Asymptotic Analysis of Tail Conditional Expectations

Transcription

1 Asympoic Analysis of Tail Condiional Expecaions Li Zhu Haijun Li June 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 ail condiional expecaion and value-a-risk for heavy-ailed scale mixures of mulivariae disribuions. Explici ail esimaes of ail condiional expecaions are obained using a convergence resul on he inegrals of ail raio for 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. MSC classificaion: 9B3, 6F5. 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 TCEs and VaR, and show ha for a large class of coninuous heavy-ailed mulivariae risks, he ail condiional expecaion 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 in par by NSF gran CMMI 8596.

2 given ha aggregaed risk exceeds a large hreshold is asympoically proporional o he value-a-risk of aggregaion. The expecaion of a non-negaive 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 > ) 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 [], and a measure for righ-ailed risk can be described by TCE p (X) := E(X X > VaR p (X)), for < p <, (.) where VaR p (X) := supx R : PrX > x} p} is known as he VaR wih confidence level p (i.e., p-quanile). I is known ha for coninuous risk variable X, he TCE equals o he wors condiional expecaion (WCE), defined as he supremum of all expecaions of X condiioning on ail evens wih probabiliy a leas p. The WCE, and hus TCE for coninuous risks, 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 [5, 8, ] 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 [7], 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 loss variable X wih disribuion funcion F has a heavy or regularly varying righ ail a wih ail index α > if is survival funcion is of he following form (see, e.g., [6] for deail), F () := PrX > } = α L(), >, α >, (.) where L is a slowly varying funcion; ha is, L is a posiive funcion on (, ) wih propery L(c)/L() =, for every c >. We use RV α in his paper o denoe he class of all regularly varying funcions wih ail index α. 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 Karamaa s heorem.

3 Proposiion.. If U() RV α wih ail index α >, hen U(x)dx RV α+ wih ail index α and U(x)dx U(), for sufficienly large. (.3) α Here and hereafer he ail equivalence noaion f() g() as a means ha f()/g() as a. An immediae consequence of applying he Karamaa s heorem o TCE wih heavy-ailed loss X RV α is illusraed as follows: TCE p (X) = E(XIX > VaR p(x)}) PrX > VaR p (X)} ( ) = VaR p (X) PrX > VaR p (X)} + PrX > x}dx PrX > VaR p (X)} VaR p(x) α α VaR p(x), as p. (.4) Tha is, TCE for any heavy-ailed loss disribuion is asympoically proporional o is VaR wih ail consan ha depends on is ail index, in a manner similar o ha for he Pareo loss F () = ( + ) α,. The ail esimae (.4) has been widely documened in he lieraure (see, e.g., []), bu he derivaion in (.4) illusraes how he Karamaa s heorem can be used o esablish iing resuls in asympoic analysis for expecaion-based risk measures, in he univariae case as well as in he mulivariae case. A risk measure ϱ(x) for loss vecor X = (X,..., X d ) 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 [4]. 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 [, 8]) was developed in [3] o derive racable 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. In his paper, we focus on he loss variables wih heavy-ailed scale mixing: X = (X,..., X d ) = (RT,..., RT d ), and R RV α, (.5) where (T,..., T d ) is any non-negaive random vecor wih some finie join momens. The class (.5) of loss disribuions is smaller han he class of all mulivariae regularly varying disribuions ha is discussed in [3], bu i covers a variey of loss disribuions, including 3

4 mulivariae Pareo disribuions and mulivariae ellipical disribuions whose ail dependence funcions are usually no explici. Using he Karamaa s heorem (Proposiion.), we esablish a convergence resul on he inegrals of ail raio for regular variaion ha faciliaes he derivaion of explici ail esimaes of TCEs for he class (.5) of loss disribuions. A disincive feaure of our approach in his paper is ha he ail esimaes for he TCE given ha aggregaed risk exceeds a large hreshold depend explicily on ail index α and join momens of variables T,..., T d. Tail Risk of Heavy-Tailed Scale Mixure of Mulivariae Disribuions In his secion, we derive he ail esimae of TCE of loss variable X given anoher loss variable X exceeds a lager hreshold when X, X are joinly disribued as ha of (.5) (d = ), and as consequence, ail esimaes of various TCEs for aggregaed risks can be obained. We also presen some explici examples o illusrae our resuls. Our mehod is based on a convergence heorem on he inegrals of ail raio for regular variaion. The following varian of he Karamaa s heorem and he represenaion of slowly varying funcions will be used in he proof of our main resul. Lemma.. Le R be a non-negaive random variable wih regularly varying survival funcion R() := PrR > } = α L(), >, α >, where L is a slowly varying funcion. Then we have c PrR > x}dx c α PrR > }, as, for every c >. (.) α Proof. According o Proposiion., R RV α implies ha R(x)dx RV c α and which is equivalen o c R(x)dx c R(c) c R(x)dx = α, c R(c), as. (.) α Since R RV α, hen R(c) c α R(), as. Plug his ino (.) and we ge he desired ail esimae (.). The slowly varying funcions behave in a conrol manner as described in he Karamaa represenaion (see page of [6]). 4

5 Lemma.. The funcion L : R + R + is slowly varying if and only if L() = c() exp s ɛ(s)ds}, >, where c : R + R + is bounded wih c() = d (, ) and ɛ : R + R is bounded wih ɛ() =. The above fundamenal properies lead o he following iing resul on he inegral of ail raio for regular variaion. Proposiion.3. Le M( ) be any finie measure on R +. If U() = α L() RV α wih ail index α > and x α+ɛ M(dx) < for some small ɛ >, hen U(/x) U() M(dx) = U(/x) U() M(dx) = x α M(dx). Proof. According o he uniform convergence heorem for regular variaion (see, e.g., Proposiion.4 in page 4 of []), U(/x)/U() converges uniformly, as, o x α on (, b) for any b >. Since M is finie, we have b U(/x) U() M(dx) = b for any b >. Observe from Lemma. ha for x, x α M(dx) (.3) U(/x) U() = x α c(/x) c() e /x s ɛ(s)ds, where c( ) and ɛ( ) are wo funcions saisfying he properies saed in Lemma.. Since c is bounded, c(/x) C < for all, x. Since c() = d (, ), we have, for sufficienly large, ha c() d η > for some small η >. Thus, for sufficienly large, U(/x) U() C d η xα e /x s ɛ(s)ds, for x, and for some η >. Since ɛ() =, hen for sufficienly large, ɛ ɛ() ɛ for some small ɛ >. Thus for sufficienly large, we have log x ɛ = ɛ log x /x s ɛ(s)ds ɛ log x = log x ɛ, or x ɛ e /x s ɛ(s)ds x ɛ for x, which implies ha for sufficienly large, U(/x) U() C d η xα+ɛ, for x, and for small ɛ >. 5

6 Since x α+ɛ M(dx) < for some small ɛ >, hen for any small δ >, here exiss a sufficienly large b > such ha C d η b x α+ɛ M(dx) δ 3, x α M(dx) I follows from (.3) ha for sufficienly large, b Therefore, for sufficienly large, U(/x) U() M(dx) b b U(/x) U() M(dx) x α M(dx) b U(/x) b U() M(dx) x α M(dx) + δ 3 + C d η and hus he i holds. b x α+ɛ M(dx) + δ 3 = δ, x α M(dx) δ 3. b b x α+ɛ M(dx) δ 3. U(/x) U() M(dx) + b x α M(dx) Proposiion.3 allows us o pass he i of ail raio hrough inegraion, which faciliaes asympoic analysis of TCEs. Proposiion.4. Le (X, X ) = (RT, RT ) be a bivariae random vecor, where (T, T ), independen of random variable R, has finie momens E(T ), E(T T α ) and E(T α+ɛ ) for some ɛ >. If R has he survival funcion R() := PrR > } = α L() RV α wih ail index α >, hen we have E(X X > ) α α E[T T α E[T α ] ], as. (.4) Proof. Observe firs ha E(T ) < and E(T T α ) < imply ha E(T T α ) <. Consider, PrX > x, X > } E(X X > ) = dx. PrX > } I follows from Breiman s heorem (see, e.g., pages 3-3 of []) and E(T α+ɛ ) < for some ɛ > ha X = RT RV α wih PrX > } E(T α ) PrR > }, for sufficienly large. (.5) 6

7 Le x = w, we have E(X X > ) = = = PrX > } PrX > } PrX > } Pr R + R + R > w PrX > w, X > }dw, R > Pr R > max w, } dwdf (, ) }} dwdf (, ), (.6) where F denoes he join disribuion of (T, T ). For he inner inegral, we have w Pr R > max, }} dw = Pr R > } dw + Pr R > w } dw = Pr R > } + For he firs summand in (.7), observe ha } } Pr R > Pr R > PrR > } = PrR > } Pr R > w } dw. (.7) = α, for any >, >, where R has he survival funcion R () := α+ L() RV α+ wih α >. Le M (B) := B R + df (, ), B R +, denoe he marginal mean measure induced by T, and M ( ) is a finie measure due o he fac ha E(T ) <. Thus by Proposiion.3 wih E(T T α ) <, } } Pr R > Pr R > R + PrR > } df (, ) = R + PrR > } M (d ) = E(T T α ). (.8) For he second summand in (.7), le x = w, and i follows from (.) ha Pr R > w } dw = PrR > x}dx α α PrR > }, as. Le R () := ha PrR > x}dx = R(x)dx RV α+ wih α >. Observe from (.3) } Pr R > w dw PrR > } R(x)dx ( = R() 7 ) R (/ ) R () α α, as,

8 and i follows from Proposiion.3 wih E(T T α ) < ha Pr R > w R + PrR > } } dw Plugging (.7) ino (.6), we have E(X X > ) + df (, ) = = PrR > } PrX > } R + Using (.5), (.8) and (.9), we obain ha E(X X > ) as desired. R(x)dx R() R (/ ) R + R () M (d ) = (α ) E(T T α ). (.9) PrR > } PrX > } R + R > w Pr R > } PrR > } df (, ) } Pr dw df (, ). PrR > } [ ( PrR > } = PrX > } E T T α + )] = α α α E [ ] T T α, ET α The ail esimaes of various TCEs can be obained immediaely from Proposiion.4. For any random vecor (X,..., X d ) = (RT,..., RT d ), observe ha (X i, X ) = (RT i, R T ) where denoes a norm on R d, and hus he following resul follows from Proposiion.4. Theorem.5. Le X = (X,..., X d ) = (RT,..., RT d ) be a random vecor, where R RV α wih α >, and T = (T,..., T d ), independen of R, has finie momens E(T i ), E(T i T α ), i d, and E( T α+ɛ ) for some ɛ >, wih respec o a norm. Then, for any i d, we have E(X i X > VaR p ( X )) α α E[T i T α ] VaR E[ T α p ( X ), as p. ] For example, ake he l norm, we have ha for any i d, as p, ( ( T d i T ) ) α i E ( X i d d X i > VaR p ( X i ) ) α E α For he l norm, we have for any i d, E ( X i X (d) > VaR p (X (d) ) ) α E(T i T α α 8 E( d j= T j) α VaR p ( ET α (d) (d) ) d ) X i. (.) VaR p ( X(d) ), as p,

9 where X (d) and T (d) denoe he larges order saisics of X,..., X d and T,..., T d respecively. In fac, as indicaed in Proposiion.4, he TCE of a homogeneous funcion of X,..., X d given ha anoher homogeneous funcion 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.4. I is worh emphasizing here ha he asympoic proporionaliy consans of he TCEs discussed in Theorem.5 depend on ail index α and also on he dependence srucure of T,..., T d. For example, consider he bivariae case wih α =, and we have for i =, E(X X + X > VaR p (X + X )) ET + ET T VaR ET + ET 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 + ET T = if ET = ET, ET + ET 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 risks 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 F () = ( + ) α for and α >, and le (T, T ) have a bivariae Marshall-Olkin exponenial disribuion funcion on [, ) [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(T i T j ) for any non-negaive inegers i, j can be calculaed explicily. I is easy o see ha VaR p (R) = ( p) /α. I follow from Breiman s heorem (see page 3-3 of []) ha PrX + X > } E(T + T ) α PrR > } for sufficienly large, which implies ha ( ) E(T + T ) α /α VaR p (X + X ), as p. p 9

10 Taking he l norm in Theorem.5 and (.) gives us he ail esimae of TCE for i =,, E ( X i X + X > VaR p (X + X ) ) α ( ) α [ (E(T ) ET i T + T + T ) α /α ], α E(T + T ) α p as p. For ineger-valued α, hese ail esimaes can be evaluaed analyically. For example, if α =, 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 E ( T T ) = (λ + λ ), E(T ) = (λ + λ ), E(T ) = (λ + λ )λ + (λ + λ )λ, λ + λ, var(t ) = (λ + λ ). (λ + λ ), where λ = λ + λ + λ. By using all he above equaions, we obain he ail esimaes: as p, E ( X X + X > VaR p (X + X ) ) VaR p (X + X ) VaR p (X + X ) ( (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

11 survival funcion of X i is hen given by F i () = E(e /R ) = ϕ(),. Assume ha ϕ() RV α wih ail index α >. The scale mixure (X,..., X d ) has regularly varying margins and Archimedean copula dependence srucure [7]. For example, if R has he gamma disribuion wih uni scale parameer and shape parameer α >, hen he Laplace ransform ϕ() = ( + ) α, leading o he so called Clayon copula dependence srucure for X. Take he l norm, and we have, as p, ( ( E ( d d X i X i > VaR p ( X i ) ) α E T d i T ) ) α i α E( ( d ) d j= T VaR p X i. j) α For example, if α =, 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 ) α E T i T i = d ( d ) α E T i T i = d d d E( T i ) α. Thus for any α >, E ( X i d d X i > VaR p ( X i ) ) α d(α ) 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 [,, 4, 9], and our ail esimae for TCE in erms of VaR complemens hese asympoic resuls. Example.8. Le Σ be a d d posiive semi-definie marix, and U = (U,..., U m ) be uniformly disribued on he uni sphere in R m. Consider he sochasic represenaion X T = (X,..., X d ) T = (µ,..., µ d ) T + RA(U,..., U m ) T, where A is an d m marix wih AA T = Σ and R > 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. The examples of ellipical disribuions include he mulivariae normal, - and logisic disribuions.

12 Wihou loss of generaliy, we choose (µ,..., µ d ) = (,..., ). Le (T,..., T d ) T = A(U,..., U d ) T, hen X = (X,..., X d ) = R(T,..., T d ). If R RV α wih ail index α >, hen X has mulivariae regularly varying ails. Take he l norm in Theorem.5 and α = for example, and we have, i d, E(X i d X i > VaR p ( d X i)) VaR p ( d X i) E(Ti+) + d j=,j i E(T i+t j+ ) d j= E(T j+ ) + d m,n=,m n E(T, as p, m+t n+ ) where T k+ := maxt k, } for k =,..., d. For he bivariae disribuion (X, X ) = (RT, RT ) where (T, T ) has he bivariae sandard normal disribuion wih correlaion coefficien ρ, we have, as p, σ + σ E(X X + X > VaR p (X + X )) VaR σ + σ + σ p (X + X ), σ + σ E(X X + X > VaR p (X + X )) VaR σ + σ + σ p (X + X ). where σ ii and σ are he variances and correlaion of (T +, T + ) wih bivariae runcaed sandard normal disribuion, and can be calculaed from correlaion coefficien ρ (see page 33 of [5]). 3 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 variable i, as measured by TCEs. The risk allocaion/decomposiion wih TCE for ellipically disribued loss vecors can be found in [6], and a general discussion on his opic can be found in []. 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 inference). These proporionaliy consans in Theorem.5 can be evaluaed using numerical inegraions. For example, if loss variables follow a mulivariae

13 -disribuion, hen he proporionaliy consan can be evaluaed hrough numerical inegraions on mulivariae normally disribued variables (see []). This is beer han fiing a disribuion o he enire daa for simulaion of he ail raio of TCE over VaR, o avoid exrapolaion from a fi ha is dominaed by he middle of he daa, and o reduce he simulaion sample size. References [] Albrecher, H., Asmussen, S. and Korschak, D. (6). Tail asympoics for he sum of wo heavy-ailed dependen risks. Exremes, 9:7 3. [] Alink, S., Löwe, M. and Wührich, M. V. (4). Diversificaion of aggregae dependen risks. Insurance: Mah. Econom., 35: [3] Alink, S., Löwe, M. and Wührich, M. V. (5). Analysis of he expeced shorfall of aggregae dependen risks, ASTIN Bullein, 35():5 43. [4] Alink, S., Löwe, M. and Wührich, M. V. (7). Diversificaion for general copula dependence. Saisica Neerlandica, 6: [5] Arzner, P., Delbaen, F., Eber, J.M. and Heah, D. (999). Coheren measures of risks. Mahemaical Finance 9:3 8. [6] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (987). Regular Variaion. Cambridge Universiy Press, Cambridge, UK. [7] Cai, J. and Li, H. (5). Condiional ail expecaions for mulivariae phase-ype disribuions. J. Appl. Prob. 4:8 85. [8] Delbaen, F. (). Coheren risk measure on general probabiliy spaces. Advances in Finance and Sochasics-Essays in Honour of Dieer Sondermann, Eds. K. Sandmann, P. J. Schönbucher, Springer-Verlag, Berlin, -37. [9] Embrechs, P., Neslehová, J. and Wührich, M. V., (9). Addiiviy properies for value-a-risk under Archimedean dependence and heavy-ailedness. Insurance: Mahemaics and Economics, 44(): [] Genz, A. and Brez, F. (9). Compuaion of Mulivariae Normal and Probabiliies. Springer, New York. 3

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