Modeling Intra and Inter Correlations in Credit Default Losses

2006

We consider a single period portfolio of n dependent credit risks that are 3275 subject to default during the period. We show that using stochastic loss given 3276 default random variables in conjunction with default correlations can give rise 3277 to an inconsistent set of assumptions for estimating the variance of the port-3278 folio loss. Two sets of consistent assumptions are provided, which it turns 3279 out, also provide bounds on the variance of the portfolio's loss. An example 3280 of an inconsistent set of assumptions is given.

The Journal of Risk Management, 2007

The risk of a credit portfolio depends crucially on correlations between latent covariates, for instance the probability of default (PD) in different economic sectors. Often, correlations have to be estimated from relatively short time series, and the resulting estimation error hinders the detection of a signal. We suggest a general method of parameter estimation which avoids in a controlled way the underestimation of correlation risk. Empirical evidence is presented how, in the framework of the CreditRisk+ model with integrated correlations, this method leads to an increased economic capital estimate. Thus, the limits of detecting the portfolio's diversification potential are adequately reflected.

The Journal of Fixed Income, 2006

Fixed-Income portfolios are increasingly susceptible to correlated default risk. Defaults of individual firms will cluster if there are common factors that affect each firm's default risk. Using a comprehensive dataset of firm-level default probabilities, we examine co-variation of default probabilities across US public non-financial firms. We observe that systematic time-variation in default risk is driven more by an economy-wide volatility factor than by changing debt levels, and therefore is closely linked to the business cycle. Specifically, both default probabilities and default correlations vary over time resulting in substantial variation in joint default risk. For example, over the latter half of the 1990s, default probabilities across the economy doubled, and correlations increased by an even greater magnitude. We provide a reduced-form framework to jointly model time variation in both default probabilities and their correlations over the business cycle. Calibration of the model demonstrates the economic importance of modeling time-variation of joint default risk; for example, our model suggests that the ex-ante probability of observing the record defaults of 2001 doubled across regimes. We also document cross-sectional differences across rating classes-default probability correlations are higher amongst higher quality issuers.

SSRN Electronic Journal, 2000

The paper analyzes a two-factor credit risk model allowing to capture default and recovery rate variation, their mutual correlation, and dependence on various explanatory variables. At the same time, it allows computing analytically the unexpected credit loss. We propose and empirically implement estimation of the model based on aggregate and exposure level Moody's default and recovery data. The results confirm existence of significantly positive default and recovery rate correlation. We empirically compare the unexpected loss estimates based on the reduced two-factor model with Monte Carlo simulation results, and with the current regulatory formula outputs. The results show a very good performance of the proposed analytical formula which could feasibly replace the current regulatory formula.

Economic Modelling , 2015

Appropriate modelling of loan default correlation capturing the fat tail distributions and non-symmetrical behaviour linked to the sensitivity of the loss correlations is a prerequisite for effective credit risk manage- ment, as banks seek to optimally allocate capital. In this study, we provide an insight to the use of copula functions, particularly addressing the key question of why Gaussian copulas caused so much instability dur- ing 2007–08. We empirically demonstrate that using an Archimedean copula, particularly the Gumbel, it is more efficient in capturing the top right hand side tail-dependencies, thereby illustrating the impact of fat- tails on non-linear parameters. This finding has significant implications for banks and their capital manage- ment requirements, particularly banks employing the Advanced Internal Rate-Based method. This is even more relevant now, with Basel III providing more detailed information as to what constitutes Tier 1, Addi- tional Tier 1 and 2 Capital.

Science Journal of Applied Mathematics and Statistics, 2015

We outline the ingredients necessary to compute the Joint Default Probability from which we obtain Default Correlation, an important risk quantity in the determination of Internal Rating Based Approach in Basel II and III documents on banking supervision and regulations. We discuss Merton's structural approach of which one key drawback is the difficulty in tracking and calibrating asset value processes and the limitations of variant models which tend to be analytically too complex and computationally intensive. We address these issues by simulating all the possible asset value processes of a firm whose asset paths we assume to be Gaussian. By generating random values that simulate all the possible asset value processes, we are able to capture all the possible default horizons within a certain macroeconomic framework. Drawing standardised normally distributed assets values of obligors we obtain a range of values of Joint Default Probabilities at a specified asset correlation from which the corresponding range of default correlations are obtained. The results is a simplified approach to the determination of default correlation, easily implementable in Excel and less analytically complicated than existing procedures.

Journal of Applied Probability

We investigate default probabilities and default correlations of Merton-type credit portfolio models in stress scenarios where a common risk factor is truncated. For elliptically distributed asset variables, the asymptotic limits of default probabilities and default correlations depend on the max-domain of attraction of the asset variables. In the regularly varying case, we derive an integral representation for multivariate default probabilities, which turn out to be strictly smaller than 1. Default correlations are in (0, 1). In the rapidly varying case, asymptotic multivariate default probabilities are 1 and asymptotic default correlations are 0.

International Journal of Risk Assessment and Management, 2010

We propose a reduced-form model for credit risk in a multivariate setting. The default intensities are linear combinations of three independent affine jump-diffusion processes representing the intensities of general, sectoral and idiosyncratic credit events. The model can be efficiently calibrated to term structures of default probabilities and conditional probabilities of default given the occurrence of common credit events. We analyse the correlation of defaults and formulate an algorithm for the exact simulation of default scenarios.

Risk and Decision Analysis, 2018

Within the new Basel regulatory and capital framework for market risks (FRTB, 2016), the Basel Committee on Banking Supervision's (BCBS's) sets out significant revisions to the market risk capital requirements framework. Key areas include moving from Value-at-Risk (VaR) to Expected Shortfall (ES). Non securitized credit positions in the trading book are also subject to a separate default risk charge (DRC) and banks using the internal model approach are required to use a two-factor model and a 99.9% VaR over a one year capital horizon to calculate the DRC capital charge. In this framework, the multi-factor Merton-type models of credit risk are topical since banks are required to measure the capital charge using Value-at-Risk (VaR) and Expected Shortfall (ES). But with this new framework, the implementation of these indicators using the semi-analytical formulas is no longer straightforward and the practice of financial institutions seems to head towards Monte Carlo simulations which are usually more time consuming. In this paper, we present a new approach for simultaneously calculating the VaR and ES of large and heterogeneous credit portfolios. VaR in the multi-factor Merton framework where its numerical implementation is carried out using a semi-analytical formula. This is of particular relevance in the Fundamental Review of Trading book framework (FRTB) since the measure of the two indicators VaR and ES are in general required: the risk measure to be employed in the market risk capital charge computation is changed from the VaR to the ES but the VaR calculation is also needed to for the validation and back-testing process. For this purpose, we use a result of Rockafellar and Uryasev [20] where they characterize the Value-at-Risk (VaR) and the Expected Shortfall (ES) as a solution of a convex optimization problem of a target function chosen in an appropriate way. The target function is built across the portfolio loss and its cumulative distribution. This has been applied by the authors to portfolio optimization. The setup of our approach requires the use of the Central Limit Theorem (CLT). Although the assumption of large portfolios ensures the classical conditions of the CLT, the heterogeneous nature of the credit portfolio can sometimes prevent the direct use of this theorem. To overcome this difficulty, the portfolio loss can be decomposed in two components with CLT assumptions verified for the first one, whereas the second is treated by a recursive method. Even if our new approach can be used in a general framework of the FRTB, we first apply it to a large and heterogeneous credit portfolio to compute the default risk charge and we explore its ability to deal with the exposure concentration. Results of several numerical tests support our quantitative analysis and confirm the theoretical aspects of our approach.

We propose a portfolio credit risk model with dependent loss given default (LGD) which allows for a reasonable economic interpretation and can easily be applied to real data. We build up a precise mathematical framework and stress some general important issues when modeling dependent LGD. Finally, we calibrate the model based on American bond data from 1982 to 2001 and compare the results with recently published alternative models.