Of couples and copulas
How one actuary's insight into life, death and love eventually brought Wall Street to its knees.
By Sam Jones
Published: April 24 2009 22:04 | Last updated: April 24 2009 22:04
Johnny Cash and June Carter met backstage at the Grand Ole Opry. It was a little like a country song: he was married, she recently divorced, and an affair ensued; both singers had young children, and Cash would have three more with his first wife before she left him in 1966, citing his drinking and carousing. Two years later, he proposed to Carter on stage and, despite having turned him down numerous times before, she accepted. They’d each found a life match.
It ended like a country song, too. In 2003, Carter died in Nashville of complications from heart surgery, and Cash followed her to the grave four months later. The heart complication for him, it seemed, was that it was broken: “It hurts so bad,” he told the audience at the last concert he would give. The pain, he said, tuning his guitar, close to tears, was “the big one. It’s the biggest.”
Cash was speaking for many a bereaved partner – and well before Johnny ever met June, scientists had noticed that cases of spouses dying in rapid succession were not at all unusual. By the 1980s, medical researchers had started writing about “stress cardiomyopathy”, or “apical ballooning syndrome”, the ungainly name for the peculiar condition whereby an individual’s brain, following an intense emotional trauma, would inexplicably release chemicals into the bloodstream that weakened the heart – in some cases, causing it quite literally to break.
The medical community was interested because it offered a chance, potentially, to intervene and prolong life. Another industry was interested in the phenomenon, too – but less to stop it and more to understand it. These were the actuaries working in life assurance. Actuarial science is the study of the statistics surrounding life and death – and the statistics surrounding the broken heart phenomenon were striking. Pages and pages of death records showed the same marked trend: that in human couples, the death of one partner significantly increases the chances of the death of the other. Dying of a broken heart – in the most general sense, not necessarily from stress cardiomyopathy – was not a rare occurrence, but something of a statistical probability. So much so that life assurers, in order to conduct their business, needed to incorporate it into their models. In a March 2008 study, Jaap Spreeuw and Xu Wang of the Cass Business School observed that in the year following a loved one’s death, women were more than twice as likely to die than normal, and men more than six times as likely. “This implies … that joint life annuities [in which payments continue at the same price until both partners die] are underpriced while last survivor annuities [in which payments increase after one partner dies] are overpriced,” concluded the authors.
Even before the definitive Cass study, however, actuaries had begun to incorporate the broken heart trend into their mathematical models calculating the chances of clients dying. How could such an ephemeral relationship be reliably captured? The actuaries, of course, relied on probability. While they could not hope to devise a model that predicted the likelihood of death from a broken heart for a specific couple, they could use statistical science to devise a fairly accurate picture across a group of people.
They borrowed from physics and devised a formula based on something called a Markov chain: a way of expressing a series of statistical events whose outcomes are dependent upon one another. In physics, Markov chain processes underlie our most basic understanding of the world around us, from the way liquid turns to gas to the way a drop of vivid ink might slowly diffuse through a glass of water. If you treated people like atoms, the actuaries reasoned, you could apply the same maths.
. . .
In the autumn of 1987, the man who would become the world’s most influential actuary landed in Canada on a flight from China. Neither Xiang Lin Li nor the handful of fellow junior academics with whom he was travelling – all from the University of Nankai – had ever been abroad before, yet they had come at the behest of the Chinese and Canadian governments to do something most unusual: study capitalism. The small band of mathematicians and statisticians would be taking business degrees at Quebec’s Laval University.
For Li, going to Canada was just the latest in a series of unlikely opportunities that had shaped his life up to then. Decades earlier, his family had suffered at the blunt end of Mao’s cultural revolution: his father, a mid-ranking police official, was precisely the kind of lowly bureaucrat that the red guard mob was intent on re-educating, and the family was uprooted to a small village in southern China. In the countryside, the chances of young Xiang Lin going to school – let alone university – were slim. But he was talented and driven, and made it not only into school, but on to Nankai, one of the country’s most prestigious institutions. Li studied economics, and had just passed his master’s examinations when the Canadians came calling. Determined to be among those sent to Quebec, he learnt French in four months – as much at home studying the language, it seemed, as he was crunching numbers.
Li’s drive did not abate abroad. Four years after arriving in Canada, he’d earned his MBA; by this stage, he had no intention of returning home. In the few years he had been away, China’s mini-glasnost period had withered. Hu Yaobang, general secretary of the Communist party and a pro-democratic reformist, had been ousted, and Chinese leader Deng Xiaoping was now wary of the liberalism genie that had been let out of the bottle. In 1989, the world had looked on as students were mowed down in Tiananmen Square. Universities such as Nankai were not exactly the safest places for ambitious young students – especially not ones returning from MBA courses in the west.
As if to make clear the break, Xiang Lin changed his name and became David Li. After graduation from Laval, he enrolled at a new university, Waterloo, near Toronto. He would now be studying actuarial science. And this wasn’t the only change: the move from genteel, francophone Montreal to the more worldly and business-oriented Toronto was profound – and deliberate. According to Jie Dai, a fellow immigrant from China and a classmate at Laval, “I clearly remember [Li] mention that if you are an actuarial guy, you can earn a lot of money.”
. . .
The big money in the 1990s, of course, was not to be found at Waterloo but in Silicon Valley, on Wall Street and in the City of London. The first of these might have been an obvious destination for a talented mathematician, but the latter two were also becoming magnets for the likes of Li. In 1984, Robert Rubin, who a decade later would become US treasury secretary, made a bold decision for his employer at the time, the investment bank Goldman Sachs. Rubin decided to hire Fischer Black, an economist and academic at the Massachusetts Institute of Technology’s Sloan School of Management. Prior to 1983, a few academics had toyed with economics and markets, but as intellectual curiosities; Black was the first of his kind – a serious academic, with publications under his belt and a tenured position to boot – to make the move to Wall Street, putting theory into practice and risking the scorn of his ivory tower colleagues.
Rubin’s bet earned Goldman many multiples of Black’s salary. At the bank, the professor pioneered the use of mathematics in pursuit of money. He was one half of the duo that came up with the Black-Scholes formula, which revolutionised Wall Street by promising to determine a rational price for market risks – a principle that would become the founding doctrine of a new field, quantitative finance. Quantitative finance’s practitioners were trying to outwit the markets, using maths to eliminate risk by first using maths to calculate it. And the numbers of those practitioners grew quickly. With the collapse of the Soviet Union, the end of the arms race and, in 1993, the cancellation by the US congress of the superconducting super collider – intended to be the world’s greatest physics experiment – particle physicists, experts in quantum mechanics and computing engineers were twiddling their thumbs. For the younger generation of newly qualified grads and PhDs, applying their expertise to finance was the obvious alternative to fighting it out for the dwindling number of jobs strictly in their fields.
Emanuel Derman – a particle physicist – was such a convert. He joined Goldman in 1985, working under Black before eventually taking over from his mentor, and recalls members of the “quant” influx being referred to as “POWs” – physicists on Wall Street. Equally accurate was the acronym used by Andrew Lo, another Wall Street quant and now a lecturer at MIT. What Wall Street was really after, said Lo in a recent lecture, was not PhDs, but PSDs: people who were “poor, smart and with a deep desire to get rich”. At Waterloo, Li fitted that description to a T. He was studying for his PhD in actuarial science, but no one expected him to go on to a career in academia. Instead, in 1997, after earning his doctorate, he took a job at one of Canada’s largest banks, Canadian Imperial Bank of Commerce (CIBC).
For graduates such as Li, joining the rough and tumble world of business could be something of a shock, even when armed with MBAs. At best, the mathematicians were semi-disparagingly referred to as quants; if they were lucky, trader colleagues might slap them on the back and call them, half in flattery, “rocket scientists”. Emanuel Derman recalls a time at Goldman Sachs when he and another quant colleague were standing on the trading floor, on either side of a central aisle, and a senior trader passed between them. The trader “winced, clutching his head with both hands as though in excruciating pain, and exclaimed, ‘Aaarrggh-hhh! The force field! It’s too intense! Let me out of the way!’”
And yet by the time Li got to New York, in 1998, the quants had taken over the asylum. In the summer of that year, Long Term Capital Management, a hedge fund run by the finest minds in quantitative finance, required a massive bailout from the federal government. But far from serving as a warning that mathematical models could get investors into serious trouble, LTCM exploded the notion of quantitative finance as a geeky, back-office support task. The fund’s might before its fall – and the fact that its failure might have left a trillion-dollar hole in the financial system – discounted the notion that traders’ instinct and experience counted more than numerical intelligence.
The quants weren’t exactly out on the trading floor, however. The best of them still spent their days writing papers, crunching numbers, applying their academic expertise to the world of business. Li had come to New York to work for a consultancy called the RiskMetrics Group, which had been spun out of JP Morgan, but he was still thinking about life, death and love. In 2000, he published a paper in the prestigious Journal of Fixed Income that gained some serious attention. In it, Li performed a most elegant trick. Borrowing from his work in actuarial science and insurance and his knowledge of the broken-heart syndrome, he attempted to solve one of Wall Street quants’ most intractable problems: default correlation.
Markets do not function in laboratory-like isolation. They are linked, correlated. It isn’t enough for any quant to try and know the probability of each individual company in his bank’s portfolio going bust; he has to know how the bankruptcy of one company – or several – might increase (or decrease) the likelihood that other companies will default. Suppose, for example, that a bank loans money to two outfits – a dairy farm and a dairy. The farm, according to ratings agencies, has a 10 per cent chance of going bust and the dairy a 5 per cent chance. But if the farm does go under, the chances that the dairy will follow will rise above 5 per cent – quickly and steeply – if the farm was its main milk supplier.
And it gets more complicated from there. How correlated are the default probabilities on bonds issued by our Irish dairy farm and those issued by a software company in Malaysia? Not at all, you might think: the businesses not only provide totally different products and services, they’re also geographically remote from each other. Suppose, though, that both companies have been lent money by the same troubled bank that is now calling in its loans.
In fact, this is exactly what sank LTCM. How correlated are Russian government bonds and those in Mexico? Not at all, according to LTCM’s model, which, it should be noted, crunched data going back a hundred years. And yet it turned out for the hedge fund that both markets were dominated by the same few investors. The 1998 financial crisis in Russia, when Boris Yeltsin’s government defaulted on its bonds, caused panic selling in Mexico as investors rushed to de-risk their portfolios.
Li realised that his insight was groundbreaking. Speaking to The Wall Street Journal seven years later, he said: “Suddenly I thought that the problem I was trying to solve [as an actuary] was exactly the problem these guys were trying to solve. Default [on a loan] is like the death of a company.” And if he could apply the broken hearts maths to broken companies, he’d have a way of mathematically modelling the effect that one company’s default would have on the chance of default for others.
. . .
When mathematicians and physicists want to describe the chances of events occurring, they often rely on a curve called a copula. The Latin root is a noun meaning a “link or tie”, and indeed, copulas connect variables in such a way that their interdependence can be plotted. Throughout his PhD at Waterloo, and at CIBC, Li had been interested in how he could use copulas to develop existing actuarial models of the broken heart syndrome. The problem with relying on Markov chains was that they painted a far too mechanical, physical – atomic, even – picture of the human lifespan. Li reasoned that with a copula that showed a probable distribution of outcomes, a more accurate, encompassing picture of the broken heart or, for that matter, the broken company, could be devised.
He decided to use a very standard type of curve – the Gaussian copula, which is better known as a bell curve, or normal distribution – to map and determine the correlation on any given portfolio of assets. In the same way that actuaries could tell their employers the chances of Johnny Cash dying soon after June Carter without knowing anything about Cash other than the fact of his recent widowhood, so quants could tell their employers the effect one company defaulting might have on another doing the same – without knowing anything about the companies themselves. From this point on, it really could be, would be, a number-crunching game.
By 2003, Li’s paper had made his name on Wall Street. By now he was director and global head of derivatives research at Citigroup, and on a bright Tuesday morning in November, he arrived at the annual Quant Congress to bask in the glory with a presentation about his work. In front of a room of hundreds of fellow quants (“not a million miles from some kind of science fiction convention”, one person who was there recalls) he ran through his model – the Gaussian copula function for default.
The presentation was a riot of equations, mathematical lemmas, arching curves and matrices of numbers. The questions afterwards were deferential, technical. Li, it seemed, had found the final piece of a risk-management jigsaw that banks had been slowly piecing together since quants arrived on Wall Street.
. . .
By 2001, correlation was a big deal. A new fervour was gripping Wall Street – one almost as revolutionary as that which had struck when the Black-Scholes model brought about the explosion in stock options and derivatives in the early 1980s. This was structured finance, the culmination of two decades of quants on Wall Street. The basic idea was simple: that banks no longer had to hold on to risks. Instead they could value them, using complex maths and modelling, then package and trade them like any other, ordinary security.
Mortgages were the prime example. Rather than make a mortgage loan and gradually collect interest over its lifespan, banks began to bundle the loans together and sell them into specially created off-balance-sheet shell companies. These companies in turn issued bonds to raise cash. And by using the modelling and maths being cranked out by quants, banks were able to tailor the structure of mortgage portfolios to ensure that bonds of varying risks could be issued to investors. The problem, however, was correlation. The one thing any off-balance-sheet securitisation could not properly capture was the interrelatedness of all the hundreds of thousands of different mortgage loans they owned. As a consequence, structured finance had remained a niche and highly bespoke practice throughout the 1990s.
On August 10 2004, however, the rating agency Moody’s incorporated Li’s Gaussian copula default function formula into its rating methodology for collateralised debt obligations, the structured finance instruments that subsequently proved the nemesis of so many banks. Previously, Moody’s had insisted that CDOs meet a diversity score – that is, that each should contain different types of assets, such as commercial mortgages, student loans and credit card debts, as well as the popular subprime debt. This was standard investing good practice, where the best way to guard against risk is to avoid putting all your eggs in one basket. But Li’s formula meant Moody’s now had a model that enabled it to gauge the interrelatedness of risks – and that traditional good practice could be thrown out of the window, since risk could be measured with mathematical certainty. No need to spread your eggs across baskets if you knew the exact odds of your one basket being dropped. A week after Moody’s, the world’s other large rating agency, Standard & Poor’s, changed its methodology, too.
CDOs built solely out of subprime mortgage debt became the rage. And using the magic of the Gaussian copula correlation model, and some clever off-balance-sheet architecture, high-risk mortgages were re-packaged into triple-A-rated investor gold. The CDO market exploded. In 2000, the total number of CDOs issued were worth somewhere in the tens of billions of dollars. By 2007, two trillion dollars of CDO bonds had been issued. And with so many investors looking to put their money in debt, that debt became incredibly cheap, fuelling a massive boom in house prices and turbo-charging the world’s economies.
. . .
The unwinding started, as we all now know, in the US subprime housing market. Defaults started to increase in late 2006. The banks weren’t worried at first. Their models assumed that the pinprick default points all over the US were not correlated. But the defaults kept coming. By early 2007, it was clear that the US subprime market had a problem and by that summer, homeowners all over the US were defaulting on their mortgages. The cheap debt made available by the finance revolution was so cheap, in fact, that the loans should never have been made. And the correlation model was still mapping the housing market as it had been in 1990s, not the grossly inflated monster it had become. The development of the model had, ironically, changed the nature of the reality it was modelling.
The losses the banks began to take against their holdings of CDOs were staggering. And as the institutions grew fearful about one another’s solvency, they stopped lending to each other. Global liquidity dried up. The rot spread from asset class to asset class, and the banks’ pain spread to the real economy. Suddenly, everything was highly correlated.
How had Li’s formula failed to anticipate this? The problem was that it assumed events tended to cluster heavily around an average – a “normal” state. In actuarial science, Li’s formula could adequately capture binary outcomes such as life or death, but in the messy world of mortgages and economics, it faltered. The range of possible outcomes here was more complicated, and indeed, random, than those facing an insurance company’s clients. Markets – particularly the mortgage market – were far more prone to extreme correlation scenarios than were insurers. Death from a broken heart, for all its poetic associations, is far easier to predict than the more prosaic, but ultimately unknowable, interrelatedness of markets.
Why did no one notice the formula’s Achilles heel? Some did. Nassim Nicholas Taleb, author of the bestselling The Black Swan – a book about the importance of considering outliers when looking at copulas – was a voluble critic of quantitative finance and Li’s formula. “The thing never worked,” he says. “Anything that relies on correlation is charlatanism.”
In 2007, David Li left Wall Street and moved back to China. He could not be contacted for this story. But two years earlier, before the financial system blew up, he did warn: “Very few people understand the essence of the model.” Harry Panjer, a professor of statistics and actuarial science who was Li’s mentor at Waterloo, strikes a balance between Taleb’s accusations and the stance of Li. Earlier this year, Panjer told The Toronto Star newspaper: “We have a saying in statistics, ‘All models are wrong, but some are useful.’” And David Li’s model was, for a period, profoundly useful.
Sam Jones is a reporter for FT Alphaville.
Copyright The Financial Times Limited 2009