Tail comonotonicity: properties, constructions, and asymptotic additivity of risk measures

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1 Tail comonotonicity: properties, constructions, and asymptotic additivity of risk measures Lei Hua Harry Joe June 5, 2012 Abstract. We investigate properties of a version of tail comonotonicity that can be applied to absolutely continuous distributions, and give several methods for constructions of multivariate distributions with tail comonotonicity or strongest tail dependence. Archimedean copulas as mixtures of powers, and scale mixtures of a non-negative random vector with the mixing distribution having slowly varying tails, lead to a tail comonotonic dependence structure. For random variables that are in the maximum domain of attraction of either Fréchet or Gumbel, we prove the asymptotic additivity property of Value at Risk and Conditional Tail Expectation. Key words. Copula, Archimedean copula, asymptotic full dependence, regularly varying, slowly varying, extreme value distributions, elliptical distributions. 1 Introduction Suppose we have bivariate loss data and hope to estimate some high-risk scenarios, say by Value at Risk (VaR) or Conditional Tail Expectation (CTE). A widely used method is to fit some parametric models based on the bivariate Student t or other parametric copula families. Then risk measures or tail dependence can be derived from these fitted models. Due to model uncertainty, all of these methods are not conservative from the viewpoint of an actuary. The middle part might influence our estimation more than the important tail part does. These traditional methods are too sensitive to the middle part which contains most of the data [33]. {leihua, harry}@stat.ubc.ca, Department of Statistics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada. Corresponding author 1

2 Given that we do not have enough information for the joint tail of losses, we conservatively assume that it is upper tail comonotonic (as defined in Section 3), and the conservativity can be justified by Hua and Joe (2012) [24]. In this way, we actually give up estimating the first order tail parameter (i.e., the traditional tail dependence parameter λ) by assuming a conservative one (i.e., λ = 1), and let the likelihood of data contribute to the estimation of the second order tail parameter. Since the first order parameter is only for an asymptotic property, the conservative assumption on it would not put too much constraint on the model. This approach will give us a more robust method of measuring risks. In the literature on actuarial science and quantitative finance, many efforts have been done to seek finer upper bounds for dependence structures. The concepts of comonotonicity, conditional comonotonicity, and most recently, the upper comonotonicity have been studied to provide theoretically tractable bounds. However, these conditions are either too strong or not tailored for the tail, and could lead to an over-conservative risk measure. Moreover, those dependence structures lack flexible distribution families that can be used to model real data. We refer to [12, 11, 8, 9] for the reference of these concepts. Tail comonotonicity, on the other hand, needs a weaker condition and requires that the degree of positive dependence approaches its maximum only when all the marginal losses go to infinity. The degree of dependence at the sub-extremal level can still be estimated from the data. This approach should better balance the requirements of safety and accuracy for risk management. Tail comonotonicity is also referred to as asymptotic full dependence or a dependence structure where the tail dependence parameter satisfies λ = 1. Although such a dependence structure is not new, we find that it has interesting properties, such as asymptotic additivity of VaR and CTE, that are analogous to the usual comonotonicity. Moreover, some parametric families illustrate its suitability for modeling loss data that may appear to have tail dependence. Among many copula families, Archimedean copulas and copulas constructed from a scale mixture of a non-negative random vector can be used to provide a tail comonotonic dependence structure. This paper is organized as follows. In section 2, some preliminary concepts and notation used throughout the paper will be presented. In Section 3, the concept of tail comonotonicity, its properties and some parametric examples will be studied. Methods of constructing copulas with tail comonotonicity or the strongest tail dependence are given in Section 4. Asymptotic additivity of VaR and CTE under the assumption of tail comonotonicity is shown in Section 5. Finally, in Section 6, we conclude the paper and propose some directions for future research. 2

3 2 Preliminaries In this section, we summarize definitions of tail dependence, some classes of copulas, regularly variation, VaR, and CTE. These are used in the remaining sections for our main new definitions and results. 2.1 Basic concepts Let X = (X 1,..., X d ) T be a random vector with distribution function F and univariate marginal distribution functions F i, i = 1,..., d. Due to Sklar s theorem [26, 32], there exists a multivariate uniform distribution function C : [0, 1] d [0, 1] such that F (x 1,..., x d ) = C(F 1 (x 1 ),..., F d (x d )). (1) When the univariate margins are all continuous, then the copula function C is uniquely determined by C(u 1,..., u d ) = F (F 1 1 (u 1 ),..., F 1 d (u d)), (2) where F 1 i is the inverse function of F i, i = 1,..., d. The corresponding survival function C is defined as C(u 1,..., u d ) = 1 + I {1,...,d},I ( 1) I C I (u i, i I), where C I is the I-margin of the copula C with I the cardinality of the set I. Then the survival copula Ĉ can be written as Ĉ(u 1,..., u d ) = C(1 u 1,..., 1 u d ). (3) As summary information of tail dependence, the lower tail dependence parameter of C is defined as λ L = lim u 0 + u 1 C(u,..., u), and the upper tail dependence parameter is defined as provided the limits exist. λ U = lim u 0 + u 1 Ĉ(u,..., u), Among methods of constructing copulas, inverse of multivariate distribution functions as (2), and Archimedean copulas are widely used. The former includes Elliptical copulas, such 3

4 as Student t and Gaussian copulas, and Archimedean copulas can be constructed as follows, C ψ (u 1,..., u d ) = ψ(ψ 1 (u 1 ) + + ψ 1 (u d )), (u 1,..., u d ) [0, 1] d, (4) where the function ψ 1 is referred to as the generator of C ψ. In McNeil and Nešlehová (2009), sufficient and necessary conditions of ψ to generate a d-dimensional copula have been given. The Laplace Transforms (LTs) of positive random variables with ψ( ) = 0 is a subclass of ψ functions in (4); such choices of ψ cover most of the useful Archimedean copulas [26]. For a positive random variable X, its LT is defined as ψ(t) := 0 e xt F X (dx), where F X (x) is the distribution function of X. Another general copula family consists of extreme value copulas. If a copula C satisfies C(u t 1,..., u t d) = C t (u 1,..., u d ) for any (u 1,..., u d ) [0, 1] d and t > 0, then we refer to C as an extreme value copula. Standard references for theory of copulas are Joe (1997) [26] and Nelsen (2006) [32]. For applications of copulas in actuarial science and quantitative risk management, we refer to Denuit et al. (2005) [10] and McNeil et al. (2005) [30]. For asymptotic analysis of tail behavior of random variables or equivalently their distribution functions, the theory of regular variation provides a powerful platform. Here we only give some most fundamental concepts. More relevant concepts and results about regular variation will be used throughout the paper, and will be introduced when they are needed. Standard references for regular variation theories are Bingham et al. (1987) [6], Resnick (1987) [36] and Geluk and de Haan (1987) [21]; and Embrechts et al. (1997) [14] and Resnick (2007) [37] are more relevant for applications in actuarial science, quantitative finance and risk management. Definition 1 A measurable function g : R + R + is regularly varying at with index α 0 (written g RV α ) if for any t > 0, g(xt) lim x g(x) = tα. (5) If equation (5) holds with α = 0 for any t > 0, then g is said to be slowly varying at and written as g RV 0. Regular variation and slow variation at 0 are defined similarly and written as RV α (0) and RV 0 (0), respectively. We use l(x) to represent a slowly varying function, and then a regularly varying function g with regular variation index α can be written as g(x) = x α l(x). When 4

5 we say that a random variable is regularly varying or X RV α, we actually mean that the corresponding survival function F X (x) := P[X > x] RV α. Some statistical quantities about a random variable X are often referred to as risk measures. Among many of them, VaR and CTE are probably the most popular risk measures. Both of them have been adopted by regulations of insurers. For example, CTE has been required for calculating the relevant risks of segregated fund in Canada (see Chapter 8 of [34]). Definition 2 Let X be a random variable representing the amount of risk, and p (0, 1) a probability level, then the corresponding VaR, denoted by VaR p (X), is defined as VaR p (X) = inf{x R F X (x) p}; and the corresponding CTE is defined as CTE p (X) = E[X X > VaR p (X)]. We refer to Denuit et al. (2005) [10] for some other relevant risk measures and their relationships. 2.2 Notation We use bold letters for vectors, such as x := (x 1,..., x d ) T. For a random variable X, the generalized inverse of its cumulative distribution function (cdf) is defined as F 1 X (p) = inf{x R : F X(x) p}. Also, note that for all x R and p [0, 1], we have (e.g., [12]) F 1 X (p) x p F X(x). (6) Some notation for asymptotic relationships: Given a limiting point t [, ], h(t) = o(g(t)) means lim t t h(t)/g(t) = 0; h(t) g(t) means lim t t (h(t)/g(t)) = 1; h(t) g(t) means lim t t (h(t)/g(t)) 1. Other notation: Cartesian product is denoted as [a, b] := [a 1, b 1 ] [a d, b d ]; let x := (x 1,..., x d ) and y := (y 1,..., y d ), then x y means that x i y i for all i = 1,..., d; A B means A is a subset but not necessarily a proper subset of B; X d = Y means that random variables X and Y are equal in distribution; R + := [0, ); R + := [0, ]; R := [, ]; 1 d = (1,..., 1) T with d components; 1 A (x) is the indicator function such that 1 A (x) = 1, if x A, and otherwise 0. 5

6 3 Definitions of tail comonotonicity and properties Let X = (X 1,..., X d ) T be a non-negative random vector, representing amounts of d losses, and the univariate marginal cdf s are all continuous and denoted as F 1,..., F d. The Fréchet space containing all the random vectors with these univariate margins is denoted as R(F 1,..., F d ). For risk management, the joint tail probability P[X 1 > x 1,..., X d > x d ] is relevant, especially when x 1,..., x d are large values. Cheung (2009) studied an upper comonotonicity structure: roughly speaking, beyond a finite threshold a, the dependence structure becomes comonotonicity, while keeping the dependence structure below the threshold flexible; this means that the distribution is not absolutely continuous. Under this dependence structure, if x a, then clearly P[X 1 > x 1,..., X d > x d ] = min{f 1 (x 1 ),..., F d (x d )} = 1 max{f 1 (x 1 ),..., F d (x d )}, which is the upper bound for such a tail probability of any X R(F 1,..., F d ). The upper bound coincides with the tail probability for the usual comonotonicity. Letting Ĉ be the survival copula of X, P[X 1 > x 1,..., X d > x d ] = Ĉ(F 1(x 1 ),..., F d (x d )). A proper threshold a might not exist in real applications. However, note that our aim of proposing a conservative dependence structure will be met if the tail probability can be well approximated by the upper bound when x is sufficiently large; that is, Ĉ(F 1 (x 1 ),..., F d (x d )) min{f 1 (x 1 ),..., F d (x d )}, x i s are sufficiently large. When x is sufficiently large, F i (x i ) s become sufficiently close to 0. But how x i s converge to the right end points of their supports will affect the joint tail probability. Considering that the usual comonotonicity is a copula property and does not depend on the margins, here we want to define a dependence concept which will not rely on margins as well. Therefore, letting u i := F i (x i ), it suffices to have that Ĉ(u 1,..., u d ) min{u 1,..., u d }, u i s are sufficiently small. To be convenient in real applications, this condition should also be satisfied with copulas Ĉ that are absolutely continuous. Of course, how u i s approach 0 will affect the above approximation. However, as long as the rates of convergence of u i s to 0 are comparable (in the sense that u i = uw i for any given 0 < w i < + ), the above approximation should be good. Then Ĉ 6

7 would be what we want if it satisfies Ĉ(uw 1,..., uw d ) min{uw 1,..., uw d }, u 0 +, w i,..., w d [0, + ), i.e., Ĉ(uw 1,..., uw d ) lim u 0 + u where the trivial case of w i = 0 for some i is also included. = min{w 1,..., w d }, w i,..., w d [0, + ), (7) Definition 3 A random vector X is said to be upper tail comonotonic if X has a copula C and its survival copula Ĉ satisfies (7); the copula C is said to be an upper tail comonotonic copula. X is said to be lower tail comonotonic if X has a copula C that satisfies lim u 0 + C(uw 1,..., uw d )/u = min{w 1,..., w d }, w i [0, + ); the copula C is said to be a lower tail comonotonic copula. Remark 1 Based on the above definition, tail comonotonicity is a concept about copula and does not rely on the marginal distributions as long as the conditions hold. In Definition 3, the random vector X is not necessarily continuous and the right end points of each univariate margin can be finite or infinite. Remark 2 In the framework of multivariate regular variation (MRV), the concept of upper tail comonotonicity is represented by the limit Radon measure (Resnick 2007 [37]) ν([0, x] c ) = (min{x 1,..., x d }) α. So asymptotic full dependence for MRV is a case of upper tail comonotonicity with the univariate margins being power laws. Recall from Joe et al. (2010) [29] that, when C is a d-variate copula with the survival function C, then lower tail dependence means that there exists a non-zero homogeneous function b of order 1 such that lim C(uw 1,..., uw d )/u = b(w 1,..., w d ), w i [0, + ), (8) u 0 + and upper tail dependence means that there exists a non-zero homogeneous function b of order 1 such that lim u 0 + C(1 uw 1,..., 1 uw d )/u = b (w 1,..., w d ), w i [0, + ). (9) 7

8 So, tail comonotonicity simply means that the lower and/or upper tail dependence functions are min{w 1,..., w d }. Also, note that for any tail dependence function b(w), we must have b(w) min(w 1,..., w d ). Some relevant properties for upper/lower tail comonotonicity are mentioned in [5], we give some alternative results in the following two propositions. Proposition 1 Suppose C is a copula. Then λ U (C) = 1 if and only if the upper tail dependence function exists and b (w 1,..., w d ) = min(w 1,..., w d ). In parallel, λ L (C) = 1 if and only if the lower tail dependence function exists and b(w 1,..., w d ) = min(w 1,..., w d ). Proof: If we take (w 1,..., w d ) = (1,..., 1), then it is obvious that λ = 1. direction, letting w = min(w 1,..., w d ) 0, then For the other C(uw,..., uw ) u w = C(uw 1,..., uw d ) u C(uw,..., uw ) lim uw 0 + uw min(uw 1,..., uw d ) u = w w lim inf u 0 + C(uw 1,..., uw d ) u C(uw 1,..., uw d ) lim sup u 0 + u w. Thus, b(w 1,..., w d ) = min(w 1,..., w d ). Similar for the upper tail. Remark 3 From Proposition 1, we can conclude that the concept of upper/lower tail comonotonicity is nothing but the tail dependence parameter λ = 1. The designation of this concept is just consistent with the names of upper/lower tail dependence. If the tail dependence function exists, then it must have the unique form. It is well known that all pairs of a random vector are comonotonic is equivalent to that the random vector is comonotonic. A parallel result also holds for upper/lower tail comonotonicity. Proposition 2 Suppose C is a d-variate copula, then any bivariate marginal copula C {ij}, i j, is upper [resp. lower] tail comonotonic if and only if C is upper [resp. lower] tail comonotonic. Proof: By Proposition 1, it suffices to prove that λ = 1. We only state the proof for lower tail comonotonicity here. Suppose U i Uniform(0, 1) with joint distribution C. Since P[U 1 u] d i=2 P[U 1 u < U i ] u C(u1 d) u 1, and pairwise lower tail comonotonicity implies that lim d u 0 + i=2 P[U 1 u < U i ]/u = 0, this implies that C is lower tail comonotonic. The other direction is due to the fact that if λ L (C) = 1, then λ L (C I ) = 1 for every marginal copula C I, 1 < I < d. 8

9 Next, we give two examples of parametric tail comonotonic copulas that are two-parameter Archimedean copulas. Example 1 (BB2 in Joe and Hu 1996 [28], Joe 1997 [26]) With LT ψ(s) = [1 + δ 1 log(1 + s)] 1/θ, the bivariate Archimedean copula is C ψ (u, v) = [ 1 + δ 1 log ( e δ(u θ 1) + e δ(v θ 1) 1)] 1/θ, θ > 0, δ > 0. Then as u 0 +, C ψ (uw 1, uw 2 ) u [ 1 + δ 1 max{δ((uw 1 ) θ 1), δ((uw 2 ) θ 1)} ] 1/θ u 1 = min(w 1, w 2 ). Thus C ψ is lower tail comonotonic. Scatterplots of the BB2 copula with parameters δ = 0.2 and θ = 0.4 or 0.2 are in Figure 1 (N = 2000); there is no upper tail dependence for this copula. Figure 1: Simulation of BB2 with/without the univariate margins being transformed to the standard Normal; in the left and middle plots δ = 0.2, θ = 0.4 and in the right plot δ = 0.2, θ = 0.2 BB2 d= 0.2 th= 0.4 (Uniform margin) BB2 d= 0.2 th= 0.4 (Normal margin) BB2 d= 0.2 th= 0.2 (Normal margin) Example 2 (BB3 in Joe and Hu 1996 [28], Joe 1997 [26]) With LT ψ(s) = exp { [δ 1 log(1 + s)] 1/θ}, the bivariate Archimedean copula is C ψ (u, v) = exp { [δ 1 log(e δũθ + e δṽθ 1)] 1/θ }, θ > 1, δ > 0, where ũ = log u, ṽ = log v. Then as u 0 +, C ψ (uw 1, uw 2 ) u exp { [δ 1 log(e δ( log[u min(w 1,w 2 )]) θ )] 1/θ } u 1 = min(w 1, w 2 ). 9

10 Thus C ψ is also lower tail comonotonic. Scatterplots of the BB3 copula with parameters δ = 0.2 and θ = 1.7 or 1.3 are in Figure 2 (N = 2000); there is also upper tail dependence for this copula and λ U = 2 2 1/θ. Figure 2: Simulation of BB3 with/without the univariate margins being transformed to the standard Normal; in the left and middle plots δ = 0.2, θ = 1.7 and in the right plot δ = 0.2, θ = 1.3 BB3 d= 0.2 th= 1.7 (Uniform margin) BB3 d= 0.2 th= 1.7 (Normal margin) BB3 d= 0.2 th= 1.3 (Normal margin) Remark 4 Looking at the plots for the lower comonotonic copulas, they appear suitable as survival copulas to be used to get conservative dependence structure for joint large losses. Although they are both lower tail comonotonic, there is not much constraint on the subextremal level. 4 Construction of tail comonotonic copulas In this section, we propose methods to construct tail comonotonic copulas based on mixing distributions with very heavy tails; very heavy means there are no moments of any positive/negative orders or the corresponding survival function is slowly varying. Precise conditions will be presented in the following subsections. 4.1 Archimedean copulas In this subsection, we will relate the tail behavior of LTs to upper/lower tail comonotonicity of corresponding Archimedean copulas. The next result says that if the LT ψ is slowly varying at the tails, then C ψ must have the tail dependence function min(w 1,..., w d ). Proposition 3 Let the Archimedean copula C ψ be based on the LT ψ satisfying ψ( ) = 0. If ψ(s) RV 0 then the lower tail dependence function exists and C ψ is lower tail comonotonic; 10

11 if 1 ψ(s) RV 0 (0 + ) then the upper tail dependence function exists and C ψ comonotonic. is upper tail Proof: By Proposition 1, we only need to prove the tail dependence parameter λ = 1. For the lower tail, letting s := ψ 1 (u), then because ψ(s) RV 0, λ L = lim u 0 + C ψ (u1 d ) u = lim u 0 + ψ(dψ 1 (u)) u ψ(ds) = lim s + ψ(s) = 1. For the upper tail, by Proposition 2, it suffices to prove the bivariate case. Let s := ψ 1 (1 u), then λ U = lim u 0 + Ĉ ψ (u, u) u [ ] [ = lim 2 + ψ(2ψ 1 (1 u)) 1 = lim 2 1 ψ(2s) ] = 1, u 0 + u s ψ(s) which finishes the proof for the upper tail. Remark 5 Write the Archimedean copula as the mixture C ψ (u 1,..., u d ) = 0 d G η (u i )df H (η), where G(u) = exp{ ψ 1 (u)} is a cdf on [0, 1] and H is a resilience random variable with LT ψ. The condition 1 ψ RV 0 (0 + ) means that the density f H (η) has a very heavy tail as η so that E[H m ] = for all m > 0. The condition ψ RV 0 means that the density f H (η) has a very heavy tail as η 0 so that E[H m ] = for all m > 0. The implication of the condition of 1 ψ at 0 follows by the proof of Lemma 1 in Hua and Joe (2011a) [23]. Similarly, we can show the implication of the condition of ψ at as follows. Suppose to the contrary that E[H m ] < for some m > 0. Let W m = (Z/H) m, where Z Exponential(1), independent of H, and H has the LT ψ. Then P[W m w] = P[Z Hw 1/m ] = 0 exp( yw 1/m )F H (dy) = ψ(w 1/m ). Then E[H m ] < and Z having all positive moments implies that E[W m ] < and thus wp[w m w] 0, as w ; that is, wψ(w 1/m ) 0, or equivalently s m ψ(s) 0, as s. It is well known that if ψ(s) RV 0 and m > 0, we must have s m ψ(s). The contradiction implies that E[H m ] = for all m > 0. It can be verified that the LTs for BB2 and BB3 are both slowly varying at +, except in the lower boundary case (i.e., θ = 1 for BB3); that is, ψ BB2 (s) = [1 + δ 1 log(1 + s)] 1/θ RV 0, θ > 0, δ > 0; ψ BB3 (s) = exp { [δ 1 log(1 + s)] 1/θ} RV 0, θ > 1, δ > 0. 11

12 Now, consider the tail behavior of the LT studied in [23]; that is, assume that a LT η satisfies η(s) T (s) = a 1 s q exp{ a 2 s r } and η (s) T (s), s, with a 1 > 0, a 2 0, (10) where r = 0 implies a 2 = 0 and q < 0, and r > 0 implies r 1 and q can be 0, negative or positive. Note that r > 1 or r < 0 is not possible because of the complete monotonicity property of a LT. This condition covers almost all of the LT families in the Appendix of [26], as well as other LT families that can be obtained by integration or differentiation. The next result is contained in the Appendix of Joe (1997) [26], where L + is the class of infinitely differentiable increasing functions of [0, ) onto [0, ), with alternating signs for derivatives. This combines some results from pages 441 and 450 of Feller (1971)[19]; the condition on φ implies that it is the LT of an infinitely divisible random variable. Lemma 4 If φ is a LT such that log φ L + and η is another LT, then ψ(s) = η( log φ(s)) is a LT. Proposition 5 Suppose the LT η(t) satisfies condition (10), and take LT of Gamma φ(s) = (1+s) 1/θ, thus ψ(s) := η( log(φ(s))) is also a LT. Then 0 r < 1 implies that ψ(s) RV 0. Proof: Clearly, log(φ(s)) = (1/θ) log(1 + s) L +, by Lemma 4, ψ(s) is a LT. Then the conclusion is straightforward by plugging log(φ(s)) into η( ) of (10) since exp( a[log(1 + s)] r ) RV 0 for a > 0 and 0 < r < 1. In the literature, risks are compared with respect to some stochastic orders, say, the usual stochastic order and the increasing convex order [31]. In our opinion, the comparisons of risks are more meaningful for high-risk scenarios. However, the conditions these stochastic orders need to satisfy are too strong to be flexible for comparing tail risks. To this end, we may define some new concepts of stochastic order that are particularly used to compare the tails of univariate/multivariate cdf s. For example, we may define a stochastic order based on asymptotic properties of distribution functions as what follows. Definition 4 (Ultimate usual stochastic orders) Let X, Y be random variables, X is said to be greater than Y in the upper ultimate usual stochastic order if there exists a finite q such that F X (t) F Y (t) for all t q, written as X st Y. Let X and Y be random vectors, then X is said to be greater than Y in the upper ultimate usual stochastic order if there exists a finite threshold q such that F X (t) F Y (t) for all t q, written as X st Y. For lower ultimate usual stochastic order, the notation is st. Then X is said to be greater than Y in the lower ultimate usual stochastic order (written as X st Y ) if there exists a finite q such that 12

13 F X (t) F Y (t) for all t q, and X is said to be greater than Y in the lower ultimate usual stochastic order if there exists a finite threshold q such that F X (t) F Y (t) for all t q, written as X st Y. The relationships can also be symbolized by using corresponding cdfs, such as, F X st F Y for X st Y. If X, Y represent amounts of losses, then X st Y implies that X is riskier than Y. If the right tail of X is heavier than that of Y, then this is a sufficient condition for X st Y. Proposition 6 Let ψ 1, ψ 2 be given LTs. If ψ 1 1 ψ 2 (s) is superadditive for sufficiently large s, then there exists a finite threshold q such that for any u q, C ψ1 (u) C ψ2 (u), i.e., C ψ1 st C ψ2. Proof: Superadditivity of ψ 1 1 ψ 2 (s) for sufficiently large s implies that there exists sufficiently large s 0 such that s 1 s 0 and s 2 s 0 lead to ψ 1 1 ψ 2 (s 1 + s 2 ) ψ 1 1 ψ 2 (s 1 ) + ψ 1 1 ψ 2 (s 2 ). Let s 1 := ψ 1 2 (u) and s 2 := ψ 1 2 (v), then u, v ψ 2 (s 0 ) implies that ψ 1 1 ψ 2 (ψ 1 2 (u) + ψ 1 2 (v)) ψ 1 1 ψ 2 (ψ 1 2 (u)) + ψ 1 1 ψ 2 (ψ 1 2 (v)); that is, ψ 2 (ψ 1 2 (u) + ψ 1 2 (v)) ψ 1 (ψ 1 1 (u) + ψ 1 1 (v)), which completes the proof. Note: using notation of Archimedean copula in Nelsen (2006) [32], the generator should be strict in order to apply Proposition 6. Remark 6 From Proposition 6, we expect that BB2 is more lower tail positive dependent than BB3 below a sufficiently low threshold, although they are both lower tail comonotonic. Let ψ 1 (s) := ψ BB2 (s) = [1+δ 1 1 log(1+s)] 1/θ 1 and ψ 2 (s) := ψ BB3 (s) = exp { [δ 1 2 log(1 + s)] 1/θ 2}, then Letting h(s) := log(1 + s)/δ 2, g(s) := ψ 1 1 ψ 2 (s) = exp{δ 1 (e θ 1(log(1+s)/δ 2 ) 1/θ 2 1)} 1. g (s) = δ 1 θ 1 h(s) 1/θ 2 e θ 1h(s) 1/θ 2 e δ 1 ( ) e θ 1 h(s)1/θ2 1 /(θ 2 (1 + s) log(1 + s)). By observing g (s), we know that g(s) is strictly increasing and ultimately strictly convex. Assume that g(s) is strictly convex as s s 0, and let y 0 := g(s 0 ) and z(s) := g(s + s 0 ) y 0, then z(s) is superadditive for s [0, ). Therefore, for x, y 0, z(x+y) z(x)+z(y); that is, 13

14 g(x+y +s 0 )+g(s 0 ) g(x+s 0 )+g(y +s 0 ). Since g(s) is strictly increasing and strictly convex as s s 0, we must have, when x, y are sufficiently large, g(x + y + 2s 0 ) g(x + y + s 0 ) + g(s 0 ). Therefore, ψ 1 1 ψ 2 (s) is ultimately superadditive. 4.2 Heavy tail mixtures Consider the heavy tail scale mixture X = (RT 1,..., RT d ), (11) where R and T i s are all non-negative random variables, R is independent of the T i s, and the dependence structure between T i s is not specified. This form covers truncated elliptical distributions. The CTE for such random vectors has been studied in Zhu and Li (2012) [39], and Hua and Joe (2011b) [22], where R is assumed to be regularly varying and second order regularly varying in the right tail, respectively. The following lemma is often referred to as the Breiman s Theorem (Breiman 1965 [7]). Although the proof in [7] is not for a general α, it can be adapted for proving a more general case where 0 α < +. To the best of our knowledge, we have not found a complete and detailed proof for this case. So we include a proof in the Appendix, and emphasize that the result for α = 0 corresponding to slowly variation of Y also holds. Lemma 7 Suppose a random variable R 0 with F R (y) RV α, (0 α < + ). T 0 is independent of R and E[T α+δ ] < for some δ > 0. Then P[RT > y] lim y P[R > y] = E[T α ]. (12) Proposition 8 Let X = (RT 1,..., RT d ) be defined as in (11). If R RV 0, E[T δ i i ] <, i = 1,..., d, for some δ i > 0, then X is upper tail comonotonic. Proof: Due to Lemma 7, F X1 (s) F Xi (s), s +, for i = 2,..., d, thus, s = F 1 X i (F Xi (s)) F 1 X i (F X1 (s)). Since F 1 X 1 (t) = F 1 X 1 (1 t), clearly, F 1 X i (F X1 (s)) s, s + ; i = 2,..., d. (13) Assuming F is the joint cdf of (T 1,..., T d ), and h i (s) := F 1 X i (F X1 (s))/s, i = 1,..., d, the 14

15 upper tail dependence parameter [ P X1 > F 1 X λ U = lim 1 (t),..., X d > F 1 X d (t) ] t 1 P [ X 1 > F 1 X 1 (t) ] = lim s + = lim s + P [ RT 1 > s,..., RT d > F 1 X d (F X1 (s)) ] /P[R > s] P[RT 1 > s]/p[r > s] R d + P [R > s (max{h 1 (s)/t 1,..., h d (s)/t d })] F (dt 1,..., dt d )/P[R > s] P[RT 1 > s]/p[r > s]. (14) Due to Lemma 7, lim s + P[RT 1 > s]/p[r > s] = 1. For the numerator of (14), choose some b(s) such as sl 0 (s) with a proper slowly varying function l 0 such that, s implies b(s), and b ɛ (s)f R (s), ɛ > 0 (15) b(s)/s 0. (16) Denote (b(s)1 d, + ] c := R d + \ (b(s)1 d, + ], then lim s + = lim s + + lim s + R d + R d + R d + =: lim s + I 1 + lim s + I 2. P [R > s (max{h 1 (s)/t 1,..., h d (s)/t d })] F (dt 1,..., dt d ) P[R > s] 1 (b(s)1d,+ ] c(t 1,..., t d ) P [R > s (max{h 1(s)/t 1,..., h d (s)/t d })] F (dt 1,..., dt d ) P[R > s] 1 (b(s)1d,+ ](t 1,..., t d ) P [R > s (max{h 1(s)/t 1,..., h d (s)/t d })] F (dt 1,..., dt d ) P[R > s] Due to (13), for any 0 < γ < 1, there exists an s 0 < such that, s > s 0 implies that h i (s) > γ for all i. Then s > s 0 implies that 1 (b(s)1d,+ ] c(t 1,..., t d ) P [R > s (max{h 1(s)/t 1,..., h d (s)/t d })] P[R > s] 1 [0,γ1d )(t 1,..., t d ) + 1 (b(s)1d,+ ] c \[0,γ1 d )(t 1,..., t d ) P [R > sγ (max{1/t 1,..., 1/t d })]. P[R > s] It is well known that P[R > s] is slowly varying implies P[R > st]/p[r > s] converges uniformly in t [a, b] as s, where 0 < a, b < (see [6]). Then P[R > sγ(max{1/t 1,..., 1/t d })]/P[R > s] 15

16 converges uniformly in t (b(s)1 d, + ] c \[0, γ1 d ) to 1. Moreover, t (b(s)1 d, + ] c \[0, γ1 d ) and (16) implies that, as s +, sγ max{1/t 1,..., 1/t d } +. Then slow variation of P[R > s] and dominated convergence theorem implies that lim I 1 = s + =: R d + R d + lim 1 (b(s)1 s + d,+ ] c(t 1,..., t d ) P [R > s (max{h 1(s)/t 1,..., h d (s)/t d })] F (dt 1,..., dt d ) P[R > s] lim 1 (b(s)1 s + d,+ ] c(t 1,..., t d )Ω(s, t 1,..., t d )F (dt 1,..., dt d ). Since (13), for any 0 < ɛ 1, ɛ 2 < 1, there exists an s 1 such that s > s 1 implies that 1 ɛ 1 h i (s) 1 + ɛ 2 for all i. Thus, for any t 1,..., t d such that min{t 1,..., t d } > 0, 1 = lim s P[R > s(1 + ɛ 2 )/ min{t 1,..., t d }] P[R > s] lim sup Ω(s, t 1,..., t d ) s 1 <s lim s P[R > s(1 ɛ 1 )/ min{t 1,..., t d }] P[R > s] = 1. Therefore, For I 2, lim I 1 = 1. s 1 (b(s)1d,+ ](t 1,..., t d ) P [R > s (max{h 1(s)/t 1,..., h d (s)/t d })] P[R > s] 1 1 (b(s)1d,+ ](t 1,..., t d ) P[R > s] 1 (b(s)1 d,+ ](t 1,..., t d ) (min{t 1,..., t d }) ɛ. P[R > s]b ɛ (s) Note that, E[T δ i i ] < implies that E[(min{T 1,..., T d }) ɛ ] < for some ɛ > 0, because E[(min{T 1,..., T d }) δ i ] t δ R d i i F (dt 1,..., dt d ) = E[T δ i i ] <. Since E[(min{T 1,..., T d }) ɛ ] < +, and P[R > s]b ɛ (s) + (due to (15)), by dominated convergence theorem, Thus, the claim is proved. lim I 2 = 0. s + The above scale mixture can be used to construct upper tail comonotonic copulas. Note that, the margins T i s are not necessarily identical and their dependence structure is not specified. An elliptical random vector X can be written as X d = µ + RAU E d (µ, Σ, φ), (17) 16

17 where the radial random variable R 0 is independent of U, U is an k-dimensional random vector uniformly distributed on the surface of the unit hypersphere S k 1 2 = {z R k z T z = 1}, A is a d k matrix with rank(a) = k and AA T = Σ that is positive semidefinite, φ is the characteristic generator. When Σ is positive definite, write ρ ij := Σ ij / Σ ii Σ jj, and ρ ij ±1 in the following to avoid trivial cases. We refer to Fang et al. (1990) [18] for a comprehensive reference for elliptical distributions. Now, we are studying whether there exist tail comonotonic elliptical distributions. Note that the T i s in Proposition 8 are all non-negative, so it does not cover the usual elliptical distributions. For tail dependence of elliptical distributions, Schmidt (2002) [38], and Hult and Lindskog (2002) [25] study the case where the radial random variable R has certain tail patterns such as regularly varying right tails. Frahm et al. (2003) [20] shows that the two different representations of tail dependence parameters in [38] and [25] for the regularly varying case are equivalent. From their results, we know that, with a regularly varying radial random variable R (tail index α < 0), the nondegenerate elliptical distributions have tail dependence parameters that are strictly less than 1. We now show that, even when P[R > s] is slowly varying in s, tail comonotonicity still does not hold. The intuitive reasoning for this result is that for bivariate elliptical distributions that have usual tail dependence, there is tail dependence in all corners/quadrants, so that the tail dependence in the upper positive quadrant is not 1. Proposition 9 Suppose X is an elliptical random vector as defined in (17) with Σ ii > 0 for i = 1,..., d, if P[R > s] RV 0, then any bivariate margins of X is upper (and lower) tail dependent, and the tail dependence parameter λ ij = 1/2 + (1/π) arcsin ρ ij. Proof: The proof in Theorem 4.3 of Hult and Lindskog (2002) [25] remains valid for this case where R is slowly varying, although in their proof, R is required to be regularly varying with a tail index α < 0. Remark 7 In the proof of Theorem 4.3 in [25], it is claimed that X d = RAU implies that ( X i X j ) ( ) d Σii 0 = R cos ϕ, Σjj ρ jj Σjj 1 ρ 2 ij sin ϕ where ϕ Uniform( π, π). Here we need to make a note that the above claim is true only when X is bivariate. For a general d-dimensional X (d > 2), radial random variables for margins will not be the same as the original R for the whole distribution. We make a more detailed argument about this in what follows. 17

18 Due to the Corollary in page 43 of Fang et al. (1990) [18], margins of elliptical distributions are still elliptical with the same characteristic generator φ. Moreover, due to Lemma 5.3 of [38], the radial random variable R depends on the dimension d of the elliptical distribution, and there exists a constant k > 0 such that R d s = kr d B, B Beta(s/2, (d s)/2), s < d, and B is independent of R d. Since a Beta distribution has moments of all orders, by Lemma 7, R s inherits the tail behavior of R d when R d is regularly varying or slowly varying, which is why the original proof in [25] can still yield a correct conclusion. Let (U 1, U 2 ) follows a bivariate copula, where U i Uniform(0, 1), i = 1, 2. From Embrechts et al. (2009) [15], the tail dependence parameters in the North East (NE) and South East (SE) for the bivariate Student t copula are the following: λ NE := lim u 1 P[U 2 > u, U 1 > u] = λ SE := lim u 1 P[U 2 1 u, U 1 > u] = π/2 (cos (π/2 arcsin ρ)/2 t)ν dt π/2 (cos t) 0 ν dt π/2 (π/2+arcsin ρ)/2 (cos t)ν dt π/2 0 (cos t) ν dt =: Λ(ν, ρ); (18) =: Λ(ν, ρ). (19) Note that λ NE + λ SE is still a conditional probability and thus 0 λ NE + λ SE 1; in fact, for the above 0 < Λ(ν, ρ) + Λ(ν, ρ) < 1 for ν > 0. Moreover, both λ NE and λ SE are decreasing in ν: for a bivariate t ν copula ([16], [27]), λ NE = 2T ν+1 ( ) (ν + 1)(1 ρ)/(1 + ρ) ; (20) λ SE = 2T ν+1 ( ) (ν + 1)(1 + ρ)/(1 ρ), (21) and it can be verified that (18) and (20) are equivalent, and (19) and (21) are equivalent (see Appendix for a direct derivation). Since T ν+1 (x) is decreasing in ν for a fixed x 0, and T ν+1 (x) is increasing in x for a fixed ν, both λ NE and λ SE are decreasing in ν. From Theorem 5.2 of Schmidt (2002) [38], we know that if there exists a bivariate tail dependent margin, then we must have 0 < lim inf x F (tx) F (x) lim sup x F (tx) F (x) 1, t 1, where F ( ) is the survival function of the radial random variable R. If R is slowly varying, then the upper bound in the above necessary condition is reached. 18

19 Although we have not proved it for now, we conjecture that there are no tail comonotonic nondegenerate elliptical distributions and the maximum tail dependence parameter for a bivariate elliptical copula is λ = 1/2 + (1/π) arcsin ρ. As ν 0 + in (18) and (19), the sum converges to [1/2 + (1/π) arcsin ρ] + [1/2 + (1/π) arcsin( ρ)] = 1 and this is the maximum possible value of lim x P( X 2 > x X 1 > x) when X 1, X 2 have a common distribution. Hence when all corners are considered, elliptical distributions with a slowly variable radial random variable have the strongest possible tail dependence. In contrast, note that Archimedean copulas based on LTs do not have tail dependence on quadrants that have different signs. 4.3 Extreme value copulas We now show that for extreme value copulas, tail comonotonicity is equivalent to comonotonicity. Proposition 10 Suppose C is an extreme value copula, then λ U = 1 C is a comonotonic copula λ L = 1. Proof: By Proposition 2, it suffices to prove the bivariate case. For any multivariate extreme value copula C, there exists a function A : [0, ) d [0, ) such that C(u 1,..., u d ) = exp{ A( log u 1,..., log u d )}, where A is convex, homogeneous of order 1 and satisfies that max{x 1,..., x d } A(x 1,..., x d ) x x d. For a bivariate extreme value copula, the function A(x, y) can be written as A(x, y) = (x + y)b( x ), where B( ) is convex and x+y max{w, 1 w} B(w) 1 for 0 w 1 (see Theorem 6.4 of Joe 1997 [26]). The upper tail dependence parameter is λ U = 2 A(1, 1). If we let λ U = 1, then B(1/2) = 1/2, also B(w) must be convex, so we have B(w) = max{w, 1 w}; that is, A(x, y) = max{x, y} and thus C(u 1, u 2 ) = exp{ A( log u 1, log u 2 )} = min{u 1, u 2 }. straightforward. The other direction is The lower tail order for a bivariate extreme value copula is κ = A(1, 1) (see Example 2 of Hua and Joe 2011a [23]). If λ L = 1, we must have the lower tail order A(1, 1) = 1. Then the subsequent argument is the same as for the upper tail. 5 Asymptotic additivity of risk measures As a dependence structure, tail comonotonicity may affect the risk measures of aggregated losses. In this section, we mainly study impacts of tail comonotonicity on additivity of commonly used risk measure such as VaR and CTE. 19

20 It is well known that VaR and CTE are additive when the loss random variables are comonotonic (see Dhaene et al [13]); that is, if (X 1,..., X d ) is comonotonic, then for all p (0, 1), VaR p ( X i ) = VaR p (X i ); CTE p ( X i ) = CTE p (X i ). The additivity property also holds for the upper comonotonicity in the sense of Cheung (2009) [9] when the probability level p associated with the risk measures is larger than a threshold specified by the upper comonotonicity structure. A natural question is whether such an additivity property can be kept asymptotically as p 1. Asymptotic super and/or sub additivity of risk measures has been studied explicitly or implicitly in several papers. For example, Embrechts et al. (2008) [17] studied asymptotic additivity properties of VaR for multivariate (d > 2) dependent loss random variables that have regularly varying survival functions (with index β) and Archimedean dependence structures. It is shown that for a probability level p < 1 and sufficiently close to 1, whether strict super or sub additivity depends on whether β < 1 or β > 1. The Archimedean copula has upper tail dependence but the upper tail dependence parameter λ < 1 (since the generator of the Archimedean copula is assumed to be regularly varying). In Section 3.3 of Alink et al. (2007) [2], an example of lower tail comonotonic copula has been shown together with Corollary 2.4 of [17] that VaR is asymptotically additive for exchangeable regularly varying dependent random variables that have an upper tail comonotonic copula. Analogous to the additivity property of VaR and CTE for a comonotonic random vector, we find that, asymptotic additivity of VaR and CTE still holds for a large class of random vectors that are relevant for quantitative risk management. We now prove some results for asymptotic additivity of VaR and CTE for random variables that are in the maximum domain of attraction (MDA) of Fréchet and Gumbel, respectively. The case for MDA of Weibull corresponds to loss random variables that are bounded above, and is not very relevant for actuarial applications, so we do not consider it here. Only upper tails of losses are to be studied as they are more relevant for risk measures; analogous results for lower tails also hold but are omitted here. It suffices to consider nonnegative random variables to study upper tails, so in the following the random vector X is assumed to be non-negative. However, for lower tails, we have simpler notation for tail comonotonicity. Therefore, instead of studying X directly, in what follows, we always define Y := X and prove the results based on Y. The following corresponding assumptions are also assigned on Y, which have corresponding natural meanings for the non-negative random vector X. 20

21 For study of tail behavior of random variables, we always assume that the distribution function is continuous (or at least continuous in the tail regions). So P[X < x] coincides with P[X x] in what follows. Assumption A. Let Y be a non-positive continuous random vector with marginal distributions F 1,..., F d defined on (, 0] such that F i ( t) RV α with α > 0, and for any i = 1,..., d. lim F i( t)/f 1 ( t) = k i, 0 < k i <, t Remark 8 If Y := X satisfies Assumption A, thus P[X i > t] RV α, then each univariate margin of X is in the MDA of Fréchet (Theorem of [14]). The following lemmas are analogous to Lemma 6.1 of [37]. They are useful in proving Propositions 13 and 14. For E R d, denote M + (E) := {µ : Radon, non-negative measures on the Borel σ-algebra E of E}, and let the lower boundary LB := {(y 1,..., y d ) : some y i = 0} and the upper boundary UB := {(y 1,..., y d ) : some y i = }, then define E 1 := [0, ] \ LB; E 2 := [, ] \ UB. Lemma 11 Suppose µ n, µ M + (E 1 ). Then, as n µ n v µ in M + (E 1 ) µ n ([y, ]) µ([y, ]) for any y [0, ) \ LB such that µ( [y, ]) = 0. Proof: The proof is similar to the proof of Lemma 6.1 of [37]. We rewrite the proof here for completeness. ( ) is due to Theorem 3.2 of [37]. For ( ), let g C + K (E 1), where C + K (E 1) := {g : E 1 R +, continuous with compact support}, then the support of g must be contained in some [y, ] such that µ( [y, ]) = 0. Since convergence on this set holds, sup n µ n ([y, ]) < and thus, sup µ n (g) sup g(x) sup µ n ([y, ]) <. n x E 1 n 21

22 This is true for any g C + K (E 1), so {µ n } is relatively compact due to (3.16) of [37]. If µ and µ are two subsequential limits, then µ and µ agree on the continuity sets [y, ]. Additionally, the rectangles of those continuity sets [y, ] constitute the π-system which generates B(K o ) the Borel σ-algebra of K o, where K o := {B E 1 : B is relatively compact, µ( B) = 0}. Then µ = µ on E 1 by Theorem 3.2 of [37] again. Remark 9 The set E 1 excludes the lower boundary, since otherwise rectangles of the form [y, ] can not determine the vague convergence. If µ puts mass on axes such as the usual normalization for asymptotic independence in Section of [37], then µ / M + (E 1 ). The condition µ( [y, ]) = 0 also requires that µ does not put mass on the upper boundary. Lemma 12 Suppose µ n, µ M + (E 2 ). Then, as n µ n v µ in M + (E 2 ) µ n ([, y]) µ([, y]) for any y (, ] \ UB such that µ( [, y]) = 0. Proof: The proof here is similar to the proof of Lemma 11 by replacing [y, ] by [, y]. Proposition 13 (Asymptotic additivity of VaR: Fréchet case) Suppose X is non-negative and upper tail comonotonic, and Y := X satisfies Assumption A. If S = X X d, then VaR p (S) VaR p (X i ), p 1. Proof: Let W = S and Y i = X i, i = 1,..., d, then (Y 1,..., Y d ) is lower tail comonotonic, and the map y P[Y i y] RV α. Moreover, since lim t F i ( t)/f 1 ( t) = k i for any i = 1,..., d, clearly, F 1 i (p) k 1/α i F1 1 (p) as p 0 + (Lemma 2.1 of [3], or Proposition 0.8(vi) of [36]). It suffices to prove that, F 1 1 W (p) F1 (p) k 1/α i, p 0 +. Also, P[W t] RV a (Proposition 7.3 of Resnick 2007 [37]) and apply Proposition 0.8(vi) again. Then it suffices to show as t, P[W t] P[Y 1 t] ( k 1/α i ) α. (22) 22

23 Under Assumption A, if Y is also lower tail comonotonic, and C is the copula, then we have for any y [0, ) \ LB, P[Y 1 ty 1,..., Y d ty d ] lim t P[Y 1 t] ( ) C F1 ( ty 1 ) F F 1 ( t) 1( t),..., F d( ty d ) F F 1 ( t) 1( t) = lim t F 1 ( t) = min{k 1 y1 α,..., k d y α }, (23) where the last equality is due to lower tail comonotonicity of C and uniform continuity of copula functions (see Nelsen 2006 [32]). Define a measure µ on E 1 as µ([y 1, ] [y d, ]) := min{k 1 y α 1,..., k d y α d }. This measure µ only puts mass on the line {(y 1,..., y d ) : k 1 y1 α = = k d y α d }. If k 1y1 α = = k d y α d =: h, then µ([y 1, ] [y d, ]) = h. Clearly, µ is Radon and every set [y, ] is a continuity set. Define then by Lemma 11 and (23), µ t ( ) := P [( Y 1/t,..., Y d /t) ], P[Y 1 t] d µ t ( ) v µ( ), t. where v is vague convergence. Define a set ( H := y [0, ) \ LB : ) y i 1. Then clearly, µ( H) = 0 and H is relatively compact. Thus by Theorem 3.2 of [37], That is, letting z i := k i y α i, as t, µ t (H) µ(h), t. P[W t] { P[Y 1 t] = µ t(h) min k 1 y1 α,..., k d y α d : = µ Lebesgue (z : } y i 1 ) ( (z/k i ) 1/α 1 = k 1/α i ) α, 23

24 which justifies (22), thus finishing the proof. Assumption B. Let Y be a non-positive continuous random vector with marginal distributions F 1,..., F d defined on (, 0] and there exists a positive measurable function a( ) such that F i ( t + a(t)s) lim t F i ( t) and lim t F i ( t)/f 1 ( t) = k i with 0 < k i < for any i = 1,..., d. = e s, s R, (24) Remark 10 Note that the condition F i ( t)/f 1 ( t) k i implies that we can take a i (t) = a(t) for all i without loss of generality, and a(t) = o(t), t. If Y := X satisfies Assumption B, then due to Theorem of [14], each univariate margin of X is in the MDA of Gumbel; that is F Xi (t + a(t)s)/f Xi (t) e s, s R. Remark 11 A mixture distribution can satisfy the tail equivalence condition F Xi (t)/f X1 (t) k i. Without loss of generality, we can let 0 < k i 1. For example, let random variables X 1 Exponential(1) and X 2 = Then F X2 (x)/f X1 (x) k 2. { Exponential(1), with probability k 2 0, with probability 1 k 2. Proposition 14 (Asymptotic additivity of VaR: Gumbel case) Suppose X is non-negative and upper tail comonotonic, and Y := X satisfies Assumption B. If S = X X d, then VaR p (S) VaR p (X i ), p 1. Proof: Let W = S and Y i = X i, i = 1,..., d, then (Y 1,..., Y d ) is lower tail comonotonic. Moreover, since lim t F i ( t + a(t)s)/f 1 ( t) = k i e s for any s R and i = 1,..., d, taking s = log k i leads to VaR p (Y i ) VaR p (Y 1 ) a( VaR p (Y 1 )) log k i, p 0 +. (25) Then it suffices to show as t, [ ] P W dt a(t) log k i P[Y 1 t]. (26) 24

25 Under Assumption B, if Y is also lower tail comonotonic, and C is the copula, then we have for any y (, ] \ UB, P[Y i t a(t) log k i + a(t)y i, i = 1,..., d] lim t P[Y 1 t] C = lim t C = lim t ( Fi ( t a(t) log k i +a(t)y i ) F 1 ( t) F 1 ( t) ( ki e y i log k i F 1 ( t), i = 1,..., d ) F 1 ( t) ) F 1 ( t), i = 1,..., d = min{e y 1,..., e y d }, (27) where the last equality is due to lower tail comonotonicity of C and uniform continuity of copula functions ([32]). Define a measure µ on E 2 as µ([, y 1 ] [, y d ]) := min{e y 1,..., e y d }. This measure µ only puts mass on the line {(y 1,..., y d ) : e y 1 = = e y d }. If e y 1 = = e y d =: h, then µ([, y1 ] [, y d ]) = h. Clearly, µ is Radon and every set [, y] is a continuity set. Define µ t ( ) := P [((Y 1 + t)/a(t) + log k 1,..., (Y d + t)/a(t) + log k d ) ], P[Y 1 t] then by Lemma 12 and (27), µ t ( ) v µ( ), t. where v is vague convergence. Define a set ( H := y (, ] \ UB : ) y i 0. Then clearly, µ( H) = 0 and H is relatively compact. Thus by Theorem 3.2 of [37], µ t (H) µ(h), t. 25

26 That is, letting z i := e y i, as t, P[W dt a(t) d log k i] P[Y 1 t] { = µ t (H) min e y 1,..., e y d : = µ Lebesgue (z > 0 : } y i 0 ) log z 0 = 1. This justifies (26) and asymptotic additivity holds. The conditions that are studied for asymptotic additivity of VaR satisfy assumptions considered in Asimit et al. (2011) [3], which investigates asymptotic proportionality between CTE and VaR. With the help of their results, we may conclude the asymptotic additivity of CTE in what follows. Proposition 15 (Asymptotic additivity of CTE: Fréchet case) Suppose X is non-negative and upper tail comonotonic, and Y := X satisfies Assumption A with α > 1. If S = X 1 + +X d, then CTE p (S) CTE p (X i ), p 1. Proof: It is well known that under Assumption A with α > 1, CTE p (X i ) 1, for i = 1,..., d (e.g., [39]). By Theorem 2.1 in [3], and Proposition 13, CTE p (X i ) α α 1 VaR p (X i ) α VaR α 1 p(x i ), p α α 1 VaR p(s) CTE p (S), p 1, which completes the proof. Proposition 16 (Asymptotic additivity of CTE: Gumbel case) Suppose X is non-negative and upper tail comonotonic, and Y := X satisfies Assumption B. If S = X X d, then CTE p (S) CTE p (X i ), p 1. Proof: It is well known that under Assumption B, CTE p (X i ) VaR p (X i ), p 1 (e.g., Section of [14]). Also, CTE p (S) VaR p (S), p 1 (see (2.15) of [3]). Therefore, by Proposition 14, CTE p (S) VaR p (S) VaR p (X i ) 26 CTE p (X i ), p 1,

27 which completes the proof. Let X := (X 1,..., X d ) T be a d-dimensional random vector. Then diversification benefit of X with respect to a risk measure Q p at level p (0, 1) is defined as 1 Q p (X X d )/ d Q p(x i ), where Q p : R R + is a quantile-based risk measure. The above asymptotic additivity results suggest that when we use tail comonotonicity as the dependence structure for such marginal distributions, we should expect that the diversification benefit will decrease to 0 as p 1. However, the speed of decay is unknown by observing the asymptotic relationship only; that is, we do not know when p is sufficiently close 1 so that the additivity relationship of the risk measures is sufficiently good. The relevant study will involve a second order approximation, and it is out of the scope of this paper. 6 Concluding remarks and future research Tail comonotonicity, like comonotonicity and upper comonotonicity, provides a bound-like dependence structure, and it is more reasonable to be used to capture information from data. Tail comonotonicity has some parallel properties of the usual comonotonicity, such as asymptotic additivity of VaR and CTE. Among many different copula families, Archimedean copulas with a mixing distribution that has no moments of any positive orders (for upper tail comonotonicity) or no moments of any negative orders (for lower tail comonotonicity), and copulas based on scale mixtures with a slowly varying non-negative random variable can be used to construct tail comonotonic copulas. However, elliptical and extreme value copulas cannot provide useful tail comonotonic copulas. Since tail comonotonicity is only an asymptotic property, it may or may not provide a conservative dependence structure for sub-extremal levels of risks. In [24], we began a study of how conservative are risk measures, under the tail comonotonic dependence structure. Moreover, benefits and limitations of using tail comonotonicity for modeling real data will be further investigated. Although asymptotic additivity properties of VaR and CTE we have proved has already covered a wide range of random vectors, an elegant proof for a most general case (note that additivity of VaR and CTE for the usual comonotonicity does not depend on margins) must be welcome. Many other topics, such as optimal portfolio design, could be studied under the assumption of tail comonotonicity. We think that tail comonotonicity is particularly useful for dealing with high dimensional data for high-risk scenarios. Due to the curse of dimensionality, it is very hard (or impossible) to accurately model the tail behavior of aggregated highdimensional risks. For dependence modeling of multivariate random variables, vine copula is promising [4, 1, 29]. By Proposition 2, it is feasible to use bivariate tail comonotonic copulas to 27