Gaussian Copula Vs. Loans Loss Assessment: A Simplified And Easy-To-Use Model

The copula theory is a fundamental instrument used in modeling multivariate distributions. It defines the joint distribution via the marginal distributions together with the dependence between variables. Copulas can also model dynamic structures. This paper offers a brief description of the copulas’ statistical procedures implemented on real market data. A direct application of the Gaussian copula to the assessment of a portfolio of loans belonging to one of the banks operating in Lebanon is illustrated in order to make the implementation of the copula simple and straightforward.


INTRODUCTION
he concept of "copulas 1 " was named by Sklar (1959). In statistics, the word copula describes the function that "joins" one dimensional distribution functions to form multivariate ones, and may serve to characterize several dependence concepts. It was initiated in the context of probabilistic metric spaces. Copulas function became a popular multivariate modeling tool in financial applications since 1999 (Embrechts, McNeil and Straumann (1999)). They are used for asset allocation, credit scoring, default risk modeling, derivative pricing, and risk management (Bouyè, Durrleman, Bikeghbali, Riboulet, and Roncalli 2000; Cherubini, Luciano, and Vecchiato 2004). They became more and more popular because as proven, the returns of financial assets are non-Gaussian and show nonlinearities; therefore, copulas became an imperative modeling device in a non-Gaussian world. They implement algorithms to simulate asset return distributions in a realistic way by modeling a multivariate dependence structure separately from the marginal distributions.
For multivariate distributions, the univariate margins and the dependence structure can be separated and the latter may be represented by a copula. The copula of a multivariate distribution whose marginals are all uniform over (0,1). Sklar's Theorem states that a d-dimensional cumulative distribution function F evaluated at point X = (x 1 , …, x d ) can be represented as where C is the copula function and F i , i = 1, ..d, are the margins. In most cases the latter function is uniquely defined by (1.1).
If F 1 , F 2 are continuous, then C is unique; otherwise, C is uniquely determined on Ran(F 1 ) x Ran(F 2 ). Conversely, if C is a copula and F 1 , F 2 are distribution functions, then the function F defined in (1) is a joint distribution function with margins F 1 and F 2 . For the proof demonstration, please refer to Nelsen (1999, p. 18). It is the converse of the Sklar's theorem that is mostly used for modeling multivariate distributions in finance. Any group Using copulas is attractive to practitioners because even under increasing transformations of the margins, copulas remain invariant. This requires the completion of two steps: modeling each univariate marginal distribution then specifying a copula that recapitulates all the dependencies between margins.
In this paper our aim is to simplify the implementation of copulas for the Lebanese banks since none of which is using copulas for risk assessment. Section 1 introduces the model that generates the copula through the Archimedian and elliptical copulas. Section 2 illustrates the use of the Gaussian copula in the assessment of a portfolio of loans belonging to one of the banks operating in Lebanon. Section 3 draws some concluding remarks.

DEFINITION AND FAMILIES OF COPULAS
The quantification of dependence in finance has led to the development of copulas. Copulas are now witnessing increasing interest in many areas of risk analysis. The rank order correlation 2 used by most Monte Carlo simulation tools is certainly a meaningful measure of dependence but is still limited in the patterns it can produce.
As described in the introduction, copulas offer an efficient flexible procedure for combining marginal distributions into multivariate distributions and are able to deduce the real correlation pattern. The purpose of this paper is not to go through the explanations and proof of the mathematics related to copulas (that can be found in any related book), we will rather simplify the concept and depict what is just needed to run an analysis using the copula.
To this end, in what follows, we use the needed formulae for a bivariate copula for simplicity purpose. Related graphs of bivariate copulas will be illustrated. Multivariate copulas can be extended based on the bivariate concept.
Because the copula of a multivariate distribution describes its dependence structure, we can use measures of dependence such as Kendall's 3 tau and Spearman's 4 rho, as well as the coefficient of tail dependence, which are copula-based. The relationship between Kendall's tau t of two variables X and Y and the copula C(u,v) of the bivariate distribution function of X and Y gives a tool for fitting a copula to a set of data. This consists in determining Kendall's tau then applying a transformation in order to get the appropriate parameters for the fitted copula.

The Copula Families
The most frequently used copula families are the elliptical and Archimedean copulas. The below table summarizes and compares the two families. Kendall's tau for a two observed sets of variables is given by: where C is the number of concordant pairs and D the number of discordant pairs. This can also be written as: This is used to measure the degree of correspondence between two variables (paired observations): Perfect correspondence between the two variables means that the coefficient has a value of 1 Perfect disagreement between the two rankings means that the coefficient has a value of -1. For all other arrangements, the value lies between -1 and 1, 0 meaning the variables are completely independent. Therefore, Kendall's tau for (X, Y ) T is simply the probability of concordance minus the probability of discordance. 4 Kendall's tau and Spearman's rho are both carried out on the ranks of the data as explained and defined in note 2. That is, for each variable separately the values are put in order and numbered. 5 The normal copula is given by: 6 The Student-t copula is an elliptical copula defined as: where v (the number of degrees of freedom) and ρ (linear correlation coefficient) are the parameters of the copula. When v is large (greater than 30) the copula converges to the Normal copula just as the Student distribution converges to the Normal. In the opposite case, the behavior of the copulas is different: the t-copula has more points in the tails than the Gaussian one and a star like shape. A Student-t copula with n = 1 is sometimes called a Cauchy copula. 7 It is an asymmetric copula, exhibiting greater dependence in the negative tail than in the positive. This copula is written as: http://www.cluteinstitute.com/ © 2012 The Clute Institute

Gumbel 9
Its generator is: where: The relationship between Kendall's tau and the Clayton copula parameter is given by: 8 It is a symmetric copula given by: Its generator is: where: The relationship between Kendall's tau and the Frank copula parameter is given by: where: 9 The Gumbel-Hougard copula is asymmetric, exhibiting greater dependence in the positive tail than in the negative. It is given by: The relationship between the linear correlation coefficient ρ and Kendall's tau t for all the elliptical copulas' families is given by: The general relationship between Kendall's tau t and the generator of an Archimedean copula a (t)for a bivariate data set is given by: 2. Advantages Different levels of correlation between the marginals can be easily determined.
Easy to be deduced.

Disadvantages
Absence of closed form expressions, impossibility to have radial symmetry.
The definition doesn't extend to a multivariate data set of n variables because there will be multiple values of tau, one for each pairing 10 .

Graphics of Copula Families
Source: Vose Software, Risk Software Specialists Its generator is: where: The relationship between Kendall's tau and the Gumbel copula parameter is given by: http://www.cluteinstitute.com/ © 2012 The Clute Institute

APPLICATIONS
The purpose of this section is to illustrate a direct example on how to implement a one factor model of the Gaussian copula (normal copula) on a portfolio of loans belonging to a Lebanese bank operating in Lebanon (Naïmy (2012)).
The portfolio is constituted of 30 different companies. We define T i (1 ≤ i ≤ N) as the date upon which the company will default. We consider that all the studied companies in the portfolio can eventually default (Naïmy (2011)) however; this cannot happen except over long periods sometimes exceeding 50 years or more.

The Model
Q i is the distribution of T i . In order to define a correlation structure among the Ti based on a one factor model of a Gaussian copula, we have transformed each variable, percentile per percentile, into a U i variable, following a standardized bivariate distribution. We followed the below factorial model (one factor model) for the correlation structure: Where F and Z i follow a normal distribution N(0,1). The link between U i and T i implies: According to equation (2), the probability of U i < U conditional to the value of F is: Equation (4) can be written for Prob(T i <T): In order to simplify the reasoning process, we can assume that the time distribution of default Q i is the same for each i and equals Q. We also assume for simplicity purpose that the copula correlation between each pair of companies is the same and equals ρ. In this case, for each pair of companies i and j, the correlation is a i a j , which means that for each i, a i = √ρ. Therefore we can rewrite equation (5) as follow: For a portfolio constituted of a huge number of loans, this equation offers a valid approximation of the proportion of companies that can default on date T. We name this proportion the default rate. When F decreases, the default rate increases. Therefore what is the cap level that this rate can reach? Since F follows a standardized bivariate distribution, the probability for F to have a value less than N -1 (Y) is Y. Therefore, there is a probability Y for the default rate to be greater than: We note that the default rate V(T,X) will not be exceeded with a probability of X. In other words, we are sure up to X% that the default rate V(T,X) will not be exceeded. The value V(T,X) is measured by substituting 1-X to Y: This output has been first developed by Vasicek (1987).
If we assume that the total value of the loan in USD is L and the recovery rate in case of default is R. The VaR of a loan over a horizon period of T and a confidence level of X is:

The Sample
We were able to analyze the loan portfolio of a Lebanese bank operating in Lebanon. For discretion purpose we will omit the name of the bank however, the below tables and graphs are enough to give an idea about the portfolio details and the financial performance of this bank. The main purpose of this section is to directly implement the Normal copula model in order to measure the expected loss on a portfolio of loan.

Results
The total amount of loan on which we applied the model is USD 56.4 millions. Based on our observation we have been able to determine a default probability over a horizon period of one year of 2%. The recovered amount in case of default was quantified by the bank at a level of 72%. Using the Vose Software 12 we deduced a copula correlation rate of 0.1.
Based on the above and according to equation (7)

CONCLUDING REMARKS
This work showed how the Gaussian copulas theory can be very powerful tool in estimating the expected losses on a portfolio of loans.
Risk managers often estimate the marginal distribution between each of the variables and face the obligation to set hypothesis on the structure of the variables' correlation. When the marginal distributions are normal, it is natural to consider that these variables follow multivariate normal distributions. In the opposite cases, copulas are therefore used. Variables are transformed into normal variables based on a percentile per percentile adjustment process. The correlation is then indirectly defined among the transformed variables. When many variables are studied, a factorial model is used by analysts. This approach allows reducing the number of correlation to be calculated. The correlations between variables are described by the correlations of variables toward the common factor. The default correlation between many firms can be modeled based on a Gaussian copula factorial model.
Despite all the benefits of copula models, specifying a copula that recapitulates all the dependencies between margins suffers from some statistical pitfalls especially for most of multivariate financial series. This is due to temporal dependencies such as serial autocorrelation, or time varying heteroskedasticity, etc. Further research must be done in order to broaden the scope of implementations to higher dimensions while considering the limitations of copulas in general.