"TAP não sobrevive se se mantiver pública", garante Pinto Luz

[atomez]

E já agora, aqui fica o paper original de David X. Li

On Default Correlation: A Copula Function Approach

Abstract

This paper studies the problem of default correlation. We first introduce a random variable called "time-until- default" to denote the survival time of each defaultable entity or financial instrument, and define the default correlation between two credit risks as the correlation coefficient between their survival times. Then we argue why a copula function approach should be used to specify the joint distribution of survival times after marginal distributions of survival times are derived from market information, such as risky bond prices or asset swap spreads. The definition and some basic properties of copula functions are given. We show that the current CreditMetrics approach to default correlation through asset correlation is equivalent to using a normal copula function. Finally, we give some numerical examples to illustrate the use of copula functions in the valuation of some credit derivatives, such as credit default swaps and first-to-default contracts.

[atomez]

Bem, isto é muito melhor que ficção científica!

Ainda por cima já em 2005 houve um pré-aviso real que o modelo não era de fiar ...

(é longo e denso mas vale a pena)

HOW A FORMULA IGNITED MARKET THAT BURNED SOME BIG INVESTORS

The Wall Street Journal
September 12, 2005


/SLICES OF RISK/

HOW A FORMULA IGNITED MARKET
THAT BURNED SOME BIG INVESTORS



Credit Derivatives Got a Boost
From Clever Pricing Model;
Hedge Funds Misused It
Inspiration: Widowed Spouses

By *MARK WHITEHOUSE*
*Staff Reporter of THE WALL STREET JOURNAL*


When a credit agency downgraded General Motors
Corp.'s debt in May, the auto maker's securities sank. But it wasn't
just holders of GM shares and bonds who felt the pain.

Like the proverbial flap of a butterfly's wings rippling into a tornado,
GM's woes caused hedge funds around the world to lose hundreds of
millions of dollars in other investments on behalf of wealthy
individuals, institutions like university endowments -- and, via pension
funds, regular folk.

[DAVID LI]

All this traces back, in a sense, to a day eight years ago when a
Chinese-born New York banker got to musing about love and death --
specifically, how people tend to die soon after their spouses do.
Therein lies a tale of how a statistician unknown outside a small
coterie of finance theorists helped change the world of investing.

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."

[PROTECTION MONEY]

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."

[atomez]

Já suspeitava que a crise financeira que se abateu sobre os mercados tinha sido provocada por um erro fundamental no método de avaliação do risco, em especial na equação Black-Scholes.

Afinal parece que o "culpado" é um tal David X. Li e a sua Cópula Gaussiana.

Este artigo da Wired explica o caso:

The Formula That Killed Wall Street

A year ago, it was hardly unthinkable that a math wizard like David X. Li might someday earn a Nobel Prize. After all, financial economists—even Wall Street quants—have received the Nobel in economics before, and Li's work on measuring risk has had more impact, more quickly, than previous Nobel Prize-winning contributions to the field. Today, though, as dazed bankers, politicians, regulators, and investors survey the wreckage of the biggest financial meltdown since the Great Depression, Li is probably thankful he still has a job in finance at all. Not that his achievement should be dismissed. He took a notoriously tough nut—determining correlation, or how seemingly disparate events are related—and cracked it wide open with a simple and elegant mathematical formula, one that would become ubiquitous in finance worldwide.

For five years, Li's formula, known as a Gaussian copula function, looked like an unambiguously positive breakthrough, a piece of financial technology that allowed hugely complex risks to be modeled with more ease and accuracy than ever before. With his brilliant spark of mathematical legerdemain, Li made it possible for traders to sell vast quantities of new securities, expanding financial markets to unimaginable levels.

His method was adopted by everybody from bond investors and Wall Street banks to ratings agencies and regulators. And it became so deeply entrenched—and was making people so much money—that warnings about its limitations were largely ignored.

Then the model fell apart. Cracks started appearing early on, when financial markets began behaving in ways that users of Li's formula hadn't expected. The cracks became full-fledged canyons in 2008—when ruptures in the financial system's foundation swallowed up trillions of dollars and put the survival of the global banking system in serious peril.

David X. Li, it's safe to say, won't be getting that Nobel anytime soon. One result of the collapse has been the end of financial economics as something to be celebrated rather than feared. And Li's Gaussian copula formula will go down in history as instrumental in causing the unfathomable losses that brought the world financial system to its knees.

...