By Bikramjit Das Singapore University of Technology and Design and. By Vicky Fasen-Hartmann Karlsruhe Institute of Technology

Transcription

1 CONDITIONAL EXCESS RISK EASURES AND ULTIVARIATE REGULAR VARIATION By Bikramjit Das Singapore University of Technology and Design and By Vicky Fasen-Hartmann Karlsruhe Institute of Technology Conditional excess risk measures like arginal Expected Shortfall and arginal ean Excess are designed to aid in quantifying systemic risk or risk contagion in a multivariate setting. In the context of insurance, social networks, and telecommunication, risk factors often tend to be heavy-tailed and thus frequently studied under the paradigm of regular variation. We show that regular variation on different subspaces of the Euclidean space leads to these risk measures exhibiting distinct asymptotic behavior. Furthermore, we elicit connections between regular variation on these subspaces and the behavior of tail copula parameters extending previous work and providing a broad framework for studying such risk measures under multivariate regular variation. We use a variety of examples to exhibit where such computations are practically applicable. 1. Introduction. In the domain of environment, insurance, finance, social networks and telecommunication, one often encounters risk factors which are heavy-tailed in nature; which means values further away from the mean have a relatively high probability of occurring than for example for exponentially-tailed distributions like normal or exponential; see Anderson and eerschaert (1998), Crovella et al. (1999), Embrechts et al. (1997), cneil et al. (25), Smith (23) for details. The joint behavior of such multi-dimensional heavy-tailed random variables are often studied under the framework of multivariate regular variation (RV); see Bingham et al. (1989), Resnick (27). In this paper we study the asymptotic behavior of risk measures pertaining to non-negative bivariate random vectors under the paradigm of RV. For a single risk factor, a popular risk measure is the Expected Shortfall (ES), also known as Conditional Tail Expectation (CTE) and Tail-Value-at-Risk (TVaR). It is widely used in practice and also incorporated in the regulatory frameworks of Basel III for banks and Solvency II for insurances. In systems with more than one variable, it is of interest to judge the risk behavior of one component given a high risk or stress in the others. Conditional excess risk measures like arginal Expected Shortfall (ES) and arginal ean Excess (E) are often useful in such cases and have been studied recently to ascertain various aspects of their tail behavior; see Cai and usta (217), Cai et al. (215), Das and Fasen-Hartmann (218), Hashorva (218). The ES is well-known in many contexts, and has been especially B. Das gratefully acknowledges support from OE Tier 2 grant OE217-T and hospitality of the Department of athematics, Karlsruhe Institute of Technology during his visit in 218. AS 2 subject classifications: Primary 6F1, 6G5, 6G7 Keywords and phrases: copula models, expected shortfall, heavy-tails, hidden regular variation, mean excess, multivariate regular variation, systemic risk 1

2 2 B. DAS AND V. FASEN-HARTANN proposed for measuring systemic risk; see Acharya et al. (21), Brownlees and Engle (215), Zhou (21). In order to define the above risk measures, recall that for a random variable Z R and p (, 1) the Value-at-Risk (VaR) at level p is the quantile function VaR 1 p (Z) = inf{x R : Pr(Z > x) p} = inf{x R : Pr(Z x) 1 p}. Note that smaller values of p lead to higher values of VaR 1 p. Let Z = (Z 1, Z 2 ) [, ) 2 denote the risk exposure of a financial institution, and Z 1 and Z 2 are the marginal risks of two risk factors. For studying the expected behavior of one risk, given that the other risk is high we particularly look at the following two risk measures: For E Z 1 < the arginal Expected Shortfall (ES) at level p (, 1) is defined as (1) ES(p) = E{Z 1 Z 2 > VaR 1 p (Z 2 )}, and the arginal ean Excess (E) at level p (, 1) is defined as (2) E(p) = E{(Z 1 VaR 1 p (Z 2 )) + Z 2 > VaR 1 p (Z 2 )}. Clearly the measures are not symmetric in their components in general and we consider the second component of Z, in this case Z 2 to be the conditioning variable. ES represents the expected shortfall of Z 1 given that Z 2 is higher than its Value-at-Risk at level 1 p, whereas E represents the expected excess of risk Z 1 over the Value-at-Risk of Z 2 at level 1 p given that the value of Z 2 is already greater than the same Value-at-Risk; see Das and Fasen- Hartmann (218) for details where the measure E is also defined. Note that the measure ES(p) is equal to the Expected Shortfall (ES) if Z 1 Z 2. In this context, we also investigate a few other extensions of the Expected Shortfall measure: (3) ES + (p) = E{Z 1 Z 1 + Z 2 > VaR 1 p (Z 1 + Z 2 )}, ES min (p) = E{Z 1 min(z 1, Z 2 ) > VaR 1 p (min(z 1, Z 2 ))}, ES max (p) = E{Z 1 max(z 1, Z 2 ) > VaR 1 p (max(z 1, Z 2 ))}, see Cai and Li (25), Cousin and Di Bernardino (214). The idea is that Z 1 + Z 2 is the aggregate risk of the institution, and min(z 1, Z 2 ) and max(z 1, Z 2 ) are the extremal risks. The risk measure ES + is associated with the Euler allocation rule; see (cneil et al., 25, Section 6.3). Further interpretations of these risk measures in finance and insurance are elaborated in Cai and Li (25). In a bivariate set-up, these conditional risk measures tend to have different limit behavior depending on how their joint tails behave. To this end, for a bivariate random vector Z = (Z 1, Z 2 ) we define asymptotic tail independence (in the upper tails) as (4) lim Pr(Z 1 > FZ p 1 (1 p) Z 2 > FZ 2 (1 p)) = where F Z i (1 p) = inf{x R : F Zi (x) 1 p} is the generalized inverse of the distribution function F Zi of Z i. Hence, the presence of asymptotic tail independence among Z 1 and Z 2 implies that it is highly unlikely for the two random variables to take extreme values together. This phenomenon aptly noted by Sibuya (196) more than half a century back, especially in the context of the very popular and useful bivariate normal distribution has been a source of

3 REGULAR VARIATION AND RISK EASURES 3 intrigue and further research by many. It has lead to the notions of tail dependence coefficient (Ledford and Tawn, 1997) and hidden regular variation (Resnick, 22). The asymptotic tail behavior of ES and its statistical inference was discussed under the assumption of regular variation and asymptotic tail dependence in Cai et al. (215). It has been observed that under certain tail conditions on (Z 1, Z 2 ) ES(p) K 1 E(p) K 2 VaR 1 p (Z 1 ) as p, for some constants K 1, K 2 >. Interestingly, in the asymptotically tail independent case these risk measures may have different rates of convergence or even converge to a constant, e.g., for independent random variables Z 1, Z 2 we have ES(p) = E(Z 1 ). For a hidden regularly varying random vector Z = (Z 1, Z 2 ) the asymptotic behavior of E and ES has been investigated in Das and Fasen-Hartmann (218), furthermore, consistent estimators for these risk measures based on methods from extreme value theory have been proposed; for asymptotic normality of ES, see Cai and usta (217). The asymptotic limits for E and ES are computed for multivariate Gaussian risks by Hashorva (218). Such risk metrics have also been studied in a time-series context in Kulik and Soulier (215), and under a copula-framework in Hua and Joe (212, 214). On the other hand, the asymptotic behavior of the risk measures defined in (3) are not particularly well-studied except in the heavy-tailed asymptotically tail dependent case; see Hua and Joe (211b), Joe and Li (211), Zhu and Li (212). Explicit formulas for ES + are given in Chiragiev and Landsman (27) for multivariate Pareto distributions, in Cai and Li (25) for multivariate phase-type distributions, in Bargès et al. (29) for light tailed risks with Farlie-Gumbel-orgenstern copula and in Landsman and Valdez (23) for elliptical distributions. In this paper we extend the work of Das and Fasen-Hartmann (218) for the asymptotic tail behavior of E and ES under regular variation in a more general framework. We derive the limit behavior of the risk measures under the assumption of multivariate regular variation on different subspaces of [, ) 2, specifically E = [, ) 2 \ {(, )} and E = (, ) 2 without specific requirement on asymptotic independence or hidden regular variation. Asymptotic limits are also obtained for the risk measures given in (3) pursuing a similar approach. oreover, we formulate the asymptotic behavior of these conditional excess risk measures under very general assumptions on the copula tail parameters showing similarities with the limit behavior for p of ES(p) and E(p) while assuming regular variation on E and E. In particular, we show that the asymptotic behavior of ES(p) depends only on the tail of Z 1 and the tail copula; neither the presence of asymptotic tail dependence, hidden regular variation nor the tail of Z 2 have any influence on the limit. We compare the conclusions for ES(p) with those of Cai and usta (217). Finally, we provide several examples for models satisfying our assumptions. The paper is organized as follows. In Section 2 we provide a brief introduction to multivariate regular variation on subspaces of [, ) 2, as well as notions of copulas and survival copulas. Section 3 addresses characteristics of models having different regular variation on E and E ; we concentrate on additive models (Weller and Cooley, 214) as often used in a systemic risk context as well as copula models (Hua and Joe, 214, Hua et al., 214). The asymptotic behavior of ES(p), E(p) and the measures in (3) under the models of Section 3 are discussed in Section 4.1. In Section 4.2, we develop sufficient conditions on the copula tail parameters to obtain the asymptotic behavior for E(p) and ES(p) as in Section 4.1 without assuming regular variation on E. Conclusions are drawn in Section 5

4 4 B. DAS AND V. FASEN-HARTANN along with ideas for future directions of research in this domain. All proofs are relegated to the Appendix. 2. Preliminaries. In this section we discuss necessary tools and definitions for multivariate regular variation and copula theory which are used in the subsequent sections. Details on regular variation defined using -convergence is available in Das et al. (213), Hult and Lindskog (26), Lindskog et al. (214) and details on copulas and survival copulas can be found in Nelsen (26). Unless otherwise stated all random variables take non-negative values and we discuss copulas and regular variation in two-dimensions. oreover, for vectors x = (x 1, x 2 ) R 2, we denote by x any suitable norm in R ultivariate Regular Variation. A measurable function f : (, ) (, ) is regularly varying at with index ρ R if lim f(tx)/f(t) = xρ t for any x > and we write f RV ρ ; in contrast, we say f is regularly varying at with index ρ if lim t f(tx)/f(t) = x ρ for any x >. In this paper, unless otherwise specified, regular variation means regular variation at infinity. A random variable Z with distribution function F Z has a regularly varying tail if F Z = 1 F Z RV α for some α. We often write Z RV α by abuse of notation. We define multivariate regular variation using -convergence; see Lindskog et al. (214). All notions are restricted to [, ) 2 and their subspaces. Suppose C C [, ) 2 where C and C are closed cones containing {(, )} R 2. Denote by (C \ C ) the class of Borel measures on C \ C which are finite on subsets bounded away from C. For functions f : [, ) 2 R, denote by µ(f) := f dµ. Then µ n µ in (C \ C ) if µ n (f) µ(f) for all continuous and bounded functions f : C \ C R whose supports are bounded away from C. Definition 2.1 (ultivariate regular variation). A random vector Z = (Z 1, Z 2 ) C is (multivariate) regularly varying on C \ C, if there exist a function b(t) and a non-zero measure ν( ) (C \ C ) such that as t, (5) ν t ( ) := t Pr (Z/b(t) ) ν( ) in (C \ C ). The limit measure has the homogeneity property: ν(ca) = c α ν(a) for some α >. We write Z RV(α, b, ν, C \ C ) and sometimes write RV for multivariate regular variation. Classically, RV is defined on the space E := [, ) 2 \{(, )} = C\C where C := [, ) 2 and C := {(, )}. Sometimes it is possible and perhaps necessary to define further regular variation on subspaces of E, since the limit measure ν as obtained in (5) concentrates on a proper subspace of E. The most likely way this happens is when Z RV(α, b, ν, E) and the limit measure ν concentrates on the co-ordinate axes implying ν((, ) 2 ) =. In such a case one may seek regular variation on the space E := (, ) 2 if it exists. This relates to the idea of hidden regular variation discussed in Resnick (22) and is partially addressed for E and ES in Das and Fasen-Hartmann (218). We briefly discuss this concept in Section 2.3. Remark 2.2. Definition 2.1 works for non-negative random variables Z F Z as well; we have Z RV α if there exists a function b(t) as t and α such that t Pr(Z/b(t) ) ν α ( ) in ([, ) \ {})

5 REGULAR VARIATION AND RISK EASURES 5 where ν α (x, ) = x α, x >. We also write Z RV(α, b), or, Z RV(α, b, (, )) Copulas and survival copulas. Copula theory is popularly used to separate out the marginal behavior of random variables from their dependence structure. In two dimensions, a copula is a distribution function on [, 1] 2 with uniformly distributed margins. Using Sklar s theorem (see Nelsen (26)), we know that for every bivariate distribution function F with marginal distribution functions F i (i = 1, 2), there exists a copula C such that (6) F (x, y) = C(F 1 (x), F 2 (y)) for (x, y) R 2. If F 1, F 2 are continuous then C is uniquely defined by C(u, v) = F (F 1 (u), F 2 (v)) for (u, v) [, 1] 2. Denoting the survival or tail distribution of F i by F i (x) := 1 F i (x), a version of (6) applies also to the joint survival function F (x, y) := Pr(Z 1 > x, Z 2 > y) = F 1 (x) + F 2 (y) (1 F (x, y)) of the bivariate random vector Z = (Z 1, Z 2 ) with distribution function F and margins F 1, F 2. In this case, there exists again a copula Ĉ, the survival copula, such that F (x, y) = Ĉ(F 1(x), F 2 (y)) for (x, y) R 2. oreover, in the bivariate case C and Ĉ are related by Ĉ(u, v) = u + v 1 + C(1 u, 1 v) for (u, v) [, 1] 2. Since we are interested in the dependence (as well as independence) in the upper tails, the behavior of the survival copula Ĉ(u, v) for u, v close to will be of significance in this paper. The relationship between the survival copula and multivariate regular variation on subspaces of [, ) 2 is discussed further in Section Asymptotic independence and hidden regular variation. It is instructive here to note certain relationships between asymptotic tail independence, survival copulas and the notion of hidden regular variation (HRV). The key results of the paper are under the general assumptions of RV on the two subspaces E and E, and more general than HRV. We call Z 1 and Z 2 to be tail-equivalent if lim t Pr(Z 1 > t)/ Pr(Z 2 > t) = K exists with K (, ). Lemma 2.3. Suppose Z = (Z 1, Z 2 ) RV(α, b, ν, E) having continuous marginals and survival copula Ĉ. Consider the following statements. (a) (Z 1, Z 2 ) are asymptotically tail independent. (b) lim p Ĉ(p, p)/p =. (c) ν((, ) (, )) =. Then (a) (b), (b) = (c), and if Z 1 and Z 2 are tail-equivalent then (c) = (a). Consequently, if (Z 1, Z 2 ) are RV, asymptotically tail independent and the margins are tail-equivalent we would approximate Pr(Z 2 > x Z 1 > x) for large thresholds x and conclude that risk contagion between Z 1 and Z 2 is absent. This conclusion may be naive and hence, the concept of hidden regular variation on E = (, ) 2 was introduced in Resnick (22). In our work, we do not assume that the marginal tails of Z are necessarily tailequivalent in order to define hidden regular variation, which is usually done in Resnick (22).

6 6 B. DAS AND V. FASEN-HARTANN Definition 2.4 (Hidden regular variation). A regularly varying random vector Z on E possesses hidden regular variation on E = (, ) 2 with index α ( α > ) if there exist scaling functions b(t) RV 1/α, b (t) RV 1/α with b(t)/b (t) and limit measures ν, ν such that Z RV(α, b, ν, E) RV(α, b, ν, E ). We write Z HRV(α, b, ν ) and sometimes write HRV for hidden regular variation. The following lemma is now easy to verify using a combination of aulik and Resnick (25), Resnick (22) and (Das and Fasen-Hartmann, 218, Lemma 1). Lemma 2.5. Let Z RV(α, b, ν, E) RV(α, b, ν, E ). Then Z HRV(α, b, ν ) iff Z is asymptotically tail independent. Remark 2.6. The key results for the paper are stated under the general assumptions of RV on the two subspaces E and E, without assuming HRV, meaning without the explicit assumption of asymptotic independence. 3. Regular variation in additive models and copula models. The aim of this paper is to exhibit the effect of assumptions of RV on the risk measures ES, E and the measures in (3). To this end, we address three different aspects of bivariate random vectors exhibiting RV. First, we look at additive models governed by the sum of two multivariate regularly varying random vectors; this is a natural way how two risks add up to create a new risk factor; much of literature has addressed behavior of sums of risks in terms of diversification, see cneil et al. (25). Secondly, we observe that assuming RV on both E and E has a certain effect on extremal risks which are functions of the original risk vector and are useful in assessing the behavior of risk measures defined in (3). Finally, we investigate regularly varying random vectors in terms of properties of their copula parameters and relate them to RV on E and E. In Section 4, we use these results to compute the asymptotic limits of the conditional excess measures ES, E and the measures in (3). Note that we concentrate on multivariate regularly varying models in a non-standard sense. Hence, Z = (Z 1, Z 2 ) RV(α) does not necessarily imply that both marginal variables have equivalent (or equal) tails; in fact, both margins need not be regularly varying either Regular variation in additive models. Regular variation properties of additive models (sometimes called mixture models) have been discussed in Weller and Cooley (214) and Das and Resnick (215) where the authors concentrate on adding two standard regularly varying models to get an additive structure with hidden regular variation. The class of models considered here are more general in the sense that the marginal tails of the additive components are not necessarily tail-equivalent or asymptotically independent. We establish the presence of RV on E in these models; under some additional assumptions the model would also exhibit HRV. We note that the presence/absence of HRV do not play a significant role in computing the limits of risk measures if RV on E is present. Theorem 3.1. Suppose Y and V are independent random vectors in [, ) 2 and Z := Y + V. Assume the following holds: (i) Y RV(α Y, b Y, ν Y, E).

7 REGULAR VARIATION AND RISK EASURES 7 (ii) V RV(α V, b V, ν V, E) RV(α V, bv, νv, E ). (iii) α V < αy + α V and lim t Pr(min(Y 1, Y 2 ) > t)/ Pr(min(V 1, V 2 ) > t) =. Then we have the following consequences: (a) If lim t b V (t)/b Y (t) = K [, ) then (b) if lim t b Y (t)/b V (t) = then Z RV(α Y, b Y, ν Y + K 1/αY ν V, E) RV(α V, b V, ν V, E ), Z RV(α V, b V, ν V, E) RV(α V, b V, ν V, E ). In Theorem 3.1, we did not assume RV on E for Y. If we do so, under certain conditions on the tail parameters, we can check that the sum Z = Y + V will still remain RV on E (along with being RV on E). We state the result without proof next. Corollary 3.2. Suppose Y and V are independent random vectors in [, ) 2 and Z := Y + V. oreover (i)-(ii) from Theorem 3.1 holds. Furthermore, suppose Y RV(α Y, by, νy, E ) where α V < α Y + α V, α V < α Y and α Y < α V. Then Z RV(α V, bv, νv, E ) Functions of regularly varying random vectors. The conditional risk measures in (3) indicate that for certain random vectors Z = (Z 1, Z 2 ), it may be necessary to ascertain the behavior of functions of the same; for example aggregate risks like Z 1 + Z 2, or extremal risks min(z 1, Z 2 ) and max(z 1, Z 2 ), not only as univariate vectors but as jointly distributed with the originals. For further explanation on such risk measures see Cai and Li (25), Cousin and Di Bernardino (214). In the following result we find the joint behavior of such entities under an RV assumption on the vector Z [, ) 2. Theorem 3.3. Suppose Z = (Z 1, Z 2 ) RV(α, b, ν, E) RV(α, b, ν, E ). Define = {(z 1, z 2 ) [, ) 2 : z 1 z 2 }, = {(z 1, z 2 ) [, ) 2 : z 1 z 2 }, L x = [, ) {}, L y = {} [, ), L xy = {(z 1, z 2 ) [, ) 2 : z 1 = z 2 }. (a) Let F = \{(, )} and F = \L x. If lim inf t Pr(Z 1 > t)/ Pr(Z 2 > t) = K (, ], then (Z 1, min(z 1, Z 2 )) RV(α, b, ν min, F) RV(α, b, ν min, F ) where ν min (A) = ν({(z 1, z 2 ) E : (z 1, min(z 1, z 2 )) A}) for A B(F), ν min (A) = ν ({(z 1, z 2 ) E : (z 1, min(z 1, z 2 )) A}) for A B(F ). (b) Let G = \ {(, )} and G = \ (L y L xy ). (i) If lim inf t Pr(Z 1 > t)/ Pr(Z 2 > t) = K (, ], then (Z 1, Z 1 + Z 2 ) RV(α, b, ν +, G) RV(α, b, ν + G, G )

8 8 B. DAS AND V. FASEN-HARTANN where ν + (A) = ν({(z 1, z 2 ) E : (z 1, z 1 + z 2 ) A}) for A B(G). (ii) If lim t Pr(Z 1 > t)/ Pr(Z 2 > t) =, Z 1 RV(β, b 1, ν 1, (, )) and lim t b (t)/b 1 (t) = K [, ) then we have where for A B(G ), (Z 1, Z 1 + Z 2 ) RV(α, b, ν +, G) RV(β, b 1, ν, G ) ν (A) = ν 1 (z 1 (, ) : (z 1, z 1 ) A}) + K 1/β ν ({(z 1, z 2 ) E : (z 1, z 1 + z 2 ) A}). (c) Let G 1 = \ L y. (i) If lim inf t Pr(Z 1 > t)/ Pr(Z 2 > t) = K (, ], then where (Z 1, max(z 1, Z 2 )) RV(α, b, ν max, G) RV(α, b, ν max G1, G 1 ) ν max (A) = ν({(z 1, z 2 ) E : (z 1, max(z 1, z 2 )) A}) for A B(G). (ii) If lim t Pr(Z 1 > t)/ Pr(Z 2 > t) =, Z 1 RV(β, b 1, ν 1, (, )) and for lim t b (t)/b 1 (t) = K [, ) then we have where for A B(G 1 ), (Z 1, max(z 1, Z 2 )) RV(α, b, ν max, G) RV(β, b 1, ν, G 1 ) ν (A) = ν 1 (z 1 (, ) : (z 1, z 1 ) A})+K 1/β ν ({(z 1, z 2 ) E : (z 1, max(z 1, z 2 )) A}) Regular variation and the survival copula. In this section we exhibit a precise connection between the behavior of the survival copula and the existence of RV on both E and E for a random vector Z [, ) 2. We also elucidate that the presence of HRV is not necessary to exhibit such a connection, and eventually in Section 4 we show that results on asymptotic limits of the risk measures also do not require the assumption of HRV. First, we introduce a generalized version of the upper tail order function (see Hua and Joe (211a, 213)) along with an upper tail order pair. The notion of upper tail order pair is related also to operator tail dependence in Li (217), and to the generalized upper tail index κ in Wadsworth and Tawn (213). Definition 3.4 (Upper Tail Order). Let F be a bivariate distribution function with survival copula Ĉ. For a given constant τ >, if there exist a real constant κ >, and a slowly varying function l at with (7) Ĉ(s, s τ ) s κ l(s) as s, the pair (κ, τ) is called an upper tail order pair of F. The upper tail order function T : E R + with respect to (κ, τ) is defined as (8) T (x, y) = lim s Ĉ(sx, s τ y) s κ l(s) provided that the limit function exists. for x, y >

9 REGULAR VARIATION AND RISK EASURES 9 Remark 3.5. Note that the pair (τ, κ) need not be unique for the definition to hold. Nevertheless, introducing the quantity τ helps in rescaling marginal tails when they are not equivalent (see Theorem 3.9 below). Since Ĉ(s, sτ ) Ĉ(1, sτ ) = s τ for s (, 1) we have κ τ and similarly we obtain κ 1 as well. Note that the existence of the upper tail order pair is not a sufficient assumption for the existence of the upper tail order function. We often provide examples fixing τ = 1, which also fixes the value of κ. The behavior of the survival copula in terms of the upper tail order function reflects upon the asymptotic upper tail dependence of a bivariate random vector and the following lemma formalizes it. Lemma 3.6. Suppose that the bivariate distribution function F with survival copula Ĉ exhibits asymptotic upper tail independence. Specifically, the upper tail order pair (κ, τ) exists with τ 1 and Ĉ(s, sτ ) s κ l(s) as s. Then lim s κ 1 l(s) =. s Remark 3.7. The classical definition of upper tail order is for τ = 1 and it is equivalent to the definition of coefficient of tail dependence in Ledford and Tawn (1997). (a) If κ = τ = 1 and lim s l(s) = K for some finite constant K, then we get asymptotic dependence in the upper tail. In this case T is the upper tail dependence function introduced in Jaworski (26). However, if κ = τ = 1 and lim s l(s) = then again we observe asymptotic tail independence. (b) The case 1 < κ < 2, τ = 1 is between tail dependence (when κ = 1 and K ) and tail independence (κ = 2) and indicates some positive tail dependence although the tails are asymptotically tail independent. It is called intermediate tail dependence by Hua and Joe (211a, 213). (c) Note that it is possible to have κ > 2 which often signifies negative tail dependence; see Example 3.8 (a) below. Example 3.8. We compute upper tail order functions and upper tail order pairs for a few well-known (survival) copula models here. (a) The Gaussian copula turns out to be one of the most famous, if not infamous copula models, especially in financial risk management; see Salmon (29). It is given by C Φ,ρ (u, v) = Φ 2 (Φ (u), Φ (v)) for (u, v) [, 1] 2, where Φ is the standard-normal distribution function and Φ 2 is a bivariate normal distribution function with standard normally distributed margins and correlation ρ. Then the survival copula satisfies: Ĉ Φ,ρ (s, s) = C Φ,ρ (s, s) s 2 ρ+1 l(s) as s. (see Ledford and Tawn (1997), Reiss (1989)). For ρ ( 1, 1) we have κ = 2/(ρ + 1) > 1 and τ = 1 so that a distribution with Gaussian copula has still some kind of dependence although it exhibits asymptotic upper tail independence. The upper tail order function is given by (see (Reiss, 1989, Example 7.2.7)) T (x, y) = x 1 ρ+1 y 1 ρ+1 for x, y >.

10 1 B. DAS AND V. FASEN-HARTANN Note that < ρ < 1 implies 1 < κ < 2 relating to positive intermediate tail dependence, ρ = implies κ = 2 which is the independent case and ρ < implies κ > 2 which is the case of negative tail dependence. (b) If the survival copula is a arshall-olkin copula then: Ĉ γ1,γ 2 (u, v) = uv min(u γ 1, v γ 2 ) for (u, v) [, 1] 2, some γ 1, γ 2 (, 1). For a fixed τ 1. The upper tail order is κ = max(τ + 1 γ 1, τ + 1 τγ 2 ), and the upper tail order function is xy 1 τγ 2, if γ 1 > τγ 2, T (x, y) = xy max(x, y 1/τ ) γ 1, if γ 1 = τγ 2, for x, y >. x 1 γ 1 y, if γ 1 < τγ 2, The arshall-olkin copula belongs to the class of extreme value copulas. This structure of T (x, y) holds in general for bivariate extreme value copulas with discrete Pickands dependence function; see (Hua and Joe, 211a, Example 2). (c) If the survival copula is a orgenstern copula with parameter 1 θ 1 then: Ĉ θ (u, v) = uv(1 + θ(1 u)(1 v)) for (u, v) [, 1] 2. Hence, for 1 < θ 1 and fixed τ 1 we get κ = τ + 1 with upper tail order function T (x, y) = xy τ for x, y >. For θ = 1 we have the upper tail order pair (κ = 3, τ = 1) with upper tail order function T (x, y) = xy(x + y) for x, y >. If we fix τ > 1 then κ = 1 + 2τ and the upper tail order function is T (x, y) = xy 2 for x, y >. (d) A copula is called an Archimedean copula if it is of the form C(u, v) = φ (φ(u) + φ(v)) for (u, v) [, 1] 2, where the Archimedean generator φ : [, 1] [, 1] is convex, decreasing and satisfies φ(1) =. In the bivariate case this is a necessary and sufficient condition for C to be a copula. If the generator φ is regularly varying near, then the lower tail is asymptotically independent and if φ is regularly varying at 1, then the upper tail is asymptotically independent (see Ballerini (1994), Capéraà et al. (2)). (Charpentier and Segers, 29, Section 4) provide tail order coefficients and functions for Archimedean copulas specifically under asymptotic tail independence. We use them to generalize results for the tail order pair here; see also Hua and Joe (211a). 1. Let φ be twice continuously differentiable with φ () <. Then the upper tail order pair is (κ = 2, τ = 1) with upper tail order function T (x, y) = xy. 2. Let φ (1) = and let the function φ (1 s) s 1 φ(1 s) be positive and slowly varying around. Then the upper tail order pair is (κ = 1, τ = 1) with upper tail order function T (x, y) = {(x + y) log(x + y) x log x y log y}/(2 log 2).

11 REGULAR VARIATION AND RISK EASURES Assume that Ĉ is an Archimedean copula with generator φ satisfying φ() =, lim s sφ (s)/φ(s) = and 1/(log φ ) is regularly varying with some index ρ 1. Then the upper tail order pair is (κ = 2 1 ρ, τ = 1) with upper tail order function T (x, y) = x 2 ρ y 2 ρ. In the context of Z having RV on both E and E, we know from Lemma 2.5 that, asymptotic upper tail independence is equivalent to the existence of HRV. On the other hand, Lemma 3.6 shows that the presence of asymptotic upper tail independence and an upper tail order provides a particular tail behavior for the survival copula. In Theorems 3.9 and 3.11 we show that if the marginal tails are equivalent up to a power-law transformation then there is a particular way in which the upper tail order function connects with the RV behavior on E and E. These results are extensions of Hua et al. (214) and Li and Hua (215) which include several examples as well. The first result is a generalization of (Hua et al., 214, Proposition 3.2) for τ = 1 and asymptotic tail independence. Theorem 3.9. Let Z RV(α, b, ν, E) RV(α, b, ν, E ) with continuous margins F 1, F 2 satisfying lim t F 2 (t)/f τ 1(t) = η (, ) where α /α τ 1. Then an upper tail order pair exists with (κ = α /α, τ), and the corresponding upper tail order function is ) ( )) T (x, y) = Kν ((x 1/α, η 1/(τα) y 1/(τα), for x, y >, where K = { ν ( (1, ) ( η 1/(τα), ))} 1 (, ). Remark 3.1. Note that the pair (b, ν ) is not uniquely defined since if Z RV(α, b, ν, E ) then Z RV(α, Kb, K α ν, E ) for < K < as well. But we can consider this to be uniqueness up to a scale. The converse of Theorem 3.9 also holds; this is an extension of (Hua et al., 214, Proposition 3.3) for τ = 1 and asymptotic tail independence.. Theorem Let Z [, ) 2 with continuous margins F 1, F 2, survival copula Ĉ and F 1 RV α. Suppose Ĉ has upper tail order pair (κ, τ) with κ τ 1 and some slowly varying function l at with lim s s κ 1 l(s) = K [, ) satisfying (7) and (8). oreover, assume that lim t F 2 (t)/f τ 1(t) = η (, ). Then with α = ακ, Z RV(α, b, ν, E) RV(α, b, ν, E ) ν ((x, ) (y, )) = T (x α, η τα y τα ) for x, y >, and some properly chosen b RV 1/α. A conclusion from this result is that RV on E is not only a consequence of the behavior of the survival copula of the joint distribution but also of the ratio of the individual marginal tails. This should not appear as a surprise, since copulas, in theory, are supposed to decouple the marginal distributions from the dependence structure of random vectors; hence the regularly varying behavior of the margins of the distribution seems necessary. For κ = τ = 1 and K > we have the asymptotic tail dependent case, but for κ = τ = 1 and K = the asymptotic tail independent case. Thus, we obtain not necessarily HRV as in (Hua et al., 214, Proposition 3.3).

12 12 B. DAS AND V. FASEN-HARTANN 4. Asymptotic behavior of conditional excess risk measures. The asymptotic behavior of ES under an assumption of multivariate regular variation (with asymptotic tail dependence) and its statistical inference was addressed in Cai et al. (215). In Das and Fasen-Hartmann (218), this work was extended for the ES under the assumption of hidden regular variation and the measure E was defined for which the behavior was studied under asymptotic independence. Consistent estimators for E and ES were also derived in these papers; moreover Cai and usta (217) have also shown asymptotic normality for ES under HRV. For multivariate Gaussian risks, the limit behavior of these risk measures have been explored in Hashorva (218). In this section, first we state results in the flavor of Theorems 1 and 2 in Das and Fasen- Hartmann (218) formulated without assuming HRV, but assuming RV on both E and E in Theorem 4.1. Furthermore, we also posit Lemma 4.2 which includes the case of asymptotically tail dependent models discussed in Cai et al. (215). These two results essentially summarize the limit behavior of E and ES under a very broad set of RV assumptions on [, ) 2. In Section 4.1 we discuss the limit behavior for additive models along with limit results for the risk measures defined in (3). Eventually, in Section 4.2 we obtain the asymptotic behavior of these measures under assumptions on the survival copula and relate them to RV on E and E. Theorem 4.1. Let Z = (Z 1, Z 2 ) [, ) 2 and Z RV(α, b, ν, E) RV(α, b, ν, E ) with α α 1 and E Z 1 <. (a) Suppose the following condition holds: (A) Then (9) lim lim t pb lim {VaR 1 p(z 2 )} E(p) = p VaR 1 p (Z 2 ) Pr(Z 1 > xt, Z 2 > t) Pr(Z 1 > t, Z 2 > t) oreover, < 1 ν ((x, ) (1, )) dx <. (b) Suppose the following condition holds: (B) Then (1) [ lim lim + t 1/ pb lim {VaR 1 p(z 2 )} ES(p) = p VaR 1 p (Z 2 ) oreover, < ν ((x, ) (1, )) dx <. ] 1 dx =. ν ((x, ) (1, )) dx. Pr(Z 1 > xt, Z 2 > t) Pr(Z 1 > t, Z 2 > t) dx =. ν ((x, ) (1, )) dx. Clearly, condition (B) implies condition (A). In the face of it, it appears that the rate of increase of ES which is governed by the function pb {VaR 1 p(z 2 )}/VaR 1 p (Z 2 ) is determined by the tail behavior of the marginal distribution F 2. However, we notice in Section 4.2 that this is not true for ES; the rate is in fact governed by the joint tail behavior of the copula of (Z 1, Z 2 ) and that of the marginal tail of F 1.

13 REGULAR VARIATION AND RISK EASURES 13 Lemma 4.2. Let Z = (Z 1, Z 2 ) [, ) 2 and Z RV(α, b, ν, E) RV(α, b, ν, E ) with α 1 and E Z 1 <. Then (A) and (B) hold and subsequently (9) and (1) hold with b = b. In particular, this means that if Z = (Z 1, Z 2 ) RV(α, b, ν, E) with α 1, E Z 1 <, lim t Pr(Z 1 > t)/ Pr(Z 2 > t) = K (, ) and Z is asymptotically tail dependent then (A) and (B) hold. Remark 4.3. in Theorem 4.1 as (11) Define the function governing the limit behavior of ES(p) and E(p) a(t) := b {VaR 1 1/t(Z 2 )}. tvar 1 1/t (Z 2 ) (a) From (Das and Fasen-Hartmann, 218, Remark 6) we know that under condition (B) the limit lim p a(1/p) = is valid and hence, under the conditions of Theorem 4.1, lim p ES(p) =. Thus, even under the presence of asymptotic upper tail independence, the tail dependence might still be strong enough for lim p ES(p) =. In contrast, if Z 1 and Z 2 are independent we have ES(p) = E(Z 1 ) (then condition (B) is not satisfied; cf (c) below). (b) Consider the case F Z2 RV α. Then (Z 1, Z 2 ) RV(α, b, ν, E) implies α α. Furthermore, if additionally (Z 1, Z 2 ) RV(α, b, ν, E ) then α α as well. In this case, a(t) RV (α α 1)/α. Therefore, a necessary condition for lim p a(1/p) = is α α + 1 and a sufficient condition is α < α + 1. Finally, α α + 1 is as well a necessary assumption for (B). (c) If Z 1 and Z 2 are independent then α = α + α and hence, α 1 and α < α + 1 is not possible. Thus, the independent case does not satisfy (B) and a scaled limit for ES(p) cannot be calculated using Theorem 4.1. (d) Under condition (A) both lim p a(1/p) = and lim p E(p) = are possible. For the independent margin case the asymptotic behavior of E(p) can be calculated using Theorem 4.1, since with α > 1 and Z 1, Z 2 independent, condition (A) is satisfied E and ES in additive models. In Section 3.1, we introduced a general additive model for multivariate regular variation and discussed in Theorem 3.3 the existence of multivariate regular variation on E. We know from Theorem 4.1 that if these models satisfy condition (B) and hence, (A), then we can compute the asymptotic limits of ES and E. The following theorem provide conditions under which condition (B) is satisfied. Theorem 4.4. holds: Suppose Y = (Y 1, Y 2 ) and V = (V 1, V 2 ) are in [, ) 2 and the following (i) Y and V are independent. (ii) V RV(α V, bv, νv, E) RV(αV, bv, νv, E ) and α V > 1. (iii) Y RV(α Y, b Y, ν Y, E) and E Y 1 <. oreover, also assume that one of the following holds: (iv1) α V < αy, OR, (iv2) α Y < α V < 1 + αy, Y 1, Y 2 are independent and lim Pr(min(Y 1, Y 2 ) > t)/ Pr(min(V 1, V 2 ) > t) =. t

14 14 B. DAS AND V. FASEN-HARTANN Then the following models satisfy condition (B): (a) Z = (Z 1, Z 2 ) = Y + V. (b) Z + = (Z 1 + Z 2, Z 1 ) and Z + = (Z 1, Z 1 + Z 2 ) if in the case of (iv2) additionally Y 1 holds. (c) Z min = (Z 1, min(z 1, Z 2 )). (d) Z max = (max(z 1, Z 2 ), Z 2 ) if in the case of (iv2) additionally Y 1 holds. Note that a direct consequence of Theorem 3.3, Theorem 4.1 and Theorem 4.4 is the ability to compute the asymptotic limits of ES, ES +, ES min and ES max as defined in (3) and summarized in the following corollary. Corollary 4.5. Let the assumptions of Theorem 4.4 hold and a(t) be defined as in (11). Then there exist finite constants K, K +, K min, K max > so that following statements hold: (a) lim p a(1/p)es(p) = K. (b) lim p a(1/p)es + (p) = K +. (c) lim p a(1/p)es min (p) = K min. (d) lim p a(1/p)es max (p) = K max E and ES in copula models. Theorem 4.1 provides conditions under which we can compute the asymptotic behavior of E and ES in an additive model which possesses multivariate regular variation on E. A question to ask here is whether a similar result would hold for heavy-tailed multivariate distributions with dependence governed by certain copulas or survival copulas. It turns out that the answer is positive and we can provide a suitable generalization of Theorem 4.1 without necessarily assuming either RV on E or a tail behavior for the distribution function F 2 of Z 2. The outcomes for E and ES require mildly different conditions and hence, are stated separately Asymptotic behavior of E for copula models. Theorem 4.6. Let Z [, ) 2 with continuous margins F 1, F 2, survival copula Ĉ, E Z 1 < and F 1 RV α for some α 1. Suppose Ĉ has upper tail order pair (κ, τ) with κ τ 1 satisfying (7) and (8) and lim t F 2 (t)/f τ 1(t) = η (, ). Also assume that (C) lim lim Ĉ(x α s, s τ ) dx = s Ĉ(s, s τ ) holds. Then there exists a function a(t) RV κα τα 1 τα and a constant K (, ) such that lim a(1/p)e(p) = K. p Remark 4.7. Let Z [, ) 2 with continuous margins F 1, F 2, E Z 1 < and F 1 RV α for some α 1. Suppose that the survival copula Ĉ of Z is either a Gaussian copula, a arshall-olkin copula or a orgenstern copula as given in Example 3.8. Then the assumptions of Theorem 4.6 hold.

15 REGULAR VARIATION AND RISK EASURES Asymptotic behavior of ES for copula models. The next result complements as well as generalizes the results of Hua and Joe (214) where the asymptotic behavior of the ES was investigated for special copula families. Theorem 4.8. Let Z [, ) 2 with continuous margins F 1, F 2, survival copula Ĉ, E Z 1 < and F 1 RV α for some α 1. Suppose Ĉ has upper tail order pair (κ, τ) with κ τ 1 satisfying (7) and (8). oreover, assume that for some continuous distribution function F2 with lim t F 2(t)/F τ 1(t) = η (, ) the asymptotic behavior (D) [ 1/ lim lim + t ] Ĉ(F 1 (xt), F 2(t)) Ĉ(F 1 (t), F 2(t)) dx = holds. Then (κ τ)α < 1 and there exists a function a(t) RV κα τα 1 τα K (, ) such that and a constant lim a(1/p)es(p) = K. p Corollary 4.9. Let Z = (Z 1, Z 2 ) [, ) 2 with survival copula Ĉ, continuous margins F 1, F 2, E Z 1 < and F 1 (t) K 1 t α for some α > 1 and constant K 1 (, ). Suppose Ĉ has upper tail order pair (κ, τ) with κ τ 1 satisfying (7) and (8). oreover, (E) [ lim lim + s ] Ĉ(x α s, s τ ) dx = Ĉ(s, s τ ) holds. Then (κ τ)α < 1 and there exists a function a(t) RV κα τα 1 τα K (, ) such that and a constant Remark 4.1. lim a(1/p)es(p) = K. p A few observations from the above results are noted below. (a) The result shows that the asymptotic behavior of the ES is determined only by the dependence structure and the tail behavior of Z 1 ; the tail behavior of Z 2 has no influence. Particularly, we see that RV on E is not a necessary assumption. (b) An analogous result for the E does not hold, since a monotone transformation of Z 2 will in fact change the E in contrast to the ES; the tail of Z 2 has an influence on the limit behavior of E. Further, note that (C) is only an assumption on the upper tail dependence in contrast to (E) where the whole dependence structure plays a role as well. (c) A result similar to Corollary 4.9 under stronger assumptions has been discussed in (Cai and usta, 217, Proposition 2.1). In their case, they assume the slowly varying function l to be a constant, x T (x, 1) to be continuous and τ = 1. (d) The copula examples in Example 3.8 satisfy (C) but not (E) and hence, Corollary 4.9 cannot be applied. However, such examples are covered in (Hua and Joe, 214, Section 3.4) for both Pareto or Weibull-margins. In these examples the rate of increase of the ES is slower than in the asymptotically tail dependent case but faster than under condition (E).

16 16 B. DAS AND V. FASEN-HARTANN Examples of copula models. The rest of this section is dedicated to construct examples of survival copulas that satisfy the assumptions of Theorem 4.8. The examples are created using the additive models as in Theorem 4.4 and Bernoulli mixture models as discussed in (Hua et al., 214, Section 5) and (Das and Fasen-Hartmann, 218, Example 2). First, we propose a result which we apply on the suggested models. Note that the models in the examples are not created using copulas a priori but we use the inherent copula structure governing the generation method in order to obtain the examples. Proposition Let Z RV(α, b, ν, E) RV(α, b, ν, E ) with continuous margins F 1, F 2, E Z 1 <, lim t F 2 (t)/f τ 1(t) = 1 for some α /α τ 1 and suppose (B) holds. Denote by Ĉ(u, v) the survival copula of Z. Furthermore, let Z = (Z1, Z 2 ) [, )2 be a random vector with survival copula Ĉ, marginal distribution function F 1 of Z1 and some continuous distribution function F2 of Z 2. Then with a(t) = b {VaR 1 1/t(Z 2 )}/{tvar 1 1/t (Z 2 )} RV (α ατ 1)/ατ we have lim a(1/p)e(z p 1 Z2 > VaR 1 p (Z2)) = where < ν ((x, ) (1, )) dx <. ν ((x, ) (1, )) dx Example Let Z = Y + V be as in Theorem 4.4 (i)-(iii)+(iv2) with continuous margins for Y 1, Y 2, V 1, V 2. Further, let Z = (Z 1, Z 2) be defined as in Proposition Then there exists a function a(t) RV (α V α Y 1)/α Y and a constant K (, ) such that lim a(1/p)e(z p 1 Z2 > VaR 1 p (Z2)) = K. Clearly, analogous results hold if Ĉ is the copula of the other examples of vectors defined in Theorem 4.4. Example This model is motivated by the Bernoulli mixture type models discussed in (Hua et al., 214, Section 5) and (Das and Fasen-Hartmann, 218, Example 2). Suppose that X 1, X 2, X 3 are independent Pareto random variables with parameters α, α and γ, respectively, where 1 < α < α < γ, α + γ > α. Let B be a Bernoulli(q) random variable with q (, 1), R = (R 1, R 2 ) be a random vector with each margin defined on [1, ) and E R α <. We also assume X 1, X 2, X 3, B, R are independent of each other. Now define Z = (Z 1, Z 2 ) = B(X 1, X 3 ) + (1 B)(R 1 X 2, R 2 X 2 ) and let Z = (Z 1, Z 2) be defined as in Proposition Then lim p 1 α E(Z p 1 Z2 > VaR 1 p (Z2)) = (1 q) E(min(x 1 R 1, R 2 )) α dx. 5. Conclusion. Our goal in this paper was to investigate certain conditional excess measures for bivariate models exhibiting heavy-tails in the margins and different dependence structures. We have observed that even without assuming hidden regular variation, and just assuming RV on the spaces E and E we are able to find asymptotic rates of convergence for the measures ES, E as well as ES +, ES min, ES max for a variety of copula models,

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