‘Basically,” says the American mathematician John Allen Paulos, looking a little sheepish, ”I was motivated by exuberance, greed and arrogance. I became infatuated. Or demented.” By April last year Professor Paulos was effectively an addict, and his drug of choice was shares in WorldCom, the telecommunications company that would shortly implode in a carnival of cooked books and executive extravagance. Intellectually, he knew — indeed, he was professionally trained to know — that ploughing money into a single ”hot” company could only end badly, but still he kept spending.
At first he invested only an unexpected cash windfall, but slowly he began to use up more and more of his savings. ”That was where the money came in,” he recalls. ”It was frighteningly disjointed from my professional life, from my normal critical stance towards things … I remember writing these stupid, futile letters to [WorldCom chief executive] Bernie Ebbers. That was probably irrational.”
This is a painful admission for a mathematician to make, but in Paulos’s case there was, at least, a silver lining in the shape of his latest book.
A Mathematician Plays the Stock Market (published by Allen Lane) is a remorselessly self-flagellating attempt to pick apart where he went wrong. More generally it shows why no amount of clever maths can guarantee you a fast buck on the stock market — and why we all seem so hardwired to believe, against all logic, that we can beat the odds. (You can’t help suspecting that Paulos was spurred on in his own foolish trading — he won’t say how much he lost, except that it was plenty — by the knowledge that he’d get a book out of it.)
Most investors who lost their shirts in the collapse of WorldCom raged, understandably, against Ebbers and his colleagues. But Paulos, a wild-haired professor at Temple University in Philadelphia, was too well schooled in logic and rationality not to rage against himself, too. He had fallen victim, he says, to several ”cognitive delusions”. For example, there was the phenomenon of confirmation bias: the mental trick whereby, having made his purchases, he cast about looking for reasons to show that he’d made a brilliant decision, while ignoring evidence that didn’t add up. And he bought more and more shares even as the price fell, demonstrating another truth: that people will take a greater risk to protect money they already have than to gain money they don’t yet have. (If you give someone £1 000, and then offer either to give them £500 more, or to flip a coin and give them either £1 000 more or nothing, depending on the outcome, most will opt for the ”safe” £500. But if you give them £2 000 and then offer to take back £500, or to flip a coin and take back either £1 000 or nothing, they’ll opt to flip the coin — even though both situations are logically identical.)
Above all, though, he fell for the fundamental illusion that the stock market is largely non-random: that with enough intelligence and the right method of analysis, you’ve got a good chance of predicting which shares will do well. Unfortunately, you just can’t.
A glance at the United States financial channel CNBC, Paulos says, ought to demonstrate instantly that something suspicious is going on. ”It still never fails to amuse me,” he says. ”You see all these talking heads — yelling heads — with seemingly impressive credentials, long histories in the market, and they keep coming to diametrically opposed conclusions, and they’re not bothered by that. You never see biologists debating fundamental issues and screaming at each other. Or mathematicians.”
Put a little less simply, randomness rules on the stock market — and in many areas of life where we pretend not to see it — because of the truth of the ”efficient market hypothesis”, an economic insight first given its name in the 1960s.
Wherever people are involved in buying and selling shares, the hypothesis contends, they’re so intent on seeking out all the information that they can about companies that the share price ends up reflecting everything that is known about a company and its prospects at that point in time. All the available information is embodied in the price already. No extra information advantage can be dug up, future prices can’t be predicted and no amount of cleverness or training can give you an edge over someone just throwing a dice.
This is a vast generalisation, of course. Exceptions occur all the time — fleeting opportunities for people to make a quick profit from under- valued shares — which is why the efficient market hypothesis, taken to extremes, looks absurd. Hence the joke about the two efficient market theorists who leave a $100 note lying in the street, on the grounds that if it represented a real possibility of profit, it would already have been picked up by someone else.
But in the long run the hypothesis is mainly true, Paulos argues, which is extremely bad news for the thousands of people who make a living on their reputations as brilliant stockbrokers or wise market commentators. They’re just lucky, he says. ”If there are a thousand investors, half of them will do better than the index [of leading shares] next year, and half of those will do better than the index the year after that. And so on and so on, and eventually just by chance, you’ll have one in a thousand who beats it consistently, and that person will appear on the cover of Fortune, or of the London Financial Times.” They will be celebrated as a master stock-picker. But it will all be down to chance.
A well-known scam that plays on similar logic concerns an enterprising businessman who launches a newsletter, predicting which way the market will go from week to week. Paulos recounts it like this: in the first week he sends 64 000 news- letters, 32 000 predicting a rise and 32 000 predicting a fall. To 32 000 people, he will have seemed to have got it right, so to 16 000, in the next week’s newsletter, he predicts a rise, and to 16 000, a fall. And so on, until eventually there are 1 000 people who think he’s called it correctly six weeks in a row. To them, he offers an expensive subscription to carry on receiving more newsletters.
In this chance-dominated environment, the main kind of guessing that we end up trying to do is not really about the future movements of prices — but about what we think others are thinking and, worse, what we think they’re thinking about what we’re thinking. This is a recipe for radical uncertainty and confusion, as demonstrated by a particularly cruel trick Paulos played on his students. At the end of test papers, he sometimes added a little box and promised an extra 10 points to those who ticked the box — provided less than half the class did so. If more than half ticked it, though, they were warned, those who did would lose 10 points. After several weeks of test papers, the number of students ticking the box stabilised at about 40% — but a different 40% each time, as each student tried to guess how every other student might act.
In the end, the curse of cognitive illusions and misunderstood logic only led Paulos to lose some of his savings. But mathematical illiteracy in general, he contends, poses far greater threats to society. Take, for example, the US government’s nascent data-mining operation, originally called Total Information Awareness and now renamed Terrorist Information Awareness: the idea is to collect so many facts about so many people —details of appearance, health, credit records, travel and more — that terrorists can be identified accurately before they strike.
Suppose that the system could become 99% accurate, correctly identifying future terrorists as terrorists 99% of the time, and harmless people as harmless 99% of the time. That might sound like a fairly reasonable deal, until you do the maths.
If there are 1 000 future terrorists, say, in a population of 300-million (roughly equivalent to that of the US), the 99%-accurate system will catch 990 of them. But it will also wrongly identify as terrorists 1% of the harmless people, which is 1% of 300-million, or 2 999 990. ”It’s frightening,” Paulos says. ”Hundreds, thousands, millions of people would have these dossiers hanging over them, and most of the people arrested or followed will be innocent.”
Perhaps what we all need, to be good shareholders and good citizens, is advanced degrees in mathematics. But then again, as Paulos’s own romance with WorldCom demonstrates, even that may not always be enough. — Â