Portfolio Credit Risk Research Articles

Dynamic Hedging of Portfolio Credit Risk in a Markov Copula Model

We devise a bottom-up dynamic model of portfolio credit risk where instantaneous contagion is represented by the possibility of simultaneous defaults. Due to a Markovian copula nature of the model, calibration of marginals and dependence parameters can be performed separately using a two-step procedure, much like in a standard static copula setup. In this sense this solves the bottom-up top-down puzzle which the CDO industry had been trying to do for a long time. This model can be used for any dynamic portfolio credit risk issue, such as dynamic hedging of CDOs by CDSs, or CVA computations on credit portfolios.

A Bottom-Up Dynamic Model of Portfolio Credit Risk: Part II: Common-Shock Interpretation, Calibration and Hedging Issues

In this paper, we prove that the conditional dependence structure of default times in the model of A Bottom-Up Model of Portfolio Credit Risk. Part I: Copula Perspective belongs to the class of Marshall-Olkin copulas. This allows us to derive a factor representation in terms of common-shocks, the latter being able to trigger simultaneous defaults in some prespecified groups of obligors. This representation depends on the current default state of the credit portfolio so that fast convolution pricing schemes can be exploited for pricing and hedging credit portfolio derivatives. As emphasized in A Bottom-Up Model of Portfolio Credit Risk: Part I: Copula Perspective, the innovative breakthrough of this dynamic bottom-up model is a suitable decoupling property between the dependence structure and the default marginals as in Dynamic Modeling of Dependence in Finance via Copulae Between Stochastic Processes (like in static models but here in a full-flesh dynamic Markov copula model). Given the fast deterministic pricing schemes of the present paper, the model can then be jointly calibrated to single-name and portfolio data in two steps, as opposed to a global joint optimization procedures involving all the model parameters at the same time which would be untractable numerically. We illustrate this numerically by results of calibration against market data from CDO tranches as well as individual CDS spreads. We also discuss hedging sensitivities computed in the models thus calibrated.

A Fast Wavelet Expansion Technique for Evaluation of Portfolio Credit Risk Under the Vasicek Multi-Factor Model

This paper presents a new methodology to compute value at risk (VaR) and the marginal VaR contribution (VaRC) in the Vasicek multi-factor model of portfolio credit loss. The wavelet approximation method can be useful to compute non-smooth distributions, often arising in small or concentrated portfolios. This paper contributes to this technique by extending the wavelet approximation method for the Vasicek one-factor model to the multi-factor model. Key features of the new algorithm presented in this paper are (i) a finite series expansion of the wavelet scaling coefficients, (ii) fast calculation methods to accelerate convergence of those series and (iii) an efficient spline interpolation method to calculate the Laplace transforms. This paper also illustrates the effectiveness of the algorithm through numerical examples.

INFORMATIONALLY DYNAMIZED GAUSSIAN COPULA

In order to dynamize the static Gaussian copula model of portfolio credit risk, we introduce a model filtration made of a reference Brownian filtration progressively enlarged by the default times. This yields a multidimensional density model of default times, where, as opposed to the classical situation of the Cox model, the reference filtration is not immersed into the enlarged filtration. In mathematical terms this lack of immersion means that martingales in the reference filtration are not martingales in the enlarged filtration. From the point of view of financial interpretation this means default contagion, a good feature in the perspective of modeling counterparty wrong-way risk on credit derivatives. Computational tractability is ensured by invariance of multivariate Gaussian distributions through conditioning by some components, the ones corresponding to past defaults. Moreover the model is Markov in an augmented state-space including past default times. After a discussion of different notions of deltas, the model is applied to the valuation of counterparty risk on credit derivatives.

Sampling Exchangeable and Hierarchical Marshall-Olkin Distributions

The seminal Marshall-Olkin distribution is given by the following d-dimensional frailty model: independent exponentially distributed random variables represent the arrival times of shocks that destroy subgroups of a vector with d components. Sampling the resulting random vector of extinction times along the lines of this stochastic representation is conceptually straightforward. Firstly, all possible shocks are sampled. Secondly, the extinction time of each component is computed. However, implementing this sampling scheme involves 2 d − 1 random variables and requires finding the minimum of a set with 2 d−1 elements for each component. Thus, the overall effort is exponentially increasing in the number of considered components, preventing the general model from being used in high-dimensional simulation studies. This problem is overcome for the subfamily of exchangeable Marshall-Olkin distributions, for which a sampling algorithm with polynomial effort in d, an upper bound is 𝒪(d 3), is presented. In a second step, it is shown how hierarchical models are constructed from exchangeable structures and how models with alternative univariate marginal laws are obtained. The presented algorithm is then adapted to sample such structures as well. Finally, a small case study in the context of portfolio credit risk is presented.

Research on Assessment Method for Credit Risk in Commercial Banks of China Based on Data Mining

Due to the huge loss to the bank caused by credit risk in the financial debt crisis of the international Banking industry in 1980s, the research on Credit Assessment Methods is becoming the central issue of the study of financial theory in China and abroad. This paper builded the assets financial evaluation system of credit risk level based on the association rules-Apriori algorithm of data mining technology, which aimed the problems and the serious shortage of risk quantification study in domestic banks credit risk management. At the same time, taking into account the actual situation of our country, this paper analyzed that there are certain difficulties to use modern credit risk measurement models to evaluation the credit risk of commercial banks. And it suggests building a credit portfolio risk measurement model suitable for China's commercial banks with using logistic regression model of data mining technology.

Macroeconomic Determinants Of Loan Portfolio Credit Risk In Banks

The credit risk is one of the main risks in commercial banks and the ability to manage it meaningly affect banks’ stability. This risk arises due to the particular reasons related with the possibility to lose loans if the debtors are not able to meet their financial obligations. When making the decisions of financing the loan applicants banks use the credit risk assessment models that allow to estimate the probability of the potential borrowers to default on their loan commitments. The main goal of managing the credit risk in banks is to compound the loan portfolio of the acceptable risk level. According to Derelioglu and Gurgen (2011) the credit risk analysis aims to decrease future losses by estimating the potential risk and eliminating the new credit proposal if the risk is higher than a defined tolerance value. In this respect, it is essential to identify the main factors causing this risk in order to manage it. When assessing the credit risk of every company banks usually analyze the financial data and some qualitative factors as the independent variables in the statistical credit risk assessment models. But changing the credit policy in banks and pricing the credits it is very important to predict the quality of loan portfolio in future. The problem can be summarized as finding the statistical methods that relates the proportion of doubtful and non-performing credits in the loan portfolio (dependent variable) with the set of explanatory variables (macroeconomic information of a country). The aim of this research is to find the macroeconomic determinants that significantly influence the changes of loan portfolio credit risk in banks and to develop the statistical model for prediction of the proportion of doubtful and non-performing loans. The scientific literature analysis results confirmed the influence of macroeconomic conditions on credit risk of debtors in banks and presented that the changes in quality of loan portfolio in banks depend on GDP, inflation, interest rates, money supply, industrial production index, current account balance and other. In empirical research 22 EU countries were grouped into 3 clusters according to their similarity in changes of the doubtful and non-performing loans percentage in banks. The set of 20 independent variables as factors determining the changes in amount of doubtful and non-performing loans was created. These variables were calculated from 9 macroeconomic indicators of 3 years. The model was developed to classify the countries into clusters applying the logistic regression, factor analysis and probit methods. The classification accuracy is 100%. The predictions of doubtful and non-performing loans indexes are based on the analysis of the scores of extracted 5 factors as new independent variables. The multiple regression and polynomial regression methods were applied for the index predictions in clusters. The developed model in this research enables to predict the percentage of doubtful and non-performing loans in banks with the average 98,06% accuracy. The research has confirmed that the amount of doubtful and non-performing loans in banks highly depends on macroeconomic changes in a country. The model can warn the banks in advance if the significant increase in the loan portfolio credit risk after 2 years is highly possible.DOI: http://dx.doi.org/10.5755/j01.ee.23.5.1890

Measuring Economic Capital Using Loss Distributions

This paper investigates the complexity involved in the quantitative measurement of Economic Capital and proposes simulation methods as a practical solution for obtaining the loss distribution of a portfolio of obligors. The paper examines a one factor model to generate loss distribution which establishes the necessary ingredients to measure the credit risk quantities in a loan portfolio. The general elements of credit risk modeling are outlined and then a specific model that employs a Monte Carlo simulation is developed. An example is provided that calculates the risk quantities in a loan portfolio from which the Economic Capital in a credit risk portfolio is obtained.

Asset correlations and bank capital adequacy

This paper addresses the estimation of confidence sets for asset correlations used in credit risk portfolio models. Research on the estimation of asset correlations using endogenous probabilities of default estimations has focused on the impact of concentration risk factors, such as firm size and industry. The empirical evidence from Italian small- and medium-size companies show that the assumptions underlying the Basel Committee regulatory capital risk weight function are not substantiated. The regulatory impact is that the capital adequacy is significantly compromised, driving an adverse selection, which favors the worst companies, and transferring the procyclical effects from firms to banks.

Empirical estimation of default and asset correlation of large corporates and banks in India

Purpose – Estimation of default and asset correlation is crucial for banks to manage and measure portfolio credit risk. The purpose of this paper is to find empirical relationship between the default and asset correlation with default probability, to understand the effect of systematic risk.

Portfolio Risk and Dependence Modeling

This paper has considered portfolio credit risk with a focus on two approaches, the factor model, and copula model. We have reviewed two models with emphasis on the joint default probably. The copula function describes the dependence structure of a multivariate random variable, in this paper, it used as a practical to simulation of generate portfolio with different copula, and we only used Gaussian and t–copula case. We generated portfolio default distributions and studied the sensitivity of commonly used risk measures with respect to the approach in modeling the dependence structure of the portfolio.

A Markov Copula Model of Portfolio Credit Risk with Stochastic Intensities and Random Recoveries

In Hedging of Portfolio Credit Risk in a Markov Copula Model, the authors introduced a Markov copula model of portfolio credit risk where pricing and hedging can be done in a sound theoretical and practical way. Further theoretical backgrounds and practical details are developed in A Bottom-Up Dynamic of Portfolio Credit Risk. Part I: Markov Copula Perspective and A Bottom-Up Dynamic of Portfolio Credit Risk. Part II: Common-Shock Interpretation, Calibration and Hedging issues where numerical illustrations assumed deterministic intensities and constant recoveries. In the present paper, we show how to incorporate stochastic default intensities and random recoveries in the bottom-up modeling framework of Hedging of Portfolio Credit Risk in a Markov Copula Model while preserving numerical tractability. These two features are of primary importance for applications like computations on credit derivatives (Valuation and Hedging of CDS Exposure in a Markov Copula Model, CVA computation for counterparty risk assessment in credit portfolios, Counterparty Risk Modeling - Collateralization, Funding and Hedging) , as is sensitive to the stochastic nature of credit spreads and random recoveries allow to achieve satisfactory calibration even for badly behaved data sets. This paper is thus a complement to Hedging of Portfolio Credit Risk in a Markov Copula Model, A Bottom-Up Dynamic of Portfolio Credit Risk. Part I: Markov Copula Perspective and A Bottom-Up Dynamic of Portfolio Credit Risk. Part II: Common-Shock Interpretation, Calibration and Hedging issues.

Fast simulations in credit risk

We consider the problem of simulating tail loss probabilities and expected losses conditioned on exceeding a large threshold (expected shortfall) for credit portfolios. Our new idea, called the geometric shortcut, allows an efficient simulation for the case of independent obligors. It is even possible to show that, when the average default probability tends to zero, its asymptotic efficiency is higher than that of the naive algorithm. The geometric shortcut is also useful for models with dependent obligors and can be used for dependence structures modeled with arbitrary copulae. The paper contains the details for simulating the risk of the normal copula credit risk model by combining outer importance sampling with the geometric shortcut. Numerical results show that the new method is efficient in assessing tail loss probabilities and expected shortfall for credit risk portfolios. The new method outperforms all known methods, especially for credit portfolios consisting of weakly correlated obligors and for evaluating the tail loss probabilities at many thresholds in a single simulation run.

Cross-Market Hedging Strategies for Credit Default Swaps under a Markov Regime-Switching Framework

This article explores strategies used for hedging credit default swap (CDS) risks. By analyzing the AIG financial crisis in 2008, the authors increase their understanding of the nature of CDSs and why AIG incurred huge losses. In addition, they investigate other financial products that can be used to hedge CDS risks. Complementary to Jarrow [2010] who suggested that government regulation should impose stricter collateral requirements and higher equity capital for CDS traders to minimize CDS seller failure, they provide feasible hedging strategies for CDS sellers instead. The authors contribute to the literature in the following aspects. First, they find that stock returns significantly affect CDS returns, depending on regimes, and their model is able to capture these switching characteristics. Second, they explain how the correlation between CDS and stock returns can be used to adjust hedge ratios based on different regime-switching situations. Finally, they find that high hedge ratios are suitable for firms with low credit quality, and low hedge ratios are suitable for firms with high credit quality. Additionally, the proposed hedging strategy is useful for companies in the financial industry, particularly for better management of portfolio credit risk.

One-Factor Copula-Based Model for Credit Portfolio

Aim of this research paper it to develop an alternative model for portfolio credit risk to those widely used - CreditRisk+ and CreditMetrics. The model aspires to patch the usual weak points of portfolio credit models and also is easy to implement. General engine relies on one-factor copula model and a concept of generating functions. Proposed model also uses only elementary inputs, which make it quick to deploy in the banking environment. The output is the usual credit VaR. We also illustrate various methods related to the implementation and incorporation of credit wrong-way-risk. As a particular case of the implementation, double t–copula is chosen in combination with a stochastic LGD feature. Finally, we show the incidence on capital requirement of various models by testing them on a sample portfolio.

Dependence properties of dynamic credit risk models

Abstract We give a unified mathematical framework for reduced-form models for portfolio credit risk and identify properties which lead to positive dependence of default times. Dependence in the default hazard rates is modeled by common macroeconomic factors as well as by inter-obligor links. It is shown that popular models produce positive dependence between defaults in terms of association. Implications of these results are discussed, in particular when we turn to pricing of credit derivatives. In mathematical terms our paper contains results about association of a class of non-Markovian processes.

Confidence sets for asset correlations in portfolio credit risk

Asset correlations are of critical importance in quantifying portfolio credit risk and economic capitalin financial institutions. Estimation of asset correlation with rating transition data has focusedon the point estimation of the correlation without giving any consideration to the uncertaintyaround these point estimates. In this article we use Bayesian methods to estimate a dynamicfactor model for default risk using rating data (McNeil et al., 2005; McNeil and Wendin, 2007).Bayesian methods allow us to formally incorporate human judgement in the estimation of assetcorrelation, through the prior distribution and fully characterize a confidence set for the correlations.Results indicate: i) a two factor model rather than the one factor model, as proposed bythe Basel II framework, better represents the historical default data. ii) importance of unobservedfactors in this type of models is reinforced and point out that the levels of the implied asset correlationscritically depend on the latent state variable used to capture the dynamics of default,as well as other assumptions on the statistical model. iii) the posterior distributions of the assetcorrelations show that the Basel recommended bounds, for this parameter, undermine the levelof systemic risk.

Stress testing a retail loan portfolio: an error correction model approach

The use of stress testing for risk monitoring has increased considerably over the last decade. Stress testing – a simulation technique used to assess the strength of a portfolio or a financial institution under unusual economic conditions – emerged as a powerful tool that was originally used in market risk. Its use has subsequently been extended into credit risk. To stress test a credit risk portfolio, practitioners focus on the key parameters that allow the risk of a credit portfolio to be assessed. These parameters, also known as Basel II parameters, are probability of default, loss given default, exposure at default and asset correlation. In this paper, using a time series approach (specifically, an error correction model), we focus on the probability of default parameter that is related to macroeconomic factors. Such an approach involves dealing with the nonstationarity of economic time series and cointegration issues. Hence, when the model is estimated, the probability of default can be simulated by measuring the effects of macroeconomic shocks applied to the model. In turn, these probabilities of default can be used to measure the impact on the probability of default under a given macroeconomic scenario, and then to improve the credit risk monitoring. The results of our study suggest that error correction models are well-suited to macroeconomic stress testing. Indeed, the fitting of the historical probability of default and the results under the stress scenarios considered here are satisfactory.

Copula-Based Portfolio Credit Risk Assessment in Infrastructure Project Financing

Project lenders are concerned about credit risk management in infrastructure project financing. According to the maxim in risk management, “you cannot manage what you cannot measure,” so project lenders want to know the exposure of their debt loss. Although many sophisticated credit risk models have been developed for corporate financing, only some relevant literature has focused on credit risk assessment for infrastructure projects. However, the models developed assume that project debt lenders invest only in one single project at a time. In fact, project lenders are inclined to be simultaneously involved in a portfolio of assets around the world rather than a single local project asset in order to diversify away idiosyncratic risks of an individual infrastructure project. Consequently, in order to meet project lenders’ needs, this article presents a copula-based model to measure the credit risk of a portfolio of projects by implementing the variance model and the double stochastic intensity model. A numerical example of two interdependent projects is shown as a case study to illustrate how to quantify this joint default risk in infrastructure project financing. <b>TOPICS:</b>Private equity, project finance, credit risk management, statistical methods

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