The document discusses David X. Li, the inventor of the Gaussian copula formula which was widely used to rate collateralized debt obligations (CDOs) containing mortgage-backed securities in the run-up to the 2008 financial crisis. While the formula helped fuel the rapid growth of the CDO market, it had significant flaws and limitations that were not properly understood. When the housing market collapsed, it revealed that CDOs given high credit ratings based on the formula were in fact highly risky and worthless. This contributed greatly to the financial crisis and global recession.

1. Shane Kaiser

2.  2007/2008 – End of Housing Bubble
◦ Marked the start of the major recession, and left
most people with feelings of wanting to find some
one to blame
◦ Most ended up initally blaming the big financial
institutions (Bear Sterns, Goldman Sachs, AIG, etc.)
◦ Many people then pointed the finger at the
formulas the big corporations were using to rate
the risk their investments
◦ The main formula being the Gaussian Copula
formula, created by the mathematician and actuary
Dr. David X. Li

3.  Inventor of this Gaussian Copula formula
 Born and grew up in China in the 1960s and
became a well known a quantitative analyst and
actuary
 In 2000, he published a paper titled “On Default
Correlation: A Copula Function Approach” which
was the first instance were he used his formula
on to rate Collateralized Debt Obligations (CDOs)
 The Financial Times called him the world’s most
influential actuary after publishing this paper

4.  CDOs are a type of structured asset-backed
security (ABS) whose value and payments are
derived from a portfolio of fixed-
income underlying assets, such as such as
bonds, loans, credit default swaps, and
mortgage-backed securities
 The first one was issued in 1987, and grew in
popularity throughout the late 1990s and
early to mid 2000s, similarly to how CDSs
grew

5.  When purchasing a CDO, there are different
levels of security, known as tranches
 The “senior” tranche gets paid first and is the
most secure but most expensive
 The lowest tranche or subordinate/equity
tranche are the riskiest but cheapest
 Investors in the tranches have the ultimate
credit risk exposure to the underlying
entities, so banks used them as a way to
transfer risk from themselves to investors

6.  On each tranche the investor has an
“attachment percentage” and a “detachment
percentage”
 When the total percentage loss of the entities
in the CDO reach the attachment
percentage, investors in that tranche start to
lose money (not get paid fully) and when the
total percentage the detachment
percentage, the investors in that tranche
won’t get paid at all

7.  Example:
◦ Tranche 1 = 0% - 5%
◦ Tranche 2 = 5% - 15%
◦ Tranche 3 = 15% - 30%
◦ Tranche 4 = 30% - 70%
 If CDO has a 3% loss, the members in Tranche 1
(the equity tranche) will absorb that loss, but the
rest of the investors will be unaffected.
 If the CDO has a 35% loss, the members of
Tranche 1 and 2 will receive no payment, Tranche
3 will lose most of its payment, and Tranche 4
(the senior tranche) will be unaffected

8.  When the paper was first published, it caught the
attention of many people, as he allegedly found a
way to came up with a way using “relatively”
simple mathematics to model the correlation
between two entities defaulting without looking
at any historical default data
 Instead of using historical default data, he used
historical prices from the CDS market
◦ The CDS market was less than a decade old at this point
 The main flaw in his assumptions was that he
trusted that the financial markets, and CDS
markets in particular, were pricing CDS’s default
risk correctly on each individual underlying

9.  Every underlying is give a certain amount of
“basis points” (each representing .01%)
 These basis points are dependent upon the
stability/riskiness of the underlying credit
 The riskier the underlying, the higher the
basis points will be.
◦ Reflect markets perception of the risk of default
over the risk free rate, almost like a percentage
chance of how likely the underlying will default
before maturity

10.  A copula is used in statistics to couple the
behavior of two or more variables and
determine if the variables are correlated
 With CDOs and portfolio/index CDSs having
so many different underlying entities at
times, a copula seemed perfect for this
situation
 There are many different kinds of copula
formulas, but Dr. Li’s Gaussian Formula was
the only one the was used to measure risk of
default

11.  P(TA<1, TB<1) = Φγ(Φ-1(A), Φ-1(B))
◦ T is the period of time
◦ Φ-1(A) and Φ-1(B) is the probabilities of if A and B
not defaulting throughout T using the inverse of a
standard normal cumulative distribution function
◦ Φγ is the copula the individual probabilities
associated with A and B to come up with a single
number, using a standard bivariate normal
cumulative distribution function of correlation
coefficient γ
◦ P(TA<1, TB<1) is the probability any a member of
both group A and B defaulting within T, seeing if
they are in fact correlated or not

12.  The industry loved it, and began selling off
more AAA rated securities than ever before
 This was because the rating agencies no
longer needed to examine the underlying
thoroughly, they just needed this one
correlation number
 If the underlying entities were considered to
not be correlated, it was considered nearly
very low risk CDO, especially for investors
looking to be a part of the senior tranche

13.  Banks began throwing all kinds of risky
underlying together in a CDO, and as long as
they didn’t have a high correlation of
defaulting, the CDO was able to get a high
rating
 As time went on the market for CDSs and
CDOs exploded
◦ The CDS market grew from $920 billion at the end
of 2001 in credit default swaps outstanding to $62
trillion by the end of 2007
◦ The CDO market grew from at $275 billion in 2000
to $4.7 trillion by 2006

14.  Before the formula:
◦ it was considered good practice to have diversify
the underlying entities in a CDO
 With the formula:
◦ if you were to found a group of home loans that
were found not to be highly correlated to
default, banks would advertise the CDO as a safe
investment with a high rating, because you know
you will never lose everything

15.  As time progressed, banks kept finding more
and more ways to take risky investments and
put them into CDOs making them appear to
be a safe investment
◦ For example, some began making CDOs made up of
the lower tranches of a group CDOs, tranche them
into a separate CDO (known as CDO squared)
◦ And as time progressed, they started creating CDO
cubed by taking the lower tranches of the CDOs
squared and making them into a CDO (CDOs of
CDOs of CDOs…)

16.  Banks began finding ways to sell off riskier and
riskier CDOs, especially ones with home loans, by
using this new rating system
 Banks also began giving out more home loans
and mortgages to riskier prospective
homeowners, knowing in the long run they can
sell off all the risk through a CDO
◦ Additionally the government was pushing banks and
incentivizing them to issue more home loans
◦ Originally banks were resistant to the governments
demands, but began to comply when they knew they
could just get rid of all the risk and receive the
government benefits

17.  Li’s formula was used to price hundreds of
billions of dollars' worth of CDOs filled with
mortgages, and a lot of them being sub-prime
◦ CDSs were less than a decade old at this point, and it
was during a period when house prices soared, which
made rates of default and default correlations very
low, giving the CDOs a high rating
◦ But when the mortgage boom ended with the bubble
popping, values of homes fall across the country
◦ People began defaulting on homes, and default
correlations started showing up, but it was too late
◦ Home loan CDOs that once had a AAA rating, became
worthless

18.  “Very few people understand the essence of the
model” – Dr. Li
 Investment banks would regularly call Dr. Li to
come in to speak about his formula he would
warn them that it was not suitable for use in risk
management or valuation.
 It was merely a method to determine if entities
are likely to default at the same time
 Banks never really listened to Dr. Li’s warnings
partly because they were making too much
money to stop what they were doing
◦ It was working for a good 6 to 7 years

19.  Bankers should have noted that very small
changes in their underlying assumptions
(such as the correlation parameter) could
result in very large changes in the correlation
number, but many of them did not truly
understand how the formal worked
◦ They were able to understand a single correlation
number, and exploited it by abuseding the rating
system

20.  http://www.wired.com/techbiz/it/magazine/17-
03/wp_quant?currentPage=all
 http://www.ft.com/cms/s/2/912d85e8-2d75-11de-9eba-
00144feabdc0.html#axzz17RdCGcza
 http://en.wikipedia.org/wiki/David_X._Li
 http://www.forbes.com/2009/05/07/gaussian-copula-david-x-li-
opinions-columnists-risk-debt.html
 http://en.wikipedia.org/wiki/Copula_(statistics)#Gaussian_copula
 http://en.wikipedia.org/wiki/Subprime_mortgage_crisis#Government_po
licies
 http://en.wikipedia.org/wiki/Collateralized_Debt_Obligations#Investors
 http://www.google.com/url?sa=t&source=web&cd=1&ved=0CBYQFjAA&
url=http%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fdownload%3Fdoi
%3D10.1.1.123.5878%26rep%3Drep1%26type%3Dpdf&rct=j&q=pricing%
20of%20CDO&ei=-3_-TI2REIGglAenzs2nCA&usg=AFQjCNG5q-
X2MPdrBSH5Kf5OM3yNJRXP4Q&cad=rja