Credit Risk Management Based on Copula Functions

Abstract

Credit risk is the major risk faced by financial institutions, so the significance of the measuring and managing credit risk is obvious. With the rapid development of the financial globalization, credit risk become more diversified and complicated. The dependence structure of financial time series is usually nonlinear, asymmetric, and time-varying. Copula functions can well describe the dependence structure and tail dependence of financial time series, which makes them widely used in the multivariate distribution of portfolio credit risk. The general objective of the doctoral thesis is to investigate the credit risk measurements combined with the copula functions. The first subgoal is to compare the original credit risk model and the credit risk model based on copulas; the second subgoal is to compare the ability of pricing of credit derivatives under copula framework; and the third subgoal is to extend the constant copulas to the dynamic copulas and verify the best copula for given portfolios. There are mainly four chapters in the thesis. Chapter 2 is the basic description of the copula theory, including elliptical copulas, Archimedean copulas, calibration of the copula parameters, and dependence measures. Chapter 3 presents the general classification of credit risk models, including the structural and reduced-form models, and the framework of the CreditMetrics™ model. The CreditMetrics™ model is applied to two different portfolios consisting of ten bonds traded on FSE from 9th October 2017 to 8th October 2018, one is a good-quality portfolio, and another is a credit-risky portfolio. We calibrate the parameters of copula functions to fit the data and obtain the correlation matrix based on copulas. The assumption of normality is also removed. The comparison of the VaR of two portfolios by the original CreditMetrics™ model and the CreditMetrics™ model based on copulas suggests that copula functions improve the accuracy of the CreditMetrics™ model for the credit-risky portfolio at any confidence levels. Chapter 4 focuses on the CDO pricing in the semi-analytic approach under copula framework. We summarize the conditional probabilities of default and the approximating distributions of the portfolio loss associated with different factor copula models. Then we study how the tranche spreads of a CDO will change according to different correlations and recovery rates based on the multinomial Gaussian copula and find that the CDO spreads increase with an increase in correlation and with a decrease in recovery rate for the senior tranche, while the converse is true for the equity tranche. As for the mezzanine tranche, it is similar to the equity tranche in the relationship between the spreads and correlations, while it is similar to the senior tranche in the relationship between the spreads and recovery rates. Besides, we compare the ability of selected factor copula models to fit the market quotes and correlation skew according to Dow Jones iTraxx Europe tranches with 5-year maturity from 5th of January 2015 to 10th of May 2016 and the NIG model with lowest absolute error sum is the best. Chapter 5 introduces dynamic copula models, including multivariate copula-ARCH model, time-varying copula model, and MRS-copula-GARCH model. We compare the negative log-likelihood of static copulas, time-varying copulas, and MRS-copula-GARCH copulas considering two stocks issued by ExxonMobil and IBM in FSE from 11th of May 2009 to 15th of March 2019 and then find that MRS-copula-GARCH copulas usually perform best. Besides, we study the dependence structure of four constituents of the FSET 100 index from 6th of June 2013 to 13th of May 2019 by the time-varying copula-GARCH models and find that the time-varying SJC copula is the optimal copula model because it can provide lowest negative log-likelihood.

Credit risk is the major risk faced by financial institutions, so the significance of the measuring and managing credit risk is obvious. With the rapid development of the financial globalization, credit risk become more diversified and complicated. The dependence structure of financial time series is usually nonlinear, asymmetric, and time-varying. Copula functions can well describe the dependence structure and tail dependence of financial time series, which makes them widely used in the multivariate distribution of portfolio credit risk. The general objective of the doctoral thesis is to investigate the credit risk measurements combined with the copula functions. The first subgoal is to compare the original credit risk model and the credit risk model based on copulas; the second subgoal is to compare the ability of pricing of credit derivatives under copula framework; and the third subgoal is to extend the constant copulas to the dynamic copulas and verify the best copula for given portfolios. There are mainly four chapters in the thesis. Chapter 2 is the basic description of the copula theory, including elliptical copulas, Archimedean copulas, calibration of the copula parameters, and dependence measures. Chapter 3 presents the general classification of credit risk models, including the structural and reduced-form models, and the framework of the CreditMetrics™ model. The CreditMetrics™ model is applied to two different portfolios consisting of ten bonds traded on FSE from 9th October 2017 to 8th October 2018, one is a good-quality portfolio, and another is a credit-risky portfolio. We calibrate the parameters of copula functions to fit the data and obtain the correlation matrix based on copulas. The assumption of normality is also removed. The comparison of the VaR of two portfolios by the original CreditMetrics™ model and the CreditMetrics™ model based on copulas suggests that copula functions improve the accuracy of the CreditMetrics™ model for the credit-risky portfolio at any confidence levels. Chapter 4 focuses on the CDO pricing in the semi-analytic approach under copula framework. We summarize the conditional probabilities of default and the approximating distributions of the portfolio loss associated with different factor copula models. Then we study how the tranche spreads of a CDO will change according to different correlations and recovery rates based on the multinomial Gaussian copula and find that the CDO spreads increase with an increase in correlation and with a decrease in recovery rate for the senior tranche, while the converse is true for the equity tranche. As for the mezzanine tranche, it is similar to the equity tranche in the relationship between the spreads and correlations, while it is similar to the senior tranche in the relationship between the spreads and recovery rates. Besides, we compare the ability of selected factor copula models to fit the market quotes and correlation skew according to Dow Jones iTraxx Europe tranches with 5-year maturity from 5th of January 2015 to 10th of May 2016 and the NIG model with lowest absolute error sum is the best. Chapter 5 introduces dynamic copula models, including multivariate copula-ARCH model, time-varying copula model, and MRS-copula-GARCH model. We compare the negative log-likelihood of static copulas, time-varying copulas, and MRS-copula-GARCH copulas considering two stocks issued by ExxonMobil and IBM in FSE from 11th of May 2009 to 15th of March 2019 and then find that MRS-copula-GARCH copulas usually perform best. Besides, we study the dependence structure of four constituents of the FSET 100 index from 6th of June 2013 to 13th of May 2019 by the time-varying copula-GARCH models and find that the time-varying SJC copula is the optimal copula model because it can provide lowest negative log-likelihood.