Credit Risk: Pricing, Measurement, and Management (Princeton Series in Finance)
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Download full text in PDF Download. Procedia Computer Science Volume 18 , , Pages Author links open overlay panel Yanbin Shen J. Van Der Weide J.
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No Downloads. Views Total views. We consider the Merton model, in which defaults and recoveries are determined by the underlying asset processes. The correlation matrix of the asset returns has to be estimated from historical time series. This is not always easy, because the correlations change in time, ie, they are non-stationary.
Since only time series up to a certain length can be used, the correlation coefficients contain a specific type of randomness, see  , . Several methods have been put forward to estimate and to reduce this "noise''. Thus, we assume that such a noise reduction has been done. The corresponding "true'' correlation coefficients and matrices are the proper input for the structural credit risk model of the Merton type that we consider. We discussed this issue of noise reduction to emphasize that the random matrix approach in that context focuses on the spectral statistics of correlation or covariance matrices, see  ,  ,  ,  , .
It is based on a very different motivation as compared to the present application. Searching for generic properties, we devised the present random matrix approach. Instead of calculating the portfolio loss distribution for a specific correlation matrix, we average over an ensemble of random correlation matrices.
Our approach transfers concepts of statistical physics. In quantum chaos, the average over an individual, long spectrum equals the average over an ensemble of random matrices, if the level number is very high. We expect that a similar self-averaging property also holds here. This line of reasoning is supported by the following consideration: The correlation coefficients are varying functions in time, because the business relations of the companies change. This implies that a correlation matrix over a somewhat longer period in time is a varying quantity, ie, it corresponds to some kind of ensemble.
In our model the average correlation level is zero and we assume that there is no branch structure in the correlations. The fluctuation strength of individual correlations is controlled by a single parameter. This ansatz allowed us to estimate generic statistical properties of the Merton model. Some features are not taken into account which are present in empirical data, such as jumps or an overall positive correlation level.
Those features are difficult to treat completely analytically. However, even in our simple setup we obtain a heavy—tailed loss distribution.
Book Review: Credit Risk Modeling: Theory and Applications - raibularestpass.gq
In this sense our model can be used to estimate a lower bound of the risk embedded in a credit portfolio. Our results clearly demonstrate that the risk in a credit portfolio is heavily underestimated if correlations are not taken into account. Even for random correlations with an average correlation level of zero, we observe very slowly decaying portfolio loss distributions.
In contrast, the probability of large losses in uncorrelated portfolios is significantly reduced within the Merton model. The results are especially relevant for CDOs, bundles of credits that are traded on equity markets. CDOs are constructed in order to lower the overall risk. The components of a CDO can be exposed to large risks. It is often believed that the CDO has a significantly lower risk. We showed that this diversification only works well if the correlations in the credit portfolio are identical to zero.
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. National Center for Biotechnology Information , U. PLoS One.
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Published online May Michael C. Renaud Lambiotte, Editor. Author information Article notes Copyright and License information Disclaimer. Competing Interests: The authors have declared that no competing interests exist. Received Dec 20; Accepted Apr This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited.
This article has been cited by other articles in PMC. Appendix S2: PDF. Appendix S3: PDF. Appendix S4: PDF. Abstract We estimate generic statistical properties of a structural credit risk model by considering an ensemble of correlation matrices. Introduction The financial crisis of — clearly revealed that an improper estimation of credit risk can lead to dramatic effects on the world's economy.
A Structural Credit Risk Model Our model is based on Merton's original model, assuming a zero-coupon bond for the debt structure of the obligor. Table 1 Input of the structural credit risk model. Open in a separate window. Average distribution of asset values For the sake of simplicity, let us first consider the case of a Brownian motion for the asset values. Figure 1.
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Figure 2. Loss distribution We now turn to the calculation of the loss distribution. Average loss distribution Now we have developed all necessary tools to model the average distribution of losses, under the assumption of random correlations and an average correlation level of zero. Homogeneous portfolios In case of a homogeneous portfolio, in which all credits have the same face value and the same variance and initial value , the weights can be simplified to. Improved approximation for a homogeneous portfolio The second order approach can be improved by approximating the individual terms of the loss distribution instead of approximating the expression as a whole, similar as discussed in .
Results We now apply the analytically developed model to a specific example. Figure 3. Figure 4. Discussion To assess the risk of a credit portfolio, it is crucial to take correlations between obligors into account. Acknowledgments M. Funding Statement The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. References 1. Princeton University Press. Journal of Political Economy 81 : — Journal of Finance 29 : — Journal of Finance 50 : 53— Review of Financial Studies 10 : — Duffie D, Singleton K Modeling the term structure of defaultable bonds.
Review of Financial Studies 12 : — Journal of Derivatives 8 : 29— Journal of Finance 31 : — Portfolio Modeling, Barclays Capital. Physical Review E 82 : Rosenow B, Weissbach R Modelling correlations in credit portfolio risk. Journal of Risk Management in Financial Institutions 3 : 16— Technical report, Morgan Guaranty Trust Company. Economic Modelling 30 : 1—9. Journal of Credit Risk 8 : 31— Journal of Risk Finance 3 : 45— Glasserman P Tail approximations for portfolio credit risk.
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Working Paper. Physica A : — Physics Reports : — Mehta M Random Matrices. Academic Press.