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Bernoulli and Tail-Dependence Compatibility

Paul Embrechts∗, Marius Hofert†and Ruodu Wang‡

July 30, 2015

Abstract

The tail-dependence compatibility problem is introduced. It raises the question whether

a given d×d-matrix of entries in the unit interval is the matrix of pairwise tail-dependence co-

efficients of a d-dimensional random vector. The problem is studied together with Bernoulli-

compatible matrices, i.e., matrices which are expectations of outer products of random vec-

tors with Bernoulli margins. We show that a square matrix with diagonal entries being 1 is

a tail-dependence matrix if and only if it is a Bernoulli-compatible matrix multiplied by a

constant. We introduce new copula models to construct tail-dependence matrices, including

commonly used matrices in statistics.

Key-words: Tail dependence, Bernoulli random vectors, compatibility, matrices, copulas, in-

surance application.

MSC2010: 60E05, 62H99, 62H20, 62E15, 62H86.

1 Introduction

The problem of how to construct a bivariate random vector (X1, X2) with log-normal

marginals X1 ∼ LN(0, 1), X2 ∼ LN(0, 16) and correlation coefficient Cor(X1, X2) = 0.5 is

well known in the history of dependence modeling, partially because of its relevance to risk

management practice. The short answer is: There is no such model; see Embrechts et al. (2002)

who studied these kinds of problems in terms of copulas. Problems of this kind were brought

to RiskLab at ETH Zurich by the insurance industry in the mid 1990s when dependence was

∗RiskLab and SFI, Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland.

Email: embrechts@math.ethz.ch†Department of Statistics and Actuarial Science, University of Waterloo, Canada.

Email: marius.hofert@uwaterloo.ca‡Department of Statistics and Actuarial Science, University of Waterloo, Canada.

Email: wang@uwaterloo.ca

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thought of in terms of correlation (matrices). For further background to Quantitative Risk Man-

agement, see McNeil et al. (2015). Now, almost 20 years later, copulas are a well established

tool to quantify dependence in multivariate data and to construct new multivariate distribu-

tions. Their use has become standard within industry and regulation. Nevertheless, dependence

is still summarized in terms of numbers (as opposed to (copula) functions), so-called measures

of association. Although there are various ways to compute such numbers in dimension d > 2,

measures of association are still most widely used in the bivariate case d = 2. A popular measure

of association is tail dependence. It is important for applications in Quantitative Risk Manage-

ment as it measures the strength of dependence in either the lower-left or upper-right tail of the

bivariate distribution, the regions Quantitative Risk Management is mainly concerned with.

We were recently asked1 the following question which is in the same spirit as the log-

normal correlation problem if one replaces “correlation” by “tail dependence”; see Section 3.1

for a definition.

For which α ∈ [0, 1] is the matrix

Γd(α) =

1 0 · · · 0 α

0 1 · · · 0 α...

.... . .

......

0 0 · · · 1 α

α α · · · α 1

(1.1)

a matrix of pairwise (either lower or upper) tail-dependence coefficients?

Intrigued by this question, we more generally consider the following tail-dependence compatibility

problem in this paper:

When is a given matrix in [0, 1]d×d the matrix of pairwise (either lower or upper)

tail-dependence coefficients?

In what follows, we call a matrix of pairwise tail-dependence coefficients a tail-dependence

matrix. The compatibility problems of tail-dependence coefficients were studied in Joe (1997).

In particular, when d = 3, inequalities for the bivariate tail-dependence coefficients have been

established; see Joe (1997, Theorem 3.14) as well as Joe (2014, Theorem 8.20). The sharpness of

these inequalities is obtained in Nikoloulopoulos et al. (2009). It is generally open to characterize

the tail-dependence matrix compatibility for d > 3.

Our aim in this paper is to give a full answer to the tail-dependence compatibility problem;

see Section 3. To this end, we introduce and study Bernoulli-compatible matrices in Section 2.

1by Federico Degen (Head Risk Modeling and Quantification, Zurich Insurance Group) and Janusz Milek

(Zurich Insurance Group)

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As a main result, we show that a matrix with diagonal entries being 1 is a compatible tail-

dependence matrix if and only if it is a Bernoulli-compatible matrix multiplied by a constant. In

Section 4 we provide probabilistic models for a large class of tail-dependence matrices, including

commonly used matrices in statistics. Section 5 concludes.

Throughout this paper, d and m are positive integers, and we consider an atomless prob-

ability space (Ω,A,P) on which all random variables and random vectors are defined. Vectors

are considered as column vectors. For two matrices A,B, B > A and B 6 A are understood as

component-wise inequalities. We let A B denote the Hadamard product, i.e., the element-wise

product of two matrices A and B of the same dimension. The d× d identity matrix is denoted

by Id. For a square matrix A, diag(A) represents a diagonal matrix with diagonal entries equal

to those of A, and A> is the transpose of A. We denote IE the indicator function of an event

(random or deterministic) E ∈ A. 0 and 1 are vectors with all components being 0 and 1

respectively, as long as the dimension of the vectors is clear from the context.

2 Bernoulli compatibility

In this section we introduce and study the Bernoulli-compatibility problem. The results

obtained in this section are the basis for the tail-dependence compatibility problem treated in

Section 3; many of them are of independent interest, e.g., for the simulation of sequences of

Bernoulli random variables.

2.1 Bernoulli-compatible matrices

Definition 2.1 (Bernoulli vector, Vd). A Bernoulli vector is a random vector X supported by

0, 1d for some d ∈ N. The set of all d-Bernoulli vectors is denoted by Vd.

Equivalently, X = (X1, . . . , Xd) is a Bernoulli vector if and only if Xi ∼ B(1, pi) for some

pi ∈ [0, 1], i = 1, . . . , d. Note that here we do not make any assumption about the dependence

structure among the components of X. Bernoulli vectors play an important role in Credit Risk

Analysis; see, e.g., Bluhm and Overbeck (2006) and Bluhm et al. (2002, Section 2.1).

In this section, we investigate the following question which we refer to as the Bernoulli-

compatibility problem:

Question 1. Given a matrix B ∈ [0, 1]d×d, can we find a Bernoulli vector X such that B =

E[XX>]?

For studying the Bernoulli-compatibility problem, we introduce the notion of Bernoulli-

compatible matrices.

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Definition 2.2 (Bernoulli-compatible matrix, Bd). A d× d matrix B is a Bernoulli-compatible

matrix, if B = E[XX>] for some X ∈ Vd. The set of all d× d Bernoulli-compatible matrices is

denoted by Bd.

Concerning covariance matrices, there is extensive research on the compatibility of covari-

ance matrices of Bernoulli vectors in the realm of statistical simulation and time series analysis;

see, e.g., Chaganty and Joe (2006). It is known that, when d > 3, the set of all compatible d-

Bernoulli correlation matrices is strictly contained in the set of all correlation matrices. Note that

E[XX>] = Cov(X) + E[X]E[X]>. Hence, Question 1 is closely related to the characterization

of compatible Bernoulli covariance matrices.

Before we characterize the set Bd in Section 2.2 and thus address Question 1, we first collect

some facts about elements of Bd.

Proposition 2.1. Let B,B1, B2 ∈ Bd. Then

i) B ∈ [0, 1]d×d.

ii) maxbii + bjj − 1, 0 6 bij 6 minbii, bjj for i, j = 1, . . . , d and B = (bij)d×d.

iii) tB1 + (1− t)B2 ∈ Bd for t ∈ [0, 1], i.e., Bd is a convex set.

iv) B1 B2 ∈ Bd, i.e., Bd is closed under the Hadamard product.

v) (0)d×d ∈ Bd and (1)d×d ∈ Bd.

vi) For any p = (p1, . . . , pd) ∈ [0, 1]d, the matrix B = (bij)d×d ∈ Bd where bij = pipj for i 6= j

and bii = pi, i, j = 1, . . . , d.

Proof. Write B1 = E[XX>] and B2 = E[Y Y >] for X,Y ∈ Vd, and X and Y are independent.

i) Clear.

ii) This directly follows from the Frechet–Hoeffding bounds; see McNeil et al. (2015, Re-

mark 7.9).

iii) Let A ∼ B(1, t) be a Bernoulli random variable independent of X,Y , and let Z = AX +

(1 − A)Y . Then Z ∈ Vd, and E[ZZ>] = tE[XX>] + (1 − t)E[Y Y >] = tB1 + (1 − t)B2.

Hence tB1 + (1− t)B2 ∈ Bd.

iv) Let p = (p1, . . . , pd), q = (q1, . . . , qd) ∈ Rd. Then

(p q)(p q)> = (piqi)d(piqi)>d = (piqipjqj)d×d = (pipj)d×d (qiqj)d×d

= (pp>) (qq>).

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Let Z = X Y . It follows that Z ∈ Vd and E[ZZ>] = E[(X Y )(X Y )>] = E[(XX>)

(Y Y >)] = E[XX>] E[Y Y >] = B1 B2. Hence B1 B2 ∈ Bd.

v) Consider X = 0 ∈ Vd. Then (0)d×d = E[XX>] ∈ Bd. Similarly for (1)d×d.

vi) Consider X ∈ Vd with independent components and E[X] = p.

2.2 Characterization of Bernoulli-compatible matrices

We are now able to give a characterization of the set Bd of Bernoulli-compatible matrices

and thus address Question 1.

Theorem 2.2 (Characterization of Bd). Bd has the following characterization:

Bd = n∑i=1

aipip>i : pi ∈ 0, 1d, ai > 0, i = 1, . . . , n,

n∑i=1

ai = 1, n ∈ N

; (2.1)

i.e., Bd is the convex hull of pp> : p ∈ 0, 1d. In particular, Bd is closed under convergence

in the Euclidean norm.

Proof. Denote the right-hand side of (2.1) by M. For B ∈ Bd, write B = E[XX>] for some

X ∈ Vd. It follows that

B =∑

p∈0,1dpp>P(X = p) ∈M,

hence Bd ⊆ M. Let X = p ∈ 0, 1d. Then X ∈ Vd and E[XX>] = pp> ∈ Bd. By

Proposition 2.1, Bd is a convex set which contains pp> : p ∈ 0, 1d, hence M ⊆ Bd. In

summary,M = Bd. From (2.1) we can see that Bd is closed under convergence in the Euclidean

norm.

A matrix B is completely positive if B = AA> for some (not necessarily square) matrix

A > 0. Denote by Cd the set of completely positive matrices. It is known that Cd is the convex

cone with extreme directions pp> : p ∈ [0, 1]d; see, e.g., Ruschendorf (1981) and Berman and

Shaked-Monderer (2003). We thus obtain the following result.

Corollary 2.3. Any Bernoulli-compatible matrix is completely positive.

Remark 2.1. One may wonder whether B = E[XX>] is sufficient to determine the distribution

of X, i.e., whether the decomposition

B =

2d∑i=1

aipip>i (2.2)

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is unique for distinct vectors pi in 0, 1d. While the decomposition is trivially unique for d = 2,

this is in general false for d > 3, since there are 2d − 1 parameters in (2.2) and only d(d+ 1)/2

parameters in B. The following is an example for d = 3. Let

B =1

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1 2 1

1 1 2

=

1

4

((1, 1, 1)>(1, 1, 1) + (1, 0, 0)>(1, 0, 0) + (0, 1, 0)>(0, 1, 0)

+ (0, 0, 1)>(0, 0, 1))

=1

4

((1, 1, 0)>(1, 1, 0) + (1, 0, 1)>(1, 0, 1) + (0, 1, 1)>(0, 1, 1)

+ (0, 0, 0)>(0, 0, 0)).

Thus, by combining the above two decompositions, B ∈ B3 has infinitely many different de-

compositions of the form (2.2). Note that, as in the case of completely positive matrices, it is

generally difficult to find decompositions of form (2.2) for a given matrix B.

2.3 Convex cone generated by Bernoulli-compatible matrices

In this section we study the convex cone generated by Bd, denoted by B∗d:

B∗d = aB : a > 0, B ∈ Bd. (2.3)

The following proposition is implied by Proposition 2.1 and Theorem 2.2.

Proposition 2.4. B∗d is the convex cone with extreme directions pp> : p ∈ 0, 1d. Moreover,

B∗d is a commutative semiring equipped with addition (B∗d,+) and multiplication (B∗d, ).

It is obvious that B∗d ⊆ Cd. One may wonder whether B∗d is identical to Cd, the set of

completely positive matrices. As the following example shows, this is false in general for d > 2.

Example 2.1. Note that B ∈ B∗d also satisfies Proposition 2.1, Part ii). Now consider p =

(p1, . . . , pd) ∈ (0, 1)d with pi > pj for some i 6= j. Clearly, pp> ∈ Cd, but pipj > p2j = minp2

i , p2j

contradicts Proposition 2.1, Part ii), hence pp> 6∈ B∗d.

For the following result, we need the notion of diagonally dominant matrices. A matrix

A ∈ Rd×d is called diagonally dominant if, for all i = 1, . . . , d,∑j 6=i |aij | 6 |aii|.

Proposition 2.5. Let Dd be the set of non-negative, diagonally dominant d× d-matrices. Then

Dd ⊆ B∗d.

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Proof. For i, j = 1, . . . , d, let p(ij) = (p(ij)1 , . . . , p

(ij)d ) where p

(ij)k = Ik=i∪k=j. It is straightfor-

ward to verify that the (i, i)-, (i, j)-, (j, i)- and (j, j)-entries of the matrix M (ij) = p(ij)(p(ij))>

are 1, and the other entries are 0. For D = (dij)d×d ∈ Dd, let

D∗ = (d∗ij)d×d =

d∑i=1

d∑j=1,j 6=i

dijM(ij).

By Proposition 2.4, D∗ ∈ B∗d. It follows that d∗ij = dij for i 6= j and d∗ii =∑dj=1,j 6=i dij 6 dii.

Therefore, D = D∗ +∑di=1(dii − d∗ii)M (ii), which, by Proposition 2.4, is in B∗d.

For studying the tail-dependence compatibility problem in Section 3, the subset

BId = B : B ∈ B∗d, diag(B) = Id

of B∗d is of interest. It is straightforward to see from Proposition 2.1 and Theorem 2.2 that BIdis a convex set, closed under the Hadamard product and convergence in the Euclidean norm.

These properties of BId will be used later.

3 Tail-dependence compatibility

3.1 Tail-dependence matrices

The notion of tail dependence captures (extreme) dependence in the lower-left or upper-

right tails of a bivariate distribution. In what follows, we focus on lower-left tails; the problem

for upper-right tails follows by a reflection around (1/2, 1/2), i.e., studying the survival copula

of the underlying copula.

Definition 3.1 (Tail-dependence coefficient). The (lower) tail-dependence coefficient of two

continuous random variables X1 ∼ F1 and X2 ∼ F2 is defined by

λ = limu↓0

P(F1(X1) 6 u, F2(X2) 6 u)

u, (3.1)

given that the limit exists.

If we denote the copula of (X1, X2) by C, then

λ = limu↓0

C(u, u)

u.

Clearly λ ∈ [0, 1], and λ only depends on the copula of (X1, X2), not the marginal distributions.

For virtually all copula models used in practice, the limit in (3.1) exists; for how to construct an

example where λ does not exist, see Kortschak and Albrecher (2009).

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Definition 3.2 (Tail-dependence matrix, Td). Let X = (X1, . . . , Xd) be a random vector with

continuous marginal distributions. The tail-dependence matrix of X is Λ = (λij)d×d, where λij

is the tail-dependence coefficient of Xi and Xj , i, j = 1, . . . , d. We denote by Td the set of all

tail-dependence matrices.

The following proposition summarizes basic properties of tail-dependence matrices. Its

proof is very similar to that of Proposition 2.1 and is omitted here.

Proposition 3.1. For any Λ1,Λ2 ∈ Td, we have that

i) Λ1 = Λ>1 .

ii) tΛ1 + (1− t)Λ2 ∈ Td for t ∈ [0, 1], i.e., Td is a convex set.

iii) Id 6 Λ1 6 (1)d×d with Id ∈ Td and (1)d×d ∈ Td.

As we will show next, Td is also closed under the Hadamard product.

Proposition 3.2. Let k ∈ N and Λ1, . . . ,Λk ∈ Td. Then Λ1 · · · Λk ∈ Td.

Proof. Note that it would be sufficient to show the result for k = 2, but we provide a general

construction for any k. For each l = 1, . . . , k, let Cl be a d-dimensional copula with tail-

dependence matrix Λl. Furthermore, let g(u) = u1/k, u ∈ [0, 1]. It follows from Liebscher (2008)

that C(u1, . . . , ud) =∏kl=1 Cl(g(u1), . . . , g(ud)) is a copula; note that

(g−1( max

16l6kUl1), . . . , g−1( max

16l6kUld)

)∼ C (3.2)

for independent random vectors (Ul1, . . . , Uld) ∼ Cl, l = 1, . . . , k. The (i, j)-entry λij of Λ

corresponding to C is thus given by

λij = limu↓0

∏kl=1 Cl,ij(g(u), g(u))

u= lim

u↓0

k∏l=1

Cl,ij(g(u), g(u))

g(u)

=

k∏l=1

limu↓0

Cl,ij(g(u), g(u))

g(u)=

k∏l=1

limu↓0

Cl,ij(u, u)

u=

k∏l=1

λl,ij ,

where Cl,ij denotes the (i, j)-margin of Cl and λl,ij denotes the (i, j)th entry of Λl, l = 1, . . . , k.

3.2 Characterization of tail-dependence matrices

In this section, we investigate the following question:

Question 2. Given a d× d matrix Λ ∈ [0, 1]d×d, is it a tail-dependence matrix?

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The following theorem fully characterizes tail-dependence matrices and thus provides a

theoretical (but not necessarily practical) answer to Question 2.

Theorem 3.3 (Characterization of Td). A square matrix with diagonal entries being 1 is a tail-

dependence matrix if and only if it is a Bernoulli-compatible matrix multiplied by a constant.

Equivalently, Td = BId.

Proof. We first show that Td ⊆ BId. For each Λ = (λij)d×d ∈ Td, suppose that C is a copula

with tail-dependence matrix Λ and U = (U1, . . . , Un) ∼ C. Let Wu = (IU16u, . . . , IUd6u).

By definition,

λij = limu↓0

1

uE[IUi6uIUj6u]

and

Λ = limu↓0

1

uE[WuW

>u ].

Since BId is closed and E[WuW>u ]/u ∈ BId, we have that Λ ∈ BId.

Now consider BId ⊆ Td. By definition of BId, each B ∈ BId can be written as B = E[XX>]/p

for an X ∈ Vd and E[X] = (p, . . . , p) ∈ (0, 1]d. Let U, V ∼ U[0, 1], U, V,X be independent and

Y = XpU + (1−X)(p+ (1− p)V ). (3.3)

We can verify that for t ∈ [0, 1] and i = 1, . . . , d,

P(Yi 6 t) = P(Xi = 1)P(pU 6 t) + P(Xi = 0)P(p+ (1− p)V 6 t)

= pmint/p, 1+ (1− p) max(t− p)/(1− p), 0 = t,

i.e., Y1, . . . , Yd are U[0, 1]-distributed. Let λij be the tail-dependence coefficient of Yi and Yj ,

i, j = 1, . . . , d. For i, j = 1, . . . , d we obtain that

λij = limu↓0

1

uP(Yi 6 u, Yj 6 u) = lim

u↓0

1

uP(Xi = 1, Xj = 1)P(pU 6 u)

=1

pE[XiXj ].

As a consequence, the tail-dependence matrix of (Y1, . . . , Yd) is B and B ∈ Td.

It follows from Theorem 3.3 and Proposition 2.4 that Td is the “1-diagonals” cross-section

of the convex cone with extreme directions pp> : p ∈ 0, 1d. Furthermore, the proof of

Theorem 3.3 is constructive. As we saw, for any B ∈ BId, Y defined by (3.3) has tail-dependence

matrix B. This interesting construction will be applied in Section 4 where we show that com-

monly applied matrices in statistics are tail-dependence matrices and where we derive the copula

of Y .

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Remark 3.1. From the fact that Td = BId and BId is closed under the Hadamard product (see

Proposition 2.1, Part iv)), Proposition 3.2 directly follows. Note, however, that our proof of

Proposition 3.2 is constructive. Given tail-dependence matrices and corresponding copulas, we

can construct a copula C which has the Hadamard product of the tail-dependence matrices as

corresponding tail-dependence matrix. If sampling of all involved copulas is feasible, we can

sample C; see Figure 1 for examples2.

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

U1

U2

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

U1

U2

Figure 1: Left-hand side: Scatter plot of 2000 samples from (3.2) for C1 being a Clayton copula

with parameter θ = 4 (λ1 = 2−1/4 ≈ 0.8409) and C2 being a t3 copula with parameter ρ = 0.8

(tail-dependence coefficient λ2 = 2t4(−2/3) ≈ 0.5415). By Proposition 3.2, the tail-dependence

coefficient of (3.2) is thus λ = λ1λ2 = 23/4t4(−2/3) ≈ 0.4553. Right-hand side: C1 as before,

but C2 is a survival Marshall–Olkin copula with parameters α1 = 2−3/4, α2 = 0.8, so that

λ = λ1λ2 = 1/2.

Theorem 3.3 combined with Corollary 2.3 directly leads to the following result.

Corollary 3.4. Every tail-dependence matrix is completely positive, and hence positive semi-

definite.

Furthermore, Theorem 3.3 and Proposition 2.5 imply the following result.

Corollary 3.5. Every diagonally dominant matrix with non-negative entries and diagonal en-

tries being 1 is a tail-dependence matrix.

Note that this result already yields the if-part of Proposition 4.7 below.

2All plots can be reproduced via the R package copula (version > 0.999-13) by calling

demo(tail compatibility).

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4 Compatible models for tail-dependence matrices

4.1 Widely known matrices

We now consider the following three types of matrices Λ = (λij)d×d which are frequently

applied in multivariate statistics and time series analysis and show that they are tail-dependence

matrices.

a) Equicorrelation matrix with parameter α ∈ [0, 1]: λij = Ii=j + αIi6=j, i, j = 1, . . . , d.

b) AR(1) matrix with parameter α ∈ [0, 1]: λij = α|i−j|, i, j = 1, . . . , d.

c) MA(1) matrix with parameter α ∈ [0, 1/2]: λij = Ii=j + αI|i−j|=1, i, j = 1, . . . , d.

Chaganty and Joe (2006) considered the compatibility of correlation matrices of Bernoulli vectors

for the above three types of matrices and obtained necessary and sufficient conditions for the

existence of compatible models for d = 3. For the tail-dependence compatibility problem that

we consider in this paper, the above three types of matrices are all compatible, and we are able

to construct corresponding models for each case.

Proposition 4.1. Let Λ be the tail-dependence matrix of the d-dimensional random vector

Y = XpU + (1−X)(p+ (1− p)V ), (4.1)

where U, V ∼ U[0, 1], X ∈ Vd and U, V,X are independent.

i) For α ∈ [0, 1], if X has independent components and E[X1] = · · · = E[Xd] = α, then Λ is

an equicorrelation matrix with parameter α; i.e., a) is a tail-dependence matrix.

ii) For α ∈ [0, 1], if Xi =∏i+d−1j=i Zj, i = 1, . . . , d, for independent B(1, α) random variables

Z1, . . . , Z2d−1, then Λ is an AR(1) matrix with parameter α; i.e., b) is a tail-dependence

matrix.

iii) For α ∈ [0, 1/2], if Xi = IZ∈[(i−1)(1−α),(i−1)(1−α)+1], i = 1, . . . , d, for Z ∼ U[0, d], then

Λ is an MA(1) matrix with parameter α; i.e., c) is a tail-dependence matrix.

Proof. We have seen in the proof of Theorem 3.3 that if E[X1] = · · · = E[Xd] = p, then Y

defined through (4.1) has tail-dependence matrix E[XX>]/p. Write Λ = (λij)d×d and note that

λii = 1, i = 1, . . . , d, is always guaranteed.

i) For i 6= j, we have that E[XiXj ] = α2 and thus λij = α2/α = α. This shows that Λ is an

equicorrelation matrix with parameter α.

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ii) For i < j, we have that

E[XiXj ] = E[ i+d−1∏

k=i

Zk

j+d−1∏l=j

Zl

]= E

[ j−1∏k=i

Zk

]E[ i+d−1∏

k=j

Zk

]E[ j+d−1∏k=i+d

Zk

]= αj−iαi+d−jαj−i = αj−i+d

and E[Xi] = E[X2i ] = αd. Hence, λij = αj−i+d/αd = αj−i for i < j. By symmetry,

λij = α|i−j| for i 6= j. Thus, Λ is an AR(1) matrix with parameter α.

iii) For i < j, note that 2(1− α) > 1, so

E[XiXj ] = P(Z ∈ [(j − 1)(1− α), (i− 1)(1− α) + 1])

= Ij=i+1P(Z ∈ [i(1− α), (i− 1)(1− α) + 1]) = Ij=i+1α

d

and E[Xi] = E[X2i ] = 1

d . Hence, λij = αIj−i=1 for i < j. By symmetry, λij = αI|i−j|=1

for i 6= j. Thus, Λ is an MA(1) matrix with parameter α.

4.2 Advanced tail-dependence models

Theorem 3.3 gives a characterization of tail-dependence matrices using Bernoulli-compatible

matrices and (3.3) provides a compatible model Y for any tail-dependence matrix Λ(= E[XX>]/p).

It is generally not easy to check whether a given matrix is a Bernoulli-compatible matrix or

a tail-dependence matrix; see also Remark 2.1. Therefore, we now study the following question.

Question 3. How can we construct a broader class of models with flexible dependence structures

and desired tail-dependence matrices?

To enrich our models, we bring random matrices with Bernoulli entries into play. For

d,m ∈ N, let

Vd×m =X = (Xij)d×m : P(X ∈ 0, 1d×m) = 1,

m∑j=1

Xij 6 1, i = 1, . . . , d,

i.e., Vd×m is the set of d × m random matrices supported in 0, 1d×m with each row being

mutually exclusive; see Dhaene and Denuit (1999). Furthermore, we introduce a transformation

L on the set of square matrices, such that, for any i, j = 1, . . . , d, the (i, j)th element bij of L(B)

is given by

bij =

bij , if i 6= j,

1, if i = j;

(4.2)

i.e., L adjusts the diagonal entries of a matrix to be 1, and preserves all the other entries. For a

set S of square matrices, we set L(S) = L(B) : B ∈ S. We can now address Question 3.

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Theorem 4.2 (A class of flexible models). Let U ∼ CU for an m-dimensional copula CU with

tail-dependence matrix Λ and let let V ∼ CV for a d-dimensional copula CV with tail-dependence

matrix Id. Furthermore, let X ∈ Vd×m such that X,U ,V are independent and let

Y = XU + Z V , (4.3)

where Z = (Z1, . . . , Zd) with Zi = 1 −∑mk=1Xik, i = 1, . . . , d. Then Y has tail-dependence

matrix Γ = L(E[XΛX>]).

Proof. Write X = (Xij)d×m, U = (U1, . . . , Um), V = (V1, . . . , Vd), Λ = (λij)d×d and Y =

(Y1, . . . , Yd). Then, for all i = 1, . . . , d,

Yi =

m∑k=1

XikUk + ZiVi =

Vi, if Xik = 0 for all k = 1, . . . ,m, so Zi = 1,

Uk, if Xik = 1 for some k = 1, . . . ,m, so Zi = 0.

Clearly, Y has U[0, 1] margins. We now calculate the tail-dependence matrix Γ = (γij)d×d of Y

for i 6= j. By our independence assumptions, we can derive the following results:

i) P(Yi 6 u, Yj 6 u, Zi = 1, Zj = 1) = P(Vi 6 u, Vj 6 u, Zi = 1, Zj = 1) = CVij (u, u)P(Zi =

1, Zj = 1) 6 CVij (u, u), where CV

ij denotes the (i, j)th margin of CV . As V has tail-

dependence matrix Id, we obtain that

limu↓0

1

uP(Yi 6 u, Yj 6 u, Zi = 1, Zj = 1) = 0.

ii) P(Yi 6 u, Yj 6 u, Zi = 0, Zj = 1) =∑mk=1 P(Uk 6 u, Vj 6 u,Xik = 1, Zj = 1) =∑m

k=1 P(Uk 6 u)P(Vj 6 u)P(Xik = 1, Zj = 1) 6 u2 and thus

limu↓0

1

uP(Yi 6 u, Yj 6 u, Zi = 0, Zj = 1) = 0.

Similarly, we obtain that

limu↓0

1

uP(Yi 6 u, Yj 6 u, Zi = 1, Zj = 0) = 0.

iii) P(Yi 6 u, Yj 6 u, Zi = 0, Zj = 0) =∑mk=1

∑ml=1 P(Uk 6 u, Ul 6 u,Xik = 1, Xjl = 1) =∑m

k=1

∑ml=1 C

Ukl (u, u)P(Xik = 1, Xjl = 1) =

∑mk=1

∑ml=1 C

Ukl (u, u)E[XikXjl] so that

limu↓0

1

uP(Yi 6 u, Yj 6 u, Zi = 0, Zj = 0)

=

m∑k=1

m∑l=1

λklE[XikXjl] = E[ m∑k=1

m∑l=1

XikλklXjl

]=(E[XΛX>]

)ij.

By the Law of Total Probability, we thus obtain that

γij = limu↓0

P(Yi 6 u, Yj 6 u)

u= lim

u↓0

P(Yi 6 u, Yj 6 u, Zi = 0, Zj = 0)

u

=(E[XΛX>]

)ij.

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This shows that E[XΛX>] and Γ agree on the off-diagonal entries. Since Γ ∈ Td implies that

diag(Γ) = Id, we conclude that L(E[XΛX>]) = Γ.

A special case of Theorem 4.2 reveals an essential difference between the transition rules

of a tail-dependence matrix and a covariance matrix. Suppose that for X ∈ Vd×m, E[X] is a

stochastic matrix (each row sums to 1), and U ∼ CU for an m-dimensional copula CU with

tail-dependence matrix Λ = (λij)d×d. Now we have that Zi = 0, i = 1, . . . , d in (4.3). By

Theorem 4.2, the tail dependence matrix of Y = XU is given by L(E[XΛX>]). One can check

the diagonal terms of the matrix Λ∗ = (λ∗ij)d×d = XΛX> by

λ∗ii =

m∑j=1

m∑k=1

XikλkjXij =

m∑k=1

Xikλkk = 1, i = 1, . . . ,m.

Hence, the tail-dependence matrix of Y is indeed E[XΛX>].

Remark 4.1. In summary:

i) If an m-vector U has covariance matrix Σ, then XU has covariance matrix E[XΣX>] for

any d×m random matrix X independent of U .

ii) If an m-vector U has uniform [0,1] margins and tail-dependence matrix Λ, then XU has

tail-dependence matrix E[XΛX>] for any X ∈ Vd×m independent of U such that each row

of X sums to 1.

It is noted that the transition property of tail-dependence matrices is more restricted than that

of covariance matrices.

The following two propositions consider selected special cases of this construction which are

more straightforward to apply.

Proposition 4.3. For any B ∈ Bd and any Λ ∈ Td we have that L(B Λ) ∈ Td. In particular,

L(B) ∈ Td and hence L(Bd) ⊆ Td.

Proof. Write B = (bij)d×d = E[WW>] for some W = (W1, . . . ,Wd) ∈ Vd and consider X =

diag(W ) ∈ Vd×d. As in the proof of Theorem 4.2 (and with the same notation), it follows that

for i 6= j, γij = E[XiiλijXjj ] = E[WiWjλij ]. This shows that E[XΛX>] = E[WW> Λ] and

B Λ agree on off-diagonal entries. Thus, L(B Λ) = Γ ∈ Td. By taking Λ = (1)d×d, we obtain

L(B) ∈ Td.

The following proposition states a relationship between substochastic matrices and tail-

dependence matrices. To this end, let

Qd×m =

Q = (qij)d×m :

m∑j=1

qij 6 1, qij > 0, i = 1, . . . , d, j = 1, . . . ,m

,

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i.e., Qd×m is the set of d×m (row) substochastic matrices; note that the expectation of a random

matrix in Vd×m is a substochastic matrix.

Proposition 4.4. For any Q ∈ Qd×m and any Λ ∈ Tm, we have that L(QΛQ>) ∈ Td. In

particular, L(QQ>) ∈ Td for all Q ∈ Qd×m and L(pp>) ∈ Td for all p ∈ [0, 1]d.

Proof. Write Q = (qij)d×m and let Xik = IZi∈[∑k−1

j=1 qij ,∑k

j=1 qij) for independent Zi ∼ U[0, 1],

i = 1, . . . , d, k = 1, . . . ,m. It is straightforward to see that E[X] = Q, X ∈ Vd×m with

independent rows, and∑mk=1Xik 6 1 for i = 1, . . . , d, so X ∈ Vd×m. As in the proof of

Theorem 4.2 (and with the same notation), it follows that for i 6= j,

γij =

m∑l=1

m∑k=1

E[Xik]E[Xjl]λkl =

m∑l=1

m∑k=1

qikqjlλkl.

This shows that QΛQ> and Γ agree on off-diagonal entries, so L(QΛQ>) = Γ ∈ Td. By taking

Λ = Id, we obtain L(QQ>) ∈ Td. By taking m = 1, we obtain L(pp>) ∈ Td.

4.3 Corresponding copula models

In this section, we derive the copulas of (3.3) and (4.3) which are able to produce tail-

dependence matrices E[XX>]/p and L(E[XΛX>]) as stated in Theorems 3.3 and 4.2, respec-

tively. We first address the former.

Proposition 4.5 (Copula of (3.3)). Let X ∈ Vd, E[X] = (p, . . . , p) ∈ (0, 1]d. Furthermore, let

U, V ∼ U[0, 1], U, V,X be independent and

Y = XpU + (1−X)(p+ (1− p)V ).

Then the copula C of Y at u = (u1, . . . , ud) is given by

C(u) =∑

i∈0,1dmin

minr:ir=1urp

, 1

maxminr:ir=0ur − p

1− p, 0P(X = i),

with the convention min ∅ = 1.

Proof. By the Law of Total Probability and our independence assumptions,

C(u) =∑

i∈0,1dP(Y 6 u,X = i)

=∑

i∈0,1dP(pU 6 min

r:ir=1ur, p+ (1− p)V 6 min

r:ir=0ur,X = i)

=∑

i∈0,1dP(U 6

minr:ir=1urp

)P(V 6

minr:ir=0ur − p1− p

)P(X = i);

the claim follows from the fact that U, V ∼ U[0, 1].

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For deriving the copula of (4.3), we need to introduce some notation; see also Example 4.1

below. In the following theorem, let supp(X) denote the support of X. For a vector u =

(u1, . . . , ud) ∈ [0, 1]d and a matrix A = (Aij)d×m ∈ supp(X), denote by Ai the sum of the

i-th row of A, i = 1, . . . , d, and let uA = (u1IA1=0 + IA1=1, . . . , udIAd=0 + IAd=1), and

u∗A = (minr:Ar1=1ur, . . . ,minr:Arm=1ur), where min ∅ = 1.

Proposition 4.6 (Copula of (4.3)). Suppose that the setup of Theorem 4.2 holds. Then the

copula C of Y in (4.3) is given by

C(u) =∑

A∈supp(X)

CV (uA)CU (u∗A)P(X = A). (4.4)

Proof. By the Law of Total Probability, it suffices to verify that P(Y 6 u |X = A) = CV (uA)CU (u∗A).

This can be seen from

P(Y 6 u |X = A)

= P( m∑k=1

AjkUk + (1−Aj)Vj 6 uj , j = 1, . . . , d

)= P(UkIAjk=1 6 uj , VjIAj=0 6 uj , j = 1, . . . , d, k = 1, . . . ,m)

= P(Uk 6 min

r:Ark=1ur, Vj 6 ujIAj=0 + IAj=1, j = 1, . . . , d, k = 1, . . . ,m

)= P

(Uk 6 min

r:Ark=1ur, k = 1, . . . ,m

)P(Vj 6 ujIAj=0 + IAj=1, j = 1, . . . , d

)= CU (u∗A)CV (uA).

As long as CV has tail-dependence matrix Id, the tail-dependence matrix of Y is not

affected by the choice of CV . This theoretically provides more flexibility in choosing the body

of the distribution of Y while attaining a specific tail-dependence matrix. Note, however, that

this also depends on the choice of X; see the following example where we address special cases

which allow for more insight into the rather abstract construction (4.4).

Example 4.1. 1. For m = 1, the copula C in (4.4) is given by

C(u) =∑

A∈0,1dCV (uA)CU (u∗A)P(X = A); (4.5)

note that X,A in Equation 4.4 are indeed vectors in this case. For d = 2, we obtain

C(u1, u2) = M(u1, u2)P(X = ( 11 )) + CV (u1, u2)P(X = ( 0

0 ))

+ Π(u1, u2)P(X = ( 10 ) or X = ( 0

1 )),

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and therefore a mixture of the Frechet–Hoeffding upper bound M(u1, u2) = minu1, u2,

the copula CV and the independence copula Π(u1, u2) = u1u2. If P(X = ( 00 )) = 0 then C

is simply a mixture of M and Π and does not depend on V anymore.

Now consider the special case of (4.5) where V follows the d-dimensional independence cop-

ula Π(u) =∏di=1 ui and X = (X1, . . . , Xd−1, 1) is such that at most one of X1, . . . , Xd−1

is 1 (each randomly with probability 0 6 α 6 1/(d− 1) and all are simultaneously 0 with

probability 1− (d− 1)α). Then, for all u ∈ [0, 1]d, C is given by

C(u) = α

d−1∑i=1

(minui, ud

d−1∏j=1,j 6=i

uj

)+ (1− (d− 1)α)

d∏j=1

uj . (4.6)

This copula is a conditionally independent multivariate Frechet copula studied in Yang

et al. (2009). This example will be revisited in Section 4.4; see also the left-hand side of

Figure 3 below.

2. For m = 2, d = 2, we obtain

C(u1, u2) = M(u1, u2)P(X = ( 1 01 0 ) or X = ( 0 1

0 1 ))

+ CU (u1, u2)P(X = ( 1 00 1 )) + CU (u2, u1)P(X = ( 0 1

1 0 ))

+ CV (u1, u2)P(X = ( 0 00 0 ))

+ Π(u1, u2)P(X = ( 0 01 0 ) or ( 0 0

0 1 ) or ( 1 00 0 ) or ( 0 1

0 0 )). (4.7)

Figure 2 shows samples of size 2000 from (4.7) for V ∼ Π and two different choices of U

(in different rows) and X (in different columns). From Theorem 4.2, we obtain that the

off-diagonal entry γ12 of the tail-dependence matrix Γ of Y is given by

γ12 = p(1,2)(1,1) + p(1,2)(2,2) + λ12(p(1,2)(2,1) + p(1,2)(1,2)),

where λ12 is the off-diagonal entry of the tail-dependence matrix Λ of U .

4.4 An example from risk management practice

Let us now come back to Problem (1.1) which motivated our research on tail-dependence

matrices. From a practical point of view, the question is whether it is possible to find one

financial position, which has tail-dependence coefficient α with each of d − 1 tail-independent

financial risks (assets). Such a construction can be interesting for risk management purposes,

e.g., in the context of hedging.

Recall Problem (1.1):

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0.0 0.2 0.4 0.6 0.8 1.0

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Y1

Y2

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0.0

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Y2

0.0 0.2 0.4 0.6 0.8 1.0

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0.8

1.0

Y1

Y2

Figure 2: Scatter plots of 2000 samples from Y for V ∼ Π and U following a bivariate (m = 2)

t3 copula with Kendall’s tau equal to 0.75 (top row) or a survival Marshall–Olkin copula with

parameters α1 = 0.25, α2 = 0.75 (bottom row). For the plots on the left-hand side, the number

of rows of X with one 1 are randomly chosen among 0, 1, 2(= d), the corresponding rows and

columns are then randomly selected among 1, 2(= d) and 1, 2(= m), respectively. For the

plots on the right-hand side, X is drawn from a multinomial distribution with probabilities 0.5

and 0.5 such that each row contains precisely one 1.

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For which α ∈ [0, 1] is the matrix

Γd(α) =

1 0 · · · 0 α

0 1 · · · 0 α...

.... . .

......

0 0 · · · 1 α

α α · · · α 1

(4.8)

a matrix of pairwise (either lower or upper) tail-dependence coefficients?

Based on the Frechet–Hoeffding bounds, it follows from Joe (1997, Theorem 3.14) that for

d = 3 (and thus also d > 3), α has to be in [0, 1/2]; however this is not a sufficient condition

for Γd(α) to be a tail-dependence matrix. The following proposition not only gives an answer

to (4.8) by providing necessary and sufficient such conditions, but also provides, by its proof, a

compatible model for Γd(α).

Proposition 4.7. Γd(α) ∈ Td if and only if 0 6 α 6 1/(d− 1).

Proof. The if-part directly follows from Corollary 3.5. We provide a constructive proof based on

Theorem 4.2. Suppose that 0 6 α 6 1/(d−1). Take a partition Ω1, . . . ,Ωd of the sample space

Ω with P(Ωi) = α, i = 1, . . . , d− 1, and let X = (IΩ1, . . . , IΩd−1

, 1) ∈ Vd. It is straightforward to

see that

E[XX>] =

α 0 · · · 0 α

0 α · · · 0 α...

.... . .

......

0 0 · · · α α

α α · · · α 1

.

By Proposition 4.3, Γd(α) = L(E[XX>]) ∈ Td.

For the only if part, suppose that Γd(α) ∈ Td; thus α > 0. By Theorem 3.3, Γd(α) ∈ BId.

By the definition of BId, Γd(α) = Bd/p for some p ∈ (0, 1] and a Bernoulli-compatible matrix Bd.

Therefore,

pΓd(α) =

p 0 · · · 0 pα

0 p · · · 0 pα...

.... . .

......

0 0 · · · p pα

pα pα · · · pα p

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is a compatible Bernoulli matrix, so pΓd(α) ∈ Bd. Write pΓd(α) = E[XX>] for some X =

(X1, . . . , Xd) ∈ Vd. It follows that P(Xi = 1) = p for i = 1, . . . , d, P(XiXj = 1) = 0 for

i 6= j, i, j = 1, . . . , d − 1 and P(XiXd = 1) = pα for i = 1, . . . , d − 1. Note that XiXd = 1,

i = 1, . . . , d − 1, are almost surely disjoint since P(XiXj = 1) = 0 for i 6= j, i, j = 1, . . . , d − 1.

As a consequence,

p = P(Xd = 1) > P( d−1⋃i=1

XiXd = 1)

=

d−1∑i=1

P(XiXd = 1) = (d− 1)pα,

and thus (d− 1)α 6 1.

It follows from the proof of Theorem 4.2 that for α ∈ [0, 1/(d − 1)], a compatible copula

model with tail-dependence matrix Γd(α) can be constructed as follows. Consider a partition

Ω1, . . . ,Ωd of the sample space Ω with P(Ωi) = α, i = 1, . . . , d−1, and let X = (X1, . . . , Xd) =

(IΩ1 , . . . , IΩd−1, 1) ∈ Vd; note that m = 1 here. Furthermore, let V be as in Theorem 4.2,

U ∼ U[0, 1] and U,V ,X be independent. Then,

Y = (UX1 + (1−X1)V1, . . . , UXd−1 + (1−Xd−1)Vd−1, U)

has tail-dependence matrix Γd(α). Example 4.1, Part 1 provides the copula C of Y in this case.

It is also straightforward to verify from this copula that Y has tail-dependence matrix Γd(α).

Figure 3 displays pairs plots of 2000 realizations of Y for α = 1/3 and two different copulas for

V .

Remark 4.2. Note that Γd(α) is not positive semidefinite if and only if α > 1/√d− 1. For

d < 5, element-wise non-negative and positive semidefinite matrices are completely positive; see

Berman and Shaked-Monderer (2003, Theorem 2.4). Therefore Γ3(2/3) is completely positive.

However, it is not in T3. It indeed shows that the class of completely positive matrices with

diagonal entries being 1 is strictly larger than Td.

5 Conclusion and discussion

Inspired by the question whether a given matrix in [0, 1]d×d is the matrix of pairwise tail-

dependence coefficients of a d-dimensional random vector, we introduced the tail-dependence

compatibility problem. It turns out that this problem is closely related to the Bernoulli-

compatibility problem which we also addressed in this paper and which asks when a given matrix

in [0, 1]d×d is a Bernoulli-compatible matrix (see Question 1 and Theorem 2.2). As a main find-

ing, we characterized tail-dependence matrices as precisely those square matrices with diagonal

entries being 1 which are Bernoulli-compatible matrices multiplied by a constant (see Question 2

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0.0 0.4 0.8

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0.8

0.0 0.4 0.8

Y4

Figure 3: Pairs plot of 2000 samples from Y ∼ C which produces the tail dependence matrix

Γ4(1/3) as given by (1.1). On the left-hand side, V ∼ Π (α determines how much weight is on

the diagonal for pairs with one component being Y4; see (4.6)) and on the right-hand-side, V

follows a Gauss copula with parameter chosen such that Kendall’s tau equals 0.8.

and Theorem 3.3). Furthermore, we presented and studied new models (see, e.g., Question 3

and Theorem 4.2) which provide answers to several questions related to the tail-dependence

compatibility problem.

The study of compatibility of tail-dependence matrices is mathematically different from

that of covariances matrices. Through many technical arguments in this paper, the reader may

have already realized that the tail-dependence matrix lacks a linear structure which is essential

to covariance matrices based on tools from Linear Algebra. For instance, let X be a d-random

vector with covariance matrix Σ and tail-dependence matrix Λ, and A be an m× d matrix. The

covariance matrix of AX is simply given by AΣA>, however, the tail-dependence matrix of AX

is generally not explicit (see Remark 4.1 for special cases). This lack of linearity can also help

to understand why tail-dependence matrices are realized by models based on Bernoulli vectors

as we have seen in this paper, in contrast to covariance matrices which are naturally realized by

Gaussian (or generally, elliptical) random vectors. The latter have a linear structure, whereas

Bernoulli vectors do not. It is not surprising that most classical techniques in Linear Algebra

such as matrix decomposition, diagonalization, ranks, inverses and determinants are not very

helpful for studying the compatibility problems we address in this paper.

Concerning future research, an interesting open question is how one can (theoretically or nu-

merically) determine whether a given arbitrary non-negative, square matrix is a tail-dependence

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or Bernoulli-compatible matrix. To the best of our knowledge there are no corresponding algo-

rithms available. Another open question concerns the compatibility of other matrices of pairwise

measures of association such as rank-correlation measures (e.g., Spearman’s rho or Kendall’s

tau); see (Embrechts et al., 2002, Section 6.2). Recently, Fiebig et al. (2014) and Strokorb et al.

(2015) studied the concept of tail-dependence functions of stochastic processes. Similar results

to some of our findings were found in the context of max-stable processes.

From a practitioner’s point-of-view, it is important to point out limitations of using tail-

dependence matrices in quantitative risk management and other applications. One possible such

limitation is the statistical estimation of tail-dependence matrices since, as limits, estimating tail

dependence coefficients from data is non-trivial (and typically more complicated than estimation

in the body of a bivariate distribution).

After presenting the results of our paper at the conferences “Recent Developments in De-

pendence Modelling with Applications in Finance and Insurance – 2nd Edition, Brussels, May

29, 2015” and “The 9th International Conference on Extreme Value Analysis, Ann Arbor, June

15–19, 2015”, the references Fiebig et al. (2014) and Strokorb et al. (2015) were brought to our

attention (see also Acknowledgments below). In these papers, a very related problem is treated,

be it from a different, more theoretical angle, mainly based on the theory of max-stable and

Tawn-Molchanov processes as well as results for convex-polytopes. For instance, our Theorem

3.3 is similar to Theorem 6 c) in Fiebig et al. (2014).

Acknowledgments

We would like to thank Federico Degen for raising this interesting question concerning tail-

dependence matrices and Janusz Milek for relevant discussions. We would also like to thank the

Editor, two anonymous referees, Tiantian Mao, Don McLeish, Johan Segers, Kirstin Strokorb and

Yuwei Zhao for valuable input on an early version of the paper. Furthermore, Paul Embrechts

acknowledges financial support by the Swiss Finance Institute, and Marius Hofert and Ruodu

Wang acknowledge support from the Natural Sciences and Engineering Research Council of

Canada (NSERC Discovery Grant numbers 5010 and 435844, respectively). Ruodu Wang is

grateful to the Forschungsinstitut fur Mathematik (FIM) at ETH Zurich for financial support

during his visits in 2013 and 2014 when this topic was brought to his attention.

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