Default Correlation Research Articles

Credit risk assessment of fixed income portfolios using explicit expressions

We propose a model to assess the credit risk features of fixed income portfolios assuming they can be characterized by two parameters: their default probability and their default correlation. We rely on explicit expressions to assess their credit risk and demonstrate the benefits of our approach in a complex leveraged structure example. We show that using expected loss as a proxy for credit risk is misleading as it does not capture the dispersion effects introduced by correlation. The implications of these findings are relevant for improving current risk management practices and for regulation purposes.

Option-Based Credit Spreads

We propose a non-parametric empirical benchmark for credit risk analysis. We build fictitious “pseudo firms” that purchase real traded assets by issuing equity and zero-coupon bonds. By no-arbitrage, these bonds equal Treasuries minus put options on the firms’ assets, whose values are all observable. We exploit our pseudo firms as a laboratory, and empirically show that their credit spreads are large and countercyclical, illiquidity and corporate frictions are unlikely determinants of bonds’ credit spreads, infrequent rating changes and idiosyncratic asset uncertainty greatly increase spreads, and, in a banking application, discount rate shocks substantially increase banks’ tail risks and default correlations.

A Structural Model with Default Correlation

In this paper, we consider a structural model where two firms following jump diffusion processes have the same Poisson process (i.e. the moment of jumps is the same) and the jump sizes are correlated. In the market, we also observe this kind of phenomenon whereby firms experience the same jumps. We first establish the characteristics of the pair of default stopping times and their joint distribution, as well as the associated martingales. Second, we consider the question of the copula which gives the correlation of the two default stopping times and determine the copula that appears in our model. Our model can be used for the pricing of credit derivatives such as the first to default, discount bond, and CDS. Our objective is to give a general model for pricing credit derivatives on two firms presented in this framework, in the presence of the special anticipated default stopping time. The technical tool we mostly use in this paper is the intensity approach in a framework of structural model.

Default prediction with dynamic sectoral and macroeconomic frailties

This paper extends the macroeconomic frailty model to include sectoral frailty factors that capture default correlations among firms in a similar business. We estimate sectoral and macroeconomic frailty factors and their effects on default intensity using the data for Japanese firms from 1992 to 2010. We find strong evidence for the presence of sectoral frailty factors even after accounting for the effects of observable covariates and macroeconomic frailty on default intensity. The model with sectoral frailties performs better than that without. Results show that accounting for the sources of unobserved sectoral default risk covariations improves the accuracy of default probability estimation.

The pricing of credit risky securities under stochastic interest rate model with default correlation

In this paper, we study the pricing of credit risky securities under a three-firms contagion model. The interacting default intensities not only depend on the defaults of other firms in the system, but also depend on the default-free interest rate which follows jump diffusion stochastic differential equation, which extends the previous three-firms models (see R.A. Jarrow and F.Yu (2001), S.Y.Leung and Y.K.Kwok (2005), A.Wang and Z.Ye (2011)). By using the method of change of measure and the technology (H. S.Park (2008), R.Hao and Z.Ye (2011)) of dealing with jump diffusion processes, we obtain the analytic pricing formulas of defaultable zero-coupon bonds. Moreover, by the “total hazard construction”, we give the analytic pricing formulas of credit default swap (CDS).

The valuation of credit default swaps including investor–counterparty–reference entity default correlation

We extend the work of Hull and White and Kettunen and Meissner and build a credit default swap (CDS) pricing model that includes default intensities and default correlation of all three involved entities, i.e., the investor (the CDS buyer), the counter-party (the CDS seller) and the underlying reference entity. We build the model in a discrete time frame, so that the user can match the timing of CDS spread payments exactly and alter cash flows if desired. We combine two octuplet trees and use swap evaluation techniques to derive a closed-form solution for the CDS spread including default correlation of all three entities. We find that the default intensity of the investor has a strong impact on the CDS spread, whereas the default correlation of the investor with other CDS entities has a minor impact. The Matlab source code of the model is provided upon request.

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.

Simulation of entrained flow gasification with advanced coal conversion submodels. Part 2: Char conversion

CFD modeling results for entrained flow coal gasification using advanced submodels for coal conversion are presented and compared to detailed experimental data. The focus of this investigation is on the accurate modeling of the char conversion process. The CBK/E and CBK/G models are used for calibrating a simplified Single Nth-Order Reaction (SNOR) kinetic model, which is suitable for CFD calculations. The kinetics of the CBK/G model are fitted using char gasification data obtained from drop tube furnace experiments. CFD simulations are performed for the BYU entrained flow gasifier fired with four different coals varying significantly in rank ranging from bituminous coal to lignite. The comparison with the experiments generally gives good agreement in terms of flame stand-off, gas composition and carbon conversion when the kinetics of the CBK/G model calibrated with drop tube experiments is used. On the other hand, lower carbon conversion is obtained using the default correlation of the CBK/G model, which only allows to depict the overall tendency of the gasification reactivity with coal rank.

Default Correlations in the Merton Model*

We examine the relationship between default probabilities and default correlations of two firms in the Merton model. We show that default correlations increase under a homogeneous increase of default probabilities. The same is true if the increase of the default probability is more pronounced for the firm with the lower likelihood of default. Default correlations may only decline if the increase of the default probability is significantly larger for the firm with higher default risk. These findings may have important implications for the assessment of credit portfolio risk, loan pricing, and capital requirements when adverse macroeconomic shocks occur.

Evaluating Double-Default Risk with a Regime Switching Hazard Rate Process and a Chaining Copula Model

Since the financial crises in 2008, the default risk of counter-parties has drawn increased attention. One way of mitigating this default risk is the so-called credit support annex (CSA). The CSA denies rules under which collateral is received -- respectively posted -- in order to protect against losses in case of the default of the counter-party. However, for credit derivatives there is still a residual risk, when the derivative's reference entity and the counter-party default at the same time. As will be outlined in due course, standard techniques known from default correlation derivatives are inappropriate to evaluate that gap risk. Hence, we will present new techniques, namely a copula model for simultaneous defaults and a regime-switching hazard rate model to cope with that evaluation problem.

Inferring Default Correlation from Equity Return Correlation

This paper presents a new approach for estimating default correlation by linking default correlation to equity return correlation while preserving the fundamental relation between default and asset correlations in the structural framework. Our hybrid model thus overcomes a long-standing empirical difficulty that default correlation estimation relies on the unobservable asset process. The empirical analysis shows that our hybrid model demonstrates a considerable improvement over the existing structural model of Zhou (2001) for the sample periods of 1970-1993 and 1990-2010. We also illustrate the difference between the two models in predicting default correlations over the period of the 2008 financial crisis.

Stressed Testing in Credit Portfolio Models

As, in light of the recent financial crises, stress tests have become an integral part of risk management and banking supervision, the analysis and understanding of risk model behaviour under stress has become ever more important. In this paper, we present a general approach to implementing stress scenarios in a multi-factor credit portfolio model and analyse asset correlations, default probabilities and default correlations under stress. We use our results to study the implications for credit reserves and capital requirements and illustrate the proposed methodology by stressing a large investment banking portfolio. Although our stress testing approach is developed in a particular credit portfolio model, the main concept - stressing risk factors through a truncation of their distributions - is independent of the model specification and can be applied to other risk types as well.

COMPUTATIONAL ASPECTS OF MONTE-CARLO SIMULATIONS OF THE FIRST PASSAGE TIME FOR MULTIVARIATE TRANSFORMED BROWNIAN MOTIONS WITH JUMPS

Many problems in science, engineering, and finance require the information on the first passage time (FPT) of a stochastic process. Mathematically, such problems are often reduced to the evaluation of the probability density of the time for such a process to cross a certain level, a boundary, or to enter a certain region. While in other areas of applications the FPT problem can often be solved analytically, in finance we usually have to resort to the application of numerical procedures, in particular when we deal with jump-diffusion stochastic processes (JDP). In this paper, we propose a Monte-Carlo-based methodology for the solution of the FPT problem in the context of multivariate (and correlated) JDPs. The developed technique provides an efficient tool for a number of applications, including credit risk and option pricing. We demonstrate its applicability to the analysis of default rates and default correlations of several different, but correlated firms via a set of empirical data.

PRICING COUNTERPARTY RISK INCLUDING COLLATERALIZATION, NETTING RULES, RE-HYPOTHECATION AND WRONG-WAY RISK

This article is concerned with the arbitrage-free valuation of bilateral counterparty risk through stochastic dynamical models when collateral is included, with possible rehypothecation. The payout of claims is modified to account for collateral margining in agreement with International Swap and Derivatives Association (ISDA) documentation. The analysis is specialized to interest-rate and credit derivatives. In particular, credit default swaps are considered to show that a perfect collateralization cannot be achieved under default correlation. Interest rate and credit spread volatilities are fully accounted for, as is the impact of re-hypothecation, collateral margining frequency, and dependencies.

Rating Shopping or Catering? An Examination of the Response to Competitive Pressure for CDO Credit Ratings

We examine whether “rating shopping” or “rating catering” is a more accurate characterization of rating agency interactions regarding collateralized debt obligations (CDOs) prior to the credit crisis. Although investors paid a premium for dual-ratings, AAA CDO tranches rated by both Moody’s and S&P defaulted more frequently than tranches rated by only one of them, which is inconsistent with pure rating shopping. Even though AAA-rated tranche sizes were typically identical, there was considerable disagreement between S&P’s and Moody’s assumptions on default correlations and even collateral quality. Rating agencies made upward adjustments beyond their model outputs when their competitor had more lenient assumptions. Finally, although S&P’s and Moody’s adjustments and disagreements were not reflected in yields at issuance, they were reflected in subsequent rating downgrades, suggesting that rating agencies later reverted towards their initial models.

A Factor Model for the Calculation of Portfolio Credit VaR

In nowadays financial institutions pay more and more attention to credit risk. The default correlation between credit instruments has a fatal influence on credit risk. In this paper, we use the factor copula model to calculate the credit VaR (value at risk) for the target portfolio. We first use the asset value obtained from Merton model to substitute the equity price which is used by most researchers. Then we employ the principal component analysis method to construct a virtual variable representing the macro factor, and estimate the VaR for a credit portfolio.

Estimation of Regulatory Credit Risk Models

This article estimates a general credit risk model with both macroeconomic and latent credit factors for Spanish banks during the period 2004-2010. The proposed framework allows to estimate with bank level data both the standard credit risk model of Basel II and generalized models. I find evidence of persistence in the credit latent factor and of a significant effect of GDP growth and interbank rates on loan default rates. The estimated default correlation is low across specifications. The model is also used to calculate the impact on the probabilities of default of stressed economic scenarios.

In This Issue

In This Issue

Modelling Default Correlations in a Two-Firm Model with Dynamic Leverage Ratios

This article provides a generalized two-firm model of default correlation, based on the structural approach that incorporates interest rate risk. In most structural models default is driven by the firms' asset dynamics. In this article, a two-firm model of default is instead driven by the dynamic leverage ratios, which combines the measure of risks of the firms' total liabilities and assets. This article investigates analytical methods and numerical tools to solve the two-dimensional first passage time problem with time-dependent parameters. We carry out a comparative analysis of the impact of model parameters and provide some insights of their effects on joint survival probabilities and default correlations.

Tetrachoric Correlation as a Measure of default Correlation

Default correlation is a crucial in risk management. The aim of this paper is to introduce a new method for measuring default correlation. In this paper we recall tetrachoric correlation coefficient which is a measure of association between two continuous variables that have each been measured on a dichotomous scale. We presented the applicability of tetrachoric correlation as a default risk correlation measure.