Dependence Structure between the Equity
Market and the Foreign Exchange Market{A
Copula Approach
Cathy Ning
February 2006
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 nd that there exists signicant upper
and lower tail dependence between the two nancial markets, and the dependence is symmetric. This nding has important implications for both global
investment management and asset pricing modeling.
1 Introduction
Studying the co-movements across nancial markets is an important issue for
risk management and portfolio management. There is a great deal of research
focusing on the co-movements of international equity markets. Chakrabarti
and Roll(2002) compare the co-movements of Asian stock markets with those
of European markets before and during the Asian crisis. They nd that
the correlations increased from the pre-crisis to the crisis period in both regions. They also nd that diversication potential was bigger in Asia than
in Europe in the pre-crisis period, but this was reversed during the crisis.
Other examples of research on the co-movements of equity markets can be
found in Karolyi and Stulz (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 correlation-based models as identied in Embrechts et al. (2002), research has started to use copulas
1
to directly model the dependence structure across nancial markets. Works
along this line include Marshal 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 nds 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 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 nancial markets and foreign exchange is an important asset in international nancial 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 exchange rates would have
important implications for their cross market diversication. There has been
extensive research (both theoretical and empirical) in the relationship and
co-movements between these two markets. Theoretical research includes the
\ow-oriented" models of exchange rates as in Dornbusch and Fischer(1980)
and the \stock oriented" models of exchange rate (see Branson(1983) and
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 new technique: copulas. The methodology we use in this paper diers in a fundamental way
from most of the methods used in the literature in analyzing dependence between the nancial markets, which is also sometimes called co-movement .We
will use dependence or co-movement interchangeably 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
2
is multifold. First, copulas are very exible in modeling dependence. Various
copulas represent dierent dependence structures between variables. Copulas allow us to separately model the marginal behavior and the dependence
structure. This property gives us more options in model specication 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 dierent copulas. However linear correlation does not give the information about tail dependence and the
symmetrical property of the dependence. Third, the copula is an alternative
dependence measure that is reliable when correlation is not. Correlation can
only be used for elliptical distributions 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(in particular 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 aect our correlation estimates, possibly rendering the numerical value of the correlation meaningless. This is not a problem of the
copula. 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 results or theoretical guidance on the dependence structure
between the stock market and the exchange rate, it requires us to be exible
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, exible, direct and easy methodology: the copula approach, can be applied to analyze the co-movements between the equity returns and the foreign exchange
3
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 nancial markets and the
risks associated with the dependence structure between the markets.
The nancial markets considered are the G5 countries (US, UK, Germany, Japan, France) which include 5 stock markets and 4 exchange rates.
We nd that there exists signicant 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 is
symmetric between the stock market and the foreign exchange market. Our
nding has important implications in cross market diversication for international investors: diversication would have limited function especially when
there are extreme shocks. This nding should also aect the pricing of assets.
In the literature, joint extreme risks have not been considered in the asset
pricing model. However, investors should be compensated for this risk. We
hope that this work will also improve our understanding of risks associated
with the extreme events and our results will lead to the possible revision of
the asset pricing models by picking up the tail dependence.
The remainder of the paper is structured as follows. Section 2 provides
a brief review of copula concepts. Section 3 species the models and the
estimation. In Section 4, we describe the data and discuss the results. Section
5 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 justied by the fundamental fact known as Sklar's(1959) theorem.
For the purpose of this paper and simplicity, we consider the bivariate case.
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)):
4
(1)
If F and G are continuous, then C is unique. Conversely, if C is a copula
and F and G are the cumulative distribution functions, then the function H
dened 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
transformations are commonly used in economics and nance. 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 and Spearman's are two scale free measures of dependence and are commonly studied with copula models. Kendall's
tau is dened as the dierence between the probability of the concordance
and the probability of the discordance:
tau(X; Y ) = P [(X X )(Y Y ) > 0] P [(X X )(Y Y ) < 0](2)
for tau 2 [ 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:
1
2
tau = 4
1
Z
0
1
2
Z
0
1
1
C (u; v)dC (u; v) 1
2
1
2
(3)
Therefore, Kendall's tau doesn't depend on marginal distributions. Comparisons between results using dierent copula functions should be based on a
common Kendall's tau.
Another useful dependence measure dened 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.
Furthermore, the tail dependence between X and Y, as one of the copula
5
properties, is invariant under strictly increasing transformation of X and Y.
The left(lower) and right(upper) tail dependence coecients are dened as
C (u; u)
;
(4)
= lim P r[G(y) ujF (X ) u] = lim
l
u
!0
u
!0
u
1 2u + C (u; u) ;
(5)
1 u
where l and r 2 [0; 1]. If l or r 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).
Dierent copulas usually represent dierent dependence structures with
the association parameters indicating the strength of the dependence. Some
commonly used copulas in economics and nance include: Gaussian copula,
student t copula, Gumbel copula, Clayton copula, and Symmetrized JoeClayton(SJC) copula.
r = ulim
P r[G(Y ) ujF (X ) u] = ulim
!1
!1
Bivariate Gaussian copula
C (u; v; ) = ( 1 (u); 1 (v); );
(6)
where 0 u; v 1 and 1 1. is the bivariate normal distribution
function with correlation coecient , and is the inverse of the univariate
normal distribution function. By Sklar's theorem, we can have
1
H (x; y) = C (F (x); G(y)) = ( 1 (F (x)); 1 (G(y)); ):
(7)
That is, we can construct bivariate distributions with non-normal marginal
distributions and the Gaussian copula.
The relationship between Kendall's tau and for Gaussian copula is:
2
(8)
tau = arcsin();
The Gaussian copula has zero tail dependence, therefore l = r = 0.
T copula
The T copula is dened as
C; (u; v) = t; (t 1 (u); t 1 (v));
6
(9)
where t; is the bivariate student t distribution with degree of freedom and the correlation coecient . t is the inverse of the univariate student
t distribution.
Its Kendall's tau can be expressed as a function of :
2
(10)
tau = arcsin():
1
The T copula has symmetric tail dependence with dependence coecient as
follows:
l = r = 2t ( ( + 1) = (1 ) = (1 + ) = ):
(11)
1 2
+1
1 2
1 2
Gumbel copula
The Gumbel copula is dened as
C (u; v) = exp( (( ln u) + ( ln v) ) ); for 2 (0; 1];
1
(12)
where a is the associate parameter.
The Kendall's tau and the associate parameter is linked by the following
equation:
= 1=(1 tau):
(13)
The Gumbel copula has no left tail dependence but positive right tail dependence. The tail dependence coecients can be written as
l = 0;
r = 2 21= :
(14)
Clayton copula
The Clayton copula is dened as:
C (u; v) = (u
+v 1) = for > 0
1
(15)
where is the associate parameter. The associate parameter can be expressed as a function of Kendall's tau as
= 2tau=(1 tau):
Clayton copula does not have right tail dependence but has left tail dependence as
l = 2 = ;
r = 0:
(16)
1
7
Symmetrized Joe-Clayton(SJC) copula
The SJC copula is a modication of the so called "BB7" copula of Joe
(1997). It is dened as
CSJC (u; vjr; l ) = 0:5 (CJC (u; vjr; l ) + CJC (1 u; 1 vjl ; r)); (17)
where CJC (u; vjr; l ) is the BB7 copula (also called Joe-Clayton copula)
dened as
CJC (u; vjr; l )
= 1 (1
n
1 (1 u)k
r
+ 1 (1 v)k
r
1
o
=r
1
) =k ; (18)
1
where k = 1=log (2 r), r = 1=log (l ); and l 2 (0; 1), r 2 (0; 1):
By construction, the SJC copula is symmetric when l =r.
2
2
3 Model Specication 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 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 ts the univariate distribution
of the daily exchange rate returns quite well. Many asset returns also show
autoregressive characteristics. As a result, the 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 that the marginal distributions are indeed not normal, we use the
8
Jarque-Bera normality tests for each series. The order of the autoregressive
terms k is determined by specifying the maximum being 10 and deleting the
insignicant (with signicant level of 5%) autoregressive terms. Hence the
marginal model can be specied as follows:
ri;t = mi +
2
i;t
= consti +
X
p
X
k
garch(p)i t p +
2
r
ARi;k ri;t k + "i;t ;
(19)
X
(20)
q
arch(q)i "i;t q :
2
nu
" iid tnu
i;t (nu 2) i;t
2
where ri;t is the returns for the ith asset at time t. i;t is the variance of "i;t.
and nu is the degree of freedom for the t distribution.
2
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 are 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 exible 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 nance. 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 satised 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 satises all the exibility requirements. Therefore, we choose SJC
copula for the joint model. More specically, the variables u; v in the SJC
9
copula are the cumulative distribution functions 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 breaks the estimation into two steps: at the rst step, we estimate
the parameters in the marginal distributions; at the second step, given the
estimated marginal parameters, we estimate the copula parameters. Next,
we give a brief discussion of the two estimation approaches.
Without loss of generality, we consider two marginals. By Sklar's theorem, we can decompose the joint distribution into its marginal distributions
and its dependence function (copula):
FXY (x; y) = C (FX (x); FY (y));
(21)
where FXY (x; y), FX (x), FY (y) are the joint CDF and marginal CDFs, while
C is the copula function. Taking derivative of above, we get
fXY (x; y) = fX (x) fY (y) c(u; v);
(22)
where f and c are density functions:
@F (x)
@F (y)
@ FXY (x; y)
,
fX (x) = X , fY (y) = Y ;
fXY (x; y) =
@x@y
@x
@y
@ C (FX (x); FY (y))
c(u; v) =
;
2
2
@u@v
with u = FX (x), v = FY (y).
Take logarithm of the above density function, we get:
LXY = LX + LY + LC ;
(23)
where LXY = log(fXY (x; y)), LX = log(fX (x)), LY = log(fY (y)), LC =
log(c(u; v)). The one stage full ML estimator is obtained by maximizing LXY .
Under the regularity conditions the ML estimator is consistent, ecient, and
asymptotically normal.
10
Note that in (23), the likelihood is composed of two positive parts: LX and
LY only involving the marginal parameters, and LC involving the marginal
and dependence parameters. Therefore, Joe and Xu (1996) proposed the
two step IFM method. In the rst step, they estimate the marginal models
by maximizing the logarithm likelihoods: LX and LY : In the second step,
given the estimated parameters for the marginal models, they estimate the
copula parameters by maximizing LC . Joe (1997) proves that under regular
conditions, the IFM estimator is consistent and asymptotic normal.
Compared with the ML, the IFM method is less computationally intensive. Moreover, the large number of parameters in the simultaneous ML
estimation could make numerical maximization of the likelihood function
dicult Since it is computationally easier to obtain the IFM estimator, it is
naturally worthwhile to compare the eciency of the IFM estimator with the
ML estimator. Joe(1997) points out that the IFM method is highly ecient
compared with the ML method. Joe and Xu (1996) compared the eciency
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 ecient, in that it attains the minimum
asymptotic variance bound. However, for the nite sample, Patton (2003)
found that the IFM was often more ecient than the ML, and in most cases
not less ecient. 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 rst 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 , and B = 1 (1 v)k ,
@ 2 CJC (u; vjr; l )
@u@v
= (AB ) r 1(1 u)k 1(1 v)k 1
f[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)g;
cJC (u; vjr; l ) =
(24)
11
where k = 1=log (2 r), r = 1=log (l ): The density of the SJC copula is
2
2
@ 2 CSJC (u; vjr; l )
;
@u@v
2
(u; vjr; l ) + @ 2CJC (1 u; 1 vjl ; r) ]:
= 0:5 [ @ CJC@u@v
@ (1 u)@ (1 v)
cSJC (u; vjr; l ) =
(25)
u;vjr;l
The expression for @ CJC@ uu;@ vjvl;r is the same as @ CJC@u@v
. But we
substitute u and v in the latter with 1 u and 1 v to get the former. Also
note that k = 1=log (2 l ); r = 1=log (r) for the former.
The copula logarithm likelihood is: LC = log(cSJC (u; vjr; l )). r and
l can be estimated by maxLC .
2
(1
(1
1
) (1
2
)
(
)
)
2
2
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 ve largest developed countries: US, UK, Germany, France and
Japan. Data start from 1991, since before this date various exchange rate
arrangements (currency snake 1970-1975, European Exchange Rate Mechanism 1979-1993) prevailed in the developed countries. The data end before
the introduction of the Euro.
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 Germany, TOTMFR$ for France,
TOTMJP$ for Japan). Foreign Exchange rates are expressed in US dollars
per local currency (The codes in Datastream: BRITPUS, WGMRKUS,FRNFRUS,
JAPYNUS).
The returns are calculated as 100 times the logarithm dierences 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.
12
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
that the stock markets are more volatile than the foreign exchange markets.
European stock markets are more volatile than the US stock market. The
skewnesses of returns are dierent from zero with most of them skewing to
the left. All returns show excess Kurtosis ranging from 5.65 to 14.80. Both
the skewness and the excess kurtosis indicate that the return series are not
normally distributed.
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 signicantly dierent from zero. The Japanese
stock -Yen pair has the highest linear correlation coecient of 0.47. The
French pair has the lowest linear dependence with the correlation coecient
being 0.29. 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 dierence between the probability of
the concordance and the probability of the discordance. The Kendall's taus
for our pairs of interest are all signicantly positive, showing the probability
of concordance is signicantly higher 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 signicantly
positive, indicating strong rank correlations. The values of taus and rhos
are consistent with each other and the linear correlation: The Japanese pair
have the strongest dependence, followed by the German pair, UK pair and
the French pair.
In order to see the dependence structure from the data. We calculate an
empirical copula table(see Knight, Lizieri and Satchell(2005)). To do this, we
rst 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
13
with lower returns in foreign exchange market. Thus we count the 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 positive upper tail
dependence exists, 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 seems symmetric except for the UK
pair.
4.2
Estimation Results of the Models
We rst 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 insignicant (with
signicant level of 5%) autoregressive 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 (equities) and Table 7 (foreign exchange rates). 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 0.0001 (not listed in table to save space).
For US stock index, both lag one and lag ve are signicant autoregressive
terms. That implies the returns of yesterday and the same day of last week
will signicantly aect the return today. Both ARCH and GARCH terms are
strongly signicant, indicating heteroscedasticity of the data. The number of
14
degrees of freedom for the t distribution is 5.29 and statistically signicant.
For the UK stock index returns, lag 5 is the only signicant autoregressive
term, indicating the weekly seasonality. Again the ARCH1 is the signicant
term. In contrast to the US stock index, it requires the GARCH2 term to
better t the data. The number of degrees of freedom of t is again small
(7.8) and signicant. For the other seven marginal models, the degrees of
freedom are all small (ranging from 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
signicant 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 signicance of the 10th autoregressive term.
The German Mark shows six day seasonality with AR6 being signicant. For
the stocks, GARCH(1,1) is able to capture the heteroscedasticity. For the
exchange rates, higher GARCH terms are required to better model the heteroskedasticity. This is reected in the signicance 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 signicant.
In order to evaluate the goodness of t of the marginal models, we obtain
the frequencies of the standardized residuals from the marginal models. The
result is presented in Table 7. Comparing frequencies in Table 8 with those
in Table 5 (from the data), we can see that the 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 t is generally very good. 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 9. All of the tail dependence coecients are
statistically signicant. The Japanese pair has the highest tail dependence
1
1 This
can be inferred from the strong signicance of the inverse of the t degree of
freedom shown in the Table.
15
coecients with l being 0.2479 and r being 0.2764. Since l (r) measures
the dependence between the stock returns and exchange rate returns when
both of them are in extremely small (large) values, the signicance of l (r)
means that the Japanese stock market crashes (booms) when the Japanese
yen depreciates (appreciates) heavily against US dollars and vice versa. This
nding is consistent with the basic international nance theory. 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 extreme cases as
investors are more sensitive to the extreme events.
Since the values of l and r are very similar for our pairs, we wish to
know whether the tail dependence is symmetric. We use a likelihood ratio
test to test the hypothesis: l =r. The test is presented in Table 10. All the
p-values from the test are greater than 0.10. Therefore we can not reject the
hypothesis that the upper and lower tail dependence is the same. This means
that the dependence between the stock market and the foreign exchange
market is the same at the time of crashing and booming. This nding is
dierent from the nding for the dependence structure between stock markets
documented in the literature: stock markets are more dependent at the time
of crashing than booming.
Given that we nd that the dependence is symmetric, we estimate the
copula model again by forcing the equality of l and r. The results are in
Table 11. Again we nd signicant tail dependence for all the pairs. The
values of the tail dependence are 0.2662 for Japanese pair, 0.2259 for the
German pair, 0.1667 for the British pair and 0.1015 for the French pair.
Note that the order of the degree of the tail dependence is consistent with
the linear correlation coecient, Kendall's tau and Spearman's rho for our
pairs.
Our main nding is the existence of extreme co-movements (tail dependence) between the stock market and the exchange rate market. The extreme
co-movements are symmetric, implying both markets booming and crashing
together. This nding improves our understanding of the market dependence. The signicance of tail dependence implies that the stock market
and exchange rate tend to experience concurrent extreme shocks. This has
important implications for diversication across these two markets during
extreme events. Hence it is also very important to investors who wish to
diversify investments globally. Furthermore, the nding is also important for
16
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 signicant in all the stock indexexchange rate pairs analyzed in this study. This nding is very important
for global investors in their portfolio management during extreme market
events. The nding also implies that the Gaussian dependence hypothesis
that underlies most modern nancial applications may be inadequate.
In most multivariate nancial models, dependence assumptions are very
important. Our study shows that the copula approach is a exible, 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 nancial modelling purposes. Picking up the tail dependence could
lead to a more realistic assessment of the linkage between nancial markets
and possibly more accurate risk management and pricing models.
17
References
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Model for Speculative Prices and Rates of Return, Review of Economics
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[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
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[5] Dornbusch, R. and S. Fischer, (1980), Exchange Rates and the Current
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2223-2261.
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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, 307-325.
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Copula Approach, forthcoming in the Applied Financial Economics.
18
[11] Joe H., (1997), Multivariate Models and Dependence Concepts, Volumn
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19
Specication of Data
Data source and data period: Daily data from Datastream from 1/1/1991 to 31/12/1998.
The data are from ve developed countries: US, UK, Germany, France and Japan.
Raw data:. stock market index (TOTMKUS for US, TOTMUK$ for UK, TOTMBD$
for Germany, 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-dierences of the index or the exchange
rate. r us, r uk, r gm, r fr, r jp are the returns of the US, UK, German, French, and the
Japanese stock market indices respectively. r pdus, r gmus, r frus, r jyus are the returns
from the British pound, German mark, French franc and the Japanese Yen expressed in
US dollars.
Table 1: Descriptive Statistics
Mean
Std. Dev
Skewness
Kurtosis
r us
11:1055
5:6535
r gm
0:0342
1:0800
0:8745
14:7964
2007
2007
2007
0:0689
0:8116
0:5831
Number of Observations
Mean
Std. Dev
Skewness
Kurtosis
Number of Observations
r uk
0:0372
0:8955
0:1913
r fr
0:0418
1:0561
0:2747
r jp
0:0101
1:3841
0:4955
9:1200
8:0921
2007
2007
r pdus
0:0075
0:6496
0:2603
6:7597
r gmus
0:0055
0:7013
0:0544
5:2416
r frus
0:0048
0:6783
0:2260
6:9365
r jyus
0:0090
0:7677
1:1714
12:7625
2007
2007
2007
2007
20
Table 2: Pearson Correlation
r us
r uk
r gm
r fr
r jp
r pdus
r gmus
r frus
r jyus
-0.0607
-0.1017
-0.1028
-0.0236
(0.0065)
(<0.0001)
0.3945
0.21946
0.22084
0.09927
0.3204
0.4157
0.3952
0.2071
0.2331
0.272654
0.290872
0.128172
0.168539
0.218283
0.200216
0.471459
(<0.0001)
(<0.0001)
(<0.0001)
(<0.0001)
(<0.0001)
(<0.0001)
(<0.0001)
(<0.0001)
(<0.0001)
(<0.0001)
(<0.0001)
*Numbers in brackets are p
(<0.0001)
values.
(<0.0001)
(0.2902)
(<0.0001)
(<0.0001)
(<0.0001)
(<0.0001)
Table 3: Kendall's Tau
r
r
r
r
r
us
uk
gm
fr
jp
r pdus
r gmus
r frus
r jyus
-0.05003
-0.08192*
-0.08297*
-0.03743
0.22446*
0.110616*
0.116384*
0.070818*
0.187499*
0.26832*
0.25884*
0.153973*
0.112601*
0.144408*
0.164057*
0.0767*
0.098649*
0.143137*
0.131483*
0.317182*
Note: * stands for p
value <0.0001 in Table 3 and Table 4.
Table 4: Spearman's Rho
r
r
r
r
r
us
uk
gm
fr
jp
r pdus
r gmus
r frus
r jyus
-0.0741
-0.12159*
-0.12294*
-0.05488
0.324051*
0.162147*
0.170088*
0.104126*
0.272933*
0.385582*
0.371116*
0.224556*
0.1657*
0.210926*
0.238784*
0.11302*
0.145626*
0.210924*
0.19339*
0.453602*
21
Table 5: Empirical Copula for Stock Returns and Exchange Rate Returns
UK Pair
1
2
3
4
5
6
7
8
9
10
Total
1
71
23
17
19
15
16
9
13
9
8
200
2
40
22
29
23
18
21
13
15
15
5
201
3
21
23
27
26
26
22
19
16
14
7
201
4
6
22
26
31
20
22
18
21
16
18
200
5
16
18
16
19
29
20
12
26
20
15
2001
6
13
17
21
15
25
20
27
23
21
19
201
7
12
16
19
19
13
15
34
19
17
16
200
8
6
18
17
17
18
13
22
26
33
21
201
9
9
15
17
21
17
16
21
28
24
33
201
10
6
17
12
10
10
16
25
14
32
59
201
7
9
200
German Pair
1
66
26
21
15
17
16
12
11
2
47
34
24
20
19
18
12
9
9
9
201
3
20
30
35
25
21
23
16
11
9
11
201
4
11
19
27
31
21
21
26
17
22
15
200
5
12
16
20
27
29
19
23
18
22
15
201
6
14
20
19
20
21
30
17
26
17
17
201
7
13
20
22
12
16
17
26
27
30
17
200
8
5
9
17
22
21
23
32
19
29
24
201
9
8
11
11
18
21
23
16
32
37
24
201
10
4
16
5
10
15
11
20
31
19
70
201
French Pair
1
59
27
19
16
7
15
18
12
13
14
200
2
31
41
17
16
19
11
19
17
15
15
201
3
21
20
24
20
30
22
19
17
15
13
201
4
16
23
20
17
26
26
23
18
19
12
200
5
14
17
25
17
24
20
21
25
22
16
201
6
19
13
19
28
21
26
22
19
20
14
201
7
10
14
23
24
21
22
21
20
25
20
200
8
12
22
20
25
19
19
25
22
17
20
201
9
8
12
21
21
21
19
18
27
27
27
201
10
12
13
16
13
21
14
24
28
50
201
10
Japanese Pair
1
66
41
25
18
17
11
8
6
8
0
200
2
37
27
28
27
19
18
16
13
10
6
201
3
24
25
25
26
32
25
13
16
12
3
201
4
22
32
29
25
20
15
13
12
200
15
19
21
20
25
2216
28
16
5
33
19
9
12
201
6
7
17
17
24
20
22
32
22
29
11
201
7
8
14
16
21
22
21
33
25
23
17
200
8
3
10
14
18
17
26
20
31
32
30
201
9
10
7
16
13
18
23
19
32
24
39
201
8
9
10
8
11
11
10
22
41
71
201
10
Table 6: Estimation of Marginal Models for Stocks
US stock
ApproxP r > jtj
<.0001
Variable
Estimate
Standard Error
t Value
Intercept
0.0733
0.0138
5.33
0.0507
0.0219
2.31
0.0209
-0.0475
0.021
-2.26
0.0236
ARCH0
0.004891
0.00214
2.29
0.0223
ARCH1
0.0453
0.008948
5.07
GARCH1
0.9481
0.0101
93.89
TDFI (DF=5.2854)
0.1892
0.0217
8.72
<.0001
<.0001
<.0001
Standard Error
t Value
ApproxP r > jtj
AR1 us
AR5 us
UK stock
Variable
Estimate
Intercept
0.0551
0.0175
3.15
-0.0478
0.0222
-2.15
0.0313
ARCH0
0.005698
0.003204
1.78
0.0753
ARCH1
0.0315
0.006955
4.54
GARCH2
0.9623
0.00891
108.01
TDFI (DF=7.8125)
0.128
0.0188
6.8
<.0001
<.0001
<.0001
Standard Error
t Value
Approx
AR5 uk
0.0016
German stock
Variable
Estimate
Intercept
0.0625
0.0189
3.31
0.0009
-0.046
0.0228
-2.02
0.0438
ARCH0
0.019
0.006401
2.96
0.003
ARCH1
0.0656
0.0119
5.5
GARCH1
0.9177
0.0141
65.3
TDFI (DF=6.5574)
0.1525
0.0151
10.09
<.0001
<.0001
<.0001
t Value
Approx
AR1 gm
French stock
Variable
Estimate
Standard Error
Intercept
-0.00526
0.0121
-0.43
0.6638
ARCH0
0.005555
0.002945
1.89
0.0592
ARCH1
0.0383
0.0109
3.52
0.0004
ARCH2
0.0335
0.0101
3.32
GARCH2
0.9202
0.0164
56.1
TDFI (DF=4.5413)
0.2202
0.0245
8.98
<.0001
<.0001
t Value
Approx
0.4129
Variable
Estimate
Standard Error
Intercept
-0.0191
0.0233
-0.82
0.0669
0.0223
3
0.0027
ARCH0
0.0298
0.009789
3.05
0.0023
ARCH1
0.0842
0.0137
6.13
GARCH1
0.9037
0.0146
61.68
TDFI (DF=5.5157)
0.1813
0.0222
8.15
23
* TDFI stands for the degrees of freedom of the t distribution
P r > jt j
0.0009
Japanese stock
AR1 jp
P r > jt j
<.0001
<.0001
<.0001
P r > jt j
Table 7: Estimation of Marginal Models for the Exchange Rates
British Pound
Variable
Estimate
Intercept
ApproxP r > jtj
Standard Error
t Value
0.007354
0.0109
0.68
0.4987
-0.0466
0.0212
-2.2
0.0277
0.0467
0.0188
2.49
0.0127
ARCH0
0.004911
0.001981
2.48
0.0132
ARCH1
0.0621
0.0133
4.68
GARCH2
0.9337
0.0126
74.32
TDFI (DF=3.9246)
0.2548
0.0286
8.9
AR1 pdus
AR10 pdus
German Mark
<.0001
<.0001
<.0001
ApproxP r > jtj
Variable
Estimate
Standard Error
t Value
Intercept
-0.00361
0.0132
-0.27
0.7839
-0.0493
0.02
-2.46
0.0139
ARCH0
0.006094
0.002943
2.07
0.0384
ARCH1
0.0486
0.0106
4.6
GARCH2
0.9417
0.0126
74.91
TDFI (DF=4.9702)
0.2012
0.0266
7.55
AR6 gmus
French Franc
<.0001
<.0001
<.0001
ApproxP r > jtj
Variable
Estimate
Standard Error
t Value
Intercept
-0.00526
0.0121
-0.43
0.6638
ARCH0
0.005555
0.002945
1.89
0.0592
ARCH1
0.0383
0.0109
3.52
0.0004
ARCH2
0.0335
0.0101
3.32
0.0009
GARCH2
0.9202
0.0164
56.1
TDFI (DF=4.5413)
0.2202
0.0245
8.98
Japanese Yen
<.0001
<.0001
ApproxP r > jtj
Variable
Estimate
Standard Error
t Value
Intercept
-0.0222
0.0131
-1.7
0.0899
-0.0472
0.0203
-2.32
0.0202
AR1 jyus
AR10 jyus
0.0445
0.0198
2.25
0.0247
ARCH0
0.0111
0.004079
2.71
0.0067
ARCH1
0.0424
0.0104
4.09
GARCH1
0.9395
0.0143
65.73
TDFI (DF=4.0355)
0.2478
0.0226
10.99
* TDFI stands for the degrees of freedom of the t distribution
24
<.0001
<.0001
<.0001
Table 8: Empirical Copula for the Standardized Residuals
UK Pair (from model)
1
2
3
4
5
6
7
8
9
10
Total
1
57
24
21
25
15
14
11
10
13
9
199
2
30
32
27
17
26
15
17
16
16
4
200
3
29
18
32
24
22
26
16
10
10
13
200
4
12
18
27
27
26
14
20
15
22
18
199
5
16
22
21
18
29
19
24
18
13
20
200
6
16
22
19
21
17
18
24
29
14
20
200
7
17
16
12
20
20
26
30
20
20
18
199
8
9
17
16
18
23
21
28
29
25
14
200
9
7
14
14
16
12
26
15
29
34
33
200
10
6
17
11
13
10
21
14
24
33
51
200
German Pair (from model)
1
61
26
25
13
13
19
19
6
10
7
199
2
37
35
30
19
22
16
13
14
5
9
200
3
22
24
34
32
23
21
15
8
9
12
200
4
19
20
28
27
21
21
18
17
21
7
199
5
12
19
18
23
31
22
21
18
21
15
200
6
15
22
17
21
17
23
17
26
26
16
200
7
11
24
18
21
10
23
18
35
19
20
199
8
8
10
10
21
25
23
31
28
26
18
200
9
10
13
12
11
24
24
25
22
35
24
200
4
7
8
11
14
8
22
26
28
72
200
10
French Pair (from model)
1
51
31
21
14
10
14
21
10
14
13
199
2
38
32
19
15
18
13
20
17
12
16
200
3
15
22
26
25
27
21
18
14
14
18
200
4
18
24
17
21
25
32
19
18
12
13
199
5
15
16
17
24
21
19
24
21
30
13
200
6
15
14
22
25
22
20
24
19
21
18
200
7
13
19
22
18
21
24
18
26
23
15
199
8
12
17
22
23
18
17
20
21
23
27
200
9
12
11
22
20
17
21
23
29
24
21
200
10
10
14
12
14
21
19
12
25
27
46
200
Japanese Pair (from model)
1
68
34
28
18
14
10
11
7
9
0
199
2
36
28
31
26
21
15
17
18
6
2
200
3
29
29
24
21
27
24
18
9
12
7
200
4
19
31
19
28
29
13
13
10
199
12
21
22
20
19
2516
27
21
5
26
26
15
12
200
6
13
15
22
23
27
21
24
20
22
13
200
7
6
12
17
14
21
22
25
27
31
24
199
8
3
13
16
19
20
25
21
27
29
27
200
9
7
7
11
21
12
30
23
26
28
35
200
10
6
10
10
9
10
10
13
27
35
70
200
Table 9: Results for the SJC copula models
Lower tail dependence (l )
(Standard error)
Upper tail dependence (r )
UK Pair
German Pair
French Pair
Japanese Pair
0.1711389
0.1844531
0.1144922
0.2479416
(0.0023492)
(0.0006275)
(0.0015026)
(0.0008858)
0.1616939
0.2441406
0.0914922
0.2763549
(Standard error)
(0.002332)
(0.000659)
(0.0014755)
(0.0009057)
Copula log likelihood
137.20
195.59
78.72
246.73
Table 10: Likelihood Ratio Test for Symmetric Tail Dependence
Pairs
p-value of likelihood ratio test
British stock and British pound
0.8064959405
German stock and German Mark
0.1636685341
French stock and French franc
0.3322778417
Japanese stock and Japanese yen
0.3961439092
Table 11: Results for the SJC copula models when l = r
UK pair
German Pair
French Pair
Japanese Pair
Tail dependence
0.1666957
0.2258588
0.101496
0.2662292
(Standard error)
(0.0168171)
(0.016647)
(0.0162581)
(0.0160155)
Copula log likelihood
137.17
194.62
78.25
246.37
26