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Gaussian copula and credit derivatives

Wallstyouth

Vice President
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."
 
Good stuff,

I attach Mr Li's famous article which changed the world. I wrote my master thesis on copulas and when I was searching for a job Mr Li invited me for an interview at Barclays. Of coarse I failed the the interview (It went really bad). That was before I started Baruch MFE program.
 

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Nice article. It explains very well the popularity of the CDS indexes such as CDX and iTraxx. The market for CDX is massively liquid, specially for those 5yr, 7yr and 10 yr. The 3yr tenor is less popular and hence very illiquid.

Compared to the time of Mr. Li, the tools today are available off the shelf. Trading desks use tools like Quantifi which comes with tons of Excel add-in that can do ton of amazing stuff.

With the introduction of CDX Series 9 tomorrow, we will see lot of interesting things.
 

Yuriy

MFE Alum
It is interesting to note that not everybody likes copulas :) If I'm not mistaken it was Schonbucher who avoided them in his book. That was pointed out by a professor at CMU who I took credit derivatives with.
 
It is interesting to note that not everybody likes copulas :) If I'm not mistaken it was Schonbucher who avoided them in his book. That was pointed out by a professor at CMU who I took credit derivatives with.

I agree, it is not the perfect model, but just like BSM model for options, despite the know flaws (multivariate normal assumptions for the Gaussian copula, resulting in base correlation skew), if everyone trade off that model, a market(s) can still flourish. Note the CDX tranche markets =D>
 
Here is some interesting development in the correlation model world

A new kind of correlation model that is causing excitement among some large banks will be added to a commercial software product for the first time. Rohan Douglas, chief executive of Quantifi, says his company has the first implementation of what has been described as a Gaussian copula model with correlated stochastic recovery.
Quantifi’s version, dubbed the correlated recovery model, is included in an upgrade unveiled this month for the structured credit analytics provider’s software.
Douglas says that at least three large banks are looking at using this kind of approach to deal with a problem that has hit the structured credit derivatives market for most of the past year. This is the fact that the standard one-factor Gaussian copula models cannot make sense of the wide spreads on super senior tranches relative to other prices. As a result, standard models can sometimes calculate deltas as being negative and correlation as greater than 100%.
Credit quants have tried to grapple with this problem in several ways. Some have come up with completely new models, such as those that take a top-down approach. However, banks have found it hard to get these models to make sense of real market prices. Another approach has been to make ad hoc adjustments to the standard model, such as reducing single name recovery assumptions across the portfolio. However, these approaches have the disadvantage of inconsistency between tranches.
The model in Quantifi’s new release approaches the problem in a different way. It uses standard base correlation and makes the usual assumptions about the average recovery rate of names in the portfolio, but it allows individual recoveries
to be correlated with the default risk on that name. In other words, it increases the dispersion of expected losses in the portfolio.
“If you think of a tranche in option terms, then implied correlation is a dial that increases the loss volatility of the portfolio,” says Douglas. “What we are doing is adding another dial that introduces an extra degree of volatility and allows you to push value up to the top tranches.”
Douglas says this makes the model backwards-compatible with existing one-factor Gaussian models. It also produces implied correlations that make better sense of current distressed prices and more sensible deltas.
 
In fact, Quantifi develop new methods because with the classic gaussian copula, the base correlation of the tranche 15-30 for example can not be calibrated with constant recovery of 0.4. Moreover, we see a price in the market for the Tranche 60-100% which must be 0. bp with a constant recovery of 0.4
Andersen and Sidenius were the first to introduce random recovery in their famous article about random factor loading
You can find the 3 main articles about stochastic recovery on my website :
http://www.quant-press.com/Credit_Gaussian_Copula.php
CDS and CDO Pricing with Stochastic Recovery , Charaf Ech-Chatbi (2008)
Pricing distressed CDOs with Base Correlation and Stochastic Recovery , M.Krekel (2008)
Optimal Stochastic Recovery for Base Correlation, S.Amraoui, S.Hitier (2008)
I think quantifi implements one of the three model but those models are main drawbacks
For example, in Krekel's article what kind of input for the function of recovery you put in our model
In The last article, random recovery depends of the correlation of the tranche...So, it's weird to not have a distribution of recovery which depends of the tranche
 
Hey everyone, while I read lots of articles about this topic I have never found anything that really shows the mathematics, when calculating default correlation using the copula. Li's paper shows how he got his formula yet I really need to know how to use it with actual numbers, does that make sense? I hope you can help me, its pretty urgent. Any help is appreciated

---------- Post added at 12:27 PM ---------- Previous post was at 12:17 PM ----------

Hi, do you think you could help me calculate default correlation (by hand) using Li's copula
 
Hey everyone, while I read lots of articles about this topic I have never found anything that really shows the mathematics, when calculating default correlation using the copula. Li's paper shows how he got his formula yet I really need to know how to use it with actual numbers, does that make sense? I hope you can help me, its pretty urgent. Any help is appreciated

---------- Post added at 12:27 PM ---------- Previous post was at 12:17 PM ----------

Hi, do you think you could help me calculate default correlation (by hand) using Li's copula

I have the same problem as copulamix, can't seem to find any articles online showing practical examples. can anyone direct me to a piece where it does?
 
From Wall Street executive who likes to remain anonymous:

Read the Creditmetrics technical paper at RiskMetrics.com. That tells exactly how to build a copula-based portfolio tool. They are the basis for credit derivatives. Note that it is an early approach that describes a means to do portfolio analytics using pairwise correlations. They propose using equity index correlations as proxies, which is no longer considered workable. It provides a glass box, however, into the workings of the model.

Be prepared. While it's not that complicated mathematically, there are a lot of moving parts.
 
What are we still discussing here? It has been made clear that the Gaussian copula, with or without stochastic recovery, does not work for credit risk since credit events are generally correlated. The main RiskMetrics approach, which is based on the same type of copula, does not work for this reason. It failed in 2008. That should have been the end of this discussion. Some have even rejected the copula approach altogether because of what happened, but that was a mistake too. It's important to know what copula to use since the results are extremely sensitive to this choice. Btw, Dr. Li has left Wall Street and is back in China.
 
Just googled and google redirected me here. This is the topic I am writing a diploma research. Does anyone know or possibly point me to any sample to read for the sake of understanding structure. It will be also helpful to get the credit derivatives examples with many actual underlyings. I also created a thread about it few days ago. Thanks in advance
 
Thank you. Im seeking a particular example of any credit derivative(no matter if it is CDO or not) with many underlings. I need it because I have to construct a copula approach on the distributions of underlings. I don't have the example what can be the underlings and what their marginal distributions are to be expected.
 

bob

Faculty (Undercover)
Perhaps this will help you get you started; it's something I posted some time ago in response to a question about t-copulas used to model nth-to-default baskets:

http://www.quantnet.com/forum/threads/student-t-copula-in-excel.3661/#post-31683

It does Monte Carlo to simulate correlated t's and uses the copula to join the (exponential) distributions of time to default of each of the names, so it's simple enough that it should be comprehensible.

For some reason, Excel's chi-inverse function is failing on this from time to time, but if the sim stops you can just end it without losing all of the accumulated samples; running further simulation just adds to the samples you already have. You have to run cleanUp if you want to start over. Sorry, just no time to make the thing behave 100% properly.

It's important to understand what a copula is--and isn't. What it is is a way of imposing a joint distribution on a bunch of known marginals in such a way that the marginals are preserved. What is isn't is a true (i.e., consistent with reality) picture of the correlation structure of the factors of interest: in a Gaussian copula, the "correlations" are not default correlations. They are pure artifacts of the method chosen. This is why for large baskets, often the approach is to choose a single correlation input to control all of the names' pairwise correlations; they aren't observable or even in any sense "real" parameters.

What this means is that the best you can hope for is to calibrate your approach to traded instruments--as has already been pointed out, this doesn't always work--and it's worth investigating what the result looks like from day to day rather than just trusting that a successful calibration results in a good model. If your calibrated correlation swings wildly over small time spans, chances are that what you're doing is overfitting instead of capturing useful information. Of course, this is one situation in which the model becomes a self-fulfilling prophecy, since if everyone is trading on a correlation that comes out of a Gaussian copula approach, then lo and behold from day to day this parameter actually does acquire some meaning, since everyone is making decisions on the basis of it.
 
Thank you very much Bob. I have just noticed this post since I haven't been alerted. I'll take those suggestions into account. I have already started writing it.
 
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