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


In the course of talking to investors about their portfolios, a key issue that our advisors face revolves around answering the question, "What is the best way to describe the distribution of stock returns — a normal distribution, lognormal or something else?" This often leads to a follow up discussion concerning another important take away: "What should investors actually do with this information?"

To answer such significant questions, we've taken the answers of two well-known financial scholars -- Eugene Fama and Kenneth French -- from a past piece on DFA's Fama/French Forum where they tackled these types of issues. In a type of questions-and-answers format, here are some important highlights and points we've gleaned from their writings:

"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.  

In a normal distribution, 99.7% of the data points should fall within three standard deviations from the mean. Let's take, for example, a globally diversified all-stock portfolio like Index Portfolio 100. For illustrative purposes, suppose only seven monthly returns (about 1%) over the last 50 years fell beyond three standard deviations from the mean -- i.e., between 15% and -13% in a month. 

Then let's say the other IFA Index Portfolios showed similar, or even more normal, results. Next, let's click in the chart's bottom left corner and extend the data back to 1928. Again, hypothetically, say this indicates that 1.7% of the monthly returns are beyond the three standard deviations. 

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 80 years.) 

As for what investors should do with this information, we could not agree more with the statement by Fama and French 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.

This article is distributed for informational purposes, and it is not to be construed as an offer, solicitation, recommendation, or endorsement of any particular security, products, or services. Data in this article is provided for illustrative purposes only, it does not represent actual performance of any client portfolio or account and it should not be interpreted as an indication of such performance. IFA utilizes standard deviation a quantification of risk, for more detailed explanation of standard deviation and other important investment terms go to IFA glossary. There is no guarantee investment strategies will be successful.  Investing involves risks, including possible loss of principal.