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A special report on the future of finance
In Plato's cave
Jan 22nd 2009
From The Economist print edition
Mathematical models are a powerful way of predicting financial markets. But they are fallible
Illustration by S. Kambayashi
ROBERT RUBIN was Bill Clinton’s treasury secretary. He has worked at the top of Goldman Sachs and Citigroup. But he made arguably the single most influential decision of his long career in 1983, when as head of risk arbitrage at Goldman he went to the MIT Sloan School of Management in Cambridge, Massachusetts, to hire an economist called Fischer Black.
A decade earlier Myron Scholes, Robert Merton and Black had explained how to use share prices to calculate the value of derivatives. The Black-Scholes options-pricing model was more than a piece of geeky mathematics. It was a manifesto, part of a revolution that put an end to the anti-intellectualism of American finance and transformed financial markets from bull rings into today’s quantitative powerhouses. Yet, in a roundabout way, Black’s approach also led to some of the late boom’s most disastrous lapses.
Derivatives markets are not new, nor are they an exclusively Western phenomenon. Mr Merton has described how Osaka’s Dojima rice market offered forward contracts in the 17th century and organised futures trading by the 18th century. However, the growth of derivatives in the 36 years since Black’s formula was published has taken them from the periphery of financial services to the core.
In “The Partnership”, a history of Goldman Sachs, Charles Ellis records how the derivatives markets took off. The International Monetary Market opened in 1972; Congress allowed trade in commodity options in 1976; S&P 500 futures launched in 1982, and options on those futures a year later. The Chicago Board Options Exchange traded 911 contracts on April 26th 1973, its first day (and only one month before Black-Scholes appeared in print). In 2007 the CBOE’s volume of contracts reached almost 1 trillion.
Trading has exploded partly because derivatives are useful. After America came off the gold standard in 1971, businesses wanted a way of protecting themselves against the movements in exchange rates, just as they sought protection against swings in interest rates after Paul Volcker, Mr Greenspan’s predecessor as chairman of the Fed, tackled inflation in the 1980s. Equity options enabled investors to lay off general risk so that they could concentrate on the specific types of corporate risk they wanted to trade.
The other force behind the explosion in derivatives trading was the combination of mathematics and computing. Before Black-Scholes, option prices had been little more than educated guesses. The new model showed how to work out an option price from the known price-behaviour of a share and a bond. It is as if you had a formula for working out the price of a fruit salad from the prices of the apples and oranges that went into it, explains Emanuel Derman, a physicist who later took Black’s job at Goldman. Confidence in pricing gave buyers and sellers the courage to pile into derivatives. The better that real prices correlate with the unknown option price, the more confidently you can take on any level of risk. “In a thirsty world filled with hydrogen and oxygen,” Mr Derman has written, “someone had finally worked out how to synthesise H2O.”
Poetry in Brownian motion
Black-Scholes is just a model, not a complete description of the world. Every model makes simplifications, but some of the simplifications in Black-Scholes looked as if they would matter. For instance, the maths it uses to describe how share prices move comes from the equations in physics that describe the diffusion of heat. The idea is that share prices follow some gentle random walk away from an equilibrium, rather like motes of dust jiggling around in Brownian motion. In fact, share-price movements are more violent than that.
Over the years the “quants” have found ways to cope with this—better ways to deal with, as it were, quirks in the prices of fruit and fruit salad. For a start, you can concentrate on the short-run volatility of prices, which in some ways tends to behave more like the Brownian motion that Black imagined. The quants can introduce sudden jumps or tweak their models to match actual share-price movements more closely. Mr Derman, who is now a professor at New York’s Columbia University and a partner at Prisma Capital Partners, a fund of hedge funds, did some of his best-known work modelling what is called the “volatility smile”—an anomaly in options markets that first appeared after the 1987 stockmarket crash when investors would pay extra for protection against another imminent fall in share prices.
The fixes can make models complex and unwieldy, confusing traders or deterring them from taking up new ideas. There is a constant danger that behaviour in the market changes, as it did after the 1987 crash, or that liquidity suddenly dries up, as it has done in this crisis. But the quants are usually pragmatic enough to cope. They are not seeking truth or elegance, just a way of capturing the behaviour of a market and of linking an unobservable or illiquid price to prices in traded markets. The limit to the quants’ tinkering has been not mathematics but the speed, power and cost of computers. Nobody has any use for a model which takes so long to compute that the markets leave it behind.
The idea behind quantitative finance is to manage risk. You make money by taking known risks and hedging the rest. And in this crash foreign-exchange, interest-rate and equity derivatives models have so far behaved roughly as they should.
A muddle of mortgages
Yet the idea behind modelling got garbled when pools of mortgages were bundled up into collateralised-debt obligations (CDOs). The principle is simple enough. Imagine a waterfall of mortgage payments: the AAA investors at the top catch their share, the next in line take their share from what remains, and so on. At the bottom are the “equity investors” who get nothing if people default on their mortgage payments and the money runs out.
Despite the theory, CDOs were hopeless, at least with hindsight (doesn’t that phrase come easily?). The cash flowing from mortgage payments into a single CDO had to filter up through several layers. Assets were bundled into a pool, securitised, stuffed into a CDO, bits of that plugged into the next CDO and so on and on. Each source of a CDO had interminable pages of its own documentation and conditions, and a typical CDO might receive income from several hundred sources. It was a lawyer’s paradise.
This baffling complexity could hardly be more different from an equity or an interest rate. It made CDOs impossible to model in anything but the most rudimentary way—all the more so because each one contained a unique combination of underlying assets. Each CDO would be sold on the basis of its own scenario, using central assumptions about the future of interest rates and defaults to “demonstrate” the payouts over, say, the next 30 years. This central scenario would then be “stress-tested” to show that the CDO was robust—though oddly the tests did not include a 20% fall in house prices.
This was modelling at its most feeble. Derivatives model an unknown price from today’s known market prices. By contrast, modelling from history is dangerous. There was no guarantee that the future would be like the past, if only because the American housing market had never before been buoyed up by a frenzy of CDOs. In any case, there are not enough past housing data to form a rich statistical picture of the market—especially if you decide not to include the 1930s nationwide fall in house prices in your sample.
Illustration by S. Kambayashi
Neither could the models take account of falling mortgage-underwriting standards. Mr Rajan of the University of Chicago says academic research suggests mortgage originators, keen to automate their procedures, stopped giving potential borrowers lengthy interviews because they could not easily quantify the firmness of someone’s handshake or the fixity of their gaze. Such things turned out to be better predictors of default than credit scores or loan-to-value ratios, but the investors at the end of a long chain of securities could not monitor lending decisions.
The issuers of CDOs asked rating agencies to assess their quality. Although the agencies insist that they did a thorough job, a senior quant at a large bank says that the agencies’ models were even less sophisticated than the issuers’. For instance, a BBB tranche in a CDO might pay out in full if the defaults remained below 6%, and not at all once they went above 6.5%. That is an all-or-nothing sort of return, quite different from a BBB corporate bond, say. And yet, because both shared the same BBB rating, they would be modelled in the same way.
Issuers like to have an edge over the rating agencies. By paying one for rating the CDOs, some may have laid themselves open to a conflict of interest. With help from companies like Codefarm, an outfit from Brighton in Britain that knew the agencies’ models for corporate CDOs, issuers could build securities with any risk profile they chose, including those made up from lower-quality ingredients that would nevertheless win AAA ratings. Codefarm has recently applied for administration.
There is a saying on Wall Street that the test of a product is whether clients will buy it. Would they have bought into CDOs had it not been for the dazzling performance of the quants in foreign-exchange, interest-rate and equity derivatives? There is every sign that the issuing banks believed their own sales patter. The banks so liked CDOs that they held on to a lot of their own issues, even when the idea behind the business had been to sell them on. They also lent buyers much of the money to bid for CDOs, certain that the securities were a sound investment. With CDOs in deep trouble, the lenders are now suffering.
Modern finance is supposed to be all about measuring risks, yet corporate and mortgage-backed CDOs were a leap in the dark. According to Mr Derman, with Black-Scholes “you know what you are assuming when you use the model, and you know exactly what has been swept out of view, and hence you can think clearly about what you may have overlooked.” By contrast, with CDOs “you don’t quite know what you are ignoring, so you don’t know how to adjust for its inadequacies.”
Now that the world has moved far beyond any of the scenarios that the CDO issuers modelled, investors’ quantitative grasp of the payouts has fizzled into blank uncertainty. That makes it hard to put any value on them, driving away possible buyers. The trillion-dollar bet on mortgages has gone disastrously wrong. The hope is that the trillion-dollar bet on companies does not end up that way too.
Almost as damaging is the hash that banks have made of “value-at-risk” (VAR) calculations, a measure of the potential losses of a portfolio. This is supposed to show whether banks and other financial outfits are being safely run. Regulators use VAR calculations to work out how much capital banks need to put aside for a rainy day. But the calculations are flawed.
The mistake was to turn a blind eye to what is known as “tail risk”. Think of the banks’ range of possible daily losses and gains as a distribution. Most of the time you gain a little or lose a little. Occasionally you gain or lose a lot. Very rarely you win or lose a fortune. If you plot these daily movements on a graph, you get the familiar bell-shaped curve of a normal distribution (see chart 4). Typically, a VAR calculation cuts the line at, say, 98% or 99%, and takes that as its measure of extreme losses.
However, although the normal distribution closely matches the real world in the middle of the curve, where most of the gains or losses lie, it does not work well at the extreme edges, or “tails”. In markets extreme events are surprisingly common—their tails are “fat”. Benoît Mandelbrot, the mathematician who invented fractal theory, calculated that if the Dow Jones Industrial Average followed a normal distribution, it should have moved by more than 3.4% on 58 days between 1916 and 2003; in fact it did so 1,001 times. It should have moved by more than 4.5% on six days; it did so on 366. It should have moved by more than 7% only once in every 300,000 years; in the 20th century it did so 48 times.
In Mr Mandelbrot’s terms the market should have been “mildly” unstable. Instead it was “wildly” unstable. Financial markets are plagued not by “black swans”—seemingly inconceivable events that come up very occasionally—but by vicious snow-white swans that come along a lot more often than expected.
This puts VAR in a quandary. On the one hand, you cannot observe the tails of the VAR curve by studying extreme events, because extreme events are rare by definition. On the other you cannot deduce very much about the frequency of rare extreme events from the shape of the curve in the middle. Mathematically, the two are almost decoupled.
The drawback of failing to measure the tail beyond 99% is that it could leave out some reasonably common but devastating losses. VAR, in other words, is good at predicting small day-to-day losses in the heart of the distribution, but hopeless at predicting severe losses that are much rarer—arguably those that should worry you most.
When David Viniar, chief financial officer of Goldman Sachs, told the Financial Times in 2007 that the bank had seen “25-standard-deviation moves several days in a row”, he was saying that the markets were at the extreme tail of their distribution. The centre of their models did not begin to predict that the tails would move so violently. He meant to show how unstable the markets were. But he also showed how wrong the models were.
Modern finance may well be making the tails fatter, says Daron Acemoglu, an economist at MIT. When you trade away all sorts of specific risk, in foreign exchange, interest rates and so forth, you make your portfolio seem safer. But you are in fact swapping everyday risk for the exceptional risk that the worst will happen and your insurer will fail—as AIG did. Even as the predictable centre of the distribution appears less risky, the unobserved tail risk has grown. Your traders and managers will look as if they are earning good returns on lower risk when part of the true risk is hidden. They will want to be paid for their skill when in fact their risk-weighted returns may have fallen.
Edmund Phelps, who won the Nobel prize for economics in 2006, is highly critical of today’s financial services. “Risk-assessment and risk-management models were never well founded,” he says. “There was a mystique to the idea that market participants knew the price to put on this or that risk. But it is impossible to imagine that such a complex system could be understood in such detail and with such amazing correctness…the requirements for information…have gone beyond our abilities to gather it.”
Every trading strategy draws upon a model, even if it is not expressed in mathematical symbols. But Mr Phelps believes that mathematics can take you only so far. There is a big role for judgment and intuition, things that managers are supposed to provide. Why have they failed?