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Asia-Pacific Financial Markets

, Volume 17, Issue 3, pp 261–302 | Cite as

Modelling Co-movements and Tail Dependency in the International Stock Market via Copulae

  • Katja Ignatieva
  • Eckhard PlatenEmail author
Article

Abstract

This paper examines international equity market co-movements using time-varying copulae. We examine distributions from the class of Symmetric Generalized Hyperbolic (SGH) distributions for modelling univariate marginals of equity index returns. We show based on the goodness-of-fit testing that the SGH class outperforms the normal distribution, and that the Student-t assumption on marginals leads to the best performance, and thus, can be used to fit multivariate copula for the joint distribution of equity index returns. We show in our study that the Student-t copula is not only superior to the Gaussian copula, where the dependence structure relates to the multivariate normal distribution, but also outperforms some alternative mixture copula models which allow to reflect asymmetric dependencies in the tails of the distribution. The Student-t copula with Student-t marginals allows to model realistically simultaneous co-movements and to capture tail dependency in the equity index returns. From the point of view of risk management, it is a good candidate for modelling the returns arising in an international equity index portfolio where the extreme losses are known to have a tendency to occur simultaneously. We apply copulae to the estimation of the Value-at-Risk and the Expected Shortfall, and show that the Student-t copula with Student-t marginals is superior to the alternative copula models investigated, as well the Riskmetics approach.

Keywords

International equity market Student-t distribution Symmetric generalized hyperbolic distribution Time-varying copula Value-at-risk World stock index 

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References

  1. Abramowitz M., Stegun I. (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New YorkGoogle Scholar
  2. Akaike H. (1974) A new look at the statistical model identification. IEEE Transactions on Automatic Control 19(6): 716–723CrossRefGoogle Scholar
  3. Andersen T., Bollerslev T., Diebold F., Ebens H. (2001) The distribution of realized stock return volatility. Journal of Financial Econometrics 61: 43–76CrossRefGoogle Scholar
  4. Angel Canela, M., & Pedreira Collazo, E. (2006). Modelling dependence in Latin American markets using copula functions. Working paper.Google Scholar
  5. Barndorff-Nielsen O. (1977) Exponentially decreasing distributions for the logarithm of particle size. Proceedings of the Royal Society London A 353: 401–419CrossRefGoogle Scholar
  6. Barndorff-Nielsen, O. (1995). Normal-inverse gaussian processes and the modelling of stock returns. Technical report, University of Aarhus.Google Scholar
  7. Barndorff-Nielsen, O., & Stelzer, R. (2004). Absolute moments of generalized hyperbolic distributions and approximate scaling of normal inverse gaussian Levy-processes. Working paper, University of Aarhus, Denmark.Google Scholar
  8. Box G., Pierce D. (1970) Distribution of residual autocorrelations in autoregressive integrated moving average time series models. Journal of the American Statistical Association 65: 1509–1526CrossRefGoogle Scholar
  9. Breymann W., Dias A., Embrechts P. (2003) Dependence structures for multivariate high-frequency data in finance. Quantitative Finance 3: 1–14CrossRefGoogle Scholar
  10. Dias, A. (2004). Copula inference for finance and insurance. Doctoral thesis.Google Scholar
  11. Dias, A., & Embrechts, P. (2008). Modelling exchange rate dependence at different time horizons. Working paper.Google Scholar
  12. Eberlein E., Keller U. (1995) Hyperbolic distributions in finance. Journal of Business 1: 281–299Google Scholar
  13. Embrechts, P., Lindskog, F., & McNeil, A. (2001a). Modelling dependence with copulas and applications to risk management. Working paper, ETH Zürich.Google Scholar
  14. Embrechts P., McNeil A., Straumann D. (2001b) Correlation and dependency in risk management: Properties and pitfalls. In: Press U. (eds) Risk management: Value at risk and beyond. M. Dempster and H. Moffatt, CambridgeGoogle Scholar
  15. Fang H., Fang K., Kotz S. (2002) The meta-elliptical distributions with given marginals. Journal of Multivariate Analysis 82(1): 1–16CrossRefGoogle Scholar
  16. Föllmer, H. & Schied A. (2004). Stochastic finance: An introduction in discrete time. Berlin: de Gruyter.Google Scholar
  17. Frahm G., Junker M., Szimayer A. (2003) Elliptical copulas: Applicability and limitations. Statistics and Probability Letters 63: 275–286CrossRefGoogle Scholar
  18. Genest C., Rivest L. (2002) Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association 88(423): 1034–1043CrossRefGoogle Scholar
  19. Granger C. (2003) Time series concept for conditional distributions. Oxford Bulletin of Economics and Statistics 65: 689–701CrossRefGoogle Scholar
  20. Hosking J. (1980) The multivariate portmanteau statistic. Journal of the American Statistical Association 75: 602–608CrossRefGoogle Scholar
  21. Hosking J. (1981) Lagrange multiplier tests of multivariate time series models. Journal of the Royal statistical society series B 43(2): 219–230Google Scholar
  22. Hu L. (2006) Dependence patterns across financial markets: a mixed copula approach. Applied Financial Economics 16: 717–729CrossRefGoogle Scholar
  23. Hult, H., & Lindskog, F. (2001). Multivariate extremes, aggregation and dependence in elliptical distributions. Working paper, Risklab.Google Scholar
  24. Hurst S., Platen E. (1997) The marginal distributions of returns and volatility. IMS Lecture Notes—Monograph Series. Hayward, CA: Institute of Mathematical Statistics 31: 301–314CrossRefGoogle Scholar
  25. Joe H. (1993) Parametric families of multivariate distributions with given margins. Journal of Multivariate Analysis 46(2): 262–282CrossRefGoogle Scholar
  26. Joe H. (1997) Multivariate models and dependence concepts. Chapman & Hall, LondonGoogle Scholar
  27. Jørgensen B. (1982) Statistical properties of the generalized inverse Gaussian distribution, Lecture Notes in Statistics. Springer, New YorkGoogle Scholar
  28. Lee T., Platen E. (2006) Approximating the growth optimal portfolio with a diversified world stock index. Journal of Risk Finance 7(5): 559–574CrossRefGoogle Scholar
  29. Lindskog, F., McNeil, A., & Schmock, U. (2001). Kendall’s tau for elliptical distributions. Working paper, Risklab.Google Scholar
  30. Ljung G., Box G. (1978) On a measure of lack of fit in time series models. Biometrika 66: 66–72Google Scholar
  31. Madan D., Seneta E. (1990) The variance gamma model for share market returns. Journal of Business 63: 511–524CrossRefGoogle Scholar
  32. McLeish D. L., Small C. G. (1988) The theory and applications of statistical inference functions. Lecture Notes in Statistics. Springer, New YorkGoogle Scholar
  33. McNeil, A., Frey, R., & Embrechts, P. (2005). Quantitative risk management: Concepts, techniques, and tools. Princeton Series in Finance.Google Scholar
  34. Morgan/Reuters (1996). Riskmetrics technical document.Google Scholar
  35. Nelsen R. (1998) An introduction to Copulas. Springer, New YorkGoogle Scholar
  36. Platen E., Heath D. (2006) A benchmark approach to quantitative finance. Springer, New YorkCrossRefGoogle Scholar
  37. Platen E., Rendek R. (2008) Empirical evidence on Student-t log-returns of diversified world stock indices. Journal of Statistical Theory and Practice 2: 233–251Google Scholar
  38. Praetz P. D. (1972) The distribution of share price changes. Journal of Business 45: 49–55CrossRefGoogle Scholar
  39. Rachev S., Han S. (2000) Portfolio management with stable distributions. Mathematical Methods of Operations Research 51: 341–352CrossRefGoogle Scholar
  40. Rachev S., Mittnik S. (2000) Stable paretian models in finance. Wiley, New YorkGoogle Scholar
  41. Solnik B., Boucrelle C., Le Y. (1996) International market correlation and volatility. Financial Analysts Journal 52(5): 17–34CrossRefGoogle Scholar
  42. Sun, W., Rachev, S., Fabobozzi, F., & Petko, S. (2006). Unconditional copula-based simulation of tail dependence for co-movement of international equity markets. Working paper.Google Scholar
  43. Wang, S. (1997). Aggregation of correlated risk portfolios. Preprint, Casuality Actuarial Society (CAS).Google Scholar
  44. Wenbo, H., & Kercheval, A. (2008). Risk management with generalized hyperbolic distributions. Working paper.Google Scholar
  45. Wu, F., Valdez, A., & Sherris, M. (2006). Simulating exchangeable multivariate archimedeancopulas and its applications. Working paper.Google Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Graduate Program “Finance and Monetary Economics,”Goethe University, House of FinanceFrankfurt am MainGermany
  2. 2.School of Finance and Economics and Department of Mathematical SciencesUniversity of Technology SydneySydneyAustralia

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