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### CDS/CDOs and the Gaussian Copula Formula

Shane Kaiser

Housing Crisis/Recession

- 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 theGaussian Copula formula, created by the mathematician and actuary Dr. David X. Li

Dr. David X. Li

- 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

Collateralized Debt Obligations

- 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

Collateralized Debt Obligations

- 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

Collateralized Debt Obligations

- 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

Collateralized Debt Obligations

- 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

Dr. Li and his formula

- 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

Credit Default Swaps (quick review)

- 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

The Gaussian Copula Formula

- 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

The Gaussian Copula Formula

- 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
- Φγis 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

Implementation of the formula

- 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

Implementation of the formula

- 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

Examples

- 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 ofhome 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

Controversial Transactions

- As time progressed, banks kept finding more and more ways to takerisky 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…)

Controversial Transactions

- 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

Controversial Transactions

- 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

Is Dr. Li to blame?

- “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

Is Dr. Li to blame?

- 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

Sources

- 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_policies
- 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

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