Optimal capital allocation with copulas

Transkript

1 Hacettepe Journal of Mathematcs and Statstcs Volume , Optmal captal allocaton wth copulas Zou We and Xe Je-hua Keywords: Abstract In ths paper, we nvestgate optmal captal allocaton problems for a portfolo consstng of derent lnes of rsks lnked by a Farle-Gumbel- Morgenstern copula, modellng the dependence between them. Based on the Tal Mean-Varance prncple, we examne the bvarate case and then the multvarate case. Explct formulae for optmal captal allocatons are obtaned for exponental loss dstrbutons. Fnally, the results are llustrated by varous numercal examples. Optmal Captal allocaton; Tal Mean-Varance; Copula 2 AMS Classcaton: 91B3; 6E5 Receved : Accepted : Do : / HJMS Introducton A fundamental queston n the elds such as quanttatve rsk management, nsurance and nance s how much money to ask from a polcyholder n a heterogeneous portfolo? A closely related queston s how to allocate a gven amount of captal between the derent classes n a portfolo. The rst queston s known as the premum calculaton problem and the second s the captal allocaton problem. The two problems are well-known and have been studed, see e.g. Laeven and Goovaerts [13], Zaks et al. [2], Frostg et al. [1], Belles-Sampera [4], Bauer and Zanjan [2,3] and references theren. In recent years, there has been growng nterest n studyng the optmal allocaton problems because they play a central role n Solvency II. For example, Dhaene et al. [9] assumed a portfolo consstng of n rsks X 1, X 2,, X n and a company wshed to allocate the total captal d = d 1d 2 d n to the correspondng rsks. They formulated captal allocaton as an optmzaton problem and provded a reasonable crteron, whch s to set the amount d to X so that the potental loss measured by some approprate School of Scence, Nanchang Insttute of Technology, Nanchang 3399, P.R.Chna School of Economcs and Trade, Nanchang Insttute of Technology, NanChang 3399, P.R.Chna Emal : jhxe@nt.edu.cn Correspondng Author.

2 45 dstance measure s as small as possble. And further, they proposed to model the captal allocaton problem by the followng optmzaton problem: n [ ] X d n mn v E ζ D, such that d = d, 1.1 d 1,,d n =1 v where the v, = 1, 2,, n, are non-negatve real numbers such that n =1 v = 1, D s a non-negatve functon and the ζ, = 1, 2,, n, are non-negatve random varables such that E[ζ ] = 1. The non-negatve real number v s a measure of exposure or busness volume of the th unt, such as revenue, nsurance premum, etc. D s a dstance measurement functon that measures the loss of allocaton. The non-negatve random varables ζ, = 1, 2,, n, are used as weght factors to the derent possble outcomes of DX d. The framework s general and ncludes many optmsaton crteron as specal cases, such as Quantle, Harcut and Covarance. A specal case of 1.1 s the followng optmzaton problem: mn d 1,,d n n v [ E X d 2], =1 such that =1 n d = d, whch s showed n Dhaene et al. [9]. The soluton to the equaton above s d = d n EX 1 n E, = 1, 2,, n. n Xu and Hu [17] generalzed ths dea and dened the followng loss functon: n L p = DX p. j=1 =1 Then, they proposed to model the captal allocaton problem by the followng optmzaton problem: mn p A P rl p t, t. 1.2 You and L [19] further studed the captal allocaton concernng mutually exchangeable random rsks. Ca and We [5] proposed some new notons of dependence wth applcatons n optmal allocaton problems. Zaks and Tsanakas [21] studed the optmal captal allocaton n a herarchcal corporate structure where stake holders at two organzatonal levels. Manesh and Khaled [14] consdered the allocatons of polcy lmts and orderng relatons for aggregate remanng clams. Note that allocaton crteron based on mnmzng the loss functon does not take nto account the the factor of varablty of loss functon. We also note that both allocaton rule 1.1 and 1.2 do not take nto account tal rsk. Xu and Mao [18] proposed a Tal Mean-Varance TMV framework to overcome ths lmtaton. In more detal, they provded the followng allocaton rule: { [ n ] n } mn d 1,,d n E =1 X d 2 S > VaR κs such that βvar =1 =1 X d 2 S > VaR κs n d = d, 1.3 where β >, S = n =1 X s the aggregate rsk, VaRκS = nfx R, F S κ s the value at rsk at level κ, < κ < 1 of S and F S = PrS s s the dstrbuton of S. Xu and Mao [18] studed the optmal captal allocatons when the rsks have multvarate ellptcal dstrbutons. =1,

3 451 In most papers descrbed above, the dependence between derent lnes of busness of a portfolo s due to constructon of a multvarate dstrbuton. Copulas are currently seen as exble and eectve tools to descrbe dependence between the random varables. In ths paper, we propose ntroducng dependence wth a copula. We use the Farle- Gumbel-Morgenstern FGM copula, lke Bargès et al. [1] and Cossette et al. [7], to descrbe the dependence. Based on the smplcty and tractablty of FGM copula, we derve the closed-form expressons for the optmal captal allocatons based on the Tal Mean-Varance model when the FGM copula represent dependence between the rsk margnals. The work of ths paper can be seen as a complement to the research on the the optmal captal allocatons based on the Tal Mean-Varance prncple and extend the former results by ntroducng dependence wth a copula. When the optmal captal allocaton problems for a portfolo consstng of derent lnes of rsks lnked by a Farle-Gumbel- Morgenstern copula are consdered, we show that, the explct captal allocaton formulae can be obtaned under the Tal Mean-Varance prncple. The rest of the paper s organzed as follows. In Secton 2, we descrbe basc dentons and propertes of the FGM copula and the exponental dstrbutons. In Secton 3, we derve explct captal allocaton formulae for the bvarate case, where both losses are exponentally dstrbuted wth the dependence lnked by the FGM copula. In Secton 4, we extend the results to the multvarate case. Some numercal examples to calculate the optmal captal allocaton are presented to llustrate the soluton procedure n Secton Dentons Here, we brey recall the characterstcs and denton of the FGM copula. The FGM copula s a perturbaton of the ndependence copula and t s not Archmedan. The FGM copula s a rst order approxmaton of the Plackett copula Nelsen [15], p.1 and of the Frank copula p.133. Among the recent applcatons of the FGM copula, we menton Bargès et al. [1] and Cossette et al. [7] who deal wth the applcaton of the TVaR-based allocaton rule usng the FGM copula. The FGM copula s appled n the context of rsk measurement by Gebzloglu and Yagc [11] and n the context of sums of dependent r.v. by Geluk and Tang [12]. The FGM copula s also used n a run context by Cossette et al. [8], Xe and Zou [16]. The FGM copula s attractve due to ts tractablty and ts smplcty. We assume the lnes of busness of an portfolo are lnked wth a FGM copula. Snce the exponental dstrbuton s a classc dstrbuton for the rsk random varables and ts practcal mathematc propertes allow explct results, we assume margnal rsks are dstrbuted as exponentals. 2.1 The bvarate case Frstly, we consder two busness lnes whose losses X 1 and X 2 follow the exponental dstrbuton: X Expλ, = 1, 2. Ther probablty densty functons pdf f X and cumulatve dstrbuton functons cdf F X are gven by f X = λ e λ x, F X = 1 e λ x, for = 1, 2. We assume that the couple X 1, X 2 has a bvarate dstrbuton dened wth FGM copula. The FGM copula s dened as follows: C F GM θ u 1, u 2 = u 1u 2 θu 1u 21 u 11 u 2,

4 452 for u 1, u 2 [, 1] [, 1]. The dependence parameter θ takes value n [1, 1], where θ < > corresponds to a negatve postve dependence relaton. The densty of the bvarate FGM copula s c F GM θ u 1, u 2 = 2 C F GM θ u 1, u 2 u 1 u 2 = 1 θ2ū 1 12ū 2 1, where ū = 1 u, = 1, 2. Then, the jont c.d.f. F X1,X 2 x 1, x 2 of X 1, X 2 wth margnals F X1 and F X2 and dened wth the FGM copula s gven by F X1,X 2 x 1, x 2 = C F GM θ F X1 x 1, F X2 x 2 = F X1 x 1F X2 x 2 θf X1 x 1F X2 x 21 F X1 x 11 F X2 x 2. The jont pdf of X 1, X 2 s f X1,X 2 x 1, x 2 = c F GM θ F X1 x 1, F X2 x 2f X1 x 1f X2 x 2 = 1 θλ 1e λ 1x 1 λ 2e λ 2x 2 θ2λ 1e 2λ 1x 1 λ 2e λ 2x 2 θλ 1e λ 1x 1 2λ 2e 2λ 2x 2 θ2λ 1e 2λ 1x 1 2λ 2e 2λ 2x The multvarate case Now, we consder n busness lnes whose losses X 1, X 2,, X n follow the exponental dstrbuton: X Expλ, = 1, 2,, n. We also assume that the n derent rsks jonted by a multvarate FGM n-copula, whch has 2 n n 1 parameters, s dened as follows: C F GM θ n u 1,..., u n = u 1 u n 1 1 p 1 < <p q n θ p1...p q ū p1 ū pq, where θ [1, 1], ū = 1 u, = 1, 2,, n. Its densty can be wrtten as n c F θ GM u 1,..., u n = 1 2ū p1 1 2ū pq 1. Then, the jont pdf of X 1,..., X n s 1 p 1 < <p q n f X1,...,X n x 1,..., x n = c F θ GM F X1 x 1,..., F Xn x nf X1 x 1 f Xn x n n = f X1 x 1 f Xn x n1 1 F Xp1 x p1 1 F Xpq x pq n = ωx 1,..., x n; λ 1,..., λ n q l= 1 p 1 < <p q n 1 p 1 < <p q n ɛ 1,...,ɛ q τ l,q 1 l ωx p1,..., x pq ; x q1,..., x n ; 2 ɛ1 λ p1,..., 2 ɛq λ pq, λ q1,, λ n, 2.2 where ωx 1, x 2,, x n; α 1, α 2,, α n = α 1e α 1x 1 α 2e α 2x2 α ne αnxn, q1,, n are the mssng ndexes of p 1,, p q to complete 1,, n and τ l,q are the sets of q- tuples composed of l zeros and q l ones,.e., τ l,q, l =, 1,, q and q = 2,, n are dened as τ,q = {1, 1,, 1 1 q}, τ 1,q = {1, 1,, 1 q,,, 1,, 1 1 q}, τ 2,q = {1, 1,,, 1 q,,,,, 1 1 q},, τ q,q = {,,, 1 q}.

5 Calculaton of the optmal captal allocaton for two rsks In ths secton, we obtan the close formed expresson of the optmal captal allocaton based on the Tal Mean-Varance TMV model for two exponentally dstrbuted rsks lnked by a FGM copula. Let X 1 and X 2 are two exponentally dstrbuted wth λ 1 and λ 2. In order to smplfy the calculaton, we assume that λ 1 λ 2, λ 1 2λ 2, and λ 2 2λ 1. Applyng the smlar technque proposed below, one can obtans the adjusted results wthout these restrctons. Theorem 3.1. Let X 1 and X 2 are two exponentally dstrbuted wth λ 1 and λ 2. The dependence structure for X 1, X 2 s dened by the bvarate FGM copula wth parameter θ [1, 1]. Then the optmal allocaton soluton d = d 1, d 2 to the TMV model { [ 2 ] 2 } mn d 1,d 2 E =1 X d 2 S > VaR κs βvar =1 X d 2 S > VaR κs such that d 1 d 2 = d, s presented as follows: 1 d 1 = 21 2β2E X 1X 2 E X 1 E X 22 E X2 1 E X2 2 2βE X 1X 2 2 E X2 1 X 2 E X3 1 E X3 2 E X 1 E X 2 1 2βE X2 1 E X2 2 d1 4βE X 1X 2 E X2 2 E X 2 E X 2 E X 1, d 2 = 21 2β2E X 1X 2 E X 1 E X 22 E X2 1 E X2 2 2βE X 1X 2 2 E X2 1 X 2 E X3 1 E X3 2 E X 1 E X 2 1 2βE X2 1 E X2 2 d1 4βE X 1X 2 E X2 1 E X 1 E X 1 E X 2, 3.2 where [ ] E Xk = E X k S > VaR κs = 1 1 κ [1θξ kvar κs; λ ; λ jθξ k VaR κs; 2λ ; λ j θξ k VaR κs; λ ; 2λ j θξ k VaR κs; 2λ ; 2λ j],, j = 1, 2, j, k = 1, 2, 3, E X2 = E [ X 2 S > VaR ] κs = 1 [1 θχvarκs; λ; λj 1 κ θχvar κs; 2λ ; λ jθχvar κs; λ ; 2λ jθχvar κs; 2λ ; 2λ j],, j = 1, 2, j, E X = E [X S > VaR κs] = 1 [1 θϕvarκs; λ; λj 1 κ θϕvar κs; 2λ ; λ j θϕvar κs; λ ; 2λ j θϕvar κs; 2λ ; 2λ j], ξ 1x; α ; α j = αjeα x x 1 α αjeαx α e αj x, α j α α j α 2 e αx α j 2 2x α ξ 2x; α ; α j = 2 α x 2 2αjeαx x 1 α 2αjeαx α e αj x, α j α α j α 2 α j α 3 ξ 3x; α ; α j = e αx α j 6 6x 3x2 α 3 α 2 α j α α x 3 3e αx α j 2 2x α 2 α x 2 α j α 2 6αjeα x x 1 α α j α 3 6αjeαx α e αj x α j α 4,,

6 454 χx; α ; α j = α je αx 2 2x α 2 α x 2 4αje α x x 1 α 2α e α j x x 1 α j α 2 α j α 3 6αjeα x α e α j x α j α 4, ϕx; α ; α j = αjeα x x 1 α α e α j x x 1 α j α j α 2 2αjeαx α e αj x α j α 3. Proof. In order to nd the optmal allocaton soluton d = d 1, d 2, we utlze the method of Lagrange multpler. Assume ld 1, d 2 = E [ X 1 d 1 2 X 2 d 2 2 S > VaR κs ] and Snce βvar X 1 d 1 2 X 2 d 2 2 S > VaR κs, Ld 1, d 2, γ = ld 1, d 2 γ d d 1 d 2. Var X 1 d 1 2 X 2 d 2 2 S > VaR κs = E [ X 1 2 X 2 2 2d 2X 2 2 S > VaR κs ] E X2 1 E X2 2 2d 2E X 2 2 4d 2 1E X2 1 E X 1 2 4d 1[2d 2E X 1X 2 E X 1 E X 2 E X3 1 E X 1 E X2 1 E X 1X 2 2 E X 1 E X2 2 ], we get Ld 1, d 2, γ d 1 = 2E X 1 2d 18βd 1E X 2 1 E X 1 2 4β[2d 2E X 1X 2 E X 1 E X 2 E X3 1 E X 1 E X2 1 E X 1X 2 2 E X 1 E X2 2 ] γ =. 3.3 Smlarly, Ld 1, d 2, γ d 2 = 2E X 2 2d 28βd 2E X 2 2 E X 2 2 4β[2d 1E X 1X 2 E X 1 E X 2 E X3 2 E X 2 E X2 2 E X 2X 2 1 E X 2 E X2 1 ] γ =, 3.4 and Ld 1, d 2, γ = d 1 d 2 d =. 3.5 γ From Eqs , the results of 3.1 and 3.2 follow mmedately. Now, we need to nd the formulae of E Xk, E X 2 and E X 1X 2. Frstly, the explct expresson of E X3 1 s calculated as follow. E X3 1 = E [ X 3 1 S > VaR κs ] = VaR κs E [ X 3 1 S = s ] f S S>VaRκSsds VaR = E [ κs X1 3 S = s ] f Ssds = fss VaR κs x3 f X1 Sx S = sdxds PrS > VaR κs 1 F SVaR κs VaR = κs x3 f x1,sx, sdxds κ Note that S = X 1 X 2. From Eq.2.1, we get x 3 f x1,sx, sdx = x 3 f x1,x 2 x, sxdx 6e λ 2 s e λ1s = 1 θλ 1λ 2 6seλ1s λ 2 λ 1 4 λ 2 λ 3s2 e λ1s 1 3 λ 2 λ s3 e λ1s 1 2 λ 2 λ 1 α j

7 6e λ 2 s e 2λ1s θ2λ 1λ 2 6se2λ1s λ 2 2λ 1 4 λ 2 2λ 3s2 e 2λ1s 1 3 λ 2 2λ s3 e 2λ1s 1 2 λ 2 2λ 1 θλ 12λ 2 6e 2λ 2 s e λ 1s 2λ 2 λ 1 4 6seλ1s 2λ 2 λ 1 3 3s2 e λ1s 2λ 2 λ 1 2 s3 e λ1s 2λ 2 λ 1 6e 2λ 2 s e 2λ1s θ2λ 12λ 2 6se2λ1s 2λ 2 2λ 1 4 2λ 2 2λ 3s2 e 2λ1s 1 3 2λ 2 2λ s3 e 2λ1s 1 2 2λ 2 2λ 1 Dene 6e λ 2 s e λ1s λ 1λ 2 6seλ1s λ 2 λ 1 4 λ 2 λ 3s2 e λ1s 1 3 λ 2 λ s3 e λ1s ds 1 2 λ 2 λ 1 VaR κs e λ 1VaR κs λ 2 6 6VaRκS 3VaRκS2 λ = 3 1 λ 2 λ 1 1 VaR κs 3 λ 2 λ 1 3e λ 1VaR κs λ 2 2 2VaRκS λ 2 λ 1 1 VaR κs 2 λ 2 λ λ2eλ 1VaR κs VaR κs 1 λ 1 6λ2eλ1VaRκS λ 1e λ2varκs λ 2 λ 1 3 λ 2 λ 1 4 = ξ 3VaR κs; λ 1; λ Pluggng Eqs.3.7 and 3.8 nto Eq.3.6, we can derve the formula of E X3 1 = 1 1 θξ3varκs; λ1; λ2 θξ3varκs; 2λ1; λ2 1 κ θξ 3VaR κs; λ 1; 2λ 2 θξ 3VaR κs; 2λ 1; 2λ 2 The formula of E X3 2 s symmetrcally gven by E X3 2 = 1 1 θξ3varκs; λ2; λ1 θξ3varκs; 2λ2; λ1 1 κ θξ 3VaR κs; λ 2; 2λ 1 θξ 3VaR κs; 2λ 1; 2λ 2. Snce the formulae of E X and E X2 can be calculated smlarly, we omt the proof. Now, we gve the formula of E X2 1 X 2. E X2 1 X 2 = E [ X 2 1 X 2 S > VaR κs ] = = VaR κs VaR κs E [ X 2 1 s X 1 S = s ] f S S>VaRκSsds VaR = E [ κs X1 2 s X 1 S = s ] f Ssds = PrS > VaR κs Note that x 2 s xf x1,sx, sdx = s By the smlar calculatons, we can derve E [ X 2 1 X 2 S = s ] f S S>VaRκSsds VaR κs x 2 f x1,x 2 x, s xdx x2 s xf x1,sx, sdxds. 1 κ x 3 f x1,x 2 x, s xdx. E X2 1 X 2 = 1 1 θχvarκs; λ1; λ2 θχvarκs; 2λ1; λ2 1 κ θχvar κs; λ 1; 2λ 2 θχvar κs; 2λ 1; 2λ 2. The formula of E X2 2 X 1 s also symmetrcally gven by E X2 2 X 1 = 1 1 κ 1 θχvarκs; λ2; λ1 θχvarκs; 2λ2; λ1 θχvar κs; λ 2; 2λ 1 θχvar κs; 2λ 2; 2λ 2.

8 456 Snce the formula of E X 1X 2 can be calculated smlarly, we also omt the proof. Ths completes the proof of ths theorem. Remark 3.1. In Theorem 3.1, we assume that λ 1 λ 2, λ 1 2λ 2, and λ 2 2λ 1. In fact, one can obtans the adjusted results wthout these restrctons. Snce the rst part on the rght-hand sde of equaton 3.6 can be rewrtten as 1 θλ 1λ 2e λ 1s 6e λ 1 λ 2 s 1 6s λ 2 λ 1 4 λ 2 λ 3s λ 2 λ s3, 1 2 λ 2 λ 1 where e λ 1λ 2 s 1 could be expanded n Talyor seres f λ 2 λ 1, e λ 1λ 2 s 1 = λ 1 λ 2s λ1 λ22 s 2 λ1 λ23 s 3 λ 1 λ 2 n s n, 2 6 n! then the lmt of the rst part on the rght-hand sde of equaton 3.6 s lm 1 θλ 1λ 2e λ 1s 6e λ 1 λ 2 s 1 6s λ 2 λ 1 λ 2 λ 1 4 λ 2 λ 3s λ 2 λ s3 1 2 λ 2 λ 1 = lm 61 θλ 1λ 2e λ 1s λ 1 λ 2 n4 s n = 1 θλ2 1 e λ1s s λ 2 λ 1 n! 4 n=4 For the case λ 1 = λ 2, the rst component of f X1,X 2 x 1, x 2 can be rewrtten as 1 θλ 2 1e λ 1x 1 x 2, the rst part on the rght-hand sde of equaton 3.6 can be smpled as x 3 1 θλ 2 1e λ1xsx dx = 1 θλ2 1 e λ1s s By the results of 3.9 and 3.1, for the case λ 1 = λ 2, we can apply the smlar technque to obtan the adjusted results. For the cases λ 1 = 2λ 2 and λ 2 = 2λ 1, one obtans the smlar results. 4. Calculaton of the optmal captal allocaton for n rsks Explct formulae for the optmal captal allocaton for rsks lnked by a FGM copula cannot only be derved n the bvarate case but even for an undened number of rsks. In order to derve the explct formulae for the optmal captal allocaton, we rst gve the followng lemma. Lemma 4.1. Let X = X 1, X 2,, X n. Then the optmal allocaton soluton d = d 1, d 2,, d n to the TMV model, { [ n ] n } mn d 1,,d n E X d 2 S > VaR κs βvar X d 2 S > VaR κs, =1 such that =1 n d = d, s presented as follows: d = A 1 z, 4.1 where A 1 = a j n n s the nverse matrx of A = 8βΣ S 2I n, Σ S s the condtonal covarance matrx of X S > VaR κs, and z = Λ δ 1, Λ δ 2,, Λ δ n wth δ = 4β n j=1 CovX2 j, X S > VaR κs 2E[X S > VaR κs], = 1, 2,, n, where Λ = d n n =1 j=1 ajδj/ n n aj. =1 j=1 =1 n=4

9 457 Proof. The dea of proof s also based on the method of Lagrange multpler. One can refer to Xu and Mao [17] for the full proof. Note that Lemma 4.1 provdes general formulae for optmal allocaton based on the TMV model. There are some quanttes, such as E[X S > Var κs], CovXj 2, X S > VaR κs, and Σ S, whch are non-trval to calculate. In order to nd E[X S > Var κs], CovXj 2, X S > VaR κs, and Σ S, for the sake of dervng the explct formulae for the optmal captal allocaton, we need to derve the key quanttes E Xk = E[X k S > VaR κs], E X2 = E[X 2 S > VaR κs], and E X = E[X S > VaR κs], k = 1, 2, 3,, j = 1, 2,, n, j. As n the bvarate case, the explct formulae for the optmal captal allocaton can also be explctly gven. In order to nd the key quanttes, we rst ntroduce the nth order dvded derence of the functon f presented n Chragev and Landsman [6]. Assume that x 1, x 2,, x n, x n1 are arbtrary ponts and x x j for j. The nth order dvded derence of the functon f on x 1, x 2,, x n, x n1 s dened as fx 1, x 2,, x n, x n1 = n1 =1 j fx. x xj In the follow proposton, we provde the explct formulae of E Xk, E X 2 and E X for n exponentally dstrbuted rsks lnked by a multvarate FGM copula. Let X = X 1, X 2,, X n, X 1, X 2,, X n are n exponentally dstrbuted wth λ 1, λ 2,, λ n. As n the bvarate case, we assume that λ λ j and λ 2λ j for j. Proposton 4.2. Let X = X 1, X 2,, X n, X 1, X 2,,X n are n exponentally dstrbuted wth parameters λ 1, λ 2,, λ n. The dependence structure for X 1, X 2,, X n s dened by a multvarate FGM copula. Then quanttes E Xk, E X 2 and E X, j,, j = 1, 2,, n, k = 1, 2, 3, are presented as follows: E Xv = 1n2 η n 1 κ { L X v VaR κs; λ 1; ; λ n q [1 l 2 q2l l= ɛ 1,,ɛ qj τ l,q2 L X v VaR κs; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n 1 p 1 < <p q n j {p 1,,p q} 1 l1 2 q2l MX v VaR κs; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n 1 l1 2 q2l NX v VaR κs; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n 1 l 2 q2l QX v VaR κs; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n ] n q 1 p 1 < < p q n {p 1,, p q} 1 p 1 < < p q n j / {p 1,, p q} l= ɛ 1,,ɛ q τ l,q1 [1 l1 2 q1l LX v VaR κs; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n 1 l 2 q1l MX v VaR κs; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n ] n q 1 p 1 < < p q n / {p 1,, p q} 1 p 1 < < p q n j {p 1,, p q} l= ɛ 1,,ɛ qj τ l,q1 [1 l1 2 q1l LX v VaR κs; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n

10 458 E Xk q1 [ l= 1 l 2 q1l NX v VaR κs; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n ] n q 1 l 2 ql l= ɛ 1,,ɛ q τ l,q 1 p 1 < < p q n / {p 1,, p q} 1 p 1 < < p q n j / {p 1,, p q} L X v VaR κs; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n }, v = 1, 2, j, 4.2 = 1n1 η n 1 κ { L X kvar κs; λ 1; ; λ n 1 p 1 < <p q n {p 1,,p q} ɛ 1,,ɛ q τ l,q1 {1 l1 2 q1l LX k VaR κs; 2 ɛ1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n 1 l 2 q1l MX kvar κs; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n }] n q [ 1 l 2 ql l= ɛ 1,,ɛ q τ l,q 1 p 1 < <p q n / {p 1,,p q} L X k VaR κs; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n ]}, k = 1, 2, 3, 4.3 where η = λ 1 λ n, q1,, n are the mssng ndexes of p 1,, p q to complete 1,, n, τ l,q are the sets of q-tuples composed of l zeros and q l ones, for l =, 1,, q and q = 2,, n, LX kx; α 1; ; α 1; α 1; ; α n and M X kx; α 1; ; α 1; α 1; ; α n are the n 2th order dvded derences of L X kx; α and M X kx; α, respectvely, and L X v x; α 1; ; α 1; α 1; ; α j1; α j1; ; α n, MX v x; α 1; ; α 1; α 1; ; α j1; α j1; ; α n, N X v x; α 1; ; α 1; α 1; ; α j1; α j1; ; α n, and Q X v x; α 1; ; α 1; α 1; ; α j1; α j1; ; α n are the n3th order dvded dfferences of L X v x; α, MX v x; α, NX v x; α and Q X v x; α, respectvely. The expressons of L X kx; α and M X kx; α are gven n Eqs and the expressons of L X v x; α, MX v x; α, NX v x; α and Q X v x; α are gven n Eqs Proof. We rst calculate the explct the formulae of E X2, j,, j = 1, 2,, n. E X2 = E [ X 2 S > VaR κs ] = VaR κs x VaR κs E [ X 2 S = s ] f S S>VaRκSsds x 2 x jf X,,Sx, x j, sdx jdx ds = κ In order to evaluate Eq.4.4, we need to derve a recursve formula for f X,,Sx, x j, s = f X,,SX x, x j, s x x j. Gven that the random varables X 1, X 1,, X n are not ndependent, we cannot separate f X,,SX x, x j, s x x j nto the product of f X x, f Xj x j and f SX s x x j. Now, we derve the recursve formula for f X,,SX x, x j, s x x j. Frst, we rewrte the f X1,...,X n x 1, x 2,, x n dened n Eq.2.2 as n f X1,...,X n x 1,..., x n = f X x f Xj x j{ω jx 1; ; x n; λ 1; ; λ n 1 p 1 < < p q n j {p 1,, p q} 1 2F X x 1 2F Xj x j[ q l= ɛ 1,,ɛ qj τ l,q2 1 l ω jx p1 ; ; x pq ; x q1 ; ; x n ; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n ]

11 n 1 p 1 < < p q n {p 1,, p q} 1 p 1 < < p q n j / {p 1,, p q} 1 2F X x [ q 459 l= ɛ 1,,ɛ q τ l,q1 1 l ω jx p1 ; ; x pq ; x q1 ; ; x n ; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n ] n q 1 2F Xj x j[ 1 p 1 < < p q n / {p 1,, p q} 1 p 1 < < p q n j {p 1,, p q} l= ɛ 1,,ɛ qj τ l,q1 1 l ω jx p1 ; ; x pq ; x q1 ; ; x n ; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n ] n q [ 1 p 1 < < p q n / {p 1,, p q} 1 p 1 < < p q n j / {p 1,, p q} l= ɛ 1,,ɛ q τ l,q 1 l ω jx p1 ; ; x pq ; x q1 ; ; x n ; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n ]}, 4.5 where ω jx 1; x 2; ; x n; α 1; α 2; ; α n = α 1e α 1x 1 α 1e α 1x 1 α 1 e α 1x 1 α j1e α j1x j1 α j1e α j1x j1 α ne αnxn. Assume that x x j x1 x x j = n l=1,l,l j Note that q=1,q l,q,q j x1 x n1 ω jx 1;... ; x n; α 1;... ; α ndx 1 dx 1 dx 1 dx j1dx j1 dx n1 α q f X,,SX x, x j, s x x j α l e α lsx x j = h js x x j; α 1;... ; α n. α q α l = f X,,X 1 X 1 X X n x, x j, s x x j = x x j x1 x x j x1 x n1 4.6 f X1,...,X n x 1,..., x n1, s x 1 x n1 dx 1 dx 1dx 1 dx j1dx j1 dx n Pluggng 4.5 nto 4.7, we have f X,,SX x, x j, s x x j { = f X x f Xj x j h js x x j; λ 1; ; λ 1; λ 1; ; λ j1; ; λ j1; ; λ n n 1 p 1 < <p q n j {p 1,,p q} l= n 1 2F X x 1 2F Xj x j q 1 l h js x x j; 2 ɛ1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n ɛ 1,,ɛ qj τ l,q2 l= 1 p 1 < < p q n {p 1,, p q} 1 p 1 < < p q n j / {p 1,, p q} 12F X x q 1 l h js x x j; 2 ɛ1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n ɛ 1,,ɛ q τ l,q1

12 46 n 1 p 1 < < p q n / {p 1,, p q} 1 p 1 < < p q n j {p 1,, p q} 12F Xj x j q 1 l h js x x j; 2 ɛ1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n l= ɛ 1,,ɛ qj τ l,q1 n l= 1 p 1 < < p q n / {p 1,, p q} 1 p 1 < < p q n j / {p 1,, p q} q 1 l h js x x j; 2 ɛ1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n. 4.8 ɛ 1,,ɛ q τ l,q Utlzng the dvdend derence presented n Chragev and Landsman [6], note that h jx; α 1; ; α 1; α 1; ; α j1; α j1; ; α n = 1 n3 α 1 α 1 α 1 α j1 α j1 α n Λx; α 1; ; α 1; α 1; ; α j1; α j1; ; α n, where Λx; α 1; ; α 1; α 1; ; α j1; α j1; ; α n s the n 3th order dvded dfference of Λx; α = e αx. Then, we get x = x x 2 x jf X x f Xj x jh js x x j; λ 1; ; λ 1; λ 1; ; λ j1; ; λ j1; ; λ ndx jdx x 2 x jf X x f Xj x j1 n3 η j Λx; λ1; ; λ 1; λ 1; ; λ j1; λ j1; ; λ ndx jdx = 1 n2 ηl X 2 s; λ 1; ; λ 1; λ 1; ; λ j1; λ j1; ; λ n, 4.9 where η j = λ 1 λ 1 λ 1 λ j1 λ j1 λ n, L X 2 s; λ = x x 2 x je λ x e λ j x j Λs x x j; λdx jdx. Let also M X 2 s; λ = x 2x 2 x je 2λ x e λ j x j Λsxx j; λdx jdx, N X 2 s; λ = x 2x 2 x je λ x e 2λ j x j Λs x x j; λdx jdx, and O X 2 s; λ = x j; λdx jdx, j,, j = 1, 2,, n. x 4x 2 x je 2λ x e 2λ j x j Λs x Gven that the rsks are exponentally dstrbuted, then L X 2, M X 2, N X 2 and can be rewrtten as O X 2 L X 2 s; λ = 2e λs λ λ 3 λ j λ 2λ 4λj 3λeλj s 2s3λ λj 2λeλs 2 λ λ j 4 λ j λ 2 λ λ j 3 λ λ 2 2se λ j s λ λ j 3 λ j λ s 2 e λs λ λ j 2 λ λ 26λ2 λ 2 j 2λλ j 3λ 2 4λ λ j 2λe λs, λ λ j 4 λ λ e λs M X 2 s; λ = 2 2λ λ 3 λ j λ 22λ 4λj 3λeλj s 2se λ j s 2 2λ λ j 4 λ j λ 2 2λ λ j 3 λ j λ s 2 e 2λ s 2λ λ j 2 2λ λ 2s32λ λj 2λe2λs 2λ λ j 3 2λ λ 2 262λ2 λ 2 j 2λλ j 3λ 2 42λ λ j 2λe 2λ s 2λ λ j 4 2λ λ 3, e λs N X 2 s; λ = 2 λ λ 3 2λ j λ 2λ 42λj 3λe2λj s 2 λ 2λ j 4 2λ j λ 2

13 2se 2λ j s λ 2λ j 3 2λ j λ s 2 e λs λ 2λ j 2 λ λ 2s3λ 2λj 2λeλs λ 2λ j 3 λ λ 2 26λ2 2λ j 2 2λ2λ j 3λ 2 4λ 2λ j 2λe λ s λ 2λ j 4 λ λ 3, e λs O X 2 s; λ = 4 2λ λ 3 2λ j λ 22λ 42λj 3λe2λj s 2 2λ 2λ j 4 2λ j λ 2 2se 2λ j s 2λ 2λ j 3 2λ j λ s 2 e 2λs 2λ 2λ j 2 2λ λ 2s32λ 2λj 2λe2λs 2λ 2λ j 3 2λ λ 2 262λ2 2λ j 2 2λ2λ j 3λ 2 42λ 2λ j 2λe 2λ s λ 2λ j 4 2λ λ 3 Utlzng the results of 4.8 and , one can rewrte x x 2 x jf x,x j,sdx jdx as x x 2 x jf x,x j,sdx jdx = 1 n2 η{l X 2 s; λ 1; ; λ n n 1 p 1 < < p q n j {p 1,, p q} q l= [L X 2 s; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n M X 2 s; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n N X 2 s; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n 1 p 1 < < p q n {p 1,, p q} 1 p 1 < < p q n j / {p 1,, p q} Q X 2 s; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n ] n q [L X 2 s; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n 1 p 1 < < p q n / {p 1,, p q} 1 p 1 < < p q n j {p 1,, p q} ɛ 1,,ɛ qj τ l,q2 1 l 2 q2l l= M X 2 s; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n ] n q [L X 2 s; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n N X 2 s; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n ] n q l= 1 p 1 < < p q n / {p 1,, p q} 1 p 1 < < p q n j / {p 1,, p q} l= ɛ 1,,ɛ q τ l,q1 1 l 2 q1l ɛ 1,,ɛ qj τ l,q1 1 l 2 q1l ɛ 1,,ɛ q τ l,q 1 l 2 ql L X 2 s; 2 ɛ1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; ; λ n. In order to calculate the explct the formulae of E X2, j, as n 4.4, the L X 2 s; λ, M X 2 s; λ, N X 2 s; λ and O X 2 s; λ terms n the equaton above must be ntegrated on s as follows. L X 2 V ; λ = V L X 2 s; λds = 2λ 4λj 3λeλ j V λ jλ λ j 4 λ j λ 2 21 V λjeλj V λ 2 j λ λj3 λ j λ 461

14 462 2 λv 2 λv eλ V λ 3 λ λj2 λ λ 21 V λ3λ λj 2λeλ V λ 2 λ λj3 λ λ 2 26λ2 λ 2 j 2λλ j 3λ 2 4λ λ j 2λe λ V λ λ λ j 4 λ λ 3 M X 2 V ; λ = 2 22λ 4λj 3λeλ j V λ j2λ λ j 4 λ j λ 2 21 V λjeλj V λ 2 j 2λ λj3 λ j λ 2 2λV 2 2λV e2λ V 2λ 3 2λ λ j 2 2λ λ 2e λv λλ λ 3 λ j λ 2, V 2λ32λ λj 2λe2λ V 2λ 2 2λ λ j 3 2λ λ 2 262λ2 λ 2 j 2λλ j 3λ 2 42λ λ j 2λe 2λ V 2λ 2λ λ j 4 2λ λ 3 2e λv λ2λ λ 3 λ j λ 2, N X 2 V ; λ = 2 2λ 42λj 3λe2λ j V 2λ jλ 2λ j 4 2λ j λ 2 21 V 2λ je 2λj V 2λ j 2 λ 2λ j 3 2λ j λ 2 λv 2 λv eλ V λ 3 λ 2λj2 λ λ 21 V λ3λ 2λj 2λeλ V λ 2 λ 2λj3 λ λ 2 26λ2 2λ j 2 2λ2λ j 3λ 2 4λ 2λ j 2λe λ V λ λ 2λ j 4 λ λ 3 2e λv λλ λ 3 2λ j λ 2, Ō X 2 s; λ = 4 22λ 42λj 3λe2λ j V 2λ j2λ 2λ j 4 2λ j λ 2 21 V 2λ je 2λj V 2λ j 2 2λ 2λ j 3 2λ j λ 2 2λV 2 2λV e2λ V 2λ 3 2λ 2λ j 2 2λ λ 21 V 2λ32λ 2λj 2λe2λV 2λ 2 2λ 2λ j 3 2λ λ λ2 2λ j 2 2λ2λ j 3λ 2 42λ 2λ j 2λe 2λ V 2e λv 2λ 2λ 2λ j 4 2λ λ 3 λ2λ λ 3 2λ j λ Fnally, we can derve the explct the formulae for E X2, j,, j = 1, 2,, n. Snce the formula of E X can be calculated smlarly, we omt the proof. We gve the expressons of L X V ; λ, MX V ; λ, NX V ; λ and ŌX V ; λ drectly, e L λv X V ; λ = λλ λ 2 λ j λ λ 3λj 2λeλj V 2 λ jλ λ j 3 λ j λ 1 V λjeλj V 2 λ 2 j λ λj2 λ j λ 21 V λeλ V λ 2 λ λj2 λ λ 3λ λj 2λeλV λ λ λ j 3 λ λ 3 e M λv X V ; λ = 2 λ2λ λ 2 λ j λ 2λ 3λj 2λeλj V 2 λ j2λ λ j 3 λ j λ 2, V λjeλ j V λ 2 j 2λ λj2 λ j λ 21 V 2λ e 2λV 6λ λj 2λe2λV 2λ 2 2λ λ j 2 2λ λ 2λ 2λ λ j 3 2λ λ, e N λv X V ; λ = 2 λλ λ 2 2λ j λ λ 6λj 2λe2λj V 2 2λ jλ 2λ j 3 2λ j λ 2 1 V 2λ je 2λ j V 2λ j 2 λ 2λ j 2 2λ j λ 21 V λeλv λ 2 λ 2λj2 λ λ 3λ λ j λe 2λ V Ō X V ; λ = 4 λ 2λ 2λ j 3 2λ λ λ 3λj λe2λj V 3 λ j2λ 2λ j 3 2λ j λ 2 3λ 2λj 2λeλV, 4.2 λ λ 2λ j 3 λ λ 3 1 V 2λ je 2λ j V 2λ j 2 2λ 2λ j 2 2λ j λ 21 V 2λ e 2λV 2λ 2 2λ 2λ j 2 2λ λ e λv λ2λ λ 2 2λ j λ

15 463 Now, we gve the formulae of E X3, = 1, 2,, n. E X3 = E [ X 3 S > VaR ] VaR κs = κs x3 f X,Sx, sdx ds κ Now, we rewrte the f X1,X 2,,X n x 1, x 2,, x n dened n Eq.2.2 as { f X1,X 2,,X n x 1, x 2,, x n = f X x ω x 1; x 2; ; x n; λ 1; λ 2; ; λ n n 1 p 1 < <p q n {p 1,,p q} 1 2F X x [ q l= ɛ 1,,ɛ q τ l,q1 1 l ω x p1 ; x p2 ; ; x pq ; x q1 ; ; x n ; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n ] n q [ 1 p 1 < <p q n / {p 1,,p q} l= ɛ 1,,ɛ q τ l,q 1 l ω x p1 ; x p2 ; ; x pq ; x q1 ; ; x n ; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n ], 4.23 where ω x 1; x 2; ; x n; α 1; α 2; ; α n = α 1e α 1x 1 α 1e α 1x 1 α 1 e α 1x 1 α ne αnxn. Assume that x x1 x x1 x n1 ω x 1;... ; x n; α 1;... ; α ndx 1 dx 1 dx 1 dx n1 n = α q α l e α lsx = h s x ; α 1; α 2; ; α n α q α l l=1,l, q=1,q l,q Utlzng the dvdend derence presented n Chragev and Landsman [6], note that h x; α 1; ; α 1; α 1; ; α n = 1 n2 α 1 α 1 α 1 α n Λx; α 1; ; α 1; α 1; ; α n, where Λx; α 1; ; α 1; α 1; ; α n s the n2th order dvded derence of Λx; α. Then, we get x 3 f X x h s x ; λ 1; ; λ 1; λ 1; ; λ ndx = 1 n1 ηl X 3 s; λ 1; ; λ 1; λ 1; ; λ j1; λ j1; ; λ n, where η = λ 1 λ 1 λ 1 λ n, L X 3s; λ = x3 e λ x Λsx; λdx. Let also M X 3s; λ = 2x3 e 2λ x Λs x; λdx. Gven that the rsks are exponentally dstrbuted, then L X 3 and M X 3 can be rewrtten as L X 3s; λ = s3 e sλ λ λ 3s2 e sλ λ λ 6sesλ 2 λ λ 6esλ 3 λ λ 6esλ 4 λ λ, s 3 e 2sλ M X 3s; λ = 2 2λ λ 3s2 e sλ λ λ 6se2sλ 2 2λ λ 6es2λ 3 2λ λ 6esλ. 4 2λ λ Then, utlzng the results of 4.23, 4.25 and 4.26, we can express x3 f X,Sx, sdx as s n x 3 f X,Sx, sdx = 1 n1 η{l X 3s; λ 1; λ 2; ; λ n 1 p 1 < <p q n {p 1,,p q}

16 464 q1 [ {1 l1 2 q1l L X 3s; 2 ɛ1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n l= ɛ 1,,ɛ q τ l,q1 n 1 l 2 q1l M X 3s; 2 ɛ 1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n }] [ q l= 1 p 1 < <p q n / {p 1,,p q} ɛ 1,,ɛ q τ l,q 1 l 2 ql L X 3 s; 2 ɛ1 λ p1 ; ; 2 ɛq λ pq ; λ q1 ; λ n ]}. In order to calculate the explct the formulae of E X3, = 1, 2,, n, as n 4.22, the L X 3s; λ and M X 3s; λ terms n the equaton above must be ntegrated on s as follows. L X 3 V ; λ = 6eV λ V L X 3 s; λds = eλ V 6 V λ 6 V λ 3 V λ λ 4 λ λ 6eV λ λ λ λ 4 λλ λ 3eV λ 2 V λ 2 V λ 4 λ λ2 M X 3 V ; λ = V λ 3 6eV λ 1 V λ e V λ λ 2 λ λ M X 3 s; λds = e2λ V 3 V λ 6 V 2λ 3 V 2λ 2 2 λ 4 2λ λ 12eV λ λ2λ λ 4 3eV 2λ 1 V λ 2 V 2λ 2λ 3 2λ λ 2 3eV 2λ 1 V 2λ e V 2λ λ 2 2λ λ 3 6eV 2λ λ 2λ λ Fnally, we can derve the explct formulae 4.3 for E X3, = 1, 2,, n. Snce the formulae of E X2 and E X can be calculated smlarly, we also omt the proof. We gve the expressons of L X 2V ; λ, MX 2V ; λ, L X V ; λ and M X V ; λ drectly, L X 2V ; λ = ev λ 2 V λ 2 V λ λ 3 λ λ 2eV λ 1 V λ e V λ λ 2 λ λ2 2eV λ λ λ λ 2eV λ 3 λλ λ, M X 2 V ; λ = ev 2λ 1 V λ 2 V 2λ 2 2 λ 3 2λ λ ev 2λ 1 V 2λ e V 2λ 2λ 2 2λ λ 2 ev 2λ λ 2λ λ 2eV λ 3 λ2λ λ, L X V ; λ = ev λ 1 V λ e V λ λ 2 λ λ ev λ λ λ λ ev λ 2 λλ λ, M X V ; λ = ev 2λ 1 V 2λ e V 2λ 2λ 2 2λ λ Ths completes the proof of ths proposton. 5. Numercal llustratons 5.1 Bvarate case e V 2λ 2λ 2λ λ 2 e V λ λ2λ λ We llustrate our results wth an example for the bvarate case. As an example, assume that rsk varables X 1 and X 2 are exponentally dstrbuted wth parameters λ 1 = 2/5 and λ 2 = 3/4. We calculate the optmal captal allocatons based on Tal Mean-Varance prncple for derent rsk levels κ, FGM copula parameters θ and derent β. The values of d = 1, 2 wth a total captal d = 4 for θ = 1,, 1, β =.1,.3,.5,.7,.9, and κ =.5,.75,.95,.99,.995 are lsted n Table 1-3. We observe that for

17 465 xed κ and θ, d 1 s a decreasng functon wth respect to β, but d 2 s an ncreasng functon wth respect to β. We also nd that for xed β and θ, d 1 s an ncreasng functon wth respect to κ, but d 2 s a decreasng functon wth respect to κ. Moreover, the larger the value of θ, the greater changes n d 1 and d 2 for derent κ and β. 5.2 Multvarate case For multvarate case, we llustrate our results wth an example for three rsk varables dependent through a trvarate FGM copula. Assume that rsk varables X 1, X 2 and X 3 are exponentally dstrbuted wth parameters λ 1 = 1/2, λ 2 = 1/3 and λ 3 = 1/5. The trvarate FGM copula wth 4 parameters can be wrtten as Cu 1, u 2, u 3 = u 1u 2u 31 θ 123ū 1ū 2ū 3 θ 12ū 1ū 2 θ 13ū 1ū 3 θ 23ū 2ū 3, where ū = 1 u, = 1, 2, 3. We calculate the optmal captal allocatons the amount d to X, = 1, 2, 3 based on Tal Mean-Varance prncple for derent rsk levels κ, FGM copula parameters and derent β. The values of d = 1, 2, 3 wth a total captal d = 12 for β =.1,.3,.5,.7,.9, κ =.5,.75,.95,.99,.995, θ 123 =.25, θ 12 =.2,.2, θ 13 =.5, and θ 23 =.6,.6 are lsted n Table 4-5. We observe that for xed κ, d 2 and d 3 are decreasng functons wth respect to β, but d 1 s an ncreasng functon wth respect to β n both cases. We also nd that for xed β, d 3 s an ncreasng functon wth respect to κ, but d 1 and d 2 s decreasng functon wth respect to κ. Moreover, the changes n d 1, d 2 and d 2 for derent κ and β n case θ 123 =.25, θ 12 =.2, θ 13 =.5 and θ 23 =.6 are greater than n case θ 123 =.25, θ 12 =.2, θ 13 =.5 and θ 23 = Concludng Remarks In ths paper, we ntroduce the use of copulas n optmal captal allocaton based on Tal Mean-Varance prncple. We obtan explct expressons for the optmal captal allocaton for exponental dstrbuted rsks lnked by a FGM copula. The handy form of the FGM copula permts a drect calculaton of the E [ X k S > VaR κs ], E[X 2 S > VaR κs], and E [X S > VaR κs] when we assume only two rsks. In the multvarate case, the dvdend derences presented n Chragev and Landsman 27 are used. Acknowledgements The authors thank the Edtor and two anonymous revewers for ther comments that helped to mprove the presentaton of ths paper. Xe's research was supported by the Natural Scence Foundaton of Chna Grant No , the Socal Scence Foundaton of JangX Provnce Grant No. 16YJ28, and the STRP of JangX Provnce Grant No. GJJ Zou's research was supported by the STRP of JangX Provnce Grant No.GJJ References [1] Bargès, M., Cossette, H. and Marceau, E. TVaR-based captal allocaton wth copulas, Insurance: Mathematcs and Economcs, 45, , 29. [2] Bauer, D. and Zanjan, G. Captal allocaton and ts dscontents, In Handbook of Insurance pp , Sprnger New York, 213. [3] Bauer, D. and Zanjan, G. The margnal cost of rsk, rsk measures, and captal allocaton, Management Scence 62, , 215.

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19 467 Table 1. Optmal captal allocatons for derent β and κ wth a total captal d = 4 and θ = 1. β κ =.5 κ =.75 κ =.95 κ =.99 κ =.995 β =.1 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = β =.3 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = β =.5 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = β =.7 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = β =.9 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = Table 2. Optmal captal allocatons for derent β and κ wth a total captal d = 4 and θ =. β κ =.5 κ =.75 κ =.95 κ =.99 κ =.995 β =.1 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = β =.3 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = β =.5 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = β =.7 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = β =.9 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = Table 3. Optmal captal allocatons for derent β and κ wth a total captal d = 4 and θ = 1. β κ =.5 κ =.75 κ =.95 κ =.99 κ =.995 β =.1 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = β =.3 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = β =.5 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = β =.7 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = β =.9 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 =

20 468 Table 4. Optmal captal allocatons wth d = 12 and θ 123 =.25, θ 12 =.2, θ 13 =.5, θ 23 =.6. β κ =.5 κ =.75 κ =.95 κ =.99 κ =.995 β =.1 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = d 3 = d 3 = d 3 = d 3 = d 3 = β =.3 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = d 3 = d 3 = d 3 = d 3 = d 3 = β =.5 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = d 3 = d 3 = d 3 = d 3 = d 3 = β =.7 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = d 3 = d 3 = d 3 = d 3 = d 3 = β =.9 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = d 3 = d 3 = d 3 = d 3 = d 3 = Table 5. Optmal captal allocatons wth d = 12 and θ 123 =.25, θ 12 =.2, θ 13 =.5, θ 23 =.6. β κ =.5 κ =.75 κ =.95 κ =.99 κ =.995 β =.1 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = d 3 = d 3 = d 3 = d 3 = d 3 = β =.3 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = d 3 = d 3 = d 3 = d 3 = d 3 = β =.5 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = d 3 = d 3 = d 3 = d 3 = d 3 = β =.7 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = d 3 = d 3 = d 3 = d 3 = d 3 = β =.9 d 1 = d 1 = d 1 = d 1 = d 1 = d 2 = d 2 = d 2 = d 2 = d 2 = d 3 = d 3 = d 3 = d 3 = d 3 =