Abstract
This paper incorporates state dependent correlations (those that vary systematically with the state of the economy) into the Vasicek default model. Other approaches to randomizing correlation in the Vasicek model have either assumed that correlation is independent of the systematic risk factor (zero state dependence) or is an explicit function of the systematic risk factor (perfect state dependence). By contrast, our approach allows for an arbitrary degree of state dependence and includes both zero and perfect state dependence as special cases. This is accomplished by expressing the factor loading as a function of an auxiliary (Gaussian) variable that is correlated with the systematic risk factor. Using Federal Reserve data on delinquency rates we use maximum likelihood to estimate the parameters of the model, and find the empirical degree of state dependence to be quite high (but generally not perfect). We also find that randomizing correlation, without allowing for state dependence, does not improve the empirical performance of the Vasicek model.
References
[1] Albanese, C., D. Li, E. Lobachevskiy, and G. Meissner (2013). A comparative analysis of correlation approaches in finance. J. Deriv. 21(2), 42–66.10.3905/jod.2013.21.2.042Search in Google Scholar
[2] Andersen, L. and J. Sidenius (2004). Extensions to the Gaussian copula: random recovery and random factor loadings. J. Credit Risk 1(1), 29–70.10.21314/JCR.2005.003Search in Google Scholar
[3] Arnold, B. C., R. J. Beaver, R. A. Groeneveld, and W. Q. Meeker (1993). The nontruncated marginal of a truncated bivariate normal distribution. Psychometrika 58(3), 471–488.10.1007/BF02294652Search in Google Scholar
[4] Azzalini, A. (1985). A class of distributions which includes the normal ones. Scand. J. Statist. 12(2), 171–178.Search in Google Scholar
[5] Basel Committee on Banking Supervision (2005). An Explanatory Note on the Basel II IRB Risk Weight Functions. Bank for International Settlements, Basel.Search in Google Scholar
[6] Batiz-Zuk, E., G. Christodoulakis, S.-H. Poon (2013). Structural credit loss distributions under non-normality. J. Fixed Income 23(1), 56–75.10.3905/jfi.2013.23.1.056Search in Google Scholar
[7] Burtschell, X., J. Gregory, and J.-P. Laurent (2007). Beyond the Gaussian copula: stochastic and local correlation. J. Credit Risk 3(1), 31–62.10.21314/JCR.2007.059Search in Google Scholar
[8] Burtschell, X., J. Gregory, and J.-P. Laurent (2009). A comparative analysis of CDO pricing models under the factor copula framework. J. Deriv. 16(4), 9–37.10.3905/JOD.2009.16.4.009Search in Google Scholar
[9] Campbell, R., K. Koedijk, and P. Kofman (2002). Increased correlations in bear markets. Financ. Anal. J. 58(1), 87–94.10.2469/faj.v58.n1.2512Search in Google Scholar
[10] Elouerkhaoui, Y. (2017). Credit Correlation. Palgrave Macmillan, Basingstoke.10.1007/978-3-319-60973-7Search in Google Scholar
[11] Gordy, M. B. (2003). A risk-factor model foundation for ratings-based bank capital rules. J. Financ. Intermed. 12(3), 199–232.10.1016/S1042-9573(03)00040-8Search in Google Scholar
[12] Gregory, J. and J.-P. Laurent (2008). Practical pricing of synthetic CDOs. In G. Meissner (Ed.), The Definitive Guide to CDOs, pp. 223–258. Risk Books, London.Search in Google Scholar
[13] Kritzman, M., Y. Li, S. Page, and R. Rigobon (2010). Principal components as a measure of systemic risk. J. Portfolio Manage. 37(4), 112–126.10.3905/jpm.2011.37.4.112Search in Google Scholar
[14] Kupiec, P. H. (2009). How well does the Vasicek-Basel AIRB model fit the data? Evidence from a long time series of corporate credit rating data. Available at http://dx.doi.org/10.2139/ssrn.1523246.10.2139/ssrn.1523246Search in Google Scholar
[15] Laurent, J.-P. and A. Cousin (2009). An overview of factor modeling for CDO pricing. In R. Cont (Ed.), Frontiers in Quantitative Finance: Volatility and Credit Risk Modeling, pp. 185–216. John Wiley & Sons, Hoboken NJ.Search in Google Scholar
[16] Lehmann, E. L. and J. P. Romano (2005). Testing Statistical Hypotheses. Third edition. Springer, New York.Search in Google Scholar
[17] Li, D. X. (2000). On default correlation: A copula function approach. J. Fixed Income 9(4), 43–54.10.3905/jfi.2000.319253Search in Google Scholar
[18] Longin, F. and B. Solnik (2001). Extreme correlation of international equity markets. J. Finance 56(2), 649–676.10.1111/0022-1082.00340Search in Google Scholar
[19] Melkuev, D. (2014). Asset Return Correlations in Episodes of Systemic Crises. Master thesis, University of Waterloo, Canada.Search in Google Scholar
[20] Metzler, A. and D. L. McLeish (2011). A multiname first-passage model for credit risk. J. Credit Risk 7(1), 35–64.10.21314/JCR.2011.122Search in Google Scholar
[21] Pimbley, J. M. (2018). T-Vasicek credit portfolio loss distribution. J. Struct. Finance 24(3), 65–78.10.3905/jsf.2018.24.3.065Search in Google Scholar
[22] Puccetti, G. and M. Scherer (2018). Copulas, credit portfolios, and the broken heart syndrome. Depend. Model. 6, 114–130.10.1515/demo-2018-0007Search in Google Scholar
[23] Pykhtin, M. (2004). Multi-factor adjustment. Risk Magazine 3, 85–90.Search in Google Scholar
[24] Rutkowski, M. and S. Tarca (2015). Regulatory capital modelling for credit risk. Int. J. Theor. Appl. Finance 18(5), Article ID 1550034, 44 pages.10.1142/S021902491550034XSearch in Google Scholar
[25] Schlösser, A. and R. Zagst (2013). The crash-NIG factor model: Modeling dependence in credit portfolios through the crisis. Eur. Actuar. J. 3, 407–438.10.1007/s13385-013-0073-9Search in Google Scholar
[26] Scott, A. and A. Metzler (2015). A general importance sampling algorithm for estimating portfolio loss probabilities in linear factor models. Insurance Math. Econom. 64, 279–293.10.1016/j.insmatheco.2015.06.001Search in Google Scholar
[27] Tarashev, N. and H. Zhu (2008). Specification and calibration errors in measures of portfolio credit risk: The case of the ASRF model. Int. J. Cent. Bank. 4(2), 129–173.Search in Google Scholar
[28] Turc, J. and P. Very (2009). Pricing CDOs with a smile: the local correlation model. In R. Cont (Ed.), Frontiers in Quantitative Finance: Volatility and Credit Risk Modeling, pp. 235–250. John Wiley & Sons, Hoboken NJ.Search in Google Scholar
[29] Vasicek, O. (2002). Loan portfolio value. Risk Magazine 12, 160–162.Search in Google Scholar
[30] Witzany, J. (2013). A Note on the Vasicek’s Model with the Logistic Distribution. Ekonomický čsopis 61(10), 1053–1066.Search in Google Scholar
© 2020 A. Metzler, published by De Gruyter
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