Setup >Credit >Copula >Time-To-Default

Time-To-Default from Obligor Asset Correlation and Credit-Curves

 

The Time to default methodology computes the time when one or a series of Obligors associated to a Credit curve and a series of Asset Correlations will actually default.

This probabilistic model is currently the most advanced model to compute correlated defaults in the Financial Industry.


The Time-To-Default is obtained by computing a Copula between the Obligor's Asset Correlation and the Obligor's associated Credit-[Default]-Curve.

 

All you need to know about Credit-Curves:

This document assumes you are familiar with Credit-[Default]-Curves. 

Credit Curves are covered in detail in the Credit Default Curve Excel Add-In. Credit Curve Document  
For further information relating to Hazards, Expected Default Probabilities, Transition Matrices, Marginal Conditional Defaults and Survival Rates,  please refer to the Credit Curve online documentation or the Credit-Curve Add-In Help file.

Simulating Time to Default
Copula Model Application

We briefly review the theorem. Then we show how it applies to coupling the Multivariate Normal distribution of Obligor / Asset Correlations to Univariate distributions of the asset's/obligor's mortality/survival  probability.


An n-dimensional copula is a multivariate distribution function, C, with uniform distributed margins in [0,1] i.e. U(0,1) which has the following properties:
a. C [0,1]n -> [0,1]. 
b. C is grounded and non-decreasing.
c. C has margins Ci which satisfy Ci(u) = C(11...u1,1.....u.. , ....1n...un) = u for all u in [0,1].

Note: These three properties have important implications which are very often misunderstood: 
In order to couple multivariate distributions to univariate distributions, both distributions must be standardized.

The most important aspect of  Copula Theory is based upon Sklar's Theorem:  

Let  F be an n-dimensional distribution function with  margins F1,...Fn. 

Then there exists an n-copula C such that for all y in Rn          

F(y1,y2,....,yn)=C(F1(y1),....Fn(yn))   [1]

If F1,...Fn are all continuous then C is uniquely defined, Otherwise C is uniquely determined within a range  F1x... x. range Fn.

Conversely, if C is an n-copula and F1,...Fn are univariate distribution functions, then the function F defined by [1] is an n-dimensional distribution function with margins F1,...Fn





 An immediate consequence of Sklar's Theorem  is C(u1,...un)=F(Fn-1(u1),...,Fn-1(u1)).

Where F1-1 ,..Fn-1  are generalized or quasi inverse of  F1,...Fn 

 


Computing Time to Default by coupling asset correlations to probabilities of default.

The Time to Default module bridges the cumulative probability embedded in the Credit Curve of  each Obligor (or Asset) to the multivariate normal distribution of  price returns simulated through a standard unitized Multivariate normal distribution 

=Is the cumulative default probability of an asset or obligor with a Rating R up to a given time period t.
=Is the default time of the asset.

  We seek the probability of an event occurring at a given time t prior to a known event T or we can seek the time t by knowing the probability.

.

Note: Pricing Application
Once we know the time of default of a given asset cash-flow, we can separate "certain" payments, including proceeds from reinvested interest from "recovered" payments (with or without stochastic recovery).


The cumulative default probability is often computed by taking the opposite probability of Survival (No survival), as well as Marginal Conditional Default Probabilities (aka. Forward default probability) or Hazard Rates. To see how these probabilities are interchangeable and lead to the same results,  see the Credit-Curve module.

Let:

= the correlation matrix between assets / obligors.
= the multivariate normal distribution function.
= the univariate standard normal cumulative distribution function.
= the univariate standard normal cumulative distribution function with uniform random
   variate u(i).

 

 

 

The Time to Default of a Given Obligor is computed by creating a dependency between (or perhaps better said Coupling) the Gaussian Copula function C(u,1,u2,u3,�Un) of a series of individual univariate functions and the standard multivariate normal function accordingly:

Correlated default events and default times are then simulated by first generating multivariate normal random variates from the unitized / normalized asset or obligor correlations yi with I=1,�n Compute u(I)=

As mentioned in properties a,b and c. data must be standardized (see unitized returns). 
. 
Once variates have been unitized and the appropriate margin thresholds defined, the Gaussian Copula can be solved by generating a multivariate normal distribution, either by Upper / Lower Cholesky partitioning, spectral decomposition, singular value decomposition (SVD), etc.

Despite common perception, fitting marginal probability densities is numerically trivial:

 This framework is valid for Both Gaussian Copulas and Student-T copulas.

 

 

Case 1: the Correlation Matrix is Positive Definite:

1 : We generate n independent p1,...,pn (normal or student-T) Variates in a Column Vector P.
2 : We decompose the Correlation Matrix C into Cholesky Coefficients. C=UTU=AAT. 
 (See Cholesky decomposition Application note). Here we assume A is Lower Triangular.
3 : We simulate a multivariate normal distribution with � + AP. 
i.e. The resulting random vector is standardized to fit the normal distribution with 
mean .0 and covariance matrix C:N(0,C). 

Case 2: the Correlation Matrix is Not Positive and - or Definite:

1 : We generate n independent p1,...,pn (normal or student-T)  Variates in a Column Vector P.
2 : We perform a spectral decomposition of the Correlation Matrix C. 
(i.e. we decompose the Matrix into eigenvalues and eigenvectors, 

we then sort both eigenvectors and eigenvalues according to the decreasing order of eigenvalues and find the proper rank of the Matrix (see principal component application note).
3 : We simulate a multivariate normal distribution with . 
The resulting random vector is normally distributed with mean .0 and covariance matrix C:N(0,C). 

 

Once we have obtained u(1),u(2),...u(n) we obtain default times from the credit curve by seeking the corresponding inverse cumulative normal density function of defaults, either via Survival, Cumulative Hazards, Expected Default Probability or Marginal conditional defaults (forward defaults) conversion.



 

Extending Gaussian Copula To Normal-Mixture or Student-T Copula

Note: The C++ Multivariate Student T AddIn implements the normal mixture method described below. 


Despite all the leverage and power provided by the Gaussian Copula model, some practitioners assume the Gaussian Copula does not mimic well enough what can be observed in the market (especially specific asset classes, such as equities or foreign exchange that are not diversified). 

As  with the normal distribution assumption of Parametric VaR, the Gaussian model does not take into account the fact that some risk factors returns are skewed and fat tailed and that asset return extreme movements  (or joint co-movements) happen more often than assumed by the normal distribution (see six sigma).

Note: The methodology to simulate a student-t distribution or a mixture model  is based  largely on the same approach, except that we must:

  • Scale the Variance covariance / correlation matrix by (v/(v-2)).
    where (v) is the degree of freedom of the distribution. P=C (v/(v-2)).

  • Either decompose the scaled  Variance/Covariance - Correlation  Matrix (if positive definite) with Cholesky technology or Perform a spectral decomposition retaining only positive definite variables.  (i.e. eigenvectors that belong to eigenvalues >0). 

  • If we want to generate a p variate vector T from a multivariate Student's T distribution with mean vector and Covariance matrix P, we must :Generate p independent Student's T random variates. we can do this either through Box Muller Polar Coordinates or by mixing a standard normal variate with a random Chi Square / Gamma distributed variate. 

  • We then apply the simulation technique to the Lower Triangular of the Cholesky coefficients  �+AT* or eigenvectors / eigenvalue square roots of the the spectral decomposition

 

 

 

 

 

 


References:
  • Sklar A (1959): Fonctions de r�partition � n dimensions et leurs marges.
    Publications de l�Institut de Statistique de l�Universit� de Paris 8, pp. 229-231.
  • CA.Schewizer B. & Sklar A. 1983. Probabilistic Metric Spaces. North-Hollan/Elsevier. New York. 
  • Joe: Multivariate Models and Dependence Concepts. Chapman & Hal. London. 
  • Nelsen, R. (1998) An introduction to copulas. Springer, New York.
  • Li David: On default correlation: a copula function approach. Journal of Fixed income 9. 43-45.
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