Basket CDS pricing with interacting intensities

We propose a factor contagion model for correlated defaults. The model covers the heterogeneous conditionally independent portfolio and the infectious default portfolio as special cases. The model assumes that the hazard rate processes are driven by external common factors as well as defaults of other names in the portfolio. The total hazard construction method is used to derive the joint distribution of default times. The basket CDS rates can be computed analytically for homogeneous contagion portfolios and recursively for general factor contagion portfolios. We extend the results to include the interacting counterparty risk and the stochastic intensity process. The authors thank two anonymous referees for several suggestions which have helped to improve the earlier versions. The authors thank Sheng Miao for help in implementation with C++, Huiqi Pan for help in implementation with Fortran, and Xiaozhou Cao for help in implementation with MAPLE. Harry Zheng thanks the London Mathematical Society for its collaborative grant support (Grant 4544 and Grant 4707).     

Moody's correlated binomial default distributions for inhomogeneous portfolios

This paper generalizes Moody's correlated binomial default distribution for homogeneous (exchangeable) credit portfolios, which was introduced by Witt, to the case of inhomogeneous portfolios. We consider two cases of inhomogeneous portfolios. In the first case, we treat a portfolio whose assets have uniform default correlation and non-uniform default probabilities. We obtain the default probability distribution and study the effect of inhomogeneity. The second case corresponds to a portfolio with inhomogeneous default correlation. Assets are categorized into several different sectors and the inter-sector and intra-sector correlations are not the same. We construct the joint default probabilities and obtain the default probability distribution. We show that as the number of assets in each sector decreases, inter-sector correlation becomes more important than intra-sector correlation. We study the maximum values of the inter-sector default correlation. Our generalization method can be applied to any correlated binomial default distribution model that has explicit relations to the conditional default probabilities or conditional default correlations, e.g. Credit Risk⁺, implied default distributions. We also compare some popular CDO pricing models from the viewpoint of the range of the implied tranche correlation.     

Valuation of collateralized debt obligations with hierarchical Archimedean copulae

Modelling portfolio credit risk is one of the crucial challenges faced by financial services industry in the last few years. We propose the valuation model of collateralized debt obligations (CDO) based on hierarchical Archimedean copulae (HAC) with up to three parameters, with default intensities calibrated to market data and with random loss given defaults that are correlated with default times. The methods presented are used to reproduce the spreads of the iTraxx Europe tranches. Our approach describes the market prices better than the standard pricing procedure based on the Gaussian distribution. We also obtain a flat correlation smile across tranches thereby solving the implied correlation puzzle.     

Modeling of Contagious Credit Events and Risk Analysis of Credit Portfolios

We present a new model of the occurence of credit events such as rating changes and defaults for risk analyses of some portfolio credit derivatives. The framework of our model is based on a so-called top-down approach. Specifically, we first consider modeling the point process of each type of credit event in the whole economy using a self-exciting intensity process. Next, we characterize the point processes of credit events in the underlying sub-portfolio using random thinning processes specified by the distribution of credit ratings in the sub-portfolio. One of the main features of our model is that the model can capture credit risk contagion simultaneously among several credit portfolios. We present a credit event simulation algorithm based on our model and illustrate an application of the model to risk analyses of loan portfolios.     

Checking for asymmetric default dependence in a credit card portfolio: A copula approach

Traditional credit risk models adopt the linear correlation as a measure of dependence and assume that credit losses are normally-distributed. However some studies have shown that credit losses are seldom normal and the linear correlation does not give accurate assessment for asymmetric data. Therefore it is possible that many credit models tend to misestimate the probability of joint extreme defaults.This paper employs Copula Theory to model the dependence across default rates in a credit card portfolio of a large UK bank and to estimate the likelihood of joint high default rates. Ten copula families are used as candidates to represent the dependence structure. The empirical analysis shows that, when compared to traditional models, estimations based on asymmetric copulas usually yield results closer to the ratio of simultaneous extreme losses observed in the credit card portfolio.Copulas have been applied to evaluate the dependence among corporate debts but this research is the first paper to give evidence of the outperformance of copula estimations in portfolios of consumer loans. Moreover we test some families of copulas that are not typically considered in credit risk studies and find out that three of them are suitable for representing dependence across credit card defaults.     

Contagion models a la carte: which one to choose?

We propose a copula contagion mixture model for correlated default times. The model includes the well-known factor, copula, and contagion models as its special cases. The key advantage of such a model is that we can study the interaction of different models and their pricing impact. Specifically, we model the default times of the underlying names in a reference portfolio to follow contagion intensity processes with exponential decay coupled with a copula dependence structure. We also model the default time of the counterparty and its dependence structure with the reference portfolio. Numerical tests show that correlation and contagion have an enormous joint impact on the rates of CDO tranches and the corresponding credit value adjustments are extremely high to compensate for the wrong-way risk.     

An Empirical Analysis of the Pricing of Collateralized Debt Obligations

We use the information in collateralized debt obligations (CDO) prices to study market expectations about how corporate defaults cluster. A three‐factor portfolio credit model explains virtually all of the time‐series and cross‐sectional variation in an extensive data set of CDX index tranche prices. Tranches are priced as if losses of 0.4%, 6%, and 35% of the portfolio occur with expected frequencies of 1.2, 41.5, and 763 years, respectively. On average, 65% of the CDX spread is due to firm‐specific default risk, 27% to clustered industry or sector default risk, and 8% to catastrophic or systemic default risk.     

Aggregating sectors in the infectious defaults model

Abstract

How to model the dependence between defaults in a portfolio subject to credit risk is a question of great importance. The infectious default model of Davis and Lo offers a way to model the dependence. Every company defaulting in this model may ‘infect’ another company causing it to default. An unsolved question, however, is how to aggregate independent sectors, since a naive straightforward computation quickly gets cumbersome, even when homogeneous assumptions are made. Here, two algorithms are derived that overcome the computational problem and further make it possible to use different exposures and probabilities of default for each sector. For an ‘outbreak’ of defaults to occur in a sector, at least one company has to default by itself. This fact is used in the derivations of the two algorithms. The first algorithm is derived from the probability generating function of the total credit loss in each sector and the fact that the outbreaks are independent Bernoulli random variables. The second algorithm is an approximation and is based on a Poisson number of outbreaks in each sector. This algorithm is less cumbersome and more numerically stable, but still seems to work well in a realistic setting.     

债券违约对流动性影响的传染效应研究

2014年以来我国信用债市场违约事件频发,信用风险的积聚可能引发债券市场流动性恶化。本文以2014―2019年交易所和银行间市场信用债为研究对象,实证考察违约事件对债券流动性影响的传染效应。研究发现:违约事件在同一发行主体的债券之间具有流动性传染效应,当公司的某期债券出现违约时,公司其他未到期债券的流动性水平显著下降;违约事件对同行业其他公司债券的流动性具有传染效应,当行业中出现债券违约事件时,行业内其他公司的债券流动性显著降低;违约事件爆发越密集或者违约事件越严重,对债券流动性的负面影响越大,而且民营企业债受到的影响要大于国有企业债,低信用等级债受到的影响要大于高信用等级债;在市场密集爆发违约事件或出现较为严重的违约事件时期,宏观流动性增加能够改善债券流动性。     

Modeling the evolution of implied CDO correlations

CDO tranche spreads (and prices of related portfolio-credit derivatives) depend on the market’s perception of the future loss distribution of the underlying credit portfolio. Applying Sklar’s seminal decomposition to the distribution of the vector of default times, the portfolio-loss distribution derived thereof is specified through individual default probabilities and the dependence among obligors’ default times. Moreover, the loss severity, specified via obligors’ recovery rates, is an additional determinant. Several (specifically univariate) credit derivatives are primarily driven by individual default probabilities, allowing investments in (or hedging against) default risk. However, there is no derivative that allows separately trading (or hedging) default correlations; all products exposed to correlation risk are contemporaneously also exposed to default risk. Moreover, the abstract notion of dependence among the names in a credit portfolio is not directly observable from traded assets. Inverting the classical Vasicek/Gauss copula model for the correlation parameter allows constructing time series of implied (compound and base) correlations. Based on such time series, it is possible to identify observable variables that describe implied correlations in terms of a regression model. This provides an economic model of the time evolution of the market’s view of the dependence structure. Different regression models are developed and investigated for the European CDO market. Applications and extensions to other markets are discussed.     

Macroeconomic Conditions and the Puzzles of Credit Spreads and Capital Structure

I build a dynamic capital structure model that demonstrates how business cycle variation in expected growth rates, economic uncertainty, and risk premia influences firms' financing policies. Countercyclical fluctuations in risk prices, default probabilities, and default losses arise endogenously through firms' responses to macroeconomic conditions. These comovements generate large credit risk premia for investment grade firms, which helps address the credit spread puzzle and the under‐leverage puzzle in a unified framework. The model generates interesting dynamics for financing and defaults, including market timing in debt issuance and credit contagion. It also provides a novel procedure to estimate state‐dependent default losses.     

Good and bad credit contagion: Evidence from credit default swaps

This study examines the intra-industry information transfer effect of credit events, as captured in the credit default swaps (CDS) and stock markets. Positive correlations across CDS spreads imply that contagion effects dominate, whereas negative correlations indicate competition effects. We find strong evidence of contagion effects for Chapter 11 bankruptcies and competition effects for Chapter 7 bankruptcies. We also introduce a purely unanticipated event, in the form of a large jump in a company's CDS spread, and find that this leads to the strongest evidence of credit contagion across the industry. These results have important implications for the construction of portfolios with credit-sensitive instruments.     

Pricing distressed CDOs with stochastic recovery

In this article, a framework for the joint modelling of default and recovery risk in a portfolio of credit risky assets is presented. The model especially accounts for the correlation of defaults on the one hand and correlation of default rates and recovery rates on the other hand. Nested Archimedean copulas are used to model different dependence structures. For the recovery rates a very flexible continuous distribution with bounded support is applied, which allows for an efficient sampling of the loss process. Due to the relaxation of the constant 40% recovery assumption and the negative correlation of default rates and recovery rates, the model is especially suited for distressed market situations and the pricing of super senior tranches. A calibration to CDO tranche spreads of the European iTraxx portfolio is performed to demonstrate the fitting capability of the model. Applications to delta hedging as well as base correlations are presented.     

From equity to default correlation with taxes

For fixed income investment, the preponderant risk is the clustering of defaults in the portfolio. Accurate prediction of such clustering depends on the knowledge of default correlation. We develop models with exogenous debt and endogenous debt to predict default correlations from equity correlations based on a self-consistent structural framework. We also examine how taxes affect the prediction of default correlations based on the two models. The empirical analysis shows that the corporate taxes tend to decrease default correlations, while personal taxes could increase or decrease default correlations. Our default correlation model with exogenous debt does a better job of predicting default correlations for high quality bonds, while the one with endogenous debt predicts more accurately for lower rated bonds. Our studies not only theoretically improve the modeling of default correlation in the structural setting but also shed new light on various aspects of default correlations and thereby help financial practitioners price credit derivatives more accurately and formulate more effective strategies to manage default risk of credit portfolios.     

Default correlation: An empirical investigation of a subprime lender

In recent years, subprime lending has grown substantially as an important sector of the credit markets. This paper is concerned with the risk management of subprime loan portfolios and the importance of default correlation in measuring that risk. Using a large portfolio of residential subprime loans from an anonymous subprime lender, we show that default correlation is substantial for this lender. In particular, the significance of default correlation increases as the internal credit rating declines. Our results suggest that lenders and regulators would be well served investing in the understanding of default correlation in subprime portfolios.     

Pricing CDOs with state-dependent stochastic recovery rates

Up to the 2007 crisis, research within bottom-up CDO models mainly concentrated on the dependence between defaults. Since then, due to substantial increases in market prices of systemic credit risk protection, more attention has been paid to recovery rate assumptions. In this paper, we use stochastic orders theory to assess the impact of recovery on CDOs and show that, in a factor copula framework, a decrease of recovery rates leads to an increase of the expected loss on senior tranches, even though the expected loss on the portfolio is kept fixed. This result applies to a wide range of latent factor models and is not specific to the Gaussian copula model. We then suggest introducing stochastic recovery rates in such a way that the conditional on the factor expected loss (or, equivalently, the large portfolio approximation) is the same as in the recovery markdown case. However, granular portfolios behave differently. We show that a markdown is associated with riskier portfolios than when using the stochastic recovery rate framework. As a consequence, the expected loss on a senior tranche is larger in the former case, whatever the attachment point. We also deal with implementation and numerical issues related to the pricing of CDOs within the stochastic recovery rate framework. Due to differences across names regarding the conditional (on the factor) losses given default, the standard recursion approach becomes problematic. We suggest approximating the conditional on the factor loss distributions, through expansions around some base distribution. Finally, we show that the independence and comonotonic cases provide some easy to compute bounds on expected losses of senior or equity tranches.     

Contagion effect on bond portfolio risk measures in a hybrid credit risk model

This paper illustrates how modelling the contagion effect among assets of a given bond portfolio changes the risk perception associated to it. This empirical work is developed in a hybrid credit risk framework that incorporates recovery rate risk. Dependence structures among firms and between external shocks affecting firms together are considered. The presence of correlations among firm leverage ratios and the interrelation between default probabilities and recovery rates produces clusters of defaults with low recovery rates. This has a major impact on standard risk measures such as Value-at-Risk and conditional tail expectation. Consequently, an appropriate measurement of the contagion has a tremendous effect on the capital requirement of many financial institutions.     

Modelling the distribution of credit losses with observable and latent factors

This paper proposes a dynamic model to estimate the credit loss distribution of the aggregate portfolio of loans granted in a banking system. We consider a sectoral approach distinguishing between corporates and households. The evolution of their default frequencies and the size of the loans portfolio are expressed as functions of macroeconomic conditions as well as unobservable credit risk factors, which capture contagion effects between sectors. In addition, we model the distributions of the Exposures at Default and the Losses Given Default. We apply our framework to the Spanish banking system, where we find that sectoral default frequencies are not only affected by economic cycles but also by a persistent latent factor. Finally, we identify the riskier sectors, perform stress tests and compare the relative risk of small and large institutions.     

Bilateral credit valuation adjustment for large credit derivatives portfolios

We obtain an explicit formula for the bilateral counterparty valuation adjustment of a credit default swaps portfolio referencing an asymptotically large number of entities. We perform the analysis under a doubly stochastic intensity framework, allowing default correlation through a common jump process. The key insight behind our approach is an explicit characterization of the portfolio exposure as the weak limit of measure-valued processes associated with survival indicators of portfolio names. We validate our theoretical predictions by means of a numerical analysis, showing that counterparty adjustments are highly sensitive to portfolio credit risk volatility as well as to the intensity of the common jump process.     

A set-valued Markov chain approach to credit default

We propose a novel credit default model that takes into account the impact of macroeconomic factors and intergroup contagion on the defaults of obligors. We use a set-valued Markov chain to model the default process, which includes all defaulted obligors in the group. We obtain analytic characterizations for the default process and derive pricing formulas in explicit forms for synthetic collateralized debt obligations (CDOs). Furthermore, we use market data to calibrate the model and conduct numerical studies on the tranche spreads of CDOs. We find evidence to support that systematic default risk coupled with default contagion could have the leading component of the total default risk.