# histogram of stock market daily changes? **URL:** \ **Category:** Factual Questions\ **Created:** [July 11, 2011, 7:24pm UTC](https://boards.straightdope.com/t/histogram-of-stock-market-daily-changes/588662 "2011-07-11T19:24:49Z")\ **Posts on this page:** 3\ **Page:** 1
**Author:** ![Machine\_Elf](https://avatars.discourse-cdn.com/v4/letter/m/82dd89/32.png) [@Machine\_Elf](https://boards.straightdope.com/u/Machine_Elf)\ **Post date:** [July 11, 2011, 7:24pm UTC](https://boards.straightdope.com/t/histogram-of-stock-market-daily-changes/588662/1 "2011-07-11T19:24:49Z")
Since the early 1900, the S&P 500 index has posted annualized gains of approximately 10%. That’s a daily average increase of about 0.06%, but of course the actual daily change can vary quite a bit from this mean. Today for example is pretty ugly: the S&P is down about 2%, with an hour to go. Some days have been much worse, the record being [Black Monday (1987)](http://en.wikipedia.org/wiki/Black_Monday_(1987)), when the S&P dropped about 20% in a single day. Ouch. Some people, like [Nassim Taleb,](http://en.wikipedia.org/wiki/Nassim_taleb) are making big money using an investment model that assumes a [“Fat tail”](http://en.wikipedia.org/wiki/Fat_tail) distribution of daily stock market fluctuations, i.e. his model assumes that large price changes happen more often than is predicted by a [normal distribution.](http://en.wikipedia.org/wiki/Normal_distribution) It seems like this should be easy to verify. Can anyone point to a histogram of stock market daily price fluctuations that shows a fat-tail distribution fits better than a normal distribution? ---
**Author:** ![OldGuy](https://avatars.discourse-cdn.com/v4/letter/o/3bc359/32.png) [@OldGuy](https://boards.straightdope.com/u/OldGuy)\ **Post date:** [July 11, 2011, 8:52pm UTC](https://boards.straightdope.com/t/histogram-of-stock-market-daily-changes/588662/2 "2011-07-11T20:52:11Z")
He’s basically right. The annual standard deviation varies of course, but a typical result would be in the high teens or low twenties. Let’s call it 16%. There are 250 odd trading days in a year. Let’s call that 256. Then the daily standard deviation is 16%/SQRT(256) = 1%. That means a 20% drop is a twenty sigma event. (Actually I’ve seen it referenced as a 19 sigma event on occasion.) You can do the math yourself\* but as I recall that a 19 sigma day has a probability something once chance in a billion (or maybe it was a million) of happening during the life of the universe (17 billion years). Here is a histogram for the last year [http://www.etfreplay.com/volatility\_histogram.aspx](http://www.etfreplay.com/volatility_histogram.aspx) \*The math isn’t simple because the standard approximations for the normal don’t work that far off in the tails. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [July 11, 2011, 10:16pm UTC](https://boards.straightdope.com/t/histogram-of-stock-market-daily-changes/588662/3 "2011-07-11T22:16:17Z")
A histogram doesn’t tell you the whole story. Take a look at [this time series plot](http://i.imgur.com/vURdf.png) I did for [this thread](http://boards.straightdope.com/sdmb/showthread.php?t=614912). As you can pretty clearly see, the volatility of returns is nowhere near constant over time. If you just do a straight-up histogram, then yeah, you’ll see fat tails, but obviously there’s a lot more going on.