What Is Not Seen

Kieran Kelly, a derivatives expert, has done some very interesting work regarding the workings of the law of large numbers, and its implications for financial theory. By studying the impact of the reverse of the law of large numbers, he delivers a devastating blow to the almost omnipresent efficient market theory. From a FT article we read that:

The reverse of the law of large numbers, as its name suggests, describes the reverse effect of the law of large numbers, which crops up everywhere in the fields of mathematics, physics, engineering and the social sciences. Put simply, the law of large numbers, or LLN, says that the larger the sample, the closer the average will approach to the expected average. Rolls of a die, for example, fluctuate wildly around the expected average of 3.5 in a small sample, but converge on the average as the sample size increases…

…In his work Mr Kelly addresses what happens to the balance of expectations when so-called rational independent entities start copying each other. Since copying reduces the independence of individual entities, Mr Kelly designed an experiment to see what happens to the balance of expectations when the LLN goes into reverse and individual behaviour starts to merge into group behaviour.

The experiment focuses on a number of individual entities who are faced with a choice between two equally likely options – the market will go up or the market will go down. Given that the future is unknown, in an unbiased market, the likelihood of the next move should be a 50-50 bet.

“We first examined what was the most probable outcome of expectations when a large sample was all acting/choosing independently,” says Mr Kelly.

“Then, we gradually reduced the number of individual entities by allowing first one individual to copy another, then two individuals to copy a third (or one to copy another and a second to copy a different other) and so on. This progressed step by step to the ultimate extreme of everyone copying everyone else, so the market as a whole is acting as one.”

They found that as individuals move toward herd behaviour, the probability distribution changes from the normal bell curve, with expectations clustered around the 50/50 level, through a tipping point, where there is an equal likelihood of a balanced or unbalanced market of expectations, to a position where a market is almost certain to be unbalanced in its expectations. At this extreme, the whole market has the same view, so the balance of expectations is polarised one way or the other.

Austrian economists have already provided lots of counterexamples and, economic reasons why the efficient market hypothesis is deeply flawed (read more).

The work of Mr Kelly adds an additional dimension to this criticism. The idea of the reverse law of large numbers, provides insight how real market agents act and interact, contrary to the efficient market theory. By themselves, recent market bubbles should be evidence enough, that phenomenon such as herding- and crowd behaviour is present in determining asset prices.