Normality Examined: Is the S&P Truly Normal?

By Paul White, CAIA, Ph.D.

“When life itself seems lunatic, who knows where madness lies? Perhaps to be too practical is madness. To surrender dreams — this may be madness. Too much sanity may be madness — and maddest of all: to see life as it is, and not as it should be!” Don Quixote, Miguel Cervantes 

Normal is practical, and, as Cervantes says, practical is madness.  Don Quixote was rational in an irrational world; the people who were “normal” were the irrational ones.  The S&P 500 is the world of investing for most people, and no interpretation of the word, “normal”, applies to it.  Very few things in finance are mathematically normal.   

Why do we (the author is included) use it so often?  The assumption of normality makes the harder problems in investing analytically tractable, meaning, one can write down the answer in “closed form” – we can write an equation!  From a scholastic perspective, closed form solutions are easier to teach than solutions which either don’t exist or can’t be written down on a single page. 

How much damage are we doing by assuming something is normal? 

The histogram above is for the monthly returns of the S&P 500 for 10 years since January 2014.   The old adage of “proof by picture” tells us that this is not a Gaussian distribution.  However, standards require us to run a statistical test on the distribution to test for normality.  If we choose the Shapiro Wilk test, we calculate p = 0.0203 < 0.05.  This is not normal.   

Despite these results, professors, practitioners, and mostly everyone involved in investments apply this assumption everywhere imaginable.  This can be useful to get an idea of the data, but it would be difficult to draw hard conclusions beyond that.  It would almost be irresponsible to invest and/or trade based on normality.  Again, we’re not pointing this out to say people are wrong, we point these things out to remind people to be careful.