In love as in equities, we are fooled by randomness

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As the year ends, the author of a weekly column should look back and acknowledge the things he got wrong. I made at least one serious mistake. I wrote that if men think about sex on average every seven seconds, the average man last thought about sex three and a half seconds ago. I should have consulted the poet Wendy Cope, who wrote that: “Bloody men are like bloody buses – / You wait for about a year / And as soon as one approaches your stop / Two or three others appear”.

Her analogy is helpful. If a bus arrives at fixed intervals of seven minutes, you will wait an average of three and a half minutes. But, as Ms Cope knows all too well, the interval is unpredictable: two buses come at once and then there is a lengthy wait for the next one. The average frequency of the bus may still be seven minutes. But if the intervals are alternately zero and 14 minutes the expected wait is now seven minutes, not three and a half. If someone is standing at a bus stop, it is more likely that they are victim of a bus which is late than the beneficiary of one that is early.

You may think this does not matter very much. But this summer the UK business secretary Vince Cable asked me to devote an entire year to thinking about equity markets. I discovered that the arithmetic of thinking about share values is the same as the arithmetic of thinking about sex. The average length of time for which buyers hold shares today is very short. But the average length of time for which shares have been held by their current owner is much longer. There are many more high frequency trades than passive investors, but passive investors hold a high proportion of outstanding shares.

The people queueing for a bus are the people whose bus has not arrived, and the people on yachts are those whose boat came in. What we see will always be influenced by the ways in which the sample studied is selected. The traders we interview are mostly successful because mostly it is the successful who are still trading – and this past success may be no guide to future performance. As the essayist Nassim Taleb has observed, we are regularly fooled by randomness, identifying skill where there was only luck, finding patterns in data when none really exist.

The sometimes counter-intuitive mathematics of variation crops up in many different places. If Persil is sometimes on special offer, the percentage increase when it goes back to its usual price will be larger than the percentage price reduction when it goes on special. If that seems an unremarkable fact, it was enough to send several hundred thousand government employees on strike a month ago.

The average of price changes shows an increase even though the price has remained the same. And perhaps you buy more, perhaps even spend more, when the good is on offer than when it is at full price. After all, that is why they put it on special. These issues pose problems for compilers of price indices, and there are different methods of handling them. That is the principal reason why the new European harmonised index of consumer prices generally increases by less than the old retail prices index. The UK chancellor George Osborne is planning to make large savings in public expenditure by shifting pension indexation from one basis to the other.

The same problem arises in measuring benchmarks and portfolio performance in equity markets. If you average a 50 per cent fall and a 100 per cent increase, you show a 25 per cent gain. But if that happened to your shares, you would – just – have recouped your initial investment. Neither method of calculation is necessarily a guarantee to the experience of real investors.

I may have made another mistake in my earlier column. Since it appeared, details have been published of a study by researchers at Ohio State University. They surveyed college students – who, one might expect, think about sex more often than the average of the population. The subjects were asked to make a note each time the topic entered their heads. Men thought about sex, on average, 19 times a day (the figure for women was only 10). Does this mean that we should correct “every seven seconds” to “on average, once every waking hour”? Only if men think about sex at absolutely regular intervals. And neither love nor equity markets are so predictable.

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