From the December 01, 2011 issue of Futures Magazine • Subscribe!

The law of large numbers and avoiding the Siren’s song

Ulysses made a pact with his men as they sailed near the Sirens’ rocky island. The Sirens lured sailors to their death through their wondrous singing that caused men to lose all rational thought. Ulysses wanted to hear their song, so he had his men bind him to the mast and he filled their ears with wax.

Ulysses knew well that a siren song is truly sweet and hard to resist but, if heeded, eventually will lead to one’s demise. Traders should beware the similar sweet song of unproven profits. It is all too easy even for the astute trader to succumb to empty promises of success while forgetting the shoals brought on by the law of large numbers.

A modern Odyssey

Meet modern day S&P trader Neil Ulysses. Like his namesake, Neil is on an arduous journey to find the perfect guru of trading. Things went swimmingly until he heard the siren song of Martin Gale.

Neil, like all other non-statisticians — that is, almost everyone in the world — thinks about the law of averages. He assumes if a coin is flipped 20 times, it will come out heads roughly 10 of them because of the law of averages. After all, he reasons, the probability on any one flip is 50/50, so about half of the flips should be heads.

Not so. It is perfectly possible, although not terribly likely, to get 15 heads and five tails. Or, even 20 heads. As any Bayesian would tell Neil gleefully, if he flipped 19 heads, the chances of that coin coming down heads again is a steadfast 50/50. There is no law of averages; only a law of large numbers.

Nevertheless, the typical person gladly will bet you the chance of 20 heads in a row is small and will stake large sums on it the more heads are flipped. This results from a psychological phenomenon known as a Taleb distribution and leads to the downfall of many a trader.

A Taleb distribution — a term coined by economists Wolf and Kay and named after Nassim Taleb — describes a return’s profile that appears at times deceptively low-risk with steady returns, but periodically experiences catastrophic drawdowns. It does not describe a statistical probability distribution, and does not have an associated mathematical formula. The term is meant to refer to an investment return’s profile in which there is a high probability of a small gain, and a small probability of an extremely large loss that more than outweighs the gains. In these situations, the expected value is (much) less than zero, but this is camouflaged by the appearance of low-risk and steady returns; think option writing.

The ultimate downside is a combination of kurtosis risk and skewness risk: Overall returns are dominated by extreme events (kurtosis), which are to the downside (skew). Ah, but how sweet the song sounds while it is working.

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