On the multidimensional extension of countermonotonicity and its applications

Insurance: Mathematics and Economics 56 (2014) 68–79

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On the multidimensional extension of countermonotonicity and its applications Woojoo Lee a , Jae Youn Ahn b,∗ a

Department of Statistics, Inha University, 235 Yonghyun-Dong, Nam-Gu, Incheon 402-751, Republic of Korea

b

Department of Statistics, Ewha Womans University, 11-1 Daehyun-Dong, Seodaemun-Gu, Seoul 120-750, Republic of Korea

article

info

Article history: Received October 2013 Received in revised form January 2014 Accepted 18 March 2014 JEL classification: C100 Keywords: Countermonotonicity Comonotonicity Minimal copula Measures of concordance Herd behavior index

abstract In a 2-dimensional space, Fréchet–Hoeffding upper and lower bounds define comonotonicity and countermonotonicity, respectively. Similarly, in the multidimensional case, comonotonicity can be defined using the Fréchet–Hoeffding upper bound. However, since the multidimensional Fréchet–Hoeffding lower bound is not a distribution function, there is no obvious extension of countermonotonicity in multidimensions. This paper investigates in depth a new multidimensional extension of countermonotonicity. We first provide an equivalent condition for countermonotonicity in 2-dimension, and extend the definition of countermonotonicity into multidimensions. In order to justify such extensions, we show that newly defined countermonotonic copulas constitute a minimal class of copulas. Two applications will be provided. First, we will study the relationships between multidimensional countermonotonicity and such well-known multivariate concordance measures as Kendall’s tau or Spearman’s rho. Second, we will give a financial interpretation of multidimensional countermonotonicity via the existing herd behavior index. © 2014 Elsevier B.V. All rights reserved.

1. Introduction The study of the dependence structure between random variables is a classical problem in statistics and other applications. A copula is an easy but essential tool to understand the dependence structures in multivariate random variables and has gained popularity in the areas of finance, insurance, hydrology and medical studies; see for example, Frees and Valdez (1998), Genest et al. (2007) and Cui and Sun (2004). Especially in insurance, people are interested in aggregate claims, X1 + · · · + Xn , where Xi is ith individual’s claim of an insurance portfolio. The dependence between individual claims is the key to understand the distribution of aggregated claims. Copula has been used as a main tool to answer such questions both in academia and industry. When we consider the dependence structure between random variables, two extreme situations can be considered. The first extreme situation is that all random variables move together in the same direction. This concept is formalized under the name of ‘‘comonotonicity’’ (Dhaene et al., 2002), and the corresponding copula is known as the Fréchet–Hoeffding upper bound. Recently, Dhaene et al. (2012) exploited the comonotonicity theory in

∗

Corresponding author. Tel.: +82 232772378; fax: +82 232773607. E-mail addresses: [email protected] (W. Lee), [email protected], [email protected] (J.Y. Ahn). http://dx.doi.org/10.1016/j.insmatheco.2014.03.002 0167-6687/© 2014 Elsevier B.V. All rights reserved.

computing the herd behavior index of a financial market. As the opposite extreme situation, we may imagine that all variables move into opposite directions. If only two variables are considered, this situation is easy to understand, i.e. one variable increases and the other variable decreases. In fact, the corresponding 2-dimensional copula is the Fréchet–Hoeffding lower bound and is said to be ‘‘countermonotonic’’. However, this countermonotonicity is hard to extend to arbitrary dimensions. For illustration, suppose that there are three variables. Can we imagine that the third variable moves in reverse to the first two variables? In fact, it is well known that, when a dimension is greater than two, the Fréchet–Hoeffding lower bound is no longer a copula and the minimum copula does not exist; see Kotz and Seeger (1992) for an example. In spite of its difficulty, the extension of the countermonotonicity to higher dimensions is still an interesting problem in insurance and other applications because it may lead to solutions for hedging or other minimization problems. There have been many variations or extensions of the countermonotonicity into multidimensions to address such minimization problems. For example, Gaffke and Rüschendorf (1981) proposed the concept of complete mixability, which explains that the sum of random variables with a given marginal distribution is constant, to solve some minimization problems; see Wang and Wang (2011) for more examples. With some restrictive marginal distributions, Cheung and Lo (2014) used the Fréchet–Hoeffding lower bound as a definition of multidimensional countermonotonicity. While use of

W. Lee, J.Y. Ahn / Insurance: Mathematics and Economics 56 (2014) 68–79

complete mixability or the Fréchet–Hoeffding lower bound to define multidimensional countermonotonicity involves various minimization problems, the usage can be somewhat limited, because these terms are defined within a certain class of marginal distributions. Furthermore, considering that comonotonicity and countermonotonicity depend on copula only in 2-dimension, it is desirable to separate the definition of countermonotonicity from marginal distributions. In this paper, we investigate a multidimensional extension of countermonotonicity. We first provide an equivalent representation of 2-dimensional countermonotonicity. This representation is appealing because it can be easily extended into any d-dimensions for d > 2, and such an extension will be called as d-countermonotonicity (d-CM). Such an extension is of interest because d-CM copulas constitute a minimal class of copulas, as will be shown later in Section 3. For further justification of d-CM, we address two specific questions; (1) do d-CM copulas indeed correspond to the minimum in a broad parametric class of copulas if we admit that the minimum copula of the entire class of copulas does not exist? and (2) do d-CM copulas achieve the maximum (or minimum) of a properly defined measure which preserves the ordering of copulas. For the first question, it is a difficult task to determine an interesting and meaningful broad class of copulas. For example, if the restriction on the class is too strict, the result will not be meaningful although it will be simple to determine whether the d-CM copula is the minimum copula in the class. Thus, as a minimal requirement for the class, the following two factors are considered: the class is large enough to be sufficiently useful in practice and small enough to have a minimum copula. Indeed Archimedean copulas satisfy these two conditions. For the former, many literatures revealed that Archimedean copulas cover a broad range of applications; see, for example, Cherubini et al. (2004) and Nelsen (2006). For the latter, McNeil and Nešlehová (2009) showed that Archimedean copulas have the minimum Archimedean copula. In this paper, we show that this minimum copula is actually d-CM. For the second question, a measure called the moment of inertia (MOI), which was developed by Kotz and Seeger (1992), is introduced to assess the extremity of the copula. Later, we show not only that MOI preserves the ordering of copulas, but also that MOI is maximized when the copula satisfies d-CM and MOI is minimized when the copula is comonotonic. Similar to the wide applicability of the comonotonicity, d-CM may also be useful in various statistical problems. Through our new definition, we first study the relations between d-CM and the minimum of popular concordance measures such as multivariate Kendall’s tau as defined in Joe (1990) and multivariate Spearman’s rho in Schmid and Schmidt (2007). As a second application, we focus on the relationships between d-CM and some recently proposed herd behavior measures (Dhaene et al., 2012, 2014; Choi et al., 2013; Lee et al., 2013). We note that they are seemingly closely related, but they have different responses to the change of marginal distribution because Lee et al. (2013)’s version is based on Spearman’s rho, and those of Dhaene et al. (2012, 2014) and Choi et al. (2013) are based on covariance. In terms of the herd behavior index in Lee et al. (2013), the comonotonicity can be interpreted as extreme herd behavior, and d-CM can be interpreted as the opposite extreme of herd behavior. In this paper, we give an example of the Middle East stock market to explain how d-CM can be interpreted. The rest of this paper is organized as follows. We first review the basic theory for copula. In Section 3, comonotonicity and countermonotonicity are introduced in 2-dimensional space. We provide an equivalent representation for the countermonotonicity in 2-dimension. In Section 4, we extend the definition of countermonotonicity into multivariate random vectors and show

69

that such extensions define a minimal class of copulas. These newly proposed concepts are useful in finding the minimum values of popular multivariate association measures such as Kendall’s tau and Spearman’s rho. The details are given in Section 5. A financial application of countermonotonicity is given in Section 6 and is followed by conclusions. 2. Notations and preliminary results Throughout this paper, we denote x = (x1 , x2 , . . . , xd ) and y = (y1 , y2 , . . . , yd ) as constant vectors in Rd and X = (X1 , X2 , . . . , Xd ), U = (U1 , . . . , Ud )

X∗ = (X1∗ , . . . , Xd∗ ),

and U∗ = (U1∗ , . . . , Ud∗ )

as d-variate random vectors defined on a probability space Ω ,



F , P . Two observations, x and y, are said to be concordant if



x1 < y1 , . . . , xd < yd , or if x1 > y1 , . . . , xd > yd . The joint distribution function of X is denoted by H (x) = P (X1 ≤ x1 , . . . , Xd ≤ xd ),

for x ∈ Rd ,

and, for x ∈ R, the marginal distribution function of Xi as Fi (x) = P (Xi ≤ x) for i = 1, . . . , d. Sets [a, b]×[a, b] · · ·×[a, b](⊆ Rd ) and (a, b) × (a, b) · · · × (a, b)(⊆ Rd ) are denoted by [a, b]d and (a, b)d , respectively. We use u = (u1 , . . . , ud ) to denote any constant vector in [0, 1]d . For any x, y ∈ Rd , we say x ≤ y if xi ≤ yi for all i = 1, . . . , d. We assume that the marginal distribution functions of X and X∗ are continuous throughout the paper. According to Sklar (1959), given H, there is a unique function C : [0, 1]d → [0, 1] which satisfies H (x) = C (F1 (x1 ), . . . , Fd (xd )). The function C is called a d-dimensional copula, which is known to be a distribution function on [0, 1]d . We use U and U∗ for random vectors whose distribution functions are copulas C and C ∗ , respectively. For more information on copulas, see Cherubini et al. (2004) or Nelsen (2006). If X1 , . . . , Xd are independent, the corresponding copula is

Π (u) =

d 

ui

for all u ∈ [0, 1]d

i =1

and it is called an independent copula. Denote  H as a joint survival function of X and, Ci# is a copula associated with a random vector

(X1 , . . . , Xi−1 , −Xi , Xi+1 , . . . , Xd ). Lastly, under the condition that the distribution function H has the same marginal distribution with the distribution function H ′ , we say H is smaller than H ′ in concordance order, H ≺ H ′ , if H (x) ≤ H ′ (x)

′ (x) and  H (x) ≤ H

for all x ∈ Rd .

Equivalently, we say that the random vector X is smaller than the random vector X′ in concordance order (X ≺ X′ ) if H ≺ H ′ , where the distribution functions of X and X′ are H and H ′ , respectively. 3. Comonotonicity and countermonotonicity Comonotonicity has gained popularity in actuarial science and finance; see for example, Dhaene et al. (2002), Cheung (2008), Nam et al. (2011). Especially, it may be useful to describe an economic crisis as shown in Dhaene et al. (2012) and Choi et al. (2013). First, we give the formal definition of comonotonicity. For any random

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W. Lee, J.Y. Ahn / Insurance: Mathematics and Economics 56 (2014) 68–79

vector X, define the support of X as a set A ⊆ Rd with P (X ∈ A)

= 1. Definition 1 (Dhaene et al. (2002)). A set B ⊆ Rd is comonotonic if, for any x, y ∈ B, either x ≤ y or y ≤ x holds. X is called comonotonic if it has comonotonic support, i.e., if P (X ∈ B) = 1 for some comonotonic set B ⊆ Rd . Theorem 1 (Dhaene et al. (2002)). For a d-variate random vector X, we have the following equivalent statements. i. X is comonotonic ii. For any x ∈ R2

P (X ≤ x) = min {F1 (x1 ), . . . , Fd (xd )}

i. f1 (s1 ) and f2 (s2 ) are strictly increasing functions at (s1 , s2 ) = ( x1 , x2 ) ii. f1 (x1 ) + f2 (x2 ) = 1 for (x1 , x2 ) on A. Proof. The necessary part is trivial by definition of countermonotonicity. Now, we prove the sufficient part. By using continuous mapping theorem and Theorem 2, we have d

F1 (X1 ) + F2 (X2 ) = F1 F1−1 (U ) + F2 F2−1 (1 − U ) ,

iii. For U ∼ Uniform(0, 1), we have d

Proposition 1. A bivariate random variable X is countermonotonic if and only if there exist non-decreasing continuous functions f1 and f2 and support, A, of X, which satisfy the following conditions





(3)





Since F1 and F2 are continuous, we have

Note that, in iii, X1 , . . . , Xd are driven by a single factor. Now we have the formal definition of countermonotonicity. Definition 2. A set A ⊆ R2 is countermonotonic if the following inequality holds

F1 F1−1 (u) + F2 F2−1 (1 − u) = 1,









d

X is called countermonotonic if it has countermonotonic support.

P (F1 (X1 ) + F2 (X2 ) = 1) = 1.

Theorem 2. For a bivariate random vector X, we have the following equivalent statements.

Bi = x ∈ R∃y ∈ R such that Fi (x) = Fi (y) and x ̸= y .





Since Fi (·) is non-decreasing function, it is easy to show that

P (X ≤ x) = max {F1 (x1 ) + F2 (x2 ) − 1, 0} .

(1)

iii. For U ∼ Uniform (0, 1), we have



For the d-variate comonotonic random vector X = (X1 , . . . , Xd ), the copula of X is uniquely expressed as where (u1 , . . . , ud ) ∈ [0, 1]d .

Bi =

−∞ ≤ ci,j < di,j ≤ ∞

for some countable index set Ji . Note that Fi (ci,j ) = Fi (di,j ) for any j ∈ Ji , and Fi (x) is strictly increasing for x ∈ Bci . Finally, for i = 1, 2, we have the following equation

P Xi ∈ Bci = 1 − P (Xi ∈ Bi )



(2)

where (u1 , u2 ) ∈ [0, 1]2 .

It is well known that every copula C is bounded in the following sense: W (u) ≤ C (u) ≤ M (u), where W (u) := max{u1 + · · · + ud − (d − 1), 0} and M (u) := min{u1 , . . . , ud }, which are called Fréchet–Hoeffding lower bounds and Fréchet– Hoeffding upper bounds, respectively. We note that M coincides with the comonotonic copula (2). While W is a countermonotonic copula for d = 2, it is not a copula in general for d ≥ 3. In fact, it is well known that there is no minimum copula for d ≥ 3; see Kotz and Seeger (1992) for an example. See Sklar (1959) and Cherubini et al. (2004) for more details. In the following proposition, we give a new equivalent representation for countermonotonicity. In the following section, we extend this equivalent expression of countermonotonicity to multidimensions.



= 1−



Fi (di,j ) − Fi (ci,j )



j∈J

For the bivariate countermonotonic random vector X, from (1), we know that the copula of X is uniquely expressed as C (u) = max{u1 + u2 − 1, 0},

 [ci,j , di,j ], j∈Ji

X = F1−1 (U ), F2−1 (1 − U ) .

C (u) = min{u1 , . . . , ud },

(5)

Since F1 and F2 are non-decreasing functions, the remaining goal is to find the support of (X1 , X2 ) such that F1 and F2 are strictly increasing. For i = 1, 2, define the set



i. X is countermonotonic ii. For any x ∈ R2

(4)

F1 (X1 ) + F2 (X2 ) = 1, which in turn implies

The following classical results are also useful.

for u ∈ [0, 1].

From (3) and (4), we conclude

(x1 − y1 )(x2 − y2 ) ≤ 0 for all x, y ∈ R2 .





for U ∼ Uniform(0, 1).

X = F1−1 (U ), . . . , Fd−1 (U ) .

d



= 1,

(6)

where the second equality holds, because Ji is a countable set. Finally, (5) and (6) produce

P X1 ∈ Bc1 , X2 ∈ Bc2 and F1 (X1 ) + F2 (X2 ) = 1 = 1,



which finishes the proof.





We can easily show that functions f1 and f2 in condition ii of Proposition 1 are not unique. Further, the condition may be replaced by f1 (x1 ) + f2 (x2 ) = c for some given c ∈ R. Note that, in this case, changing constant c will change functions f1 and f2 . 4. Multidimensional extension of countermonotonicity The key equation to represent countermonotonicity in Proposition 1 is f1 (x1 ) + f2 (x2 ) = 1,

W. Lee, J.Y. Ahn / Insurance: Mathematics and Economics 56 (2014) 68–79

which tells that the decrease of the second term is accompanied by the increase of the first term and can be directly extended to multivariate random vectors as follows: f1 (x1 ) + f2 (x2 ) + · · · + fd (xd ) = 1.

(7)

This describes how elements, x1 , . . . , xd , systematically move in different directions. With this equation, we give a formal definition of countermonotonicity in d-dimensions, and the property of (7) will be explored in detail.

is a strictly increasing function on (0, 1), i = 1, . . . , d. Hence, by defining the support of X∗ as B∗2 := xFi∗ (xi ) ∈ (0, 1), for i = 1, . . . , d ,

 



we know that Fi−1 ◦ Fi∗ is strictly increasing on B∗1 ∩ B∗2 . Finally, from (8), we can show that

P (X ∈ B) = P



F1−1 ◦ F1∗ (X1∗ ), . . . , Fd−1 ◦ Fd∗ (Xd∗ ) ∈ B = 1



P X ∗ ∈ B∗3 = 1 with

i. f1 (s1 ), . . . , fd (sd ) are strictly increasing functions at (s1 , . . . , sd ) = (x1 , . . . , xd )  d ii. j=1 fi (xi ) = 1

P X ∗ ∈ B∗1 ∩ B∗2 ∩ B∗3 = 1,

The case where only the subset of X satisfies Definition 3 is also interesting in practice. Definition 4. For any integer m, satisfying d > m ≥ 2, a random vector X is called partially m-CM if there exists a subset, {i1 , . . . , im }(⊆ {1, . . . , d}) with i1 < · · · < im such that





B∗3 := xi  F1−1 ◦ F1∗ (X1∗ ), . . . , Fd−1 ◦ Fd∗ (Xd∗ ) ∈ B .

 





Hence,





and g1 (x∗1 ), . . . , gd (x∗d ) are strictly increasing functions at x∗ ∈ B∗1 ∩ B∗2 ∩ B∗3 . In summary, we have the nondecreasing functions g1 , . . . , gd and support B∗ := B∗0 ∩ B∗1 ∩ B∗2 ∩ B∗3 of X∗ which satisfy the following conditions i. g1 (s1 ), . . . , gd (sd ) are strictly increasing functions at (s1 , . . . , sd ) = (x∗1 , . . . , x∗d ) ii.

d

j=1

gi (x∗i ) = 1

for any (x∗1 , . . . , x∗d ) ∈ B∗ . This completes the proof.



To illustrate the d-CM, the following example is given.

Xi1 , Xi2 , . . . , Xim ,





i.e., there exists a support B∗3 of X∗ such that

Definition 3. A d-variate random vector X will be called dcountermonotonic (d-CM) or d-CM with (f1 , . . . , fd ) if there exist non-decreasing continuous functions f1 , . . . , fd on R and a support, B, of X, which satisfy the following conditions

for any (x1 , . . . , xd ) ∈ B. Equivalently, we say that the distribution function H is d-CM if X is d-CM.

71



is m-countermonotonic.

Example 1. Let X to be a d-variate random vector satisfying

While the definition of comonotonicity can be characterized by a copula only as explained in Section 3, the definition of d-CM seems to be involved with the marginal distributions as well as copula. The following lemma shows that the definition of d-CM also can be characterized by a copula only.

d 

Lemma 1. Let X and X∗ be random vectors from the distribution functions H = C (F1 , . . . , Fd ) and

H ∗ = C (F1∗ , . . . , Fd∗ ),

respectively, where marginal distribution functions, F1 , . . . , Fd , are possibly different from marginal distribution functions, F1∗ , . . . , Fd∗ . Then X is d-CM if and only if X∗ is d-CM. Proof. Since X is d-CM, there exist non-decreasing continuous functions f1 , . . . , fd and a support, B, of X, which satisfy i and ii in Definition 3. Since Xi and Xi∗ are continuous, we have d Xi = Fi−1 ◦ Fi∗ (Xi∗ )

(8)

for i = 1, . . . , d, which in turn implies

 P



d



gi (Xi∗ ) = 1

= 1,

i =1

with gi := fi ◦ Fi−1 ◦ Fi∗ for i = 1, . . . , d. Hence, there exists the support B∗0 of X∗ such that d 

 ∗

gi xi

= 1,

for x∗ = (x∗1 , . . . , x∗d ) ∈ B∗0 .

i=1

Now, we want to find another support of X∗ , where g1 (·), . . . , gd (·) are strictly increasing. As in the proof of Proposition 1, there exists a support B∗1 of X∗ such that F1∗ , . . . , Fd∗ are strictly increasing functions on B∗1 . Further, since Fi is a continuous function, Fi−1 (·)

Xi = c ,

for some c ∈ R.

(9)

i=1

By defining f1 , . . . , fd in (7) as identity functions, which are strictly increasing on R, it is apparent that X is d-CM. In (9), we note that if one of Xi is increasing, the sum of the remaining components is decreasing. We also note that (9) is the key condition in the definition of complete mixability. We refer to Gaffke and Rüschendorf (1981) for the formal definition of complete mixability. One can argue that complete mixability is a special version of d-CM which requires a further condition on the marginal distributions. While complete mixability is an useful concept which can be applied to various minimization problems such as convex minimization problems of an aggregated sum, not every marginal distribution is d-completely mixable (Wang and Wang, 2011; Puccetti et al., 2012). The following is one of such examples. We show that while the marginal distribution F is not completely mixable, but it is possible to construct a 2-CM random vector (X1 , X2 ) whose marginal distributions are F . Example 2. Let F be a distribution function of exponential random variables with mean λ > 0, and X1 and X2 be random variables with a common distribution function F . Since an exponential distribution is not symmetric, by Puccetti et al. (2012), F is not 2-complete mixable. That is, for any given constant c ∈ R, there is no bivariate random vector X such that

P (X1 + X2 = c ) = 1. On the other hand, if we define X1 = F −1 (U ) and X2 = F −1 (1 − U ) for an uniform(0, 1) random variable, we can easily check that (X1 , X2 ) are 2-CM with (f1 , f2 ) satisfying F (F −1 (U )) + F (F −1 (1 − U )) = F (X1 ) + F (X2 )

= f1 (X1 ) + f2 (X2 ) =1

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where f1 (x) := F (x) and f2 := F (x) are strictly increasing function on x ∈ (0, ∞). Using the same idea, one can easily show that any random variables X1 and X2 with possibly different marginal distributions can also be 2-CM. While the results in Example 2 can be anticipated from Proposition 1, we can use a similar logic to show that for any given marginal distributions, F1 , . . . , Fd , there exists d-CM random vector X whose marginals are F1 , . . . , Fd . In other words, for any given marginal distributions F1 , . . . , Fd , we can always find a random vector satisfying the following conditions. i. X has marginal distributions, F1 , . . . , Fd . ii. X is d-CM.

Proof. Let g be any strictly increasing function on A ⊆ (0, 1) and Lebesgue measure A to be 1. Then it is easy to show that g (x) is strictly increasing for all x ∈ (0, 1). Since U is d-CM with (f1 , . . . , fd ), we can conclude that fi ’s are strictly increasing on (0, 1) for any i = 1, . . . , d. Hence, assuming the following condition for some constant c ∈ R

 P

 fi (Ui ) = c

it is enough to show

P

d 

 fi (Ui ) = c

P

d 

F i ( Xi ) =

j =1

d 2

= 1.

∗

(11)

i =1

The condition C ∗ ≺ C implies

P U∗ ≤ u ≤ P (U ≤ u)





= P ((f1 (U1 ), . . . , fd (Ud )) ≤ (f1 (u1 ), . . . , fd (ud )))   d d   ≤P fi (Ui ) ≤ f i ( ui )

Definition 5. A d-variate random vector X is strict d-CM if



= 1,

i =1



The following special version of d-CM is particularly useful for some minimization problem; one such example can be found in Section 4.2. When functions, fi for i ∈ {1, . . . , d}, in Definition 3 are the inverse of marginal functions, X will be called strict d-CM, and the formal definition is given as follows.

d 

i =1

 = 1.



(10)

=P c≤

i=1 d 

 f i ( ui ) ,

i =1

Equivalently, we say that H is strict d-CM if X is strict d-CM. It is a simple exercise to show that strict d-CM is d-CM. Hence strict d-CM is also invariant with marginal distributions, i.e., strict d-CM is a property of copula only. So for the properties of strict d-CM, it is enough to consider a d-variate random vector U whose distribution is a copula C , instead of considering all the random vectors with all possible marginal distributions. For the random vector U, condition (10) is equivalent to

 P

d  j=1

Ui =

d 2

for all u ∈ [0, 1]d , which in turn derives

P U∗ ≤ u = 0 if





d 

fi (ui ) < c .

(12)

i=1

By defining two sets



   d Rc := u ∈ [0, 1]  fi (ui ) < c , d

 = 1.

i =1

and

The existence of a strict d-CM copula can be easily shown. When d is even, we can easily construct strict d-CM copula, U, with U1 = · · · = Ud/2 = V

 Qc :=

and Ud/2+1 = · · · = Ud = 1 − V ,

where V is a uniform[0, 1] random variable. For odd d, it is enough to show the existence of a strict 3-CM copula. Once a strict 3-CM copula is constructed, we can use a similar technique as in the case of even d to construct a strict d-CM copula. As shown in Nelsen and Úbeda-Flores (2012), we can define a strict 3-CM random vector, U, by having uniformly distributed mass on the edges of a triangle with vertices (0, 1/2, 1), (1/2, 1, 0), (1, 0, 1/2). A similar technique for the construction of a strict 3-countermonotonic copula can be found in Rüschendorf and Uckelmann (2002).

  d

u ∈ [0, 1]d 

fi (ui ) < c

i =1

 and u1 , . . . , ud are rational numbers , we have the following set equality



d  x ∈ [0, 1]  f i ( xi ) < c



d

=

 

x ∈ [0, 1]d x < u





u∈Rc

i =1

=



x ∈ [0, 1]d x < u ,





u∈Qc

4.1. d-countermonotonicity as a minimal class of copulas Since there is no minimum copula available, we do not expect d-CM copulas to be the minimum copula. Then, to have necessary qualifications of countermonotonicity in multidimensions, it is natural to ask the minimality of d-CM copulas. In the following theorem, we show that the class of d-CM copulas can be regarded as a minimal class. Theorem 3. Let the random vectors U and U∗ be random vectors whose distribution functions are copulas C and C ∗ , respectively. Further assume that U is d-CM with (f1 , . . . , fd ). If we consider the random vector U∗ satisfying U∗ ≺ U, then U∗ is also d-CM with (f1 , . . . , fd ).

where the last equality holds because the set of points in [0, 1]d having all coordinates rational is dense in [0, 1]d . Hence, since Qc is countable set, we have

 P

d  i =1

 fi (Ui ) < c ∗





   d = P U ∈ x ∈ [0, 1]  fi (xi ) < c i=1      ∗ d =P U ∈ x ∈ [0, 1] x < u ∗

d

u∈Qc

≤



P U


∗



u∈Qc

= 0.

(13)

W. Lee, J.Y. Ahn / Insurance: Mathematics and Economics 56 (2014) 68–79

73

Fig. 1. 3D plot of (u1 )0.5 + (u2 )0.5 + (u3 )0.5 = 2 from two different angles.

Using the same logic, we have

 P

d 

 fi (Ui∗ ) > c

= 0,

i =1

which, in turn, concludes (11) along with (13).



So far, we have used the equivalence condition for the countermonotonicity in Proposition 1 to extend the concept of countermonotonicity to multivariate random vectors, and show that d-CM can be regarded as a minimal class of copulas. Since there is no minimum copula for d ≥ 3, the minimality is the best result that we can achieve. In the following section, we show that there exists a minimum copula in the Archimedean copula class, and the copula is d-CM. Further, we show that the strict d-CM can function as a minimum class of copulas from a geometrical point of view.

As already discussed, there is no minimum copula for d ≥ 3, and this implies that the set of all copulas is too wide to have a minimum element. Thus, one immediate solution is to restrict the class of copulas. Then, what is a broad and interesting class of copulas which contains the minimum? In particular, if such a class of copulas and its minimum copula exists, the relationship between the minimum copula and d-CM is of interest. Archimedean copulas have been studied in the context of probability metric spaces, see Schweizer and Sklar (1983) for details, and later used in a variety of applications. Especially in finance and actuarial science, Archimedean copulas become popular (Frees and Valdez, 1998; Hennessy and Lapan, 2002; Müller and Scarsini, 2005). The reason is not only because they are computationally tractable and have a simple algebraic form, but also because they provide a wide choice in models; see Nelsen (2006), Cherubini et al. (2004) and McNeil and Nešlehová (2009) for details of Archimedean copulas. In the following two propositions, we focus on the relation between d-CM copulas and the minimum copula of Archimedean copulas. We start with the existence of the minimum of Archimedean copulas. Let C be a d-

CdL ≺ C , where the Archimedean copula, CdL , is defined as





1

1

Theorem 4. Archimedean copula, CdL , is d-CM. Proof. Let U be the random vector whose distribution is the 1 copula, CdL , and define fi (x) = x d−1 on x ∈ [0, 1] for i = 1, . . . , d. To show that U is d-CM, and it is enough to show that

P (f1 (U1 ) + · · · , fd (Ud ) = d − 1)

  1 1 = P (U1 ) d−1 + · · · + (Ud ) d−1 = d − 1

4.2. Some multidimensional countermonotonicity examples

Proposition 2 (McNeil and Nešlehová (2009)). dimensional Archimedean copula, then

Since CdL coincides with the Fréchet–Hoeffding lower bound in 2-dimension, it is natural to investigate the countermonotonicity of CdL . Fig. 1 shows a 3D plot where the support of CdL lies at d = 3. From the figure, we can observe that, when one component is increasing, the sum of the other two components tends to decrease, which is typical behavior of d-CM random vectors. In the following proposition, we prove that the minimum of Archimedean copulas, CdL , is indeed d-CM.

= 1.

(14)

Note that (f1 , . . . , fd ) are strictly increasing functions on x ∈ (0, 1), and satisfy conditions in Definition 4. Since CdL (u) = 0 for any u satisfying 1

1

(u1 ) d−1 + · · · + (ud ) d−1 < (d − 1), one can show that





1

1

P (U1 ) d−1 + · · · + (Ud ) d−1 < d − 1 = 0,

by using the similar logic as in the proof of Theorem 3. Then, the following observation



1

1

E (U1 ) d−1 + · · · + (Ud ) d−1 and (15) can conclude (14).



= d − 1, 

Now, we show that the strict d-CM can function as a minimum from a geometrical point of view. To define a minimum of multivariate distribution functions, Kotz and Seeger (1992) introduced the concept of moment of inertia (MOI) for multivariate distributions. To give the formal definition of MOI, consider first the distance of a point a = (a1 , . . . , ad ) from the curve {F1 (x1 ) = · · · = Fd (xd )}: d(a; F1 , F2 , . . . , Fd )

d−1

CdL (u) := max (u1 ) d−1 + · · · + (ud ) d−1 − (d − 1), 0

.

(15)

=

inf

 { (a1 − y1 )2 + · · · + (ad − yd )2 }.

y∈{x|F1 (x1 )=···=Fd (xd )}

74

W. Lee, J.Y. Ahn / Insurance: Mathematics and Economics 56 (2014) 68–79

Let X be a random vector with distribution H = C (F1 , . . . , Fd ). The MOI is defined as

5. Measures of concordance and multidimensional countermonotonicity

MOI (H ) := E (d2 (X; F1 , F2 , . . . , Fd )),

As a statistical application, multidimensional countermonotonicity is useful to compute the minimum of popular concordance measures such as Kendall’s tau and Spearman’s rho. We first review measures of association in 2-dimension and their extension to multidimensions. Then, the relationships between the minimum of such measures and multidimensional countermonotonicity will be investigated in detail.

where the expectation is taken with respect to X. If all the probability mass is concentrated on the curve {x|F1 (x1 ) = · · · = Fd (xd )}, MOI becomes 0, and this corresponds to the comonotonic random variables which imply extreme positive dependence. Thus, the basic geometric idea of finding an extreme negative dependence is to assign the most probability to points away from the curve {x|F1 (x1 ) = · · · = Fd (xd )}, and this corresponds to maximizing the MOI. In line with this, Kotz and Seeger (1992) considered MOI as a generalized measure of negative dependence. In view of physics, MOI can be interpreted as a measure of resistance of the random vector X to a change of angular velocity when it is rotating around the curve. The application in Kotz and Seeger (1992) seems very limited. They found a normal density function achieving the maximal MOI among the multivariate normal distributions with the same marginal distributions, and interpreted such a multivariate normal distribution as a minimal element among the class. In the following theorem, we generalize their results to an arbitrary group of copulas. Specifically, we show that the MOI can be used as a generalized measure of negative dependence among copulas by showing that the MOI preserves the ordering of copulas. Furthermore, we also observe that comonotonicity and strict d-CM are at the both ends of inequalities, respectively.

5.1. Review: measures of concordance Kendall’s tau and Spearman’s rho are the two most popular measures of association, defined by Kruskal (1958), Lehmann (1966, 1975), to measure concordance in the following sense. Definition 6 (Kendall’s Tau and Spearman’s Rho). Let X and X∗ be independent and identically distributed bivariate random vectors, each with joint distribution H. Then Kendall’s tau, τ2 (H ) is defined as

  τ2 (X) = 4 · P X1 ≤ X1∗ , X2 ≤ X2∗ − 1  =4 C (u)dC (u) − 1. [0,1]2

Spearman’s rho, ρ2 (H ), is defined as

ρ2 (H ) = 12

 [0,1]2

C (u)du − 3.

Theorem 5. Let U be a random vector whose distribution is a copula C . Then,

Scarsini (1984) proposed a set of axioms where any reasonable measures of association in 2-dimension need to satisfy.

i. If copulas C1 and C2 satisfy C1 ≺ C2 , then MOI (C1 ) ≥ MOI (C2 ).

Definition 7. Let X and X∗ be bivariate random vectors with distribution functions H = C (F1 , F2 ) and H ∗ = C ∗ (F1∗ , F2∗ ), respectively. A numeric measure κ(H ) (or κ(X)) is a measure of concordance if it satisfies the following axioms.

ii. MOI (C ) is minimized if and only if U is comonotonic. iii. MOI (C ) is maximized if and only if U is strict d-CM. Proof. Note that D := {x|F1 (x1 ) = · · · = Fd (xd )} = {xx1 = · · · = xd }. As the normal case shown in Kotz and Seeger (1992), the direct computation of MOI gives



MOI(C ) =

d−1 12

−

2 d i
cov(Ui , Uj ).

(16)

1

 0

1



P (U1 ≤ u1 , U2 ≤ u2 ) 0

− P (U1 ≤ u1 ) P (U2 ≤ u2 ) du1 du2  1 1 1 = P (U1 ≤ u1 , U2 ≤ u2 ) du1 du2 − , 0

lim κ(Hi ) = κ(H );

i→∞

Using a similar method to Lemma 2 of Lehmann (1966), we derive the following equation cov(U1 , U2 ) =

S1. −1 ≤ κ (H ) ≤ 1, κ (M (F1 , F2 )) = 1, κ (Π (F1 , F2 )) = 0, and κ (W (F1 , F2 )) = −1; S2. If C ≺ C ∗ , then κ(H ) ≤ κ(H ∗ ); S3. If a sequence of joint distribution functions, {H1 , H2 , . . .}, converges pointwise to H, then

4

0

from which, along with (16), we can easily prove parts i and ii.  Since MOI(C ) is maximized if and only if i


cov(Ui , Uj ) ≥ 0,

i
we can deduce that the minimum of i


S4. κ(Hπ ) = κ(H ), where Hπ is a distribution function of the bivariate random vector, (Xπ (1) , Xπ (2) ), and π is any permutation on {1, 2}; S5. κ(C1# (F1 , F2 )) = κ(C2# (F1 , F2 )) = −κ(H ). It is well known that Kendall’s tau and Spearman’s rho are measures of concordance, see for example Nelsen (2006). Extension of the bivariate Kendall’s tau and Spearman’s rho to multidimensions has received much attention; see Joe (1990), Kendall and Smith (1940), Schmid and Schmidt (2007) for details. Definition 8 (Kendall’s Tau and Spearman’s Rho). Let X be a d-variate random vector with joint distribution H = C (F1 , . . . , Fd ). Then Kendall’s tau, τd (H ) and Spearman’s rho, ρd (H ), are defined as

τd (H ) =

1 2d−1 − 1

 2

d



 [0,1]d

C (u)dC (u) − 1

(17)

and

ρd (H ) =

d+1 2d − (d + 1)



d





2

[0,1]d

C (u)du − 1

W. Lee, J.Y. Ahn / Insurance: Mathematics and Economics 56 (2014) 68–79

respectively. One can also define pairwise version of Kendall’s tau, Td , and Spearman’s rho, Rd , as 1

Td (H ) =  

 

d 2

τ2 (Hi,j )

and

   1 ρ2 (Hi,j ) Rd (H ) =  

(18)

respectively, where Hi,j refers to the bivariate marginal distribution of H which corresponds to the ith and jth margin. For convenience, we call τd , ρd , Td and Rd multivariate Kendall’s tau, multivariate Spearman’s rho, pairwise Kendall’s tau and pairwise Spearman’s rho, respectively. Scarsini (1984) extends the concept of bivariate measures of concordance in Definition 7 to multivariate random vectors as follows. Definition 9. Let X and X∗ be d-variate random vectors with distribution functions H = C (F1 , . . . , Fd ) and H ∗ = C ∗ (F1∗ , . . . , Fd∗ ), respectively. A numeric measure κ(H ) (or κ(X)) is a measure of concordance if it satisfies the following axioms. A1. (Normalization) κd (H ) ≤ 1, κd (M (F1 , . . . , Fd )) = 1 and κn (Π (F1 , . . . , Fd )) = 0; A2. (Monotonicity) If C ≺ C ∗ , then κd (H ) ≤ κd (H ∗ ); A3. (Continuity) If a sequence of joint distribution functions, {H1 , H2 , . . .}, converges pointwise to H, then limi→∞ κd (Hi ) = κd (H ); A4. (Permutation Invariance) κd (Hπ ) = κd (H ), where Hπ is a distribution function of the d-variate random vector, (Xπ (1) , . . . , Xπ(d) ), and π is any permutation on {1, . . . , d}; A5. (Duality) κd (H ) = κd ( H ); A6. (Transition Property) There exists a constant rd such that rd κd−1 H(i) = κd (H ) + κd Ci# (F1 , . . . , Fd ) ,



Proof. Let X = (X1 , . . . , Xd ) be a d-CM random vector and X∗ be its independent copy. From (17), we know that showing τd (X) = L1,d is equivalent to showing

P X1 ≤ X1∗ , . . . , Xd ≤ Xd∗ = 0.





i


Theorem 6. For a given random vector, X = (X1 , . . . , Xd ), we have the following. i. τd (X) = L1,d if X is d-CM. ii. τd (X) = L1,d if X is partially m-countermonotonic, where m is a positive integer less than d.



i
d 2

75





for any i ∈ {1, . . . , d},

Since X is d-CM, we may consider non-decreasing continuous functions, f1 , f2 , . . . , fd , satisfying condition i and ii in Definition 3 on a set A ⊆ Rd with P (X ∈ A) = 1. Since X∗ is an independent copy of X, they have the same probability distribution function. Hence, we also have P (X∗ ∈ A) = 1, and obviously f1 , f2 , . . . , fd , satisfies condition i and ii in Definition 3 on the set A ⊆ Rd . Then

P X1 ≤ X1∗ , . . . , Xd ≤ Xd∗





  ≤ P f1 (X1 ) ≤ f1 (X1∗ ), . . . , fd (Xd ) ≤ fd (Xd∗ )   d−1 d−1   ∗ ∗ ≤P f i ( Xi ) ≤ fi (Xi ), fd (Xd ) ≤ fd (Xd )  =P

i =1

i =1

d−1 

d−1 

fi (Xi ) ≤

i=1

 fi (Xi ), fd (Xd ) ≤ fd (Xd ), X ∈ A, X ∈ A ∗

∗

∗

i=1

= P(c − fd (Xd ) ≤ c − fd (Xd∗ ), fd (Xd ) ≤ fd (Xd∗ ), X ∈ A, X∗ ∈ A) = 0,

(19)

where the last equality holds because X is continuous. From (19), we prove the part i. Now we show the part ii. First, assume that X is partially m-CM with

(Xi1 , . . . , Xim ), is m-CM. Similar to part i, it is enough to show

where H(i) is a distribution function of the (d − 1)-variate random vector

P X1 ≤ X1∗ , . . . , Xd ≤ Xd∗ = 0.

(X1 , . . . , Xi−1 , Xi+1 , . . . , Xd ).

Now we may conclude (20) with the following equality





P X1 ≤ X1∗ , . . . , Xd ≤ Xd∗ ≤ P Xi1 ≤ Xi∗1 , . . . , Xim ≤ Xi∗m ,



Dolati and Úbeda-Flores (2006) and Schmid and Schmidt (2007) showed that τd , ρd , Td and Rd are measures of concordance. 5.2. Measures of concordance and multidimensional countermonotonicity In this section, we study some extreme properties of multivariate countermonotonicity by observing when the minimum of various measures of concordance are achieved. First, we provide the relation between multivariate Kendall’s tau and d-CM. For the minimum of d-variate Kendall’s tau, Genest et al. (2011) and ÚbedaFlores (2005) showed that

τd (X) ≥ −

1 2d−1

−1

,

and the minimum on the right is denoted by L1,d := −

1 2d−1 − 1

.

Genest et al. (2011) and Úbeda-Flores (2005) provide some examples where the minimum, L1,d , is achieved. The following theorem shows that d-CM is a sufficient condition to achieve the minimum of d-variate Kendall’s tau.

(20)







(21)

and the fact that

(Xi1 , . . . , Xim ), is m-CM.



The converse of Theorem 6 is not true, i.e., τd (X) = − 2d−11 −1 does not imply that X is d-CM. We give such an example below. Example 3. Let U[0,1] , U[0,1/2] , and U[1/2,1] be independent uniform distributions on the interval [0, 1], [0, 1/2], and [1/2, 1], respectively. Further, define a random vector X as X1 = U[0,1] ,

 X3 =

 X2 =

U1/2,1 ; −X 1 + 1 ;

−X1 + 1; U[0,1/2] ;

if 0 ≤ X ≤ 1/2 if 1/2 < X ≤ 1,

and

if 0 ≤ X ≤ 1/2 if 1/2 < X ≤ 1,

and X∗ to be an independent copy of X. Then we can easily show that

P X1 < X1∗ , X2 < X2∗ , X3 < X3∗ = 0,





which implies τ3 (X) = − 13 , and that X is not 3-CM.

76

W. Lee, J.Y. Ahn / Insurance: Mathematics and Economics 56 (2014) 68–79

The minimum  of the d-variate Kendall’s tau, τd , is achieved if and only if [0,1]d C (u)dC (u) is zero in (17). Thus, for two independent samples X and X∗ from the distribution, H, τd (X) = L1,d if and only if

P X1 < X1∗ , . . . , Xd < Xd∗ + P X1 > X1∗ , . . . , Xd > Xd∗ = 0,









which implies that the measure of the comonotonic domain is 0. The proof is trivial from the definition. Now we investigate the relations between strict d-CM and measures of concordance. For the d-variate Kendall’s tau, by Theorem 6, any strict d-CM X satisfies

τd (X) = L1,d , because strict d-CM implies d-CM. For the pairwise Spearman’s rho, Nelsen and Úbeda-Flores (2012) studied the case when Rd (X) attains the minimum in d = 3. Then next theorem is the generalization of Nelsen and Úbeda-Flores (2012) to any dimension d ≥ 2. Theorem 7. For any d-variate random vector, X, we have the following inequality Rd (X) ≥ −

1 d−1

,

(22)

where equality holds if and only if X is a strict d-CM random variable. For convenience, we define the minimum of pairwise Spearman’s rho in (22) as L3,d := −

1 d−1

.

To determine when the minimum of pairwise Kendall’s tau is achieved, we first consider the case, d = 3. As shown in Nelsen (1996) and Genest et al. (2011), using the inclusion–exclusion principle, one can show that 3-variate Kendall’s tau τ3 (X) coincides with pairwise Kendall’s tau T3 (X), that is,

τ3 (X) = T3 (X). Then, using Theorem 6, it can be shown that the minimum of τ3 (X) is achieved if X is (strict) 3-CM. In the following example, we show that (strict) d-CM generally does not imply the minimum of pairwise Kendall’s tau for d > 3, which is the anticipated result from the fact that there is no minimum copula available for d ≥ 3. In other words, not all concordance measures achieve the minimum with a (strict) d-CM condition. Example 4. Consider the 6-variate random vector U, whose distribution function is a copula C , with 3

U1 + U2 + U3 =

and

2

U4 + U5 + U6 =

3 2

,

and (U1 , U2 , U3 ) is independent to (U4 , U5 , U6 ). Hence, we observe that X is strict 6-CM, but T6 (X) = −1/30 which is greater than L2,6 = −1/5. The following theorem provides a simpler proof of (23) and gives an equivalent condition to achieve the minimum of pairwise Kendall’s tau.

Proof. First, observe the following classical result

Theorem 8. Let X and X∗ be two independent d-variate random vectors from the same distribution function, H.

ρ2 (Xi , Xj ) = ρ2 (Ui , Uj ) = 12 Cov(Ui , Uj ),

Td (X) ≥ L2,d , where the inequality holds if and only if

where U := (F1 (X1 ), . . . , Fd (Xd )), which in turn implies Rd (X) = Rd (U). If we set c =

d , 2

P

i=1

d 3

−

d

i =1

2

2

+

1

 

6

2

d

Rd (U) +

i
1

 

d−1

2

2

d

+

d

2

4

P

d



Ui =

i =1

d 2



I Xi < Xi∗ − I Xi > Xi∗



= −1, 0, or 1 = 1.

.

Td (X) ≥ −

= 1. 

.

  d

2

Td (X),

1 d−1

,

where equality holds if and only if P (Sd = 0) = 1. Since Sd cannot be 0 for odd d, consider the following

1 2⌊(d + 1)/2⌋ − 1

2⌊(d + 1)/2⌋ − 1



we have



1



Since

E Sd = d + 2

.

(23)

For convenience, we define the minimum in the above inequality (23) as L2,d = −

d   

 2

Now we wonder whether (strict) d-CM achieves the minimum of pairwise Kendall’s tau or other measures of concordance. Kendall and Smith (1940) and Genest et al. (2011) give the following inequality for any random vector X. Td (X) ≥ −

− I Xi > Xi

∗



i=1

,

1 Further, the equality Rd (U) = − d− holds if and only if 1





Proof. Let Sd =

which leads to the inequality 1

∗

i=1

i=1

=



I Xi < Xi

we have

 2    d d d     2 2 E Ui − c  = E Ui − 2c Ui Uj + c Ui + 2

Rd (U) ≥ −

 d   

.

E (Sd − [I {Sd = 1} − I {Sd = −1}])2



  =d+2

d

2



Td (X) + E (I {Sd = 1} − I {Sd = −1})2





− 2E [Sd I {Sd = 1} − Sd I {Sd = −1}] .

(24)

From (24), we have 1 Td (X) ≥ − , d where the equality holds if and only if P (Sd = 1 or − 1) = 1.



W. Lee, J.Y. Ahn / Insurance: Mathematics and Economics 56 (2014) 68–79

6. Application to the herd behavior index Herd behavior is a term often used in financial literature to describe the comovement of members in a group without a planned direction. It is an important economic phenomenon, especially in the context of the recent financial crises. Various studies about herd behavior were conducted among a number of researchers over several years (Chang et al., 2000; Sushil and Sunil, 2001; Skintzi and Refenes, 2005). In this section, we first review the herd behavior indices of Dhaene et al. (2012), Choi et al. (2013) and Lee et al. (2013). The herd behavior index of Lee et al. (2013) is particularly of interest, because it is the only one satisfying the monotonicity axiom, axiom A2 of Definition 9. Then, we study the relationship between this herd behavior index and multidimensional countermonotonicity using a Middle East stock market example. We first define some notations. Consider a financial market with d stocks and let X be d aggregated stock prices at a fixed time t + 1 assuming that the current time is fixed at t. For given X, define the market index S as the sum of the d aggregated stock prices: S=

d 

Xi .

i =1

Following Dhaene et al. (2012), X is comonotonic if and only if d

X = (FX−11 (U ), . . . , FX−11 (U )), where U is a uniform [0, 1] random variable. Under a comonotonic aggregated stock prices assumption, define Sc =

d 

wi FX−i 1 (U ).

i =1

For convenience, comonotonic aggregated stock prices at time t are denoted by Xc = (X1c , . . . , Xdc ). Based on the observation that the variance of the market index is maximized when the aggregated stock prices are comonotonic, Dhaene et al. (2012) defined the herd behavior index (HIX) as the ratio of variance of the market index to that of the index under the comonotonic assumption: HIX (X) =

Var[S ] Var[S c ]

.

As specified in Choi et al. (2013), HIX is sensitive to marginal distributions. To reduce such marginal distribution effects, Choi et al. (2013) gave a revised version of HIX (RHIX) by subtracting variance terms in both the numerator and denominator of RHIX, as shown below.



cov Xi , Xj



cov Xic , Xjc



Definition 10 (Lee et al., 2013). For a given random vector X, Spearman’s rho type of the Herd Behavior Index (SIX) is defined as

 i̸=j

SIX (X) := 

.

  ρ 2 Xi , Xj

 . ρ2 Xic , Xjc

i̸=j

Since

  ρ2 Xic , Xjc = 1 for all i, j ∈ {1, . . . , d}, SIX can be expressed as

 SIX (X) =

  ρ2 Xi , Xj

i̸=j d 2

−−→ W (·) := (W1 (·), . . . , Wd (·))T , and W1 , . . . , Wd are d independent Wiener processes on a local time interval [0, T ]. It is assumed that −−→ the stock price vector X (t ) follows the Gaussian process on [0, T ] defined as

  −−→ −−→ −−→ − → X (t ) = X (0) ∗ exp µ − 1/2diag[6] · t + 2W (t ) for 0 ≤ t ≤ T ,

(25)

where ∗ denotes componentwise product of two vectors and, 6 := → 22T is called a covolatility matrix, and − µ and 2 are a vector in d R and a d × d matrix respectively. Since, as illustrated in Lee et al. (2013), SIX during the time interval [0, T ] can be denoted by

 SIX[0,T ] :=

i̸=j

  d 2

 =

  ρ2 Xi (T ), Xj (T )

i̸=j



6

arcsin π

√ 6i,j



6i,i 6j,j

  d 2

i̸=j

While RHIX is known to reduce the effect of marginal distributions, one may be interested in the herd behavior indices which are free of marginal effects. Lee et al. (2013) proposed a new herd behavior index which is free of the marginal distributions, by replacing covariance terms in RHIX with a concordance measure such as Spearman’s rho, ρ2 . Hence the new herd behavior is interpreted as the ratio of average Spearman’s rho of market indices to average Spearman’s rho of indices under the comonotonic assumption. In the following definition, we provide a simplified version of the herd behavior index from Lee et al. (2013).

,

 

which is the same as the pairwise Spearman’s rho in (18), which in turns implies that the pairwise Spearman’s rho can be interpreted as a new herd behavior index. Furthermore, pairwise Spearman’s rho is a measure of concordance and hence satisfies the monotonicity axiom. As specified in Theorem 7, the minimum of pairwise Spearman’s rho is achieved if and only if X is strict d-CM, and the maximum of pairwise Spearman’s rho is achieved if and only if X is comonotonic. Having SIX as the herd behavior index, countermonotonicity can be interpreted as the strongest herd behavior, while strict d-CM can be interpreted its opposite. While comonotonic movement is observable during the recession, as specified in Dhaene et al. (2012) and Choi et al. (2013), strict d-CM of stock prices has not been reported, although we can observe it in practice. As an illustration, we provide Example 5 below, where a strict d-CM movement of stock prices in the Middle East is observed. Let



i̸=j

RHIX (X) = 

77

,

(26)

the estimation of SIX[0,T ] can be made from the estimation of 6. For the estimation of 6 for the given interval [0, T ], let

Λ := {τ0 , . . . , τn } be n+1 equi-distant time grids such that τ0 = 0 and τn = T , and let − → ri be the ith log-return vector during [τi−1 , τi ]. Then, from (25), it − → n is easy to check that ri i=1 is a sequence of independent Gaussian random vectors with a mean vector and a covariance matrix

−  T → µ − 1/2 diag [6] ·

n

and

6·

T n

,

78

W. Lee, J.Y. Ahn / Insurance: Mathematics and Economics 56 (2014) 68–79

(a) Pairwise Spearman’s rho and F1 (X1 ) + . . . + Fd (Xd ).

(b) Stock indices. Fig. 2. Pairwise Spearman’s rho in the Middle East.

respectively. Thus, the estimation of 6 can be made from a sample covariance matrix of the log-return vectors. While, in (25), we set the interval to be [0, T ] for the simplicity of the notation, it can be any time interval of interest. Hence the above estimation procedure for SIX can be repeated by moving the time window to the right to get Fig. 2(a). Here the length of time window is set to be relatively large, 21 weeks, for the purpose of reducing the noise effect. Detailed estimation methods including the weak convergence of the estimator can be found in Lee et al. (2013). Note that we are using a realized version of estimator based on the observed stock prices, whereas Dhaene et al. (2012) uses an implied version using the prices of vanilla options. Example 5. In this example, the herd behaviors in the Middle East are analyzed. First, assuming that the current time is t, consider the representative stock indices at time t +1, with a unit of one week, of four middle east countries (Jordan, Oman, Qatar and Saudi Arabia): X = (X1 , . . . , X4 ). All data are from Morgan Stanley Capital International (MSCI). As observed from Fig. 2, the pairwise Spearman’s rho is close to the minimum, −1/3, during the period between July of 2005 and February of 2006. Hence, it is interesting to see whether the behavior of stock indices are similar to the strict d-countermonotonicity movements. By defining

that multivariate countermonotonicity can be interpreted as the counterpart of the extreme herd behavior as specified in Lee et al. (2013). In this paper, we provide an example with real financial data, which shows the multivariate countermonotonicity movements. Various further research directions can be obtained from this paper. For example, in the definition of d-CM, it is not shown how to construct f1 , . . . , fd . To perform more specific investigation on the mathematical properties of d-CM, such construction will be helpful. In studying the herd behavior indices, the conceptual link between the comonotonicity and agents’ behaviors in an economic system seems obvious. However, it is not obvious how the countermonotonicity conceptualizes agents’ behaviors. We only understand that the systematic behavior leading to d-CM gives the opposite extreme value of the herd behavior indices. By investigating this further, we can increase our understanding of the herd behavior. Acknowledgments For Woojoo Lee, this work was supported by Inha University Research Grant. For Jae Youn Ahn, this work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (2013R1A1A1076062) and Ewha Womans University Research Grant of 2013.

ψ := F1 (X1 ) + · · · + F4 (X4 ), we note that the stock indices, X, are strict d-CM if and only if P (ψ = c ) = 1 for some c > 0 during the period of interest. During the period between July of 2005 and February of 2006, we can observe that ψ is flat, which supports that the stock indices, X, are strict 4-CM. As shown in the Fig. 2, stock markets before July of 2005 in the Middle East were in an unprecedented booming period, and short after February of 2006, the stock market crashed. It is interesting that we can identify strict d-CM movement of stock prices at the end of boom period, where people still believe in the increase of stock prices while the stock market is actually ready to fall. 7. Conclusions In this paper, we extend the concept of countermonotonicity into multidimensional space and show that newly defined countermonotonic copulas constitute a minimal class of copulas. We found that the definition is also useful in various statistical applications. In particular, we study the relationship between multidimensional countermonotonicity and conditions to achieve the minimums of concordance measures such as multivariate Kendall’s tau and Spearman’s rho. In addition to these, by using the pairwise Spearman’s rho as a herd behavior index, we review

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