Copula Methods and the Analysis of Credit Risk

SSRN Electronic Journal, 2000

ABSTRACT This paper aims to provide an introductory review of copulae and their potential application finance, in particular in capturing the dependence between financial assets that follow non-gaussian distributions and hence for modelling credit risk, pricing options and portfolio design. We briefy review the failure of methods based on multivariate normality and where dependency is measured by correlation. Using a copula approach enables us to measure the different relationships that may exist between financial assets in different ranges of their behaviour- for instance do assets exhibit similar dependency patterns in the tails of their distributions as they do around their means? We review different measures of association and show that when these dependency measures can be expressed simply as functions of a copula they will be invariant to strictly monotone transformations of the random variables and hence the units in which we choose to express our data. The standard Pearson Correlation statistic is not in general invariant to scale changes in the data. We then discuss several statistical issues relating to the estimation of Copulae and their empirical application through both parametric and non parametric methods. An important issue lies in the statistical discrimination between copulae and we propose a encompassing framework based on simulation for discriminating between what are effectively separate statistical families. We then consider several applications, modelling default risk, tail dependence, quantile regression and portfolio design for non-gaussian assets.

Copula functions have proven to be extremely useful in describing joint default and survival probabilities. We overview the state of the art and point out some modelling issues, such as the choice of a speci…c copula and of the amount of dependence, as well as re-mapping of models. We also discuss how to simplify credit models using factors. The survey leads us to focus on historical versus risk neutral dependence. We present an original methodology for inferring risk neutral dependence without using a single factor, provide an application to market data and explore its impact on the fees of some credit derivatives. The …rst part of the paper is a review of the copula techniques developed so far in order to evaluate credit risk in default-no default models. The review proceeds as follows: we …rst resume some basics about structural and intensity- based models, then recall how and under which conditions two models of these classes can be re-mapped into each other, using the latent va...

We would like to thank an anonymous referee for providing many helpful suggestions.

2007

The aim of this paper is to study the impact of structure of dependency on the pricing of multi-name credit derivatives such as collateralised debt obligations (CDO). The correlation between names defaulting has an effect on the value of the basket credit derivatives. We present a copula based simulation procedure for pricing CDO under different structure of dependency and assessing the influence of different price drivers (correlation, hazard rates and recovery rates) on modelling portfolio losses. Gaussian copulas and Monte Carlo simulation are widely used to measure the default risk in basket credit derivatives. Many studies have shown that many distributions have fatter tails than those captured by the normal distribution. We use distributions with fat tails such as the t-student distribution. The paper has several practical implications that are of value for financial hedgers and engineers, financial regulators, central banks, and financial risk managers.

Buletin Ekonomi Moneter dan Perbankan

We develop a joint default probability index to signal potential systemic risks in the highly concentrated Indonesian banking industry. To build the index, we estimate bank-level tail risks using monthly bank financial reports. We use the copula approach to derive the joint multivariate dependencies at the bank level, as reflected in the monthly financial reports. Our results, which are based on a sample of 104 banks fromDecember 2003 to April 2020, show joint multivariate dependencies at the bank level suggesting that the standard univariate normal distribution is unsuitable for capturing tail risks of individual banks. Our index accurately captures the global financial crisis of 2007-2008 indicating that it is a valid joint default probability index. Further, our index also signaled a higher degree of joint default before the COVID-19 outbreak in2020, suggesting that it is a good indicator of potential systemic risk in the economy.

2017

This paper considers the problem of measuring the credit risk in portfolios of loans, bonds, and other instruments subject to possible default under multi-factor models. Due to the amount of the portfolio, the heterogeneous effect of obligors, and the phenomena that default events are rare and mutually dependent, it is difficult to calculate portfolio credit risk either by means of direct analysis or crude Monte Carlo under such models. To capture the extreme dependence among obligors, we provide an efficient simulation method for multi- factor models with a normal mixture copula that allows the multivariate defaults to have an asymmetric distribution, while most of the previous literature focuses on simulating one dimensional cases. To this end, we first propose a general account of an importance sampling algorithm based on a two-parameter exponential embedding. Note that this innovative tilting device is more suitable for the multivariate normal mixture model than traditional one-...

Journal of …, 2006

The multivariate modeling of default risk is a crucial aspect of the pricing of credit derivative products referencing a portfolio of underlying assets, and the evaluation of Value at Risk of such portfolios. This paper proposes a model for the joint dynamics of credit ratings of several firms. Namely, individual credit ratings are modeled by univariate continuous time Markov chain, while their joint dynamic is modeled using copulas. A by-product of the method is the joint laws of the default times of all the firms in the portfolio. The use of copulas allows us to incorporate our knowledge of the modeling of univariate processes, into a multivariate framework. The Normal and Student copulas commonly used in the literature as well as by practitioners do not produce very different estimates of default risk prices. We show that this result is restricted to these two two basic copulas. That is, for any other family of copula, the choice of the copula greatly affects the pricing of default risk.

International Journal of Applied Management …, 2010

The aim of this paper is to study the impact of structure of dependency on the pricing of multi-name credit derivatives such as collateralised debt obligations (CDO). The correlation between names defaulting has an effect on the value of the basket credit derivatives. We present a copula based simulation procedure for pricing CDO under different structure of dependency and assessing the influence of different price drivers (correlation, hazard rates and recovery rates) on modelling portfolio losses. Gaussian copulas and Monte Carlo simulation are widely used to measure the default risk in basket credit derivatives. Many studies have shown that many distributions have fatter tails than those captured by the normal distribution. We use distributions with fat tails such as the t-student distribution. The paper has several practical implications that are of value for financial hedgers and engineers, financial regulators, central banks, and financial risk managers.

2007

This paper deals with the impact of structure of dependency and the choice of procedures for rareevent simulation on the pricing of multi-name credit derivatives such as n th to default swap and Collateralized Debt Obligations (CDO). The correlation between names defaulting has an effect on the value of the basket credit derivatives. We present a copula based simulation procedure for pricing basket default swaps and CDO under different structure of dependency and assessing the influence of different price drivers (correlation, hazard rates and recovery rates) on modelling portfolio losses.

arXiv: Methodology, 2020

Understanding the dependence relationship of credit spreads of corporate bonds is important for risk management. Vine copula models with tail dependence are used to analyze a credit spread dataset of Chinese corporate bonds, understand the dependence among different sectors and perform conditional inferences. It is shown how the effect of tail dependence affects risk transfer, or the conditional distributions given one variable is extreme. Vine copula models also provide more accurate cross prediction results compared with linear regressions. These conditional inference techniques are a statistical contribution for analysis of bond credit spreads of investment portfolios consisting of corporate bonds from various sectors.