Bounds for the Sum of Dependent Risks and Worst. Value-at-Risk with Monotone Marginal Densities

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1 Bouds for the Sum of Depedet Risks ad Worst Value-at-Risk with Mootoe Margial Desities RUODU WANG, LIANG PENG AND JINGPING YANG Jue 23, 202 Abstract I quatitative risk maagemet, it is importat ad challegig to fid sharp bouds for the distributio of the sum of depedet risks with give margial distributios, but a uspecified depedece structure. These bouds are directly related to the problem of obtaiig the worst Value-at-Risk of the total risk. Usig the idea of the complete mixability, we provide a ew lower boud for ay give margial distributios ad give a ecessary ad sufficiet coditio for the sharpess of this ew boud. For the sum of depedet risks with a idetical distributio, which has either a mootoe desity or a tail-mootoe desity, the explicit values of the worst Value-at-Risk ad the bouds o the distributio of the total risk are obtaied. Some examples are give to illustrate the ew results. Keywords: Complete mixability, Mootoe desity, Sum of depedet risks, Value-at- Risk. Mathematics Subject Classificatios (2000): 60E05, 60E5 JEL Classificatios: G0 School of Mathematics, Georgia Istitute of Techology, Atlata, GA , USA. ruodu.wag@math.gatech.edu School of Mathematics, Georgia Istitute of Techology, Atlata, GA , USA. peg@math.gatech.edu Correspodig author. LMEQF, Departmet of Fiacial Mathematics, Ceter for Statistical Sciece, Pekig Uiversity, Beijig, 0087, Chia. yagjp@math.pku.edu.c

2 Itroductio Let X = (X,, X ) be a risk vector with kow margial distributios F,, F, deoted as X i F i, i =,, ad let S = X + + X be the total risk. For the purpose of risk maagemet, it is of importace to fid the best-possible bouds for the distributio of the total risk S whe the depedece structure is uspecified, amely m + (s) = if{p(s < s) : X i F i, i =,, }, () ad M + (s) = sup{p(s < s) : X i F i, i =,, }. (2) See Embrechts ad Puccetti [6] for discussios o such problems i risk maagemet. Sice techiques for hadlig M + (s) are very similar to those for m + (s), we shall focus o m + (s) i this paper. First let us review some kow results o m + (s). Rüschedorf [] foud m + (s) whe all margial distributios have the same uiform or biomial distributio; Deuit et al. [] ad Embrechts et al. [2] used copulas to yield the so-called stadard bouds, which are o loger sharp for 3, ad discussed some applicatios; Embrechts ad Puccetti [4] provided a better lower boud whe all margial distributios are the same ad cotiuous, ad some results whe partial iformatio o the depedece structure is available; Embrechts ad Höig [3] provided a geometric iterpretatio to highlight the shape of the depedece structures with the worst VaR scearios; Embrechts ad Puccetti [5] exteded this problem to multivariate margial distributios ad provided results similar to the uivariate case. I summary, for 3, exact bouds were oly foud for the homogeous case (F = = F = F) i Rüschedorf [] where F is uiform or biomial ad i Wag ad Wag [4] where F has a mootoe desity o its support ad satisfies a mea coditio. Besides the above results o m + (s), Rüschedorf [] associated a equivalet dual optimizatio problem with the bouds for a geeral fuctio 2

3 of X,, X istead of the total risk S. The bouds m + (s) ad M + (s) directly lead to the sharp bouds o quatile-based risk measures of S. A widely used measure is the so-called Value-at-Risk (VaR) at level α, defied as VaR α (S ) = if{s R : P(S s) α}. The boud o the above VaR is called the worst Value-at-Risk sceario. Derivig sharp bouds for the worst VaR is of great iterest i the recet research of quatitative risk maagemet; see Embrechts ad Puccetti [6] ad Kaas et al. [8] for more details. I this paper, we first provide a ew lower boud o m + (s), which is easy to calculate. Usig the idea of the joitly mixable distributios, we give a ecessary ad sufficiet coditio for this boud to be the true value of m + (s). See Sectio 2 for details. I Sectio 3 we employ a special class of copulas to fid m + (s) ad the worst Value-at-Risk whe all the margial distributios are idetical ad have a mootoe or tail-mootoe desity. The methods are illustrated by some examples. Some coclusios are draw i Sectio 4. Some proofs are put i the appedix. 2 Bouds for the sum with geeral margial distributios Throughout, we idetify probability measures with the correspodig distributio fuctios. Let X = (X,, X ) ad S = X + + X. For ay distributio F, we use F (t) = if{s R : F(s) t} to deote the (geeralized) iverse fuctio ad deote by F a the coditioal distributio of F o [F (a), ) for a [0, ), i.e., F a (x) = max { F(x) a a, 0} for x R. It is straightforward to check that for u [0, ], F a (u) = F (( a)u + a). I additio, we defie F (x) = lim a F a (x). I this paper, o specific probability space is assumed ad discussios are focused o distributios, sice m + (s) oly depeds o s ad the distributios F,, F. 3

4 2. Geeral bouds I this sectio, we will give a geeral lower boud o m + (s). Before showig this boud, we eed some defiitios ad lemmas. Defiitio 2.. The radom vector X = (X,, X ) with margial distributios F,, F is called a optimal couplig for m + (s) if P(X + + X < s) = m + (s). It is kow that the optimal couplig for m + (s) always exists (see the itroductio i Rüschedorf [2] for istace). The followig lemma is Propositio 3(c) of Rüschedorf [], which will be used later. Lemma 2.. Suppose F,, F are cotiuous. The there exits a optimal couplig X = (X,, X ) for m + (s) such that {S s} = {X i F i (m + (s))} for each i =,,. Next we itroduce the cocept of completely mixable ad joitly mixable distributios. Defiitio 2.2. (completely mixable ad joitly mixable distributios). A uivariate distributio fuctio F is -completely mixable (-CM) if there exist idetically distributed radom variables X,, X with the same distributio F such that P(X + + X = C) = (3) for some C R. 2. The uivariate distributio fuctios F,, F are joitly mixable (JM) if there exist radom variables X,, X with distributio fuctios F,, F respectively, such that (3) holds for some C R. The defiitio of the CM distributios is formally give i Wag ad Wag [4] although the cocept has bee used i variace reductio problems earlier (see Gaffke ad Rüschedorf 4

5 [7], Kott ad Smith [9], Rüschedorf ad Uckelma [3]). Some examples of -CM distributios iclude the distributio of a costat (for ), uiform distributios (for 2), Normal distributios (for 2), Cauchy distributios (for 2), biomial distributios B(, p/q) with p, q N (for = q), bouded mootoe distributios o [0, ] with /m E(X) /m (for m). See Wag ad Wag [4] for more details of the CM distributios. The cocept of JM distributios is first itroduced i this paper as a geeralizatio of the CM distributios. Obviously, F,, F are JM distributios whe F = = F = F ad F is -CM. The followig propositio gives a ecessary coditio for JM distributios ad the coditio is sufficiet for ormal distributios. The proof is give i the appedix. Propositio Suppose F,, F are JM with fiite variace σ 2,, σ2. The max i σ i 2 σ i. (4) 2. Suppose F i is N(µ i, σ 2 i ) for i =,,. The F,, F are JM if ad oly if (4) holds. i= Remark 2.. Due to the complexity of multivariate distributioal problems, it remais ukow ad extremely difficult to fid geeral sufficiet coditios for the JM distributios. Before presetig the mai results o the relatioship betwee the bouds o m + (s) ad the joitly mixable distributios, we defie the coditioal momet fuctio Φ(t) which turs out to play a importat role i the problem of fidig m + (s). Suppose X i F i for i =,,. Defie for t (0, ), ad Φ(t) = i= E(X i X i Fi (t)) Φ() = lim t Φ(t), Φ(0) = lim t 0+ Φ(t). Obviously Φ(t) is icreasig ad cotiuous whe F i, i =,, are cotiuous. Defie Φ (x) = if{t [0, ] : Φ(t) x} 5

6 for x Φ() ad Φ (x) = for x > Φ(). Theorem 2.3. Suppose the distributios F,, F are cotiuous. () We have m + (s) Φ (s); (5) (2) For each fixed s Φ(0), the equality m + (s) = Φ (s) (6) holds if ad oly if the coditioal distributios F,a,, F,a are joitly mixable, where a = Φ (s). Proof. () It is trivial to prove the result whe Φ(0) =. So we assume Φ(0) <. Note that from Lemma 2. we kow that there exists a optimal couplig X = (X,..., X ) for m + (s) such that {S s} = {X i Fi (m + (s))} for each i =,,. Hece s E[S S s] = i= E[X i X i Fi (m + (s))] = Φ(m + (s)), which implies (5). (2) Suppose X = (X,, X ) is a optimal couplig for m + (s) such that {S s} = {X i Fi (m + (s))} for each i =,,. Whe m + (s) = Φ (s), it follows from the proof of part () that E(S S s) = s, which implies that the coditioal distributios of X,, X o the set {S s} are JM, i.e., the coditioal distributios F,a,, F,a are JM. Coversely, assume that F,a,, F,a are JM. The there exist Y F,a,, Y F,a such that Y + + Y = E(Y + + Y ) = Φ(a) s. Let X i = F i (U) {U a} + Y i {U>a}, (7) 6

7 where U U[0, ] ad is idepedet of (Y,, Y ). The it is easy to verify that X i has the distributio fuctio F i for i =,, ad m + (s) P(S < s) a = Φ (s). The other iequality m + (s) Φ (s) is show i part (). Remark It is see from the proof that the cotiuity of F i ca be removed. I a recet preprit, Puccetti ad Rüschedorff [0] established Theorem 2.3 idepedetly, where the equivalet form sup{p(s > s), X F,, X F } Φ (s) is proved without assumig the cotiuity of F i. 2. The optimal couplig is give i (7). Although the existece of such Y,, Y is guarateed by the mixable coditio, fidig Y,, Y remais quite challegig. For example, whe the margial distributios F i are idetical ad completely mixable, the depedece structure of radom variables Y,, Y may ot be uique ad is hard to be specified i geeral as discussed i Wag ad Wag [4]. 2.2 Bouds for the sum with idetical margial distributios defie Here we cosider m + (s) i the homogeeous case, i.e. F = = F F. For X F, ψ(t) = E(X X F (t)) for t (0, ), ψ() = lim t ψ(t), ψ(0) = lim t 0+ ψ(t), ψ (x) = if{t [0, ] : ψ(t) x} 7

8 for x ψ() ad ψ (x) = for x > ψ(). The followig result follows from Theorem 2.3 immediately. Corollary 2.4. Suppose F = = F F ad F is cotiuous. () We have m + (s) ψ (s/). (8) (2) For each fixed s ψ(0), the equality m + (s) = ψ (s/) (9) holds if ad oly if the coditioal distributio fuctio F a is -completely mixable, where a = ψ (s/). Next we compare the boud i (8) with the boud obtaied i Embrechts ad Puccetti [4], which is m + (s) if r [0,s/) s ( )r r ( F(t))dt s r for s > 0. (0) Propositio 2.5. The boud (0) is greater tha or equal to the boud (8). Moreover, both are equal if ad oly if F () < ad a solutio to the ifimum if r [0,s/) s ( )r r ( F(t))dt s r () lies i [0, s F () ]. The proof of the above Propositio is give i the appedix. Differet from the boud i Embrechts ad Puccetti [4], Theorem 2.3 deals with a more geeral case, where the radom variables X,, X do ot eed to be idetically distributed ad positive. Moreover, the boud i Theorem 2.3 is easier to calculate. Obviously, the bouds i Corollary 2.4 ad i Embrechts ad Puccetti [4] are the same ad both are sharp whe the coditioal distributio F a is completely mixable. A compariso of the two bouds is give i 8

9 Figure 3.2 i Sectio 3 whe the margial distributios have ifiite support (see also Remark 3.2). Note that ifiite support geerally implies that the mixable coditio i Theorem 2.3 ad Corollary 2.4 does ot hold. 3 Bouds for idetically distributed risks with mootoe desities I this sectio, we ivestigate the homogeeous case whe F = = F = F ad F has either a mootoe desity or a tail-mootoe desity o its support. Sice the case of = is trivial, we assume 2. Whe the distributio F with support o [0, ] has a decreasig desity ad satisfies the regular coditio ψ(t) t + t, Wag ad Wag [4] showed that m + (s) = ψ (s/), which ow becomes a corollary of Theorem 2.3. Whe the support of the distributio F is ubouded, the mixable coditio i Theorem 2.3 ad Corollary 2.4 is ot satisfied (see Propositio 2.(7) i Wag ad Wag [4]), i.e., the boud ψ (s/) is ot sharp. I this sectio, we fid a formula for calculatig the boud m + (s) for ay distributio with a mootoe desity or a tail-mootoe desity, ad obtai the correspodig correlatio structure. This partially aswers the questio of optimal couplig for m + (s), which has remaied ope for decades. As a direct applicatio, the bouds o VaR α (S ) are obtaied as well. 3. Prelimiaries To calculate m + (s) for F havig a mootoe margial desity, we first review the costructio of copula Q F ( 2) i Wag ad Wag [4], where F is a distributio fuctio with a icreasig (o-decreasig) desity. More specifically, for some 0 c / ad radom vector (U,, U ) with uiform margial distributios o [0,], we say (U,, U ) Q F (c) if (a) for each i =,,, give U i [0, c], we have U j = ( )U i, j i; (b) F (U ) + + F (U ) is a costat whe ay oe of U i s lies i (c, ( )c). 9

10 Deote Q F = Q F (c ) where c is the smallest possible c such that Q F (c) exists. Note that c = 0 if ad oly if F is -CM. Defie H(x) = F (x) + ( )F ( ( )x) for F with a o-decreasig desity. (2) The the smallest possible c for F with a icreasig desity is ad for ay covex fuctio f, c = mi{c [0, ] : H(t)dt ( c)h(c)} (3) c mi E f (X + + X ) = E QF f ( F (U ) + + F (U ) ). (4) X,,X F Note that Q F may ot be uique. The existece of Q F ad details of the above results ca be foud i Sectio 3 of Wag ad Wag [4]. For F with a decreasig desity ( 2), we defie Q F (c) similarly as follows. For some 0 c /, we say (U,, U ) Q F (c) if (a ) for each i =,,, give U i [ c, ], we have U j = ( )( U i ), j i; (b ) F (U ) + + F (U ) is a costat whe ay oe of U i lies i (( )c, c). Defie H(x) = ( )F (( )x) + F ( x) for F with a decreasig desity. (5) As for the distributio of Z with a decreasig desity, the distributio of Z has a icreasig desity, thus the above properties hold for F with a decreasig desity. That is, the smallest possible c for F with a decreasig desity is c = mi{c [0, ] : H(t)dt ( c)h(c)}. (6) c Ad for a distributio F with a decreasig desity ad ay covex fuctio f the equatio (4) holds. 0

11 Although m + (s) = mi X,,X F E( {S <s}), the above results ca ot be applied directly to solve m + (s) sice the idicator fuctio (,s) ( ) is ot a cocave fuctio. Here we propose to fid m + (s) for F with a mootoe margial desity based o the followig properties of Q F. Propositio 3.. Suppose F admits a mootoe desity o its support.. If (U,, U ) Q F (c) ad F has a icreasig desity, the {Ui (c, ( )c)} = {U (c, ( )c)} a.s. for i =,,. 2. If X,, X F with copula Q F, the H(U/) {U c } + H(c ) {U>c }, c > 0; S = X + + X = E(X ), c = 0 (7) for some U U[0, ]. The proof of Propositio 3. is give i the appedix. For more details of the copula Q F, see Wag ad Wag [4]. 3.2 Mootoe margial desities Now we are ready to give a computable formula for m + (s). I the followig we defie a fuctio φ(x) which works similarly as Φ(x) i the mixable case. For F with a decreasig desity ad a [0, ], defie H a (x) = ( )F (a + ( )x) + F ( x) (8) for x [0, a ] ad c (a) = mi{c [0, ( a)] : ( a) H a (t)dt ( ( a) c)h a(c)}. (9) c

12 Write φ(a) = H a (c (a)) if c (a) > 0, ψ(a) if c (a) = 0. (20) O the other had, for F with a icreasig desity ad a [0, ], defie H a (x) = F (a + x) + ( )F ( ( )x), (2) c (a) = mi{c [0, ( a)] : ( a) H a (t)dt ( ( a) c)h a(c)} (22) c ad φ(a) = H a (0) if c (a) > 0, ψ(a) if c (a) = 0. (23) Some probabilistic iterpretatio of the fuctios H a (x) ad φ(a) is give i the followig remark. Techical details are put i Lemma 3.2 later. Remark 3.. Suppose Y,, Y F a with copula Q F a. By (7) we have H(U/) {U c } + H( c ) {U> c }, c > 0, Y + + Y = E(Y ), c = 0 for some U U[0, ], where H(x) ad c are H(x) ad c defied i (2), (3), (5) ad (6) by replacig F with F a. It is easy to check that H(x) = H a (( a)x), c = c (a)/( a) ad H( c ) = H a (c (a)). For c (a) > 0, later we will show that H a (x), x [0, c (a)] attais its miimum value at H a (c (a)) for F a with a decreasig desity ad at H a (0) for F a with a icreasig desity. Therefore, the miimum possible value of Y + + Y is mi H a(x) {c (a)>0} + E(Y ) {c (a)=0} = φ(a). x [0,c (a)] Thus, P(Y + +Y φ(a)) =, which leads to P(S < φ(a)) a by settig X i = F (V) {V a} + Y i {V>a} where V U[0, ] is idepedet of Y,, Y. This suggests m + (s) φ (a), i.e., φ (a) is potetially a optimal boud. I order to prove the optimality of φ (a), more details of the fuctios H a (x) ad φ(a) are give i the followig lemma, whose proof is put i the appedix. 2

13 Lemma 3.2. Suppose F admits a mootoe desity. (i) If F has a decreasig desity, the give a [0, ), H a (x) is decreasig ad differetiable for x [0, c (a)]. (ii) If F has a icreasig desity, the give a [0, ), H a (x) is icreasig ad differetiable for x [0, c (a)]. (iii) If F has a decreasig desity, the φ(a) = E[F (V a )] where V a U[a + ( )c (a), c (a)]. (iv) For ay radom variables U,, U U[a, ] ad 0 a < b, we have E(F (U i ) A) < E[F (V b )] for i =,,, where V b is defied i (iii) ad A = i= {U i [a, c (b)]}. (v) Suppose Y,, Y F a with copula Q F a, the P(Y + + Y φ(a)) =. (vi) φ(a) is cotiuous ad strictly icreasig for a [0, ). Sice φ(a) is cotiuous ad strictly icreasig, its iverse fuctio φ (a) exists. Put φ (t) = 0 if t < φ(0) ad φ (t) = if t > φ(). Theorem 3.3. Suppose the distributio F(x) has a decreasig desity o its support ad φ(a) is defied i (20), or the distributio F(x) has a icreasig desity o its support ad φ(a) is defied i (23). The we have m + (s) = φ (s). Proof. (a) We first prove m + (s) φ (s). Write a = φ (s). For i =,,, let Y,, Y F a with copula Q F a ad X i = F (V) {V a} + Y i {V>a} where V U[0, ] is idepedet of Y,, Y. It is easy to check that X i F ad by Lemma 3.2(v), m + (s) P(S < φ(a)) = P(S φ(a)) P(Y + + Y φ(a))p(v > a) = a. Thus m + (s) φ (s). 3

14 (b) Next we prove m + (s) φ (s) whe F(x) has a decreasig desity. Suppose a = m + (s) < φ (s) = b ad X = (X,, X ) is a optimal couplig for m + (s) such that {S s} = {X i F (a)} for each i =,,. Hece there exist U a,,, U a, U[a, ] such that F (U a, ) + + F (U a, ) s with probability. By Lemma 3.2(iii) ad (iv), we have s E[ F (U a,i ) A] < E(F (V b )) = φ(b) = s. i= This leads to a cotradictio. Thus m + (s) = φ (s). (c) Fially we prove m + (s) φ (s) whe F(x) has a icreasig desity. I this case F () <. Write a = m + (s) ad let X = (X,, X ) be a optimal couplig for m + (s) such that {S s} = {X i F (a)} for each i =,,. It is clear that P(S < F (a) + ( )F () + ɛ S s) P(X i < F (a) + ɛ X i F (a)) > 0 for ay ɛ > 0. Note that P(S < s S s) = 0 ad thus s F (a) + ( )F () = H a (0). This shows s H a (0). The iequality s ψ(a) is give by Theorem 2.3. Hece s φ(a) ad a φ (s). The proof of the above theorem suggests costructig the optimal correlatio structure as follows. I both cases, for a = φ (s) let U a,,, U a, U[a, ] with copula Q F a ad U U[0,] is idepedet of (U a,,, U a, ). Defie U i = U a,i {U a} + U {U<a} (24) for i =,,. The P(F (U ) + + F (U ) < s) = φ (s). 4

15 Remark The copula Q F plays a importat role i derivig bouds for the covex miimizatio problem (4) ad the m + (s) problem with mootoe margial desities. Note that Q F may ot be uique, hece the structure (24) may ot be uique. Also, o the set {S < s}, the depedece structure of X,, X ca be arbitrary. 2. The value φ (s) is accurate eve whe E(max{X, 0}) =. Whe the distributio F a is -CM, Theorem 3.3 gives the sharp boud Φ (s) i Theorem Whe a radom variable X has a mootoe desity, X has a mootoe desity too. Hece the above theorem also solves the similar problem M + (s) = sup P(S < s) = if P( S s) = if P( S < s), X i P X i P X i P where P has a mootoe desity. 4. Figure 3. shows the sketch of a optimal couplig for F with a decreasig desity, some a > 0 ad c (a) > 0. Here U,, U U[0, ] ad P(F (U ) + + F (U ) < s) = φ (s). (i) Whe U i [0, a], U i is arbitrarily coupled to all other U j i Part A. (ii) Whe U i [a, a + ( )c (a)], U i is coupled to other U j, j i i Part B ad Part D. For j i, either U i a = ( )( U j ) or U j = U i. (iii) Whe U i [a + ( )c (a), c (a)], U i is coupled to all other U j, j i i Part C, ad F (U ) + + F (U ) = φ(a). It is the completely mixable part. (iv) Whe U i [ c (a), ], U i is coupled to other U j, j i i Part B. For j i, U j a = ( )( U i ). 5. Figure 3.2 shows the real values of m + (s) i Theorem 3.3 ad the lower boud ψ (s/) i Theorem 2.3 for the Pareto(2,) distributio. We also calculate the boud (0) i 5

16 Embrechts ad Puccetti [4] (see Sectio 2.2). It turs out that i this case the real values are equal to the boud (0), which suggests that the boud (0) i [4] may be sharp for Pareto distributios. a + ( )c (a) a = m + (s) = φ (s) F (s/) c (a) 0 A B C D I this part U i [0,a] ad U i is arbitrarily coupled to all other U j i A. I this part U i [a + ( )c (a), c (a)], U i is coupled to all other U j, j i i C, ad F (U ) + + F (U ) = φ(a). It is the completely mixable part. 0 I this part U i [a,a + ( )c (a)], U i is coupled to other U j, j i i B ad D. For j i, either U i a = ( )( U j) or U j = U i. I this part U i [ c (a),] ad U i is coupled to other U j, j i i B. For j i, U j a = ( )( U i). 0 Figure 3.: Sketch of the optimal couplig 3.3 Tail-mootoe margial desities For the distributio F with desity p(x), we say p(x) is tail-mootoe, if for some b R, p(x) is decreasig for x > b or p(x) is icreasig for x < b. We are particularly iterested i the case whe p(x) is tail-decreasig (p(x) is decreasig for x > b) sice the risks are usually positive radom variables. For most risk distributios the tail-decreasig property is satisfied. For example, the Gamma distributio with shape parameter α for α > ad the F-distributio with d, d 2 degrees of freedom with d > 2 have a tail-decreasig desity, but do ot have a mootoe desity. I the VaR problems, oe is cocered with the tail behavior of the distributio. From the proof of Theorem 3.3, iformatio o the left tail of F does ot play ay role i the calculatio 6

17 m + (s) for Pareto distributios, = 3, α = 2, θ = real value φ ( ) (s) lower boud ψ ( ) (s/) φ ( ) (s) ad ψ ( ) (s/) s Figure 3.2: m + (s) ad ψ (s/) for a Pareto distributio of m + (s). Based o this observatio, we have the followig theorem, which solves m + (s) for F with tail-decreasig desity ad some large s. Theorem 3.4. Suppose the desity fuctio of F is decreasig o [b, ), ad φ(a) is defied i (20). The for s φ(f(b)), m + (s) = φ (s). Proof. Sice the desity fuctio of F is decreasig o [b, ), the coditioal distributio F F(b) has a decreasig desity. Note that H a (x), c (a) ad φ(a) oly deped o the coditioal distributio F a, hece they are well defied for F(b) a. Sice s φ(f(b)), φ (s) F(b) ad the coditioal distributio F φ (s) has a decreasig desity. Theorem 3.4 follows from the same argumets as i the proof of Theorem 3.3, where o coditio o the distributio of X i o {X i < F (φ (s))} is used. 7

18 3.4 The worst Value-at-Risk scearios The Value-at-Risk (VaR) is a importat risk measure i risk maagemet; see Embrechts ad Puccetti [6] ad refereces therei. Recall that VaR is the α-quatile of the distributio, i.e., VaR α (S ) = F S (α) = if{s R : F S (s) α}, (25) where F S is the distributio of S. Typical values of the level α are 0.95, 0.99 or eve As metioed i Embrechts ad Puccetti [6], baks are cocered with a upper boud o VaR( d i= X i ) whe the correlatio structure betwee X = (X,, X d ) is uspecified. Fidig the bouds o the VaR is equivalet to fidig the iverse fuctio of m + (s) (ote that m + (s) is o-decreasig). Usig Theorem 3.3 ad Theorem 3.4, we are able to obtai the explicit value of the upper boud o the VaR, amely, the worst Value-at-Risk. The proof follows directly from the fact that sup Xi F, i VaR α(s ) = m + (α) whe m + (s) is cotiuous ad strictly icreasig. Corollary 3.5. Suppose that the desity fuctio of the margial distributio F is decreasig o [b, ) ad φ(a) is defied i (20). The for α F(b), the worst VaR of S = X + + X is sup VaR α (S ) = m + (α) = φ(α). (26) X i F, i I particular, (26) holds for all α if the margial distributio F has decreasig desity o its support ad a optimal correlatio structure is give by (24). For arbitrary margial distributios F,, F, Theorem 2.3 gives a upper boud for the worst-var problem as follows. Corollary 3.6. For arbitrary margial distributios, where Φ(α) is defied i Sectio 2. sup VaR α (S ) m + (α) Φ(α), (27) X i F i,i=,, 8

19 Figure 3.3 shows the explicit worst-var i (26) ad the upper boud i (27) for the distributio Pareto(4,) ad 0.9 α worst-var α (S) for Pareto distributio, = 3, α = 4, θ = Real value of worst-var Upper Boud o worst-var 3 worst-varα(s) α Figure 3.3: Worst-VaR for a Pareto distributio 3.5 Examples Here we give some examples to show how to compute m + (s). Example 3.. Assume that X U[0, ], the uiform distributio o [0,]. The p(x) =, F(x) = x, x [0, ], F (t) = t, t [0, ]. Further we have c (a) = 0 for all 0 a ad φ(t) = ψ(t) = E(X X > t) = (+t) 2 for t [0, ]. Thus ( ) 2s m + (s) = φ (s) =. + This result ideed is the same as that i Rüschedorf []. Oe optimal correlatio structure is also give i Rüschedorf ad Uckelma [3]. 9

20 Example 3.2. Assume that X Pareto(α, θ), α >, θ > 0 with desity fuctio p(x) = αθ α x α, x θ. The ( x ) α F(x) =, x θ, F (t) = θ( t) /α, t [0, ]. θ Further we have that c (a) is the smallest c [0, ( a)] such that α α (( a ( )c) /α c /α ) ( ( a) c)(( )( a ( )c) /α + c /α ). The umerical values of m + (s) for two Pareto distributios ad = 3 are plotted i Figure 3.4. A possible correlatio structure is give i (24) m + (s) for Pareto distributios, = 3 α = 2, θ = α = 3, θ = m+(s) s Figure 3.4: m + (s) for Pareto distributios Example 3.3. Assume that X Gamma(α, λ), α, λ > 0 with desity fuctio The p(x) = λα Γ(α) xα e λx. F(x) = γ(α, λx), x > 0, 20

21 where γ(α, t) = t 0 Γ(α) xα e λx dx is the lower icomplete Gamma fuctio. Further c (a) is the smallest c [0, ( a)] such that α λ (γ(α +, λf ( c)) γ(α +, λf (a + ( )c))) ( ( a) c)h a(c), which ca be calculated umerically. The umerical values of m + (s) for two Gamma distributios ad = 3 are plotted i Figure 3.5. A possible correlatio structure is give i (24) m + (s) for Gamma distributios, = 3 α =, λ = α = /2, λ = /2 0.7 m+(s) s Figure 3.5: m + (s) for Gamma distributios 4 Coclusios I this paper, we provide a ew lower boud for m + (s) with ay give margial distributios, ad give a ecessary ad sufficiet coditio for its sharpess i terms of the joit mixability. Whe the margial distributios have a commo mootoe desity, the explicit value of m + (s) ad the worst Value-at-Risk are obtaied. We also exted these results to distributios with a commo tail-mootoe desity. 2

22 Appedix Proof of Propositio The case = is trivial. For 2, by the defiitio of JM distributios, there exist X F,, X F such that Var(X + + X ) = 0. Sice Var(X + X X ) Var(X ) Var(X X ) σ σ i, i=2 we have 2σ i= σ i 0. Similarly, we ca show that 2σ k i= σ i 0 for ay k =,,, i.e., (4) holds. 2. We oly eed to prove the part for 2. Without loss of geerality, we assume σ σ 2 σ. Let X = (X,, X ) be a multivariate Gaussia radom vector with kow margial distributios F,, F ad a uspecific correlatio matrix Γ. We wat to show there exists a correlatio matrix Γ such that Var(X + + X ) = 0. Let T be the correlatio matrix of (X 2,, X ) ad Y = X X. Defie f (T) = Var(X ) Var(Y). Obviously f (T) is a cotiuous fuctio of T with caoical distace measure. It is easy to check that f (T) = σ i=2 σ i 0 whe X 2 = σ 2 Z + µ 2,, X = σ Z + µ for some Z N(0, ). Sice σ σ 2 σ, we also have f (T) = σ i=2 ( ) i σ i 0 whe X i = ( ) i σ i Z + µ i for i = 2,,. Hece there exists a correlatio matrix T 0 such that f (T 0 ) = 0. With the correlatio matrix of (X 2,, X ) beig T 0, we defie X = Y + E(Y) + µ. Hece X N(µ, σ 2 ) ad Var(X + + X ) = 0, which imply that F,, F are JM. 22

23 Proof of Propositio 2.5. Whe s F (), we have m + (s) = = ψ (s/). Suppose s < F (). Obviously if r [0,s/) s ( )r r ( F(t))dt s r if r [0,s/) r ( F(t))dt. (28) s r ( For r [0, s ), from r ( F(t))dt s r ) = 0, we have g(r) := ( F(r))(s r) + Suppose r = r satisfies (29). The r ( F(t))dt = 0. (29) F(r ) = r { F(t)}dt s r. (30) Note that r always exists sice g is cotiuous, g(0) = s + µ < 0, F(s/) < ad Itegratio by parts leads to ad hece { F(r )}(s r ) + g(s/) = r s/ { F(t)}dt > 0. { F(t)}dt = s{ F(r )} + s( F(r )) = E(X X > r )( F(r )), r tdf(t) = 0, i.e., F(r ) = ψ (s/). (3) Therefore, the boud i (0) is greater tha or equal to the boud i (8) by (28), (30) ad (3). Note that the boud (0) is strictly greater if F () =. For provig the secod part of Propositio 2.5, we oly eed to cosider the case of s < F () < sice m + (s) = = ψ (s/) whe F () s <. Cosider the problem () ad if r [0,s/) F () r ( F(t))dt. (32) s r 23

24 From the above proof, we ca see that r is the uique solutio to (32). Therefore, the bouds (8) ad (0) are equal if ad oly if () ad (32) are equal. Sice ( s F ) () g ( = F we have r [0, s F () ]. Thus ( s F )) () F () s + s F () ( F(t))dt 0, if r [0,s/) F () r ( F(t))dt s r = if r [0, s F () ] = if r [0, s F () ] F () r s ( )r r ( F(t))dt s r ( F(t))dt. s r Therefore, the bouds (8) ad (0) are equal if ad oly if a solutio to () lies i [0, s F () ]. Proof of Propositio 3... By (a) i Sectio 3., for ay i j, U i [0, c] U j [ ( )c, ]. Hece A i := {U i [0, c]} {U j [ ( )c, ]} =: B j ad P(A i A j ) = 0. As a cosequece, i j A i B j. Note that P( i j A i ) = ( )c = P(B j ). Thus i j A i = B j a.s. ad i= A i = A j B j = {U j [0,c] [ ( )c,]} a.s. which imply that {U j (c, ( )c)} = ( i= A i ) c a.s. for j =,,. 2. We oly prove the case whe F has a icreasig desity. Whe c = 0, (7) follows from the defiitio of Q F. Next we assume c > 0. Write D j = A j B j ad X j = F (U j ), U j U[0,] for j =,,. First ote that by coditio (b) i Sectio 3., for ay j =,,, F (U ) + + F (U ) is a costat o the set D c j. This costat equals its 24

25 expectatio, which is E(F (U ) + + F (U ) D c j ) = E(F (U ) D c ) = = = = c c c c = H(c ). ( )c c F (x)dx F (x)dx + c c F (x)dx + c c c H(x)dx ( )c F (x)dx c F ( ( )t)d( )t The last equality holds because (3) ad c H(x)dx = ( c )H(c ) for c > 0. Therefore, almost surely S = F (U ) + + F (U ) = F (U i ) D + F (U i ) D c = = = = i= i= F (U i ) j= A j + H(c ) D c i= F (U i )( A j ) + H(c ) D c i= j= [F (U j ) + ( )F ( ( )U j )] A j + H(c ) D c j= H(U j ) A j + H(c ) D c. j= 25

26 Sice c / ad the sets A,, A ad D c are disjoit, we have P( H(U j ) A j + H(c ) D c < t) j= = P(H(U ) {U c } < t) + P(H(c ) D c < t) = P(H(U /) {U c } < t) + P(H(c ) {U >c } < t) = P(H(U /) {U c } + H(c ) {U >c } < t). Hece there exists a U U[0,] such that H(U j ) A j + H(c ) D c j = H(U/) {U c } + H(c ) {U>c }. j= Proof of Lemma 3.2 (i) Uder the assumptio of F, F (x) is covex ad differetiable. Thus H a (x) is covex ad differetiable. The defiitio of c (a) shows that the average of H a (x) o [c (a), ( a)] is H a (c (a)) if 0 < c (a) < a, amely ( a) H a (t)dt = H a (c (a)). ( a) c (a) c (a) With H a (x) beig covex, we have H a(c (a)) 0 ad so H a(x) 0 o [0, c (a)]. Here H a(x) deotes H a (x)/ x. Note that for > 2, H a( a ) = (( )2 )(F ) ( a ) > 0 implies ( a) c H a (t)dt ( ( a) c)h a(c) for some c < a, thus c (a) < a always holds. For = 2, H a(x) 0 o [0, a ] sice H a( a ) = 0 ad H is covex. (ii) It follows from similar argumets as i (i). 26

27 (iii) Suppose c (a) > 0. By the cotiuity of H a (x) w.r.t. x ad (9), we kow that c (a) satisfies ( a) c (a) H a (t)dt = ( ( a) c (a))h a (c (a)). Note that for ay c [0, ( a)], ( a) c H a (t)dt = = = ( a) c a+ ( a) a+( )c c a+( )c ( )F (a + ( )t)dt + ( a) c F (t)dt + F (t)dt ( a) F (t)dt. c F ( t)dt Thus it follows from the defiitio of c (a) that H a (c (a)) = E[F (V a )]. For the case c (a) = 0, it is obvious that ψ(a) = φ(a) = E[F (V a )]. (iv) Note that i a give probability space, for ay measurable set B with P(B) > 0 ad cotiuous radom variable Z with cdf G, we have E(Z B) E[Z Z G ( P(B))]. To see this, deote the coditioal distributio of Z o B by G ad the coditioal distributio o {Z G ( P(B))} by G 2. The we have G 2 (x) = = P(Z x, G(Z) P(B)) P(B) max{g(x) + P(B), 0} P(B) P(Z x, B) = G (x), x R, (33) P(B) which implies that for U U[0,], E(Z B) = E[G (U)] E[G 2 (U)] = E[Z Z G ( P(B))]. (34) Sice A = i= {U i [a, c (b)]}, we have P(A) c (b) a > 0 ad U i c (b) o 27

28 A. By defiig Z = F (U i ) {Ui c (b)} + F (a) {Ui > c (b)}, it follows from (34) that E[F (U i ) A] = E[Z A] E[Z Z F ( c (b) ( a)p(a))] E[F (U i ) U i [ c (b) ( a)p(a), c (b)]] E[F (U i ) U i [a + ( )c (b), c (b)]] < E[F (U i ) U i [b + ( )c (b), c (b)]] = E(F (V b )). (35) (v) It follows from (i), (ii) ad the argumets i Remark 3.. (vi) We first prove the case whe F has a decreasig desity. Sice H a (x) is covex w.r.t. x ad differetiable w.r.t. a, the defiitio of c (a) implies that c (a) is cotiuous. Hece φ(a) = E[F (V a )] is cotiuous. Suppose U a,,, U a, U[a, ] with copula Q F a. The F (U a, ),, F (U a, ) F a ad have copula Q F a too. By (v), we have F (U a, ) + + F (U a, ) φ(a). (36) Thus from (35) ad (36) we have φ(a) E[ F (U a,i ) A] < E(F (V b )) = φ(b). i= Next we prove the case whe F has a icreasig desity. The cotiuity of c (a) comes from the same argumets as above. By defiitio, H a (0) ad ψ(a) are cotiuous ad icreasig fuctios of a. So we oly eed to show that whe c (a) approaches 0, H a (0) ψ(a) approaches 0 too. Suppose that as a a 0, c (a) 0 ad c (a) 0 for a 0 ɛ < a < a 0 ad ɛ > 0. The ( a) 0 H a (t)dt ( a 0)H a0 (0), 28

29 which implies that ψ(a) = a a F (a + t)dt = ( a) H a (t)dt H a0 (0) a 0 as a a 0. Together with the cotiuity of H a (0) ψ(a) we kow H a (0) ψ(a) 0 as a a 0. Ackowledgmet. We thak Co-editor Professor Kerry Back, a associate editor ad two reviewers for their helpful commets which sigificatly improved this paper. Wag s research was partly supported by the Bob Price Fellowship at the Georgia Istitute of Techology. Peg s research was supported by NSF Grat DMS Yag s research was supported by the Key Program of Natioal Natural Sciece Foudatio of Chia (Grats No. 3002). Refereces [] Deuit, M., Geest, C., Marceau, É. (999). Stochastic bouds o sums of depedet risks. Isurace: Mathematics ad Ecoomics 25, [2] Embrechts, P., Höig, A. ad Juri, A. (2003). Usig copulae to boud the value-at-risk for fuctios of depedet risks. Fiace ad Stochastics 7, [3] Embrechts, P ad Höig, G. (2006). Extreme VaR scearios i higher dimesios. Extremes 9, [4] Embrechts, P ad Puccetti, G. (2006a). Bouds for fuctios of depedet risks. Fiace ad Stochastics 0, [5] Embrechts, P. ad Puccetti, G. (2006b). Bouds for fuctios of multivariate risks. Joural of Multivariate Aalysis 97,

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