# Copula Correlations

The design of the copula functions is to easily simplify statistical problems. The global financial crisis of 2007 disgraced the flexible copula functions introduced to finance in the year 2000 despite being enthusiastically embraced. The popularity of copulas was due to their ability to easily solve complex problems by correlating multiple assets with a single function.

Multiple univariate distributions are joined by copula functions to form a single multivariate distribution. Objectively, to form a unit-dimensional one, an \(n\)-dimensional function on the interval \(\left[ 0,1 \right] \) is transformed by copula function \(C\) as shown below:

$$ C:\quad { \left[ 0,1 \right] }^{ n }\rightarrow \left[ 0,1 \right] $$

Given a univariate uniform distribution \({ G }_{ i }\left( { U }_{ i } \right) \in \left[ 0,1 \right] \) with \({ U }_{ i }={ U }_{ 1 }….,{ U }_{ n }\), and \(i\epsilon N\). Then \(\exists \) a copula function \(C\) such that:

$$ C\left[ { G }_{ 1 }\left( { U }_{ 1 } \right) ,…,{ G }_{ n }\left( { U }_{ n } \right) \right] ={ F }_{ n }\left[ { F }_{ n }^{ -1 }\left( { G }_{ 1 }\left( { U }_{ 1 } \right) \right) ,…,{ F }_{ n }^{ -1 }\left( { G }_{ n }\left( { U }_{ n } \right) \right) ;{ \rho }_{ F } \right] $$

\({ G }_{ i }\left( { U }_{ i } \right)\) is marginal distribution, \({ F }_{ n }\) the joint cumulative distribution function, \({ F }_{ i }^{ -1 }\) the inverse of \({ F }_{ i }\) and \({ \rho }_{ F }\) correlation structure of \({ F }_{ n }\) . Uniqueness in \(C\) is s due to continuity in \({ F }_{ i }^{ -1 }\left( { G }_{ i }\left( { U }_{ i } \right) \right)\).

The categorization of the numerous copula functions in one-parameter is the Gaussian and the Archimedean Copula family, with Gumbel, Clayton and Frank copulas being the most applied.

## The Gaussian Copula

In the \(n\)-variate case the definition of the Gaussian copula which is denoted as \({ C }_{ G }\) goes as:

$$ { C }_{ G }\left[ { G }_{ 1 }\left( { U }_{ 1 } \right) ,…,{ G }_{ n }\left( { U }_{ n } \right) \right] ={ M }_{ n }\left[ { N }^{ -1 }\left( { G }_{ 1 }\left( { U }_{ 1 } \right) \right) ,…,{ N }^{ -1 }\left( { G }_{ n }\left( { U }_{ n } \right) \right) ;{ \rho }_{ M } \right] $$

The joint \(n\)-variate cumulative standard normal distribution is \({ M }_{ n }\).\({ \rho }_{ M }\) is the \(n\times n\) symmetric positive-definite correlation matrix belonging to \({ M }_{ n }\), with \({ N }^{ -1 }\) being the inverse of a univariate standard normal distribution.

The uniformity in \({ G }_{ X }\left( { U }_{ X } \right)\) implies that \({ N }^{ -1 }\left( { G }_{ X }\left( { U }_{ X } \right) \right)\) are standard normal with the standard univariate being \({ M }_{ n }\) .

At a fixed time \(t\), the cumulative default probabilities \(Q\) for entity \(i\),\({ Q }_{ i }\left( t \right) \), was defined by David Li (2000) as marginal distributions.

Therefore:

$$ { C }_{ GD }\left[ { Q }_{ i }\left( t \right) ,…,{ Q }_{ n }\left( t \right) \right] ={ M }_{ n }\left[ { N }^{ -1 }\left( { Q }_{ 1 }\left( t \right) \right) ,…,{ N }^{ -1 }\left( { Q }_{ n }\left( t \right) \right) :{ \rho }_{ M } \right] \quad \quad \quad \quad \quad \left( a \right) $$

Which is the definition of Gaussian default time copula \({ C }_{ GD }\).

In the above equation, \({ Q }_{ i }\left( t \right)\) percentile is mapped by \({ N }^{ -1 }\) to a univariate standard normal distribution percentile. Therefore, the abscise values of the standard normal distribution are \( { N }^{ -1 }\left( { Q }_{ i }\left( t \right) \right)\). The multivariate normal distribution’s correlation structure applied with correlation matrix \({ \rho }_{ M }\) joins \({ N }_{ i }^{ -1 }\left( { Q }_{ i }\left( t \right) \right) \) to a single \(n\)-variate distribution \({ M }_{ n }\). \({ M }_{ n }\) gives the probability of \(n\) correlated defaults at time \(t\).

# Simulating the Correlated Default Time for Multiple Assets

In this subtopic, we use the Gaussian copula to determine a portfolios default for an asset that is correlated to the default time for other assets. Deriving sample \({ M }_{ n }\left( ˖ \right) \) from multivariate copula [r.h.s of equation (\(a\)) in the Gaussian case], \({ M }_{ n }\left( ˖ \right) \epsilon \left[ 0,1 \right] \), is crucial in deriving thedefault time \(\tau \) for asset \(i\) correlated to all other assets’, \(i = 1\),…,\(n\), default times. Through the default correlation matrix \({ \rho }_{ M }\) of the standard normal \(n\)-variate distribution \({ M }_{ n }\) , the default correlation is included in the sample.

\({ Q }_{ i }\left( { \tau }_{ i } \right) \), which is the cumulative individual default probability \(Q\) of asset \(i\) at time \( \tau \) is equated to sample (˖) from \({ M }_{ n }\) ,\({ M }_{ n }\left( ˖ \right) \). Hence:

$$ { M }_{ n }\left( ˖ \right)={ Q }_{ i }\left( { \tau }_{ i } \right) \quad \quad \quad \quad I $$

$$ \Rightarrow { \tau }_{ i }={ Q }_{ i }^{ -1 }\left( { M }_{ n }\left( ˖ \right) \right) \quad \quad \quad \quad II $$

Since there is no closed-form solution for equations \(I\) and \(II\), sample \({ M }_{ n }\left( ˖ \right)\) is first taken and applied to equation \(I\) and then equated to \({ Q }_{ i }\left( { \tau }_{ i } \right) \) using search models like the Newton-Raphson, to find the solution.

The correlation matrix is an input of the \(n\)-variate standard normal distribution \({ M }_{ n }\) ,and therefore, default correlation with a portfolio’s other assets is included in the estimated default time of asset \(i\).

To further illustrate this, we consider a random drawing from \({ M }_{ n }\left( ˖ \right)\) as 35%. The 35% is equated to the market given function \({ Q }_{ i }\left( { \tau }_{ i } \right) \) and the expected default time for asset \(i\),\({ \tau }_{ I }\) is determined as displayed in the following figure with an assumption of \({ \tau }_{ i }\) = 5.5 years. To find the estimate for \({ \tau }_{ i }\) , the process is repeated many times with each \({ \tau }_{ i }\) of every simulation averaged.

# Practice Question

1) Which of the following is the correct definition of a copula?

- A function linking a univariate marginal distribution to the full multivariate distribution
- A mapping of default probabilities of assets at a given time
- A model assigning correlation structures to multivariate distributions
- All of the above

The correct answer is **A**.

If we are provided with marginal distributions \({ G }_{ 1 }\left( { U }_{ 1 } \right)\) to \({ G }_{ n }\left( { U }_{ n } \right)\) then exist a copula function allowing the linking of marginal distributions \({ G }_{ 1 }\left( { U }_{ 1 } \right)\) to \({ G }_{ n }\left( { U }_{ n } \right)\) through an inverse function ┌^{-1} and the joining of abscise values ┌^{-1}\(\left( { G }_{ 1 }\left( { U }_{ 1 } \right) \right) \) to an n-variate function ┌_{n}[┌_{1}^{-1}\(\left( { G }_{ 1 }\left( { U }_{ 1 } \right) \right)\),….,┌_{n}^{-1}\(\left( { G }_{ n }\left( { U }_{ n } \right) \right) \)] with a correlation structure \({ \rho }_{ F }\).