PORTFOLIO RISK AND DEPENDENCE MODELING: APPLICATION OF FACTOR AND COPULA MODELS

We consider portfolio credit risk modeling with a focus on two approaches, the factor model, and the copula model. While other models have received greater scrutiny, both factor and cupola models have received little attention although these are appropriate for rating-based portfolio risk analysis. We review the two models with emphasis on the joint default probability. The copula function describes the dependence structure of a multivariate random variable. In this paper, it is used as a practical to simulation of generate portfolio with different copula, we only use Gaussian and t–copula case. And we generate portfolio default distributions and study the sensitivity of commonly used risk measures with respect to the approach in modeling the dependence structure of the portfolio.


Introduction
There is a need to understand components of portfolio risk and their interaction. The Basel Committee for Banking Supervision in its Basel proposed (BIS, 2001) to develop an appropriate framework for a global financial regulation system. Several portfolio credit risk models developed in the industry have been made public since then. Examples are: CreditMetrics (Gupton et al., 1997), CreditRisk+ (Credit Suisse Financial Products, 1997) and Credit Portfolio View (Wilson 1997a;1997b). Others systems remain proprietary, such as KMV's Portfolio Manager (Kealhofer, 1996). Although the models appear quite different on the surface, recent theoretical work has shown an underlying mathematical equivalence among them (Gordy, 2000; and Koyluoglu and Hickman, 1998).
The credit portfolio models to obtain portfolio loss distributions, which are statistical models, can be classified as based on credit rating systems; See Crouhy et al. (2001) for exact description and discussion of the various models. Frey and McNeil (2001) study the mathematical properties of the models and consider the modeling of dependent defaults in large credit portfolios using latent variable models and mixture models. Crouhy et al. (2000) compared and reviewed models on benchmark portfolio using credit migration approach, the structural approach, the actuarial approach, and McKinsey approach. However, few studies have attempted to investigate aspects of portfolio risk based on rating-based credit risk models. Gordy (2000) offered a comparative anatomy of two especially influential benchmarks for credit risk models, the Risk Metrics Group's Credit Metrics and Credit Suisse Financial Product's . Kiesel et al. (1999) employ a mark-to-market model and stress the importance of stochastic changes in credit spreads associated with market values, an aspect also highlighted in Hirtle et al. .
The aim of this paper is to contribute to the understanding of the performance of ratingbased credit portfolio models, long ignored in the field. We apply a default-mode model to assess the effect of changing dependence structure within the portfolio. First, in the ensuing section, we discuss about the copula model as one of the dependency approaches within the portfolio.
Second, we describe a factor model by focusing on the effects of default dependence model within the portfolio. Finally, in the penultimate section, we simulated types of copula model with different degree of freedom within the portfolio.

Copula Modelling
An overview of basic copula uses in structural systems and models is provided in this section.
Copulas provide a natural way to study and measure dependence between random variables.
Suppose we have specified a portfolio of obligors, with default times . The variable of default of obligor , at time t, is donated as . The probability space is . This space has filtration : (1) For the joint default probability at time t, evaluated at time 0, as The survival property as We take for granted the copula definition as a joint distribution function with uniform margins, which implies that and take for granted the fundamental Sklar's theorem, in terms of a copula and the marginal distribution functions : The joint survival probability with survival copula, and the marginal survival functions : Factor copula is, In the credit risk case, since the variables are default time, the copula represents default dependence. It is donated as , The default probability at time is, The marginal default probability can be easily computed to be (11) Then, And is the instantaneous return on assets, which equates the riskless rate under the risk neutral measure. The joint default probability of assets is (13) Where, is the distribution function of a standard normal vector with correlation matrix R. the marginal default probabilities is follows ) To study the effect of different copula on default correlation, we use the following examples of copula (further details on these copula can be found in Embrechts et al., 2001).
(i) Gaussian copula: Where, denotes the joint distribution function of the -variety normal with linear correlation matrix ,and the inverse of the distribution function of the univariate standard normal.
(ii) A student copula: Where is the standardized multivariate Student's distribution, with correlation matrix and degrees of freedom, While is the inverse of the corresponding margin. Gumbel copula: Where . This class of copula is a sub-class of the class of Archimedean copula.
According to the table[1], joint default probabilities of two obligors are represented through three types of obligors with individual default probabilities corresponding to rating classes.as you will see that and Gumbel copula have higher joint default probabilities than the Gaussian copula.
The joint default probabilities of two Obligors are represented through three types of obligors with individual default probabilities corresponding to rating classes.

Factor Modelling
Another popular approach to default modeling allows us to switch to the so called product copula. The reduction technique, which is widely adopted for the evaluation of losses in highdimensional portfolios, with hundreds of obligors (see for instance Laurent and Gregory (2003)), is the standard approach of (linear) factorization, or transformation into a Bernoulli factor model.
In the typical portfolio analysis the vector is embedded in a factor model, which allows for easy analysis of correlation, the typical measure of dependence. We assume that the underlying variables are driven by a vector of common factors.
Where is dimensional normal vector, and is independent normally distributed random variables. Here is obligor to factor , i.e. the so-called factor loading and is volatility of the risk contribution. The default indicators of the obligor are independent Bernoulli variables, with probability: Where is cut-off point for default obligor . The individual default probabilities are, And the joint default probability is, If we denote by the correlation of the underlying latent variables and by the default correlation of obligors and , then we obtain the default correlation formula Under assumption above, we obtain the joint default probability, Where is bivariate normal density with correlation coefficient .

Simulation Results of Copula Model
Here, we want to generate portfolios with given marginal and the above copula. we only use Gauss and copula case . We looking for random sample generation for this mean we obtain the generation of an -variety normal with liner correlation matrix ,to take realizations from a Gaussian copula we simply have to transform the marginal: • Set • To generate random varieties from the -copula we assume the random vector X act the stochastic process With Where Z and Y are independent, and then X is distributed with mean and covariance matrix we assume , while the stochastic process is still valid the parameters has to change for . We will have algorithm (this is algorithm in Embrechts et al. (2001)): • Set • Set • .
We can replace the with in order to have multivariate distribution with -copula and normal marginal, to obtain the -copula . Further analyses of the same data are plotted in Figure 2. These are scatterplots of a normal sample. In an adjacent plot next to each sample, we present a second plot as the contour ellipses for mu =# (3, 2) and sigma = # (1,-1.5) ~# (-1.5, 4) with different size.

Portfolio
For our first simulation exercise, we assume that the underlying variables are normally distributed within a single factor framework, i.e. and in formula as follow: They are constant and are chosen so that the correlation for the underlying latent variables is (Kiesel et al., 1999. Note that we use three rating classes, named A, B, and C with default probabilities 0.005, 0.05, and 0.15 roughly corresponding to default probabilities from standard rating classes (Ong, 1999). To generate different degrees of tail correlation, we link the individual assets together using a Gaussian, a and a -copula.
The information in table 2, 3 and 4 represent the effect tail-dependence has on the high quintiles of highly-rated portfolios at different quintiles: The copula is more than three-times larger than the corresponding quintile for the Gaussian copula. The same effect can be observed for lower rated portfolios although not quite with a similar magnitude. We assume the second factor, i.e. in (4), for a sub-portfolio of 100 obligors increasing the correlation of the latent variables within the sub-portfolio to 0.    for this reasaning we want to shows the effects of increased correlation within parts of the portfolio; we change the factor loading within parts of our portfolio. These results are shown in tables 7, 9 and 10.
As expected, the results in Tables 5, 6, 7 show increase in the quantiles due to the increased correlation within the portfolio. However, comparing the three tables we will see that the sensitivity of the portfolio loss quantiles is higher with regard to the underlying copula than to the correlation within the portfolio.

Conclusions
To investigate the riskiness of credit-risky portfolios is one of the big challenging in financial mathematics. An important thing for a model of credit-risky portfolios is the dependence structure of the underlying obligors. We studied two approaches, a factor structure, and the direct specification of a copula. We generated portfolio default distributions and studied the sensitivity of commonly used risk measures with respect to the approaches in modeling the dependence structure of the portfolio using as a rating-based approach using cupola mathematics.
The simulation results indicate that the degree of tail dependence of the underlying copula plays a major role. That is identified as a credit risk. The copula modeling links the underlying variables together, which is of crucial importance especially for portfolios of highly-rated obligors.
Author Information: Arsalan Azamighaimasi is a faculty member in the Department of Management, Wuhan University of Technology, China. He may be contacted at E-mail: Arsalan.azami2011@gmail.com.