Asymptotic Analysis of Multivariate Tail Conditional Expectations

Transcription

1 Asympoic Analysis of Mulivariae Tail Condiional Expecaions Li Zhu Haijun Li Ocober 0 Revision: May 0 lzhu@mah.wsu.edu, Deparmen of Mahemaics, Washingon Sae Universiy, Pullman, WA 9964, U.S.A. lih@mah.wsu.edu, Deparmen of Mahemaics, Washingon Sae Universiy, Pullman, WA 9964, U.S.A. This auhor is suppored by NSF grans CMMI and DMS

2 Absrac Tail condiional expecaions refer o he expeced values of random variables condiioning on some ail evens and are closely relaed o various coheren risk measures. In he univariae case, he ail condiional expecaion is asympoically proporional o he value-a-risk, a popular risk measure. The focus of his paper is on asympoic relaions beween he mulivariae ail condiional expecaion and value-a-risk for heavy-ailed scale mixures of mulivariae disribuions. Explici ail esimaes of mulivariae ail condiional expecaions are obained using he mehod of regular variaion. Examples involving mulivariae Pareo and ellipical disribuions, as well as applicaion o risk allocaion are also discussed. Key words and phrases: Tail risk allocaion, ail condiional expecaion, coheren risk, regular variaion, mulivariae Pareo disribuion, ellipical disribuion. JEL classificaion: C0, G3. MSC000 classificaion: 9B30, 60F05.

3 Inroducion The ail condiional expecaion (TCE) used in risk analysis describes he expeced amoun of risk ha could be experienced given ha risk facors exceed some hreshold values. TCEs are closely relaed o various coheren risk measures ha are preferable han he Value-a- Risk (VaR), a risk measure ha is widely used bu fails o saisfy he coherency principle. In his paper, we sudy he asympoic relaions beween he mulivariae TCEs and VaR, and show ha for a large class of coninuous risk facors ha follow mulivariae heavy-ailed disribuions, he ail condiional expecaion given ha aggregaed risk exceeds a large hreshold is asympoically proporional o he value-a-risk of aggregaion. The expecaion of a random variable X condiioning on a ail even X > } has a variey of inerpreaions in reliabiliy and risk modeling. In reliabiliy modeling, E(X X > ) for a non-negaive lifeime X is known as he mean residual lifeime []. In insurance and finance, E(X X > ) is known as he mean excess loss of a loss variable X [0], and a risk measure for righ-ailed losses can be described by TCE p (X) := E(X X > VaR p (X)), for 0 < p <, (.) where VaR p (X) := supx R : PX > x} p} is known as he VaR wih confidence level p (i.e., p-quanile). In his paper, risk facors are inerpreed as loss variables. I is known ha for coninuous loss X, he TCE equals he wors condiional expecaion (WCE), which is defined as he supremum of all expecaions of X condiioning on various ail evens wih probabiliy a leas p. The WCE, and hus TCE for coninuous losses, arise naurally via he dualiy heory from coheren risk measures ha saisfy four fundamenal operaing axioms: () monooniciy, () subaddiiviy, (3) posiive homogeneiy and (4) ranslaion invariance (see [3, 8, 0] for deails). In he univariae case, a coheren risk measure ϱ(x) for loss X corresponds o he amoun of exra capial requiremen ha has o be invesed in some secure insrumens so ha he resuling posiion ϱ(x) X is accepable o regulaors/supervisors. The coheren risk measures, such as TCE, overcome he shorcomings of VaR ha violaes he subaddiiviy principle and ofen underesimaes ail risk. I can be shown ha for coninuous losses, TCE is he average of VaR over all confidence levels greaer han p, focusing more han VaR does on exremal losses. Thus, TCE is more conservaive han VaR a he same level of confidence (i.e., TCE p (X) VaR p (X)) and provides an effecive ool for analyzing ail risks. For ligh-ailed loss disribuions, such as normal and phase-ype disribuions [5], TCE and VaR a he same level p of confidence are asympoically equal as p. I is precisely he heavy-ailedness of loss disribuions ha differeniaes TCE and VaR in analyzing ail risks. Formally, a non-negaive random variable X wih disribuion funcion F has a heavy

4 or regularly varying righ ail a wih ail index > 0 if is survival funcion is of he following form (see, e.g., [4] for deails), F () := PX > } = L(), > 0, > 0, (.) where L is a slowly varying funcion; ha is, L is a posiive funcion on (0, ) wih propery L(c)/L() =, for every c > 0. We use RV in his paper o denoe he class of all regularly varying funcions wih ail index. For any random variable X, wrie X = X + X, where X + := maxx, 0} and X := max X, 0}. In his siuaion, he noion of regular variaion (.) can be applied o X + (righ ail of X) or X (lef ail of X). Noe ha a regularly varying funcion behaves as a power funcion asympoically, and in paricular, any regularly varying funcion inegraes in he way as ha of a power funcion, as is shown in he well-known Karamaa s heorem (see, e.g., [], page 5). Proposiion.. If U() RV wih ail index >, hen U(x)dx RV + wih ail index > 0 and U(x)dx U(), for sufficienly large. (.3) Here and hereafer he ail equivalence noaion f() g() as a means ha f()/g() as a. An immediae consequence of applying Karamaa s heorem o TCE for a loss variable X having a regularly varying righ ail wih ail index > is illusraed as follows. For any p sufficienly close o so ha VaR p (X) > 0, we have ha X > VaR p (X)} = X + > VaR p (X)} due o he fac ha X = X + if X 0, and hus TCE p (X) = E(XIX > VaR p(x)}) = E(X +IX + > VaR p (X)}) PX > VaR p (X)} PX + > VaR p (X)} = PX + IX + > VaR p (X)} > x}dx PX + > VaR p (X)} 0 ( ) = VaR p (X)PX + > VaR p (X)} + PX + > x}dx PX + > VaR p (X)} VaR p(x) VaR p(x), as p, (.4) where IB} denoes he indicaor funcion of se B. Tha is, TCE for any heavy-ailed loss disribuion is asympoically proporional o is VaR wih asympoic consan ha depends on is ail index, in a manner similar o ha for he Pareo loss disribuion F () = (+), 0. The ail esimae (.4) has been documened in he lieraure (see, e.g., page 83 of [0]), bu he derivaion in (.4) illusraes how Karamaa s heorem and is mulivariae

5 exensions can be used o develop iing resuls in asympoic analysis of coheren risk measures, in he univariae case as well as in he mulivariae case. A risk measure ϱ(x) for a d-dimensional loss vecor X corresponds o a subse of R d consising of all he deerminisic porfolios x such ha he modified posiions x X is accepable o regulaors/supervisors. The coherency principles for mulivariae risk measures ha are similar o ha in he univariae case, and mulivariae TCEs were sudied in [5]. Noe, however, ha mulivariae TCEs are subses of R d, which ofen lack racable expressions. A mulivariae regular variaion mehod based on ail dependence funcion (see [3, 8]) was developed in [4] o derive racable asympoic bounds for mulivariae TCEs, bu hese bounds are expressed in erms of univariae inegrals of ail dependence funcions and hus sill cumbersome for loss disribuions wihou explici expressions of ail dependence funcions, such as ellipical disribuions. In his paper, we focus on he loss variables wih heavy-ailed scale mixing: X = (X,..., X d ) := (RT,..., RT d ), and R 0, (.5) where R has a regularly varying survival funcion wih ail index, and (T,..., T d ), independen of R, is any random vecor wih some finie join momens. Here and hereafer, denoes he marix ranspose. The class (.5) of loss disribuions is a sub-class of all mulivariae regularly varying disribuions ha is discussed in [4], bu i covers a variey of loss disribuions, including mulivariae Pareo disribuions and mulivariae ellipical disribuions whose ail dependence funcions are usually no explici. Uilizing he regular variaion propery of R, we esablish he explici ail esimaes of TCEs for he class (.5) of loss disribuions. In conras o [4], a disincive feaure resuled from he approach used in his paper is ha he ail esimaes for he TCE given ha aggregaed loss exceeds a large hreshold depend explicily on ail index and join momens of variables T,..., T d. The res of he paper is organized as follows. In Secion, we firs prove, via Karamaa s heorem, a convergence heorem on he inegral of he raio of regularly varying funcions, and hen esablish he ail esimae for mulivariae condiional expecaions of loss variables (.5). In Secion 3, we discuss he asympoic properies of TCEs for mulivariae ellipical disribuions, and in paricular, compare he ail esimaes of TCEs obained from our asympoic approach and ones derived from he exac TCE formulas obained in [6] for ellipical disribuions. Mulivariae TCEs of Heavy-Tailed Scale Mixures In his secion, we derive he ail esimae of TCE of loss variable X given ha anoher loss variable X exceeds a lager hreshold when X, X are joinly disribued as ha of (.5) (d = ), and as a consequence, ail esimaes of various TCEs for aggregaed risk facors can 3

6 be obained. The following varians of Karamaa s heorem and Breiman s heorem will be used in he proof of our main resul. Lemma.. Le R be a non-negaive random variable wih regularly varying survival funcion U() := PR > } = L(), > 0, >, where L is a slowly varying funcion. Then we have PR > x}dx c PR > }, as, for every c > 0. (.) c Proof. According o Proposiion., U() RV implies ha c U(x)dx c U(c), as. (.) On he oher hand, U(c) c U(), as. Plug his ino (.) and we ge he desired ail esimae (.). Proposiion.. Le M( ) be any finie non-negaive measure on R +. If U() = L() RV wih ail index > 0 and x +ɛ M(dx) < for some small ɛ > 0, hen 0 0 U(/x) U() M(dx) = 0 x M(dx) = 0 U(/x) U() M(dx). Proof. Le R 0 be a random variable wih survival funcion U(). Since M( ) is finie, we can consruc a non-negaive random variable X, independen of R, wih disribuion F (x) = PX x} := M([0, x]) M(0}) M(R + ) M(0}), x R +. Clearly, X has a finie momen of order greaer han. I follows from Breiman s heorem (see, e.g., pages 3-3 of []) ha PRX > } PR > } = E(X ). (.3) U(/x) Tha is, df (x) = x df (x), which yields he desired i by canceling 0 U() 0 consan M(R + ) M(0}) on boh sides. Proposiion. allows us o pass he i of ail raio hrough inegraion, which faciliaes asympoic analysis of TCEs. Proposiion.3. Le (X, X ) = (RT, RT ) be a bivariae random vecor, where (T, T ), independen of a random variable R 0, has finie momens E(T ), E(T T+) and E(T+ +ɛ ) for some ɛ > 0, where T + := maxt, 0}. If R has he survival funcion U() := PR > } = L() RV wih ail index >, hen we have E(X X > ) 4 E[T T + ] E[T+], as. (.4)

7 Proof. Observe firs ha E(T ) < and E(T T +) < imply ha E(T T + ) <. Since X > if and only if X + := maxx, 0} > for any > 0, we have, for > 0, E(X X > ) = E(X X + > ) = E(X + X + > ) E(X X + > ) (.5) where X + = maxx, 0} and X = max X, 0}. Consider, for > 0, E(X + X + > ) = E(X +IX + > )}) PX + > } PX + > x, X + > } = dx. PX + > } 0 I follows from Breiman s heorem (see, e.g., pages 3-3 of []) and E(T +ɛ + ) < for some ɛ > 0 ha X + = RT + has a regularly varying survival funcion wih ail index and Le x = w, we have PX + > } E(T +)PR > }, for sufficienly large. (.6) E(X + X + > ) = PX + > } = P PX + > } R + 0 = PX + > } 0 R R > w PX + > w, X + > }dw } dwdf (, ), R > P R > max w, }} dw df (, ), (.7) where F denoes he join disribuion of (T +, T + ), where T i+ = maxt i, 0}, i =,. For he inner inegral, we have }} w P R > max dw = 0 P R > = P R > }, } dw + + P P R > w } dw R > w For he firs summand in (.8), observe ha for any > 0, > 0, } P R > PR > } = L( ) + L( ) = = L() + L() } dw. (.8) P R > } PR > } =, where R has he survival funcion U () := + L() RV + wih > 0. Le M (B) := B R + df (, ), for any Borel subse B R +, denoe he marginal mean 5

8 measure induced by T +, and M ( ) is a finie measure due o he fac ha E(T ) <. Thus by Proposiion. wih E(T T+) <, } } P R > P R > R + PR > } df (, ) = R + PR > } M (d ) = E(T + T+ ). (.9) For he second summand in (.8), le x = w, and i follows from (.) ha P R > w } dw = PR > x}dx PR > }, as. Le U () := ha R + PR > x}dx = U(x)dx RV + wih > 0. Observe from (.3) } P R > w dw PR > } U(x)dx ( = U() ) U (/ ) U () and i follows from Proposiion. wih E(T T+) < ha P R > w PR > } } dw Plugging (.8) ino (.7), we have E(X + X + > ) + df (, ) = = PR > } PX + > } R + Using (.6), (.9) and (.0), we obain ha E(X + X + > ) U(x)dx U(), as, U (/ ) R + U () M (d ) = ( ) E(T + T + ). (.0) PR > } PX + > } R + R > w [ PR > } = PX + > } E T + T+ Using he similar argumens, we also have E(X X + > ) Observe ha T = T + T, we have E(X X + > ) as desired. P R > } PR > } df (, ) } P dw df (, ). PR > } ( = + )] E [ ] T T+. ET+ E(X + X + > ) E(X X + > ) = 6 = = E [ ] T + T+. ET+ E [ ] T T+, ET+

9 Remark.4.. Noe ha he momen condiion in Proposiion.3 is raher mild. For example, if T and T have finie marginal momens of any order, hen he momen condiion in Proposiion.3 holds for any ail index >.. Take T = T in Proposiion.3, and we have from (.4) ha, as, E(X X > ) E[T T + ] E[T +] = E[T T + ] E[T +] =, due o he fac ha T T + = 0 almos surely. This is exacly he same as (.4). The ail esimaes of various TCEs can be obained immediaely from Proposiion.3. Le ψ : R d R be a sricly increasing homogeneous funcion wih ψ(cx) = cψ(x) for c 0 and x R d. For example, any linear funcion of he form ψ(x) = d a ix i, defined on R d, where a i > 0, i d, is sricly increasing and homogeneous. Noe ha such a funcion mus saisfy ha ψ(0) = 0 and VaR p (ψ(x)) for heavy-ailed X, as p. For any random vecor (X,..., X d ) = (RT,..., RT d ) wih R 0, observe ha (X i, ψ(x)) = (RT i, Rψ(T )), and hus he following resul follows from Proposiion.3. Theorem.5. Le X = (X,..., X d ) = (RT,..., RT d ) be a random vecor, where he survival funcion of R, U() RV wih >, and T = (T,..., T d ), independen of R, has finie marginal momens E(T k i ), k, i d. Then, for any i d, we have E(X i ψ(x) > VaR p (ψ(x))) where ψ + (T ) := maxψ(x), 0}. E[T i ψ+ (T )] VaR E[ψ+(T p (ψ(x)), as p, )] For example, if ψ(x) = d x i represens he linear aggregaion of losses, hen we have ha for any i d, as p, E ( X i d d X i > VaR p ( X i ) ) ( ( E T d i T i ) + E( d j= T j) + ) ( d ) VaR p X i, (.) where ( d j= T j) + := max d j= T j, 0}. As anoher example, le X (d) and T (d) denoe he larges order saisics of (X,..., X d ) and (T,..., T d ) respecively, and we have for any i d, E ( X i X(d) > VaR p (X (d) ) ) E(T i T (d+) ) ET (d+) VaR p ( X(d) ), as p, where T (d+) := maxt (d), 0}. In fac, as indicaed in Proposiion.3 and Theorem.5, he TCE of a homogeneous funcion of X,..., X d given ha anoher homogeneous funcion 7

10 of he variables exceeds is VaR can be also asympoically expressed in erms of he join momens of hese funcions. For example, le X (k) denoe he k-h larges order saisic of X,..., X d, hen he ail esimaes of E(X i X (k) > ) and E(X (i) X (k) > ) can be obained using Proposiion.3. I is worh emphasizing here ha he asympoic proporionaliy consans of he TCEs discussed in Theorem.5 depend on ail index and also on he dependence srucure of T,..., T d. For example, consider he bivariae case wih T 0, T 0 and =, and we hen have for i =, E(X X + X > VaR p (X + X )) ET + E(T T ) VaR ET + E(T T ) + ET p (X + X ), as p. Le ρ be he correlaion coefficien of (T, T ). For he fixed marginal disribuions of T and T, ρ is increasing if and only if E(T T ) is increasing. Thus, we have ET + E(T T ) = if ET = ET, ET + E(T T ) + ET is increasing in ρ if ET < ET, is decreasing in ρ if ET > ET. Thus, even he marginal disribuions are fixed, merely changing he dependence srucure of mulivariae risk facors could change he asympoic proporion of he TCEs wih respec o he VaR. Example.6. Consider a bivariae Pareo disribuion of Marshall-Olkin ype for random vecor (X, X ) = (RT, RT ). Le R have he survival funcion U() = PR > } = ( + ) for 0 and >, and le (T, T ) have a bivariae Marshall-Olkin exponenial disribuion funcion on [0, ) [9], namely, T = mine, E }, T = mine, E }, where E, E, E are independen and have he exponenial disribuions wih parameers λ, λ, λ respecively. All he join momens E(TT i j ) for any non-negaive inegers i, j can be calculaed explicily. I is easy o see ha VaR p (R) = ( p) /. I follow from Breiman s heorem (see page 3-3 of []) ha PX + X > } E(T + T ) PR > } for sufficienly large, which implies ha ( ) E(T + T ) / VaR p (X + X ), as p. p I follows from (.) ha he ail esimae of TCE for i =, is given by, E ( X i X + X > VaR p (X + X ) ) ( ) ] [ (E(T ) E[T i T + T + T ) / ], E(T + T ) p 8

11 as p. For ineger-valued, hese ail esimaes can be evaluaed analyically. For example, if =, hen E ( X X + X > VaR p (X + X ) ) VaR p (X + X ) [ET + E(T T )] VaR ET + E(T T ) + ET p (X + X ) ( ) ET + E(T T ) + ET /. p Since E(T ) = we obain ha λ + λ, var(t ) = E(T ) = var(t ) + (ET ) = We also know from [9] ha where λ = λ + λ + λ. esimaes: as p, E(T T ) = E ( X X + X > VaR p (X + X ) ) VaR p (X + X ) VaR p (X + X ) (λ + λ ), E(T ) = (λ + λ ), E(T ) = (λ + λ )λ + (λ + λ )λ, λ + λ, var(t ) = (λ + λ ). (λ + λ ), Using all he above momen expressions, we obain he ail ( (λ +λ + + ) λ(λ +λ ) λ(λ +λ ) (λ +λ + + +, ) λ(λ +λ ) λ(λ +λ ) (λ +λ ) (λ +λ ) λ(λ +λ ) λ(λ +λ ) (λ +λ ) p ) /. Noe ha for he bivariae Pareo disribuion of Marshall-Olkin ype, he correlaion of X and X is decreasing in λ. I is eviden ha VaR p (X +X ) is asympoically increasing in he correlaion of X and X, whereas he asympoic monooniciy of TCE wih respec o he correlaion is more suble and also depends on he marginal parameers λ and λ. Example.7. Consider he mixure model (.5) where T,..., T d are independenly and exponenially disribued random variables wih uni mean, and R is a sricly posiive random variable wih he Laplace ransform of R being given by ϕ(). The marginal survival funcion of X i is hen given by F i () = E(e /R ) = ϕ(), 0. Assume ha ϕ() RV wih ail index >. The scale mixure X = (X,..., X d ) has regularly varying margins and Archimedean copula dependence srucure [7]. For example, if R has he gamma disribuion wih uni 9

12 scale parameer and shape parameer > 0, hen he Laplace ransform ϕ() = ( + ), leading o he so called Clayon copula dependence srucure for X. The ail esimae of TCE given ha he linear aggregaion of losses exceeds a large hreshold can be calculaed explicily via (.). For example, if =, hen E ( X i d d X i > VaR p ( X i ) ) d + d + d VaR ( d ) p X i = d VaR ( d ) p X i, due o he fac ha d T i has a gamma disribuion wih uni scale parameer and shape parameer d. In fac, since T,..., T d are independenly and idenically disribued, we have ( ) ( ) ( d ) E T i T i = d ( d ) E T i T i = d d d E( T i ). Thus for any >, E ( X i d d X i > VaR p ( X i ) ) d( ) VaR p ( d ) X i. Noe ha he ail esimaes for VaR p ( d X i) in erms of he marginal VaR are discussed in [,, 9] and he references herein, and our ail esimae for TCE in erms of VaR complemens hese asympoic resuls. 3 TCE for Mulivariae Ellipical Disribuions Le Σ be a d d posiive-semidefinie marix, and U = (U,..., U m ) be uniformly disribued on he uni sphere in R m. Consider he sochasic represenaion X = (X,..., X d ) = (µ,..., µ d ) + RA(U,..., U m ), where A is a d m marix wih AA = Σ and R > 0 is a random variable independen of U. The disribuion of X is known as a d-dimensional ellipical (conoured) disribuion wih dispersion marix Σ and is one of mos widely used radially symmeric mulivariae disribuions [0]. The examples of ellipical disribuions include he mulivariae normal, and logisic disribuions. Since we are ineresed in ail behaviors of X, we may choose (µ,..., µ d ) = (0,..., 0) wihou loss of generaliy: X = (X,..., X d ) = RA(U,..., U m ). (3.) 0

13 The characerisic funcion of X in (3.) can be wrien in he form as E(e i X ) = Ψ( Σ/), where Ψ : R + R is called he characerisic generaor. If R has a densiy and Σ is posiive-definie, hen he densiy of X exiss and can be wrien in he following form: f(x) = c d Σ / g d (x Σ x/), where g d : R + R + is called he densiy generaor and c d is he normalizing consan. For example, for he mulivariae normal disribuion, g d (x) = exp x}, x 0. For he mulivariae disribuion, ( g d (x) = + x ) p, x 0, (3.) k p where he parameer p > d/ and k p is some consan ha may depend on p (see, e.g., page 8 of [0]). Noe ha hese densiy generaors do no depend on d, as is he case for many oher ellipical disribuions. I can be shown (see, e.g., Theorem.9 in [0]) ha he densiy h of R and g d are relaed as follows: h(r) = πd/ Γ(d/) rd g d (r ), r 0, (3.3) and in paricular, h(r) = πg (r )/Γ(/). If E(R ) < and Σ is posiive-definie, hen he covariance marix Cov(X) = d E(R )Σ. Le S := d X i for an ellipically disribued random vecor (X,..., X d ) wih dispersion marix Σ = (σ i,j ). Since any affine ransform of an ellipically disribued random vecor is also ellipical, he random vecor (X i, S) is ellipical wih dispersion marix ( ) Σ i,s = σ i σ i,s where σ i = σ i,i, σ i,s = d j= σ i,j, and σ S = d d j= σ i,j. The explici expression of TCE of X i given ha S exceeds a hreshold is obained in [6] as follows. Theorem 3.. Le X = (X,..., X d ) be defined as ha in (3.) wih dispersion marix Σ and densiy generaor g d. If E(R) < and Σ is posiive-definie, hen, for i d, where = VaR p (S). E(X i S > ) = σ i,s σ S σ i,s σs g /σs (x)dx /σ S g (x /)dx, (3.4) The ail esimae of TCE of X i given ha S exceeds a large hreshold can now be obained from Theorem 3. via Karamaa s heorem (see Proposiion.).

14 Corollary 3.. Le X = (X,..., X d ) be defined as ha in (3.) wih dispersion marix Σ and densiy generaor g d. If Σ is posiive-definie and he densiy h(r) of R exiss and is regularly varying wih ail index +, >, hen, for i d, E(X i S > VaR p (S)) σ i,s VaR σs p (S), as p. (3.5) Proof. Observe firs ha since >, E(R) <. I follows from (3.3) ha h(r) = πg (r )/Γ(/), and hus he densiy generaor g (r) RV + and g (r ) RV (+). Applying Proposiion. o (3.4), we have, as, g /σs (x)dx /σ S g (x /)dx g σs ( /σs ) σ S g ( /σs ) =. σ S Plug his ail esimae ino (3.4), we have where = VaR p (S) as p. E(X i S > ) σ i,s σs Remark For any random vecor (X, X ) wih a bivariae ellipical disribuion, i follows from Lemma in [6] and he similar argumens as hese in he proof of Corollary 3. ha as p, E(X X > VaR p (X )) σ, VaR p (X ). (3.6) σ, In fac, he ail esimae (3.5) can be derived from (3.6) since (X i, S) has a bivariae ellipical disribuion.. I follows from Proposiion. ha he densiy h(r) RV (+) implies ha he survival funcion of R, PR > } = h(r)dr RV, which is he condiion used in Theorem.5. Conversely, he condiion ha PR > } = h(r)dr RV implies ha h(r) RV (+) provided ha h(r) is asympoically monoone. This resul, due o E. Landau, has a relaion wih he von Mises condiion (see Proposiion.5 in []). 3. Noe ha he proofs of boh Theorem 3. and Corollary 3. require he exisence of he join densiy funcion. In conras, he ail esimaes for ellipical disribuions via Theorem.5 do no require such an assumpion. The ail esimae for E(X i S > ), as, can be also obained using Theorem.5. Le (T,..., T d ) = A(U,..., U m ), hen X = (X,..., X d ) = R(T,..., T d ). If R has a

15 regularly varying survival funcion wih ail index >, hen X has mulivariae regularly varying ails. I follows from (.) ha for i d, as p, ( ( E ( X i S > VaRp (S) ) E T d i T ) ) i + E( ( ) d j= T VaR p S, j) + In he bivariae case, his ail esimae becomes: for i =,, ( ( ) ) E ( X i X + X > VaR p (X + X ) ) E T i T + T + ( ) VaR E(T + T ) p X + X. (3.7) + In conras, (3.5) has he following expression in he bivariae case: for i =,, E ( X i X + X > VaR p (X + X ) ) σ i, + σ i, ( ) VaR p X + X. (3.8) σ, + σ, + σ, + σ, To illusrae he fac ha he asympoic proporionaliy consans in (3.5) and (.) are approximaely same for ellipical disribuions, we simulaed samples wih simple size n = 00, 000 from he bivariae disribuion and compared wo proporionaliy consans in (3.7) and (3.8) via simulaion. The bivariae disribuions are sampled for various degrees of freedom ν > (d.f.) and covariance marices Σ = (σ i,j) of he corresponding bivariae normal disribuions. Noe ha for he disribuions,. he ail index = ν for is join survival funcion (see, e.g., [6]),. (T,..., T d ) in (.) has he normal disribuion wih zero mean vecor and covariance marix Σ = (σ i,j), and 3. Σ = ν νd E(R )Σ, where R / has he inverse gamma disribuion wih parameer ν/. Since σ i,j and σ i,j are proporionally relaed, (3.5) and (3.8) can be also expressed in he exacly same way in erms of σ i,js. The comparisons of he asympoic proporionaliy consans in (3.5) and (.) for various parameers are lised in Table and, where σ, = and σ, =.5. Noe ha he asympoic proporionaliy consans in he hird row of Tables - depend only on Σ and are independen of d.f. ν =. We also numerically compared he asympoic proporionaliy consans in (3.6) and (.4). The comparisons for bivariae disribuions wih various parameers are lised in Tables 3-4, where σ, = σ, =. In conras o Tables -, we included he facor /( ) in Tables 3-4. I is eviden ha he asympoic proporionaliy consans in (3.5) and (.) are decreasing in ail index only hrough /( ) for ellipical disribuions. On he oher hand, he asympoic proporionaliy consans in (3.6) and (.4) are increasing as correlaion increases. 3

16 σ E(T (T +T ) + ) E(T +T ) + σ, +σ, σ, +σ, +σ, +σ, Difference Table : Comparison of Approximaion for Bivariae Disribuion wih d.f. = =. σ E(T (T +T ) + ) E(T +T ) + σ, +σ, σ, +σ, +σ, +σ, Difference Table : Comparison of Approximaion for Bivariae Disribuion wih d.f. = = 7 4 Concluding Remarks One moivaion o sudy ail esimaes of TCE of loss variables given ha aggregaed risk exceeds a large hreshold is ha such ail esimaes can be applied o evaluae ail risk allocaion/decomposiion. Tha is, he ail esimae of E(X i d X i > VaR p ( d X i)) provides he conribuion o he oal ail risk aribuable o he i-h risk facor X i, as measured by TCEs. The risk allocaion/decomposiion wih TCE for ellipically disribued loss vecors and for mulivariae Pareo porfolios can be found in [6, 7, 4], and a general discussion on his opic can be found in [0]. The ail condiional variance for ellipical disribuions is sudied in [3]. We have shown ha he TCE of a loss variable given ha aggregaed risk exceeds a large hreshold is asympoically proporional o he VaR of aggregaed risk as he confidence level approaches. The proporionaliy consan can be expressed explicily in erms of he ail index and join momens of mixed variables wih heavy-ailed scale mixing. Several examples of loss variables involving mulivariae Pareo and ellipical disribuions have been explicily calculaed o illusrae our resuls. If he mixure model (.5) fis daa, hen he model parameers can be esimaed (see [, ] for deails on saisical inference). These proporionaliy consans in Theorem.5 can be evaluaed using numerical inegraion or simulaion. For example, if loss variables follow a mulivariae disribuion as is he case in he numerical examples discussed in Secion 3, hen he proporionaliy consans can be evaluaed hrough numerical inegraion or simulaion on mulivariae normally disribued variables (see []). Acknowledgmens: The auhors would like o sincerely hank anonymous referees for heir 4

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