#### Black Swans and Brown Turkeys

The term Black Swan was introduced by Taleb to denote an event not only unexpected but not even considered within the realm of possibility – something so unexpected that it completely changes the terms of discussion. But the term is at risk of being over-used, and most sightings of a Black Swan turn out to be nothing more than Brown Turkeys – events that are indeed unexpected but not exceptional probabilistically speaking. Like wild turkeys these events are hard to find and rarely seen, particularly if you’re out hunting for Thanksgiving dinner. And like wild turkeys these events are out there and they do turn up, seemingly out of nowhere and always most inconveniently, say around that blind curve on the downhill mountain-bike ride. And they’re all brown. It just seems that we humans are not particularly good at intuiting probabilities (see, for example, Mlodinow’s *The Drunkard’s Walk: How Randomness Rules Our Lives*) and we often mistake the merely unexpected for the truly exceptional.

#### Stock Market for 2008

Consider the stock market performance for 2008, which many people will want to claim as a “Black Swan” event – out of the realm of all possibility. In fact it is nothing of the sort. The S&P index (capital appreciation) fell by 38.49% for the year. Really bad and quite unexpected, but by no means out of the realm of possibility:

- The 2008 return of -38.49% is only third out of the 83 years from 1926 – behind 1937 (-38.59%) and 1931 (-47.07%). And it was not just the depression that saw big annual drops – in 1974 the S&P fell 29.72%.
^{1} - Probabilistically we can say the 2008 return was unexpected: based on history the probability of experiencing such a low return in any year is probably below 1% – less than 1 chance in 100.
^{2}But the chance of the lowest return over 83 years being as low or lower is roughly 50%, so in the larger context the 2008 return can’t be claimed as particularly unusual.^{3} - Someone looking only at recent history might have thought the likelihood of such a large fall was much lower than 1%. But even doing so and ignoring the large changes of the 1930s, a fall of the magnitude of 2008 would not be particularly unusual when considering a period of many years.
^{4}And furthermore ignoring the experience of the 1930s and 1940s is irresponsible: as George Santayana said, “Those who cannot remember the past are condemned to repeat it.”

- Using data from Ibbotson Associate’s SBBI yearbook.
- Using data for the period 1926-2007 (i.e. excluding 2008) and if we assume that returns are normal (i.e. ignoring the fact that returns may be fat-tailed and thus possibly estimating the probability as lower than it actually is) the probability of a single year’s return being below -38.59% is about 0.00828: The annual mean and standard deviation for 1926-2007 are 7.41% and 19.15%, and P[Normal Variate < (-.3849 – .0741)/.1915] = .00828. But if we consider the 83 years from 1926 to 2008 the probability that the lowest return over those 83 years is below -38.49% is 0.49841: P[lowest < (-.3849 – .0741)/.1915] = 1 – (1-.00828)^83.
- A careful reader might wish to examine log changes rather than returns (percent changes) because when considering large changes such as 2008 or 1931 the fact that returns cannot be less than -100% may matter. For log changes the 2008 change was -48.59%, the annual mean and standard deviation for 1926-2007 are 5.42% and 19.29%, and P[Normal Variate < ( .4859 – .0542)/.1929] = .00255. The probability that the lowest return over 83 years is below -48.59%, however, is 0.19127: P[lowest < (-.3859 – .0542)/.1929] = 1 – (1-.00255)^83. Still no reason to say that 2008 was extraordinary.
- Using only data 1950-2007 the annual mean and standard deviation are 9.25% and 16.22%, and P[Normal Variate < (-.3849 – .0925)/.1622] = .00162. The probability that the lowest return over 83 years is below -38.49%, however, is 0.12596: P[lowest < (-.3849 – .0925)/.1622] = 1 – (1-.00162)^83.