default correlation presentation

8/7/2019 default correlation presentation

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Default Correlation:

Gaussian Copula andCredit Crisis of 2008-2009

Emily Fero

April 29, 2009


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Default Correlation

What is the likelihood that if one

company defaults, another will

default soon after?


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Correlation Correlation measures the degree to which the

probability of one event happening moves in syncwith the probability of another event happening.

In terms of default correlation,

Zero correlation means that the default of one

company has no bearing on the default ofanother company the companies arecompletely independent of each other.

Perfect positive correlation means that if onecompany defaults, the other will automatically

follow suit. Perfect negative correlation means that if one

company defaults the other one will certainly not.

(Lucas)


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Factors of Default Correlation

macroeconomic environment: good

economy = low number of defaults

same industry or geographic area:companies can be similarly or inversely

affected by an external event

credit contagion: connections between

companies can cause a ripple effect

(Lucas)


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Why do we want to know about

default correlation? Pricing tranches of CDOs.

Collateralized debt obligation is a pool of a

debt, like mortgages, corporate bonds, orcredit default swaps. They are sliced into

tranches that pay high premiums for the risky

tranche and low premiums for the triple-A

rated tranche. Credit default swap insures against the

default of a bond.

(W

hitehouse, Salmon)


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David Li

Statistician who moved over to business

Worked at a credit derivative market in 1997

and knew about the need to measure default

correlation Colleagues in actuarial science working on

solution for death correlation, a function

called the copula

Default is like the death of a company, so we

should model this the same way as we model

human life (Li)

(Whitehouse, Salmon)


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Copula

Latin word that means to fasten or fit

Bridge between marginal distributions and

a joint distribution. In the case of death correlation, the

marginal distribution is made up of

probabilities of time until death for one

person, and joint distribution shows the

probability of two people dying in close

succession.


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(http://www.mathworks.com/access/helpdesk/help/toolbox/stats/copula_17.gif)

Joint distribution

Marginal distributions


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Skylars Theorem (1959)

If you have a joint distribution function

along with marginal distribution functions,

then there exists a copula function that

links them; if the marginal distributions are

continuous, then the copula is unique.

(Meneguzzo and Vecchiato 43)

(http://www.mathworks.com/access/helpdesk/help/toolbox/stats/copula_14.gif)


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Gaussian copula

Assumes that if the marginal probability

distributions are normal, then the joint probability

distribution will also be normal.

J

C is the copula function of two normal distributions,

2 is the multivariate normal distribution function with

correlation coefficient , and

^(-1) is the inverse of the cumulative univariate

normal distribution functions, u and v

(Jabbour et al. 32, Li 14)

C(u,v) = 2(-1(u), -1(v), ), -1 1


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From this definition of the Gaussiancopula, it is clear that for the case of

pricing CDOs, we will need two other

pieces of information aside from thechoice in copula:

the normal marginal distribution

functions

a correlation coefficient


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Marginal Probability Distributions

Characterized by default probabilities.

Can approximate default probabilities from ratingagencies, from market prices, or from a theoreticalapproach described by Merton (Li 10-11).

Obtained from market prices because they reflect themarkets perceptions about the credit-worthiness of acompany.

Series of calculations to go from the market price of a

default swap or defaultable corporate bonds to obtain thecumulative probability distribution for time-to-default (Li(1998)).

Normalize cumulative probability distributions by takingtheir inverse normal to create a new distribution.

x1=N-1[Q1(t1)] (Hull 514)


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Copula Correlation Number Specifies the shape of the multivariate distribution

Zero correlation = circular

Positive or negative correlation

= ellipse

The correlation number is always independent of themarginals (Hull 515).

Assumptions

one-to-one relationship between asset correlationand default correlation based on the definition of

default as an asset falling below a certain value. Correlation number is always positive (Li 11-12).

The correlation number is an extremely important factorin this model because it determines the information youget out of the model.

(http://www.mathworks.com/access/helpdesk/help/toolbox/stats/gmdistribution_fit1.gif)


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Marginal distributions(default probability distributions from market data)

+

Correlation number

(estimated by asset correlation)

+

Choice of copula

(Gaussian / normal copula)

=

Fully defined multivariate distribution of the

probability of defaulting within time T


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Sampling

Take samples of pairs (u,v) from this joint

distribution, where u and v are univariate

normal probability distributions.

Map sample pairs back to the blank joint

distribution to get an ordered pair of time-

to-default values.

(Hull 515)


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Default Correlation Use sample pairs to calculate default correlation in joint

distribution.

Binomial correlation measure uses a set of rules to change

time-to-default values into either the value 1 or 0, so that if a

company defaults within time T, it is assigned the variable 1,

and otherwise it is assigned a 0. Once the variable ischanged, the following equation can be used to calculate

default correlation:

where QA(T) is the cumulative probability that A will default by time T.

PAB(T) is equal to M[u, v, AB], which is the probability that, in a bivariate normal

distribution where the correlation between the variables is , the first variable is less

than u and the second variable is less than v. This calculation relies on the Gaussian

copula model.

(Hull 516).


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Why we use the copula model approach

The reason we use the copula model to calculate defaultcorrelation is that we cannot observe it. It seems probable

that there is a close relationship between asset correlation

and default correlation, but we do not know what it is.

Therefore, if we input default probability distributions into a

determined structure with a correlation number that isindependent of the marginals, we will be rewarded with a

fully defined joint probability distribution. We have to

interpret this information, though, because we have

converted the marginal probability distributions to a normaldistribution. If we take samples from this defined

multivariate distribution and convert them back to default

probabilities, we can determine the correlation between

the marginals that was induced by the asset correlation we

first inputted.


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Credit Crisis Size of CDS and CDO markets, leverage

Inputs

Asset correlation

CDS spreads: CDSs are relatively new andonly existed when housing prices were on the

rise

Copula choice

Heavy dependence on a single model

Everyone using it

Each assuming its perfect

(Whitehouse, Salmon)


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Future of the Model

Schnbucher and Schubert devised a

method that allows the default correlation

to be dynamic (2001) (Meneguzzo and Vecchiato 41)

Fit of different copulas, like Student-t

copula (Meneguzzo and Vecchiato 41, Jabbour et al. 32)

Implied correlation from new credit

derivative indices (Jabbour et al. 43)


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Works CitedSpecial thanks to Dr. Jesse Frey for his patience while explaining copulas.

Hull, John C. Options, Futures, and Other Derivatives. 7th ed. Upper Saddle River, NJ: Pearson/Prentice Hall, 2008. pp

497-516.

Jabbour, George M., Marat V. Kramin, and Stephen D. Young. Nth-to-default swaps: valuation and analysis.

Managerial Finance. Vol. 35, No 1, 25-47 (2009). Emerald Group Publishing Limited. 27 April 2009.

Li, D. X. (2000). On default correlation: A copula function approach. RiskMetrics Group. 27 April 2009.

Lucas, Douglas. (2004). Default Correlation: From Definition to Proposed Solutions. UBS: CDO Research. 27 April

2009.

Meneguzzo, Davide, andWalter Vecchiato. Copula Sensitivity in Collateralized Debt Obligations and Basket Default

Swaps. The Journal of Futures Markets Vol. 24, No 1, 37-70 (2004).Wiley InterScience. 27 April 2009.

Multivariate Modeling. The MathWorks, Inc.: Statistics Toolbox. 27 April 2009.

Salmon, Felix. Recipe for Disaster: The Formula that KilledWall Street.Wired Magazine. Issue 17-03. 23 Feb 09. 27

April 2009.

Whitehouse, Mark. How a Formula Ignited Market That Burned Some Big Investors. Wall Street Journal Online. 12

Sept 2005: Page A1. 27 April 2009. < http://www. nowandfutures.com/download/credit_default_swaps_WSJ_

news20050912.pdf>

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