Time-varying joint distribution through copulas

The analysis of temporal dependence in multivariate time series is considered. The dependence structure between the marginal series is modelled through the use of copulas which, unlike the correlation matrix, give a complete description of the joint distribution. The parameters of the copula function vary through time, following certain evolution equations depending on their previous values and the historical data. The marginal time series follow standard univariate GARCH models. Full Bayesian inference is developed where the whole set of model parameters is estimated simultaneously. This represents an essential difference from previous approaches in the literature where the marginal and the copula parameters are estimated separately in two consecutive steps. Moreover, a Bayesian procedure is proposed for the estimation of several measures of risk, such as the variance, Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) of a portfolio of assets, providing point estimates and predictive intervals. The proposed copula model enables to capture the dependence structure between the individual assets which strongly influences these risk measures. Finally, the problem of optimal portfolio selection based on the estimation of mean-variance, mean-VaR and mean-CVaR efficient frontiers is also addressed. The proposed approach is illustrated with simulated and real financial time series.

[1]  Dean Fantazzini.  The effects of misspecified marginals and copulas on computing the value at risk: A Monte Carlo study , 2009, Comput. Stat. Data Anal..

[2]  Hedibert Freitas Lopes,et al.  Copula, marginal distributions and model selection: a Bayesian note , 2008, Stat. Comput..

[3]  M. Rockinger,et al.  The Copula-GARCH model of conditional dependencies: An international stock market application , 2006 .

[4]  Andrew J. Patton.  Estimation of multivariate models for time series of possibly different lengths , 2006 .

[5]  Oriol Roch,et al.  Testing the bivariate distribution of daily equity returns using copulas. An application to the Spanish stock market , 2006, Comput. Stat. Data Anal..

[6]  L. Tierney.  Markov Chains for Exploring Posterior Distributions , 1994 .

[7]  Phhilippe Jorion.  Value at Risk: The New Benchmark for Managing Financial Risk , 2000 .

[8]  J. C. Rodríguez,et al.  Measuring financial contagion:a copula approach , 2007 .

[9]  Gerhard Stahl,et al.  Applied Quantitative Finance , 2002 .

[10]  Rüdiger Kiesel,et al.  Sensitivity analysis of credit portfolio models , 2002 .

[11]  Yan Liu,et al.  Efficient estimation of copula-GARCH models , 2009, Comput. Stat. Data Anal..

[12]  Stephen L Taylor,et al.  The Euro and European Financial Market Integration , 2004 .

[13]  L. Bauwens,et al.  Multivariate GARCH Models: A Survey , 2003 .

[14]  G. Pflug,et al.  Value-at-Risk in Portfolio Optimization: Properties and Computational Approach ⁄ , 2005 .

[15]  T. Bollerslev,et al.  A CONDITIONALLY HETEROSKEDASTIC TIME SERIES MODEL FOR SPECULATIVE PRICES AND RATES OF RETURN , 1987 .

[16]  Paul Embrechts,et al.  Dynamic copula models for multivariate high-frequency data in finance , 2004 .

[17]  Luc Bauwens,et al.  Bayesian Inference on GARCH Models Using the Gibbs Sampler , 1998 .

[18]  Ioannis D. Vrontos,et al.  Full Bayesian Inference for GARCH and EGARCH Models , 2000 .

[19]  R. Nelsen.  An Introduction to Copulas , 1998 .

[20]  S. Resnick,et al.  Extreme Value Theory as a Risk Management Tool , 1999 .

[21]  Peter Winker,et al.  THE HIDDEN RISKS OF OPTIMIZING BOND PORTFOLIOS UNDER VAR , 2007 .

[22]  F. Longin,et al.  Extreme Correlation of International Equity Markets , 2000 .

[23]  R. Rockafellar,et al.  Optimization of conditional value-at risk , 2000 .

[24]  Bill Ravens,et al.  An Introduction to Copulas , 2000, Technometrics.

[25]  William T. Shaw.  New methods for simulating the Student T-distribution - Direct use of the inverse cumulative distribution function , 2005 .

[26]  Y. Tse,et al.  A Multivariate Generalized Autoregressive Conditional Heteroscedasticity Model With Time-Varying Correlations , 2002 .

[27]  R. Engle.  Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation , 1982 .

[28]  Robert N. McCauley,et al.  The Euro and European Financial Markets , 1997 .

[29]  María Concepción Ausín,et al.  Bayesian estimation of the Gaussian mixture GARCH model , 2007, Comput. Stat. Data Anal..

[30]  P. Dellaportas,et al.  Contagion tests via copula threshold models , 2005 .

[31]  Andrew J. Patton.  Modelling Asymmetric Exchange Rate Dependence , 2006 .

[32]  P. Embrechts,et al.  Correlation: Pitfalls and Alternatives , 1999 .

[33]  Teruo Nakatsuma,et al.  Bayesian analysis of ARMA–GARCH models: A Markov chain sampling approach , 2000 .

[34]  Söhnke M. Bartram,et al.  The Euro and European Financial Market Dependence , 2007 .

[35]  Anne-Catherine Favre,et al.  Bayesian copula selection , 2006, Comput. Stat. Data Anal..

[36]  Berthold Schweizer,et al.  Probabilistic Metric Spaces , 2011 .

[37]  Ling Hu.  Dependence patterns across financial markets: a mixed copula approach , 2006 .