Copula-GARCH versus dynamic conditional correlation: an empirical study on VaR and ES forecasting accuracy
Tóm tắt
In this paper, we analyze the accuracy of the copula-GARCH and Dynamic Conditional Correlation (DCC) models for forecasting the value-at-risk (VaR) and expected shortfall (ES) of bivariate portfolios. We then try to answer two questions: First, does the correlation-based DCC model outperform the copula models? Second, how can the optimal model for forecasting portfolio risk be identified via in-sample analysis? We address these questions using an extensive empirical study of 1,500 bivariate portfolios containing data on stocks, commodities and foreign exchange futures. Furthermore, we propose to use linear discriminant analysis estimated from descriptive statistics on bivariate data samples as independent variables to identify a parametric model yielding optimal portfolio VaR and ES estimates. In particular, we try to answer the question whether the quality of a parametric model’s VaR and ES estimates is driven by common data characteristics. The results show that the proposed use of linear discriminant analysis is superior to both the Kullback-Leibler Information Criterion and several copula goodness-of-fit tests in terms of overall classification accuracy. Furthermore, the results show that the quality of the DCC model’s VaR and ES estimates is positively correlated with the portfolio marginals’ volatility, while the opposite is true for the elliptical copulas. For the Archimedean copulas in particular, the excess kurtosis of the marginals has a significant positive influence on quality of the VaR and ES estimates.
Tài liệu tham khảo
Aas K, Berg D (2009) Models for construction of multivariate dependence—a comparison study. Eur J Financ 15:639–659
Aas K, Czado C, Frigessi A, Bakken H (2009) Pair-copula constructions of multiple dependence. Insur Math Econ 44:182–198
Alexander C, Sheedy E (2008) Developing a stress testing framework based on market risk models. J Bank Financ 32:2220–2236
Ausin MC, Lopes HF (2010) Time-varying joint distribution through copulas. Comput Stat Data An 54:2383–2399
Bartram S, Taylor S, Wang Y-H (2007) The Euro and European financial market dependence. J Bank Financ 51:1461–1481
Berg D (2009) Copula goodness-of-fit testing: an overview and power comparison. Eur J Financ 15:675–701
Breymann W, Dias A, Embrechts P (2003) Dependence structures for multivariate high-frequency data in finance. Quant Financ 3:1–14
Cappiello L, Engle RF, Sheppard K (2006) Asymmetric dynamics in the correlations of global equity and bond returns. J Financ Economet 4:537–572
Chen X, Fan Y (2006) Estimation and model selection of semiparametric copula-based multivariate dynamic models under copula misspecification. J Econom 135:125–154
Chen S, Poon S-H (2011) Modeling international financial markets contagion: using copula and risk appetite. In: Robert K (eds) Financial contagion: the viral threat to the wealth of nations, 1st edn. Wiley, New York, pp 45–56
Christoffersen P (1998) Evaluating interval forecasts. Int Econ Rev 39:841–862
Christoffersen P, Pelletier D (2004) Backtesting value-at-risk: a duration-based approach. J Financ Economet 2:84–108
Deheuvels P (1979) La fonction de dépendance empirique et ses propriétés—un test non paramétrique d’indépendance. Académie Royale de Belgique Bulletin de la Classe des Sciences 5e Série 65:274–292
Deheuvels P (1981) A nonparametric test for independence. Université de Paris, Institut de Statistique, Paris
Di Clemente A, Romano C (2005) Measuring portfolio value-at-risk by a copula-EVT-based approach. Studi Economici 85:29–57
Diks C, Panchenko V, van Dijk D (2010) Out of sample comparison of copula specifications in multivariate density forecasts. J Econ Dyn Control 34:1596–1609
Embrechts P, McNeil A, Straumann D (2002) Correlation and dependence in risk management: properties and pitfalls. http://www.math.ethz.ch/~strauman/preprints/pitfalls.pdf. Accessed 24 April 2012
Engle RF (2002) Dynamic conditional correlation: a simple class of multivariate generalized autoregressive conditional heteroskedasticity models. J Bus Econ Statist 20:339–350
Engle RF, Kelly B (2008) Dynamic equicorrelation. NYU Stern School of Business. http://pages.stern.nyu.edu/~rengle/Dynamic%20Equicorrelation.pdf. Accessed 24 April 2012
Fantazzini D (2008) Dynamic copula modelling for value at risk. Front Financ Econ 31:161–180
Fantazzini D (2009) The effects of misspecified marginals and copulas on computing the value at risk: a Monte Carlo study. Comput Stat Data An 53:2168–2188
Fantazzini D (2009) Market risk management for emerging markets: evidence from the Russian stock market. In: Gregoriou G (ed) Emerging markets: performance, analysis and innovation. Chapman and Hall/CRC Finance, London, pp 533–554
Fantazzini D (2009) Value at risk for high-dimensional portfolios: a dynamic grouped-T copula approach. In: Gregoriou G (ed) The var implementation handbook. McGraw-Hill, New York, pp 253–282
Fischer M, Köck C, Schlüter S, Weigert F (2009) An empirical analysis of multivariate copula models. Quant Financ 7:839–854
Genest C, Rémillard B, Beaudoin D (2009) Goodness-of-fit tests for copulas: a review and a power study. Insur Math Econ 44:199–213
Genest C, Rémillard B (2008) Validity of the parametric bootstrap for goodness-of-fit testing in semiparametric models. Ann de l’Inst Henri-Poincaré 44:1096–1127
Genest C, Quessy J-F, Rémillard B (2006) Goodness-of-fit procedures for copula models based on the probability integral transform. Scand J Stat 33:337–366
Genest C, Rivest L-P (1993) Statistical inference procedures for bivariate archimedean copulas. J Am Stat Assoc 88:1034–1043
Hafner CM, Reznikova O (2010) Efficient estimation of a semiparametric dynamic copula model. Comput Stat Data An 54:2609–2627
Hansen PR, Lunde A (2005) A forecast comparison of volatility models: does anything beat a GARCH(1,1)?. J Appl Econom 20:873–889
Hobæk Haff I, Aas K, Frigessi A (2010) On the simplified pair-copula construction—simply useful or too simplistic? J Multivar Anal 101:1296–1310
Hoeffding W (1940) Scale invariant correlation theory. Schriften Math Inst Univ Berlin 5:181–233
Hsu CP, Huang CW, Chiou WJP (2011) Effectiveness of copula-extreme value theory in estimating value-at-risk: empirical evidence from Asian emerging markets. Rev Quant Financ Acc. doi:10.1007/s11156-011-0261-0
Huang YC, Lin BJ (2004) Value-at-risk analysis for Taiwan stock index futures: fat tails and conditional asymmetries in return innovations. Rev Quant Financ Acc 22:79–95
Joe H (1997) Multivariate models and dependence concepts. Chapman & Hall/CRC, London
Jondeau E, Rockinger M (2006) The Copula-GARCH model of conditional dependencies: an international stock market application. J Int Money Financ 25:827–853
Junker M, May A (2005) Measurement of aggregate risk with copulas. Economet J 8:428–454
Kim G, Silvapulle M, Silvapulle P (2007) Comparison of semiparametric and parametric methods for estimating copulas. Comput Stat Data An 51:2836–2850
Kole E, Koedijk KCG, Verbeek M (2007) Selecting copulas for risk management. J Bank Financ 31:405–2423
Li DX (2000) On default correlation: a copula function approach. J Fixed Income 9:43–54
Lin CH, Chien CC, Chen S (2006) Incorporating the time-varying tail-fatness into the historical simulation method for portfolio value-at-risk. Rev Pac Basin Financ Mark Policies 9:257–274
Liu Y, Luger R (2009) Efficient estimation of copula-GARCH models. Comput Stat Data An 53:2284–2297
Lu C, Tse Y, Williams M (2012) Returns transmission, value at risk, and diversification benefits in international REITs: evidence from the financial crisis. Rev Quant Finan Acc. doi:10.1007/s11156-012-0274-3
Malevergne Y, Sornette D (2003) Testing the gaussian copula hypothesis for financial assets dependencies. Quant Financ 3:231–250
Manner H, Reznikova O (2012) A survey on time-varying copulas: specification, simulations and estimation. Economet Rev 31:654–687
Markwat T, Kole E, van Dijk D (2009) Time variation in asset return dependence: strength or structure? Erasmus University Rotterdam. http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1460648. Accessed 24 April 2012
Marsh T, Pfleiderer P (2012) “Black Swans” and the financial crisis. Rev Pac Basin Financ Mark Policies. doi:10.1142/s0219091512500087
McNeil A, Frey R, Embrechts P (2005) Quantitative risk management. Princeton University Press, Princeton
Nelsen RB (2006) An introduction to copulas. Lect Notes Stat, 2nd edn, vol 139. Springer, New York
Nikoloulopoulos AK, Joe H, Li H (2010) Vine copulas with asymmetric tail dependence and applications to financial return data. Comput Stat Data An. doi:10.1016/j.csda.2010.07.016
Palaro HP, Hotta LK (2006) Using conditional copula to estimate value at risk. J Data Sci 4:93–115
Patton A (2006) Modelling asymmetric exchange rate dependence. Int Economic Rev 47:527–556
Pritsker M (2006) The hidden dangers of historical simulation. J Bank Financ 30:561–582
Q (2011) Are copula-GoF-tests of any practical use? Empirical evidence for stocks, commodities and FX futures. Q Rev Econ Financ 51:173–188
Rosenblatt M (1952) Remarks on a multivariate transformation. Ann Math Stat 23:470–472
Savu C, Trede M (2008) Goodness-of-fit tests for parametric families of archimedean copulas. Quant Financ 8:109–116
Sklar A (1959) Fonctions de répartition à n dimensions et leurs marges. Publications de l’Institut de Statstique de l’Université de Paris 8:229–231
Vuong Q (1989) Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica 57:307–333
Wong WK (2008) Backtesting trading risk of commercial banks using expected shortfall. J Bank Financ 32:1404–1415