شواهد گوناگون از بازارهای سرمایه بین المللی بر اساس مقادیر افراطی و تصادفی

Tail dependence plays an important role in financial risk management and determination of whether two markets crash or boom together. However, the linear correlation is unable to capture the dependence structure among financial data. Moreover, given the reality of fat-tail or skewed distribution of financial data, normality assumption for risk measure may be misleading in portfolio development. This paper proposes the use of conditional extreme value theory and time-varying copula to capture the tail dependence between the Australian financial market and other selected international stock markets. Conditional extreme value theory enables the model adequacy and the tail behavior of individual financial variable, while the time-varying copula can fully disclose the changes of dependence structure over time. The combination of both proved to be useful in determining the tail dependence. The empirical results show an outperformance of the model in the analysis of tail dependence, which has an important implication in cross-market diversification and asset pricing allocation.

The international financial markets have become closely integrated since regulations and barriers have been gradually removed over the past years. This provides investors an opportunity to optimize portfolios by higher returns and lower risk. One of the problems of measuring the risk of a portfolio is to model joint distribution of individual risk factor returns affecting portfolio during a specific period. The joint probability distribution plays a key part in determining dependency among variables. The normal distribution has long dominated the multivariate analysis of statistical modeling in the area of financial risk management. The joint normal distribution was usually assumed in risk analysis because the association between any two random variables can be fully described given the normal marginal distributions and the correlation coefficient which measures the dependence between variables. However, due to the existence of skewness and kurtosis in financial data, the heavy tails abnormality has been found in financial time series data (Dacorogna and Picctet, 1997, Danielsson and Vries, 1997 and Danielsson and Vries, 2000). Zivot and Wang (2006) explored the problem of abnormality in BMW and Siemens returns showing that normal distribution cannot capture tail dependence. The alternative multivariate distributions have been investigated in response to the fat-tail problem which might not fully capture dependence structure which expresses the co-movement between variables. The answer to the problem resorts to Sklar (1959) copula theorem to separate marginal effects from dependence structure. Using the copula model has several advantages in modeling dependence compared to using the joint distribution directly. First, copula functions are very flexible in modeling dependence since they allow to separately model marginals and the corresponding dependence structure. Second, copula functions enable us to directly model tail dependence; and they signify not only the degree but also the structure of the dependence. Third, copula functions are invariant under transformations of the data while the linear correlation is not. Therefore, its flexibility in model specification enable the use of any specified marginal distributions to join with any available copulas, thus creating complex non-normal distributions which can fully capture dependency. Importantly, some copulas can help model tail dependence or co-movements among financial variables. Patton (2006) has extended copula theory to time-varying conditional copula theory, which allow copula parameters to vary over time in terms of evolution equations. This model proves to be informative in analyzing the dynamics of dependence structure, providing better understanding of the fluctuation of dependence over time. Modeling dependence structure using copula is introduced by a number of authors, including Fantazzini (2007), He (2003), Mendes and Souza (2004), Canela and Collazo (2005), Giacomini and Hardle (2005), Junker and May (2005), Hu (2006), Ane and Labidi (2005), Rosenberg and Schuermann (2006), Ozun and Cifter (2007), Rodriguez (2007), among others. The marginal distributions are modeled separately and then combine with a copula function, which is a multivariate distribution with uniform marginals. According to Nelsen (2006), there is a unique copula representing the dependence structure for every multivariate distribution with continuous marginals. Therefore, modeling marginal distributions before joining them with copula is also an important step since adequate univariate distributions would bring better results for dependence structure after using copula functions. There are many methods modeling univariate distributions in statistics literature, in which extreme value distributions have been used most in risk analysis because they are based on sound statistical theory and offer parametric form for the tail of a distribution (McNeil and Frey, 2000). With interest of modeling tail dependence for risk management, the paper adopts an approach of combining conditional extreme value theory (C-EVT) and copula theory to capture the dependence structure between financial variables. C-EVT is based on the idea that the conditional return distribution can be easily constructed from the estimated distribution of the residuals and estimates of conditional mean and volatility (GARCH). As results in McNeil and Frey (2000) and Ghorbel and Trabelsi (2007), using C-EVT-based methods will reflect two stylized facts exhibited by most financial return series, namely stochastic volatility and the fat-tailedness of conditional return distributions. The dependence structure, then, will be figured out after joining C-EVT marginals with any available copulas. Recently, He (2003) used EVT-copula model to capture dependence between equities and found that the copula-base model outperforms others in risk analysis, especially Value-at-risk (VaR) computation. Clemente and Romano (2003) used copula-EVT approach to model operational risk with insurance loss data. They found that the model effectively captures the right tail of the distribution which further implies the risk measures, avoiding every possible underestimate of risk. When it comes to copula applications in financial market, there are also several papers using copula to capture tail dependence. Hu (2006) used mixed copula model to find the dependence between some markets such as the US, UK, Japan, Hong Kong and concluded that the model perform better in risk analysis and asset pricing. Palaro and Hotta (2006) used the copula model to figure out the dependence structure between NASDAQ and S&P 500 stock indexes, and also give the same conclusion for the effectiveness of copula in portfolio management. Considering the usefulness of C-EVT and time varying copula functions into account, this paper examines the dependence structure across financial markets, including Australia, the US, UK, Hong Kong, Taiwan, and Japan. The proposed combination brings an efficient performance of risk analysis and also portfolio management. To the best of our knowledge, this is the first attempt to apply conditional EVT and time-varying copula model in determining the dependence structure across financial markets. The result from empirical study demonstrates an out-performance of the model in the analysis of tail dependence which has important implications in cross-market diversifications, asset pricing and allocation. The structure of the rest of this paper is as follows. In the next section, we briefly review the basic concepts of extreme value and copula theories. In Section 3, the model and estimation method is introduced. The Gaussian copula and Symmetrized Joe-Clayton (SJC) copula is also presented in this section. Then, Section 4 presents and analyzes the empirical results of estimated marginal distributions, copula parameters and dependence structure among the studied financial markets. Section 4 contains some concluding remarks and suggests directions for further research.

In this paper, we used C-EVT and TVC to examine the dependence structure across financial markets, including Australia, the US, UK, Japan, Hong Kong and Taiwan. In order to make full use of copula functions, we used C-EVT, that is, GPD, to correctly model tails of the above financial markets’ distributions. The GPDs adequately described the tails behavior of these financial markets. Two different copula functions, Gaussian and SJC copulas, are used to analyze the dependence structure among these markets in pairs. Further, the copulas dependence parameters are allowed to vary over time in ARMA-type evolution equations. This helps figure out the time path of general dependence structure from Gaussian copula, and of upper and lower tail dependence from SJC copula. The constant and time-varying parameters are then analyzed in order to find the best fitted dependence structure between these financial markets. There are some interesting remarks and implications drawn from the analysis. First of all, examining the dependence structure among financial markets, we find that the Australian market has left tail dependence with the US market, meaning that the downturn of the US market will likely affect the Australian one. Also, it shows both upper tail and lower tail dependence with the UK, Japan and Hong Kong markets. It implies that both types of extreme events in these markets will influence the Australian financial market, especially the strongest influence from the Hong Kong market with tail dependence of approximately 0.25. Interestingly, the lower and upper tail dependence measures of the Australian and Taiwanese markets are similar, implying that the dependence between these two market is the same in the time of booming or crashing. This finding, to the best of our knowledge, has not been documented in previous research on the Australian market. Furthermore, the lower tail dependence measure of the Australian and UK markets is double that of upper tail, and this indicates that in the downturns period, the UK market will be much more correlated with the UK market compared to that in the upturns time. It is in line with previous findings that stock markets are more correlated in the crashing time. This is also the case of the UK market with the Hong Kong and Japanese markets, when the lower tail measure is double the upper tail one. The left tail dependence is also seen between the UK market and Taiwanese market; the US market and the Hong Kong, Taiwanese and Japanese markets. The other market pairs such as the Australia and Japan, the UK and US, the Japan–Hong Kong, the Taiwan–Hong Kong all show both right and left tail dependence with a bit greater left tail measure compared to the right tail measure. Interestingly, the opposite case can be seen in the pair of the Japan and Taiwan in which the measure of right tail is somewhat larger than that of the left tail. Thus, these two markets are more correlated during the booming time as compared to the crashing time. Secondly, the comparison between constant and time-varying copulas suggests that time-varying copula provides informative insights on dynamics of dependence structure over time. It also shows that time-varying dependence structure can fluctuate around its constant dependence structure and be volatile in different ranges. In almost all cases analyzed using AIC and log-likelihood in the paper, time-varying copula proves to be the best fitted models in disclosing the dependence structure between financial markets such as in cases of the Australia and UK, the Taiwan and Hong Kong, and the Japan and Hong Kong during the Asian Financial crisis 1997–1998. Therefore, using time-varying copula will make it possible to correctly capture the changes of dependence structure over time. Thirdly, our results suggest that the copula approach is an informative and flexible method to model dependence structure, especially in risk management and asset pricing when joint distributions should be taken into consideration. Also, this finding has important implications for portfolio management or diversification across financial markets during extreme events. Additionally, it implies that normality assumption underlying in most financial applications would be inadequate when analyzing dependence or assessing linkages between financial data. Finally, we expect that time-varying copula model would be very helpful in computing value-at-risk or expected shortfall, allocating assets, modeling credit risk, and event studies over time. However, these are not within the scope of this paper and are left for further investigation.