Monthly Stock Returns: One Fat Tail and a Dash of Skewness?

Monthly Stock Returns: One Fat Tail and a Dash of Skewness?

June 23, 2014 Uncategorized
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(Last Updated On: September 23, 2014)

Been thinking a lot about risk and return these days.

Even started skimming through Fama’s old book “Foundations of Finance.” The book is available for free:

Fama has some interesting charts and ideas regarding the statistics of monthly stock returns.

I went ahead and build a simple chart of the realized monthly return distribution for the S&P 500 from Jan 1927 through May 2014. (blue)

I also ran a simulation for 15,000 trials that are normally distributed with a mean of 94bps and a standard deviation of 5.53% (the actual values for monthly returns during the sample period).

The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.
The results are hypothetical results and are NOT an indicator of future results and do NOT represent returns that any investor actually attained. Indexes are unmanaged, do not reflect management or trading fees, and one cannot invest directly in an index. Additional information regarding the construction of these results is available upon request.

Typically, we’ve heard that monthly stock returns have “fat tails.” This statement is a bit misleading.

Based on the chart above the data provide the following conclusions:

  • Realized stock returns have one fat tail (the low return tail).
  • Realized stock returns are skewed to the right (negative skew).

To summarize, stock returns don’t necessarily have “fat tails,” rather, stock returns have one fat tail and negative skew…a bit different.

Does this jive with findings from others? The result surprised me a bit…


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Definitions of common statistics used in our analysis are available here (towards the bottom)




About the Author

Wesley R. Gray, Ph.D.

After serving as a Captain in the United States Marine Corps, Dr. Gray earned a PhD, and worked as a finance professor at Drexel University. Dr. Gray’s interest in bridging the research gap between academia and industry led him to found Alpha Architect, an asset management that delivers affordable active exposures for tax-sensitive investors. Dr. Gray has published four books and a number of academic articles. Wes is a regular contributor to multiple industry outlets, to include the following: Wall Street Journal, Forbes, ETF.com, and the CFA Institute. Dr. Gray earned an MBA and a PhD in finance from the University of Chicago and graduated magna cum laude with a BS from The Wharton School of the University of Pennsylvania.


  • Tom Austin

    Wes,
    I like the broad market wide analysis. Thanks for sharing it. Did you include dividends?

    Historical analysis in texts shows negative skew for stocks (about -.62 for large cap World stocks and -.7 to -.8 for LArge US stocks. Small cap US and world stocks show less negative skew…around -.22 to -.3). (see, for example, Investments – by Bode, Kane and Marcus…this was the Wharton standard text at some point when I went).

    These ‘skewnesses’ are not always stable. For example Large stocks has only -.12 skewness from 1941 to 1965. It all depends on ‘left tail’ extreme events.

    The ‘typical’ monthly ‘actual returns’ are actually very positively skewed, not negatively. That’s what the graph shows. It’s the weight of the left tails (without a corresponding offsetting right rail) that leads to this negative skewness.

    In fact, the graph actually seems to show that:

    a) That big left tail, big loss months exist and are real and non-normal.
    b) ‘Average’ monthly returns tend to be leptokurtic…’centrally clustering’ near the mean…between -1% & and +6% much more than we expect by a normal distribution alone. The ‘average’ monthly return is heavily positively skewed. This is the meat of monthly SP500 returns in ‘normal’ historical markets.

    These findings both a) make sense intuitively – mostly – although I am surprised by the lack of a heavier extreme right tail during times of big innovation (i.e. tech boom)…and they also b) mostly mesh with what I’ve read elsewhere. I haven’t actually run monthly returns for the whole history.

    I reran this analysis for past 14 years (since 1/1999). Nearly all months land between -4% and +4%. The left tail is again much ‘heavier’ than the right tail. No huge ‘up months’ that match these extreme down months.

    Best,
    Tom

  • Thanks, Tom. This does include dividends.

    So sounds like your 14 yrs look about the same? That’s good to hear. I did this analysis fairly quickly and it seemed straightforward, but the results didn’t fit my “intuition” and I was suspect something might be goofy.

  • Tom Austin

    Wes. My 14 year test was quick, too. But…it shows a big left tail (tech bubble then 2008). Your graph looks ‘skewed to the right’ with the outlier left tail bucket. That’s visually a positive skew. And that’s how most sub-periods look. In general the size of the extreme loss / left bucket (i.e. left tail) at some point in most 25 year cycles tends to dominate the mean clustering ‘normal market’ periods…leading to the negative skew results. The negative skew tends to be more persistent than the kurtosis. (for example, kurtosis has been negative since 1966 or so…and was -.57 from 1981 to 2005 for World Large Cap stocks…and 3.01 from 1931 to 1955). Over shorter periods, it’s not ‘certain’ this will happen…but it’s been the tendency in world equity markets over past 80 years or so.

  • Jason Carver

    I see the graph the same way as you, Tom.

    Perhaps this could be used to justify and calculate stop-loss protections. So if a month swings negative out of the tight central cluster, the chance that it’s headed into >=15% loss is significantly higher than a smaller negative result. For example, eye-balling the data, it looks like the chance of a 11-14% loss adds up to roughly the same chance as >=15%.

    Of course, intra-month data would be needed to substantiate that hypothesis.