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.
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References
Abramowitz M., Stegun I. (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. Dover, New York
Akaike H. (1974) A new look at the statistical model identification. IEEE Transactions on Automatic Control 19(6): 716–723
Andersen T., Bollerslev T., Diebold F., Ebens H. (2001) The distribution of realized stock return volatility. Journal of Financial Econometrics 61: 43–76
Angel Canela, M., & Pedreira Collazo, E. (2006). Modelling dependence in Latin American markets using copula functions. Working paper.
Barndorff-Nielsen O. (1977) Exponentially decreasing distributions for the logarithm of particle size. Proceedings of the Royal Society London A 353: 401–419
Barndorff-Nielsen, O. (1995). Normal-inverse gaussian processes and the modelling of stock returns. Technical report, University of Aarhus.
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.
Box G., Pierce D. (1970) Distribution of residual autocorrelations in autoregressive integrated moving average time series models. Journal of the American Statistical Association 65: 1509–1526
Breymann W., Dias A., Embrechts P. (2003) Dependence structures for multivariate high-frequency data in finance. Quantitative Finance 3: 1–14
Dias, A. (2004). Copula inference for finance and insurance. Doctoral thesis.
Dias, A., & Embrechts, P. (2008). Modelling exchange rate dependence at different time horizons. Working paper.
Eberlein E., Keller U. (1995) Hyperbolic distributions in finance. Journal of Business 1: 281–299
Embrechts, P., Lindskog, F., & McNeil, A. (2001a). Modelling dependence with copulas and applications to risk management. Working paper, ETH Zürich.
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, Cambridge
Fang H., Fang K., Kotz S. (2002) The meta-elliptical distributions with given marginals. Journal of Multivariate Analysis 82(1): 1–16
Föllmer, H. & Schied A. (2004). Stochastic finance: An introduction in discrete time. Berlin: de Gruyter.
Frahm G., Junker M., Szimayer A. (2003) Elliptical copulas: Applicability and limitations. Statistics and Probability Letters 63: 275–286
Genest C., Rivest L. (2002) Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association 88(423): 1034–1043
Granger C. (2003) Time series concept for conditional distributions. Oxford Bulletin of Economics and Statistics 65: 689–701
Hosking J. (1980) The multivariate portmanteau statistic. Journal of the American Statistical Association 75: 602–608
Hosking J. (1981) Lagrange multiplier tests of multivariate time series models. Journal of the Royal statistical society series B 43(2): 219–230
Hu L. (2006) Dependence patterns across financial markets: a mixed copula approach. Applied Financial Economics 16: 717–729
Hult, H., & Lindskog, F. (2001). Multivariate extremes, aggregation and dependence in elliptical distributions. Working paper, Risklab.
Hurst S., Platen E. (1997) The marginal distributions of returns and volatility. IMS Lecture Notes—Monograph Series. Hayward, CA: Institute of Mathematical Statistics 31: 301–314
Joe H. (1993) Parametric families of multivariate distributions with given margins. Journal of Multivariate Analysis 46(2): 262–282
Joe H. (1997) Multivariate models and dependence concepts. Chapman & Hall, London
Jørgensen B. (1982) Statistical properties of the generalized inverse Gaussian distribution, Lecture Notes in Statistics. Springer, New York
Lee T., Platen E. (2006) Approximating the growth optimal portfolio with a diversified world stock index. Journal of Risk Finance 7(5): 559–574
Lindskog, F., McNeil, A., & Schmock, U. (2001). Kendall’s tau for elliptical distributions. Working paper, Risklab.
Ljung G., Box G. (1978) On a measure of lack of fit in time series models. Biometrika 66: 66–72
Madan D., Seneta E. (1990) The variance gamma model for share market returns. Journal of Business 63: 511–524
McLeish D. L., Small C. G. (1988) The theory and applications of statistical inference functions. Lecture Notes in Statistics. Springer, New York
McNeil, A., Frey, R., & Embrechts, P. (2005). Quantitative risk management: Concepts, techniques, and tools. Princeton Series in Finance.
Morgan/Reuters (1996). Riskmetrics technical document.
Nelsen R. (1998) An introduction to Copulas. Springer, New York
Platen E., Heath D. (2006) A benchmark approach to quantitative finance. Springer, New York
Platen E., Rendek R. (2008) Empirical evidence on Student-t log-returns of diversified world stock indices. Journal of Statistical Theory and Practice 2: 233–251
Praetz P. D. (1972) The distribution of share price changes. Journal of Business 45: 49–55
Rachev S., Han S. (2000) Portfolio management with stable distributions. Mathematical Methods of Operations Research 51: 341–352
Rachev S., Mittnik S. (2000) Stable paretian models in finance. Wiley, New York
Solnik B., Boucrelle C., Le Y. (1996) International market correlation and volatility. Financial Analysts Journal 52(5): 17–34
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.
Wang, S. (1997). Aggregation of correlated risk portfolios. Preprint, Casuality Actuarial Society (CAS).
Wenbo, H., & Kercheval, A. (2008). Risk management with generalized hyperbolic distributions. Working paper.
Wu, F., Valdez, A., & Sherris, M. (2006). Simulating exchangeable multivariate archimedeancopulas and its applications. Working paper.
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Ignatieva, K., Platen, E. Modelling Co-movements and Tail Dependency in the International Stock Market via Copulae. Asia-Pac Financ Markets 17, 261–302 (2010). https://doi.org/10.1007/s10690-010-9116-2
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DOI: https://doi.org/10.1007/s10690-010-9116-2