Are S&P 500 Returns Normally Distributed?

Does a normal distribution, or bell curve, accurately model returns for the S&P 500? Below are plots of 47 years or return data sorted by daily and monthly return plotted on a linear and log scale.

Another way to look at the data is by asking what is the error from the actual and expected number (for a normal distribution) of occurrences of returns outside a given range.

Daily Change Occurrences Expected Error

> +/- 1% 1810 2486 37.4%

> +/- 3% 135 32 -76.3%

> +/- 5% 31 0 -100%

> +/- 9% 4 0 -100%

Monthly Change Occurrences Expected Error

> +/- 1% 406 421 3.6%

> +/- 3% 235 254 8.2%

> +/- 5% 120 133 10.9%

> +/- 9% 22 23 3.3%

Conclusion: The monthly returns of the S&P 500 are fairly accurately described by a normal distribution. The daily returns of the S&P 500 do not closely follow a normal distribution.

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Disclosure:

The information presented here is the opinion of the author and may quickly become outdated and is subject to change without notice. All material presented in this article are compiled from sources believed to be reliable, however accuracy cannot be guaranteed. No person should make an investment decision in reliance on the information presented here.

The information presented here is distributed for education purposes only and is not an offer to buy or sell or a solicitation of an offer to buy or sell any security or participate in any particular trading strategy.

Performance data showing past performance results is no guarantee of future returns.