Stats › Best and worst days › Fat tails
Fat tails in daily returns
Most of finance quietly assumes stock returns follow a bell curve. Risk models, option pricing, portfolio optimisers, the lot. It's an assumption people make because it's easy to work with, not because anyone actually believes it. The question this page tries to answer is how wrong it is.
Not slightly wrong, it turns out. Extreme days show up several times more often than the bell curve says they should, and that holds across the Dow, across fifteen other countries, and across more than a century. The page has the observed counts against the expected ones, what 19 October 1987 looks like when you measure it in standard deviations, why that measurement has more than one correct answer, and the part people usually skip: the tails are just as fat on the way up.
I built this because 'fat tails' is one of those phrases that gets used as a vague warning and rarely as a number. It's much more useful as a number. It also comes with a warning of its own. This page deliberately leaves out a couple of famous statistics whose original sources couldn't be retrieved, and it names them in the notes rather than repeating them. Four sources, all linked at the bottom, and the two that matter most are free PDFs. Read them if any of this changes your thinking.
Between 1900 and 2006 the Dow recorded 461 days that moved more than three standard deviations. A normal distribution predicts 79. Across fifteen international markets and 160,278 trading days the ratio is the same: 163 observed against 29 expected, on average, per market. 19 October 1987 was a 21.1 standard deviation event on the Dow using full sample volatility, with a normal probability of about four in ten to the ninety ninth. Kurtosis of S&P 500 daily returns is 31.1 where a normal distribution gives 3. The tails are fat in both directions: the smallest of the ten best Dow days was 8.6 standard deviations above the mean.
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Below this point there are 9 sections, 1 chart and 4 named sources, roughly 2000 words of it. Every figure carries the source it came from and the date the data is from.
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