ABSTRACT KEYWORDS. Comonotonicity, dependence, correlation, concordance, copula, multivariate. 1. INTRODUCTION

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1 MEASURING COMONOTONICITY IN M-DIMENSIONAL VECTORS BY INGE KOCH AND ANN DE SCHEPPER ABSTRACT In this contribution, a new easure of coonotonicity for -diensional vectors is introduced, with values between zero, representing the independent situation, and one, reflecting a copletely coonotonic situation. The ain characteristics of this coefficient are exained, and the relations with coon dependence easures are analysed. A saple-based version of the coonotonicity coefficient is also derived. Special attention is paid to the explanation of the accuracy of the convex order bound ethod of Goovaerts, Dhaene et al. in the case of cash flows with Gaussian discounting processes. KEYWORDS Coonotonicity, dependence, correlation, concordance, copula, ultivariate.. INTRODUCTION One of the recurring issues in applications of ultivariate systes in the literature, deals with the dependence in a ultivariate syste. In the first place it is of course necessary to understand, to quantify and to analyse dependence structures, but for any applications an iportant question is actually how to anage systes with coplex dependence structures. Broadly speaking, we can distinguish two classes of answers to this question in the current research. Firstly, there is an increasing interest in odelling the real or unknown dependence structure of ultivariate vectors, by eans of copulas e.g. These odels can then be used as starting points toward calculations and estiations for quantities iediately related to the underlying dependence structure. Secondly, another group of researchers ake use of extree types of dependencies so as to derive approxiate results for related quantities, without specifying the aybe coplex real underlying dependence structure. These two classes of ethods both benefit of course fro research on the analysis of dependence structures and on dependence easures, see e.g. Lehann (966), Schweizer & Wolff (98), Wolff (980), Drouet Mari & Kotz (200), Scarsini (984), Joe (990, 993, 997), and any others. Astin Bulletin 4(), doi: 0.243/AST by Astin Bulletin. All rights reserved.

2 92 I. KOCH AND A. DE SCHEPPER In the financial and actuarial research e.g., both approaches are popular and widespread, each of the of course with its own advantages and disadvantages. On the one hand we have the copula odels, ost of the tie resulting in a rather coplete description of the dependence structures under investigation. However, next to the fact that this ethod can be quite tie consuing, one of the difficult probles is how to deterine the optial copula aong all alternatives. On the other hand, we can use one of the approxiation ethods. Such a ethod does not lead to a coplete picture of the dependence structure, but it creates the possibility to derive upper and lower bounds, often by eans of easy calculations. An approxiation technique which is frequently used in the current financial and actuarial research, is based on a ethod of Kaas et al. (2000, 2002) and Dhaene et al. (2002a,b). These authors define bounds in convex order sense, calculated by eans of coonotonic vectors. If this technique is applied e.g. to present values of cash flows with stochastic interest rates, it turns out that the approxiate results are very accurate. In fact this is not copletely unexpected, since copounded rates of return for successive periods are strongly utually dependent and thus not far away fro coonotonicity. This brings us to the question whether it is possible to easure the degree of coonotonicity in an arbitrary ultivariate vector. With such a easure it is possible to predict whether or not an approxiation as suggested by Goovaerts, Dhaene et al. would be reasonable for a given vector. If so, it can also be used to quantify how reasonable such an approxiation would be. In this paper, we show how such a easure can be constructed and calculated for arbitrary -diensional vectors, and we will use our coonotonicity coefficient to explain why in the case of cash flows the convex bounds reveal such effective approxiations. We also present an alternative definition of this coefficient in case the dependence structure can be described by a copula, which akes it possible to link the approxiations of the second approach to the odelling techniques of the first approach. Finally, we deonstrate how the coonotonicity coefficient can be estiated based on real data. In order not to coplicate the forulae and the calculations, throughout this paper we will work in a continuous setting. However, an extension to the discrete case is certainly possible. The paper is organised as follows. In section 2 we recall soe definitions and notions on distributions and copulas, and we repeat the ost coon bivariate dependence easures. Section 3 is the ost iportant part, we introduce our coonotonicity coefficient and we discuss its ajor properties. Afterwards in section 4, we present several nuerical illustrations, including the situation of a cash flow with a Gaussian discounting process. Finally, in section 5 we derive a saple based version of the coonotonicity coefficient, and we present an illustration on epirical data. Conclusions can be found in section 6.

3 MEASURING COMONOTONICITY IN M-DIMENSIONAL VECTORS PRELIMINARIES The first topic we briefly describe in this section is the Fréchet space, which is an essential concept when investigating the dependence structure in ultivariate vectors. Definition. (Fréchet space) Let F,, F be arbitrary univariate distribution functions. The Fréchet space R (F,, F ) consists of all the -diensional (distribution functions F X of) rando vectors X = (X,, X ) having F,, F as arginal distribution functions, or F i (x) = Prob(X i x) for x! and i =,...,. If it is clear fro the context, we will use the short notation R. For an unabiguous definition of our coonotonicity coefficient, we will have to restrict ourselves to positive dependent vectors; we will coe back to this issue in section 3. A weak for of positive dependence in a ultivariate context is positive lower orthant dependence. Definition 2. (PLOD) The rando vector X = (X,, X ) is said to be positive lower orthant dependent (PLOD) if Prob(X x,, X x ) $ % i = Prob(X i x i ), 6(x,, x )!. Starting fro this definition we define the qualified Fréchet space as the subset of the Fréchet space with all positive lower orthant dependent vectors. Definition 3. (qualified Fréchet space) Let F,, F be arbitrary univariate distribution functions. The qualified Fréchet space R + (F,, F ) consists of all the -diensional (distribution functions F X of) rando vectors X = (X,, X ) having F,, F as arginal distribution functions and which are PLOD. If it is clear fro the context, we will use the short notation R +. Two iportant and at the sae tie extree eleents of a Fréchet space are the coonotonic and the independent vector. Definition 4. (Coonotonic vector and independent vector) For every Fréchet space R (F,, F ), we define the independent vector X I = ( X I I,, X ) as the vector with distribution F I X (x,, x ) = % i = Fi( xi), and the coonotonic vector X C = (X C C,, X ) as the vector with distribution F C X (x,, x ) = in i = Fi ( xi). Both vectors belong to the qualified Fréchet space. In Denneberg (994) and Dhaene et al. (2002a), an alternative characterisation is presented for coonotonicity, by showing that a vector X = ( X,, X ) is coonotonic if for non-decreasing functions t,, t and for a rando variable Z it is true that X = d (t (Z),, t (Z)). Note that with the inverse function defined as F i ( p) = inf{x! F i (x) $ p}, p! [ 0,], the coonotonic vector can be easily constructed as X C = ( F (U ),, F (U )), with U a rando variable, uniforly distributed on [0,]. For the construction of our coonotonicity coefficient, we will rely on the following lea.

4 94 I. KOCH AND A. DE SCHEPPER Lea. (Bounds for the Fréchet space) For any vector X = (X,, X ) with distribution F X of a given qualified Fréchet space R +, it is true that F X I F X F X C. For arbitrary vectors of the larger Fréchet space R, only the upper bound holds. In recent financial research, copulas have grown into a very popular tool for dependence odelling issues. In its essence, a copula is nothing else than a ultivariate distribution function with unifor arginal distributions. Or, alternatively, it is the result of the ultivariate distribution after rescaling the arginals to unifor distributions. As such, copulas are ostly seen as that part of the joint distribution which captures the dependence structure. In this section, we only provide soe of the ost iportant facts on copulas. For a ore coprehensive study, we refer to Nelsen (2006) and Cherubini et al. (2004). Definition 5. (Copula) An -diensional copula C is a function C : [0,] "[0,], non-decreasing and right-continuous, with the following properties: C(u,, u i, 0, u i +,, u ) = 0 C(,,, u i,,, ) = u i D a, b D a2, b 2 D a, b C(u,, u ) $ 0 if (a,, a ) < (b,, b )! [ 0,], where the differences are defined as D ai, b i C(u,, u ) = C(u,, u i, b i, u i +,, u ) C(u,, u i, a i, u i +,, u ), and this for all (u,, u )! [ 0, ], and for all i =,,. One of the ost fundaental results about copulas is suarised in the following theore. Theore (Sklar s Theore) For any -diensional distribution function H with arginals F,, F, there exists an -diensional copula C for which H(x,, x ) = C(F (x ),, F (x )). Conversely, this copula C can be calculated as C(u,, u ) = H(F (u ),, F (u )). Reark that the copula C of Theore is unique if the functions F,, F are all continuous; if not, C is only uniquely deterined on Ran F Ran F. Note that we don t need copula functions for the definition of our coonotonicity coefficient in the next section, but we will use the for illustration purposes, and also in order to copare our (ultivariate) dependence easure with the coonly used (bivariate) easures. In particular, we will use two extree copulas and three Archiedean copula failies, which are coonly used in financial applications: Independent copula I % u i i = C (u) =

5 MEASURING COMONOTONICITY IN M-DIMENSIONAL VECTORS 95 Coonotonic copula C C (u) = inu i = i Clayton copula faily (dependence is concentrated in the lower tails) / -a - a Ca( u) = c ui n+, a > 0 i = Gubel copula faily (dependence is concentrated in the upper tails) / a Ca( u) = exp * < a ln( u ) kf 4, a $ i = i a Frank copula faily (syetry between the dependence in lower and upper tails) % -au = i i Ca( u) _ e i = a ln f + a, - - p ( e ) a! with u = (u,, u )! [0,]. A copula faily {C a : a! A} is called positively ordered, if for all u! [0,] and for all a a 2! A it is true that C a (u) C a2 (u). Finally, we recall here the ost coon bivariate dependence easures, written by eans of copulas. Pearson correlation coefficient rp ( X, X - - 2) = ( C( u, u2) uu2) df ( u) df2 ( u2) ; Var( X ) Var( X ) Spearan rank correlation coefficient rs ( X, X ) = 2 C( u, u ) du du 3 = 2 ( C( u, u ) u u ) du du ; Kendall s t ( X, X ) = 4 C ( u, u ) dc( u, u ) ; t Gini s correlation coefficient GX (, X) = 2 ` u + u u u j dc( u, u) ; Bloqvist s correlation coefficient b( X, X2 ) = 4 C(/2, /2)

6 96 I. KOCH AND A. DE SCHEPPER Coefficient of upper tail dependence l U ( X, X ) = 2u+ C ( u, u ) li. u " u 2 All of these easures were originally liited to two diensions, but possible extensions to a ore-diensional setting for soe of the can be found e.g. in Wolff (980), Joe (990, 997) and Schidt (2002). 3. THE COMONOTONICITY COEFFICIENT We will now introduce our easure that accounts for the degree of coonotonicity, the strongest possible positive dependence, in an arbitrary rando vector. As entioned before, copulas receive uch attention nowadays. Although there is no doubt about the iportance of this concept, we ay not forget that not all the inforation on the ultivariate distribution function is incorporated in the copula. Therefore we have chosen to build our easure on the joint distribution function rather than solely on the copula function. As such we use all the available inforation fro the argins and the dependence structure, and not the liited and partial inforation provided by the copula. This eans that our coefficient does not define a concordance easure, but the choice to include inforation about the argins can be very relevant e.g. in financial questions such as the Value-at-Risk, where a good coprehension of the coplete dependence structure, including the arginals, is of great iportance. Note that we will return to copulas and their erits when it turns to properties and illustrations. 3.. Definition As it will be clear fro the definition below, the coonotonicity coefficient k will be defined as the ratio of two hypervolues. The nuerator describes the hypervolue between the distribution of the vector under investigation and the distribution of the independent case, while the denoinator, inserted to noralise the coefficient, corresponds to the (axiu possible) hypervolue, i.e. between the distributions of the coonotonic and the independent case, which are the two extrees. As we are interested in the ratio of the two hypervolues and not in the particular values of both hypervolues, we suggest to work with principal value integrals in case the hypervolues are diverging when considered separately. Definition 6. (Coonotonicity coefficient) Let F X C and F X I be the distribution of the coonotonic and independent vector of the qualified Fréchet space R + (F,, F ). For any vector X of R + (F,, F ) with joint cdf F X, the coonotonicity coefficient k(x ) is defined as:

7 MEASURING COMONOTONICITY IN M-DIMENSIONAL VECTORS 97 k( X) = g g ^ ^ F X C F X I ( x) FX ( x) hdx, ( x) ( x) hdx FX I () where the integration is perfored over the whole doain of X. If the hypervolues diverge when taken separately, k(x ) is defined as k( X) ( x) ( x) h dx ( x), ( x) h dx I g F X FX Ga ( ) = li a " + 3 g F C I ^ X FX Ga ( ) ^ (2) provided the liit exist, and with G(a) depending on the doain of X as follows: if X is defined on, then G(a) = [ a, a] ; if X is defined on +n, then G(a) = [/a, a] ; if X is defined on [A, +3 [... [A, +3 [, then G(a) = [A, a]... [A, a]; if X is defined on a finite doain, then G(a) equals the whole doain. Note that, when both nuerator and denoinator converge separately, expression (2) reduces to (). In the case of continuous distribution functions, there exists a unique copula C X for which F X (x,, x ) = C X (F (x ),, F (x )). Inserting this into the forula for k in () or in (2) results in the following equivalent expression for the coonotonicity coefficient: k( X) = f 0 0 f C ( X f 0 0 % u, f, u ) u p df ( u ) f df ( u ) i = i i % = i i = i - - f in u u p df ( u ) f df ( u ) - - (3) or k( X) = li " + 3 a g Ha ( ) g c - - u, f, u) ui df ( u) fdf ( u) i =, f in u u p df ( u ) fdf ( u ) C ( X Ha ( ) % i = i i = % i - - (4) with H(a) [0,] depending on the doain of X. E.g. for G(a) = [ a, a], the integration now ust be perfored over the area H(a) = [F ( a), F (a)] [ F ( a), F (a)].

8 98 I. KOCH AND A. DE SCHEPPER 3.2. Basic Characteristics In general, a concordance or dependence easure should satisfy a nuber of desirable characteristics, as entioned e.g. in Schweizer & Wolff (98), Scarsini (984), or Drouet Mari & Kotz (200). In the next theore, we suarise the results for k with regard to these axios. A proof is straightforward. Theore 2. The coonotonicity coefficient as defined in Definition 6 satisfies the following properties:. k(x ) is defined for any vector X of a qualified Fréchet space R k(x ) takes values in the range [0,]. 3. k(x ) is syetrical in the coponents X i, i =,,, of the vector X. 4. k(x ) = 0 if and only if X = (X, X 2,, X ) has independent coponents. 5. k(x ) = if and only if X = (X, X 2,, X ) has coonotonic coponents. The second ite in this theore is responsible for the restriction in the definition of our coonotonicity coefficient to vectors that are PLOD, or positive lower orthant dependent. Indeed, if vectors belonging to R inus R + are allowed, both positive and negative results are possible for k. This will cause a twofold proble. Firstly, the range of values for k is not finite anyore at the lower side, since the noralising hypervolue of the denoinator no longer corresponds to extree cases, the independent vector only aking up a lower bound for eleents of R +. Secondly, and this sees to be uch ore iportant, the interpretation of k is no longer unabiguous. If we allow for vectors outside the qualified Fréchet space, a positive value no longer exclusively belongs to globally positive dependent vectors. If e.g. one of the coponents of a ultivariate vector is negatively dependent on the other coponents, the overall result could still be positive. Note that also any other dependence easures suffer fro this drawback, since they all easure soe kind of average dependence Properties of the Coonotonicity Coefficient Next to the basic characteristics, we now investigate a few other properties of the coonotonicity coefficient k, related to the perforance of the easure. For ore details, we refer to Koch & De Schepper (2006). Proposition. For any qualified Fréchet space R + and for any vector X = (X,, X ) of R +, for which the dependence structure can be described by a copula belonging to a positively ordered copula faily, the coonotonicity coefficient k(x ) increases with the paraeter of the faily. This follows iediately fro equation (3) or (4), and it guarantees that k is a consistent dependence easure. An illustration can be found in Table,

9 MEASURING COMONOTONICITY IN M-DIMENSIONAL VECTORS 99 TABLE EVOLUTION OF k FOR INCREASING VALUES OF a n = 2, Gubel copula, standard noral arginal distributions a k(x) n = 3, Clayton copula, exponential arginal distributions a k(x) n = 4, Frank copula, unifor [0,] arginal distributions a k(x) where for different diensions the values of k are copared with the values of the paraeter for three different copulas. Proposition 2. For any (bivariate) vector X = (X, X 2 ) of R 2 + with both arginals Unifor [0,] distributions, the coonotonicity coefficient k(x, X 2 ) equals the Spearan s correlation coefficient r s (X, X 2 ). For any (-variate) vector X = (X,, X ) of R + with all arginals Unifor [0,] distributions, the coonotonicity coefficient k(x) equals the ultivariate Spearan s correlation coefficient r J (X) as entioned in Joe (990). Proof: For = 2, with ( in( u, u2) uu 2) dudu2 = 2 / 0 0, this iediately follows fro a coparison of equation (3) with the definition of the Spearan s correlation coefficient in section For > 2, a cobination of g ( ini = ui % i ui) du... du = 0 0 = 2 ( + ) and the coefficient rj ( X ) = a g C( u,, u) du du 2 ( ) f f k leads to the result. Q.E.D. As a consequence, the coefficient k can be seen as an extension of the Spearan s correlation coefficient, in the situation where the true arginal distributions are not taken into account. Proposition 3. For any (bivariate) vector X = (X, X 2 ) of R 2 + for which it is true that ^Min{ u, v} uvhdf ( u) df ( v) = Var( X ) Var( X ), 2 the coonotonicity coefficient k(x, X 2 ) equals the Pearson correlation coefficient r p (X, X 2 ).

10 200 I. KOCH AND A. DE SCHEPPER Proof: This result can be deonstrated by coparing the expression for the coonotonicity coefficient in the bivariate case (see equation (3)) with the definition of the Pearson correlation coefficient (see section 2). Q.E.D. The condition entioned in Proposition 3 is satisfied e.g. if both arginal distributions are unifor, noral or exponential distributions, not necessarily with the sae distribution paraeters. The condition is not satisfied e.g. in the case of lognoral arginal distribution functions. Proposition 4. For any rando variable X and for any set of onotone increasing and invertible real functions g,, g, the coonotonicity coefficient k of the vector (X, g (X ),, g (X )) is equal to. Proof: Starting fro the alternative definition of coonotonicity, see section 2, this result is straightforward. Q.E.D. This property establishes an iportant difference between the Pearson correlation coefficient and the coonotonicity coefficient k, the latter being capable of capturing the dependence between a variable X and its transfored value g(x) for a broad class of functions g, and not only for linear transforations as it is the case for the Pearson correlation. A striking exaple is the situation where X and X 2 are copared. Indeed, while in general r p (X, X 2 )!, the desirable result does hold for the coonotonicity coefficient, or k(x, X 2 ) =. Proposition 5. For any vector X = (X,, X ) of R + for which all the arginal distributions are exponential distributions, X i + (l i ), or for which all the arginal distributions are noral distributions, X i + N [ i, s i2 ], and where the copula C as defined in Theore is functionally independent of the paraeters of the arginal distributions, the coonotonicity coefficient k(x) is also independent of the paraeters l i or i and s i. Proof: This result can be verified by eans of a substitution of the arginal distributions into equation (3). Q.E.D. Cobining this proposition with proposition 3, we see that for certain classes of arginal distribution functions, the coonotonicity coefficient k can also be seen as a kind of ultivariate extension of the Pearson correlation. Proposition 6. For any set of rando vectors X and X,, X k belonging to the sae qualified Fréchet space R + (F,, F ), for which the distribution function of X can be written as a convex su of the distribution functions of X,, X k, k k F X = / i = a i F Xi, a i! [0,] and / i = ai =, the coonotonicity coefficient of X equals the convex su of the corresponding coonotonicity coefficients, k(x) = k / i ai ( ). = k X i Proof: The decoposition as entioned in property 6 can be rewritten as F X F I X = / i k = ai [ FX I X ]. A substitution in equation (2) proves the result. Q.E.D. i F i

11 MEASURING COMONOTONICITY IN M-DIMENSIONAL VECTORS 20 Corollary. For any vector X = (X,, X ) of R + for which the joint distribution function F X can be written as a convex su of F X C and F X I, or F X = af X C + ( a) F X I, a! [0,], the coonotonicity coefficient k(x) is equal to a. This last corollary states a crucial result, since it deonstrates that our coonotonicity coefficient is copletely in harony with how one would interpret the value of a in the decoposition of an arbitrary distribution into a convex cobination of the independent and coonotonic extrees. Note that if in a bivariate case the joint distribution function can be decoposed in this way, the coonotonicity coefficient is also equal to the coefficient of upper tail dependence. 4.. Gaussian Discounting Processes 4. NUMERICAL ILLUSTRATIONS The first illustration refers to the investigation of present values with stochastic interest rates, which in fact was the iediate cause of the developent of our coonotonicity coefficient. Consider a cash flow of future payents with present value Yi ( ) = / z( ti ) e -, i = Vt with tie points 0 < t < t 2 < < t < t = t, with z(t i ) the deterinistic payent at tie t i, and with e Y i the stochastic discounting factor over the tie period [0, t i ]. If the discounting process is odelled by eans of Gaussian processes, we start with a rando vector X = (X, X 2,, X n ) with independent and norally distributed coponents X i, each representing the rate of return for a (sall) tie period [t i, t i ]. The copounded rate of return for the whole period [0, t i ] can then be written by Y i = X + + X i. Although all coponents in these sus are independent, the rando vector Y = (Y, Y 2,, Y n ) clearly consists of strongly dependent variables. The calculation of the exact distribution of the (stochastic) present value above is very hard, and in ost cases even ipossible. Goovaerts et al. (2000), FIGURE : Evolution of k(y ) with increasing diension n.

12 202 I. KOCH AND A. DE SCHEPPER Kaas et al. (2000), Dhaene et al. (2002a) suggested to work with an upper bound in convex order sense, replacing the vector Y by its coonotonic counterpart, which corresponds to the strongest possible dependence structure. The calculations needed for this upper bound are uch easier than they are for the exact distribution, and yet the upper bound turns out to be a very accurate approxiation in case the payents z(t i ) are all non-negative (see Kaas et al. (2000) and Goovaerts et al. (2000)). We will show now that this good result can be explained by quantifying the dependence in the vector Y by eans of the coonotonicity coefficient k. In the case where all the coponents X i are (independent and) standard norally distributed, the coponents Y i correspond to noral distributions with zero ean and with variance equal to i, and analytical results for the coonotonicity coefficient are possible. Note that in this case Proposition 5 is not applicable since the joint distribution function is not independent of the paraeters of the arginal distributions. When n is equal to 2, the coonotonicity coefficient of the vector Y = (Y,Y 2 ) can be calculated as: k( Y, Y 2 ) = For n = 3, the calculations are uch ore lengthy, finally resulting in k ( Y, Y, Y3) 4 2 = $ d + + n 2 3 Figure indicates how k(y ) increases with the diension of the vector Y. In the case where the coponents X i are (independent and) identically but ore generally norally distributed, X i + N [, s 2 ], the analytical calculations becoe extreely coplex. However, nuerical calculations suggest that the values of the noral paraeters do not have any effect on the resulting coonotonicity coefficient of the vector Y. In any case, the high value of k for the vector Y confirs that due to the construction of the cuulative discounting process, the coponents of the vector Y indeed are very strongly positive dependent. This explains why in such discounting issues an approxiation of the original vector by its coonotonic counterpart indeed akes up a very good approach Contourplots For the interpretation or visualisation of k, contourplots constitute a nice and efficient tool. A first advantage is that they can be used to get an idea about the eaning of the absolute nuerical value of k, just by coparing the contourplots for the vector under investigation with the contourplots for the independent and coonotonic counterparts.

13 MEASURING COMONOTONICITY IN M-DIMENSIONAL VECTORS 203 FIGURE 2: Contourplots for n = 2 unifor [0,] arginal distribution. In Figure 2, we present contourplots for a bivariate exaple. Consider a vector X = (X, X 2 ), with unifor arginal distributions. Part (a) and (b) of the figure refer to the extree situations (k = 0 and k = ), while in part (c) and (d) we show the contourplot in case the dependence structure is odelled by eans of a Clayton copula with different paraeters. Coparing the different pictures, we see that the situation with a rather low value of k is clearly ore siilar to the independent situation, whereas the situation with a rather high value of k uch ore resebles the coonotonic situation. In order to show the influence of the arginals, Figure 3 repeats the sae type of contourplots, but now for a vector with lognoral [0, (0.2) 2 ] arginals. As in Figure 2, for a large value of k, the picture tends to the coonotonic picture, while for a sall value of k, the siilarity with the independent picture is apparent. FIGURE 3: Contourplots for n = 2 Lognoral [0, (0.2) 2 ] distributions. This technique is also workable in ore than 2 diensions, by looking at all possible two-diensional projections. This is illustrated in Figure 4, for a vector X = (X, X 2, X 3 ), with standard noral arginal distributions. In this second exaple, the extrees are copared to a Clayton copula dependence structure. Note that with a Clayton copula, the dependence is concentrated in the lower tails. Graph (c) indeed reveals that in the lower tails the situation is tending towards a ore coonotonic situation, while for the upper tails the contours are ore siilar to the independent case.

14 204 I. KOCH AND A. DE SCHEPPER FIGURE 4: Contourplots for n = 3 standard noral arginal distributions. Secondly, contourplots can also be used to analyse different dependence structures ending up with the sae value of the coonotonicity coefficient k. An elaborated illustration is provided in Figure 5, where contourplots are depicted for 5 different joint distributions, with five different arginal distributions (unifor on [0,], standard noral, exponential [], lognoral [0,(0.2) 2 ], gaa [5,] ), and three copula failies (Clayton, Frank and Gubel), all corresponding to a situation with k equal to It can be observed that there is a large degree of siilarity in these plots, although they are created by eans of rather different odels.

15 MEASURING COMONOTONICITY IN M-DIMENSIONAL VECTORS 205 FIGURE 5: Contourplots for different bivariate dependence structures with k(x ) = 0.80.

16 206 I. KOCH AND A. DE SCHEPPER 4.3. Coparison of the Coonotonicity Coefficient with Classic Bivariate Measures Assue now that we are working in a bivariate environent, investigating the dependence structure in a vector X = (X, X 2 ). We already entioned soe connections between our coonotonicity coefficient and classic association easures in the propositions in section 3. If the arginal distributions of X and X 2 are both unifor [0,], k is equal to the Spearan rank coefficient r s (see Proposition 2); if the arginal distributions satisfy a special condition, then k is equal to the Pearson correlation r p (see Proposition 3). In Table 2 we copare k with the classic dependence easures in a particular bivariate exaple not satisfying the previous specifications. The first variable is assued to be lognorally distributed, X + LogN [0.04, (0.2) 2 ], for the second one we have chosen a noral distribution, X 2 + N [0.04, (0.2) 2 ]; the dependence is odelled by eans of a Frank copula with paraeter a. As expected, all the dependence easures increase with the paraeter value of the copula a, but for each of the this occurs in a distinct way. This is of course a consequence of the definitions of each of these dependence easures, all easuring another aspect of the actual dependence. Let us exaine in particular the relation between the coonotonicity coefficient k and Kendall s t on the one hand and between k and Spearan s rank coefficient on the other hand. The popularity of Archiedean copulas (e.g. Clayton, Frank, Gubel) causes Kendall s t to be the doinant dependence easure in a copula setting. This is due to the nice atheatical connection that exists between the generator of the Archiedean copula and t (see Genest & MacKay (986)). Although t is undeniable atheatical convenient, t is not necessary suitable for easuring coonotonicity. Furtherore, Kendall s tau is a concordance easure and k is not, since we take the arginal distributions into account. Clearly both easures are not equal. In the exaple used in Table 2, we see that k is saller than t for sall values of a, corresponding to situations with TABLE 2 COMPARISON OF k(x ) WITH CLASSIC DEPENDENCE MEASURES a k(x, X 2 ) Pearson Spearan Kendall Bloqvist

17 MEASURING COMONOTONICITY IN M-DIMENSIONAL VECTORS 207 a weak dependence between X and X 2, and that k doinates t for higher values of a and thus for a stronger dependent vector. FIGURE 6: Relation between coonotonicity coefficient k and Kendall s t. FIGURE 7: Relation between coonotonicity coefficient k and Spearan s rank. In Figure 6, the relation between k (on the horizontal axis) and t (on the vertical axis) is displayed for a few different situations. Five types of arginal distributions are cobined with the Frank copula, which is syetric in the upper and lower tail, and with the Clayton copula, for which the dependence is concentrated in the lower tails. As it can be observed fro both pictures, there is no general relationship between both easures. As entioned earlier, k can be seen as an extension of Spearan s rank taking into account the effect of the arginals. In Figure 7, the relation between k (on the horizontal axis) and Spearan s rank (on the vertical axis) is depicted, for the sae set of odels as in figure 6. Note that for unifor arginals, both easures are identical (cf. Proposition 2).

18 208 I. KOCH AND A. DE SCHEPPER 5.. Calculation 5. THE COMONOTONICITY COEFFICIENT SAMPLE VERSION Suppose we want to estiate the coonotonicity coefficient for variables, for which we have N coupled observations, not necessarily independent, suarized in a data atrix Y (diension N ). The epirical distribution can be written as N X ( x,, x) f = / I( Yij xj) N i = j = F % for the epirical versions of the distribution for the independent and coonotonic vectors of the sae Fréchet space, we have C X I X F ( x, f, x) = in$ FX ( x), f, FX ( x). F ( x, f, x) = FX ( x) $ f$ FX ( x). ; Now, define j := in N i = Y ij and M j := ax N i = Y ij, and denote Y (k) j for the k-th ordered observation for the j-th variable. As a consequence, Y () j = j and Y (N) Mj j = M j for any j =,...,, and I( Y ij xj ) d x j = M j Y ij for any i =, j..., N, j =,...,. If we replace the distribution functions in Definition 6 by their epirical versions, we get a saple-based version for the coonotonicity coefficient. Definition 7. (Saple-based Coonotonicity coefficient) For a given data atrix Y with N observations for variables, and with the notations above, the saplebased coonotonicity coefficient k(x ) is defined as: N / % Mj Yij ` j j Y N % e NM / ij o i = j = N j = i = k( X ) =. N N (i) / % Mj Yj a k % NM Yij N e j / o i = j = N j = i = N (5) Note that, as the epirical distribution function is a consistent estiator of the real cuulative distribution function, the saple-based coonotonicity coefficient k is also a consistent estiator of the coonotonicity coefficient k. In practical applications, it is necessary to test whether the data are positive lower orthant dependent, before the calculation of the estiate for the coonotonicity coefficient is perfored. We suggest to rely on the results of Denuit & Scaillet (2004) or Scaillet (2005) for this purpose. A bootstrap procedure can be used in order to estiate the error on the saple-based coonotonicity coefficient.

19 MEASURING COMONOTONICITY IN M-DIMENSIONAL VECTORS Epirical illustration In order to illustrate the possibilities of the saple-based coonotonicity coefficient in ore diensions, we consider seven world arket indices: S&P 500 (U.S.); Dow Jones Industrial Average (U.S.); Nikkei index (Japan); FTSE 00 index (U.K.); CAC index (France); DAX index (Gerany); SMI index (Switzerland). We included several European indices so as to create a eaningful subset. We look at the onthly values, particularly at the closing prices at the last trading day of each onth, for the period January 2000 to Noveber 2009 (9 data points). In Figure 8, the evolution of tie is shown for the 7 rescaled indices; Table 3 suarizes soe descriptive statistics for these indices. FIGURE 8: Rescaled values of the 7 investigated indices. TABLE 3 DESCRIPTIVE STATISTICS OF MONTHLY INDEX VALUES S&P DJIA Nikkei FTSE CAC DAX SMI Jan Nov iniu (Feb 2009) (Feb 2009) (Feb 2009) (Jan 2003) (Mar 2003) (Mar 2003) (Mar 2003) axiu (Oct 2007) (Oct 2007) (Mar 2000) (Oct 2007) (Aug 2000) (Dec 2007) May 2007)

20 20 I. KOCH AND A. DE SCHEPPER Figure 8 nicely illustrates the (well-known) finding that there is a high level of dependence between the different arket indices. This is confired by the correlation atrix, which is given in 4 below. Testing for positive lower orthant dependence (see Scaillet (2005)) results in a p-value of., reassuring that the calculation of the coonotonicity coefficient is allowed. TABLE 4 CORRELATION MATRIX OF SELECTED INDICES S&P DJIA Nikkei FTSE CAC DAX SMI S&P DJIA Nikkei FTSE CAC DAX SMI This correlation atrix refers to pair-wise correlations, and it does not give a global easure for the group of indices; it also easures solely the linear correlations. The advantage of calculating the saple-based coonotonicity coefficient for the whole group or for a subset, is the fact that this approach akes it possible to get a uch copleter idea about the real dependence between the seven indices. The calculation of k results in the following values: for the whole group: k = ; for the US selection (S&P, DJIA): k = ; for the European selection (FTSE, CAC, DAX, SMI): k = Note that the value for k(s&p, DJIA) is slightly higher than the correlation between the two indices. This is due to the fact that the coonotonicity coefficient not only easures the linear interdependence, but that it easures the degree of (overall) coonotonicity, see also Proposition 4. It is also interesting to copare the estiated value for the coonotonicity coefficient for the whole period , with the results for subperiods, e.g. the periods before and after the 2007 financial crisis: for the whole period: k = ; for the period until June 2007: k = ; for the period starting fro July 2007: k = It can be observed that the estiated values for the coonotonicity coefficients are rather high in all cases, illustrating again the high degree of (positive) dependence between the different arket indices.

21 MEASURING COMONOTONICITY IN M-DIMENSIONAL VECTORS 2 An estiate for the error of the saple-based coonotonicity coefficients can be found through a bootstrap procedure. As we are faced with tie series data with serial correlation and possibly also with heteroskedasticity, we use a block bootstrap in order to iprove the perforance. We illustrate the reliability of the k estiate for the 7 arket indices over the period by eans of a block bootstrap with 5000 subsaples, where each of the subsaples is coposed of 24 overlapping blocks with block length 5 for the first siulation, and of 40 overlapping blocks with block length 3 for the second siulation. Figure 9 shows the histogra of the bootstrapped k estiates, together with the noral approxiation. The left panel corresponds to a block length of 5, the right panel to a block length of 3. Results for the ean and standard deviation and for 90% confidence intervals can be found in Table 5. FIGURE 9: Histogra of bootstrap estiates for k for the index data. TABLE 5 DESCRIPTIVE STATISTICS FOR THE BOOTSTRAP SUBSAMPLES TO BE COMPARED TO A VALUE FOR THE SAMPLE-BASED COMONOTONICITY COEFFICIENT k = block length of 5 block length of 3 ean value k = k = standard deviation s k = s k = non paraetric C.I. ( ; ) ( ; ) norality based C.I. ( ; ) ( ; ) Note that a saller block length results in a lower standard deviation (but higher bias) and that a larger block length results in a lower bias (but higher standard deviation). The choice of the block length is in line with the nuerical exaples in Hall et al. (995); as our saple consists of 9 data points, a block length of 5 iplies a nuber of blocks equal to 24, and an block length of 3 iplies a nuber of blocks equal to 40.