Q&A with IFA: Are Monthly Stock Market Returns Normally Distributed?


Question: What is the best way to describe the distribution of stock returns—a normal distribution, lognormal, or something else? What should investors do with this information?

Note: The question above was taken from DFA’s Fama/French Forum. Professors Fama and French gave this response:

“Distributions of daily and monthly stock returns are rather symmetric about their means, but the tails are fatter (i.e., there are more outliers) than would be expected with normal distributions. (This topic takes up half of Gene’s [Fama’s] 1964 PhD thesis.) In the old literature on this issue, the popular alternatives to the normal distributions were non-normal symmetric stable distributions (which are fat-tailed relative to the normal) and t-distributions with low degrees of freedom (which are also fat-tailed). The message for investors is: expect extreme returns, negative as well as positive.”

We would elaborate on their response by noting that the normal distribution is relevant for monthly returns over the last 50 years, as seen in the chart below. Click the buttons along to the top to see different portfolios, buttons in bottom left corner for different time periods and buttons in the bottom right corner to display standard deviation lines. For the S&P 500 index over the last 50 years (600 months), 99.3% on the monthly returns were within 3 standard deviations from the mean. In a normal distribution, 99.7% of the data points should fall within 3 standard deviations from the mean. On a globally diversified portfolio of 100% stock indexes, like Index Portfolio 100, only 7 monthly returns (about 1%), over the last 50 years, fell beyond 3 standard deviations from the mean (between 15% and -13% in a month). The other index portfolios show similar, or even more normal results. If the data is extended back to 1928, there are 1.7% of the monthly returns beyond the 3 standard deviations (click 88 years in the bottom left corner). However, in the years of the great depression stock market speculators could margin their accounts 10:1, where today it is 1:1. 

Also see this article on Why Does This Machine Simulate Market Returns?

On yearly returns, even extreme years such as 2008 (when the S&P 500 dropped by 37%) can be readily explained by a normal distribution. In case you were wondering, 2008 was about a 2.5 standard deviation event that could happen about once in eighty years. 

As for what investors should do with this information, we could not agree more with Fama and French’s statement that investors should expect extreme returns, with negative and positive returns having equal probabilities from every fair price. But the main lesson from recognizing that monthly returns are random and normally distributed is to not try to time the market.