Financial Principles

Multi-Step Credit Defaults

Coupling Asset Correlations to Credit Curve Cumulative Default Probabilities

This application note is part of RiskServers C++ Copula Excel® Add-In

Synopsis:

The paper explains how Risksvr™ maps obligor correlations to Default Time from the Cumulative probability of Defaults available from the obligor’s Credit Default Curve.

Copula Technology Review

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 propertiess:
a. C [0,1]ⁿ -> [0,1].
b. C is grounded and non-decreasing.
c. C has margins Ci which satisfy C_i(u) = C(1₁...u₁,1_.....u_{.. , ....}1_n...u_n) = u for all u in [0,1].

Note: These three properties have important implications which are often misunderstood:
In order to couple multivariate distributions to univariate distributions, all data 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,...F_n. Then there exists an n-copula C such that for all y in Rⁿ

F(y_1,y_2,....,y_n)=C(F1(y₁),....Fn(y_n)) [1]

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

Conversely, if C is an n-copula and F₁,...F_n are univariate distribution functions, then the function F defined by [1] is an n-dimensional distribution function with margins F₁,...F_nAn immediate consequence of Sklar's Theorem is

C(u1,...un)=F(F_n^-1(u₁),...,F_n^-1(u₁)).

Where F₁^-1 _,..F_n^-1 are generalized or quasi inverse of F₁,...F_n

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

The TimeToDefault function couples the cumulative probability of the Obligor’s(or Asset’s) cumulative probability to the multivariate density of price returns simulated through a standardized unitized Multivariate 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.

From this we can compute the probability of an event occurring at a given time t prior to a known event T. Alternatively we can seek the time t by knowing the probability.

.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.

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 the Gaussian (or Student T) Copula function C(u,1,u2,u3,…Un) of a series of individual univariate functions and the standard multivariate normal (mixture) function accordingly:

Correlated default events and default times are then simulated by first generating multivariate normal (mixture) random variates from standardized / unitized 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) to fit the distribution assumption.
.
Once variates have been standardized, the Gaussian (Student T) Copula can be solved by generating multivariate normal (mixture) distributions, either by Upper / Lower Cholesky factorisation, spectral decomposition, singular value decomposition (SVD), etc.

Despite common perception, coupling marginal probability densities between multivariate and univariate dimensions is conceptually trivial.

On the other hand, most practitioners seem at odds with the intricate details that are necessary to maintain consistency between distributions that are being coupled.

A Generic Multivariate Framework:

This framework is valid for Both Gaussian Copulas and Student-T copulas. It can easily be extended to include other copula models too.

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=U^TU=AA^T.^{(See Cholesky decomposition Application note). Here we assume A is Lower Triangular.}
3	:	We simulate a multivariate normal distribution with µ + AP. i.e. The resulting random vector is standardized to fit the normal distribution with mean µ and covariance matrix C:N(µ,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 µ and covariance matrix C:N(µ,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 Through Survival, Cumulative Hazards, Expected Default Probability or Marginal conditional defaults (forward defaults) conversion.

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

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.

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 µ+AT^* or eigenvectors / eigenvalue square roots of the the spectral decomposition

References:

Sklar A. (1959): Fonctions de répartition à n dimensions et leurs marges,Publications de l’Institut de Statistique de l’Université 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.