Estimation of the marginal expected shortfall: the mean when a related variable is extreme

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1 Estimatio of the margial expected shortfall: the mea whe a related variable is extreme Jua-Jua Cai Delft Uiversity of Techology Laures de Haa Erasmus Uiversity Rotterdam Uiversity of Lisbo Joh H.J. Eimahl Tilburg Uiversity Che Zhou De Nederladsche Ba Erasmus Uiversity Rotterdam March 7, 23 Abstract. Deote the loss retur o the equity of a fiacial istitutio as X ad that of the etire maret as Y. For a give very small value of p >, the margial expected shortfall MES is defied as EX Y > Q Y p, where Q Y p is the p-th quatile of the distributio of Y. The MES is a importat factor whe measurig the systemic ris of fiacial istitutios. For a wide oparametric class of bivariate distributios, we costruct a estimator of the MES ad establish the asymptotic ormality of the estimator whe p, as the sample size. Sice we are i particular iterested i the case p = O/, we use extreme value techiques for derivig the estimator ad its asymptotic behavior. The fiite sample performace of the estimator ad the adequacy of the limit theorem are show i a detailed simulatio study. We also apply our method to estimate the MES of three large U.S. ivestmet bas. Ruig title. Margial expected shortfall. Key words ad phrases. Asymptotic ormality, coditioal tail expectatio, extreme values. Address for correspodece: Joh H.J. Eimahl, Dept. of Ecoometrics & OR ad CetER, Tilburg Uiversity, P.O. Box 953, 5 LE Tilburg, The Netherlads. j.h.j.eimahl@uvt.l Research is partially ported by ENES-Project PTDC/MAT/277/29. Views expressed do ot ecessarily reflect the official positio of De Nederladsche Ba

2 Itroductio A importat factor i costructig a systemic ris measure for the fiacial idustry is the cotributio of a fiacial istitutio to a systemic crisis measured by the Margial Expected Shortfall MES. The MES of a fiacial istitutio is defied as the expected loss o its equity retur coditioal o the occurrece of a extreme loss i the aggregated retur of the fiacial sector. Deote the loss of the equity retur of a fiacial istitutio ad that of the etire maret as X ad Y, respectively. The the MES is defied as EX Y > t, where t is a high threshold such that p = P Y > t is extremely small. I other words, the MES at probability level p is defied as MESp = EX Y > Q Y p, where Q Y is the quatile fuctio of Y. Notice that i applicatios the probability p is at a extremely low level that ca be eve lower tha /, where is the sample size of historical data that are used for estimatig the MES. It is the goal of this paper to establish a ovel estimator of MESp ad to uravel its asymptotic behavior. The mai result establishes the asymptotic ormality of our estimator for a large class of bivariate distributios, which maes statistical iferece for the MES feasible. We also show through a simulatio study that the estimator performs well ad that the limit theorem provides a adequate approximatio for fiite sample sizes. The MES has bee studied uder the ame Coditioal Tail Expectatio CTE, or TCE i statistics ad actuarial sciece. The defiitio of CTE i a uivariate cotext is the same as that of the tail value at ris. Mathematically, it is give by EX X > Q X p where Q X is the quatile fuctio of X. I case X has a cotiuous distributio, this is also called the expected shortfall. Compared to the MES, it ca be viewed as the special case that Y = X. The cocept of CTE has bee defied more geerally i a multivariate setup. It is possible to have the coditioig evet defied by aother, related radom variable Y exceedig its high quatile. I that case, the CTE coicides with the MES. A few studies show how to calculate the CTE whe the joit distributio of X, Y follows specific parametric models. For example, Ladsma ad Valdez 23 ad Kostadiov 26 deal with elliptical distributios with heavytailed margials. Cai ad Li 25 studies the CTE for multivariate phase-type distributios. Veric 26 cosiders sewed-ormal distributios. Compared to these studies, our approach I Acharya et al. 22, the probability of such a extreme tail evet is specified as that happe oce or twice a decade or less, whereas the estimatio is based o daily data from oe year. 2

3 does ot impose ay parametric structure o X, Y. A comparable result i the literature is the approach i Joe ad Li 2, where uder multivariate regular variatio, a formula for calculatig the CTE is provided. The multivariate regularly varyig distributios form a subclass of our model. Note that we do ot mae ay assumptio o the margial distributio of Y. It should be emphasized, however, that we focus o the statistical problem of estimatig the MES ad studyig the performace of the estimator i cotrast to these papers where oly probabilistic properties of the MES are studied. I Acharya et al. 22 a estimator for the MES is provided assumig a specific liear relatioship betwee X ad Y. The estimatio procedure there ca be see as a special case of the preset oe. A similar settig has bee adopted i Browlees ad Egle 22, where a oparametric erel estimator of the MES is proposed. Such a erel estimatio method, however, performs well oly if the threshold for defiig a systemic crisis is ot too high: the tail probability level p should be substatially larger tha /. Such a method caot hadle extreme evets, that is p < /, which is particularly required for systemic ris measures. The paper is orgaized as follows. Sectio 2 provides the mai result: asymptotic ormality of the estimator. I Sectio 3, a simulatio study shows the good performace of the estimator. A applicatio o estimatig the MES for U.S. fiacial istitutios is give i Sectio 4. The proofs are deferred to Sectio 5. 2 Mai Results Let X, Y be a radom vector with a cotiuous distributio fuctio F. Deote the margial distributio fuctios as F x = F x, ad F 2 y = F, y with correspodig tail quatile fuctios give by U j =, F j j =, 2, where deotes the left-cotiuous iverse. The the MES at a probability level p ca be writte as := EX Y > U 2 /p. The goal is to estimate based o idepedet ad idetically distributed i.i.d. observatios, X, Y,, X, Y from F, where p = p as. We adopt the bivariate EVT framewor for modelig the tail depedece structure of X, Y. Suppose for all x, y [, ] 2 \ {+, + }, the followig limit exists: lim tp F X x/t, F 2 Y y/t =: Rx, y. t 3

4 The fuctio R completely determies the so-called stable tail depedece fuctio l, as for all x, y, lx, y = x + y Rx, y; see Drees ad Huag 998; Beirlat et al. 24, Chapter 8.2. For the margial distributios, we assume that oly X follows a distributio with a heavy right tail: there exists γ > such that for x >, U tx lim t U t = xγ. 2 The it follows that F is regularly varyig with idex γ ad γ is the extreme value idex. We first focus o X beig positive, the we cosider X R. Throughout, there is o assumptio, apart from cotiuity, o the margial distributio of Y. 2. X Positive Assume X taes values i,. The followig limit result gives a approximatio for. Propositio. Suppose that ad 2 hold with < γ <. The, lim p U /p = R x /γ, dx. I Joe ad Li 2, Theorem 2.4, this result is derived uder the stroger assumptio of multivariate regular variatio. Next, we costruct a estimator of based o the limit give i Propositio. Let be a itermediate sequece of itegers, that is,, /, as. By Propositio ad a stregtheig of 2 see coditio b below, we have that as, U /p U / θ For estimatig, it thus suffices to estimate γ ad θ. We estimate γ with the Hill 975 estimator: γ θ. 3 p ˆγ = log X i+, log X,, 4 i= where is aother itermediate sequece of itegers ad X i,, i =,..., is the i-th order statistic of X,..., X. 4

5 By regardig the -th order statistic Y, of Y,..., Y as a estimator of U 2 /, we costruct a oparametric estimator of θ to the highest values of Y : ˆθ = which is the average of the selected X i correspodig X i IY i > Y,. 5 i= Combiig 3, 4 ad 5, we estimate by ˆγ ˆ = ˆθ. 6 p We prove the asymptotic ormality of ˆ uder the followig coditios. a There exist β > γ ad τ < such that, as t, tp F X < x/t, F 2 Y < y/t Rx, y <x< x β = Ot τ. b There exist ρ < ad a evetually positive or egative fuctio A with lim t A t = such that U tx/u t x γ lim t A t As a cosequece, A is regularly varyig with idex ρ. = x γ xρ ρ. Coditios a ad b are atural secod-order stregtheigs of ad 2, respectively. We further require the followig coditios o the itermediate sequeces ad. c As, A /. d As, = O α for some α < mi 2τ 2τ+, 2γ ρ 2γ ρ +ρ. To characterize the limit distributio of ˆ, we defie a mea zero Gaussia process W R o [, ] 2 \{, } with covariace structure EW R x, y W R x 2, y 2 = Rx x 2, y y 2, i.e., W R is a Wieer process. Set Θ = γ W R, + Rs, ds γ W R s, ds γ, ad Γ = γ W R, + s W R s, ds. It will be show see Propositio 3 ad 24 that ˆθ ad ˆγ are asymptotically ormal with Θ ad Γ as limit, respectively. The followig theorem gives the asymptotic ormality of ˆ. 5

6 Theorem. Suppose coditios a d hold ad γ, /2. Assume d := r := lim log d [, ]. If lim log d =, the, as,, ˆθp mi log d d { Θ + rγ, if r, r Θ + Γ, if r >, where VarΘ = γ 2 b2 Rs, ds 2γ, VarΓ = γ 2 ad CovΓ, Θ = γ γ + br, γ γ + bs γ γ γ l s Rs, s ds with b = Rs, ds γ. 2.2 X Real p ad I this sectio, X taes values i R, that is, we do ot restrict X to be positive. Defie X + = maxx, ad X = X X +. Besides the coditios of Theorem, we require two more coditios: e E X /γ < ; f As, = o p 2τ γ. It ca be show that coditio e, together with a, esure that EX + Y > U 2 /p, as p. Therefore, we estimate with ˆγ ˆ = p X i IX i >, Y i > Y,, 7 i= with ˆγ as i Sectio 2.. Observe that whe X is positive, this defiitio coicides with that i 6. As stated i the followig theorem, the asymptotic behavior of the estimator remais the same as that for positive X. Theorem 2. Uder the coditios of Theorem ad coditios e ad f, as, {, ˆθp d Θ + rγ, if r ; mi log d r Θ + Γ, if r >, where r, Θ ad Γ are defied as i Theorem. 3 Simulatio Study I this sectio, a simulatio ad compariso study is implemeted to ivestigate the fiite sample performace of our estimator. We geerate data from three bivariate distributios. 6

7 A trasformed Cauchy distributio o, 2 defied as X, Y = Z 2/5, Z 2, where Z, Z 2 is a stadard Cauchy distributio o R 2 with desity 2π + x2 + y 2 3/2. It follows that γ = 2/5 ad Rx, y = x + y x 2 + y 2, x, y. It ca be show that this distributio satisfies coditios a ad b with τ =, β = 2, ad ρ = 2. We shall refer to this distributio as trasformed Cauchy distributio. Studet-t 3 distributio o, 2 with desity + x2 + y 2 5/2, x, y >. fx, y = 2 π 3 We have γ = /3, Rx, y = x+y x4/3 + 2 x2/3 y 2/3 +y 4/3 x 2/3 +y 2/3, τ = /3, β = 4/3 ad ρ = 2/3. A trasformed Cauchy distributio o the whole R 2 defied as X, Y = Z 2/5 I Z + Z /5 I Z <, Z 2 I Z + Z /3 2 I Z <. We have γ = 2/5, Rx, y = x/2 + y x 2 /4 + y, τ =, β = 2, ad ρ = 2. We shall refer to this distributio as trasformed Cauchy distributio 2. We draw 5 samples from each distributio with sample sizes = 2, ad = 5,. Based o each sample, we estimate for p = /5, /5, or /,. Besides the estimator give by 7, we costruct two other estimators. Firstly, for p, a empirical couterpart of, give by ˆθ emp = p X i IY i > Y p,, 8 i= is studied, where deotes the iteger part. Secodly, exploitig the relatio i Propositio ad usig the empirical estimator of R give by Rx, y = i= IX i > X x,, Y i > Y y, ad the Weissma 978 estimator of U /p give by Û/p = dˆγ X,, we defie a alterative EVT estimator as = Û/p =dˆγ X, Rx, dx ˆγ raxi + ˆγ IY i > Y,. 9 i= 7

8 Trasformed Cauchy Distributio =2 =3 =3 Trasformed Cauchy Distributio =5 =4 = θ^p θ^emp θ^p2 2 θ^p3 3 θ^p θ^emp θ^p2 2 θ^p Studet t_3 Distributio =2 =5 =5 Studet t_3 Distributio =5 =8 = θ^p θ^emp θ^p2 2 θ^p3 3 θ^p θ^emp θ^p2 2 θ^p Trasformed Cauchy Distributio 2 =2 =8 = Trasformed Cauchy Distributio 2 =5 =5 = θ^p θ^emp θ^p2 2 θ^p3 3 θ^p θ^emp θ^p2 2 θ^p Figure : Boxplots o ratios of estimates ad true values. Each plot is based o 5 samples with sample size =2, or 5, from the trasformed Cauchy distributios, 2 or Studet-t 3 distributio. The estimators are ˆ of 7, of 9 ad ˆθ emp of 8; p = /5 p, /5, p2 ad /, p3. The compariso of the three estimators is show i Figure, where we preset boxplots of the ratios of the estimates ad the true values. For all three distributios, the empirical estimator 8

9 uderestimates the MES ad is cosistetly outperformed by the EVT estimators. Additioally, it is ot applicable for p < /. The two EVT estimators, ad ˆ, both perform well. Their behavior is similar ad remais stable whe p chages from /5 to /,. The results for the trasformed Cauchy distributio are the best amog the three distributios, as the medias of the ratios are closest to oe ad the variatios are smallest. Next, we ivestigate the ormality of the estimator, ˆ, with p = /. For r <, the asymptotic ormality of ˆ i Theorem ca be expressed as ˆθp d Θ + rγ, or equivaletly, log ˆθp d Θ + rγ. Notice that the limit distributio is a cetered ormal distributio. Write σp 2 = VarΘ + rγ with r = log p. We compare the distributio of log ˆ, with the limit distributio N, σ 2 p. Table reports the stadardized mea of log ˆ, i.e., the average value divided by σ p, ad betwee Table : Stadardized mea ad stadard deviatio of log ˆ = 2, = 5, p = /2, p = /5, Trasformed Cauchy distributio Studet-t 3 distributio Trasformed Cauchy distributio The umbers are the stadardized mea of log ˆ ad betwee bracets, the ratio of the stadard deviatio ad σ p, based o 5 estimates with = 2, or 5, ad p = /. bracets, the ratio of the sample stadard deviatio ad σ p. As idicated by the umbers, the mea ad stadard deviatio of log ˆ are both close to that of the limit distributio. After the umerical assessmet o the parameters, we illustrate the ormality of log ˆ. Figure 2 shows the desities of the N, σ 2 p-distributio ad the histograms of log ˆ, based o 5 estimates. The ormality of the estimates is ported by the large overlap betwee the histograms ad the areas uder the desity curves. Hece we coclude that the limit theorem provides a adequate approximatio for fiite sample sizes. 9

10 Trasformed Cauchy Distributio, =2 Trasformed Cauchy Distributio, =5 Desity Desity logθ^p logθ^p Studet t_3 Distributio, =2 Studet t_3 Distributio, =5 Desity Desity logθ^p logθ^p Trasformed Cauchy Distributio 2, =2 Trasformed Cauchy Distributio 2, =5 Desity Desity logθ^p logθ^p Figure 2: Histograms of log ˆ for p = /, based o 5 samples with sample size =2, or 5, from the trasformed Cauchy distributios, 2 or Studet-t 3 distributio. The choices of ad are the same as i Figure. The curves are the desities of the N, σ 2 p-distributio. 4 Applicatio I this sectio, we apply our estimatio method to estimate the MES for some fiacial istitutios. We cosider three large ivestmet bas i the U.S., amely, Goldma Sachs GS,

11 Morga Staley MS ad T. Rowe Price TROW, all of which have a maret capitalizatio greater tha 5 billio USD as of the ed of Jue 27. The dataset cosists of the loss returs i.e., mius log returs o their equity prices at a daily frequecy from July 3, 2 to Jue 3, 2. 2 Moreover, for the same time period, we extract daily loss returs of a value weighted maret idex aggregatig three marets: NYSE, AMES ad Nasdaq. We use our method to estimate the MES, EX Y > U 2 /p, where X ad Y refer to the daily loss returs of a ba equity ad the maret idex, respectively ad p = / = /253, that correspods to a oce per decade systemic evet. Hill Estimator of γ γ^ GS MS TROW 5 5 Figure 3: The Hill estimates of the extreme value idices of the daily loss returs o the three equities. Sice X may tae egative values i.e. positive returs of the equities of the bas, it is ecessary to apply the estimator for the geeral case as defied i 7. For that purpose, we first verify two of the coditios required for the procedure. First of all, the assumptio that γ < /2 2 The choice of the bas, data frequecy ad time horizo follows the same setup as i Browlees ad Egle 22.

12 is cofirmed by the plot of the Hill estimates i Figure 3. Secodly, sice the estimatio relies o the approximatio of EX + Y > U 2 /p, it is importat to chec that high values of Y do ot coicide with egative values of X, geerally. Ituitive empirical evidece for this is preseted i Figure 4. It plots the loss returs of the equity prices agaist the maret idex. The horizotal lies idicate the 5-th largest loss of the idex. As oe ca see, from the upper parts of the plots, the largest 5 losses of the idex are mostly associated with losses X >. Idex Idex Idex GS MS....2 TROW Figure 4: The poits are the daily loss returs of the three equity prices ad the maret idex. The horizotal lies idicate the 5-th largest loss of the maret idex. The vertical lies, at, distiguish the occurrece of losses ad profits. Hece, we ca apply our method to obtai the estimates of γ ad MESp = for the Table 2: MES of the three ivestmet bas Ba ˆγ ˆθp Goldma Sachs GS Morga Staley MS T. Rowe Price TROW Here ˆγ is computed by taig the average of the Hill estimates for [7, 9]. ˆ is give i 7 with = 253, = 5, ad p = / = /253. three bas, see Table 2. It follows that i case of a oce per decade maret crisis, we estimate 2

13 that o average the equity prices of Goldma Sachs ad T. Rowe Price drop about 25% ad Morga Staley falls eve about 45% o that day. 5 Proofs Proof of Propositio Recall that for a o-egative radom variable Z, Hece, EZ = P Z > xdx. U /p = = The limit relatios ad 2 implies that p P X > x, Y > U dx 2/p U /p p P X > U /px, Y > U 2 /pdx. lim p p P X > U /px, Y > U 2 /p = Rx /γ,. Hece, we oly have to prove that the itegral i ad the limit procedure p ca be iterchaged. This is esured by the domiated covergece theorem as follows. Notice that for x, p P X > U /px, Y > U 2 /p mi, p F U /px. For < ε < /γ, there exists pε see Propositio B..9.5 i de Haa ad Ferreira 26 such that for all p < pε ad x >, Write p F U /px 2x /γ +ε. hx = {, x ; 2x /γ +ε, x >. The h is itegrable ad p P X > U /px, Y > U 2 /p hx o [, for p < pε. Hece we ca apply the domiated covergece theorem to complete the proof of the propositio. Next, we prove Theorem. The geeral idea of the proof is described as follows. It is clear that the asymptotic behavior of ˆ results from that of ˆγ ad ˆθ. The asymptotic ormality of 3

14 ˆγ is well-ow, see, e.g., de Haa ad Ferreira 26. To prove the asymptotic ormality of ˆθ, write ˆθ = X i IY i > U 2 /e, i= where e = F 2Y, P, as. Hece, with deotig θ y := y i= X iiy i > U 2 /y, we first ivestigate the asymptotic behavior of θ y for y [/2, 2]. The, by applyig the result for y = e ad cosiderig the asymptotic behavior of e, we obtai the asymptotic ormality of ˆθ. Lastly, together with the asymptotic ormality of ˆγ, we prove that of ˆ. To obtai the asymptotic behavior of θ y, we itroduce some ew otatio ad auxiliary lemmas. Write R x, y := P F X < x/, F 2 Y < y/. A pseudo o-parametric estimator of R is give as T x, y := i= I F X i < x/, F 2 Y i < y/. 3 The followig lemma shows the asymptotic behavior of the pseudo estimator. The limit process is characterized by the aforemetioed W R process. For coveiet presetatio, all the limit processes ivolved i the lemma are defied o the same probability space, via the Sorohod costructio. However, they are oly equal i distributio to the origial oes. The proof of the lemma is aalogous to that of Propositio 3. i Eimahl et al. 26 ad is thus omitted. Lemma. Suppose holds. For ay η [, /2 ad T positive, with probability, T x, y R x, y W R x, y x,y,t ] x η, T x, x W R x, x,t ] x η, T, y y W R, y. y,t ] The followig lemma shows the boudedess of the W R process with proper weighig fuctio. It follows from, for istace, a modificatio of Example.8 i Alexader 986 or that of Lemma 3.2 i Eimahl et al. 26. Lemma 2. For ay T > ad η [, /2, with probability, W R x, y W R x, y <x T,<y< x η < ad <x<,<y<t y η <. y η 3 It is called pseudo estimator because the margial distributio fuctios are uow. 4

15 Next, deote s x = F U /x γ for x >. From the regular variatio coditio 2, we get that s x x as. The followig lemma shows that whe hadlig proper itegrals, s x ca be substituted by x i the limit. Lemma 3. Suppose 2 holds. Deote g as a bouded ad cotiuous fuctio o [, S [a, b] with < S ad a < b <. Moreover, pose there exist η > γ ad m > such that gx, y <x S, a y b x η m. If S < +, we further require that < S < S. The, lim a y b S gs x, y gx, ydx γ =. Furthermore, pose gx, y gx 2, y x x 2 holds for all x, x 2 < S ad a y b. Uder coditios b ad d, we have that lim a y b S gs x, y gx, ydx γ =. 2 Proof of Lemma 3 We prove ad 2 for S = S =. The proof for < S < S < + is similar but simpler. For ay < ε <, deote T ε = ε /γ. It follows from 2 ad Propositio B.. of de Haa ad Ferreira 26 that ad lim lim <x <x T ε s x =, x γ +η 2η s x x =. With δε = ε /η γ, we have that gs x, y gx, y dx γ a y b δ gs x, y gx, y dx γ T + gs x, y gx, y dx γ a y b δ + gs x, y gx, y dx γ m T δ x γ +η 2 + x η dx γ + δ γ δ x T a y b 5 gs x, y gx, y + 2ε x< a y b gx, y

16 c ε /2 + δ γ δ x T a y b gs x, y gx, y + 2ε x< a y b gx, y, where c is a fiite costat. Hece, follows from the uiform cotiuity of g o [δ, T ] [a, b] ad the boudedess of g o [, + [a, b]. Next we prove 2. Deote T = A / ρ. By the Lipschitz property of g, gs x, y gx, y dx γ a y b T s x x dx γ + 2 x< a y b α 2 α < γ ρ ρ. Thus for ay ε gx, y T γ. 3 It is thus ecessary to prove that both terms i the right had side of 3 are o/. For the secod term, coditio d implies that, γ ρ ρ, as, we have that γ ρ +ε ρ γ ρ γ ρ +ε = O ρ α +ε ρ 2, α 2 α which leads to T γ = A / γ ρ. 4 For the first term, otice that for x, T ] ad < ε < γ ρ, whe is large eough, U /x γ U / T γ = U / A / γ ρ which implies that U /x γ γ ρ ε, + as. Hece we ca apply Theorems ad B.3. i de Haa ad Ferreira 26 to coditio b ad obtai that for sufficietly large, s x x A / xx ρ γ ρ x ρ maxx ε, x ε. Thus, we get that T s x x dx γ A / T c 2 A / T ρ γ +ε =c 2 A / γ ε ρ x x ρ γ ρ + x ρ maxx ε, x ε c 3 dx γ ρ γ ρ +ε, 5 6

17 with c 2 ad c 3 some positive costats. Agai, by coditio d, as, c 3 ρ γ +ε ρ. Hece, 2 is proved by combiig 3, 4 ad 5. With those auxiliary lemmas, we obtai the asymptotic behavior of θ y Propositio 2. Suppose ad 2 hold with < γ < /2. The, θ y θ y + W R s, yds γ P. U / y as follows. Proof of Propositio 2 Recall s x = F U /x γ, x >. Similar to, yθ y = = = P X > s, Y > U 2/yds P F X < F s, F 2 Y < y/ds R F s, y ds = U / R s x, ydx γ. 6 Similarly, y θ y = + = U / T s x, ydx γ. For ay T >, we have U / y θ y T T =: I T + I 2, T + I 3, T. yθ y + W R x, ydx γ W R x, ydx γ T s x, y R s x, y dx γ W R x, ydx γ + T s x, y R s x, y dx γ T T s x, y R s x, y W R x, ydx γ It suffices to prove that for ay ε >, there exists T = T ε such that ad = T such that for ay >, P I T > ε < ε, 7 P I 2, T > ε < ε; 8 P I 3, T > ε < ε. 9 7

18 Firstly, for the term I T, by Lemma 2 with η =, there exists T = T ε such that The for ay T > T, P P I T > ε P Thus 7 holds provided that T > T. Next we deal with the term I 2, T. W R x, y > T γ ε < ε. <x<, y 2 W R x, y > T γ ε < ε. x>t, Let P be the probability measure defied by F X, F 2 Y ad P the empirical probability measure defied by F X i, F 2 Y i i. We have P I 2, T > ε = P T s x, y R s x, y dx γ > ε T P T s x, y R s x, y > εt γ = P =: p 2. x>t, x>t, { P P, s x, y } > εt γ / Defie S = {[, ], 2/}, the P S = 2/. Now by Iequality 2.5 i Eimahl 987, there exists a costat c ad a fuctio ψ with lim t ψt =, such that 2 εt γ / Choose T 2 ε such that c exp p 2 c exp =c exp ε2 T γ P S ε2 T γ ψ 8 ψ εt γ /2 2 εt γ / P S. ε. The, for ay T > T 2, c exp ε. Furthermore, we ca choose = T such that for >, ψ εt γ /2 2 ε2 T γ 6 > /2. Therefore, for T > T 2 ε ad > T, we have p 2 < ε, which leads to 8 provided that T > T 2 ad >. Lastly, we deal with I 3, T. We have that P I 3, T > ε 8

19 P + P =: p 3 + p 32. T T T s x, y R s x, y W R s x, ydx γ > ε/2 W R s x, y W R x, ydx γ > ε/2 We first cosider p 3. Notice that for ay T, there exists 2 = 2 T such that for all > 2,s T < T +. Hece, for > 2 ad ay η γ, /2, T s, y R s, y W R s, y p 3 P <s T + s η T s x η dx γ > ε/2 Notice that by, as, T s x η dx γ γ η γ T η γ. Together with Lemma, there exists 3 T > 2 T such that for > 3 T, p 3 < ε/2. The, we cosider p 32. Applyig Lemma 2, with the aforemetioed η γ, /2, there exists λ = λη, ε such that P <x<, W R x, y x η λ ε/3. 2 Moreover, W R x, y is cotiuous o, [/2, 2], see Corollary. i Adler 99. Hece applyig 2 ad with g = W R, S = T ad S = T +, we have that there exists a 4 = 4 T such that for > 4, p 32 < ε/2. Thus, 9 holds for ay T ad > max 3 T, 4 T. To summarize, choose T = T ε > maxt, T 2, ad defie T = max j 4 j T. We get that for the chose T ad ay >, the three iequalities 7 9 hold, which completes the proof of the propositio. Next, we proceed with the secod step: establishig the asymptotic ormality of ˆθ. Propositio 3. Uder the coditio of Theorem, we have ˆθ d Θ. θ Proof of Propositio 3 Observe that lim sufficiet to show that ˆθ U / θ θ U / Rs /γ, ds. Therefore it is P Θ Rs /γ, ds. 9

20 Recall e = F 2Y,. Hece, with probability, ˆθ e θ e θ Θ Rs /γ, ds U / = U / + e θ e e θ e U / =: J + J 2. e θ e + W R s, ds γ θ W R, γ = e θ e, we thus have that We prove that both J ad J 2 coverge to zero i probability as. Firstly, we deal with J. By Lemma ad T, e =, we get that Rs /γ, ds e P W R,, 2 which implies that lim P e > /4 =. Hece, with probability tedig to, J y θ y yθ y + W y < R s, yds γ /4 U / + W R s, y W R s, ds γ. y < /4 The first part coverges to zero i probability by Propositio 2. For the secod part, otice that for ay ε >, < δ < ad η γ, /2, P W R s, y W R s, ds γ > ε y < /4 δ P W R s, y W R s, ds γ > ε/2 y < /4 + P W R s, y W R s, ds γ > ε/2 y < /4 δ W R s, y P <s, s η > εη γ δ γ η 4γ + P W R s, y W R s, δ γ > ε/2 s>, y < /4 2.

21 =: p + p 2. For ay fixed ε, Lemma 2 esures that there exists a positive δε such that for all δ < δε, we have that p < ε. The, for ay fixed δ, there must exists a positive iteger δ such that for > δ we ca achieve that p 2 < ε, because we have that as, W R s, y W R s, a.s, s>, y < /4 see Corollary. i Adler 99. Hece we proved that J P as. Next we deal with J 2. We first prove a o-stochastic limit relatio: as, R s x, y Rx, ydx γ. 22 Coditio a implies that as, Hece, as, R x, y Rx, y τ <x< x β = O. <x< R s x, y Rs x, ydx γ R x, y Rx, y x β s x β dx γ τ /2 =O s x β dx γ /2 dx γ The last step follows from the followig two facts. Firstly, coditio d esures that = O α with α < 2τ 2τ. Secodly, we have that which is a cosequece of. /2 lim s x β dx γ = /2 x β dx γ <,. To complete the proof of relatio 22, it is still ecessary to show that as, Rs x, y Rx, ydx γ. This is achieved by applyig 2 to the R fuctio which satisfies the Lipschitz coditio: Rx, y Rx 2, y x x 2, for x, x 2, y. Hece, we proved the relatio 22. 2

22 ad Combiig 6 ad 22, we obtai that θ U / = e θ e U / = Rs x, dx γ = R s x, e dx γ = From the homogeeity of the R fuctio, for y >, we have that Hece, we get that Rx, ydx γ = y γ Rx, dx γ + o, 23 Rx, e dx γ + o P. Rx, dx γ. e θ e = e γ U / θ + o P. By applyig 2, Propositio ad the Cramér s delta method, we get that, as, e θ e U / θ = e γ θ U / + o P P γ W R, which implies that to J 2 P. The propositio is thus proved. ad ˆγ to obtai the proof of Theo- Fially, we ca combie the asymptotic relatios o ˆθ rem. Proof of Theorem Write ˆθ dˆγ p = d γ ˆθ θ We deal with the three factors separately. d γ θ =: L L 2 L 3. Rs /γ, ds. Firstly, hadlig L uses the asymptotic ormality of the Hill estimator. Uder coditios b ad c, we have that, as, ˆγ γ P Γ; 24 see, e.g., Example 5..5 i de Haa ad Ferreira 26. As i the proof of Theorem of de Haa ad Ferreira 26, this leads to log d L Γ P. 25 Secodly, the asymptotic behavior of the factor L 2 is give by Propositio 3. 22

23 that Lastly, for L 3, by coditio b ad Theorem i de Haa ad Ferreira 26, we have U /p U /d γ. A / ρ Together with the fact that as, A / implied by coditio d, we get that U /p U /d γ = o Followig the same reasoig of 23 for p /, we have Combiig this with 26, we have L 3 = θ /U / /U /p U /d γ U /p 26 U /p Rs /γ, ds = o. = + o. 27 Combiig the asymptotic relatios 25, 27 ad Propositio 3, we get that ˆ =L L 2 L 3 = + log d log d Γ + o P + Θ + o P + o = log d Γ + Θ log d + o P + o P. The covariace matrix of Θ, Γ follows from the straightforward calculatio. Proof of Theorem 2 Write θ + p := EX + Y > U 2 /p. The, Hece, it suffices to prove that ˆ + = + o. θ + p ˆ = ˆ θ + p θ+ p. follows the asymptotic ormality stated i Theorem ad We first show that X +, Y satisfies coditios a ad b of Sectio 2.. Let F be the distributio fuctio of X + ad Ũ = F. It is obvious that U t = Ũt, for t > F. Hece X + satisfies coditio b. Before checig coditio a for X +, Y, we prove that, as t, tp X <, F 2 Y < /t = Ot τ

24 Observe that coditio a implies that y Rt, y = Ot τ. 29 Because of the homogeeity of R, we have Rct, = Ot τ for ay c,. Hece, 28 is proved by tp X <, F 2 Y < /t = tp X >, F 2 Y < /t = tp F X < F, F 2 Y < /t Rt F, + tp F X < F, F 2 Y < /t Rt F, =Ot τ. Now we show that X +, Y satisfies coditio a, that is, as t, tp F X + < x/t, F 2 Y < y/t Rx, y x β = O t τ. 3 <x< Firstly, observe that for < x t F, { F X + < x/t} = { F X + < x/t} = { F X < x/t}. Hece, the uiform covergece i 3 o, t F ] [/2, 2] follows from the fact that X, Y satisfies coditio a. Secodly, for x > t F, we have F X + < x/t. Therefore, t F <x< = t F <x< tp F X + < x/t, F 2 Y < y/t Rx, y y Rx, y y Rt F, y = O t τ, where the last relatio follows from 29. This completes the verificatio of 3. As a result, Theorem applies to ˆ. + Next we show that θp + Hölder s iequality, coditio e ad 28, = + o. By Propositio, EX Y > U 2 /p = p EX IX <, Y > U 2 /p θ + p U /p R x /γ, dx. By 24

25 p E X /γ γ P X <, Y > U2 /p γ =Op + τ γ. Coditio b ca be writte as: U txtx γ U tt γ lim t A tu tt γ = xρ ρ. It follows from Theorem B.2.2 i de Haa ad Ferreira 26 that U /p = Opγ, as p. Hece by coditio f, = + EX Y > U 2 /p p + τ γ = + O θ + p θ + p U /p = + O p τ γ = + o. Refereces V.V. Acharya, L.H. Pederse, T. Philippo, ad M. Richardso. Preprit, 22. Measurig systemic ris. R.J. Adler. A Itroductio to Cotiuity, Extrema, ad Related Topics for Geeral Gaussia Processes. Istitute of Mathematical Statistics Lecture Notes-Moograph Series, 99. K.S. Alexader. Sample moduli for set-idexed Gaussia processes. The Aals of Probability, 4:598 6, 986. J. Beirlat, Y. Goegebeur, J. Segers, ad J. Teugels. Statistics of Extremes: Theory ad Applicatios. Wiley, Chichester, 24. C. Browlees ad R. Egle. Volatility, correlatio ad tails for systemic ris measuremet. Preprit, 22. J. Cai ad H. Li. Coditioal tail expectatios for multivariate phase-type distributios. Joural of Applied Probability, 42:8 825, 25. L. de Haa ad A. Ferreira. Extreme Value Theory: a Itroductio. Spriger Verlag, 26. H. Drees ad X. Huag. Best attaiable rates of covergece for estimators of the stable tail depedece fuctio. Joural of Multivariate Aalysis, 64:25 47,

26 J.H.J. Eimahl. Multivariate Empirical Processes. CWI Tract 32, Mathematisch Cetrum, Amsterdam, 987. J.H.J. Eimahl, L. de Haa, ad D. Li. Weighted approximatios of tail copula processes with applicatio to testig the bivariate extreme value coditio. The Aals of Statistics, 34: , 26. B.M. Hill. A simple geeral approach to iferece about the tail of a distributio. The Aals of Statistics, 3:63 74, 975. H. Joe ad H. Li. Tail ris of multivariate regular variatio. Methodology ad Computig i Applied Probability, 3:67 693, 2. K. Kostadiov. Tail approximatio for credit ris portfolios with heavy-tailed ris factors. Joural of Ris, 8:8 7, 26. Z.M. Ladsma ad E.A. Valdez. Tail coditioal expectatios for elliptical distributios. North America Actuarial Joural, 7:55 7, 23. R. Veric. Multivariate sew-ormal distributios with applicatios i isurace. Isurace: Mathematics ad Ecoomics, 38:43 426, 26. I. Weissma. Estimatio of parameters ad larger quatiles based o the largest observatios. Joural of the America Statistical Associatio, 73:82 85,