On multivariate extensions of the conditional Value-at-Risk measure

On Mul i a ia e Ex ensions o he Condi ional Value-a -Risk measu e Di Be na dino, E.a, Fe n´andez-Ponce, J.M.b,∗, Palacios-Rod ´ıguez, F.b, Rod ´ıguez-G i˜nolo, M.R.c aCNAM, Pa is, D´epa emen IMATH, Labo a oi e C´ed ic EA4629, 292 Rue Sain -Ma in, Pa is 75003, F ance bUni e sidad de Se illa, Depa amen o de Es ad´ıs ica e In es igaci´on Ope a i a, Calle Ta ia sin n´ume o, 41012 Se illa, Espa˜na cUni e sidad Pablo de Ola ide, Depa amen o de Econom´ıa, M´e odos Cuan i a i os e His o ia Econ´omica, Ca e e a U e a, km 1, 41013 Se illa, Espa˜na Abs ac CoVaR is a sys emic isk measu e p oposed by Ad ian and B unne meie [1] able o measu e a inancial ins i u ion’s con ibu ion o sys emic isk and i s con ibu ion o he isk o o he inancial ins i u ions. CoVaR s ands o condi ional Value-a -Risk, i.e. i indica es he Value a Risk o a inancial ins i u ion ha is condi ional on a ce ain scena io. In his pape , wo al e na i e ex ensions o he classic uni a ia e Condi ional Value-a -Risk a e in oduced in a mul i a ia e se ing. The wo p oposed mul i a ia e CoVaRs a e cons uc ed om le el se s o mul i a ia e dis ibu ion unc ions ( esp. o mul i a ia e su i al dis ibu ion unc ions). These ec o - alued measu es ha e he same dimension as he unde lying isk po olio. Se e al cha ac e iza ions o hese new isk measu es a e p o ided in e ms o he copula s uc u e and s ochas ic o de ings o he ma ginal dis ibu ions. In e es ingly, hese esul s a e consis en wi h exis ing p ope ies on uni a ia e isk measu es. Fu he mo e, compa isons be ween exis en isk measu es and he p oposed mul i a ia e CoVaR a e de eloped. Illus a ions a e gi en in he class o A chimedean copulas. Es ima ion p ocedu e o he mul i a ia e p oposed CoVaRs is illus a ed in simula ed s udies and insu ance eal da a. Keywo ds: Copulas and dependence, Le el se s o dis ibu ion unc ions, Mul i a ia e isk measu es, S ochas ic o de s, Value-a -Risk. In oduc ion A isk-based app oach o supe ision and egula ion o he inancial sec o is gaining g ound in bo h eme ging and indus ialized coun ies. As pa o his app oach, egula o s need o measu e, moni o , and manage ma ke isk. Value-a -Risk (VaR) is one measu e being explo ed ∗Co esponding au ho : e p[email p o ec ed] P ep in submi ed o Else ie F iday 31s Oc obe , 2014 o his pu pose. One o he mos impo an sec o s in which his p ac ice has been adop ed is he pension und indus y. As he ecen inancial c isis has shown, isks a e gene ally di icul o measu e and o manage. This becomes c ucial in he case o pensions, whe e people ely on hei sa ings o inance hei old age. Risk is a complex no ion and can ake on a ied o ms wi h di e se applica ions. In he con ex o ading i ms, managing isk has been adi ionally achie ed by he in oduc ion o Value-a -Risk (VaR) h esholds on he po olio isk accumula ed by ade s. O e ecen decades, his p oblem has been handled mos ly in a uni a ia e e sion. Mo eo e , he isk alloca ion p oblem only in ol es in e nal isks associa ed wi h businesses in he subsidia ies. Howe e , he sol abili y o inancial ins i u ions could also be a ec ed by ex e - nal isks whose sou ces canno be con olled. These isks may also be s ongly he e ogeneous in na u e and di icul o di e si y away. One can hink, o ins ance, o sys emic isk o con agion e ec s in a s ongly in e connec ed sys em o inancial companies. In he las decade, much esea ch has been de o ed o isk measu es and many mul idimensional ex ensions ha e been in es iga ed. On heo e ical g ounds, Jouini e al. [23] p opose a class o se - alue cohe en isk measu es. Unsu p isingly, he main di icul y ega ding mul i a ia e gene aliza ions o isk measu es is he ac ha ec o p eo de s a e, in gene al, pa ial p eo de s. In o de o gene alize he Value-a -Risk measu e, Emb ech s and Pucce i [15], Nappo and Spizzichino [30], and P ´ekopa [32] use he no ion o quan ile cu e which is de ined as he bounda y o he uppe -le el se o a dis ibu ion unc ion o he lowe -le el se o a su i al unc ion. Cousin and Di Be na dino [6] in oduce wo al e na i e ex ensions o he classic uni a ia e Value-a -Risk in a mul i a ia e se ing. The p oposed measu es, which a e based on he de ini ions o mul i a ia e quan iles in Emb ech s and Pucce i [15], a e eal- alued ec o s wi h he same dimension as he conside ed po olio o isks. This ea u e can be conside ed ele an om an ope a ional poin o iew. Bo h measu es sa is y he posi i i e homogenei y and ansla ion in a iance p ope y. Cousin and Di Be na dino [7] p opose wo ex ensions o he classic uni a ia e Condi ional-Tail-Expec a ion (CTE ) in a mul i a ia e se ing. These mul i a ia e ex ensions in Cousin and Di Be na dino [6] and Cousin and Di Be na dino [7] a e cons uc ed om le el se s o mul i a ia e dis ibu ion unc ions and mul i a ia e su i al dis ibu ion unc ions, espec i ely. This le el se s app oach is also used in his pape . Ano he ecen in e es ing isk measu e is ha o he CoVaR, which s ands o Condi ional Value-a -Risk. CoVaR is a sys emic isk measu e p oposed by Ad ian and B unne meie [1] ha measu es a inancial ins i u ion’s con ibu ion o sys emic isk and i s con ibu ion o he isk o o he inancial ins i u ions. In he o iginal unidimensional model, he CoVaR (o a pa icula bank, po olio o asse , e c.) indica es he Value-a -Risk o a inancial ins i u ion which is condi ional on a ce ain (s ess) scena io. Assume now ha Xj ep esen s asse e u ns o he inancial sys em (o bank j) and Xi ep e- sen s he asse e u ns o bank i. The CoVaRj|i αcan hen be de ined by: P[Xj≤CoVaRj|i α|Xi= VaRq(Xi)] = α, o α∈(0,1),(1) whe e VaRα(Xi) is he quan ile unc ion o he andom a iable Xia isk-le el α, i.e., VaRα(Xi) = 2 in {x∈R:FXi(x)≥α}. Equa ion (1) implici ly de ines he CoVaR o he bank jwhich is condi ional on bank ibeing a i s α%-VaR le el (see Ad ian and B unne meie [1]). CoVaR in (1) ep esen s one o he majo h eads in he cu en egula o y and scien i ic discus- sion o sys emic isks. In he li e a u e, se e al al e na i e de ini ions o CoVaR can be ound (see Gi a di and E g¨un [19] and Goodha and Sego iano [21]). S a ing om (1), we can also conside he CoVaR gi en by CoVaRj α(X) = VaRα(L|Xj≥VaRα(Xj)), whe e he inancial sys em is ep esen ed ia he o al isk L=X1+. . .+Xd, i.e., he agg ega ed o al isk o he i m ne wo k and he componen jo he ec o X= (X1, . . . , Xd) ep esen s he isk exposu e o he company j. In his pape , wo new mul i a ia e gene aliza ions o CoVaR based on he mul i a ia e quan- ile se ings o Emb ech s and Pucce i [15], Cousin and Di Be na dino [6], and Cousin and Di Be na dino [7] a e in oduced. These p oposed CoVaR measu es can be use ul in he analy- sis o mul iple inancial ins i u ions all oge he in he sys emic con ex . Se e al p ope ies ha e been ob ained. In pa icula , he posi i e homogenei y and ansla ion p ope y a e shown. The beha iou o he componen s o he p oposed CoVaR ec o s wi h es- pec o he uni a ia e VaR o ma gins and o he mul i a ia e VaR in Cousin and Di Be na dino [6] is also analysed. We also s udy how hese measu es a e in luenced by a change in ma ginal dis ibu ions, by a change in dependence s uc u e, and by a change in isk le el. Ad ian and B unne meie [1] de ined a sys emic isk measu e, called ∆CoVaR, as he di e ence be ween he VaR o he ins i u ion j(o inancial sys em) condi ional on he dis ess o a pa icula inancial ins i u ion i(see (1)) and he VaR o he ins i u ion j. ∆CoVaR and o he in e es ing sys emic isk measu es a e in oduced and ga he ed in Mainik and Schaanning [24]. The in-dep h s udy o ∆CoVaR sys emic isk measu es using he mul i a ia e CoVaR p oposed in his pape goes beyond he scope o he p esen wo k. A mo e p ac ical analysis on sys emic isks using mul i a ia e ∆CoVaR measu es is cu en ly in p epa a ion. The pape is o ganized as ollows. In Sec ion 1, he piecewise-linea weigh ed loss unc ion which, can be used o gene alize se e al isk measu es, is in oduced. Mo eo e , some no a ions, ools, and echnical assump ions a e gi en. In Sec ion 2, p ope ies o in a iance o he p oposed mul i a ia e CoVaR a e shown. Fu he mo e, we analyse how hese mul i a ia e measu es beha e when he ma ginal isks o he copula s uc u es inc ease wi h espec o s ochas ic o de s (see Sec ion 3). Illus a ions and p ope ies o he A chimedean copula class a e p esen ed in Sec ion 4. In Sec ion 5, es ima ion p ocedu e o he mul i a ia e p oposed CoVaRs is illus a ed in simula ed s udies and insu ance eal da a. Conclusion discusses open p oblems and possible di ec ions o u u e wo k. 3 1. P elimina ies and De ini ions Le Xbe a non-nega i e andom a iable wi h dis ibu ion unc ion FXand quan ile unc ion a le el ωin [0,1] gi en by QX(ω) = in {x:FX(x)≥ω}. No e ha he quan ile unc ion is also de ined as a Value-a -Risk in he economics li e a u e and deno ed as VaRω(X) (see also (1)). Le L1(Ω,A, P) be he se o all andom a iables wi h ini e expec a ions. Assuming ha Xis a andom a iable o L1, he Weigh ed Loss unc ion (WL) is de ined by LX(x;ω) = ωE[(X−x)+] + (1 −ω)E[(X−x)−] o all x∈Rand ω∈[0,1],(2) whe e x+= max{x, 0}and x−= max{−x, 0}. No e ha i Xis a non-nega i e andom a iable, hen LX(x;ω) = ωE[X] o all x < 0. This unc ion has a key ole in an ac ua ial con ex . Indeed, i ep esen s he expec ed cos o he einsu ance company, called ne p emium, whe e Xdeno es he isk o he insu ance com- pany. I he insu ance company p e e s no o bea all he isk, passes on pa s o he isk o a einsu ance company. The pa e ained by he o iginal insu ance company is usually called he e en ion. A s op-loss con ac es ablishes a ixed e en ion x(see Sec ion 8.3 in M¨ulle and S oyan [29]). This means ha he maximum isk o he insu ance company is x. Thus, i X > x hen, he einsu ance company will ake o e X−x. This class o con ac s is use ul o p o ec companies om insol ency due o excessi e claims. In an ac ua ial con ex , he h eshold xis o en called he deduc ible o p io i y (see Sec ion 1.7.1 in Denui e al. [11]). Ce ain in e es ing p ope ies o he WL unc ion in (2) a e now ecalled. The p ope ies (P1)- (P6) a e i ially ob ained by he same a gumen s as hose used by Mu˜noz P´e ez and S´anchez- G´omez [27] o p o e he p ope ies o he dispe sion unc ion. (P1) I holds ha LX(x;ω) = ωZ+∞ x ¯ F( ) d + (1 −ω)Zx −∞ F( ) d . (P2) Le CFdeno e he se o con inui y poin s o FXand X∈ L1. Then FX(x) = L0 X(x;ω) + ω, ∀x∈CFand x≥0 whe e L0 Xis he de i a i e o LXwi h espec o x. (P3) The WL unc ion is di e en iable and i s de i a i e has, a mos , a coun able numbe o discon inui y poin s. (P4) LX(x;ω) is a con ex unc ion on R+. (P5) limx→+∞L0 X(x;ω)=1−ω; and limx→−∞ L0 X(x;ω)=0. (P6) limx→+∞[LX(x;ω)−(1 −ω)x] = −(1 −ω)E[X]. 4 (P7) Finally, VaRω(X) = a g min x∈R+ LX(x;ω), o w∈[0,1], wi h VaR0(X) = xF−and VaR1(X) = xF+, whe e xF+and xF−a e, espec i ely, he igh and le endpoin s o F, such ha xF+= sup{x∈R:F(x)<1}and xF−= in {x∈ R:F(x)>0}. I is easy o see ha P ope ies (P1)-(P7) uniquely cha ac e ize a WL unc ion, i.e., i LX(x;ω) is a unc ion ha sa is ies P ope ies (P1)-(P7) abo e, hen he e exi s a unique dis ibu ion unc ion which has LX(x;ω) as i s WL unc ion. The e o e, i uniquely de e mines a p obabili y measu e PFon B( he σ- ield o Bo el se on R). An in e es ing in e p e a ion o he WL unc ion is ha 2 LX(x; 1/2) is he L1-dis ance be ween FXand Fx, whe e Fxis he dis ibu ion unc ion o he degene a e andom a iable a he poin x∈R(Mu˜noz P´e ez and S´anchez-G´omez [27]). I is also in e es ing o ema k ha LX(x; 1) is he well-known s op-loss unc ion o X, and ha LX(x; 0) could be in e p e ed as he s op-gain unc ion o X. Consequen ly, he WL unc ion is a weigh ing o bo h unc ions in e ms o x. Now, le X= (X1, . . . , Xd) be a non-nega i e d-dimensional andom ec o 1. Cousin and Di Be na dino [6] de ined, unde ce ain egula i y condi ions, he mul i a ia e Lowe -O han Value-a -Risk a p obabili y le el αas he d-dimensional ec o VaRα(X) = E[X|F(X) = α], o α∈(0,1), whe e Fis he dis ibu ion unc ion o X. Pa icula ly, he i- h componen o his ec o i ially e i ies VaRi α(X) = LXi|F(X)=α(0; 1).(3) Using P ope y (P7), ou pu pose is now o gi e a new mul i a ia e app oach o he classic Condi ional Value-a -Risk model (see CoVaR in (1)) which, as in oduced p e iously, is de ined as he VaR o a inancial ins i u ion, condi ional on a ce ain scena io (see Ad ian and B unne meie [1]). In his case, he app oach is based on he condi ional scena io being a es ic ion o bo h inancial ins i u ions. Thus, in gene al, no ela ionship exis s be ween he wo CoVaRs. F om now on, assume ha X= (X1, . . . , Xd) is a non-nega i e absolu ely-con inuous andom ec o (wi h espec o Lebesgue measu e λon Rd) wi h dis ibu ion unc ion Fand su i al unc ion F. Fu he mo e, he mul i a ia e dis ibu ion unc ion Fis assumed o be pa ially s ic ly-inc easing2such ha E(Xi)<∞ o i= 1, . . . , d. Such Fis said o e i y he egula i y 1We es ic ou sel es o Rd +because, in ou applica ions, componen s o d−dimensional ec o s co espond o andom losses and a e hen alued in R+. 2A unc ion F(x1,...,xn) is pa ially s ic ly-inc easing on Rd + 0i he unc ion o one a iable g(·) = F(x1,...,xj−1,·, xj+1,...,xd) a e s ic ly-inc easing. 5 condi ions. No e ha i Fis he su i al unc ion o X, and F e i ies he egula i y condi ions, hen Fis a pa ially s ic ly-dec easing unc ion. Unless s a ed o he wise, he dimension o he ec o s is d, and he null ec o o dimension dwill be deno ed by 0, and he uni y ec o o dimension dby 1. The e o e, he o de ≤be ween ec o s will be conside ed componen -wise. Th oughou he pape , gi en a andom a iable o a ec o Xand any e en A,X|Ais deno ed as he andom a iable o ec o whose dis ibu ion is he condi ional dis ibu ion o Xgi en A. E en ually, he equali y in law is gi en by d =. Se e al use ul de ini ions o s ochas ic o de s a e now ecalled. Fu he de ails, equi alen de i- ni ions and applica ions may be ound in Shaked and Shan hikuma [37], M¨ulle [28], and Joe [22]. De ini ion 1.1. Le Xand Ybe wo andom a iables wi h dis ibu ion unc ions FXand FY espec i ely. Xis said o be smalle han Yin he usual s ochas ic o de , deno ed by X≤s Y, i FX(x)≥FY(x), o all x∈R. De ini ion 1.2 (Supe modula unc ion).A unc ion :Rd→Ris said o be supe modula i , o any x,y∈Rd, i sa is ies (x) + (y)≤ (x∧y) + (x∨y), whe e he ope a o s ∧and ∨deno e coo dina e-wise minimum and maximum espec i ely. De ini ion 1.3 (Supe modula O de ).Le Xand Ybe wo d−dimensional andom ec o s. Xis said o be smalle han Ywi h espec o he supe modula o de (deno ed by X≤sm Y) i E( (X)) ≤E( (Y)), o all supe modula unc ions :Rd→R, p o ided he expec a ions exis . In De ini ion 1, om he discussion abo e, a mul i a ia e gene aliza ion o he CoVaR measu e is now in oduced. De ini ion 1 (Mul i a ia e Lowe -O han CoVaR).Conside a andom ec o Xwhich sa is ies he egula i y condi ions. Fo α∈(0,1), we de ine he mul i a ia e lowe -o han CoVaR a p obabili y le el αby CoVaRα,ω(X) = VaRω(X|X∈∂L(α)) =    VaRω1(X1|X∈∂L(α)) . . . VaRωd(Xd|X∈∂L(α))   ,(4) 6 whe e ω= (ω1, . . . , ωd)is a ma ginal isk ec o wi h ωi∈[0,1], o i= 1, . . . , d, and ∂L(α)is he bounda y o he se L(α) := {x∈Rd +:F(x)≥α}. The e o e, CoVaRα,ω(X) =    VaRω1(X1|F(X) = α) . . . VaRωd(Xd|F(X) = α)   .(5) In a simila way, he mul i a ia e uppe -o han CoVaR can be de ined. De ini ion 2 (Mul i a ia e Uppe -O han CoVaR).Conside a andom ec o Xwhich sa is- ies he egula i y condi ions. Fo α∈(0,1), we de ine he mul i a ia e uppe -o han CoVaR a p obabili y le el αby CoVaRα,ω(X) = VaRω(X|X∈∂L(α)) =    VaRω1(X1|X∈∂L(α)) . . . VaRωd(Xd|X∈∂L(α))   ,(6) whe e ω= (ω1, . . . , ωd)is a ma ginal isk ec o wi h ωi∈[0,1], o i= 1, . . . , d, and ∂L(α)is he bounda y o he se L(α) := {x∈Rd +:F(x)≤1−α}. The e o e, CoVaRα,ω(X) =    VaRω1(X1|F(X)=1−α) . . . VaRωd(Xd|F(X) = 1 −α)   .(7) Rema k 1.1. Using he same no a ion and amewo k o De ini ions 1 and 2, we can also conside a modi ied e sion o he mul i a ia e uppe and lowe CoVaR p oposed in Equa ions (4) and (6). Indeed, conside a inancial ins i u ion Xiand he i m ne wo k wi hou Xi, i.e., (X1, . . . , Xi−1, Xi+1, . . . , Xd) := Xd−1. The ollowing modi ied e sion o he lowe CoVaR in De ini ion 1 can he e o e be p oposed: CoVaRi α,ω(X) = VaRωi(Xi|F(Xd−1) = α), whe e Fd−1is he (d−1)-dimensional dis ibu ion unc ion associa ed o he ec o Xd−1. Ana- logously, a modi ied e sion o he uppe CoVaR in De ini ion 2 can be : CoVaRi α,ω(X) = VaRωi(Xi|F(Xd−1)=1−α), whe e Fd−1is he su i al (d−1)-dimensional dis ibu ion unc ion associa ed o he ec o Xd−1. I should be bo ne in mind ha , using his modi ied e sions, when d= 2 and ωi=α, CoVaRα,ω(X)and CoVaRα,ω(X)become he classic CoVaR in (1). 7 The ollowing in e p e a ion o ou measu es can be conside ed. The i h componen o mul i- a ia e lowe -o han CoVaR o X( esp. mul i a ia e uppe -o han CoVaR o X) co esponds o he poin x∗ ha minimizes he WL unc ion o he associa ed i h ma ginal gi en ha X s ands in he α−le el cu e o i s mul i a ia e dis ibu ion unc ion ( esp. mul i a ia e su i al dis ibu ion unc ion). I is wo h men ioning ha unde egula i y condi ions, ∂L(α) ( esp. ∂L(α)) is he α-le el cu e ( esp. (1 −α)-le el cu e) o F( esp. F) (see o ins ance Di Be na dino e al. [12], Cue as e al. [8]). This means ha he e is no pla eau in he g aph o F o each le el α. The e o e, egula i y condi ions gua an ee ha he minimize x∗is unique o each componen i= 1, . . . , d. T i ially, gi en ha ou CoVaRs a e he minimize s o sui able expec ed losses (see (P7)), hey he e o e e i y he elici abili y p ope y. This p ope y was s udied by Gnei ing [20], while Bellini and Bignozzi [4] sugges ed a sligh ly mo e es ic i e de ini ion. Recen ly, Emb ech s and Ho e [13] s a ed ha elici abili y is a e y impo an p ope y o a isk measu e since i p o ides a na u al me hodology o pe o m back es ing. Ziegel [41] has also s udied he connec ions be ween elici abili y and cohe ence p ope ies o isk measu es. Mo eo e , he sol ency o an insu ance company depends on he equency o la ge claims. One o he ad an ages o wo king wi h he quan ile unc ion is ha his unc ion is mo e obus o ex eme alues han o he cen al endency measu es. 2. P ope ies o he mul i a ia e CoVaR In his sec ion, he aim is o analyse he lowe -o han and uppe -o han CoVaR in oduced in De ini ions 1 and 2 in e ms o classic sui able p ope ies o isk measu es (see, o ins ance, A zne e al. [2], Denui e al. [11]). We ocus on in a iance p ope ies (see Sec ion 2.1). Fu he mo e, in Sec ion 2.2, he ela ionships be ween ou CoVaR, he uni a ia e VaR, and he mul i a ia e VaR in oduced by Cousin and Di Be na dino [6] a e analysed. In Sec ion 2.3, some comono onic dependence p ope ies o ou measu es a e in es iga ed. 2.1. In a iance p ope ies The ollowing esul s (P oposi ion 2.1 and Co olla y 2.1) a e now in oduced, which will be cen al in p o ing in a iance p ope ies o ou isk measu es. P oposi ion 2.1. Le he unc ion hbe such ha h(x1, . . . , xd)=(h1(x1), . . . , hd(xd)). Le ω be a ec o in [0,1]dand α∈(0,1). (1) I h1, . . . , hda e non-dec easing unc ions, hen, o i= 1, . . . , d, CoVaRi α,ω(h(X)) = VaRωi(hi(Xi)|F(X) = α). 8 (2) I h1, . . . , hda e non-inc easing unc ions, hen, o i= 1, . . . , d, CoVaRi α,ω(h(X)) = VaRωi(hi(Xi)|F(X) = α). P oo . By De ini ion 1, CoVaRi α,ω(h(X)) = VaRωi(hi(Ti)) = a g min x∈[hi(VaRα(Xi)),+∞)ωiE[(hi(Ti)−x)+] + (1 −ωi)E[(hi(Ti)−x)−], whe e hi(Ti)=[hi(Xi)|Fh(X)(h(X)) = α], o i= 1, . . . , d. Since Fh(X)(y1, . . . , yd) = F(h−1 1(y1), . . . , h−1 d(yd)) i h1, . . . , hda e non-dec easing unc ions, F(h−1 1(y1), . . . , h−1 d(yd)) i h1, . . . , hda e non-inc easing unc ions, hen CoVaRi α,ω(h(X)) = VaRωi(hi(Xi)|F(X) = α) i h1, . . . , hda e non-dec easing unc ions, VaRωi(hi(Xi)|F(X) = α) i h1, . . . , hda e non-inc easing unc ions. As in P oposi ion 2.1, a simila esul can also be ob ained o he mul i a ia e uppe -o han CoVaR, by in e changing Fwi h F. F om P oposi ion 2.1, one can i ially ob ain he ollowing p ope y which links he mul i a ia e uppe -o han CoVaR and lowe -o han CoVaR. Co olla y 2.1. Le hbe a linea unc ion such ha h(x1, . . . , xd) = (h1(x1), . . . , hd(xd)). Le ωbe a ec o in [0,1]dand α∈(0,1). (1) I h1, . . . , hda e non-dec easing unc ions, hen CoVaRα,ω(h(X)) = h(CoVaRα,ω(X)) and CoVaRα,ω(h(X)) = h(CoVaRα,ω(X)). (2) I h1, . . . , hda e non-inc easing unc ions, hen CoVaRα,ω(h(X)) = h(CoVaR1−α,1−ω(X)) and CoVaRα,ω(h(X)) = h(CoVaR1−α,1−ω(X)). The ollowing esul p o es he posi i e homogenei y and in a iance ansla ion p ope ies o isk measu es in De ini ions 1 and 2. 9 As a esul , any andom ec o U= (U1, . . . , Ud) which ollows an A chimedean copula wi h gene a o φcan be ep esen ed as a de e minis ic unc ion o C(U) and an independen andom ec o S= (S1, . . . , Sd) uni o mly dis ibu ed on he uni simplex, i.e., (U1, . . . , Ud)d = (φ−1(S1φ(C(U))), . . . , φ−1(Sdφ(C(U)))).(10) Co olla y 4.1. Le Xbe a d-dimensional andom ec o wi h an A chimedean copula wi h gene a o φand α∈(0,1). The e o e, CoVaRi α,ω(X) = VaRωihF−1 Xi(φ−1(Siφ(α)))i, o i= 1, . . . , d, (11) whe e ω∈[0,1]dand Siis a andom a iable wi h Be a(1, d −1) dis ibu ion. P oo . No e ha Xis dis ibu ed as (F−1 X1(U1), . . . , F−1 Xd(Ud)), whe e U= (U1, . . . , Ud) ollows an A chimedean copula Cwi h gene a o φ. Consequen ly, each componen i= 1, . . . , d o he mul i a ia e isk measu e in oduced in De ini ion 1 can be exp essed as CoVaRi α,ω(X) = a g min x∈[VaRα(Xi),+∞)ωiE[(Ti−x)+] + (1 −ωi)E[(Ti−x)−], whe e Ti= [F−1 Xi(Ui)|C(U) = α]. Mo eo e , om ep esen a ion (10), he ollowing ela ion is e i ied [U|C(U) = α]d = (φ−1(S1φ(α)), . . . , φ−1(Sdφ(α))),(12) since Sand C(U) a e s ochas ically independen . The esul comes om he ac ha he andom ec o S ollows a symme ic Di ichle dis ibu ion. No e ha , by using (12), he ma ginal dis ibu ions o Ugi en C(U) = αcan be exp essed in a e y simple way, ha is, P(Uk≤u|C(U) = α) = 1−φ(u) φ(α)d−1 o 0 <α<u<1,and any k= 1, . . . , d. (13) Co olla y 4.2. Le Xbe a d-dimensional andom ec o wi h an A chimedean su i al copula wi h gene a o φand α∈(0,1). The e o e, CoVaRi α,ω(X) = VaRωihF−1 Xi(φ−1(Siφ(1 −α)))i o i= 1, . . . , d, (14) whe e ω∈[0,1]dand Siis a andom a iable wi h Be a(1, d −1) dis ibu ion. The p oo is simila o Co olla y 4.1 and is he e o e omi ed he e. F om (11) and (14), analy ical exp essions o he lowe -o han and he uppe -o han CoVaR o a ec o X= (X1, . . . , Xd) wi h a pa icula A chimedean copula a e now de i ed. Assume 16 ha Xiis uni o mly-dis ibu ed on [0,1], o i= 1, . . . , d. Since A chimedean copulas a e exchangeable, he componen s o CoVaRα,ω(X) ( esp. CoVaRα,ω(X)) a e equal in he case whe e ω1=. . . =ωd. Fu he mo e, i is also possible o ob ain exp essions o he uppe - o han CoVaRα,ω o ˜ X= (1 −X1,...,1−Xd) since, by using Co olla y 2.1: CoVaRi α,ω(˜ X)=1−CoVaRi 1−α,1−ω(X). 4.1. Analy ical exp essions o CoVaR measu es o A chimedean copulas In he ollowing, Co olla y 4.1 is illus a ed o some commonly used A chimedean copula amilies (see Example 4.1, 4.2, 4.3). Example 4.1 (Bi a ia e Clay on amily).In Table 1 (le ), he bi a ia e andom ec o (X, Y ) is conside ed wi h uni o m ma ginal dis ibu ions and a Clay on copula wi h pa ame e θ≥ −1 is conside ed. One can eadily show ha ∂CoVaR1 α,ω ∂θ ≤0and ∂CoVaR1 α,ω ∂θ ≥0, o θ≥ −1, α ∈(0,1) and ω∈[0,1]. Hence, he componen s o he mul i a ia e CoVaR ( esp. CoVaR) a e dec easing ( esp. in- c easing) unc ions o he dependence pa ame e θ. In e es ingly, in he comono onic case, bo h mul i a ia e isk measu es CoVaR and CoVaR co espond o he ec o composed o he uni a i- a e VaR a le el αassocia ed wi h each componen . These p ope ies a e illus a ed in Figu e 1 whe e uppe and lowe CoVaR a e plo ed as unc ions o he isk le el ω o di e en alues o dependence pa ame e θand o a ixed le el α. No e ha , when he pa ame e θinc eases, he lowe CoVaR ends o dec ease. Con e sely, he uppe bound o he uppe CoVaR is ep e- sen ed by he pe ec posi i e dependence case. The la e empi ical beha iou s will be o mally con i med in he ollowing (see Co olla y 4.4). θCoVaR1 α,ω,θ(X, Y ) (−1,∞)1 + 1 αθ−1(1 −ω1)−1/θ −1 1 −(1 −ω1)(1 −α) 0α1−ω1 1α (1−α)(1−ω1)+α ∞α θCoVaR1 α,ω,θ(X, Y ) [−1,1) 1−θ 1−θ(1−α) α(1−ω1)−θ 0α1−ω1 Table 1: CoVaR1 α,ω(X, Y ), o a bi a ia e Clay on copula (le ) and a bi a ia e Ali-Mikhail-Haq copula ( igh ). 17 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Lowe −o han CoVaR ω θ=−1 θ=0 θ=1 θ=5 θ=∞ 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Uppe −o han CoVaR ω θ=−1 θ=0 θ=1 θ=5 θ=∞ Figu e 1: Beha iou o CoVaR1 α,ω(X, Y ) (le ) and CoVaR1 α,ω(1 −X, 1−Y) ( igh ) wi h espec o he isk le el ω o di e en alues o dependence pa ame e θand o α= 0.7. He e, (X, Y ) is a bi a ia e andom ec o wi h uni o m ma ginal dis ibu ions and a Clay on copula wi h pa ame e θ≥ −1. Example 4.2 (Bi a ia e Ali-Mikhail-Haq amily).Table 1 ( igh ) illus a es he analy ical ex- p essions o CoVaR o he i s componen o a bi a ia e andom ec o wi h uni o m ma ginal dis ibu ions and a Ali-Mikhail-Haq copula, o θ∈[−1,1). Recall ha bi a ia e A chimedean copulas can be ex ended o d−dimensional copulas, wi h d > 2, on he condi ion ha he gene a o φis a d−mono one unc ion in [0,∞) (see McNeil and Neˇsleho ´a [26]). The bi a ia e Gumbel amily can be gene alized in dimension d, o θ≥1 (see Example 4.25 in Nelsen [31]). Example 4.3 (3−dimensional Gumbel amily).In his case, analy ical exp essions o he i s componen o lowe CoVaR o a 3−dimensional andom ec o (X1, X2, X3)wi h uni o m ma ginal dis ibu ions and a Gumbel copula, o θ≥1a e p o ided in Table 2. 18 θCoVaR1 α,ω,θ(X1, X2, X3) [1,∞)α(1−√ω1)1/θ 1α(1−√ω1) ∞α Table 2: CoVaR1 α,ω(X1, X2, X3) o a 3−dimensional Gumbel copula. 4.2. Illus a ions o some p ope ies o A chimedean copulas In he ollowing, some heo e ical p ope ies p esen ed in Sec ion 2 a e illus a ed in he la ge class o d−dimensional A chimedean copula. Fi s ly, using Co olla y 4.1, an illus a ion o P oposi ion 2.4 in he Clay on copula case is p o ided. Example 4.4. Assume ha Xis a bi a ia e andom ec o wi h uni o m ma ginal dis ibu ions and Clay on copula. The dis ibu ion unc ion o Xis he e o e gi en by: F(x1, x2) = hmax{x−θ 1+x−θ 2−1,0}i−1/θ , o θ∈[−1,∞) {0}and (x1, x2)∈[0,1]2. Then, by s aigh o wa d compu a ion, one can ob ain, o α∈(0,1) and ω1∈[0,1], VaR1 α(X) = θ θ−1 αθ−α αθ−1,and CoVaR1 α,ω(X) = 1 + 1 αθ−1(1 −ω1)−1/θ , whe e VaR1 α(X)is he i s -componen lowe VaR p oposed by Cousin and Di Be na dino [6]. Consequen ly, bo h measu es coincide in ω∗=α−θ−θ θ−1 αθ−α αθ−1−θ[α−θ−1]−1. Fo a ixed α= 0.6we ob ain he esul s ga he ed in Figu e 2. VaRα(X) ep esen s he case ha he comple e isk o he insu ance company is einsu ed by ano he company (x= 0) (see Cousin and Di Be na dino [6]). The insu ance company gi es he o al weigh o he expec ed cos o he einsu ance company, ha is, es ablishes ω= 1. By con as , CoVaR de ines he minimum e en ion o he insu ance company gi en a weigh ω∈[0,1] o he expec ed cos o he einsu ance company. Fo ins ance, o θ= 2, i can be obse ed in Figu e 2 ha VaR1 0.6(X) = 0.75 and he cu -o poin is ω∗= 0.56. Simila ly, analy ical exp essions o mul i a ia e uppe CoVaR and compa isons wi h he associa ed VaRα(X)(see Cousin and Di Be na dino [6]) can be ob ained. Co olla y 4.3 p o es ha assump ions o P oposi ion 2.7 a e au oma ically sa is ied in he la ge class o d-dimensional A chimedean copulas. 19 0.0 0.2 0.4 0.6 0.8 1.0 0.6 0.7 0.8 0.9 1.0 Lowe −o han CoVaR ω θ=−0.99 θ=0.01 θ=2 θ=4 θ=10 Figu e 2: VaR1 α(X) and CoVaR1 α,ω(X). He e, (X, Y ) is a bi a ia e andom ec o wi h uni o m ma ginal dis i- bu ions and a Clay on copula wi h pa ame e θ≥ −1, and α= 0.6. Co olla y 4.3. Conside a d-dimensional andom ec o X, which sa is ies he egula i y con- di ions, wi h ma ginal dis ibu ions FXi, o i= 1, . . . , d, copula Cand su i al copula C. (1) I Cis a d-dimensional A chimedean copula, hen CoVaRi α,ω(X)is a non-dec easing unc- ion o αwi h ω∈[0,1]d. (2) I Cis a d-dimensional A chimedean copula, hen CoVaRi α,ω(X)is a non-dec easing unc- ion o αwi h ω∈[0,1]d. P oo . Le Ui=FXi(Xi), U= (U1, . . . , Un), Vi=FXi(Xi) and V= (V1, . . . , Vn). Since C is he copula o X, hen Uis dis ibu ed as C. I Cis an A chimedean copula, om (13), P(Ui> u|C(U) = α) is a non-dec easing unc ion o α. Simila ly, P(Vi> u|C(V) = 1 −α) is a non-dec easing unc ion o α. The esul s a e he e o e i ially de i ed om P oposi ion 2.7. In he ollowing, an illus a ion o P oposi ion 3.1 is p o ided in he A chimedean case. Example 4.5. Th ee di e en andom ec o s (X, Yi), o i= 1,...,3a e conside ed wi h he same bi a ia e Clay on copula wi h dependence pa ame e 2, such ha X∼Exp(1),Y1∼Exp(2),Y2∼Bu (5,1),Y3∼F ´eche (4). Since Y1≤s Y2≤s Y3, om P oposi ion 3.1, hen CoVaR2 α,ω(X, Y1)≤CoVaR2 α,ω(X, Y2)≤CoVaR2 α,ω(X, Y3), 20 o any ω∈[0,1]2and α∈(0,1). The esul s a e ga he ed in Figu e 3. I should also be emphasised ha , by Co olla y 3.1, he i s componen s o he mul i a ia e lowe -o han CoVaR and uppe -o han CoVaR o he ou ec o s coincide. 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Dis ibu ion Func ions x F(x) Exp(2) F éche (4) Bu (5, 1) 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 Lowe −O han CoVaR ω Exp(2) F éche (4) Bu (5, 1) Figu e 3: Le : Dis ibu ion unc ions o andom a iables Yi, o i= 1,...,3, wi h Y1∼Exp(2), Y2∼Bu (5,1) and Y3∼F ´eche (4). Righ : CoVaR2 α,ω(X, Yi) o i= 1,...,3, wi h he same copula Clay on wi h pa ame e 2, X∼Exp(1), Y1∼Exp(2), Y2∼Bu (5,1), Y3∼F ´eche (4) and α= 0.8. The ollowing ema k will be use ul in Co olla y 4.4. Rema k 4.1. Le Uand U∗be wo andom ec o s wi h copula Cand C∗, espec i ely, and wi h uni o m ma ginal dis ibu ions. I is easy o p o e ha U≤sm U∗implies C(u)≤C∗(u), o u∈[0,1]d(Sec ion 6.3.3 in Denui e al. [11]). In addi ion, o Gumbel, F ank, Clay on, and Ali-Mikhail-Haq amilies, i can be shown ha an inc ease o θyields an inc ease o dependence in he sense o he supe modula o de (see examples in Wei and Hu [39], Joe [22]). As a consequence, in hese cases, θ≤θ∗⇒C(u)≤C∗(u), o u∈[0,1]d.(15) Co olla y 4.4. Le Xbe a d−dimensional andom ec o sa is ying he egula i y condi ions wi h copula Cand su i al copula C. I Cis a d−dimensional A chimedean copula ha sa is ies P ope y (15) in Rema k (4.1), each componen o CoVaRα,ω(X)is a dec easing unc ion o θ, wi h α∈(0,1) and ω∈[0,1]d. I Cis a d−dimensional A chimedean copula ha sa is ies P ope y (15) in Rema k (4.1), each componen o CoVaRα,ω(X)is a inc easing unc ion o θ, wi h α∈(0,1) and ω∈[0,1]d. I should be no ed ha , o ins ance o Gumbel, F ank, Clay on and Ali-Mikhail-Haq amilies, assump ions o Co olla y 4.4 a e sa is ied. The eade is e e ed, o ins ance, o he beha iou o he lowe and uppe CoVaR wi h espec o he copula pa ame e θp esen ed in Figu e 1. 21 P oo . We conside wo A chimedean copulas o he same amily, Cθ(associa ed o ec o U) and Cθ∗(associa ed o ec o U∗) wi h gene a o φθand φθ∗such ha θ≤θ∗. By P oposi ion 3.2, we ha e o p o e ha [U∗ i|Cθ∗(U∗) = α]≤s [Ui|Cθ(U) = α] holds o i= 1, . . . , d. On he o he hand, om Eq. (13), i is eadily ob ained ha [U∗ i|Cθ∗(U∗) = α]≤s [Ui|Cθ(U) = α] o any α∈(0,1) ⇔φθ∗ φθ is a dec easing unc ion. Finally, by aking in o accoun Rema k 4.1 o Clay on, F ank, Gumbel and Ali-Mikhail-Haq amilies, he unc ion φθ∗ φθis dec easing when θ≤θ∗. The e o e, om P oposi ion 3.2, an inc ease o he pa ame e θyields a dec ease in each componen o CoVaRα,ω(X). The second s a emen is ob ained i ially using he same a gumen s. 4.3. A weak subaddi i i y ail p ope y in he A chimedean cases The addi i i y o ou CoVaR is p o ided in Sec ion 2.3 in a comono onic dependence ec o ial case (see P oposi ion 2.6 o π-comono onic ec o s). In he ollowing, he aim is o s udy he condi ion o a copula o ob ain subaddi i i y inequali ies o ou lowe CoVaR . To his end, as in he uni a ia e case (see Dan´ıelsson e al. [9]), we ocus on he ails o he conside ed mul i a ia e dis ibu ion. In he ollowing, wo no ions o egula a ia ion a e applied. A measu able unc ion U:R→R is egula ly a ying a ∞wi h index ρ(deno ed by U∈RVρ), i i holds ha lim →∞ U( x) U( )=xρ, o any eal numbe x > 0. Also, a andom ec o Xwi h join dis ibu ion unc ion Fis said o be mul i a ia e egula ly a ying (X∈MRV ) i he e exis s a Radon measu e νon [0,∞] {0}, such ha lim →∞ 1−F( x) 1−F( 1)=ν([0,x]c), o all poin s x∈[0,∞) {0}, which a e con inui y poin s o he unc ion ν([0,·]c). Obse e also ha o any non-nega i e MRV andom ec o X, i s non-degene a e uni a ia e ma gins Xi ha e egula ly a ying igh ails, ha is, Fi( ) := −βL( ), ≥0, whe e β > 0 is he ma ginal hea y- ail index and L( ) is a slowly a ying unc ion, i.e. L(x )/L( )→1 as → ∞ o any x > 0. Fu he de ails abou egula a ia ion can be ound in Resnick [34], Resnick [35] and Emb ech s e al. [14]. The e o e in his se ing, he ollowing esul can be ob ained. F om now on, he ollowing no a ion is conside ed. Le Xbe a bi a ia e andom ec o wi h dis ibu ion unc ion F, A chimedean copula Cand wi h same ma gins FXi,i= 1,2. Le us deno e Ti= [Xi|F(X) = α], o α∈(0,1), i= 1,2. Theo em 4.1. Assume ha φis wice di e en iable and ha (φ◦FX1)∈RV−β,β > 0. Then T:= (T1, T2)∈MRV . 22 P oo . Fi s ly, he copula o andom ec o Tis compu ed. No e ha F(x1, x2) = φ−1(φ(FX1(x1)) + φ(FX2(x2))). Fo simplici y, he uni a ia e andom a iable F(X1, X2) is deno ed by V. Simila ly o Theo em 1 in Wang and Oakes [38], we ob ain P[V≤α, X1≤x1, X2≤x2] = (α−φ(α) φ0(α)+φ(F(x1,x2)) φ0(α),i 0 < α ≤F(x1, x2); 0,i α > F(x1, x2). (16) By s aigh o wa d calcula ion, i can be shown ha he dis ibu ion unc ion o Tis de ined as FT(x1, x2) = (P[V=α,X1≤x1,X2≤x2] P(V=α),i 0 < α ≤F(x1, x2); 0,i α > F(x1, x2), =(1−φ(F(x1,x2)) φ(α),i 0 < α ≤F(x1, x2); 0,i α > F(x1, x2), (17) whe e P(V=α) is he densi y in αo andom a iable V. On he o he hand, o i= 1,2, FTi(xi) = (1−φ(FXi(xi)) φ(α),i α≤FXi(xi); 0,i α > FXi(xi), and F−1 Ti(wi) = ((φ◦FXi)−1(φ(α)(1 −wi)),i 0 < wi≤1; 0,i wi= 0. The e o e, he copula o he andom ec o Tis CT(u1, u2) = FT(F−1 T1(u1), F−1 T2(u2)) = (u1+u2−1,i u1+u2≥1; 0,o he wise. I is now shown ha T∈MRV by Theo em 3.2 in Weng and Zhang [40]. The e o e, condi ions (C1) and (C2) o Theo em 3.2 in Weng and Zhang [40] a e p o ed. As a esul o ha (φ◦FX1)∈ RV−β,β > 0, we i ially ob ain FT1∈RV−β,β > 0 (C1). In addi ion, since Xhas he same ma gins hen, lim →∞ FT2( ) FT1( )= 1, 23 ha is, FT1and FT2ha e equi alen ails. (C2) Finally, he lowe ail dependence unc ion o he su i al copula o T, λ2(u1, u2) = lim →0+ CT( u1, u2) , is equal o 0. Due o ha and conside ing (C1) and (C2), by Theo em 3.2 in Weng and Zhang [40], T∈MRV . Rema k 4.2. No e ha , i (φ◦FX1)∈RV−β,β > 1, by applying Theo em 4.1 and P oposi ion 1 in Dan´ıelsson e al. [9] o T, hen he VaR o Tis subaddi i e su icien ly deep in he ail egions. In his case, a weak subaddi i i y o he p oposed mul i a ia e lowe CoVaR is ob ained, ha is, since VaRω(Ti) = CoVaRi α,ω(X), hen VaRω(T1+T2)<CoVaR1 α,ω(X) + CoVaR2 α,ω(X) (18) su icien ly deep in ail egions. Now, an illus a ion o Rema k 4.2 is p esen ed (see Figu e 4 and Example 4.6 below). 0.2 0.4 0.6 0.8 1.0 0 5 10 15 20 25 30 CoVaR1ω, α(X)+CoVaR2ω, α(X) and VaRω(T1+T2) ω CoVaR1ω, α(X)+CoVaR2ω, α(X) VaRω(T1+T2) 0.0 0.2 0.4 0.6 0.8 1.0 0 5 10 15 CoVaR1ω, α(X)+CoVaR2ω, α(X) and VaRω(T1+T2) ω CoVaR1ω, α(X)+CoVaR2ω, α(X) VaRω(T1+T2) Figu e 4: CoVaR1 α,ω(X) + CoVaR2 α,ω(X) and VaRω(T1+T2) o Xwi h X1∼X2∼Pa e o(2) and a Gumbel copula wi h θ= 2, as in Example 4.6, o α=ω(le panel) and o α= 0.75 ( igh panel). Example 4.6. In his example, a bi a ia e andom ec o , X, wi h X1∼X2∼Pa e o(2) and a Gumbel copula, θ= 2, is conside ed. Analy ical exp essions o CoVaRi α,ω(X),i= 1,2a e 24 ob ained. In addi ion, VaRω(T1+T2)is calcula ed by nume ic app oxima ion. The ob ained esul s a e ga he ed in Figu e 4: o ω=α∈(0,1) (see Figu e 4, le ) and o α= 0.75, ω∈(0,1) (see Figu e 4, igh ). I can be easily obse ed ha (18) is e i ied o la ge ω. 5. Es ima ion Semipa ame ic es ima o s by assuming A chimedean copula o he p oposed mul i a ia e Co- VaRs a e gi en in his sec ion. Mo eo e , illus a ions wi h simula ed and insu ance eal da a a e p o ided. Fi s ly, le assume ha Xhas an A chimedean copula s uc u e. The gene a o o an A chi- medean copula depends on he dependence pa ame e θo he copula (see, e.g., Table 4.1. in Nelsen [31]). Consequen ly, a semipa ame ic es ima o o he gene a o is ob ained by conside- ing a maximum pseudo-likelihood es ima o o he dependence pa ame e θassocia ed wi h his gene a o . Following hese conside a ions and using Equa ion (11), we in oduce a semipa ame- ic es ima o o he mul i a ia e lowe CoVaR (see De ini ion 5.1) by using a semipa ame ic es ima ion o θand he empi ical quan ile es ima ion. De ini ion 5.1. Le Xbe a d−dimensional andom ec o wi h A chimedean copula wi h ge- ne a o φθand α∈(0,1). A semipa ame ic es ima o o he i−componen o he mul i a ia e lowe CoVaR is de ined as CoVaRi α,ω(X) = d VaRωihˆ F−1 Xi(φ−1 ˆ θn(Siφˆ θn(α)))i, o i= 1, . . . , d, (19) whe e ω∈[0,1]d,Siis a andom a iable wi h Be a(1, d −1) dis ibu ion, d VaRω(X)is he empi ical es ima o o VaRω(X),φˆ θnis he semipa ame ic es ima o o φθand ˆ F−1 Xiis he empi ical es ima o o F−1 Xi o i= 1, . . . , d. Secondly, le assume ha Xhas an A chimedean su i al copula s uc u e. F om Equa ion (14), we in oduce a semipa ame ic es ima ion o mul i a ia e uppe CoVaR (see De ini ion 5.2) using he semipa ame ic es ima ion o he gene a o o he A chimedean su i al copula and he empi ical es ima ion o he quan ile unc ions. De ini ion 5.2. Le Xbe a d−dimensional andom ec o wi h A chimedean su i al copula wi h gene a o φθand α∈(0,1). A semipa ame ic es ima o o he i−componen o he mul i a ia e uppe CoVaR is de ined as CoVaR i α,ω(X) = d VaRωiˆ F−1 Xi(φ−1 ˆ θn(Siφˆ θn(1 −α))), o i= 1, . . . , d, (20) whe e ω∈[0,1]d,Siis a andom a iable wi h Be a(1, d −1) dis ibu ion, d VaRω(X)is he em- pi ical es ima o o VaRω(X),φˆ θnis he semipa ame ic es ima o o φθand ˆ F−1 Xi he empi ical es ima o o F−1 Xi o i= 1, . . . , d. 25 2 4 6 8 10 12 14 4 6 8 10 12 Loss ALAE da a in loga i hmic scale Loss ALAE 2 4 6 8 10 12 14 4 6 8 10 12 Loss ALAE da a in loga i hmic scale Loss ALAE 2 4 6 8 10 12 14 4 6 8 10 12 Loss ALAE da a in loga i hmic scale Loss ALAE Figu e 9: Loss ALAE da a in log scale, bounda y o es ima ed le el se s (∂L(α), ed line), bounda y o es ima ed le el se s (∂L(α), blue line), empi ical quan ile o Loss da a (do ed black line), empi ical quan ile o ALAE da a (do ed black line), CoVaRα,ω (s a s) and CoVaRα,ω (solid ci cles) wi h (α= 0.75, ω = 0.9) (le panel); (α= 0.9, ω = 0.95) (cen e panel); (α= 0.95, ω = 0.98) ( igh panel). The posi i e homogenei y and ansla ion in a iance p ope ies a e shown o he wo p oposed mul i a ia e CoVaR. The ela ions be ween he uni a ia e VaR and ou CoVaR a e also analysed as well as he ela ions be ween he mul i a ia e VaR p oposed by Cousin and Di Be na dino [6] and ou mul i a ia e CoVaR. In e es ingly, bo h mul i a ia e CoVaRs coincide wi h he uni- a ia e VaR when a comono onic andom ec o is conside ed, and hey e i y he addi i i y p ope y unde π-comono onic condi ions. The beha iou o he mul i a ia e CoVaR wi h es- pec o he isk le el, he usual s ochas ic o de o ma ginal dis ibu ions, and he dependence s uc u e a e s udied. Unsu p isingly, he e ec in he mul i a ia e lowe CoVaR ( esp. uppe CoVaR) wi h espec o a change in he isk le el, a change in he dependence s uc u e, o he usual s ochas ic o de o ma ginal dis ibu ions, ends o be he same as o he mul i a ia e lowe VaR ( esp. uppe VaR) p oposed in Cousin and Di Be na dino [6]. Impo an esul s and analy ical exp essions o ou mul i a ia e isk measu es a e ob ained o andom ec o s wi h A chimedean copulas. In pa icula , ce ain subaddi i i y inequali y is p esen ed in he A chimedean case unde egula a ia ion condi ions. Mo eo e , unde A chimedean copula condi ion, es ima o s o he wo p oposed mul i a ia e CoVaRs a e p o ided in simula ed da a and insu ance eal da a. In a u u e pe spec i e, quan ile eg ession es ima ions in ex eme heo y o he wo mul i a ia e CoVaRs can be s udied by adap ing he wo ks by Di Be na dino e al. [12] and by Daouia e al. [10]. Ano he app oach could in ol e he e alua ion o he p oposed measu es in ce ain mul idimensional po olios and he compa ison be ween he esul s o hese measu es and he esul s o mul i a ia e exis en measu es (see Cousin and Di Be na dino [6], Cousin and Di Be na dino [7] and Cai and Li [5]). Acknowledgemen s The au ho s wish o hank he Edi o and he wo anonymous e e ees 32 whose commen s helped o imp o e a p e ious e sion o his pape . This wo k was pa ly suppo ed by a g an om Jun a de Andaluc´ıa (Spain) o esea ch g oup (FQM- 328) and by a p e-doc o al con ac (Palacios Rod ´ıguez, F.) om he “V Plan P opio de In es igaci´on” o he Uni e si y o Se ille. Re e ences [1] Ad ian, T. and B unne meie , M. K. (2011). CoVaR. NBER Wo king Pape s 17454, Na ional Bu eau o Economic Resea ch, Inc. [2] A zne , P., Delbaen, F., Ebe , J. M., and Hea h, D. (1999). Cohe en measu es o isk. Ma h. Finance, 9(3):203–228. [3] Ba be, P., Genes , C., Ghoudi, K., and R´emilla d, B. (1996). On Kendall’s p ocess. J. Mul i a ia e Anal., 58(2):197–229. [4] Bellini, F. and Bignozzi, V. (2013). Elici able isk measu es. A ailable a SSRN 2334746. [5] Cai, J. and Li, H. (2005). Condi ional ail expec a ions o mul i a ia e phase- ype dis ibu ions. Jou nal o Applied P obabili y, 42(3):810–825. [6] Cousin, A. and Di Be na dino, E. (2013). On mul i a ia e ex ensions o Value-a -Risk. Jou nal o Mul i a ia e Analysis, 119:32 – 46. [7] Cousin, A. and Di Be na dino, E. (2014). On mul i a ia e ex ensions o Condi ional-Tail-Expec a ion. In- su ance: Ma hema ics and Economics, 55:272 – 282. [8] Cue as, A., Gonz´alez-Man eiga, W., and Rod ´ıguez-Casal, A. (2006). Plug-in es ima ion o gene al le el se s. Aus alian & New Zealand Jou nal o S a is ics, 48(1):7–19. [9] Dan´ıelsson, J., Jo gensen, B., Samo odni sky, G., Sa ma, M., and De V ies, C. (2013). Fa s ails, Va and subaddi i i y. Jou nal o Econome ics, 172:283–291. [10] Daouia, A., Ga des, L., and Gi a d, S. (2013). On ke nel smoo hing o ex emal quan ile eg ession. Be nouilli., 19(5B):2557. [11] Denui , M., Dhaene, J., Goo ae s, M., and Kaas, R. (2005). Ac ua ial Theo y o Dependence Risks. Wiley. [12] Di Be na dino, E., Lalo¨e, T., Maume-Deschamps, V., and P ieu , C. (2011). Plug-in es ima ion o le el se s in a non-compac se ing wi h applica ions in mul i a iable isk heo y. ESAIM: P obabili y and S a is ics, h p://dx.doi.o g/10.1051/ps/2011161. [13] Emb ech s, P. and Ho e , M. (2013). S a is ics and Quan i a i e Risk Managemen o Banking and Insu - ance. ETH Zu ich. [14] Emb ech s, P., Kl¨uppelbe g, C., and Mikosch, T. (1997). Modelling ex emal e en s o insu ance and inance. Sp inge . Be lin. [15] Emb ech s, P. and Pucce i, G. (2006). Bounds o unc ions o mul i a ia e isks. Jou nal o Mul i a ia e Analysis, 97(2):526–547. [16] F ees, E. and Valdez, E. (1998). Unde s anding ela ionships using copulas. No h Ame . Ac ua . J., 2:1–25. 33 [17] Genes , C., Ghoudi, K., and Ri es , L. (1995). A semipa ame ic es ima ion p ocedu e o dependence pa ame e s in mul i a ia e amilies o dis ibu ions. Biome ika, 82(3):543–552. [18] Genes , C. and Ri es , L.-P. (2001). On he mul i a ia e p obabili y in eg al ans o ma ion. S a is . P obab. Le ., 53(4):391–399. [19] Gi a di, G. and E g¨un, A. (2013). Sys emic isk measu emen : Mul i a ia e GARCH es ima ion o CoVaR. Jou nal o Banking and Finance., 37:3169. [20] Gnei ing, T. (2011). Making an e alua iong poin o escas s. Jou nal o Ame ican S a is ical Associa ion. [21] Goodha , C. and Sego iano, M. (2009). Banking s abili y measu es. IMF Wo king Pape s., pages 1 – 54. [22] Joe, H. (1997). Mul i a ia e Models and Dependence Concep s. Chapman & Hall, London. [23] Jouini, E., Meddeb, M., and Touzi, N. (2004). Vec o - alued cohe en isk measu es. Finance S och., 8(4):531–552. [24] Mainik, G. and Schaanning, E. (2014). On dependence consis ency o CoVaR and some o he sys emic isk measu es. S a is ics & Risk Modeling., 31(1):49–77. [25] McNeil, A., F ey, R., and Emb ech s, P. (2005). Quan i a i e Risk Managemen : Concep s, Techniques, and Tools. P ince on Se ies in Finance. P ince on Uni e si y P ess, New Je sey. [26] McNeil, A. J. and Neˇsleho ´a, J. (2009). Mul i a ia e A chimedean copulas, d-mono one unc ions and l1- no m symme ic dis ibu ions. The Annals o S a is ics, 37:3059–3097. [27] Mu˜noz P´e ez, J. and S´anchez-G´omez, A. (1990). A cha ac e iza ion o he dis ibu ion unc ion: The dispe sion unc ion. S a is ics & P obabili y Le e s, 10(3):235 – 239. [28] M¨ulle , A. (1997). S op-loss o de o po olios o dependen isks. Insu ance Ma h. Econom., 21(3):219–223. [29] M¨ulle , A. and S oyan, D. (2002). Compa ison Me hods o S ochas ic Models and Risks. Wiley Se ies in P obabili y and S a is ics. John Wiley & Sons Inc. [30] Nappo, G. and Spizzichino, F. (2009). Kendall dis ibu ions and le el se s in bi a ia e exchangeable su i al models. In o ma ion Sciences, 179:2878. [31] Nelsen, R. (2006). An In oduc ion o Copulas. Sp inge Se ies in S a is ics. Sp inge . [32] P ´ekopa, A. (2012). Mul i a ia e Value a Risk and ela ed opics. Annals o Ope a ion Resea ch, 193:49. [33] Pucce i, G. and Sca sini, M. (2010). Mul i a ia e comono onici y. Jou nal o Mul i a ia e Analysis, 101(1):291 – 304. [34] Resnick, S. I. (2007). Hea y-Tail Phenomena. P obabilis ic and S a is ical Modeling. Sp inge Se ies in Ope a ions Resea ch and Financial Enginee ing. [35] Resnick, S. I. (2008). Ex eme Values, Regula Va ia ion and Poin P ocesses. Sp inge . New Yo k. [36] Se ling, R. (1980). App oxima ion Theo ems o Ma hema ical S a is ics. John Wiley & Sons, New Yo k. [37] Shaked, M. and Shan hikuma , J. (2007). S ochas ic O de s. Sp inge Se ies in S a is ics. Physica-Ve lag. [38] Wang, A. and Oakes, D. (2008). Some p ope ies o he Kendall dis ibu ion in bi a ia e A chimedean copula models unde censo ing. S a is ics & P obabili y Le e s, 78(16):2578 – 2583. 34 [39] Wei, G. and Hu, T. (2002). Supe modula dependence o de ing on a class o mul i a ia e copulas. S a is . P obab. Le ., 57(4):375–385. [40] Weng, C. and Zhang, Y. (2012). Cha ac e iza ion o mul i a ia e hea y- ailed dis ibu ion amilies. Jou nal o Mul i a ia e Analysis. [41] Ziegel, J. (2014). Cohe ence and elici abili y. a Xi :1303.1690 3. 35