Dependence Structure between the Equity Market and. the Foreign Exchange Market A Copula Approach

Transcription

1 Dependence Structure between the Equity Market and the Foreign Exchange Market A Copula Approach Cathy Ning 1 Ryerson University October Corresponding author: Cathy Ning, Department of Economics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada. Phone: (416) ext. 6181, Fax: (416) , qning@ryerson.ca.

2 Abstract This paper investigates the dependence structure between the equity market and the foreign exchange market by using copulas. In particular, the Symmetrized Joe-Clayton (SJC) copula is used to directly model the underlying dependence structure. We find that there exists significant upper and lower tail dependence between the two financial markets, and the dependence is symmetric. This finding has important implications for both global investment management and asset pricing modeling.

3 1 Introduction Studying the co-movements across financial markets is an important issue for the risk management and portfolio management. There is a great deal of research focusing on the comovements of international equity markets. Chakrabarti and Roll(2002) compare the comovements of Asian stock markets with those of European markets before and during the Asian crisis. They find that the correlations increased from the pre-crisis to the crisis period in both regions. They also find that diversification potential was bigger in Asia than in Europeinthepre-crisisperiod,butthiswasreversed during the crisis. Other examples of researchontheco-movementsofequitymarketscanbefoundinkarolyiandstulz(1996), Longin and Solnik (2001), Forbes and Rigobon(2002). The methodology they use is along the line of correlations and conditional correlations. Since the limitations of correlationbased models as identified in Embrechts et al. (2002), research has started to use copulas to directly model the dependence structure across financial markets. Works along this line include Mashal and Zeevi (2002), Hu (2003) and Chollete, Pena, and Lu (2005), who report asymmetric extreme dependence between equity returns, i.e., the stock markets crash together but do not boom together. While the above literature focuses on the dependence structure and co-movements in equity markets via copulas, Patton (2005) also employs copulas to model the asymmetric exchange rate dependence. He finds that the mark-dollar and Yen-dollar exchange rates are more correlated when they are depreciating against the U.S. dollar than when they are appreciating. While there is extensive literature studying the co-movements between the international 1

4 equity markets and some literature on modeling the dependence structure between the exchange rates via copulas, there is no literature on using copulas to study the co-movements across these two markets. We consider both equities and foreign exchanges in our study since the foreign exchange market is by volume one of the largest financial markets and foreign exchange is an important asset in international financial portfolios. In the literature, Giovannini and Jorion (1989) include foreign exchanges as assets in their portfolios. For global investors who wish to diversify portfolios internationally, the co-movements and dependence structure between assets in their portfolios such as equities and foreign exchanges would have importance implications for their cross market diversification. There has been extensive research (both theoretical and empirical) in the relationship and co-movements between these two markets. Theoretical research includes the flow-oriented models of exchange rates (Dornbusch and Fischer, 1980) and the stock oriented models of exchange rate (Branson, 1983; Frankel, 1983). All these models argue that the stock market impacts the exchange rate and vice versa. Empirical study of the interaction or causality relationship between the stock price and the exchange rate leads to mixed results (positive correlation, negative correlation, existence of causality or nonexistence of causality, causality one way or the other). In this paper, we endeavour to investigate the dependence between the equity returns and the exchange rate returns, by using a relatively new technique: copulas. The methodology we use in this paper differs in a fundamental way from most of the methods used in the literature in analyzing dependence between the financial markets, which is also sometimes 2

5 called co-movement.we will use dependence or co-movement alternatively in this paper. A copula is a function that connects the marginal distributions to restore the joint distribution. The advantage of using copulas in analyzing the co-movement concerned is multifold. First, copulas are very flexible in modelling dependence. Various copulas represent different dependence structure between variables. Copulas allows to separately model the marginal behavior and the dependence structure. This property gives us more options in model specification and estimation. Second, the copula is a more informative measure of dependence between variables than linear correlation. Copulas tell us not only the degree of the dependence but also the structure of the dependence. The copula function can directly model the tail dependence. It is a succinct and exact representation of the dependencies between underlying variables, irrespective of their marginal distributions. Moreover, the copula can easily model the asymmetric dependence by specifying different copulas. But linear correlation does not give the information about tail dependence and the symmetry property of the dependence. Third, the copula is an alternative dependence measure that is reliable when correlation is not. Correlation is only for elliptical distribution with the normal distribution being a special case. Copulas do not require normality of the variables of the interest. This is especially useful when we try to model the dependence between asset returns.(especially from high frequency data), which are usually not normally distributed. Finally, the copula function is invariant to transformations of the underlying variables while the correlation is not. Transformation of our data can affect our correlation estimates. It means that the numerical value of the correlation might be meaningless. This is not a problem of the copula. 3

6 The same copula function can be used for both the prices and the logarithm of the prices. The copula theorem allows us to decompose the joint distributions into k marginal distributions, which characterize the single variables of interest (stock returns or exchange returns in our case), and a copula, which completely describes the dependence between the k variables. As there is not any empirical result or theoretical guidance on the dependence structure between the stock market and the exchange rate, it requires us to be flexible in specifying the copula models. We employ the AR-t-GARCH models for the marginal distributions of each stock index and exchange rate, and choose the Symmetrized Joe-Clayton copula in Patton (2005) for the dependence structure since this copula allows for asymmetric tail dependence and nests symmetry as a special case. The main contribution of this paper is to show how an informative, flexible, direct and easy methodology: the copula approach, can be applied to analyze the co-movements between the equity returns and the foreign exchange returns. The questions we intend to answer are: what is the dependence structure between these two assets? Is there any extreme value dependence? Is the dependence symmetric or asymmetric? By answering these questions, we hope to better understand the co-movements of financial markets and the risks associated with the dependence structure between the markets. The financial markets considered are G5 countries (US, UK, Germany, Japan, France) which include 5 stock markets and 4 exchange rates. We find that there exists significant positive tail dependence between the stock market and the foreign exchange market in each country. Unlike the co-movements between international stock markets, the tail dependence 4

7 is symmetric between the stock market and the foreign exchange market. Our finding has important implications in cross market diversification for international investors: diversification would have limited function especially when there are extreme shocks. This finding should also affect the pricing of assets. In the literature, joint risks at tails have not been considered in the asset pricing model. However, investors should be compensated for this risk. We hope that this work would also improve our understanding of risks associated with the extreme events and our result would lead to the possible revision of the asset pricing models by picking up the tail dependence. The remaining paper is structured as follows. Section 2 provides a brief review of copula concept. Section 3 specifies the models and the estimations. In Section 3, we describe the data and discuss the results. Section 4 concludes. 2 The Copula Concept and Measures of Dependence A copula is a multivariate cumulative distribution function whose marginal distribution is uniform on the interval [0,1] The importance of the copula is that it can capture the dependence structure of a multivariate distribution. This is justified by the fundamental fact known as Sklar s(1959) theorem. For the purpose of this paper and simplicity, we consider the bivariate case. 5

8 Sklar s Theorem. Let H be a joint distribution function with margins F and G. Then there exists a copula C such that for all x, y in R, H(x, y) =C(F (x),g(y)). (1) If F and G are continuous, then C is unique; otherwise, C is uniquely determined on Ran- FxRanG. Conversely, if C is a copula and F and G are the cumulative distribution functions, then the function H defined by (1) is a joint distribution function with margins F and G. From Sklar s theorem, we see that a joint distribution can be decomposed into its univariate marginal distributions, and a copula, which captures the dependence structure between the variables X and Y. As a result, copulas allow us to model the marginal distributions and the dependence structure of a multivariate random variable separately. One of the key properties of copulas is that they are invariant under increasing and continuous transformations. This property is very useful as transformation is commonly used in economics and finance. For example, the copula does not change with returns or logarithm of returns. This is not true for the correlation, which is only invariant under linear transformations. In addition to linear correlations, there are various other measures of dependence, among which Kendall s τ and Spearman s ρ are two scale free measures of dependence and are commonly studied with copula models. Kendall s tau is defined as the difference between 6

9 the probability of the concordance and the probability of the discordance: tau(x, Y ) = P [(X 1 X 2 )(Y 1 Y 2 ) > 0] P [(X 1 X 2 )(Y 1 Y 2 ) < 0] (2) for tau [ 1, 1]. Kendall s tau represents rank correlations, i.e., the relations between the rankings, instead of the actual value of the observations. The higher the tau value, the stronger is the dependence. The relation between Kendall s tau and the copula is as follows: tau =4 Z 1 Z C(u, v)dc(u, v) 1 (3) Therefore, Kendall s tau doesn t depend on marginal distributions. Comparisons between results using different copula functions should be based on a common Kendall s tau. Another useful dependence measure defined by copulas is the tail dependence, which measures the probability that both variables are in their lower or upper joint tails. Intuitively, upper(lower) tail dependence refers to the relative amount of mass in the upper(lower) quantile of the distribution. An important property of a copula is that it can capture the tail dependence. This is not a linear correlation can capture. Furthermore, the tail dependence between X and Y, as one of the copula properties, is invariant under strictly increasing transformation of X and Y. The left(lower) and right(upper) tail dependence coefficients are defined as C(u, u) λ l = lim Pr[G(y) u F (X) u] = lim, (4) u 0 u 0 u 7

10 1 2u + C(u, u) λr = lim Pr[G(Y ) u F (X) u] = lim, (5) u 1 u 1 1 u where λ l and λr [0, 1]. Ifλ l or λr are positive, X and Y are said to be left (lower) or right (upper) tail dependent. Further examination of copulas and measures of dependence can be found in Joe (1997) and Nelsen (1999). Different copulas usually represent different dependence structure with the association parameters indicating the strength of the dependence. Following we present some commonly used copulas in economics and finance: Gaussian copula, student t copula, Gumbel copula, Clayton copula, and Symmetrized Joe-Clayton(SJC) copula. Bivariate Gaussian copula C(u, v; ρ) =Φ ρ (Φ 1 (u), Φ 1 (v),ρ), (6) where 0 u, v 1 and 1 ρ 1. Φ ρ is the bivariate normal distribution function with correlation coefficient ρ, andφ 1 is the inverse of the univariate normal distribution function. By Sklar s theorem, we can have H(x, y) =C(F (x),g(y)) = Φ ρ (Φ 1 (F (x)), Φ 1 (G(y)),ρ). (7) That is we can construct bivariate distributions with non-normal marginal distributions and the Gaussian copula. 8

11 The relationship between Kendall s tau and ρ for Gaussian copula is: tau = 2 arcsin(ρ), (8) π Gaussian copula has zero tail dependence, therefore λ l = λr =0. Tcopula The T copula is defined as C υ,ρ (u, v) =t υ,ρ (t 1 υ (u),t 1 υ (v)), (9) where t υ,ρ is the bivariate student t distribution with degree of freedom υ and the correlation coefficient ρ. t 1 υ istheinverseoftheunivariatestudenttdistribution. Its Kendall s tau can be expressed as a function of ρ: tau = 2 arcsin(ρ). (10) π The T copula has symmetric tail dependence with dependence coefficient as follows: λ l = λr =2t υ+1 ( (υ +1) 1/2 (1 ρ) 1/2 (1 + ρ) 1/2 ). (11) Gumbel copula The Gumbel copula is defined as 9

12 C α (u, v) =exp( (( ln u) α +( ln v) α ) 1 α ), for α (0, 1], (12) where a is the associate parameter. The Kendall s tau and the associate parameter is linked by the following equation: α =1/(1 tau). (13) The Gumbel copula has no left tail dependence but positive right tail dependence. The tail dependence coefficients can be written as λ l =0, λr =2 2 1/α. (14) Clayton copula The Clayton copula is defined as: C α (u, v) =(u α + v α 1) 1/α for α>0 (15) where α is the associate parameter. The associate parameter can be expressed as a function of Kendall s tau as α =2tau/(1 tau). 10

13 Clayton copula does not have right tail dependence but has left tail dependence as λ l =2 1/α, λr =0. (16) Symmetrized Joe-Clayton(SJC) copula The SJC copula is a modification of the so called BB7 copula of Joe (1997). It is defined as C SJC (u, v λr, λ l )=0.5 (C JC (u, v λr, λ l )+C JC (1 u, 1 v λ l,λr)+u + v 1), (17) where C JC (u, v λr, λ l ) is the BB7 copula (also called Joe-Clayton copula) defined as C JC (u, v λr, λ l ) = 1 (1 n 1 (1 u) k r + 1 (1 v) k r 1 o 1/r) 1/k, (18) where k =1/log 2 (2 λr), r = 1/log 2 (λ l ), and λ l (0, 1), λr (0, 1). By construction, the SJC copula is symmetric when λ l =λr. 3 Model Specification and Estimation In order to study the dependence structure between the bivariate variables, i.e., the stock return series and the foreign exchange return series, we need to specify three models: the models for the marginal distribution of each stock and exchange rate, and the model for the 11

14 joint distribution of the two series by copula. 3.1 Marginal Models It is well documented in the literature that the daily asset returns show fat-tails and heteroscedasticity. As usual, the error variance is unknown and must be estimated from the data. The generalized autoregressive conditional heteroscedasticity (GARCH) model is a common approach to model time series with heteroscedastic errors. Besides, Bollerslev (1987) among others, has found that the student s t distribution fits the univariate distribution of the daily exchange rate returns quite well. Many asset returns also show autoregressive characteristic. As a result, AR(k)-t-GARCH(p,q) model has been documented to be successful in capturing these stylized facts of asset return series. This type of model and its variants have been used in Bollerslev (1987) and Patton (2005). We adopt this model for our return series. To verify the marginal distributions are indeed not normal, we use the Jarque-Bera statistic normality tests for each series. The autoregressive terms k is determined by specifying the maximum being 10 and deleting the insignificant (with significant level of 5%) autoregressive terms. Hence the marginal model can be specified as follows: r i,t = m i + X k AR i,k r i,t k + ε i,t, (19) σ 2 i,t = const i + X p r nu σ 2 i,t (nu 2) ε i,t iid t nu garch(p) i σ 2 t p + X q arch(q) i ε 2 i,t q. (20) 12

15 where r i,t is the returns for the ith asset at time t. σ 2 i,t is the variance of ε i,t.andnuisthe degree of freedom for the t distribution. 3.2 Joint Models In the literature, it is well documented that equity markets crash together but do not boom together, indicating a lower tail dependence. Since in the literature there is not any empirical results about the dependence structure between the stock market and the foreign exchange market, it requires the selection for the copula to be flexible in modeling the tail dependence in both directions, and the asymmetric dependence should be allowed while the symmetric dependence should be a special case. The Gaussian and T copulas are most commonly used in economics and finance. However, they are not suitable to use in our case. Gaussian copula forces zero tail dependence and T copula imposes symmetric tail dependence. There may exist asymmetric dependence in our variables. Moreover, T copula requires that the degrees of freedom for the marginals are the same. This constraint is not satisfied in our data. As to the asymmetric tail dependent copulas, the Gumbel copula does not have left tail dependence but has positive right tail dependence. On the other hand, the Clayton copula has positive left tail dependence but zero right tail dependence. Therefore, neither of them are suitable for our modeling. The Symmetrized Joe-Clayton (SJC) copula allows both upper and lower tail dependence and the symmetric dependence is a special case, hence it satisfies all the flexibility requirements. Therefore, we choose SJC copula for the joint model. More specifically, the variables u, v in the SJC copula are the cumulative distribution functions 13

16 of the standardized residuals from the marginal models. 3.3 Estimation There are usually two approaches to estimate a parametric copula model, namely one stage full maximum likelihood (ML) and inference for the margins (IFM). The ML approach jointly estimates the parameters in the marginal models and the parameters of the copula model simultaneously. The IFM method is to break the estimation into two steps: at a first step, estimate the parameters in the marginal distributions; at a second step, given the estimated margin s parameters. estimate the copula parameters. Next, we give a brief discussion of the two estimation approaches. Without loss of generality, we consider two margins. By Sklar s theorem, we can decompose the joint distribution into its marginal distributions and its dependence function (copula): F XY (x, y) =C(F X (x),f Y (y)), (21) where F XY (x, y), F X (x), F Y (y) arethejointcdfandmarginalcdfs,whilec is the copula function. Taking derivative of above, we get f XY (x, y) =f X (x) f Y (y) c(u, v), (22) 14

17 where f and c are density functions: f XY (x, y) = 2 F XY (x, y) x y c(u, v) = 2 C(F X (x),f Y (y)), u v, f X (x) = F X(x) x, f Y (y) = F Y (y), y with u = F X (x), v = F Y (y). Take logarithm of the above density function, we get: L XY = L X + L Y + L C, (23) where L XY =log(f XY (x, y)), L X =log(f X (x)), L Y =log(f Y (y)), L C =log(c(u, v)). The one stage full ML estimator is obtained by maximizing L XY. Under the regularity conditions the ML estimator is consistent, efficient, and asymptotically normal. Note that in (23), the likelihood is composed by two positive parts: L X and L Y only involving the margin parameters, and L C involving the margin and dependence parameters. Therefore, Joe and Xu (1996) proposed the two step IFM method. In the first step, they estimate the marginal models by maximizing the logarithm likelihoods: L X and L Y. In the second step, given the estimated parameters for the marginal models, they estimate the copula parameters by maximizing L C. Joe (1997) proves that under regular conditions, the IFM estimator is consistent and asymptotic normal. Compared with the ML, the IFM method is less computational intensive. Moreover, the large number of parameters in the simultaneous ML estimation could make numerical 15

18 maximization of the likelihood function difficult Since it is computational easier to obtain the IFM estimator, it is naturally worthwhile to compare the efficiency of the IFM estimator with the ML estimator. Joe(1997) points out that the IFM method is highly efficient compared with the ML method. Joe and Xu (1996) compared the efficiency of the IFM with the ML by simulation. They found that the ratio of the mean square errors of the IFM estimator to the MLE is close to 1. Theoretically, ML estimator should be the most efficient, in that it attains the minimum asymptotic variance bound. However, for the finite sample, Patton (2003) found that the IFM was often more efficient than the ML, and in most cases not less efficient. As a result, IFM is the main estimation method employed in estimating the copula models. Since our models involve a large number of parameters, we adopt the IFM method for our estimation as well. We first estimate the marginal AR(k)-t-GARCH(p,q) models by maximum likelihood. Then we estimate the copula parameters given the estimated parameters in the marginal models. The densities of the Joy Clayton copula and the SJC copula are derived respectively as follows. Let A =1 (1 u) k,andb =1 (1 v) k, c JC (u, v λr, λ l ) = 2 C JC (u, v λr, λ l ) u v = (AB) r 1 (1 u) k 1 (1 v) k 1 {[1 (A r + B r 1) 1/r ] 1+1/k (A r + B r 1) 2 1/r (1 + r)k +[1 (A r + B r 1) 1/r ] 2+1/k (A r + B r 1) 2 2/r (k 1)},(24) 16

19 where k =1/log 2 (2 λr), r = 1/log 2 (λ l ). The density of the SJC copula is c SJC (u, v λr, λ l ) = 2 C SJC (u, v λr, λ l ), u v = 0.5 [ 2 C JC (u, v λr, λ l ) u v + 2 C JC (1 u, 1 v λ l,λr) ]. (1 u) (1 v) (25) The expression for 2 C JC (1 u,1 v λ l,λr) isthesameas 2 C JC (u,v λr,λ l ). But we substitute u and (1 u) (1 v) u v v in the latter with 1 u and 1 v to get the former. Also note that k =1/log 2 (2 λ l ), r = 1/log 2 (λr) for the former. The copula logarithm likelihood is: L C = log(c SJC (u, v λr, λ l )). λr and λ l can be estimated by maxl C. 4 Data and the Discussion of Results 4.1 Data We use daily data from Datastream from 1/1/1991 to 31/12/1998. The data are from the five largest developed countries: US, UK, German, France and Japan. Data start from 1991 since before that exchange rate arrangements (currency snake , European Exchange Rate Mechanism ) prevail in the developed countries. And data end before the introduction of Euro. 17

20 The stock market index from each country should represent the stock market of that country. We use the Datastream calculated stock market indices expressed in US dollars from each country (The codes are: TOTMKUS for US, TOTMUK$ for UK, TOTMBD$ for German, TOTMFR$ for France, TOTMJP$ for Japan). Foreign Exchange rates are expressed in US dollars per local currency (The codes in Datastream: BRITPUS, WGM- RKUS,FRNFRUS, JAPYNUS). The returns are calculated as 100 times the logarithm differences of the indices or the exchange rates between the day t and the day t-1. r_us, r _uk, r_gm, r_fr, r_jp are the returns of US, British, German, French, Japanese stock market index respectively. r_pdus, r_gmus, r_frus, r_jyus are the returns from the British pound, German mark, French franc and Japanese Yen expressed in US dollars. Table 1 gives the summary statistics of the returns. The table shows that all the means of the returns are small relative to their standard deviations. For example, the mean of the British stock index returns is 0.037, while its standard deviation is 0.89, indicating relative high risks. The standard deviations of the stock index returns (ranging from 0.81 to 1.38) are larger than those of exchange rate returns (ranging from 0.64 to 0.77), indicating more volatile of the stock markets than the foreign exchange markets. European stock markets are more volatile than US stock market. The skewnesses of returns are different from zero with most of them skewing to the left. All returns show excess Kurtosis ranging from 5.65 to Both the skewness and the excess kurtosis indicate that the return series are not normally distributed. 18

21 In Table 2, we present the linear correlations between return series. We can see that the pairwise correlations between the stock index returns and the exchange rate returns are all significantly different from zero. The Japanese stock -Yen pair has the highest linear correlation coefficient of The French pair has the lowest linear dependence with the correlation coefficient being The correlations are all positive, indicating the increase (decrease) of the local stock market is associated with the appreciation(deprecation) of the local currency. In Table 3 and Table 4, the two measures of rank dependence, namely, the Kendall s Tau and Spearman s Rho are presented respectively. Kendall s tau measures the difference between the probability of the concordance and the probability of the discordance. The Kendall s taus for our pairs of interest are all significantly positive, showing the probability of concordance is significantly larger than the probability of discordance. Spearman s rho also measures the rank correlation between variables. The Spearman s rhos for the pairs in each country in Table 4 are all significantly positive, indicating strong rank correlations. The values of taus and rhos are consistent with each other and the linear correlation: Japanese pair has the strongest dependence, followed by German pair, UK pair and French pair. In order to see the dependence structure from the data. We use a frequency table. To do this, we first rank the pair of return series in ascending order and then we divide each series evenly into 10 bins. Bin 1 includes the observations with the lowest values and bin10 includes observations with the highest values. We want to know how the values of one series are associated with the values of the other series, especially whether lower returns in stock market is associated with lower returns in foreign exchange market. Thus we count the 19

22 numbers of observations that are in cell(i, j). The dependence information we can obtain from the frequency table is that: if the two series are perfectly positively correlated, we would see most observations lie on the diagonal; if they are independent, then we would expect that the numbers in each cell are about the same; If the series are perfectly negatively correlated, most observations should lie on the diagonal connecting the upper-right corner and the lowerleft corner; If there is positive lower tail dependence between the two series, we would expect that more observations in cell(1,1). If there exists positive upper tail dependence, we would expect large number in cell(10,10). We present the dependence table in Table 5. For the UK pair, cell(1,1) is 71, which means out of 2007 observations, there are 71 occurrences when both British pound and UK stock returns lie in their respective lowest 10th percentiles (1/10th quantile). Cell(10,10) for UK pair is 59, indicating 59 occurrences when both series are in their highest 10th percentiles (9/10th quantile) respectively. Numbers in other cells of the UK pair are much smaller than those in these two cells. This is the evidence of both upper and lower tail dependence. Comparing cell(1,1) and cell(10,10) of the German, French and Japanese pairs, we see quite obvious evidence of both upper and lower tail dependence. And the dependence seem symmetric except the UK pair. 4.2 Estimation Results of the Models We first estimate the marginal models: the AR(k)-t-GARCH(p,q) type models for each asset return series. k is set to 10, and the insignificant (with significant level of 5%) autoregressive 20

23 terms are deleted. We experiment on GARCH terms up to p=2 and q=2. The estimates of the marginal models are presented in Table 6. We test the normality of the error term in the AR equation and the null hypothesis of normality is strongly rejected for all series with the p values of the Jarque-Bera statistic being less than (not listed in table to save space). For US stock index, both lag one and lag five are significant autoregressive terms. That implies the returns of yesterday and the same day of last week will significantly affect the return today. Both ARCH and GARCH terms are strongly significant, indicating heteroscedasticity of the data. The number of degrees of freedom for the t distribution is 5.29 and statistically significant. 1 For the UK stock index returns, lag 5 is the only significant autoregressive term, indicating the weekly seasonality. Again the ARCH1 is the significant term. Different from the US stock index, it requires the GARCH2 term to better fit the data. The number of degrees of freedom of t is again small (7.8) and significant. For the other seven marginal models, the degrees of freedom are all small (ranging from 3.9 to 7.5), indicating that the error terms are not normal. Also note that the degrees of freedom parameter of the equity is bigger than that of the exchange rate in each pair. This indicates that t copula is not suitable for modeling the dependence since it requires the equality of the degrees of freedom of the margins. The AR1 term is significant in the models of the German and Japanese stock returns, British Pound and Japanese Yen returns. The British Pound and Japanese Yen also show biweekly pattern with the significance of the 10th autoregressive term. The German Mark shows six day seasonality with AR6 being significant. For the stocks, GARCH(1,1) 1 This can be inferred from the strong significanceoftheinverseofthetdegreeoffreedomshowninthe Table. 21

24 is able to capture the heteroscedasticity. For the exchange rates, higher GARCH terms are required to better model the heteroskedasticity. This is reflected in the significance of the GARCH2 term for the British Pound, the German Mark and the French Franc. For the French Franc, the ARCH2 is also found to be significant. In order to evaluate the goodness of fit of the marginal models, we obtain the frequencies of the standardized residuals from the marginal models. The result is presented in Table 7. ComparingfrequenciesinTable7withthoseinTable5(fromthedata),wecanseethatthe pairs from the marginal models keep the same dependence pattern as the data: observations mass at both upper (relatively larger number in cell(10,10)) and lower tail (larger number in cell (1,1)). The fit is generally very well. For example, for the Japanese pair, cell(1,1) from the data is 66, while it is 68 from the standardized residuals of the model; cell(10,10) is 70 from the data against 71 from the model. Cell(10,10) for the German pair is 70 from the data against 72 from the model. We then estimate the joint copula models. The estimation of Lower tail dependence, upper tail dependence and the copula log likelihood for all the pairs is provided in Table 8. All of the tail dependence coefficients are statistically significant. Japanese pair has the highest tail dependence coefficients with λ l being and λr being Since λ l (λr) measures the dependence between the stock returns and exchange rate returns when both of them are in extremely small (large) values, the significance of λ l (λr) means that the Japanese stock market crashes (booms) when the Japanese yen depreciates (appreciates) heavily against US dollars and vice versa. This finding is consistent with the basic international finance theory. 22

25 When a country s stock market is booming, investors believe that it is a good place for investment, therefore they will purchase that country s currency to buy stocks there. Hence the demand of the currency increases, which leads to the appreciation of the currency. This phenomenon happens in the extreme cases as investors are more sensitive to the extreme events. Since the values of λ l and λr are very similar for our pairs, we wish to know whether the tail dependence is symmetric. We use likelihood ratio test to test the hypothesis: λ l =λr. The test is presented in Table 9. All the p-values from the test are greater than Therefore we can not reject the hypothesis that the upper and lower tail dependence is the same. This meansthatthedependencebetweenthestockmarketandtheforeignexchangemarketisthe same at the time of crashing and booming. This finding is different from the finding for the dependence structure between stock markets documented in the literature: stock markets aremoredependentatthetimeofcrashingthanbooming. Given that we find that the dependence is symmetric, we estimate the copula model again by forcing the equality of λ l and λr. The results are in Table 10. Again we find significant tail dependence for all the pairs. The values of the tail dependence are for Japanese pair, for the German pair, for the British pair and for the French pair. Note that the order of the degree of the tail dependence is consistent with the linear correlation coefficient, Kendall s tau and Spearman s rho for our pairs. Our main finding is the existence of extreme co-movements (tail dependence) between the stock market and the exchange rate market. The extreme co-movements are symmetric, 23

26 implying both markets boom and crash together. This finding improves our understanding of the market dependence. The significance of tail dependence implies that the stock market and exchange rate tend to experience concurrent extreme shocks. This has important implications for diversification across these two markets at extreme event. Hence it is also very important to investors who wish to diversify investments globally. Furthermore, the finding is also important for asset pricing since we should then take into account the joint tail risk when pricing. 5 Conclusion In this paper, we examine the extreme co-movements between the stock market and the exchange rate market by directly modeling their dependence structure via the use of the symmetrized Joe Clayton (SJC) copula. The symmetric tail dependence is found to be significant in all the stock index-exchange rate pairs analyzed in this study. This finding is very important for global investors in their portfolio management at extreme market event. The finding also implies that the Gaussian dependence hypothesis that underlies most modern financial applications may be inadequate. In most multivariate financial models, dependence assumptions are very important. Our study shows that the copula approach is a flexible, informative and direct method to model dependence structure. Our results and the property of SJC copula also suggest that this model could be well suited for various financial modelling purposes. Picking up the tail dependence could lead to a more realistic assessment of the linkage between financial markets 24

27 and possibly more accurate risk management and pricing models. 25

28 Acknowledgement I wish to thank John Knight, Stephen Sapp, and Tony Wirjanto for their helpful suggestions and comments. I also would like to thank participates in the 2006 Canadian Economics Association annual meeting for thier comments. All the remaining errors are mine. References [1] Bollerslev, Tim, (1987), A Conditional Heteroskedastic Time Series Model for Speculative Prices and Rates of Return, Review of Economics and Statistics, 69, [2] Branson, W. H., (1983), Macroeconomic Determinants of Real Exchange Risk, in Managing Foreign Exchange Risk, R. J. Herring ed., Cambridge: Cambridge University Press. [3] Chollete, Victor de la Pena, and Ching-Chih Lu (2005), Comovement of International Financial Markets, Working paper. [4] Chakrabarti, R. and Roll R. (2002), East Asia and Europe during the 1997 Asian Collapse: a Clinical Study of a Financial Crisis, Journal of Financial Markets 5 (2002) [5] Dornbusch, R. and S. Fischer, (1980), Exchange Rates and the Current Account, American Economic Review, 70(5),

29 [6] Embrechts, P., A. Mcneil and D. Straumann (2002) Correlation and Dependence in Risk Management; Properties and Pitfalls, in Risk Management: Value at Risk and Beyond, ed. M.A.H. Dempster, Cambridge University Press, Cambridge, pp [7] Forbes K. and R. Rigobon (2002) No Contagion, Only Interdependence: Measuring Stock Market Co-movements, Journal of Finance, 57(5): [8] Frankel, J. A., (1983), Monetary and Portfolio-Balance Models of Exchange Rate Determination, in Economic Interdependence and Flexible Exchange Rates, J. S. Bhandari and B. H. Putnam eds., Cambridge: MIT Press. [9] Giovannini, A. and Jorion, P (1989), The Time Variation of Risk and Return in the Foreign Exchange and Stock Markets. The Jornal of Finance, Vol.44, No.2, [10] Hu, L (2005), Dependence Patterns across Financial Markets: a Mixed Copula Approach, forthcoming in the Applied Financial Economics. [11] Joe H., (1997), Multivariate Models and Dependence Concepts, Volume 37 of Monographs on Statistics and Applied Probability, Chapman and Hall, London. [12] Karolyi, G. A. and Stulz, R. M., 1996, Why do Markets Move Together? An Investigation of U.S.-Japan Stock Return Comovements, Journal of Finance, 51(3): [13] Longin, F. and B. Solnik (2002) Extreme Correlation of International Equity Markets, Journal of Finance, 56(2):

30 [14] Marshal R. and A. Zeevi (2002) Beyond Correlation: Extreme Co-movements between Financial Assets, Working Paper, Columbia Business School. [15] Nelson, R.B. (1999) An introduction to Copulas, Springer, New York. [16] Patton, A.L. (2006), Modelling Asymmetric Exchange Rate Dependence, International Economic Review, 47(2), [17] Sklar, A., 1959, Fonctions de répartition á n dimensions et leurs marges, Publications de l Institut de Statistique de l Université de Paris, 8:

31 Co-movements across Stock Market and Foreign Exchange Market Data: Source and data period: Daily data from Datastream from 1/1/1991 to 31/12/1998. The data are from five developed countries: US, UK, German, France and Japan. Data start from 1991 since before that exchange rate arrangements prevail in the developed countries. Raw data:. stock market index (TOTMKUS for US, TOTMUK$ for UK, TOTMBD$ for German, TOTMFR$ for France, TOTMJP$ for Japan) in US dollars. Foreign Exchange rate in US dollars per local currency (BRITPUS, WGMRKUS,FRNFRUS, JAPYNUS) Returns: The returns are 100 times the log-differences of the index or the exchange rate. r_us, r _uk, r_gm, r_fr, r_jp are the returns of US, UK, German, France, Japan stock market indices respectively. r_pdus, r_gmus, r_frus, r_jyus are the returns from the British pound, German mark, French franc and Japanese Yen expressed in US dollars. Table 1 Descriptive Statistics r_us r_uk r_gm r_fr r_jp r_pdus r_gmus r_frus r_jyus Mean Std. Dev Skewness Kurtosis Number of Obs Table 2 Pearson Correlation _NAME_ r_pdus r_gmus r_frus r_jyus r_us (0.0065) (<0.0001) (<0.0001) (0.2902) r_uk (<0.0001) (<0.0001) (<0.0001) (<0.0001) r_gm (<0.0001) (<0.0001) (<0.0001) (<0.0001) r_fr (<0.0001) (<0.0001) (<0.0001) (<0.0001) r_jp (<0.0001) (<0.0001) (<0.0001) (<0.0001) *Numbers in the bracket are p-values. 29

32 Table 3 Kendall s Tau _NAME_ r_pdus r_gmus r_frus r_jyus r_us * * r_uk * * * * r_gm * * * * r_fr * * * * r_jp * * * * * p value < Table 4 Spearman s Rho _NAME_ r_pdus r_gmus r_frus r_jyus r_us * * r_uk * * * * r_gm * * * * r_fr * * * * r_jp * * * * Procedure to construct empirical copula (tail dependence): We first rank the returns in ascending order and then group each return series in 10 bins. The bin 1 includes the returns with the lowest values and bin10 includes returns with the highest values. We want to know how values of one series are associated with values of the other series, especially whether lower returns in stock market is associated with lower returns in foreign exchange market. Thus we count the numbers of observations that are in cell(i, j). For example, cell(1,1) in the first table is 71, which means out of 2007 observations, there are 71 occurrences when both foreign exchange and stock returns lie in their respective lowest 10 th percentiles (1/10th quantile). If there is lower tail dependence between the two series, we would expect that more observations in cell(1,1). If there exists upper tail dependence, we would expect large number in cell(10,10). Table 5 Empirical Copula for Stock Returns and Exchange Rate Returns UK Pair rank_pdus rank_uk Total Total ,007 30

33 German Pair rank_gmus rank_gm Total Total ,007 French Pair rank_frus rank_fr Total Total ,007 Japanese Pair rank_jyus rank_jp Total Total ,007 31

34 Table 6 Estimation of Marginal Models The models used are AR(k) GARCH(p,q) type of model. The maximum autoregressive order is set to 10. Backward elimination is specified to delete the insignificant (with significant level of 5%) autoregressive terms. r_us Variable Estimate Standard Error t Value Approx Pr> t Intercept <.0001 AR1_us AR5_us ARCH ARCH <.0001 GARCH <.0001 TDFI (Inverse of t DF=5.2854) <.0001 r_uk Variable Estimate Standard Error t Value Approx Pr> t Intercept AR5_uk ARCH ARCH <.0001 GARCH <.0001 TDFI (Inverse of t DF=7.8125) <.0001 r_gm Variable Estimate Standard Error t Value Approx Pr> t Intercept AR1_gm ARCH ARCH <.0001 GARCH <.0001 TDFI (Inverse of t DF=6.5574) <.0001 r_fr Variable Estimate Standard Error t Value Approx Pr> t Intercept ARCH ARCH <.0001 GARCH <.0001 TDFI(Inverse of t DF=7.5075) <

35 r_jp Variable Estimate Standard Error t Value Approx Pr> t Intercept AR1_jp ARCH ARCH <.0001 GARCH <.0001 TDFI (Inverse of t DF=5.5157) <.0001 r_pdus Variable Estimate Standard Error t Value Approx Pr> t Intercept AR1_pdus AR10_pdus ARCH ARCH <.0001 GARCH <.0001 TDFI (Inverse of t DF=3.9246) <.0001 r_gmus Variable Estimate Standard Error t Value Approx Pr> t Intercept AR6_gmus ARCH ARCH <.0001 GARCH <.0001 TDFI (Inverse of t DF=4.9702) <.0001 r_frus Variable Estimate Standard Error t Value Approx Pr> t Intercept ARCH ARCH ARCH GARCH <.0001 TDFI (Inverse of t DF=4.5413) <.0001 r_jyus Variable Estimate Standard Error t Value Approx Pr> t Intercept AR1_jyus AR10_jyus ARCH ARCH <.0001 GARCH <.0001 TDFI (Inverse of t DF=4.0355) <

36 Table 7 Empirical Copula for Standardized Residuals (Since the maximum lags in the marginal model are 10, the first 10 observations are deleted as for the UK case, residuals are all missing values. Therefore, the total # of observations are less than the data by 10). rank_rsd_pdus rank_uk Total Total ,997 rank_rsd_gmus rank_gm Total Total ,997 rank_rsd_frus rank_fr Total Total ,997 34

37 rank_rsd_jyus rank_jp Total Total ,997 Table 8 Results for the SJC copula models r_uk & r_pdus r_gm & r_gmus r_fr & r_frus r_jp & r_jyus Lower tail dependence ( λ l ) ( ) ( ) ( ) ( ) Upper tail dependence ( λ r ) ( ) ( ) ( ) ( ) Copula log likelihood Table 9 Likelihood Ratio Test for Symmetric Tail Dependence Pairs p-value of likelihood ratio test British stock and British pound German stock and German Mark French stock and French franc Japanese stock and Japanese yen Table 10 Results for the SJC copula models when λ l = λ r r_uk & r_pdus r_gm & r_gmus r_fr & r_frus r_jp & r_jyus Tail dependence ( ) ( ) ( ) ( ) Copula log likelihood