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Transcript of 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>