elliptical copulas: applicability and limitations

Statistics & Probability Letters 63 (2003) 275–286

Elliptical copulas: applicability and limitations

Gabriel Frahma ;∗, Markus Junkera, Alexander Szimayerb

aResearch Center caesar, Financial Engineering, Ludwig-Erhard-Allee 2, 53175 Bonn, GermanybUWA Business School, Crawley, Australia

Received October 2002; received in revised form January 2003

Abstract

We study copulas generated by elliptical distributions. We show that their tail dependence can be simplycomputed with default routines on Student’s t-distribution given Kendall’s � and the tail index. The copulafamily generated by the sub-Gaussian �-stable distribution is unable to cover the size of tail dependenceobserved in 8nancial data.c© 2003 Elsevier Science B.V. All rights reserved.

MSC: 62H05; 62H20; 62P05

Keywords: Copula; Elliptical distribution; Kendall’s tau; Students’s t-distribution; Sub-Gaussian alpha-stable distribution;Tail dependence

1. Introduction

Elliptical distribution families are widely applied in statistics and econometrics, especially in8nance. For example, Owen and Rabinovitch (1983) discuss the impact of elliptical distributionson portfolio theory, and Kim (1994) studies the CAPM for this class of distributions. The normaldistribution is the archetype of elliptical distributions in 8nance. Besides this distribution family,other elliptical distribution families became more important. Student’s t-distribution is emphasized,e.g., by Blattberg and Gonedes (1974). The sub-Gaussian �-stable distribution is a special case ofthe extensive stable distribution class discussed by Rachev and Mittnik (2000). Portfolio selectionwith sub-Gaussian �-stable distributed returns is analyzed by Ortobelli et al. (2002).

∗ Corresponding author. Tel.: +49-2-28-9656-258; fax: +49-2-28-9656-111.E-mail addresses: [email protected] (G. Frahm), [email protected] (M. Junker), [email protected]

(A. Szimayer).URL: http://www.caesar.de/8nancialengineering

0167-7152/03/$ - see front matter c© 2003 Elsevier Science B.V. All rights reserved.doi:10.1016/S0167-7152(03)00092-0

mailto:[email protected]



http://www.caesar.de/financialengineering

276 G. Frahm et al. / Statistics & Probability Letters 63 (2003) 275–286

In this note we study the dependence structure generated by elliptical distributions. We focuson the multivariate Student’s t-distribution and the multivariate sub-Gaussian �-stable distributionsince these distribution families are frequently applied for modeling 8nancial data. This subjectis relevant with respect to risk analysis in 8nance. Here, the probability of joint extremal eventshas to be determined because the aggregate portfolio risk increases seriously in case of noticeabledependence of the particular assets in crash situations. Therefore, besides rank correlation measures,like Kendall’s �, the concept of tail dependence is of main interest. The copula concept is utilized toextract the dependence structure of a given distribution by the copula function. Thus, it is obtaineda separation of the marginal distributions of the particular random components and the dependencestructure that is contained in the copula. Recent developments on elliptical distributions concerningthe dependence structure can be found in Hult and Lindskog (2001), and Schmidt (2002).For copulas generated by elliptically distributed random vectors we discuss a general relationship

between Kendall’s �, the tail index of the underlying elliptical distribution, and the tail dependenceof the generated copula. Given Kendall’s � and the tail index � the tail dependence can be sim-ply computed with default routines on Student’s t-distribution. Applying these results we 8nd thatsub-Gaussian �-stable copulas are not suitable for modeling the dependence structure of 8nancialrisk. Empirical investigations have shown that the tail dependence of stocks is smaller than thoseprovided by the sub-Gaussian �-stable distribution assumption.

2. Copulas and dependence measures

An axiomatic de8nition of copulas is to be found in Joe (1997) and Nelsen (1999). Accordingto this a copula is a d-dimensional distribution function C : [0; 1]d → [0; 1]. Owing to our interest incopula families we have to study copulas generated by speci8c classes of distributions as follows:

De�nition 2.1 (Copula of a random vector X ): Let X = (X1; : : : ; Xd)′ be a random vector with mul-tivariate distribution F and continuous margins F1; : : : ; Fd. The copula of X (of the distribution F ,respectively) is the multivariate distribution C of the random vector

U = (F1(X1); : : : ; Fd(Xd))′:

Due to the continuity of the margins F1; : : : ; Fd every random variable Fi(Xi) = Ui is standarduniformly distributed, i.e. Ui ∼ U (0; 1). Thus the copula of a continuous random vector X hasuniform margins, and

C(u1; : : : ; ud) = F(F←1 (u1); : : : ; F←d (ud)) (1)

holds, where

F←i (ui) := inf{x:Fi(x)¿ui}; i = 1; : : : ; d

are the marginal quantile functions. That is to say by copula separation only the dependence structureis extracted from F .In general the multivariate distribution F contains parameters that do not aLect the copula of F ,

and other parameters aLects the copula and possibly the margins. We want to call the latter type ofparameters “copula parameters”.

G. Frahm et al. / Statistics & Probability Letters 63 (2003) 275–286 277

Let � be a parameter vector (�1; : : : �n)′ ∈Rn and F(·; �) a continuous multivariate distributionwith copula C(·; �). Let IC ⊂ I = {1; : : : ; n} be an index-set that contains all k for which at leastone u∈ [0; 1]d exists, such that

@C(u; �)@�k

= 0:So IC contains all copula parameter indices.Suppose a d-dimensional distribution family is generated by a multivariate distribution F∗(·; �0)

with continuous margins F∗1 (·; �0); : : : ; F∗d (·; �0) and d continuous and strictly monotone increas-ing marginal transformations h1(·; �1); : : : ; hd(·; �d), where the parameters �1; : : : ; �d may be somereal-valued vectors:

F(x1; : : : ; xd; �) = F∗(h1(x1; �1); : : : ; hd(xd; �d); �0) (2)

with

�= (�0; �1; : : : ; �d)′:

Lemma 2.2. Only the vector �0 contains copula parameters.

Proof. The lemma follows from the fact that any copula is invariant under strictly monotone in-creasing transformations h1(·; �1); : : : ; hd(·; �d). Thus also the parameters �1; : : : ; �d cannot aLect thecopula.

So the parameters �1; : : : ; �m are canceled down through copula separation and �0 still remains.We call the distribution F∗(·; �0) the “underlying distribution” of C(·; �), and C(·; �) the “copulagenerated” by F∗(·; �0). In particular the copula generated by an elliptical distribution will be called“elliptical copula”.AOne marginal transformations are often applied for constructing distribution families, more pre-

cisely location-scale-families. The location-scale-family generated by the multivariate distribution F∗contains all distributions

(x1; : : : ; xd)′ → F(x1; : : : ; xd; �) = F∗(x1 − �1�1

; : : : ;xd − �d�d

; �0

)with given parameter vector �0, variable location parameters �1; : : : ; �d and scale parameters �1; : : : ; �d.So this distribution family is generated by aOne marginal transformations and the location and scaleparameters are not copula parameters.Let us turn towards the dependence structure in F(·; �). Kendall’s � is an appropriate dependence

measure for monotonic dependence.

De�nition 2.3 (Kendall’s �): Let the bivariate random vector (X ; Y )′ be an independent copy of(X; Y )′. Kendall’s � is

�(X; Y ) := P((X − X )(Y − Y )¿ 0)− P((X − X )(Y − Y )¡ 0):

Kendall’s � is a rank correlation, so

�(X; Y ) = �(FX (X ); FY (Y ))


holds, i.e. Kendall’s � is completely determined by the copula of (X; Y )′ and so it depends only onthe copula parameters of the distribution of (X; Y )′.In addition to monotonic dependence, which is measured by rank correlation, 8nancial data is

likely to exhibit lower tail dependence, that is to say a high probability of extreme simultaneouslosses.

De�nition 2.4 (Tail dependence): Let C be the copula of (X; Y )′. The lower tail dependence is

�L(X; Y ) := limu↘0

P(Y 6F←Y (u) |X 6F←X (u)) = limu↘0

C(u; u)u

: (3)

The upper tail dependence of X and Y is

�U(X; Y ) := limu↗1

P(Y ¿F←Y (u) |X ¿F←X (u)) = limu↗1

1− 2u+ C(u; u)1− u

;

respectively.

Since tail dependence is de8ned by means of copula, beside Kendall’s � also �L and �U dependonly on copula parameters.

3. Elliptical distributions

3.1. Characterizations

De�nition 3.1 (Elliptical distribution): The d-dimensional random vector X has an elliptical distri-bution if and only if the characteristic function t → E(exp(it′X )) with t ∈Rd has the representation

t → �g(t; �; ; #) = exp(it′�)g(t′ t;#):

Here g(·;#) : [0;∞[→ R; #∈Rm, �∈Rd, and is a symmetric positive semide8nite d×d-matrix.

The parameter vector � is a location parameter, and the matrix determines the scale and thecorrelation of the random variables X1; : : : ; Xd. The function g(·;#) constitutes the distribution familyand is called “characteristic generator”, whereas # is a parameter vector that determines the shape,in particular the tail index of the distribution.For example the characteristic function of the sub-Gaussian �-stable distribution (Ortobelli et al.,

2002) is

t → exp(it′�) exp

(−(12t′ t)�=2)

; 0¡�¡ 2;

so the characteristic generator is x → exp(−( 12 x)�=2) with shape parameter �.


Now let

=

�11 · · · �1d...

. . ....

�d1 · · · �dd

; � :=

�1 0 · · · 0

0 �2...

.... . .

...

0 · · · · · · �d

; # :=

#11 · · · #1d...

. . ....

#d1 · · · #dd

with

�i :=√�ii; i = 1; : : : ; d

and

#ij :=�ij�i�j

; i; j = 1; : : : ; d; (4)

so that = �#� and �g(·; �; ; #) ≡ �g(·; �; �; #; #). Consider the multivariate elliptical distributionF∗g (·; #; #) corresponding to the characteristic function �g(·; 0; 1; #; #). The characteristic functioncorresponding to the multivariate distribution

(x1; : : : ; xd)′ → F∗g

(x1 − �1�1

; : : : ;xd − �d�d

; #; #)

with �∈Rd and �1; : : : ; �d ¿ 0 is

t → exp(it′�)�g(�t; 0; 1; #; #) = exp(it′�)g((�t)′#(�t);#)

= exp(it′�)g(t′(�#�)t;#)

=�g(t; �; ; #):

Thus the location-scale-family generated by a standard elliptical distribution is elliptical, too. Thisis emphasized by a stochastic representation of the elliptical class which is stated by the followingtheorem:

Theorem 3.2 (Fang et al., 1990): The d-dimensional random vector X is elliptically distributed withcharacteristic function �g(·; �; ; #) and rank( ) = r6d if and only if there exist a r-dimensionalrandom vector U , uniformly distributed on the unit sphere surface

{u∈ [− 1; 1]r: ‖u‖= 1};a nonnegative random variable R independent of U , and a d × r-matrix

√ with

√ √ ′= ,

such that

X d= � +R√ U: (5)

Due to the transformation matrix√ the uniform random vector U produces elliptically contoured

density level surfaces, whereas the “generating random variable” R (Schmidt, 2002, p. 6) gives thedistribution shape, in particular the tailedness of the distribution.


With the reparametrization (4) the equation√ = �

√# holds, and with it

X d= � + �R√#U:

3.2. Dependence structure

The standard density of the d-dimensional sub-Gaussian �-stable distribution can be obtainedthrough multivariate Fourier-transformation and is

f∗�;#(x) =1

(2')d

∫Rd�stable(t; 0; 1; #; �) exp(−it′x) dt′

=1

(2')d

∫Rdexp

(−(12t′#t)�=2)

cos(t′x) dt′; 0¡�¡ 2:

The copula generated by a d-dimensional sub-Gaussian �-stable distribution is

C�(u1; : : : ; ud) = F∗�;#(F∗←�;1 (u1); : : : ; F

∗←�;1 (ud));

where F∗�;# is the multivariate standard distribution function

F∗�;#(x) =∫]−∞; x]

f∗�;#(w) dw′

with ] − ∞; x] := ] − ∞; x1] × · · · × ] − ∞; xd], and F∗←�;1 is the inverse of the univariate standarddistribution function

F∗�;1(x) =∫ x

−∞f∗�;1(w) dw:

Anymore the standard density of the d-dimensional multivariate Student’s t-distribution with )degrees of freedom is

f∗t (x) =*(()+ d)=2)*()=2)

√1=|#|()')d

exp

(−12log(1 +

x′#−1x)

))+d); )¿ 0:

For continuous elliptical distributions there is a straight link between Kendall’s � and Pearson’s #:

Theorem 3.3 (Lindskog et al., 2001). Let X be an elliptical distributed random vector with char-acteristic function �g(·; �; �; #; #). For two continuous components of X , Xi and Xj, Kendall’s� is

�(Xi; Xj) =2'arcsin(#ij): (6)

That is to say Kendall’s � depends only on # and neither the characteristic generator nor the shapeof the distribution aLects the rank correlation (see also Fang et al., 2002).We know that the speci8ed distributions are heavy tailed, i.e. the marginal survival functions RFi

exhibit a power law with tail index �¿ 0:

RFi(x) = �i(x)x−�; x¿ 0; (7)


for i=1; : : : ; d. Here �1; : : : ; �d are slowly varying functions. Note that elliptically contoured distribu-tions are symmetric (heavy) tailed, i.e. the tail index of the upper tail RFi is the same for the lowertail Fi, too.For the multivariate Student’s t-distribution the tail index corresponds to the number of the degrees

of freedom ), and for the multivariate sub-Gaussian �-stable distribution the tail index equals to�. Furthermore in the elliptical framework the lower tail dependence is equal to the upper taildependence. The following theorem connects the tail index with the tail dependence of ellipticaldistributions:

Theorem 3.4 (Schmidt, 2002). Let X be an elliptically distributed random vector with character-istic function �g(·; �; �; #; #) and tail index �. For two components of X , Xi and Xj, the taildependence is

�(Xi; Xj; �; #ij) =

∫ f(#ij)0

u�√u2−1du∫ 1

0u�√u2−1 du

; �¿ 0; f(#ij) =

√1 + #ij2

: (8)

So the tail dependence in elliptical copulas is a function #�→�, where the tail index � of the

underlying elliptical distribution family results from the characteristic generator g and the shapeparameter #. Given the matrix # the tail dependence is a function �

#→ �, and due to Theorem 3.3also the relation � �→ � holds for a given matrix of Kendall’s �.

Remark 3.5. The tail index is a property of the elliptical distribution family from which the copulais extracted from, whereas the tail dependence concerns the copula by itself. By Sklar’s theorem(Nelsen, 1999) it is possible to construct new multivariate distributions with arbitrary margins,providing a speci8c copula. In this case � is generally not the tail index of the proposed marginaldistributions but still a copula parameter.

Substituting the integration variable u in Eq. (8) by cos(v) leads to the following equivalentrepresentation of the tail dependence of two elliptically distributed random variables Xi and Xj(observed by Hult and Lindskog, 2001):

�(Xi; Xj; �; #ij) =

∫ '=2g(#ij)

cos�(v) dv∫ '=20 cos�(v) dv

; g(#ij) = arccos

(√1 + #ij2

):

Due to relation (6) we can substitute #ij by sin(�ij('=2)), and we get

�(Xi; Xj; �; �ij) =

∫ '=2h(�ij)


; h(�ij) ='2

(√1− �ij2

): (9)

Thus for the limiting case �= 0 the tail dependence is an aOne function of Kendall’s �:

lim�↘0

�(Xi; Xj; �; �ij) =1 + �ij2

: (10)


Fig. 1. Tail dependence barriers for elliptical copulas as a function of #. The range of possible tail dependence for �¡ 2is marked dark-grey.

Remark 3.6. Since every random variable X1; : : : ; Xd has the same tail index � which is given bythe generating random variable R, the tail dependence �ij of each bivariate combination (Xi; Xj)′is uniquely determined by �ij. Thus modeling the tail dependence structure of elliptical copulasespecially for higher dimensions is restricted by the set {(�; �)∈ [0; 1]× [− 1; 1]: �= �(�; �)} giventhe tail index parameter �.

The tail dependence of a bivariate t-distributed random vector (X; Y )′ with ) degrees of freedomcan be obtained by computation of (3) and is

�=2Rt)+1

(√)+ 1

√1− #1 + #

)

=2Rt)+1

(√)+ 1

√1− sin(�('=2))1 + sin(�('=2))

); )¿ 0; (11)

where Rt)+1 is the survival function of the univariate Student’s t-distribution with ) + 1 degrees offreedom (see, e.g., Embrechts et al., 2002).Since Eq. (11) holds for all )¿ 0, where ) corresponds to the tail index � of X and Y , and

Theorem 3.4 states that the tail dependence of two elliptically distributed random variables dependsonly on #ij and �, Eq. (8) can be replaced by

�ij =2Rt�+1

(√�+ 1

√1− #ij1 + #ij

)

=2Rt�+1

(√�+ 1

√1− sin(�ij('=2))1 + sin(�ij('=2))

); �¿ 0: (12)

Student’s t-distribution is a default routine in statistics software and is tabulated in many textbooks(Johnson et al., 1995). So it is more convenient to use formula (12) than (8). In Fig. 1 we plot thebarriers of tail dependence as a function of # for any elliptical copula allowing for �¿ 0. The range


Fig. 2. Tail dependence barriers for elliptical copulas as a function of �. The range of possible tail dependence for �¡ 2is marked dark-grey.

of possible tail dependence in the special case �¡ 2, which holds for the sub-Gaussian �-stablecopula, is marked explicitly.An investigation of several stocks from the German and the US market shows that the lower

tail dependence ranges from 0 to 0.35, whereas Kendall’s � takes values in between 0 to 0.4,approximately (Junker, 2002). With formula (9) we can plot the tail dependence barriers as afunction of Kendall’s �, see Fig. 2. Note that for the limit case � = 0 the tail dependence is anaOne function of �, as stated by (10). We see that the sub-Gaussian �-stable copula restricts thescope of possible tail dependence too much. The dependence structure generated by the multivariatesub-Gaussian �-stable distribution is not suitable for modeling 8nancial risk because the providedrange of � has only a small intersection with the empirical results.

Appendix A. Derivations

A.1. Multidimensional sub-Gaussian standard density

The standard characteristic function of the d-dimensional sub-Gaussian �-stable distribution is

�stable(t; 0; 1; #; �) = exp

(−(12t′#t)�=2)

; t = (t1; : : : ; td)′: (A.1)

Hence �stable is an even function due to the quadratic form contained in (A.1). Furthermore theintegral∫

Rdexp

(−(12t′#t)�=2)

dt′

converges absolutely, that is to say the Fourier-transform

f∗�;#(x) =1

(2')d

∫Rdexp

(−(12t′#t)�=2)

exp(−it′x) dt′ (A.2)

exists.


At 8rst we consider the bivariate case. Since

exp(−it′x) = exp(−it1x1) exp(−it2x2)= (cos(t1x1)− i sin(t1x1)) (cos(t2x2)− i sin(t2x2))= (cos(t1x1) cos(t2x2)− sin(t1x1) sin(t2x2))− i(cos(t1x1) sin(t2x2) + sin(t1x1) cos(t2x2));

the real part of exp(−it′x) is an even function whereas its imaginary part is an odd function oft = (t1; t2)′. Because the characteristic function is even also the real part of the integrand

exp

(−(12t′#t)�=2)

exp(−it′x)

is even, and the imaginary remains odd, too. Hence the imaginary part can be eliminated from theFourier-transform (A.2), and with the addition formula (A.4) we get

f∗�;#(x) =1

(2')2

∫R2exp

(−(12t′#t)�=2)

cos(t′x) dt′:

This can be extended analogously for d¿ 2 dimensions.

A.2. Equivalent representation of Eq. (8)

Once again Eq. (8) is

�(Xi; Xj; �; #ij) =

∫ f(#ij)0

u�√u2−1 du∫ 1

0u�√u2−1 du

; �¿ 0; f(#ij) =

√1 + #ij2

:

Substituting u by cos(v) leads to

�(Xi; Xj; �; #ij) =

∫ arccos(f(#ij))arccos(0)

cos�(v)√cos2(v)−1 (−sin(v)) dv∫ arccos(1)

arccos(0)cos�(v)√cos2(v)−1 (−sin(v)) dv

=

∫ arccos(f(#ij))'=2

cos�(v)√−1√1−cos2(v) (−sin(v)) dv∫ 0

'=2cos�(v)√−1√1−cos2(v) (−sin(v)) dv

:

Since√1− cos2(v) = sin()),

�(Xi; Xj; �; #ij) =− 1

i

∫ arccos(f(#ij))'=2 cos�(v) dv

− 1i

∫ 0'=2 cos

�(v) dv=

∫ '=2arccos(f(#ij))


:


Now #ij is substituted by sin(�ij('=2)) which leads to the lower limit

arccos(f(#ij)) = arccos(f(sin(�ij'2

)))= arccos

(√1 + sin(�ij('=2))

2

):

We consider the addition formula

sin(’± �) = sin(’) cos(�)± cos(’) sin(�): (A.3)

Since sin(�('=2) = sin(('=2)− (1− �)('=2)) it follows from (A.3) that

sin(�'2

)= sin

('2

)cos((1− �)

'2

)− cos

('2

)sin((1− �)

'2

)= cos

((1− �)

'2

):

Now we are searching for a function h ful8lling

h(�) = arccos

(√1 + sin(�('=2))

2

)= arccos

(√1 + cos((1− �) '=2)

2

);

but without trigonometric components. That is to say

cos((1− �)

'2

)=2 cos2(h(�))− 1 = cos2(h(�))− (1− cos2(h(�)))

= cos2(h(�))− sin2(h(�)); ∀�∈ [− 1; 1]:Now we consider the addition formula

cos(’∓ �) = cos(’) cos(�)± sin(’) sin(�): (A.4)

For ’= � formula (A.4) becomes cos(2’) = cos2(’)− sin2(’), and thuscos((1− �)

'2

)= cos(2h(�)):

Hence

h : � → '2

(1− �2

)

and thus

�(Xi; Xj; �; �ij) =

∫ '=2h(�ij)


; h(�ij) ='2

(1− �ij2

):

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elliptical copulas: applicability and limitations

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