Asymptotics for Risk Capital Allocations based on Conditional Tail Expectation

Transcription

1 Asymptotics for Risk Capital Allocations based on Conditional Tail Expectation Alexandru V. Asimit Cass Business School, City University, London EC1Y 8TZ, United Kingdom. Edward Furman Department of Mathematics and Statistics, York University, Toronto, Ontario M3J 1P3, Canada. Qihe Tang 1 Department of Statistics and Actuarial Science, University of Iowa, 241 Schaeffer Hall, Iowa City, IA 52242, USA. qihe-tang@uiowa.edu Raluca Vernic Department of Mathematics and Informatics, Ovidius University of Constanta, Constanta, Romania. rvernic@univ-ovidius.ro Abstract. An investigation of the iting behavior of a risk capital allocation rule based on the Conditional Tail Expectation CTE risk measure is carried out. More specifically, with the help of general notions of Extreme Value Theory EVT, the aforementioned risk capital allocation is shown to be asymptotically proportional to the corresponding Value-at- Risk VaR risk measure. The existing methodology acquired for VaR can therefore be applied to a somewhat less well-studied CTE. In the context of interest, the EVT approach is seemingly well-motivated by modern regulations, which openly strive for the excessive prudence in determining risk capitals. Keywords and phrases: Asymptotic dependence and independence; Capital allocation; Conditional Tail Expectation; Extreme Value Theory; Heavytailed distributions; Value-at-Risk. Mathematics Subject Classification: Primary 91B30; Secondary 91G10, 62H20, 60E05 1 Corresponding author. Phone: ; fax:

2 2 1. Introduction Let X denote an insurance risk. Speaking more formally, X is a non-negative random variable defined on the probability space Ω, F, Pr and possessing a distribution function F x := PrX x and a tail function F x := 1 F x, x R. A risk measure is generally formulated as a functional, Q, from the space of distribution functions to [0, ]. Similarly see, e.g., Bühlmann, 1980, we can consider the functional Q as from X, the space of insurance risks, to [0, ], which we indeed often do in the sequel. Certainly, F establishes a meaningful ordering of X and, hence, it can be interpreted as a risk measure. However, for the sake of risk capital determination, it is desirable for the risk measure Q[F ] to take on monetary units. Thus, Q[F ] = F x is naturally replaced with, e.g., its inverse, bringing us to the notion of Value-at-Risk VaR. Namely, let q 0, 1 denote the confidence level required by regulations. Then VaR q [X] := inf{x : F x q} establishes arguably the most popular risk measure, which has been a cornerstone of the financial risk measurement of the last century. We note in passing that the Solvency II Accord designed by the EU Commission sets q = over a one-year time horizon. Noticeably, the recent financial instability and, as a result, regulators inclination to excessive prudence in determining risk capital requirements have to a certain extent enfeebled VaR s status. In this respect, the so-called tail-based risk measurement has emerged as a natural tool for quantifying insurance risks while emphasizing the adverse effect of low probability but high severity tail events. Thereby, a more pessimistic Conditional Tail Expectation CTE risk measure is defined, for q 0, 1, as CTE q [X] := E[X X > VaR q [X]]. CTE is known as a coherent risk measure over the space of continuous random variables. It belongs to both the distorted and weighted risk measures see Wang, 1996; Dhaene et al., 2006; Furman and Zitikis, 2008a. Practically, CTE has already replaced VaR in regulatory requirements of, e.g., Canada, Israel and Switzerland. We note in passing that the current practice in the aforementioned countries is q = 0.99 over a one-year time horizon. Let X i X, i = 1,..., d, denote d N insurance risks and let S d := X X d denote the aggregate risk. Then evaluating VaR q [S d ] and CTE q [S d ] is a somewhat basic phase of the modern risk capital framework. Indeed, while it is of pivotal importance to

3 determine the overall risk capital requirement for an insurance company, it is of consequent interest to decompose the aforementioned capital into the associated risk sources. To this end, the functional Q is naturally generalized beyond the conditional state independence, to a risk capital allocation functional, A, from the space of the Cartesian product of X with itself to [0, ], and such that A[X i, X i ] = Q[X i ], i = 1,..., d; see, e.g., Furman and Zitikis 2008b. It should be noted that apart from purely regulatory interest, the functional A is often employed for, e.g., profitability analysis, pricing and quality control. Various functional forms of A have been proposed in the literature, with the allocation based on the CTE risk measure, formulated as E[X i S d > VaR q [S d ]], i = 1,..., d, 1.1 being arguably the most popular. See Section 6.3 of McNeil et al for related discussions on this allocation as a consequence of the Euler principle, as well as Dhaene et al for optimality studies of interest. Although 1.1 is quite elegant and satisfies many desirable properties, its analytic tractability for generally distributed and possibly dependent X 1,..., X d remains seldom feasible. To emphasize the point, we refer the reader to Panjer and Jia 2001, Landsman and Valdez 2003, Valdez and Chernih 2003, Cai and Li 2005, Furman and Landsman 2005, 2006, 2008, Chiragiev and Landsman 2007, Vernic 2006, 2011 and Dhaene et al for analytic expressions for 1.1 under specific multivariate distributions of a multi-line business and/or a portfolio of risks. In this paper, we follow a different route. Namely, as the excessive prudence of the current regulatory framework requires a confidence level close to 1, the notion of Extreme Value Theory EVT becomes appropriate. We therefore study the asymptotic behavior of capital allocations defined in 1.1 as q 1, when X 1,..., X d are asymptotically dependent or asymptotically independent. Following Section 5.2 of McNeil et al. 2005, the asymptotic independence between two random variables X i and X j with distribution functions F i and F j is defined as Pr F j X j > q F i X i > q = 0, q 1 while the asymptotic dependence is defined via this relation with a positive it. However, in this paper we slightly relax the notion of asymptotic dependence and define it as inf q 1 Pr F j X j > q F i X i > q > The notion of asymptotic dependence in higher dimensions is an obvious generalization of the two-dimensional definition above. It is known that, for both Fréchet and Gumbel 3

4 4 cases, the CTE and VaR of a single risk are proportional for a high confidence level see Asimit and Badescu, Therefore, not surprisingly our main results show that capital allocations, as described by 1.1, are asymptotically proportional to VaR q [S d ] with a readily calculable coefficient of proportionality. This allows for the utilization of the VaR-related machinery when dealing with the risk capital allocation based on CTE. The remainder of the paper is organized as follows. The main results under asymptotic dependence and asymptotic independence are formulated and proved in Sections 2 and 3, respectively. Relevant examples are discussed in Section 4, while certain simulation studies verifying the accuracy of the main results are carried out in Section 5. Finally, Section 6 concludes the paper. 2. Main results under asymptotic dependence From now on, we consider a multi-line insurance business consisting of d non-negative risk variables X 1,..., X d. Denote X = X 1,..., X d and S d = d i=1 X i. Unless otherwise stated, all it relationships hold as t or q 1, letting the relations speak for themselves. For two positive functions a and b, we write a cb to mean strong equivalence, i.e., a /b = c for some positive constant c, and we write a b to mean weak equivalence, i.e., 0 < inf a /b sup a /b <. We also denote inf a /b 1 and sup a /b 1 by a b and a b, respectively. To develop the main results of this paper we extensively employ the EVT techniques. A distribution function F is said to belong to the Maximum Domain of Attraction MDA of a non-degenerate distribution function G, written as F MDAG, if there are some a n > 0 and b n R for n N such that F n a n x + b n = Gx. n Due to the Fisher-Tippett theorem see Fisher and Tippett, 1928, and Gnedenko, 1943, G is of one of the following three types: Fréchet type: Φ α x = exp { x α }, x > 0, α > 0, Gumbel type: Λx = 1 exp { e x }, < x <, Weibull type: Ψ α x = exp { x α }, x 0, α > 0. Since distributions from MDAΨ α have finite upper endpoints while we are interested in risk variables with unbounded supports, in this paper we shall consider the Fréchet and Gumbel cases only. Another important notion that is crucial for establishing our main results is vague convergence. Let {µ n, n 1} be a sequence of measures on a locally compact Hausdorff

5 space B with countable base. Then µ n converges vaguely to some measure µ, written as µ v n µ, if for all continuous functions f with compact support we have f dµ n = f dµ. n B A thorough background on vague convergence is given by Kallenberg 1983 and Resnick Fréchet case. The next assumption is sufficient for our first main result. Assumption 2.1. Let X be a non-negative random vector with marginal distributions F 1,..., F d such that PrX 1 > tx 1,..., X d > tx d F 1 t B := H F x exists for all x = x 1,..., x d [0, ] d \ {0}, where H F is assumed to be a nondegenerate function and 0 is the vector of zeroes. This assumption implies that the marginal distribution functions are tail equivalent. That is, 0 < Fj t F i t <, 2.1 for all 1 i, j d. In addition, there exists some α > 0 such that, for each 1 i d, the distribution function of X i is regularly varying with index α, written as X i R α, F i tx F i t = x α, x > Note that X i R α is equivalent to X i MDAΦ α see Resnick, Moreover, Pr X 1 /t,..., X d /t F 1 t holds on [0, ] d \ {0}, where the measure µ is given by The function H F v µ 2.3 µ x 1, ] x d, ] := H F x. 2.4 satisfies certain properties see Resnick, 1987, and one of the most important is continuity on the set 0, d, but not necessarily on the boundary of its domain. The non-degeneracy assumption ensures that the measure µ does not put any mass on the boundary of the domain. Recall Proposition 0.8vi of Resnick 1987, which in our current context can be easily restated as: 5

6 6 Lemma 2.1. Let X 1 and X 2 be two random variables, both belonging to R α for some α > 0. Then, for 0 < c <, we have Pr X 2 > t c Pr X 1 > t if and only if VaR q [X 2 ] c 1/α VaR q [X 1 ]. With the help of Lemma 2.1 we can easily verify that Assumption 2.1 describes an asymptotic dependence case. Lemma 2.2. Under Assumption 2.1 the components of X = X 1,..., X d are pairwise asymptotically dependent. Proof. As an illustration we consider relation 1.2 for i, j = 1, 2. By relations 2.1 and 2.2 and Lemma 2.1, there is some c 2 > 0 such that VaR q [X 2 ] c 1/α 2 VaR q [X 1 ]. As q 1, or, equivalently, as t = VaR q [X 1 ], we have Pr F 2 X 2 > q F 1 X 1 > q = Pr F 1X 1 > q, F 2 X 2 > q Pr F 1 X 1 > q Pr X 1 > VaR q [X 1 ], X 2 > VaR q [X 2 ] Pr X 1 VaR q [X 1 ] Pr X 1 > t, X 2 > 2c 1/α 2 t Pr X 1 > t H F 1, 2c 1/α 2, 0,..., 0 > 0. This proves relation 1.2 for i, j = 1, 2. We are now able to provide asymptotic expressions for the risk capital allocation for a multi-line insurance business. Noticeably, switching the context to a portfolio consisting of a i units of X i for all 1 i d, similar results as given in the next theorem can be obtained by replacing the measure µ from 2.4 with µ a x 1, ] x d, ] = Pra 1 X 1 > tx 1,..., a d X d > tx d Pra 1 X 1 > t = H F x 1 /a 1,..., x d /a d a α 1. Theorem 2.1. Let X be a random vector satisfying Assumption 2.1. If X 1 R α with α > 1 then we have, for all 1 k d, where, with µ defined by 2.4, C k = E [X k S d > VaR q [S d ]] C k VaR q [S d ], 1 µx : x α 1 k > µ µ x : x 0 k > z, d i=1 x i > 1 dz x :. d i=1 x i > 1

7 Due to Assumption 2.1, it follows that Pr S d > t µ x : x i > 1 i=1 F 1 t; for details, see Proposition 7.3 of Resnick Thus, S d R α, which gives that d i=1 C k = α ; see Balkema and de Haan 1974, Alink et al. 2005, or Asimit and α 1 Badescu Note 2.1. After the majority of this work had been done we became aware of a forthcoming paper by Joe and Li Their Remark 2.32 suggests that our Theorem 2.1 is a consequence of their Theorem 2.2. However, this statement is not true because their proof is based on the vague convergence property see relation 2.3, which can be used only for µ-continuous sets. The reasoning behind applying the vague convergence property is given in the proof of Theorem 2.1, for completeness. It is conveyed that the asymptotic independence case see Theorem 3.1 requires an altered argumentation. Proof. We first note that E [X k S d > t] = 0 t For the first part of 2.6, I 1 t, we have 1 I 1 t = t 0 Pr X k > z S d > t dz = Pr X k > z, S d > t PrX k > z dz + 0 Pr S d > t t Pr S d > t dz = I 1 t + I 2 t. 2.6 PrX k > tz, S d > t 1 µx : x k > z, d dz t PrS d > t 0 µx : d i=1 x i > 1 i=1 x i > 1 dz, 2.7 which is a consequence of relations 2.3 and 2.5, the Dominated Convergence Theorem and Proposition A2.12 of Embrechts et al Note that the latter proposition can be applied since µ {x : x k > z, d i=1 x i > 1} = 0 for all z 0. In fact, Assumption 2.1 implies that µ {x i = z, x 1,..., x i 1, x i+1..., x d A} = 0 for all z > 0 and any relatively compact set A in [0, ] d 1 \{0}. This is still true for z = 0 under the additional condition that the set A is bounded away from {0}. Moreover, the proof of Theorem 3.2 of Kortschak and Albrecher 2009 justifies the fact that no mass is put by the measure µ over the line d i=1 x i = 1 and any neighborhood around.

8 8 For the second part of relation 2.6, I 2 t, we have I 2 t = t PrX k > t PrX 1 > t PrX 1 > t Pr S d > t 1 PrX k > tz PrX k > t dz t α 1 µ where in the last step above we have applied the relations 2.5 and PrX k > t µx : x k > 1 PrX 1 > t µx : x k > 1 x : d i=1 x i > 1, 2.8 due to 2.3, and made use of the Dominated Convergence Theorem as justified by Proposition 0.8 of Resnick Plugging 2.7 and 2.8 into 2.6 yields E [X k S d > t] C k t. The proof is complete. As a consequence of Theorem 2.1, we now express the capital allocations in terms of the VaR of the reference risk. The proof simply uses Lemma 2.1 and relation 2.5 and for this reason is omitted. Corollary 2.1. Under the conditions of Theorem 2.1, it holds for all 1 k d that E [X k S d > VaR q [S d ]] C k µ x 1/α : x i > 1 VaR q [X 1 ], where the constants C k are defined as in Theorem Gumbel case. The Gumbel case is further investigated in the presence of asymptotic dependence. i=1 Since we are only interested in risks with unbounded supports, all individual risks are assumed to have an infinite upper endpoint. It is well known see Embrechts et al., 1997 that if F MDAΛ, then there exists a positive, measurable function a such that F t + ats F t = e s 2.9 for any s R. In addition, the latter holds locally uniformly in s see Resnick, Recall that the auxiliary function a satisfies at = ot and is such that the relation a t + atx at = holds locally uniformly in x. Moreover, this auxiliary function can be chosen as the mean excess function of F, i.e., at = t F s F t ds. See Section 3.3 of Embrechts et al for more details. The next assumption is sufficient for our main results of this subsection.

9 Assumption 2.2. Let X = X 1,..., X d be a non-negative random vector with marginal distributions F 1,..., F d such that PrX 1 > t + atx 1,..., X d > t + atx d F 1 t := H G x exists for all x R d, where H G is assumed to be a non-degenerate function. Assumption 2.2 implies that X Pr 1 t,..., X d t at at F 1 t holds on [, ] d \ { } for the measure ν given by ν x 1, ] x d, ] := H G x, v ν 2.11 where { } =,...,. Moreover, the distribution functions are tail equivalent with F i t F 1 t = νx : x i > 0, 1 i d, 2.12 and, therefore, all belong to MDAΛ. One choice for the function a from Assumption 2.2 is given by the auxiliary function corresponding to the random variable X 1 as described in relation 2.9. In fact, all marginal distributions admit the same auxiliary function. We now show that the non-degeneracy of H G in Assumption 2.2 ensures that the random vector X has the asymptotic dependence property, which is a key property in describing the tail probability of S d see Section 4.1 of Klüppelberg and Resnick, Lemma 2.3. Under Assumption 2.2 the components of X = X 1,..., X d are pairwise asymptotically dependent. Proof. As in the proof of Lemma 2.2, we consider relation 1.2 for i, j = 1, 2. By 2.12 and 2.9 with F replaced by F 1, we have 9 F 2 t + ats F 1 t = νx : x 2 > 0e s. Fix some s large enough such that the right-hand side above is smaller than 1. Then it holds for all large t that F 2 t + ats < F 1 t, from which it follows that VaR F1 t[x 2 ] t + ats.

10 10 Analogously to the proof of Lemma 2.2, as q 1, or, equivalently, as t = VaR q [X 1 ], Pr F 2 X 2 > q F 1 X 1 > q Pr X 1 > VaR q [X 1 ], X 2 > VaR q [X 2 ] Pr X 1 VaR q [X 1 ] Pr X 1 > t, X 2 > VaR F1 t[x 2 ] Pr X 1 t Pr X 1 > t, X 2 > t + ats Pr X 1 t νx : x 1 > 0, x 2 > s > 0, where in the last step we used the fact Pr X 1 t Pr X 1 > t due to Corollary 1.6 of Resnick This proves relation 1.2 for i, j = 1, 2. Note 2.2. Due to Assumption 2.2, it holds that Pr S d > dt ν x : x i > 0 F 1 t, 2.13 which is a consequence of the vague convergence in 2.11 and Proposition 4.1 of Klüppelberg and Resnick Note that Assumption 2.2 allows us to withdraw the equal marginal distributions assumption from their proposition. Thus, i=1 Pr S d > t + a t/d ds Pr S d > t = e s 2.14 holds for any s R and relation 2.9 is satisfied by S d with an auxiliary function ãt := at/dd. The latter implies that S d MDAΛ, which gives that E [S d S d > t] t Relation 2.15 agrees with the next theorem, which provides the asymptotic expressions for the capital allocations for a multi-line insurance business of d asymptotically dependent risks that belong to MDAΛ. Theorem 2.2. Let X be a random vector satisfying Assumption 2.2. Then it holds for all 1 k d that E [X k S d > VaR q [S d ]] 1 d VaR q[s d ]. Proof. The first step of the proof is developed as t E [X k S d > dt] = + Pr X k > z S d > dt dz = I 1 t + I 2 t The second integral can be easily reduced to 0 t I 2 t t PrX k > z dz Pr S d > dt

11 The change of variables z = t + atv leads to t PrX k > z dz = at PrX k > t 0 Pr X k > t + atv dv = ot PrX k > t due to the Dominated Convergence Theorem, relations 2.9 and at = ot. Thus, the latter relation, together with 2.13 and 2.17, concludes that I 2 t = ot It only remains to investigate the first term of 2.16, which is equal to t+d 1ats t I 1 t = t + Pr X k z S d > dt dz t+d 1ats where s is a negative number. 0 = t I 11 t, s I 12 t, s, 2.19 Recall that a is a positive function. It holds for every s < 0 that I 11 t, s t as a result of 2.13 and Thus, Pr X k t + d 1ats S d > dt d Pr i=1,i k X i > d 1t d 1ats Pr S d > dt ν x : d i=1,i k x i > 0 ν x : e s, 2.20 d i=1 x i > 0 I 11 t, s sup s t = 0. As before, the change of variables z = t + d 1atv yields 0 I 12 t, s = d 1at Pr X k t + d 1atv S d > dt dv s 0 Pr d s i=1,i k X i > d 1t d 1atv dv d 1at Pr S d > dt d Pr i=1,i k X i > d 1t d 1at s Pr S d > dt ν x : d i=1,i k x i > 0 d 1at s ν x : = ot, 2.21 d i=1 x i > 0 due to at = ot and relations 2.13 and

12 12 A combination of relations 2.19, 2.20 and 2.21 yields that I 1 t t. Plugging this and 2.18 into 2.16 concludes the proof. Recall that every distribution F from MDAΛ with an infinite upper endpoint also has a rapidly varying tail see Embrechts et al., 1997, page 148, written as F R, characterized by { F tx 0, x > 1, F t =, 0 < x < 1. The following result is similar to Lemma 2.1 but for the rapid variation case: Lemma 2.4. Let X 1 and X 2 be two random variables with distribution functions F 1 and F 2 in R. If F1 lt F 2 t for some l > 0 then, for every m > 0, as q = 1 p 1, VaR q [X 1 ] lvar 1 mp [X 2 ] Proof. Let 0 < ε < 1 be arbitrarily fixed. As p 0, there is some function bp 0 such that F εlvar 1 mp [X 2 ] = bp F ε/2lvar 1 mp [X 2 ] bp F ε/2var 1 mp [X 2 ] bpmp. Similarly, as p 0, there is some function cp such that F 1 1 εlvar 1 mp [X 2 ] = cp F 1 1 ε/2lvar 1 mp [X 2 ] cp F 2 1 ε/2var 1 mp [X 2 ] cpmp. Thus, for all 0 < p < 1 sufficiently close to 0, we have F εlvar 1 mp [X 2 ] < p < F 1 1 εlvar 1 mp [X 2 ]. It follows that 1 εlvar 1 mp [X 2 ] VaR q [X 1 ] 1 + εlvar 1 mp [X 2 ]. By the arbitrariness of ε, this leads to relation Similar to the Fréchet case, Theorem 2.2 allows us to express the capital allocations in terms of the reference risk measure, VaR q [X 1 ]. Finally, relation 2.13, Lemma 2.4 and the fact that S d R imply the following result:

13 13 Corollary 2.2. Under the conditions of Theorem 2.2, as q = 1 p 1 it holds that E [X k S d > VaR q [S d ]] VaR 1 p/k [X 1 ], 2.23 where K = ν x : d i=1 x i > 0. Lemma 2.4 shows that VaR q [X 1 ] as a function of p is slowly varying as p 0 see also Proposition 0.8v of Resnick, Therefore, the right-hand side of 2.23 can be changed to VaR 1 mp [X 1 ] for every constant m > 0. However, in view of relation 2.13, the most rational choice for m should be m = 1/K. We have thus obtained an appealing expression for the CTE-based risk capital allocations for a multi-line insurance business consisting of asymptotically dependent risks that are not extremely heavy tailed. The fact that all risks belong to MDAΛ allows us to conclude that the marginal risk capitals under CTE are equal for conservative scenarios. It is interesting that this remains true without assuming exchangeable risks, a situation in which the allocations are obviously equal at any degree of safeness. 3. Main results under asymptotic independence There has been a particular interest in understanding the tail behavior of the sum of asymptotically independent random variables with heavy tails see Albrecher et al., 2006, Ko and Tang, 2008, Asmussen and Rojas-Nandayapa, 2008, Geluk and Tang, 2009, and Mitra and Resnick, A similar pattern for the CTE-capital allocations is expected. The results are less homogeneous under asymptotic independence, and various assumptions are provided in this case. The Fréchet and Gumbel cases exhibit fundamentally different behaviors for the capital allocations and, therefore, analyses are made separately Fréchet case. We first restrict our attention to a portfolio of risks with regularly varying tails. Two overlapping sets of assumptions are made, but neither of them is a consequence of the other. The first assumption is a natural extension of Assumption 2.1 under the asymptotic independence setting. Assumption 3.1. Let X be a non-negative random vector with marginal distributions F 1,..., F d such that PrX 1 > tx 1,..., X d > tx d F 1 t := µ I x 1, ] x d, ] exists for all x [0, ] d \ {0}, where µ I is a Radon measure such that µ I 0, ] d = 0.

14 14 Since the iting measure is Radon, Assumption 3.1 suggests that all marginal distributions are regularly varying tailed with possibly different indexes, α i, such that α 1 = 1 i n α i. If these indexes are not equal, then the measure µ I does not put mass on the vast majority of the boundary of its domain, [0, ] d \ {0}. In such a case, the problem becomes trivial and only components with indexes equal to α 1 have contributions to extreme events related to the aggregate portfolio. Therefore, without loss of generality it is further assumed that all marginal distributions belong to MDAΦ α, since otherwise the main results remain unchanged. Then, for each 1 i d, there exists a positive constant c i such that and F i t c i F1 t 3.1 Pr X 1 /t,..., X d /t PrX 1 > t v µ I 3.2 holds on [0, ] d \ {0}. Furthermore, the measure µ I puts mass only on the coordinate axes due to the fact that µ I 0, ] d = 0. Thus, for all z > 0, µ I x : x 1 = = x i 1 = x i+1 = = x d = 0, x i z, ] = c i z α. It is interesting to outline the link between Assumption 3.1 and asymptotic independence. Under Assumption 3.1, following the proof of Lemma 2.2 we can easily verify that the components of X are pairwise asymptotically independent. As for the inverse statement, for simplicity consider X = X 1, X 2. Assume that X 1 and X 2 belong to MDAΦ α, have strongly equivalent tails as in 3.1 and are asymptotically independent. For this case, the asymptotic independence is equivalent to Then for all positive x 1 and x 2 we have Thus, PrX 1 > tx 1, X 2 > tx 2 PrX 1 > t PrX 1 > tx 1, X 2 > tx 2 PrX 1 > t Pr X 2 > t X 1 > t = 0. Assumption 3.1 describes a situation in which PrX 1 X 2 > tx 1 tx 2 PrX 1 > t Pr S d > t F 1 t = o1. x α 1, x 1 > 0, x 2 = 0, = c 2 x α 2, x 1 = 0, x 2 > 0, 0, otherwise. c i ; 3.3 i=1

15 see Lemma 2.1 of Davis and Resnick 1996 or Proposition 7.3 of Resnick The first main result of this subsection is now given for a multi-line insurance business consisting of asymptotically independent risks with regularly varying tails. Theorem 3.1. Let X be a random vector satisfying Assumption 3.1. If X 1 R α with α > 1, then for all 1 k d we have E [X k S d > VaR q [S d ]] where the constants c i are given by 3.1. Proof. Similar to the proof of Theorem 2.1, E [X k S d > t] = t 0 α c k α 1 d i=1 c VaR q [S d ], i Pr X k > z, S d > t dz + Pr S d > t t PrX k > z Pr S d > t dz = I 1 t + I 2 t Now, for any 0 < z 1 we have PrX k > tz, S d > t PrX 1 > t = PrX k > t PrX 1 > t + Prtz < X k t, S d > t PrX 1 > t c k + µ I x : z < x k 1, x i > 1 = c k, i=1 where Proposition A2.12 of Embrechts et al and the vague convergence property { in 3.2 are applied over the µ I -negligible set x : z < x k 1, } d i=1 x i > 1. The latter, relation 3.3 and the Dominated Convergence Theorem yield that I 1 t = t 1 0 PrX k > tz, S d > t dz PrS d > t As in the proof of Theorem 2.1, the second term in 3.4 satisfies Putting together 3.4, 3.5 and 3.6, we obtain c k c k d i=1 c i t. 3.5 I 2 t 1 α 1 d i=1 c t. 3.6 i E [X k S d > t] α c k α 1 d i=1 c t, i which completes the proof. An even simpler expression for the CTE-capital allocations is given without proof in the next corollary, which is a consequence of Theorem 3.1, Lemma 2.1 and relation 3.3.

16 16 Corollary 3.1. Under the conditions of Theorem 3.1, it holds for all 1 k d that 1 E [X k S d > VaR q [S d ]] α α 1 α 1 c k c i VaR q [X 1 ], where the constants c i are given by 3.1. Our next assumption is motivated by the work of Asimit and Badescu Assumption 3.2. Let X be a nonnegative random vector with marginal distribution functions F 1,..., F d. i=1 Assume that there are some measurable and bounded functions h i : 0, 0, such that, for distinct i, j {1,..., d}, the relation Pr X j > t X i = x F j th i x 3.7 holds uniformly for 0 x <. In addition, for d 3, it is also assumed that Pr X j > t, X k > t X i = x = o Fj t + F k t h i x 3.8 holds uniformly for 0 x < for distinct i, j, k {1,..., d}. The uniformity of relation 3.7 is understood as sup Pr X j > t X i = x 1 F j th i x = 0 0 x< while the uniformity of relation 3.8 as sup 0 x< Pr X j > t, X k > t X i = x Fj t + F k t h i x The recent work of Li et al discussed the verification of the uniformity of relation 3.7 and the boundedness of the functions h i. Clearly, the uniformity property implies that E [h i X] = 1. For the dependence structures discussed by Asimit and Badescu 2010 and Li et al with strongly equivalent tails, both Assumptions 3.1 and 3.2 are satisfied. It is not difficult to prove that a bivariate random vector with strongly equivalent tails and dependence structure given by the Marshall-Olkin copula see, e.g., Nelsen, 1999, page 46 satisfies only Assumption 3.1. It will be later seen in Note 3.1 that within a particular yet fairly general setting, Assumption 3.2 may provide a refinement for the capital allocations given by Theorem 3.1. Therefore, neither of Assumptions 3.1 and 3.2 is dominated by the other one. We first establish that random vectors satisfying Assumption 3.2 have pairwise asymptotically independent components. = 0.

17 17 Lemma 3.1. Relation 3.7 implies that X i and X j are asymptotically independent. Proof. By definition, Pr F j X j > q F i X i > q Pr F ix i > q, F j X j q Pr F i X i > q VaR q[x = j Pr F ] i X i > q X j = x Pr X j dx. Pr F i X i > q Hence, by the uniformity of 3.7, the right-hand side above converges to h j x Pr X j dx = 0. q 1 VaR q[x j ] Therefore, X i and X j are asymptotically independent. The next result is crucial for developing our second main result of this subsection. Lemma 3.2. Let X be a random vector satisfying Assumption 3.2 such that each component has a regularly varying tail. Then the relation Pr i=1,i k X i > t X k = x h k x holds uniformly for 0 x < for all 1 k d. F i t i=1,i k Proof. The proof below proceeds for d 3 and k = 1, but it can easily be adjusted so that it is valid for d 2 and k 1. We first derive an asymptotic lower bound. Since each X i is nonnegative, by Bonferroni s inequality we have, uniformly for 0 x <, Pr X i > t X 1 = x d Pr X i > t X 1 = x Pr X i > t X 1 = x 2 i j d h 1 x F i t, Pr X i > t, X j > t X 1 = x

18 18 where in the last step we applied both 3.7 and 3.8. Next, we derive the corresponding asymptotic upper bound. For an arbitrarily fixed constant 0 < ε < 1, we have Pr X i > t X 1 = x d d Pr X i > 1 εt X 1 = x + Pr X i > t, X i 1 εt X 1 = x = I 1 t, ε + I 2 t, ε. 3.9 Let α i > 0 be the regularly varying index of F i. Then relation 3.7 gives that I 1 t, ε Pr X i > 1 εt X 1 = x h 1 x j=2 1 ε α i Fi t Similarly, by 3.8, d I 2 t, ε = Pr X i > t, X j > t d d 1, X i 1 εt X 1 = x j=2 Pr X i > t, X j > t d 1, X j 1 εt X 1 = x = o Pr j=2 2 i j d Pr h 1 x,i j X i > X i > εt, X j > t d 1 X 1 = x εt d 2, X j > t d 1 X 1 = x F i t Substituting 3.10 and 3.11 into 3.9 and noticing the arbitrariness of ε, we obtain Pr X i > t X 1 = x h 1 x F i t. The proof is complete. Now we are ready to state the second main result of this subsection. Theorem 3.2. Let X be a random vector satisfying Assumption 3.2 such that each component X i has a regularly varying tail with index α i > 1. In addition, if each h i is a regularly varying function, then Pr S d > t F i t 3.12 i=1

19 19 and E [X k S d > t] E [X kh k X k ] d F i=1,i k i t + t F k t α k d F i=1 i t α k All examples presented in the work of Li et al give the regular variation property of the functions h i. If relation 3.1 holds for each 1 i d, then one may use Theorem 3.2 to recover Corollary 3.1. Note 3.1. Let us assume that there exist finite constants c i, 1 i d, such that F i t c i F1 t for all 1 i d 1 and F d t t F 1 tc d. Thus, α i = α 1 for all 1 i d 1 and α d = α 1 1. Suppose α 1 > 2 so that X d has a finite mean as well. In addition, Pr S d > t F d t, which together with Lemma 2.1, implies that VaR q [S d ] VaR q [X d ]. Some algebraic manipulations lead to for all 1 k d 1 and E [X k S d > t] = E [X k h k X k ] + α 1 c k α 1 1 c d α d E [X d S d > t] t α d 1. Obviously, these relations generate the asymptotic capital allocation estimates for all 1 k d 1 and E [X k S d > VaR q [S d ]] = E [X k h k X k ] + α 1 c k α 1 1 c d E [X d S d > VaR q [S d ]] On the contrary, in this particular setting, Theorem 3.1 gives that α d α d 1 VaR q X d E [X k S d > VaR q [S d ]] = o VaR q [X d ], 1 k d 1, while the d th capital allocation is given as in relation The advantage of using Theorem 3.2 over Theorem 3.1 becomes transparent. Proof. The proof is provided only for k = 1 as the extensions to all other values of k are obvious. Similar to the proof of Theorem 2.1, E [X 1 S d > t] = = x Pr X 1 dx S d > t 0 x Pr S 0 d X 1 > t x X 1 = x F 1 dx Pr S 0 d X 1 > t x X 1 = x F 1 dx. 3.15

20 20 We first investigate the numerator from By Lemma 3.2, for every ε > 0, there is some t 0 > 0 such that, for all t t 0 and x > 0, 1 εh 1 x F i t Pr S d X 1 > t X 1 = x 1 + εh 1 x F i t According to this constant t 0, for all t t 0 we split the integral into three parts as 3 t t0 t I i t = x Pr S d X 1 > t x X 1 = x F 1 dx i=1 Clearly, and 0 + t t 0 + I 2 t t I 3 t = t t xf 1 dx 3.18 t t 0 xf 1 dx = o I 3 t By 3.16, I 1 x 1 ε d t t0 xh 0 1 x F 1 + ε i t xf 1 dx Define a distribution F by F dx = xh 1 x E [X 1 h 1 X 1 ] F 1dx, where E [X 1 h 1 X 1 ] is a finite positive normalizing constant. Clearly, F has a regularly varying tail as well. Thus, by Lemma of Embrechts et al see also Proposition 1.2 or Theorem 2.1 of Cai and Tang, 2004, t t0 xh 1 x F i t xf 1 dx 0 t = E [X 1 h 1 X 1 ] F i t xf dx 0 t t t 0 = 1 + o1 E [X 1 h 1 X 1 ] Fi t + F t E [X 1 h 1 X 1 ] F t o1 F t = 1 + o1 E [X 1 h 1 X 1 ] F i t o1 F t. Substituting this into 3.20, then substituting 3.18, 3.19 and 3.20 into 3.17 and noticing the arbitrariness of ε, we obtain 3 I i t E [X 1 h 1 X 1 ] i=1 E [X 1 h 1 X 1 ] F i t + t α 1 xf 1 dx F i t + t α 1 1 F 1 t. 3.21

21 Using a similar argumentation and keeping in mind E [h 1 X 1 ] = 1, one sees that the denominator on the right-hand side of 3.15 can be approximated as 0 Pr S d X 1 > t x X 1 = x F 1 dx i=1 21 F i t, 3.22 which is equivalent to By substituting 3.21 and 3.22 into 3.15, we obtain 3.13 and the proof is complete Gumbel case. In the second part of this section, some asymptotic results are obtained for the case in which the risks belong to MDAΛ and are unbounded and asymptotically independent. The class MDAΛ essentially contains all distributions with rapidly varying tails. Moderately heavy-tailed distributions such as lognormal and Weibull as well as light-tailed distributions such as exponential and gamma are members of MDAΛ. For more information, see Embrechts et al It has been seen that asymptotic capital allocations are closely related to the tail behavior of the aggregate risk. A moderately heavy-tailed distribution from the Gumbel family is also known as a distribution from the subexponential class S for which two independent, identically distributed and nonnegative copies, X 1 and X 2, satisfy PrX 1 + X 2 > t 2 PrX 1 > t. For a distribution function F MDAΛ, conditions on its auxiliary function a under which F S are available in Goldie and Resnick 1988 and Hashorva et al The mainstream study of the tail probability of the sum of asymptotically dependent and asymptotically independent random variables has focused on the subexponential case. Mitra and Resnick 2009 investigated the problem from a different perspective. They considered dependent random variables from MDAΛ but not necessarily subexponential. Following the work of Mitra and Resnick 2009 we propose a set of conditions below: Assumption 3.3. Let X be a positive random vector with marginal distributions F 1,..., F d. Assume that F 1 MDAΛ with an auxiliary function a as defined in 2.9. In addition, for each 1 i d, there exists a non-negative constant c i such that F i t/ F 1 t c i. Furthermore, assume that, for all 1 i j d, and Pr X i > t, X j > atx F 1 t Pr X i > L ij t, X j > L ij t F 1 t = 0 for all x > 0, 3.23 = 0 for some L ij >

22 22 If c i > 0 for all 1 i d, then Assumption 3.3 indeed describes an asymptotic independence case. For simplicity, we only verify this for d = 2. Note that relation 3.23 with i, j = 1, 2 trivially implies relation 3.25 below since at = ot. Lemma 3.3. Let X = X 1, X 2 be a random vector with marginal distributions F 1 and F 2. Assume that F 1 MDAΛ with an auxiliary function a, that F 2 t/ F 1 t c 2 for some constant c 2 > 0 and that the relation Pr X 1 > t, X 2 > tx F 1 t = holds for some 0 < x < 1. Then X 1 and X 2 are asymptotically independent. Proof. Note that both F 1 and F 2 belong to MDAΛ since c 2 > 0. Thus, for every s R, F 2 t + ats F 1 t = c 2 e s. Fix some s small enough such that the right-hand side above is larger than 1. Then it holds for arbitrarily fixed y x, 1 and for all large t that F 2 tx F 2 ty + atys > F 1 ty. It follows that VaR F1 ty[x 2 ] > tx. Analogously to the proof of Lemma 2.2, as q 1, or, equivalently, as t = VaR q [X 1 ], Pr F 2 X 2 > q F 1 X 1 > q Pr X 1 VaR q [X 1 ], X 2 VaR q [X 2 ] Pr X 1 > VaR q [X 1 ] Pr X 1 t, X 2 VaR F1 ty[x 2 ] F 1 t Pr X 1 = t + Pr X 1 > t, X 2 > tx F 1 t 0, where in the last step we used the fact Pr X 1 = t = o F1 t due to Corollary 1.6 of Resnick Therefore, X 1 and X 2 are asymptotically independent. Corollary 2.2 of Mitra and Resnick 2009 shows that, under Assumption 3.3, Pr S d > t F 1 t c i This result aligns with other asymptotic results for the sum of asymptotically independent subexponential random variables and provides a conspicuous step ahead in understanding the extreme behavior of the sum of dependent random variables for the Gumbel case. The next lemma provides useful information for the proof of Theorem 3.3 below. i=1

23 23 Lemma 3.4. Let X be a random vector satisfying Assumption 3.3. y R, Then, for every and Pr X k > atx, S d X k > t + aty F 1 t PrX k atx, S d > t + aty F 1 t = = { e y d i=1,i k c i, x 0, 0, x > 0, { e y d i=1,i k c i, x > 0, 0, x Proof. Our first remark is that relation 3.27 holds for non-positive values of x due to In addition, the proof of Corollary 2.2 of Mitra and Resnick 2009 shows that, for every x > 0, Pr X k > atx, S d X k > t = o F1 t Now, relation 2.10 implies that, for arbitrarily fixed 0 < ε < 1 and all large t, Thus, for any x > 0 and y R, 1 εat a t + aty 1 + εat. Pr X k > atx, S d X k > t + aty F 1 t Pr X k > a t + aty x 1+ε, S d X k > t + aty F 1 t Pr X k > a t + aty x 1+ε, S d X k > t + aty F 1 t + aty e y 0, which is a consequence of relations 2.9 and Therefore, 3.27 is proved. The proof of the second constituent is facilitated by some vague convergence properties. Let M < y be fixed. Then, by relation 3.27, Pr X k, S d X k t at at F 1 t holds on [, ] [M, ] for the measure µ k given by µ k dx, dy := i=1,i k v µ k c i e y ɛ 0 dxdy, where ɛ 0 denotes the Dirac measure. It is useful to note that in order to fully justify the latter vague convergence property, one should obtain similar results to 3.27 for other compact sets, which can be simply verified since 3.26 holds. Note that 3.28 is trivial for non-positive values of x and for this reason we assume x > 0. Denote A := {x 1, x 2 : x 1 x, x 1 + x 2 > y}. The measure µ k puts mass over the

24 24 set A only on the line {x 1 = 0, x 2 > y} and, therefore, µ k A = 0. Thus, Proposition A2.12 of Embrechts et al allows us to generate the conclusion PrX k atx, S d > t + aty Pr = F 1 t F 1 t The proof is complete. X k, S d X k t at at = µ k A = e y i=1,i k Now, we are able to provide the last main result of this section. c i. A Theorem 3.3. Let X be a random vector satisfying Assumption 3.3. Then the relation E [X k S d > VaR q [S d ]] = C k + o1 VaR q [S d ] 3.30 holds for all 1 k d, where C k = c k d i=1 c i 1. Note that the constant C k above can take value 0 for which reason we did not write relation 3.30 as an equivalence. Proof. Similar to the proof of Theorem 2.2, at E [X k S d > t] = + 0 t at + = I 1 t + I 2 t + I 3 t. t Pr X k > z S d > t dz Now, I 1 t at = ot. The change of variables z = t + ats yields that I 3 t = at 0 Pr X k > t + ats S d > t ds at Pr X k > t + ats ds Pr S d > t 0 1 c i at e s ds i=1 = ot, as a result of relations 2.9 and 3.26, the Dominated Convergence Theorem and the fact that at = ot. It remains to justify Some useful bounds for I 2 t are as follows: 0 I 2 t = C k + o1 t t at Pr X k > t S d > t I 2 t t Pr X k > at S d > t. 3.32

25 The left-hand side above is equal to t at Pr X k > t S d > t = t at PrX k > t PrS d > t = C k + o1 t, due to relation A similar argumentation and relation 3.28 help us to find the asymptotic behavior of the right-hand side of 3.32, as Pr X k > at S d > t = C k. 25 Thus, relation 3.31 holds, which completes the proof. As before, Theorem 3.3 indicates that capital allocations can be related only to the reference risk measure, VaR q [X 1 ]. Finally, relation 3.26, Lemma 2.4 and the fact that S d R conclude the following result: Corollary 3.2. Let X be a random vector satisfying Assumption 3.3. Then, as q = 1 p 1, E [X k S d > VaR q [S d ]] = C k + o1 VaR 1 p/ d i=1 c i [X 1]. The same as in Corollary 2.2, the right-hand side above can be changed to VaR 1 mp [X 1 ] for every constant m > 0, but the most rational choice for m should be m = 1/ d i=1 c i. 4. Examples We first provide some examples under which Assumptions 2.1, 2.2 and 3.1 are satisfied. We start out by giving some information regarding the copula concept further details can be found in Nelsen, It is well known that the dependence structure associated with the distribution of a random vector can be characterized in terms of a copula. A bivariate copula is a two-dimensional distribution function defined on [0, 1] 2 with uniformly distributed marginals. Due to Sklar s theorem see Sklar, 1959, if X 1, X 2 has continuous marginal distributions, then there exists a unique copula, C, such that PrX 1 x 1, X 2 x 2 = C PrX 1 x 1, PrX 2 x 2. Similarly, the survival copula, Ĉ, is defined as the copula corresponding to the joint tail function satisfying Note that every copula satisfies PrX 1 > x 1, X 2 > x 2 = Ĉ PrX 1 > x 1, PrX 2 > x 2. u 1 + u Cu 1, u 2 u 1 u 2.

26 26 Recall that W u 1, u 2 := u 1 + u and Mu 1, u 2 := u 1 u 2 are known as the comonotonic and counter-monotonic copulae, which respectively correspond to the strongest and weakest possible dependence structures that may occur between two random variables. The comonotonic respectively, counter-monotonic dependence structure arises when one random variable is a non-decreasing respectively, non-increasing function of the other. An appealing class of copulae is the Archimedean one. By definition, an Archimedean copula C is given by Cu 1, u 2 = ϕ 1 ϕu 1 + ϕu 2, where ϕ : [0, 1] [0, ], called the generator of Cu 1, u 2, is a strictly decreasing and convex function with 0 < ϕ 0 and ϕ 1 = 0. The function ϕ 1 is the pseudoinverse of ϕ, and by convention ϕ 1 t = 0 if t > ϕ0. A strict generator satisfies ϕ0 =. Juri and Wüthrich 2003 developed a set of sufficient conditions on the generator of an Archimedean copula under which the joint concomitant extreme events are characterized. That is, if then ϕ1 xu u 0 ϕ1 u = xβ, 1 < β <, 4.1 Ĉux 1, ux 2 u 0 u = x 1 + x 2 x β 1 + x β 2 1/β. Now, under the assumption that both risks belong to MDAΦ α and have strongly equivalent tail probabilities, with c = c F > 0, we have PrX 1 > tx 1, X 2 > tx 2 PrX 1 > t PrX 2 > t PrX 1 > t = c 4.2 1/β = x α 1 + c F x α 2 x αβ 1 + c β F x αβ As expected, if condition 4.1 is satisfied and both risks belong to MDAΛ such that relation 4.2 holds with c = c G > 0, then PrX 1 > t + atx 1, X 2 > t + atx 2 PrX 1 > t = e x 1 + c G e x 2 e βx 1 + c β G e βx 2 1/β, where the auxiliary function a is defined as in 2.9 corresponding to X 1. Similarly, assume now that the survival copula is Archimedean with a strict generator satisfying u 0 ϕxu ϕu = x β, 0 β, 4.4

27 27 where β = implies that Now, Ĉux 1, ux 2 u 0 u = { ϕxu, x < 1, u 0 ϕu = 0, x > 1. 0, β = 0, 1/β x β 1 + x β 2, 0 < β <, for all x1, x 2 > 0. x 1 x 2, β =. 4.5 The work of Juri and Wüthrich 2003 justifies the above for 0 < β <, while Lemma 4.1 below proves 4.5 for β = 0 and β =. The same paper shows that Assumptions 2.1, 2.2 and 3.1 are satisfied for finite positive values of β, but one can easily extend this to the remaining cases. Specifically, if relation 4.2 holds with c = c F > 0 and X 1 MDAΦ α, then and PrX 1 > tx 1, X 2 > tx 2 PrX 1 > t PrX 1 > tx 1, X 2 > tx 2 PrX 1 > t = 1/β x αβ 1 + c β F xαβ 2, if 0 < β <, x α 1 c F x α 2, if β =, = 0, otherwise. x α 1, if β = 0, x 1 > 0, x 2 = 0, c F x α 2, if β = 0, x 1 = 0, x 2 > 0, In addition, if relation 4.2 holds with c = c G > 0 and X 1 MDAΛ, then PrX 1 > t + atx 1, X 2 > t + atx 2 1/β e βx 1 + c 2 β G = eβx, if 0 < β <, PrX 1 > t e x 1 c G e x 2, if β =. 4.6 Finally, note that a non-strict generator for a survival Archimedean copula gives 4.6 since it is certain that concomitant extreme events are impossible to occur. A comprehensive list of copulae that satisfy the conditions explained in 4.1 and 4.4 is included in Charpentier and Segers The vast majority of Archimedean copulae that satisfy 4.4 with β = 1, together with relation 4.2 with c = c F > 0 and X 1 MDAΦ α, gives 4.6, which designs the framework defined in Assumption 3.1. Additional examples outside the Archimedean world can be found in Asimit and Jones 2008 and Kortschak and Albrecher The next lemma develops some asymptotic results that are useful in justifying the extreme behavior of a bivariate random vector with an Archimedean survival copula.

28 28 Lemma 4.1. Let C, be a copula such that the survival copula is Archimedean with a strict generator satisfying 4.4. Then Ĉux 1, ux 2 u 0 u = { 0, β = 0, x 1 x 2, β =, 4.7 holds for any positive x 1 and x 2, and u 0 Ĉ u, fu fu = is true for any 0 < β and any positive measurable function fu = ou. Proof. We first prove relation 4.7 for β = 0. Recall that 4.4 holds locally uniformly. Therefore, for any 0 < ε < 1/2, x 1, x 2 > 0 and u sufficiently small, we have ϕux 1 + ϕux 2 > 21 εϕu. Now, ϕ0 = implies that ϕ 1 is rapidly varying at see Proposition 0.8v, Resnick, Thus, Ĉux 1, ux 2 u = ϕ 1 ϕux 1 + ϕux 2 u ϕ 1 21 εϕu ϕ 1 ϕu 0, u 0. The case in which β = follows in a similar manner. Proposition 0.8v of Resnick 1987 implies that ϕ 1 is slowly varying at. As before, we obtain that Ĉux 1, ux 2 u ϕ 1 2ϕ ux 1 ux 2 ϕ 1 ϕ ux 1 ux 2 x 1 x 2 x 1 x 2, u 0, 4.9 since ϕux 1 ϕux 2 ϕ ux 1 ux 2 holds for all u, x 1, x 2 > 0. The upper bound is straightforward since Cx 1, x 2 Mx 1, x 2, which concludes the first part of this lemma. To verify relation 4.8, notice that fu = ou yields Ĉ u, fu fu u fu fu 1. First assume that 0 < β <. For any ε > 0 and u sufficiently small, we have ϕ fu/ε > ϕu. Thus, Ĉ u, fu fu ϕ 1 ϕ fu + ϕ fu ε fu ε β + 1 1/β, u 0, due to 4.5. By taking ε 0, the lower and upper bounds coincide in this setting. For β =, the lower bound follows by a similar reasoning as used in relation 4.9, which concludes 4.8. The proof is complete.

29 Examples regarding Assumption 3.2 have been already discussed. Multiple examples regarding Assumption 3.3 have been provided by Mitra and Resnick We now indicate a wide class of distributions under which conditions required by Assumption 3.3 are verified. We consider an asymmetric class of copulae studied by Khoudraji 1995 see also Genest et al., If C, is an Archimedean copula then C k,l u 1, u 2 := u 1 k 1 u 1 l 2 Cu k 1, u l 2, k, l 0, 1, 4.10 defines another copula, which we call a transformed asymmetric Archimedean copula. It can be easily seen that for any dependence structure, the copula C k,l, describes an asymptotically independent scenario in the upper tail. Proposition 4.1. Let X 1, X 2 be a bivariate non-negative random vector whose marginal distributions are identical to F MDAΛ with auxiliary function a as described in 2.9. The survival copula is assumed to be given by a transformed asymmetric Archimedean copula C k,l, as defined in i If ϕ is a strict generator satisfying 4.4 with 0 < β and the distribution F is such that F b xat F c t = { 0, b > c,, b c, holds for all b, c, x > 0, then conditions 3.23 and 3.24 hold ii If ϕ is a non-strict generator, then conditions 3.23 and 3.24 hold for any distribution function F MDAΛ with at. The assumption of a non-strict generator can be easily relaxed. Specifically, conditions 3.23 and 3.24 still hold whenever PrX 1 > t 0, X 2 > t 0 = 0 for some t 0 > 0 together with the compulsory condition of strongly equivalent tails with an unbounded auxiliary function. This is the case if the underlying survival copula is given by an Archimedean copula or a transformed asymmetric Archimedean copula with non-strict generators. The counter-monotonic dependence structure reflects a similar extreme behavior. Proof. A non-strict generator excludes joint extreme events with probability one, which together with the fact that a is unbounded, indicates that, for any x 1, x 2, L > 0 and all large t, Pr X 1 > t, X 2 > x 2 at = Pr X 1 > x 1 at, X 2 > t = Pr X 1 > Lat, X 2 > Lat = 0. Evidently, conditions 3.23 and 3.24 are satisfied in this case.

30 30 It is further assumed that ϕ is a strict generator. We start out the verification of condition 3.23 by noticing that PrX 1 > x 1, X 2 > x 2 = F 1 k x 1 F 1 l x 2 Ĉ F k x 1, F l x 2 and that Ĉ F k t, F l xat { F l xat, l > k, F k t, l k, where the latter is a direct implication of relations 4.8 and Thus, { F 1 k t Pr X 1 > t, X 2 > xat F xat = o F t, l > k, F t F 1 l xat = o F t, l k, which implies To verify condition 3.24, without loss of generality we assume k l. If k < l then F l Lat = o F k Lat for any L > 0, which, together with 4.8 and 4.11, gives Pr X 1 > Lat, X 2 > Lat = F 2 k l Lat Ĉ F k Lat, F l Lat F 2 k Lat If l = k then = o F t. Pr X 1 > Lat, X 2 > Lat = F 2 2k Lat Ĉ F k Lat, F k Lat 2 1/β F 2 k Lat = o F t, where the second step is due to 4.5 and the last step due to Thus, the proof is complete. Two moderately heavy-tailed distributions satisfy the sufficient condition defined in The first example is the lognormal distribution with parameters µ R and σ > 0, F x = F log x µ SN, x > 0, σ where F SN denotes the distribution function of the standard normal distribution. The auxiliary function is given by ax = σ2 x log x µ see Embrechts et al., 1997, page 150. The

31 31 second example is a distribution function F with a tail { e log x γ, x > 1, F x = 0, x 1, x where γ > 1. Its auxiliary function is given by ax =. For these two examples, γlog x γ 1 the verification of 4.11 is straightforward and therefore is omitted. We cannot draw a conclusion that all distributions from the intersection MDAΛ S satisfy the requirements of Proposition 4.1. Recall that a Weibull distribution has a tail F x = e cxτ, x 0, c > 0, 0 < τ < 1, with an auxiliary function ax = c 1 τ 1 x 1 τ see Embrechts et al., 1997, page 150. Hence, it holds for all k, l, x > 0 that F k xat F l t =. 5. Simulation study and numerical results We perform a simulation study on the results derived in Corollary 2.1. A portfolio of two risks is considered and the individual loss random variables, X 1 and X 2, are Pareto distributed with distribution function F x; λ, α = x λ α, x 0, where λ equals 100, 000 for X 1 and 150, 000 for X 2, while α is the same for both X 1 and X 2 and it will be assigned values 2.5 and 3. The portfolio dependence structure is assumed to be given by the Gumbel copula { Cu 1, u 2 = exp log u 1 β + log u 2 β } 1/β, β 1. This copula belongs to the Archimedean family with a strict generator ϕu = log u β. Clearly, the Gumbel copula has the asymptotic dependence property for β values greater than 1 since relation 4.1 is satisfied. The measure µ in relation 2.4 is defined with H F calculated in 4.3 and relation 4.2 holds with c = c F = 3/2 α. The parameter β is chosen to be 2, 3 and 5. Note that the strength of dependence for the Gumbel copula increases as β increases. The asymptotic constants appearing in Corollary 2.1, C 1 and C 2, are numerically computed by using the formulae C 1 + C 2 = α/α 1 and C 1 = µ x α 1 0 1, x 2 : x 1 > z, x 1 + x 2 > 1 dz. µ x 1, x 2 : x 1 + x 2 > 1