Copula in cosmology?

A new paper co-authored by one of my collaborators: http://arxiv.org/abs/0909.5187

From Finance to Cosmology: The Copula of Large-Scale Structure

Robert J. Scherrer, Andreas A. Berlind, Qingqing Mao, Cameron K. McBride (Vanderbilt University)

Any multivariate distribution can be uniquely decomposed into marginal (1-point) distributions, and a function called the copula, which contains all of the information on correlations between the distributions. The copula provides an important new methodology for analyzing the density field in large-scale structure. We derive the empirical 2-point copula for the evolved dark matter density field. We find that this empirical copula is well-approximated by a Gaussian copula. We consider the possibility that the full n-point copula is also Gaussian and describe some of the consequences of this hypothesis. Future directions for investigation are discussed.

I don't understand the gravitational dynamics well enough to guess whether we should expect the dark matter distribution today to be described by Gaussian copula. The authors insert a number of caveats, referring to the use of copula to price mortgage securities :-)

See here for some correspondence with Scherrer on the topic of genius and modern science, and here for some popular coverage of the paper we wrote.

MacKenzie on the credit crisis

Edinburgh sociology professor Donald MacKenzie wrote what I feel is the best history (so far) of modern finance and derivatives. In this article in the London Review of Books, he tackles the current credit crisis. Highly recommended.

On Gaussian copula (cognitive limitations restrict attention to an obviously oversimplified model; big brains were worried from the start):

Correlation is by far the trickiest issue in valuing a CDO. Indeed, it is difficult to be precise about what correlation actually means: in practice, its determination is a task of mathematical modelling. Over the past ten years, a model known as the ‘single-factor Gaussian copula’ has become standard. ‘Single-factor’ means that the degree of correlation is assumed to reflect the varying extent to which fortunes of each debt-issuer depend on a single underlying variable, which one can interpret as the health of the economy. ‘Copula’ indicates that the mathematical issue being addressed is the connectedness of default risks, and ‘Gaussian’ refers to the use of a multi-dimensional variant of the statistician’s standard bell-shaped curve to model this connectedness.

The single-factor Gaussian copula is far from perfect: even before the crisis hit, I wasn’t able to get a single insider to express complete confidence in it. Nevertheless, it became a market Esperanto, allowing people in different institutions to discuss CDO valuation in a mutually intelligible way. But having a standard model is only part of the task of understanding correlation. Historical data are much less useful here. Defaults are rare events, and producing a plausible statistical estimate of the extent of the correlation between, say, the risk of default by Ford and by General Motors is difficult or impossible. So as CDOs gained popularity in the late 1990s and early years of this decade, often the best one could do was simply to employ a uniform, standard figure such as 30 per cent correlation, or use the correlation between two corporations’ stock prices as a proxy for their default correlations.

Ratings, indices and implied correlation:

However imperfect the modelling of CDOs was, the results were regarded by the rating agencies as facts solid enough to allow them to grade CDO tranches. Indeed, the agencies made the models they used public knowledge in the credit markets: Standard & Poor’s, for example, was prepared to supply participants with copies of its ‘CDO Evaluator’ software package. A bank or hedge fund setting up a standard CDO could therefore be confident of the ratings it would achieve. Creators of CDOs liked that it was then possible to offer attractive returns to investors – which are normally banks, hedge funds, insurance companies, pension funds and the like, not private individuals – while retaining enough of the cash-flow from the asset pool to make the effort worthwhile. As markets recovered from the bursting of the dotcom and telecom bubble in 2000-2, the returns from traditional assets – including the premium for holding risky assets – fell sharply. (The effectiveness of CDOs and other credit derivatives in allowing banks to shed credit risk meant that they generally survived the end of the bubble without significant financial distress.) By early 2007, market conditions had been benign for nearly five years, and central bankers were beginning to talk of the ‘Great Stability’. In it, CDOs flourished.

Ratings aside, however, the world of CDOs remained primarily one of private facts. Each CDO is normally different from every other, and the prices at which tranches are sold to investors are not usually publicly known. So credible market prices did not exist. The problem was compounded by one of the repercussions of the Enron scandal. A trader who has done a derivatives deal wants to be able to ‘book’ the profits immediately, in other words have them recognised straightaway in his employer’s accounts and thus in the bonus that he is awarded that year. Enron and its traders had been doing this on the basis of questionable assumptions, and accounting regulators and auditors – the latter mindful of the way in which the giant auditing firm Arthur Andersen collapsed having been prosecuted for its role in the Enron episode – began to clamp down, insisting on the use of facts (observable market values) rather than mere assumptions in ‘booking’ derivatives. That credit correlation was not observable thus became much more of a problem.

From 2003 to 2004, however, the leading dealers in the credit-derivatives market set up fact-generating mechanisms that alleviated these difficulties: credit indices. These resemble CDOs, but do not involve the purchase of assets and, crucially, are standard in their construction. For example, the European and the North American investment-grade indices (the iTraxx and CDX IG) cover set lists of 125 investment-grade corporations. In the terminology of the market, you can ‘buy protection’ or ‘sell protection’ on either an index as a whole or on standard tranches of it. A protection seller receives fees from the buyer, but has to pay out if one or more defaults hit the index or tranche in question.

The fluctuating price of protection on an index as a whole, which is publicly known, provides a snapshot of market perceptions of credit conditions, while the trading of index tranches made correlation into something apparently observable and even tradeable. The Gaussian copula or a similar model can be applied ‘backwards’ to work out the level of correlation implied by the cost of protection on a tranche, which again is publicly known. That helped to satisfy auditors and to facilitate the booking of profits. A new breed of ‘correlation traders’ emerged, who trade index tranches as a way of taking a position on shifts in credit correlation.

Indices and other tranches quickly became a huge-volume, liquid market. They facilitated the creation not just of standard CDOs but of bespoke products such as CDO-like structures that consist only of mezzanine tranches (which offer combinations of returns and ratings that many investors found especially attractive). Products of this kind leave their creators heavily exposed to changes in credit-market conditions, but the index market permitted them to hedge (that is, offset) this exposure.

Quants and massive computational power (one wonders whether the mathematics and computers did nothing more than lend a spurious air of technicality to untrustworthy basic assumptions):

With problems such as the non-observability of correlation apparently adequately solved by the development of indices, the credit-derivatives market, which emerged little more than a decade ago, had grown by June 2007 to an aggregate total of outstanding contracts of $51 trillion, the equivalent of $7,700 for every person on the planet. It is perhaps the most sophisticated sector of the global financial markets, and a fertile source of employment for mathematicians, whose skills are needed to develop models better than the single-factor Gaussian copula.

The credit market is also one of the most computationally intensive activities in the modern world. An investment bank with a big presence in the market will have thousands of positions in credit default swaps, CDOs, indices and similar products. The calculations needed to understand and hedge the exposure of this portfolio to market movements are run, often overnight, on grids of several hundred interconnected computers. The banks’ modellers would love to add as many extra computers as possible to the grids, but often they can’t do so because of the limits imposed by the capacity of air-conditioning systems to remove heat from computer rooms. In the City, the strain put on electricity-supply networks can also be a problem. Those who sell computer hardware to investment banks are now sharply aware that ‘performance per watt’ is part of what they have to deliver.

Collapse of rating agency credibility:

The rating agencies are businesses, and the issuers of debt instruments pay the agencies to rate them. The potential conflict of interest has always been there, even in the days when the agencies mainly graded bonds, which generally they did quite sensibly. However, the way in which the crisis has thrust the conflict into the public eye has further threatened the credibility of ratings. ‘In today’s market, you really can’t trust any ratings,’ one money-market fund manager told Bloomberg Markets in October 2007. She was far from alone in that verdict, and the result was cognitive contagion. Most investors’ ‘knowledge’ of the properties of CDOs and other structured products had been based chiefly on ratings, and the loss of confidence in them affected all such products, not just those based on sub-prime mortgages. Since last summer, it has been just about impossible to set up a new CDO.

Illiquid assets, difficulty of mark to market:

Over recent months, banks have frequently been accused of hiding their credit losses. The truth is scarier: such losses are extremely hard to measure credibly. Marking-to-market requires that there be plausible market prices to use in valuing a portfolio. But the issuing of CDOs has effectively stopped, liquidity has dried up in large sectors of the credit default swap market, and the credibility of the cost of protection in the index market has been damaged by processes of the kind I’ve been discussing.

How, for example, can one value a portfolio of mortgage-backed securities when trading in those securities has ceased? It has become common to use a set of credit indices, the ABX-HE (Asset Backed, Home Equity), as a proxy for the underlying mortgage market, which is now too illiquid for prices in it to be credible. However, the ABX-HE is itself affected by the processes that have undermined the robustness of the apparent facts produced by other sectors of the index market; in particular, the large demand for protection and reduced supply of it may mean the indices have often painted too uniformly dire a picture of the prospects for mortgage-backed securities. One trader told the Financial Times in April that the liquidity of the indices had become very poor: ‘Trading is mostly happening on interdealer screens between eight or ten guys, and this means that prices can move wildly on very light volume.’ Yet because the level of the ABX-HE indices is used by banks’ accountants and auditors to value their multi-billion dollar portfolios of mortgage-backed securities, this esoteric market has considerable effects, since low valuations weaken banks’ balance sheets, curtailing their capacity to lend and thus damaging the wider economy.

Josef Ackermann, the head of Deutsche Bank, has caused a stir by admitting ‘I no longer believe in the market’s self-healing power.’ ...

David X. Li

I've been seeing a lot of hits from searches on "Gaussian Copula" or "David X. Li" lately. Li is a quant who developed the Gaussian Copula model used in pricing of CDOs. See here for a post I did on him and the model back in 2005, including some predictions that it might all end in tears :-/

David X. Li is no relation to David X. Cohen of the Simpsons and Futurama, although Cohen did study physics at Harvard :-)

Below is some advice for quants from Li.

How to Become a Successful Financial Engineer?

David X. Li

Actuaries might have been the most quantitative people in the financial industry for a long time. However this unique position has been severely challenged since about a decade or two ago when the investment banks began to hire so-called "rocket scientists" – people with Ph.D. degree in physics and other quantitative fields. Nowadays you can find quantitative analysts or "quants" working in various functions in investment banks and large commercial banks, such as trading, risk management and portfolio management. You may even meet some quants working in some traditional quant-free zones, such as auditing department of a large bank. As the insurance and banking businesses converge it is hard to imagine that traditional approaches to risk measurement – like actuaries work on the insurance side and quants work on the banking side – can still be applied in isolation.

This change poses a challenge to our profession. However there is nothing to panic about. Many problems in practice might need combined skills in actuarial science and financial engineering to solve.

This article outlines some basic skills people should have in order to become a successful financial engineer. It is more from an investment bank angle since that is where the author knows the most.

Mathematical Training

Karl Marx once said ``A subject can only become a science after it successfully uses mathematics". Finance might be the most successful area to use mathematics. The mathematics used in finance ranges from basic mathematics, such as numerical analysis, calculus, and statistics to more advanced ones, stochastic processes and stochastic differential equations, non-linear optimization etc. The areas we use the most in practice are numerical analysis and statistics. Whoever wants to have a career as financial engineer should try to have a solid training in mathematics in school. Wall Street tends to hire people with strong quantitative skills. The general philosophy is that we can teach you finance, but we don’t have time to teach you mathematics. We use a lot of mathematics at the graduate level, so it is to your advantage to take some graduate mathematics courses before you walk out of school. Many practitioners go back to school to learn more mathematics after a few years working in industry. It is hard to learn all the necessary mathematics, but you can pick up other useful mathematical tools or concepts once you have built a solid foundation. I like the saying by Professor Elias Shiu at the University of Iowa "Learning Mathematics is like rowing, if you don’t push forward you’d be pushed backward". Just keep rowing!

On the other hand, don’t indulge yourself into abstract mathematics. Mathematics is just a tool. So you should keep the mentality of an applied mathematician. We find it usually takes longer time for a pure mathematician to feel his "role" in practice than a physicist or an applied mathematician. One famous applied mathematician once said, "Mathematics in every one of its applied areas is a good servant, but a bad master". At the end we are to solve practical problems, not a mathematical game or theorem.

Financial Training

The theory of finance has emerged as a prominent science with the Nobel Prize award to Harry Markowitz, William Sharpe, Merton Miller, Robert Merton and Myron Scholes. This theory attempts to understand how financial markets work, how to make them more efficient, and how they should be regulated. These revolutionary theories in the latter half of the twentieth century have created waves of financial innovation in Wall Street. There are essentially two revolutions. The first revolution, which was the introduction of quantitative methods to the black art of equity fund management, began with the 1952 publication of his Ph.D. dissertation ``Portfolio Selection" by Harry Markowitz. The second revolution in finance began with the 1973 publication of the solution by Fischer Black and Myron Scholes (in consultation with Robert Merton) to the option pricing problem. The Black-Scholes formula brought to the finance industry the modern methodology of martingales and stochastic calculus, methodology which enables investment banks to produce, price and hedge an endless variety of "derivative securities."

There are mainly two aspects of the financial theory, economic and engineering. At the early stage it is essential to understand each instruments from an engineering point of view. But you will find it is important to understand the "big picture" or the economic aspect of the theory later on. There are many books on derivative securities. The key is to fully understand each instrument thoroughly. A good reference book is John Hull’s book. Another important aspect is to understand the market conventions, which you can only learn from a practitioner’s book or some software manuals. These conventions are very important since a one-day miscount could result in a 40-bp (basis point) difference, which is usually larger than many derivative transaction profit margins.

IT Training

Computers have played an important role in our daily life, more so in the life of a financial engineer. Of course you can rely on the IT people in your company to implement your models. But it usually takes a longer time for you to explain the problem than if you just write something simple and get the answer yourself. The package people use the most is the Excel spreadsheet. Other packages, which allow you to do rapid development, are MatLab, Mathematica or Splus. If you can program in C++, Visual Basic for Application or Java, that will be even better. You should pick up these skills at the early stage of your career, which will make your life much easier later on.

In summary, you need a combined set of skills to be a successful financial engineer. Many skills could be developed while you are still in college. Just keep in mind, financial engineering skills is just one set of skills for you to be successful in your business life. Many other skills, such as communication skills and leadership, may play a more important role in your career.

David X. Li is a partner in the RiskMetrics Group where he concentrates on risk management research, product development and structural marketing. Previously he worked for banks in the areas of risk management and credit derivative trading. He taught actuarial science and finance at university briefly before he left academia. Mr. Li has a Ph. D. degree in statistics from the University of Waterloo and master's degrees in economics, finance and actuarial science from the famous NanKai. He is an Associate of the Society of Actuaries (SOA) and an elected Council Member of the Investment Section of the SOA.

Gaussian copula and credit derivatives

This WSJ article describes a mathematical innovation that helped create the now huge market for credit derivatives. Credit derivatives let banks, hedge funds and other investors trade the risk associated with credit defaults (i.e. bankruptcy of bond issuers). Just as in previous derivatives markets, things didn't take off until a simple model for pricing became widely accepted. The model itself is almost certainly too simple, but is (hopefully) improved in proprietary ways by sophisticated traders and researchers. On the plus side, credit derivatives make bond markets more liquid and efficient, allowing risk to be transferred to those most willing to bear it. On the downside, we have yet another ill-understood casino running, with trillions of dollars in play. A few years ago I looked at the Vasicek model for default probabilities (which forms the basis of the KMV methodology), and boy did it look rough. This all looks a lot like the CMO market, where traders blow up with regularity.

The banker, David Li, came up with a computerized financial model to weigh the likelihood that a given set of corporations would default on their bond debt in quick succession. Think of it as a produce scale that not only weighs a bag of apples but estimates the chance that they'll all be rotten in a week.

The model fueled explosive growth in a market for what are known as credit derivatives: investment vehicles that are based on corporate bonds and give their owners protection against a default. This is a market that barely existed in the mid-1990s. Now it is both so gigantic -- measured in the trillions of dollars -- and so murky that it has drawn expressions of concern from several market watchers. The Federal Reserve Bank of New York has asked 14 big banks to meet with it this week about practices in the surging market.

The model Mr. Li devised helped estimate what return investors in certain credit derivatives should demand, how much they have at risk and what strategies they should employ to minimize that risk. Big investors started using the model to make trades that entailed giant bets with little or none of their money tied up. Now, hundreds of billions of dollars ride on variations of the model every day.

"David Li deserves recognition," says Darrell Duffie, a Stanford University professor who consults for banks. He "brought that innovation into the markets [and] it has facilitated dramatic growth of the credit-derivatives markets."

The problem: The scale's calibration isn't foolproof. "The most dangerous part," Mr. Li himself says of the model, "is when people believe everything coming out of it." Investors who put too much trust in it or don't understand all its subtleties may think they've eliminated their risks when they haven't.

The story of Mr. Li and the model illustrates both the promise and peril of today's increasingly sophisticated investment world. That world extends far beyond its visible tip of stocks and bonds and their reactions to earnings or economic news. In the largely invisible realm of derivatives -- investment contracts structured so their value depends on the behavior of some other thing or event -- credit derivatives play a significant and growing role. Endless trading in them makes markets more efficient and eases the flow of money into companies that can use it to grow, create jobs and perhaps spread prosperity.

But investors who use credit derivatives without fully appreciating the risks can cause much trouble for themselves and potentially also for others, by triggering a cascade of losses. The GM episode proved relatively minor, but some experts say it could have been worse. "I think this is a baby financial mania," says David Hinman, a portfolio manager at Los Angeles investment firm Ares Management LLC, referring to credit derivatives. "Like a lot of financial manias, it tends to end with some casualties."

Mr. Li, 42 years old, began his journey to this frontier of capitalist innovation three decades ago in rural China. His father, a police official, had moved the family to the countryside to escape the purges of Mao's Cultural Revolution. Most children at the young Mr. Li's school didn't go past the 10th grade, but he made it into China's university system and then on to Canada, where he collected two master's degrees and a doctorate in statistics.

In 1997 he landed on the New York trading floor of Canadian Imperial Bank of Commerce, a pioneer in the then-small market for credit derivatives. Investment banks were toying with the concept of pooling corporate bonds and selling off pieces of the pool, just as they had done with mortgages. Banks called these bond pools collateralized debt obligations.

They made bond investing less risky through diversification. Invest in one company's bonds and you could lose all. But invest in the bonds of 100 to 300 companies and one loss won't hurt so much.

The pools, however, didn't just offer diversification. They also enabled sophisticated investors to boost their potential returns by taking on a large portion of the pool's risk. Banks cut the pools into several slices, called tranches, including one that bore the bulk of the risk and several more that were progressively less risky.

Say a pool holds 100 bonds. An investor can buy the riskiest tranche. It offers by far the highest return, but also bears the first 3% of any losses the pool suffers from any defaults among its 100 bonds. The investor who buys this is betting there won't be any such losses, in return for a shot at double-digit returns.

Alternatively, an investor could buy a conservative slice, which wouldn't pay as high a return but also wouldn't face any losses unless many more of the pool's bonds default.

Investment banks, in order to figure out the rates of return at which to offer each slice of the pool, first had to estimate the likelihood that all the companies in it would go bust at once. Their fates might be tightly intertwined. For instance, if the companies were all in closely related industries, such as auto-parts suppliers, they might fall like dominoes after a catastrophic event. In that case, the riskiest slice of the pool wouldn't offer a return much different from the conservative slices, since anything that would sink two or three companies would probably sink many of them. Such a pool would have a "high default correlation."

But if a pool had a low default correlation -- a low chance of all its companies stumbling at once -- then the price gap between the riskiest slice and the less-risky slices would be wide.

This is where Mr. Li made his crucial contribution. In 1997, nobody knew how to calculate default correlations with any precision. Mr. Li's solution drew inspiration from a concept in actuarial science known as the "broken heart": People tend to die faster after the death of a beloved spouse. Some of his colleagues from academia were working on a way to predict this death correlation, something quite useful to companies that sell life insurance and joint annuities.

"Suddenly I thought that the problem I was trying to solve was exactly like the problem these guys were trying to solve," says Mr. Li. "Default is like the death of a company, so we should model this the same way we model human life."

His colleagues' work gave him the idea of using copulas: mathematical functions the colleagues had begun applying to actuarial science. Copulas help predict the likelihood of various events occurring when those events depend to some extent on one another. Among the best copulas for bond pools turned out to be one named after Carl Friedrich Gauss, a 19th-century German statistician.

Mr. Li, who had moved over to a J.P. Morgan Chase & Co. unit (he has since joined Barclays Capital PLC), published his idea in March 2000 in the Journal of Fixed Income. The model, known by traders as the Gaussian copula, was born.

"David Li's paper was kind of a watershed in this area," says Greg Gupton, senior director of research at Moody's KMV, a subsidiary of the credit-ratings firm. "It garnered a lot of attention. People saw copulas as the new thing that might illuminate a lot of the questions people had at the time."

To figure out the likelihood of defaults in a bond pool, the model uses information about the way investors are treating each bond -- how risky they're perceiving its issuer to be. The market's assessment of the default likelihood for each company, for each of the next 10 years, is encapsulated in what's called a credit curve. Banks and traders take the credit curves of all 100 companies in a pool and plug them into the model.

The model runs the data through the copula function and spits out a default correlation for the pool -- the likelihood of all of its companies defaulting on their debt at once. The correlation would be high if all the credit curves looked the same, lower if they didn't. By knowing the pool's default correlation, banks and traders can agree with one another on how much more the riskiest slice of the bond pool ought to yield than the most conservative slice.

"That's the beauty of it," says Lisa Watkinson, who manages structured credit products at Morgan Stanley in New York. "It's the simplicity."

It's also the risk, because the model, by making it easier to create and trade collateralized debt obligations, or CDOs, has helped bring forth a slew of new products whose behavior it can predict only somewhat, not with precision. (The model is readily available to investors from investment banks.)

The biggest of these new products is something known as a synthetic CDO. It supercharges both the returns and the risks of a regular CDO. It does so by replacing the pool's bonds with credit derivatives -- specifically, with a type called credit-default swaps.

The swaps are like insurance policies. They insure against a bond default. Owners of bonds can buy credit-default swaps on their bonds to protect themselves. If the bond defaults, whoever sold the credit-default swap is in the same position as an insurer -- he has to pay up.

The price of this protection naturally varies, costing more as the perceived likelihood of default grows.

Some people buy credit-default swaps even though they don't own any bonds. They buy just because they think the swaps may rise in value. Their value will rise if the issuer of the underlying bonds starts to look shakier.

Say somebody wants default protection on $10 million of GM bonds. That investor might pay $500,000 a year to someone else for a promise to repay the bonds' face value if GM defaults. If GM later starts to look more likely to default than before, that first investor might be able to resell that one-year protection for $600,000, pocketing a $100,000 profit.

Just as investment banks pool bonds into CDOs and sell off riskier and less-risky slices, banks pool batches of credit-default swaps into synthetic CDOs and sell slices of those. Because the synthetic CDOs don't contain any actual bonds, banks can create them without going to the trouble of purchasing bonds. And the more synthetic CDOs they create, the more money the banks can earn by selling and trading them.

Synthetic CDOs have made the world of corporate credit very sexy -- a place of high risk but of high potential return with little money tied up.

Someone who invests in a synthetic CDO's riskiest slice -- agreeing to protect the pool against its first $10 million in default losses -- might receive an immediate payment of $5 million up front, plus $500,000 a year, for taking on this risk. He would get this $5 million without investing a dime, just for his pledge to pay in case of a default, much like what an insurance company does. Some investors, to prove they can pay if there is a default, might have to put up some collateral, but even then it would be only 15% or so of the amount they're on the hook for, or $1.5 million in this example.

This setup makes such an investment very tempting for many hedge-fund managers. "If you're a new hedge fund starting out, selling protection on the [riskiest] tranche and getting a huge payment up front is certainly something that's going to attract your attention," says Mr. Hinman of Ares Management. It's especially tempting given that a hedge fund's manager typically gets to keep 20% of the fund's winnings each year.

Synthetic CDOs are booming, and largely displacing the old-fashioned kind. Whereas four years ago, synthetic CDOs insured less than the equivalent of $400 billion face amount of U.S. corporate bonds, they will cover $2 trillion by the end of this year, J.P. Morgan Chase estimates. The whole U.S. corporate-bond market is $4.9 trillion.

Some banks are deeply involved. J.P. Morgan Chase, as of March 31, had bought or sold protection on the equivalent of $1.3 trillion of bonds, including both synthetic CDOs and individual credit-default swaps. Bank of America Corp. had bought or sold about $850 billion worth and Citigroup Inc. more than $700 billion, according to the Office of the Comptroller of the Currency. Deutsche Bank AG, whose activity the comptroller doesn't track, is another big player.

Much of that money is riding on Mr. Li's idea, which he freely concedes has important flaws. For one, it merely relies on a snapshot of current credit curves, rather than taking into account the way they move. The result: Actual prices in the market often differ from what the model indicates they should be.

Investment banks try to compensate for the shortcomings of the model by cobbling copula models together with other, proprietary methods. At J.P. Morgan, "We're not stupid enough to believe [the model] is omniscient," said Andrew Threadgold, head of market risk management. "All risk metrics are flawed in some way, so the trick is to use a lot of different metrics." Bank of America and Citigroup representatives said they use various models to assess risk and are constantly working to improve them. Deutsche Bank had no comment.

As with any model, forecasts investors make by using the model are only as good as the inputs. Someone asking the model to indicate how CDO prices will act in the future, for example, must first offer a guess about what will happen to the underlying credit curves -- that is, to the market's perception of the riskiness of individual bonds over several years. Trouble awaits those who blindly trust the model's output instead of recognizing that they are making a bet based partly on what they told the model they think will happen. Mr. Li worries that "very few people understand the essence of the model."

Consider the trade that tripped up some hedge funds during May's turmoil in GM securities. It involved selling insurance on the riskiest slice of a synthetic CDO and then looking to the model for a way to hedge the danger that the default risk would increase. Using the model, investors calculated that they could offset that danger by buying a double dose of insurance on a more conservative slice.

It looked like a great deal. For selling protection on the riskiest slice -- agreeing to pay as much as $10 million to cover the pool's first default losses -- an investor would collect a $3.5 million upfront payment and an additional $500,000 yearly. Hedging the risk would cost the investor a mere $415,000 annually, the price to buy protection on a $20 million conservative piece.

But the model's hedge assumed only one possible future: one in which the prices of all the credit-default swaps in the synthetic CDO moved in sync. They didn't. On May 5, while the outlook for most bond issuers stayed about the same, two got slammed: GM and Ford Motor Co., both of which Standard & Poor's downgraded to below investment grade. That event caused a jump in the price of protection on GM and Ford bonds. Within two weeks, the premium payment on the riskiest slice of the CDO, the one most exposed to defaults, leapt to about $6.5 million upfront.

Result: An investor who had sold protection on the riskiest slice for $3.5 million had a paper loss of nearly $3 million. That's because if the investor wanted to get out of the investment, he would have to buy a like amount of insurance from somebody else for $6.5 million, or $3 million more than he was getting.

The simultaneous investment in the conservative slice proved an inadequate hedge. Because only GM and Ford saw their default risk soar, not the rest of the bond world, the pricing of the more conservative slices of the pool didn't rise nearly as much as the riskiest slice. So there wasn't much of an offsetting profit to be made there by reselling that insurance.

This wasn't really the fault of the model, which was designed mainly to help price the tranches, not to make predictions. True, the model had assumed the various credit curves would move in sync. But it also allowed for investors to adjust this assumption -- an option that some, wittingly or not, ignored.

Because numerous hedge funds had made the same credit-derivatives bet, the turmoil they faced spilled over into stock and bond markets. Many investors worried that some hedge funds might have to dump assets to cover their losses, so they sold, too. (Some hedge funds also suffered from a separate bad bet, which relied on GM's bond and stock prices moving in tandem; it went wrong when GM shares rallied suddenly as investor Kirk Kerkorian said he would bid for GM shares.)

GLG Credit Fund told its investors it lost about 14.5% in the month of May, much of that on synthetic CDO bets. Writing to investors, fund manager Jean-Michel Hannoun called the market reaction to the GM and Ford credit downgrades too improbable an event for the hedge fund's risk model to capture. A GLG spokesman declines to comment.

The credit-derivatives market has since bounced back. Some say this shows that the proliferation of hedge funds and of complex derivatives has made markets more resilient, by spreading risk.

Others are less sanguine. "The events of spring 2005 might not be a true reflection of how these markets would function under stress," says the annual report of the Bank for International Settlements, an organization that coordinates central banks' efforts to ensure financial stability. To Stanford's Mr. Duffie, "The question is, has the market adopted the model wholesale in a way that has overreached its appropriate use? I think it has."

Mr. Li says that "it's not the perfect model." But, he adds: "There's not a better one yet."

Subprime timeline

Let me repeat: the meltdown is not a black swan event. Many predicted it long ago. See my post from 2005 (lots of details on CDO pricing, copula, how hedge funds could make a short term profit by taking on default risk).

From the Times:

All through last year, Jim Melcher saw the signs of a rapidly deteriorating American housing market — riskier mortgages, rising delinquencies and more homes falling into foreclosure. And with $100 million in assets at his hedge fund, Balestra Capital, he was in a position to do something about it.

So in October, as mortgage-backed bonds were still flying high, he bet $10 million that these bonds would plunge in value, using complex derivatives available to any institutional investor. As his gamble began to pay off in the first months of 2007, Mr. Melcher, a money manager based in New York, plowed the profits into ever bigger wagers that the mortgage crisis would worsen further, eventually risking some $60 million of the fund’s money.

“We saw the opportunity of a lifetime, and since then events have unfolded on schedule,” he said. Mr. Melcher’s flagship fund has since doubled in value, even as this summer’s market turmoil cost other investors billions, forced the closing of several major hedge funds and pushed the stock market down 7 percent since mid-July. This week, Mr. Melcher is heading to Paris for a vacation with his wife.

The extent of the turmoil has stunned much of Wall Street, but as Mr. Melcher’s case makes clear, there were ample warning signs that a financial time bomb in the form of subprime mortgages was ticking quietly for months, if not years. ...

...On Friday, the Federal Reserve was forced into a surprise cut of the discount rate it charges banks to borrow money, a move that steadied shaky stock and credit markets and reassured investors, bankers and traders who were reeling from a month of market turmoil. And for the first time, the Fed bluntly acknowledged that the credit crisis posed a threat to economic growth.

“Until recently, there was a lot of denial, but this is a big deal,” said Byron R. Wien, a 40-year veteran of Wall Street who is now chief investment strategist at Pequot Capital. “Now the big question is: Will this spill over into the broader economy?”

The answer to that question will be revealed over the coming months. But the cast of characters who missed signals like the rise of delinquencies and foreclosures is becoming easier to identify. They include investment banks happy to sell risky but lucrative mortgage debt to hedge funds hungry for high interest payments, bond rating agencies willing to hope for the best in the housing market and provide sterling credit appraisals to debt issuers, and subprime mortgage brokers addicted to high sales volumes.

What is more, some of these players now find themselves in a dual role as both enabler and victim, like the legions of individual borrowers who were convinced that their homes could only keep rising in value and were confident that they could afford to stretch for the biggest mortgage possible.

“All of the old-timers knew that subprime mortgages were what we called neutron loans — they killed the people and left the houses,” said Louis S. Barnes, 58, a partner at Boulder West, a mortgage banking firm in Lafayette, Colo. “The deals made in 2005 and 2006 were going to run into trouble because the credit pendulum at the time was stuck at easy.”

Oddly, the credit analysts at brokerage firms now being pummeled were among the Cassandras whose warnings were not heeded. “I’m one guy in a research department, but many people in our mortgage team have been suggesting that there was froth within the market,” said Jack Malvey, the chief global fixed income strategist for Lehman Brothers. “This has really been progressing for quite some time.”

A nice timeline below (larger original here):

CDOs, auctions and price discovery

How is Treasury going to buy up CDOs and other mortgage backed securities? What is the price discovery mechanism? I've heard discussion of a reverse auction process, in which the government offers a price and owners of the assets decide whether to accept the bid.

But this makes the problem sound much easier than it is. There are no simple or uniform categories for these securities -- no two are exactly alike. I imagine Treasury is going to have to do a lot of homework before each auction, perhaps aided by some sophisticated professionals (Bill Gross of PIMCO recently offered his team's services). Data on each security is available from ratings agencies like S&P and Moody's but presumably one would supplement this with additional information. After some initial analysis Treasury could set a conservative bound (i.e., using pessimistic estimates of future default rates and home prices) on the value of each security in units of the original face value (this one is worth at least 25 cents on the dollar, this is one, 45 cents, etc.). Then, they can publish a list of securities in a particular value category (without, of course, giving out the actual value estimate) and conduct a reverse auction covering all the assets on the list.

If they can get the assets below the value estimate, great for taxpayers like you and me. If banks (hedge funds? pension funds? foreign banks? who is really holding all this stuff?) won't sell at prices below the bound, and the auction heads above that price, Treasury should start demanding warrants or equity stakes on some sliding scale. In other words, the bid keeps getting higher, but at some point Treasury starts asking for not only the particular CDO but some additional warrants or stock. (This could also be done on a sliding scale from the beginning of the auction -- Treasury gets an additional x percent of the bid in warrants, where x increases with price.) The equity stake is compensation for the government for having to having to overpay for the security. At this price there is an (expected) flow of funds from taxpayers to recapitalize the seller, but at least we are getting equity in return. It is claimed that there is a range of values (roughly 20 percent of current market prices) over which the seller would be getting more at auction than the market is currently offering, but the government is still getting a good deal on the asset (expects to make money even under conservative assumptions).

Will it work? Who knows, but at least it may restore some confidence to credit markets.

Here are some old posts that really get into the nitty gritty of what is inside a typical CDO. You'll see that I've been covering credit securities since 2005 :-)

anatomy of a cdo

deep inside the subprime crisis

mackenzie on the credit crisis

gaussian copula and credit derivatives

Here's a recent NYTimes article that gives a peek into the complexity of structured finance.

NYTimes: ...Consider the Bear Stearns Alt-A Trust 2006-7, a $1.3 billion drop in the sea of risky loans. Here’s how it worked:

As the credit bubble grew in 2006, Bear Stearns, then one of the leading mortgage traders on Wall Street, bought 2,871 mortgages from lenders like the Countrywide Financial Corporation.

The mortgages, with an average size of about $450,000, were Alt-A loans — the kind often referred to as liar loans, because lenders made them without the usual documentation to verify borrowers’ incomes or savings. Nearly 60 percent of the loans were made in California, Florida and Arizona, where home prices rose — and subsequently fell — faster than almost anywhere else in the country.

Bear Stearns bundled the loans into 37 different kinds of bonds, ranked by varying levels of risk, for sale to investment banks, hedge funds and insurance companies.

If any of the mortgages went bad — and, it turned out, many did — the bonds at the bottom of the pecking order would suffer losses first, followed by the next lowest, and so on up the chain. By one measure, the Bear Stearns Alt-A Trust 2006-7 has performed well: It has suffered losses of about 1.6 percent. Of those loans, 778 have been paid off or moved through the foreclosure process.

But by many other measures, it’s a toxic portfolio. Of the 2,093 loans that remain, 23 percent are delinquent or in foreclosure, according to Bloomberg News data. Initially rated triple-A, the most senior of the securities were downgraded to near junk bond status last week. Valuing mortgage bonds, even the safest variety, requires guesstimates: How many homeowners will fall behind on their mortgages? If the bank forecloses, what will the homes sell for? Investments like the Bear Stearns securities are almost certain to lose value as long as home prices keep falling.

“Under the current circumstances it’s likely that you are going to take a loss on these loans,” said Chandrajit Bhattacharya, a mortgage strategist at Credit Suisse, the investment bank.

The Bear Stearns bonds are just one example of the kind of assets the government could buy, and they are by no means the most complicated of the lot. Wall Street took bonds like those of Bear Stearns and bundled and rebundled them into even trickier investments known as collateralized debt obligations, or C.D.O.’s

“No two pieces of paper are the same,” said Mr. Feltus of Pioneer Investments.

On Wall Street, many of these C.D.O.’s have been selling for pennies on the dollar, if they are selling at all. In July, Merrill Lynch, struggling to bolster its finances, sold $31 billion of tricky mortgage-linked investments for 22 cents on the dollar. Last November, Citadel, a large hedge fund in Chicago, bought $3 billion of mortgage securities and other investments for 27 cents on the dollar.

But Citigroup, the financial giant, values similar investments on its books at 61 cents on the dollar. Citigroup says its C.D.O.’s are relatively high quality because they were created before lending standards weakened in 2006.

A big challenge for Treasury officials will be deciding whether to buy the troubled investments near the values at which the banks hold them on their books. That would help minimize losses for financial institutions. Driving a hard bargain, however, would protect taxpayers.

Mark to market

The almost collapse of two Bear Stearns hedge funds investing in mortgage-backed securities is sending a tremor through Wall St. A last minute bailout by creditor Merrill means that $800 million in CDOs is about to be marked to market. In other words, some complex, illiquid securities are about to have a meaningful price, as opposed to the theoretical value on the books. Insert words about Long Term Capital Management and "systemic risks" here.

``Nobody wants to look at the truth right now because the truth is pretty ugly,'' Castillo said. ``Where people are willing to bid and where people have them marked are two different places.''

I've written several posts about CDOs here. This and this are getting a lot of traffic right now. Word is that Gaussian copula models are **way** off from real market prices :-)

June 21 (Bloomberg) -- Merrill Lynch & Co.'s threat to sell $800 million of mortgage securities seized from Bear Stearns Cos. hedge funds is sending shudders across Wall Street.

A sale would give banks, brokerages and investors the one thing they want to avoid: a real price on the bonds in the fund that could serve as a benchmark. The securities are known as collateralized debt obligations, which exceed $1 trillion and comprise the fastest-growing part of the bond market.

Because there is little trading in the securities, prices may not reflect the highest rate of mortgage delinquencies in 13 years. An auction that confirms concerns that CDOs are overvalued may spark a chain reaction of writedowns that causes billions of dollars in losses for everyone from hedge funds to pension funds to foreign banks. Bear Stearns, the second-biggest mortgage bond underwriter, also is the biggest broker to hedge funds.

``More than a Bear Stearns issue, it's an industry issue,'' said Brad Hintz, an analyst at Sanford C. Bernstein & Co. in New York. Hintz was chief financial officer of Lehman Brothers Holdings Inc., the largest mortgage underwriter, for three years before becoming an analyst in 2001. ``How many other hedge funds are holding similar, illiquid, esoteric securities? What are their true prices? What will happen if more blow up?''

Shares Fall

Shares of Bear Stearns, the fifth-biggest U.S. securities firm by market value, and Merrill, the third-largest, led a decline in financial company stocks yesterday, and the perceived risk of owning their bonds jumped on concerns losses related to subprime home loans may be bigger than initially thought. Both companies are based in New York.

The perceived risk of owning corporate bonds jumped to the highest in nine months today. Contracts based on $10 million of debt in the CDX North America Crossover Index rose as much as $10,000 in early trading today to $178,500, according to Deutsche Bank AG. They retraced to $171,500 at 8:28 a.m. in New York.

U.S. Securities and Exchange Commission Chairman Christopher Cox said yesterday that the agency's division of market regulation is tracking the turmoil at the Bear Stearns fund. ``Our concerns are with any potential systemic fallout,'' Cox said in an interview.

Bankers and money managers bundle securities into a CDO, dividing it into pieces with credit ratings as high as AAA. The riskiest parts have no rating because they are first in line for any losses. Investors in this so-called equity portion expect to generate returns of more than 10 percent.

Fivefold Increase

CDOs were created in 1987 by bankers at now-defunct Drexel Burnham Lambert Inc., the home of one-time junk-bond king Michael Milken. Sales reached $503 billion in 2006, a fivefold increase in three years. More than half of those issued last year contained mortgages made to people with poor credit, little loan history, or high debt, according to Moody's Investors Service.

New York-based Cohen & Co. was the biggest issuer of CDOs last year. It has formed 36 CDOs since 2001, including 15 worth a total of $14 billion in 2006, according to newsletter Asset-Backed Alert.

Not since 1994 have mortgages with past due payments been so high, according to first-quarter data compiled by the Federal Deposit Insurance Corp., the agency that insures deposits at
8,650 U.S. banks. Lehman analysts estimated in April that the collateral backing CDOs had fallen by $25 billion.

``The big question is whether these forced liquidations represent a tipping point in the market,'' said Carl Bell, who helps manage $63 billion in fixed-income assets as head of the
structured-credit team at Boston-based Putnam Investments. It ``may put pressure on other hedge funds pursuing similar strategies'' as the Bear Stearns funds, he said.

Biggest Names

The Bear Stearns funds are run by senior managing director Ralph Cioffi. One of the funds, the 10-month old High-Grade Structured Credit Strategies Enhanced Leverage Fund, lost 20 percent this year, the New York Post reported. Officials at Bear Stearns and Merrill declined to disclose the losses.

The funds had borrowed at least $6 billion from the biggest names on Wall Street. Aside from Merrill, other creditors included Goldman Sachs Group Inc., Citigroup Inc., JPMorgan Chase
& Co. and Bank of America Corp. All of the firms are based in New York, except Bank of America, which is based in Charlotte, North Carolina.

As the funds faltered, Merrill sought to protect itself by seizing the assets that were used as collateral for its loans. JPMorgan planned to sell assets linked to its credit lines before
reaching agreement with Bear Stearns to unwind the loan, people with knowledge of the negotiations said yesterday.

Bear Stearns was still in talks late yesterday with creditors to the funds to rescue the funds, said the people, who declined to be identified because the negotiations are private.

Russell Sherman, a Bear Stearns spokesman, and Jessica Oppenheim, a spokeswoman for Merrill, declined to comment.

`Pretty Ugly'

Merrill's decision yesterday to accept bids on $800 million of bonds it took as collateral for its loans further stifled trading in CDO securities, said David Castillo, who trades asset-backed, commercial-mortgage and CDO bonds in San Francisco at Further Lane Securities.

``Nobody wants to look at the truth right now because the truth is pretty ugly,'' Castillo said. ``Where people are willing to bid and where people have them marked are two different places.''

The perceived risk of holding Bear Stearns bonds jumped to a three-month high, according to traders betting on the creditworthiness of companies in the credit-default swaps market.

Contracts based on $10 million of its bonds rose $5,800 to $45,500, according to composite prices from London-based CMA Datavision. An increase in the five-year contracts suggests
deterioration in the perception of credit quality. Contracts on Merrill jumped $4,700 to $33,000, CMA prices show.

Long-Term Capital

Shares of Bear Stearns fell for a fourth day, declining 19 cents to $143.01 at 9:32 a.m. in New York Stock Exchange composite trading. The stock was down 12 percent this year before
today, compared with the 0.4 percent advance of the Standard & Poor's 500 Financials Index. Merrill dropped 20 cents to $87.48 and Citigroup fell 13 cents to $53.31.

The reaction to the Bear Stearns situation is reminiscent of Long-Term Capital Management LP, which lost $4.6 billion in 1998.

Lenders including Merrill and Bear Stearns met and agreed to take a stake in the Greenwich, Connecticut-based fund and slowly sold the assets to limit the impact of its collapse.

``We're not surprised to find the principal circle of players is pretty interconnected,'' said Roy Smith, professor of finance at New York University Stern School of Business and
former head of Goldman's London office. ``What we're looking for is whether the interconnection creates a negative domino effect: Whether Hedge Fund A creates a problem for other hedge funds,
which in turn creates a problem for the prime brokers that are lending to them.''