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Risk Aggregation with Copula for Banking Industry

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Applications + Practical Conceptualization + Mathematics = fruitful Innovation

Part of the book series: Mathematics for Industry ((MFI,volume 11))

Abstract

This paper surveys several applications of parametric copulas to market portfolios, credit portfolios, and enterprise risk management in the banking industry, focusing on how to capture stressed conditions. First, we show two simple applications for market portfolios: correlation structures for returns on three stock indices and a risk aggregation for a stock and bond portfolio. Second, we show two simple applications for credit portfolios: credit portfolio risk measurement in the banking industry and the application of copulas to CDO valuation, emphasizing the similarity to their application to market portfolios. In this way, we demonstrate the importance of capturing stressed conditions. Finally, we introduce practical applications to enterprise risk management for advanced banks and certain problems that remain open at this time.

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Notes

  1. 1.

    For differences in unconditional and conditional approaches, see [7, 14].

  2. 2.

    Some information criteria, such as BIC and AIC, are used to select the optimal copula. Both criteria are calculated based on log-likelihood, with certain penalties applied by number of parameters. We adopt BIC for the criteria, which imposes more penalties in number of parameters than AIC. BIC is calculated by \(-2 l( \xi )+p\ln N\), where \(l( \xi )\) is the maximum log-likelihood, p the number of parameters, and N the sample size. The model with the lowest BIC is selected.

  3. 3.

    We use generic interest rates calculated by Bloomberg for 5-year interest rates of government bonds.

  4. 4.

    Joint density contours with standard Gaussian margins are visual representations of the various dependencies in the center and the tail area (see [8]). If the copula is Gaussian, the contour is elliptical (see Fig. 1a).

  5. 5.

    The mixed-Gaussian copula is not always selected by BIC. For example, Yoshiba [22] shows t copula is selected by BIC for data related to the Euro crisis and the post-bubble period in Japan. As for more complicated copulas, we can construct, as examples, mixed-t or mixed-Gaussian-t copulas. An examination of these copulas is left for the future.

  6. 6.

    99 % VaR is the 99th percentile for a portfolio loss distribution. A 97.5 % ES is the average of the losses in the 2.5 % tail of the loss distribution. If the portfolio profit–loss distribution is Gaussian, a 97.5 % ES nearly equals 99 % VaR.

  7. 7.

    For vine copulas, see the handbook [12].

  8. 8.

    For HAC, see [18].

  9. 9.

    The tranche with [a %, b %] covers the portfolio loss, while the loss rate is in [a %, b %].

  10. 10.

    Credit spread s is calculated as \(s=-\ln (1-EL)/T\), where EL is the expected loss for the tranche and T is 5-year maturity.

References

  1. Azzalini, A., Capitanio, A.: Distributions generated by perturbation of symmetry with emphasis on a multivariate skew \(t\)-distribution. J. Roy. Stat. Soc. Ser. B 65(2), 367–389 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Basel Committee on Banking Supervision: Joint Forum “Developments in Modelling Risk Aggregation” 21 Oct 2010

    Google Scholar 

  3. Basel Committee on Banking Supervision: Basel III: A Global Regulatory Framework for More Resilient Banks and Banking Systems. Revised version June 2011

    Google Scholar 

  4. Brockmann, M., Kalkbrener, M.: On the aggregation of risk. J. Risk 12(3), 45–68 (2010)

    Google Scholar 

  5. Burtschell, X., Gregory, J., Laurent, J.P.: A comparative analysis of CDO pricing models under the factor copula framework. J. Deriv. 16(4), 9–37 (2009)

    Article  Google Scholar 

  6. Frey, R., McNeil, A., Nyfeler, M.: Copulas and credit models. Risk 14(10), 111–114 (2001)

    Google Scholar 

  7. Isogai, T.: Benchmarking unconditional VaR and ES calculation methods: a numerical simulation with fitted truncated stable distribution. In: Bank of Japan Working Paper Series No. 14-E-1 (2014)

    Google Scholar 

  8. Joe, H.: Multivariate Models and Dependence Concepts. Chapman & Hall, London (1997)

    Google Scholar 

  9. Jorion, P.: Value at Risk: The New Benchmark for Managing Financial Risk, 3rd edn. McGraw-Hill, New York (2006)

    Google Scholar 

  10. Kawaguchi, M., Yamanaka, S., Tashiro, Y.: Improvement of credit concentration risk evaluation using copulas. Secur. Anal. J. 52(3), 43–51 (2014)

    Google Scholar 

  11. Klaassen, P., van Eeghen, I.: Economic Capital: How It Works and What Every Manager Needs to Know. Elsevier Science, New York (2009)

    Google Scholar 

  12. Kurowicka, D., Joe, H.: Dependence Modeling: Vine Copula Handbook. World Scientific Publishing, Singapore (2010)

    Google Scholar 

  13. Li, D.: On default correlation: a copula function approach. J. Fixed Income 9(4), 43–54 (2000)

    Article  Google Scholar 

  14. McNeil, A.J., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton University Press, Princeton (2005)

    Google Scholar 

  15. Merton, R.C.: On the pricing of corporate debt: the risk structure of interest rates. J. Financ. 29(2), 449–470 (1974)

    Google Scholar 

  16. Nelsen, R.B.: An Introduction to Copulas, 2nd edn. Springer, New York (2006)

    Google Scholar 

  17. Rosenberg, J.V., Schuermann, T.: A general approach to integrated risk management with skewed, fat-tailed risks. J. Financ. Econ. 79(3), 569–614 (2006)

    Article  Google Scholar 

  18. Savu, C., Trede, M.: Hierarchies of Archimedean copulas. Quant. Financ. 10(3), 295–304 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Shintani, K., Yamada, T., Yoshiba, T.: Correlation structures between assets’ returns in financial crisis: evaluation using copulas. Kin’yu-kenkyu 20(3), 89–122 (2010) (in Japanese)

    Google Scholar 

  20. Tsafack, G.: Asymmetric dependence implications for extreme risk management. J. Deriv. 17(1), 7–20 (2009)

    Article  Google Scholar 

  21. Tozaka, H., Yoshiba, T.: Commentary on concrete applications of copula in financial practices. Kin’yu-kenkyu 24(S-2), 115–162 (2005) (in Japanese)

    Google Scholar 

  22. Yoshiba, T.: Risk aggregation by a copula with a stressed condition. In: Bank of Japan Working Paper Series No. 13-E-12 (2013)

    Google Scholar 

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Acknowledgments

For their helpful comments, the author would like to thank Satoshi Yamashita and the participants of Forum “Math-for-Industry” 2014, held in Fukuoka from October 27 to 31, 2014. The views expressed in this paper are those of the author and do not necessarily reflect the official views of the Bank of Japan.

Author information

Authors and Affiliations

  1. Institute for Monetary and Economic Studies, Bank of Japan, Tokyo, Japan

    Toshinao Yoshiba

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  1. Toshinao Yoshiba
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Correspondence to Toshinao Yoshiba .

Editor information

Editors and Affiliations

  1. CSIRO Computational Informatics, Canberra, Aust Capital Terr, Australia

    Robert S. Anderssen

  2. La Trobe University, Melbourne, Victoria, Australia

    Philip Broadbridge

  3. Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan

    Yasuhide Fukumoto

  4. Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan

    Kenji Kajiwara

  5. Kyushu University, Fukuoka, Japan

    Tsuyoshi Takagi

  6. Leiden University, Leiden, The Netherlands

    Evgeny Verbitskiy

  7. Institute of Mathematics for Industry, Kyushu University, Fukuoka, Japan

    Masato Wakayama

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Yoshiba, T. (2016). Risk Aggregation with Copula for Banking Industry. In: Anderssen, R., et al. Applications + Practical Conceptualization + Mathematics = fruitful Innovation. Mathematics for Industry, vol 11. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55342-7_21

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  • DOI: https://doi.org/10.1007/978-4-431-55342-7_21

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  • Publisher Name: Springer, Tokyo

  • Print ISBN: 978-4-431-55341-0

  • Online ISBN: 978-4-431-55342-7

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