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), when the S&P dropped about 20% in a single day. Ouch.
Some people, like Nassim Taleb, are making big money using an investment model that assumes a “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.
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?
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).
A histogram doesn’t tell you the whole story. Take a look at this time series plot I did for this thread. 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.
