independent if »U equals 0. Further, we call »U the upper tail-dependence

coe¬cient (TDC). See Schmidt and Stadtm¨ller (2002) for a generalization to

u

multivariate tail dependence. Similarly, we de¬ne the lower tail-dependence

coe¬cient by

’1 ’1

»L := lim+ P{X1 ¤ F1 (v) | X2 ¤ F2 (v)}. (2.2)

v’0

We embed the tail-dependence concept within the copula theory. An n-dimen-

sional distribution function C : [0, 1]n ’ [0, 1] is called a copula if it has

one-dimensional margins which are uniformly distributed on the interval [0, 1].

Copulae are functions that join or ”couple” an n-dimensional distribution func-

tion F to its corresponding one-dimensional marginal-distribution functions

2.1 Tail dependence and copulae 35

’1 ’1

Figure 2.2: »U (v) = P{X1 > F1 (v) | X2 > F2 (v)} for a bivariate t-

distribution with ρ = ’0.8, ’0.6, . . . , 0.6, 0.8. The exact formula

for »U > 0 is given in Section ??.

Fi , i = 1, . . . , n, in the following way:

F (x1 , . . . , xn ) = C{F1 (x1 ), . . . , Fn (xn )}.

We refer the reader to the monographs of Nelsen (1999) or Joe (1997) for more

information on copulae.

The following representation shows that tail dependence is a copula property.

Thus many copula features transfer to the tail-dependence coe¬cient, for ex-

ample the invariance under strictly increasing transformations of the margins.

If X is a continuous bivariate random vector, then

1 ’ 2v + C(v, v)

»U = lim’ , (2.3)

1’v

v’1

C(v,v)

where C denotes the copula of X. Analogously, »L = limv’0+ holds for

v

the lower tail-dependence coe¬cient.

36 2 Tail dependence

2.2 Calculating the tail-dependence coe¬cient

2.2.1 Archimedean copulae

Referring to H¨rdle, Kleinow and Stahl (2002) Archimedean copulae are im-

a

plemented and investigated within the Value at Risk framework. Therefore

and because Archimedean copulae form an important class of copulae which

are easy to construct and have nice properties, we explore the tail-dependence

coe¬cient for this family of copulae. A bivariate Archimedean copula has the

form C(u, v) = φ[’1] φ(u) + φ(v) for some continuous, strictly decreasing, and

convex function φ : [0, 1] ’ [0, ∞] such that φ(1) = 0 and the pseudo-inverse

function φ[’1] is de¬ned by

φ’1 (t), 0 ¤ t ¤ φ(0),

φ[’1] (t) =

φ(0) < t ¤ ∞.

0,

We call φ strict if φ(0) = ∞. In that case φ[’1] = φ’1 . Regarding the existence

of tail dependence for Archimedean copulae we can show that:

1. Upper tail-dependence implies φ (1) = 0

and »U = 2 ’ limv’1’ (φ[’1] —¦ 2φ) (v),

2. φ (1) < 0 implies upper tail-independence,

3. φ (0) > ’∞ or a non-strict φ implies lower tail-independence,

4. Lower tail-dependence implies φ (0) = ’∞, »L = limv’0+ (φ’1 —¦ 2φ) (v),

and a strict φ.

Table 2.1 lists various Archimedean copulae in the same ordering as in Table 2.1

in H¨rdle, Kleinow and Stahl (2002) or in Nelsen (1999) and the corresponding

a

upper and lower TDC.

The quantlet TailCoeffCopula derives the TDC for a large number of bivari-

ate copula-functions. The inputs of this quantlet are the type of copula and

the copula parameters. The type of copula is speci¬ed by an integer, between 1

and 34, listed in the following tables. For instance copula=1 - Pareto-Clayton,

copula=4 - Gumbel-Hougaard, and copula=5 - Frank. The result of the appli-

cation is assigned to the vector (lTDC,uTDC) and contains the lower and upper

TDC, respectively. We refer the reader to the VARCopula quantlet for related

2.2 Calculating the tail-dependence coe¬cient 37

copula calculations.

(lTDC,uTDC) = TailCoeffCopula (copula,parameters)

Calculates the lower and upper tail-dependence coe¬cient for var-

ious copulae

2.2.2 Elliptically contoured distributions

In this section, we calculate the tail-dependence coe¬cient for elliptically con-

toured distributions (brie¬‚y: elliptical distributions). Well-known elliptical

distributions are the multivariate normal distribution, the multivariate t-distri-

bution, the multivariate logistic distribution, and the multivariate symmetric

generalized-hyperbolic distribution.

Elliptical distributions are de¬ned as follows: Let X be an n-dimensional ran-

dom vector and Σ ∈ Rn—n be a symmetric positive semi-de¬nite matrix. If

X ’ µ, for some µ ∈ Rn , possesses a characteristic function of the form

φX’µ (t) = ¦(t Σt) for some function ¦ : R+ ’ R, then X is said to be

0

elliptically distributed with parameters µ (location), Σ (dispersion), and ¦.

Let En (µ, Σ, ¦) denote the class of elliptically contoured distributions with the

latter parameters. We call ¦ the characteristic generator.

The density function, if existent, of an elliptically contoured distribution has

the following form:

f (x) = |Σ|’1/2 g(x ’ µ) Σ’1 (x ’ µ), x ∈ Rn , (2.4)

for some function g : R+ ’ R+ , which we call the density generator. Observe

0 0

that the name ”elliptically contoured” distribution is related to the elliptical

contours of the latter density. For a more detailed treatment of elliptical distri-

butions see Fang, Kotz, and Ng (1990), Cambanis, Huang, and Simon (1981).

Bingham and Kiesel (2002) and Bingham, Kiesel, and Schmidt (2002) propose a

semi-parametric approach for ¬nancial modelling by estimating the parametric

components (µ, Σ) and non-parametric components (density generator g) of an

elliptical distribution. Regarding the tail-dependence coe¬cient, it su¬ces to

consider the tail behavior of the density generator.

38

Table 2.1: Tail-dependence coe¬cient (TDC) for various Archimedean copulae. Numbers correspond to table

4.1 in Nelsen (1999), p. 94.

Number C(u, v) upper-TDC lower-TDC

φθ (t) θ∈

& Type

(1) Pareto 0 for θ > 0 2’1/θ

max [u’θ + v ’θ ’ 1]’1/θ , 0 t’θ ’ 1)/θ [’1, ∞)\{0}

for θ > 0

1/θ