×

Testing the Gaussian copula hypothesis for financial assets dependences. (English) Zbl 1408.62177

Summary: Using one of the key properties of copulas that they remain invariant under an arbitrary monotonic change of variable, we investigate the null hypothesis that the dependence between financial assets can be modelled by the Gaussian copula. We find that most pairs of currencies and pairs of major stocks are compatible with the Gaussian copula hypothesis, while this hypothesis can be rejected for the dependence between pairs of commodities (metals). Notwithstanding the apparent qualification of the Gaussian copula hypothesis for most of the currencies and the stocks, a non-Gaussian copula, such as the Student copula, cannot be rejected if it has sufficiently many ‘degrees of freedom’. As a consequence, it may be very dangerous to embrace blindly the Gaussian copula hypothesis, especially when the coefficient of correlation between the pairs of assets is too high, such that the tail dependence neglected by the Gaussian copula can became large, leading to the ignoring of extreme events which may occur in unison.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62H05 Characterization and structure theory for multivariate probability distributions; copulas
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Andersen, J V and Sornette, D. 2001. Have your cake and eat it too: increasing returns while lowering large risks!. J. Risk Finance, 2: 70-82.
[2] Anderson, T W and Darling, D A. 1952. Asymptotic theory of certain ‘goodness of fit’ criteria. Ann. Math. Statistics, 23: 193-212. · Zbl 0048.11301
[3] Ang, A and Chen, J. 2002. Asymmetric correlations of equity portfolio. J. Financial Econ., 63: 443-94.
[4] Baig, T and Goldfajn, I. 1998. Financial markets contagion in the Asian crisis. Mimeo, International Monetary Fund,
[5] Basle Committee on Supervision Banking. 2001. The New Basle Accord,
[6] Breymann, W, Dias, A and Embrechts, P. 2003. Dependence structures for multivariate high-frequency data in finance. Quant. Finance, 3: 1-14. · Zbl 1408.62173
[7] Chen, K and Lo, S-H. 1997. On a mapping approach to investigating the bootstrap accuracy. Probab. Theory Relat. Fields, 107: 197-217. · Zbl 0868.60033
[8] Cherubini, U and Luciano, E. 2000. Bivariate option pricing with copulas. Appl. Math. Finance, 9: 69-86. · Zbl 1013.91050
[9] de Vries, C G. 1994. “Stylized facts of nominal exchange rate returns”. In The Handbook of International Macroeconomics, Edited by: Van der Ploeg, F. Oxford: Blackwell.
[10] Efron, B and Tibshirani, R. 1986. Bootstrap method for standard errors, Confidence intervals and other measures of statistical accuracy. Stat. Sci., 1: 54-77. · Zbl 0587.62082
[11] Embrechts, P, Hoeing, A and Juri, A. 2003. Using copulae to bound the value-at-risk for functions of dependent risk. Finance Stochastics, 7: 145-67. · Zbl 1039.91023
[12] Embrechts, P, McNeil, A J and Straumann, D. 1999. Correlation: pitfalls and alternatives. Risk, 12: 69-71.
[13] Embrechts, P, McNeil, A J and Straumann, D. 2001. “Correlation and dependency in risk management: properties and pitfalls”. In Value at Risk and Beyond, Edited by: Dempster, M. Cambridge: Cambridge University Press.
[14] Frees, E and Valdez, E. 1998. Understanding relationships using copula. North Am. Actuarial, J., 2: 1-25. · Zbl 1081.62564
[15] Genest, C. 1987. Frank’s family of bivariate distributions. Biometrika, 74: 549-55. · Zbl 0635.62038
[16] Genest, C and MacKay, R. 1986. The joy of copulas. Am. Stat., 40: 280-3.
[17] Genest, C and Rivest, J P. 1993. Statistical inference procedures for bivariate Archimedean copulas. J. Am. Stat. Assoc., 88: 1034-43. · Zbl 0785.62032
[18] Gopikrishnan, P, Meyer, M, Amaral, L A N and Stanley, H E. 1998. Inverse cubic law for the distribution of stock price variation. Eur. Phys. J.B, 3: 139-40.
[19] Gouriéroux, C and Jasiak, J. 1999. Truncated local likelihood and non-parametric tail analysis (unpublished work).
[20] Guillaume, D M, Dacorogna, M M, Davé, R R, Muller, J A, Olsen, R B and Pictet, O V. 1997. From the bird eye to the microscope: a survey of the new stylized facts of the intra-daily foreign exchange markets. Finance Stoch., 1: 95-129. · Zbl 0889.90021
[21] Haas, C N. 1999. On modeling correlated random variables in risk assessment. Risk Anal., 19: 1205-14.
[22] Joe, H. 1993. Parametric families of multivariate distributions with given marginals. J. Multivariate Anal., 46: 262-82. · Zbl 0778.62045
[23] Kaminsky, G L and Schmukler, S L. 1999. What triggers market jitters? A chronicle of the Asian crisis. J. Int. Money Finance, 18: 537-60.
[24] Klugman, S A and Parsa, R. 1999. Fitting bivariate loss distribution with copulas. Insur. Math. Econ., 24: 139-48. · Zbl 0931.62044
[25] KMV-Corporation. 1997. Modelling default risk. Technical Document,
[26] Laherrère, J and Sornette, D. 1999. Stretched exponential distributions in nature and economy: ‘fat tails’ with characteristic scales. Eur. Phys. J.B, 2: 525-39.
[27] Li, D X. 1999. The valuation of basket credit derivative. Credit Metrics Monitor, : 34-50. (April)
[28] Li, D X. 2000. On default correlation: a copula function approach. J. Fixed Income, 9: 43-54.
[29] Lindskog, F. 1999. Modelling dependence with copulas (chapter of forthcoming book). http://www.risklab.ch/Papers.html#MTLindskog
[30] Longin, F and Solnik, B. 2001. Extreme correlation of international equity markets. J. Finance, 56: 649-76.
[31] Lux, L. 1996. The stable paretian hypothesis and the frequency of large returns: an examination of major German stocks. Appl. Financial Econ., 6: 463-75.
[32] Malevergne, Y, Pisarenko, V and Sornette, D. 2003. “Empirical distributions of log-returns: between the stretched-exponential and the power law? (working paper)”. In Preprint physics/0305089 · Zbl 1134.91551
[33] Malevergne, Y and Sornette, D. 2002a. Minimizing extremes. Risk, 15(November): 129-32.
[34] Malevergne, Y and Sornette, D. 2002b. Tail dependence of factor models (working paper). · Zbl 1093.62098
[35] Markovitz, H. 1959. Portfolio Selection: Efficient Diversification of Investments, New York: Wiley.
[36] Mashal, R and Zeevi, A J. 2002. “Beyond correlation: extreme co-movements between financial assets”. Columbia Business School working paper. http://www.columbia.edu/∼rm586/
[37] McNeil, A and Frey, R. 2000. Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. J. Empirical Finance, 7: 271-300.
[38] Muzy, J F, Delour, J and Bacry, E. 2000. Modelling fluctuations of financial time series: from cascade process to stochastic volatility model. Eur. Phys. J.B, 17: 537-48.
[39] Muzy, J F, Sornette, D, Delour, J and Arnéodo, A. 2001. Multifractal returns and hierarchical portfolio theory. Quant. Finance, 1: 131-48. · Zbl 1405.91564
[40] Nelsen, R B. 1998. An Introduction to Copulas (Springer Lecture Notes in Statistics vol 139), New York: Springer. · Zbl 1152.62030
[41] Pagan, A. 1996. The econometrics of financial markets. J. Empirical Finance, 3: 15-102.
[42] Patton, A. 2001. “Estimation of copula models for time series of possibly different lengths (working paper)”. 01-17. University of Calfornia, Discussion Paper.
[43] RiskMetrics-Group. 1997. CreditMetrics. Technical Document,
[44] Rosenberg, J V. 1999. Semiparametric pricing of multivariate contingent claims (working paper).
[45] Sornette, D, Andersen, J V and Simonetti, P. 2000a. Portfolio theory for ‘fat tails’. Int. J. Theor. Appl. Finance, 3: 523-35. · Zbl 0973.91046
[46] Sornette, D, Malevergne, Y and Muzy, J F. 2003. What causes crashes?. Risk, 16(February): 67-71.
[47] Sornette, D, Simonetti, P and Andersen, J V. 2000b. phi^q-field theory for portfolio optimization: ‘fat tails’ and non-linear correlations. Phys. Rep., 335: 19-92.
[48] Starica, C. 1999. Multivariate extremes for models with constant conditional correlations. J. Empirical Finance, 6: 515-53.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.