In Pursuit of the Unknown (57 page)

The banks behaved like one of those cartoon characters who wanders off the edge of a cliff, hovers in space until he looks down, and only then plunges to the ground. It all seemed to be going nicely until the bankers asked themselves whether multiple accounting with non-existent money and overvalued assets was sustainable, wondered what the real value of their holdings in derivatives was, and realised that they didn't have a clue. Except that it was definitely a lot less than they'd told shareholders and government regulators.

As the dreadful truth dawned, confidence plummeted. This depressed the housing market, so the assets against which the debts were secured started to lose their value. At this point the whole system became trapped in a positive feedback loop, in which each downward revision of value caused it to be revised even further downward. The end result was the loss of about 17 trillion dollars. Faced with the prospect of the total collapse of the world financial system, trashing depositors' savings and making the Great Depression of 1929 look like a garden party, governments were forced to bail out the banks, which were on the verge of bankruptcy. One, Lehman Brothers, was allowed to go under, but the loss of confidence was so great that it seemed unwise to repeat the lesson. So taxpayers stumped up the money, and a lot of it was real money. The banks grabbed the cash with both hands, and then tried to pretend that the catastrophe hadn't been their fault. They blamed government regulators, despite having campaigned against regulation: an interesting case of ‘It's your fault: you let us do it.'

How did the biggest financial train wreck in human history come about?

Arguably, one contributor was a mathematical equation.

The simplest derivatives have been around for a long time. They are known as futures and options, and they go back to the eighteenth century at the Dojima rice exchange in Osaka, Japan. The exchange was founded in 1697, a time of great economic prosperity in Japan, when the upper classes, the samurai, were paid in rice, not money. Naturally there emerged a class of ricebrokers who traded rice as though it were money. As the Osaka merchants strengthened their grip on rice, the country's staple food, their activities had a knock-on effect on the commodity's price. At the same
time, the financial system was beginning to shift to hard cash, and the combination proved deadly. In 1730 the price of rice dropped through the floor.

Ironically, the trigger was poor harvests. The samurai, still wedded to payment in rice, but watchful of the growth of money, started to panic. Their favoured ‘currency' was rapidly losing its value. Merchants exacerbated the problem by artificially keeping rice out of the market, squirrelling away huge quantities in warehouses. Although it might seem that this would increase the monetary price of rice, it had the opposite effect, because the samurai were treating rice as a currency. They could not eat anything remotely approaching the amount of rice they owned. So while ordinary people starved, the merchants stockpiled rice. Rice became so scarce that paper money took over, and it quickly became more desirable than rice because it was possible actually to lay hands on it. Soon the Dojima merchants were running what amounted to a gigantic banking system, holding accounts for the wealthy and determining the exchange rate between rice and paper money.

Eventually the government realised that this arrangement handed far too much power to the rice merchants, and reorganised the Rice Exchange along with most other parts of the country's economy. In 1939 the Rice Exchange was replaced by the Government Rice Agency. But while the Rice Exchange existed, the merchants invented a new kind of contract to even out the large swings in the price of rice. The signatories guaranteed to buy (or sell) a specified quantity of rice at a specified future date for a specified price. Today these instruments are known as futures or options. Suppose a merchant agrees to buy rice in six months' time at an agreed price. If the market price has risen above the agreed one by the time the option falls due, he gets the rice cheap and immediately sells it at a profit. On the other hand, if the price is lower, he is committed to buying rice at a higher price than its market value and makes a loss.

Farmers find such instruments useful because they actually want to sell a real commodity: rice. People using rice for food, or manufacturing foodstuffs that use it, want to buy the commodity. In this sort of transaction, the contract reduces the risk to both parties – though at a price. It amounts to a form of insurance: a guaranteed market at a guaranteed price, independent of shifts in the market value. It's worth paying a small premium to avoid uncertainty. But most investors took out contracts in rice futures with the sole aim of making money, and the last thing the investor wanted was tons and tons of rice. They always sold it before they had to take delivery. So the main role of futures was to fuel
financial speculation, and this was made worse by the use of rice as currency. Just as today's gold standard creates artificially high prices for a substance (gold) that has little intrinsic value, and thereby fuels demand for it, so the price of rice became governed by the trading of futures rather than the trading of rice itself. The contracts were a form of gambling, and soon the contracts themselves acquired a value, and could be traded as though they were real commodities. Moreover, although the amount of rice was limited by what the farmers could grow, there was no limit to the number of contracts for rice that could be issued.

The world's major stock markets were quick to spot an opportunity to convert smoke and mirrors into hard cash, and they have traded futures ever since. At first, this practice did not of itself cause enormous economic problems, although it sometimes led to instability rather than the stability that is often asserted to justify the system. But around the year 2000, the world's financial sector began to invent ever more elaborate variants on the futures theme, complex ‘derivatives' whose value was based on hypothetical future movements of some asset. Unlike futures, for which the asset, at least, was real, derivatives might be based on an asset that was itself a derivative. No longer were banks buying and selling bets on the future price of a commodity like rice; they were buying and selling bets on the future price of a
bet
.

It quickly became big business. In 1998 the international financial system traded roughly $100 trillion in derivatives. By 2007 this had grown to one quadrillion US dollars. Trillions, quadrillions. . . we know these are large numbers, but how large? To put this figure in context, the total value of all the products made by the world's manufacturing industries, for the last thousand years, is about 100 trillion US dollars, adjusted for inflation. That's one tenth of one year's derivatives trading. Admittedly the bulk of industrial production has occurred in the past fifty years, but even so, this is a staggering amount. It means, in particular, that the derivatives trades consist almost entirely of money that does not actually exist – virtual money, numbers in a computer, with no link to anything in the real world. In fact, these trades
have
to be virtual: the total amount of money in circulation, worldwide, is completely inadequate to pay the amounts that are being traded at the click of a mouse. By people who have no interest in the commodity concerned, and wouldn't know what to do with it if they took delivery, using money that they don't actually possess.

You don't need to be a rocket scientist to suspect that this is a recipe for
disaster. Yet for a decade, the world economy grew relentlessly on the back of derivatives trading. Not only could you get a mortgage to buy a house: you could get more than the house was worth. The bank didn't even bother to check what your true income was, or what other debts you had. You could get a 125% self-certified mortgage – meaning you told the bank what you could afford and it didn't ask awkward questions – and spend the surplus on a holiday, a car, plastic surgery, or crates of beer. Banks went out of their way to persuade customers to take out loans, even when they didn't need them.

What they thought would save them if a borrower defaulted on their repayments was straightforward. Those loans were secured on your house. House prices were soaring, so that missing 25% of equity would soon become real; if you defaulted, the bank could seize your house, sell it, and get its loan back. It seemed foolproof. Of course it wasn't. The bankers didn't ask themselves what would happen to the price of housing if hundreds of banks were all trying to sell millions of houses at the same time. Nor did they ask whether prices could continue to rise significantly faster than inflation. They genuinely seemed to think that house prices could rise 10–15% in real terms every year, indefinitely. They were still urging regulators to relax the rules and allow them to lend even more money when the bottom dropped out of the property market.

Many of today's most sophisticated mathematical models of financial systems can be traced back to Brownian motion, mentioned in
Chapter 12
. When viewed through a microscope, small particles suspended in a fluid jiggle around erratically, and Einstein and Smoluchowski developed mathematical models of this process and used them to establish the existence of atoms. The usual model assumes that the particle receives random kicks through distances whose probability distribution is normal, a bell curve. The direction of each kick is uniformly distributed – any direction has the same chance of happening. This process is called a random walk. The model of Brownian motion is a continuum version of such random walks, in which the sizes of the kicks and the time between successive kicks become arbitrarily small. Intuitively, we consider infinitely many infinitesimal kicks.

The statistical properties of Brownian motion, over large numbers of trials, are determined by a probability distribution, which gives the likelihood that the particle ends up at a particular location after a given time. This distribution is radially symmetric: the probability depends only
on how far the point is from the origin. Initially the particle is very likely to be close to the origin, but as time passes, the range of likely positions spreads out as the particle gets more chance to explore distant regions of space. Remarkably, the time evolution of this probability distribution obeys the heat equation, which in this context is often called the diffusion equation. So the probability spreads just like heat.

After Einstein and Smoluchowski published their work, it turned out that much of the mathematical content had been derived earlier, in 1900, by the French mathematician Louis Bachelier in his PhD thesis. But Bachelier had a different application in mind: the stock and option markets. The title of his thesis was
Théorie de la speculation
(‘Theory of Speculation'). The work was not received with wild praise, probably because its subject-matter was far outside the normal range of mathematics at that period. Bachelier's supervisor was the renowned and formidable mathematician Henri Poincaré, who declared the work to be ‘very original'. He also gave the game way somewhat, by adding, with reference to the part of the thesis that derived the normal distribution for errors: ‘It is regrettable that M. Bachelier did not develop this part of his thesis further.' Which any mathematician would interpret as ‘that was the place where the mathematics started to get really interesting, and if only he'd done more work on that, rather than on fuzzy ideas about the stock market, it would have been easy to give him a much better grade.' The thesis was graded ‘honorable', a pass; it was even published. But it did not get the top grade of ‘très honorable'.

Bachelier in effect pinned down the principle that fluctuations of the stock market follow a random walk. The sizes of successive fluctuations conform to a bell curve, and the mean and standard deviation can be estimated from market data. One implication is that large fluctuations are very improbable. The reason is that the tails of the normal distribution die down very fast indeed: faster than exponential. The bell curve decreases towards zero at a rate that is exponential in the
square
of
x
. Statisticians (and physicists and market analysts) talk of two-sigma fluctuations, three-sigma ones, and so on. Here sigma (
σ
) is the standard deviation, a measure of how wide the bell curve is. A three-sigma fluctuation, say, is one that deviates from the mean by at least three times the standard deviation. The mathematics of the bell curve lets us assign probabilities to these ‘extreme events', see
Table 3
.

Table 3
Probabilities of many-sigma events.

The upshot of Bachelier's Brownian motion model is that large stock market fluctuations are so rare that in practice they should never happen.
Table 3
shows that a five-sigma event, for example, is expected to occur about six times in every 10 million trials. However, stock market data show that they are far more common than that. Stock in Cisco Systems, a world leader in communications, has undergone ten 5-sigma events in the last twenty years, whereas Brownian motion predicts 0.003 of them. I picked this company at random and it's in no way unusual. On Black Monday (19 October 1987) the world's stock markets lost more than 20% of their value within a few hours; an event this extreme should have been virtually impossible.