Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-29T07:47:48.582Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic behaviour of multivariate default probabilities and default correlations under stress

Published online by Cambridge University Press:  24 March 2016

N. Packham ,
M. Kalkbrener  and
L. Overbeck
M. Kalkbrener
Affiliation:
Deutsche Bank AG, Taunusanlage 12, 60325 Frankfurt am Main, Germany. Email address: michael.kalkbrener@db.com
L. Overbeck
Affiliation:
Institute of Mathematics, University of Giessen, Arndtstrasse 2, 35392 Giessen, Germany. Email address: ludger.overbeck@math.uni-giessen.de
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Financial risk managementcredit portfolio modellingstress testingelliptic distributionmax-domain of attraction
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 1 , March 2016 , pp. 71 - 81
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abdous, B., Fougères, A.-L. and Ghoudi, K. (2005). Extreme behaviour for bivariate elliptical distributions. Canad. J. Statist. 33, 317–334. CrossRefGoogle Scholar
[2]Balakrishnan, N. and Hashorva, E. (2011). On Pearson–Kotz Dirichlet distributions. J. Multivariate Analysis 102, 948–957. CrossRefGoogle Scholar
[3]Berman, S. M. (1982). Sojourns and extremes of stationary processes. Ann. Prob. 10, 1–46. CrossRefGoogle Scholar
[4]Berman, S. M. (1983). Sojourns and extremes of Fourier sums and series with random coefficients. Stoch. Process. Appl. 15, 213–238. CrossRefGoogle Scholar
[5]Berman, S. M. (1992). Sojourns and Extremes of Stochastic Processes. Wadsworth & Brooks/Cole, Pacific Grove, CA. CrossRefGoogle Scholar
[6]Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge University Press. CrossRefGoogle Scholar
[7]Bonti, G., Kalkbrener, M., Lotz, C. and Stahl, G. (2006). Credit risk concentrations under stress. J. Credit Risk 2, 115–136. CrossRefGoogle Scholar
[8]Breiman, L. (1965). On some limit theorems similar to the arc-sin law. Theory Prob. Appl. 10, 323–331. CrossRefGoogle Scholar
[9]Crosbie, P. J. and Bohn, J. R. (2002). Modelling default risk. Working paper, KMV Corporation. Google Scholar
[10]De Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York. CrossRefGoogle Scholar
[11]Duellmann, K. and Erdelmeier, M. (2009). Crash testing German banks. Internat. J. Central Banking 5, 139–175. Google Scholar
[12]Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. For Insurance and Finance. Springer, Berlin. CrossRefGoogle Scholar
[13]Fougères, A.-L. and Soulier, P. (2010). Limit conditional distributions for bivariate vectors with polar representation. Stoch. Models 26, 54–77. CrossRefGoogle Scholar
[14]Gupton, G. M., Finger, C. C. and Bhatia, M. (1997). CreditMetrics: Technical Document. JP Morgan, New York. Google Scholar
[15]Hashorva, E. (2006). Gaussian approximation of conditional elliptic random vectors. Stoch. Models 22, 441–457. CrossRefGoogle Scholar
[16]Hashorva, E. (2009). Conditional limit results for type I polar distributions. Extremes 12, 239–263. CrossRefGoogle Scholar
[17]Hult, H. and Lindskog, F. (2002). Multivariate extremes, aggregation and dependence in elliptical distributions. Adv. Appl. Prob. 34, 587–608. CrossRefGoogle Scholar
[18]Jacobsen, M., Mikosch, T., Rosínski, J. and Samorodnitsky, G. (2009). Inverse problems for regular variation of linear filters, a cancellation property for σ-finite measures and identification of stable laws. Ann. Appl. Prob. 19, 210–242. CrossRefGoogle Scholar
[19]Kalkbrener, M. and Packham, N. (2015). Correlation under stress in normal variance mixture models. Math. Finance 25, 426–456. CrossRefGoogle Scholar
[20]Kalkbrener, M. and Packham, N. (2015). Stress testing of credit portfolios in light- and heavy-tailed models. J. Risk Manag. Financial Institutions 8, 34–44. Google Scholar
[21]Klüppelberg, C., Kuhn, G. and Peng, L. (2008). Semi-parametric models for the multivariate tail dependence function—the asymptotically dependent case. Scand. J. Statist. 35, 701–718. CrossRefGoogle Scholar
[22]McNeil, A. J., Frey, R. and Embrechts, P. (2005). Quantitative Risk Management. Princeton University Press. Google Scholar
[23]Merton, R. C. (1974). On the pricing of corporate debt: the risk structure of interest rates. J. Finance 29, 449–470. Google Scholar
[24]Schmidt, R. (2002). Tail dependence for elliptically contoured distributions. Math. Meth. Operat. Res. 55, 301–327. CrossRefGoogle Scholar