chart tracking the day-to-day percentage change in the S&P 500 in the modern era for the U.S. stock market.

Here, we can see that over the past 18,661 trading days for which we have data, the average day-to-day change in the S&P 500 is +0.04%. We can also see that the index has changed from its previous trading day's closing value by 0.99% (one standard deviation) or less on 79.1% of all the trading days from 3 January 1950 through 1 March 2024.

When we expand that range of trading day-to-day volatility out to two standard deviations, or daily changes of a little under 2% or less from the mean, we find that 95.4% of daily percentage changes in the index fall within that wider range. By statistical definition, things get more interesting when we look at trading day-to-day changes that are bigger than that. But even so, we find 98.6% of the daily volatility of the S&P 500 falls within a limit of three standard deviations of the mean daily change.

Of the 18,661 trading days for which we have data, we find just 122 (0.65%) where the S&P 500 increased by more than three standard deviations and 140 (0.75%) where the decreased by more than three standard deviations with respect to the mean daily change.

We also can see that larger volatility tends to happen in clusters, with big negative changes and big positive changes in close proximity to each other.

As for the best and worst days recorded by the S&P 500 index over this period, the worst negative day of trading was a 20.5% decline on 19 October 1987, in a market event now known as Black Monday 1987.

The best positive day was a 11.6% increase on 13 October 2008, which occurred in a cluster of volatility that, on the whole, was negative because it coincided with the collapse of the U.S. automotive industry during the "Great Recession" of 2008-09 and its associated stock market crash.

Perhaps the most interesting characteristic of the S&P 500 is the concentration of day-to-day changes in its value that fall within one standard deviation of the mean. If the day-to-day variation of the index were accurately described by a normal Gaussian distribution, we would expect to see around 68.2% of all changes within that range. Instead, there are far more small changes than would be expected if that hypothesis held. At the same time, there are more "large" changes that would be expected if a normal distribution applied, which is what market analysts mean when they describe stock prices as having "fat tails".

Both these properties are characteristics of a Lévy Distribution, which is another kind of stable distribution.

References

Yahoo! Finance. S&P 500 Historical Data. [Online Database]. Accessed 2 March 2024.

Volker Ziemann. Bubbles, Crashes, Fat Tails and Lévy-Stable Distributions. Physics and Finance. pp 113-143. DOI: 10.1007/978-3-030-63643-2_9. 19 January 2021.

Takumi Fukunaga and Ken Umeno. Universal Lévy’s stable law of stock market and its characterization. [ArXiv Preprint: PDF Document.] 20 February 2018.

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