On capital allocation by minimizing multivariate risk indicators

Transcription

1 On capital allocation by minimizing mltivariate risk indicators Véroniqe Mame-Deschamps, Didier Rllière, Khalil Said To cite this version: Véroniqe Mame-Deschamps, Didier Rllière, Khalil Said. On capital allocation by minimizing mltivariate risk indicators. Eropean Actarial Jornal, Springer, 216. <hal > HAL Id: hal Sbmitted on 13 Nov 214 HAL is a mlti-disciplinary open access archive for the deposit and dissemination of scientific research docments, whether they are pblished or not. The docments may come from teaching and research instittions in France or abroad, or from pblic or private research centers. L archive overte plridisciplinaire HAL, est destinée a dépôt et à la diffsion de docments scientifiqes de nivea recherche, pbliés o non, émanant des établissements d enseignement et de recherche français o étrangers, des laboratoires pblics o privés.

2 ON CAPITAL ALLOCATION BY MINIMIZING MULTIVARIATE RISK INDICATORS V. MAUME-DESCHAMPS, D. RULLIÈRE, AND K. SAID Abstract. The isse of capital allocation in a mltivariate context arises from the presence of dependence between the varios risky activities which may generate a diversification effect. Several allocation methods in the literatre are based on a choice of a nivariate risk measre and an allocation principle, others on optimizing a mltivariate rin probability or some mltivariate risk indicators. In this paper, we focs on the latter techniqe. Using an axiomatic approach, we stdy its coherence properties. We give some explicit reslts in mono periodic cases. Finally we analyze the impact of the dependence strctre on the optimal allocation. Introdction The calclation of the reglatory economic capital, which is called the Solvency Capital Reqirement in insrance, is well controlled and its methodology is almost imposed by the spervisory athorities of the sector. Nevertheless, the allocation of this capital may be considered as an internal exercise for each company, and constittes a management choice whose sccess is a key factor for firm performance optimization. It can also be seen as an indicator of its good governance, especially for the mlti branches firms. The literatre on the sbject of the capital allocation methods is very rich. Several principles have been proposed over the last twenty years, the most important and most stdied are the Shapley method, the Amann-Shapley method and the Eler s method. The Shapley method is based on cooperative game theory. It is described in detail in Denalt s paper (21) [11], where it is proved that this method, originally sed to allocate the total cost between players in coalitional games context, can be easily adapted to solve problem of the overall risk allocation between segments. Tasche devoted two papers [24] and [25], to the description of the Eler method, which is also fond in the literatre nder the name of the gradient method. Eler method is based on the idea of allocating capital according to the infinitesimal marginal impact of each risk. This impact corresponds to the increase obtained on the overall risk, yielding an infinitely small increment in the risk i. The Eler method is very present in the literatre, several papers analyze its properties (RORAC compatibility [7], [27], Coherence [6],...) nder different assmptions (Tasche (24) [23], Balog (211) [3]). Its fame is de to the existence of economic argments that can jstify its se to develop allocation rles. Finally, Amann-Shapley method is a continos generalization of Shapley method, its principle is based on the vale introdced by Amann and Shaplay in game theory. Denalt [11] analyzes this method and its application to capital allocation. Date: November 13, Mathematics Sbject Classification. 62H, 62P5, 91B3. Key words and phrases. Mltivariate risk indicators, dependence modeling, coherence properties, risk theory, optimal capital allocation. 1

3 These three capital allocation principles rely on different risk measres, and the coherence of the allocation method depends on the properties of the selected risk measre. Several papers deal with capital allocation coherence based on the properties of the risk measre sed, we qote as examples, Fischer (23) [13], Bsh and Dorfleitner (28) [6], and Kalkbrener (29) [17]. Other techniqes have been proposed more recently for bilding optimal allocation methods, by minimizing some mltivariate rin probabilities, especially those defined by Cai and Li (27) [8], or by minimizing some new mltivariate risk indicators. In this context, Cénac et al [9], [1] defined three types of indicators, which take into accont both the rin severity at the branch level, and the impact of the dependence strctre on this local severity. In the one-period case, these indicators can be considered as special cases of a general indicator family introdced in Dhaene et al (212) [12]. Allocation by minimization of some real mltivariate risk indicators can be sed in a more general framework for modeling systemic risk, and also to determine the priorities of optimal stop-loss treaties that a reinsrer can offer according to the risk portfolios of its insrer cstomers, knowing that he covers his overall portfolio with a Stop-Loss treaty of priority. This allocation techniqe can also help to measre the performance of calclating the grops capital reqirement in the Swiss Solvency Test (SST), which provides a consistent framework both for legal branches and grop solvency capital reqirement. The article is organized as follows. The first section is devoted to some definitions and notations that will be sed throghot the rest of the article. Next, we stdy the coherence properties of the allocation by minimization of some specific indicators, and for a general choice of the penalties fnctions in Section 2. Section 3 is a presentation of some explicit formlas obtained for special model cases, the asymptotic behavior of the optimal allocation is also discssed in this section. In Section 4, we treat the qestion of the impact of the dependence strctre on the composition of the optimal allocation, throgh the analysis of comonotonic cases and sing some models of bivariate dependence with coplas. 1. Optimal allocation In a mltivariate risk framework, we consider a vectorial risk process X p = (X1, p..., X p d ), where X p k corresponds to the losses of the kth bsiness line dring the p th period. We denote by Rp k the p reserve of the k th line at time p, so: Rp k = k Xk, l where k R + is the initial capital of the k th bsiness line, then = d is the initial capital of the grop, and d is the nmber of bsiness lines. Cénac et al (212) [1] defined the two following mltivariate risk indicators, given penalty fnctions g k : the indicator I: n I ( 1,..., d ) = E g k (Rp) k 1 {R k p <} 1 d {, j=1 p=1 Rj p>} the indicator J: n J( 1,..., d ) = E g k (Rp) k 1 {R k p <} 1 d {, j=1 p=1 Rj p<} 2

4 g k : R R + are C 1, convex fnctions with g k () =, g k (x) for x <, k = 1,..., d. They represent the cost that each branch has to pay when it becomes insolvent while the grop is solvent for the I indicator, or while the grop is also insolvent in the case of the J indicator. They proposed to allocate some capital by minimizing these indicators. The idea is to find an allocation vector ( 1,..., d ) that minimizes the indicator sch as = d, where is the initial capital that need to be shared among all branches. The indicator I represents the expected sm of penalty amonts of local rins, knowing that the grop remains solvent. In the case of the indicator J, the local rin severities are taken into accont only in the case of grop insolvency. By sing optimization stochastic algorithms, we may estimate the minimm of these risk indicators. Cénac et al (212) [1] propose a Kiefer-Wolfowitz version of mirror algorithm as a convergent algorithm nder general assmptions to find optimal allocation minimizing the indictor I. This algorithm is effective to solve the optimal allocation problem, especially, for a large nmber of bsiness lines, and for allocation over several periods Definitions and notations. In this paper, we focs on the case of allocations on a single period (n = 1), since new reglation rles, sch as Solvency 2, reqires a jstified allocation over a period of one year, and also to make a first comptational approach, knowing that an annal allocation will be a more efficient choice for an insrer, that will allow it to integrate the changes that occrred at its risk portfolio and its dependence strctre dring a year of operation. The following notations are sed: The risk X k corresponds to the losses of the k th branch dring one period. It is a positive random variable in or context. is the initial capital of the firm. U d = {v = (v 1,..., v d ) [, ] d, d i=1 v i = } is the set of possible allocations. We define an allocation as an application R + (R + ) d. For all i {1,..., d} let α i = i, then, d i=1 α i = 1. 1 d = {α = (α 1,..., α d ) [, 1] d, d i=1 α i = 1} is the set of possible allocation percentages α i = i /. For ( 1,..., d ) U, d we define the reserve of the k th bsiness line at the end of the period is: R k = k X k, where k represents the part of capital allocated to the k th branch. The aggregate sm of risks is: S = d i=1 X i, and let S i = d j=1;j i X j for all i {1,..., d}. F Z is the distribtion fnction of a random variable Z, FZ is its srvival fnction and f Z its density fnction. Definition 1.1 (Optimal allocation:). Let X be a positive random vector of R d, R + and K X : U d R + a mltivariate risk indicator associated to X and. An optimal allocation is defined by: ( 1,..., d ) arg inf (v 1,...,v d ) U d {K X (v 1,..., v d )}. For risk indicators of the form K X (v) = E[S(X, v)], for a scoring fnction S : R +d R +d R +, this definition can be seen as an extension in a mltivariate framework of the elicitability concept. Elicitability has been introdced by Gneiting (211)[15], and stdied recently for nivariate risk measres, by Bellini and Bignozzi (213)[4], Ziegel (214)[28] and Steinwart et al (214)[22], for examples. 3

5 We denote by A X1,...,X d () = ( 1,..., d ), the optimal allocation of the amont on the d risky branches in U d. Assmptions: Throghot this paper, we will se the following assmptions: H1: The risk indicator admits an niqe minimm in U d. H2: The fnctions g k are differentiable and sch that for all k {1,..., d}, g k( k X k ) admits a moment of order one, and (X k, S) has a joint density distribtion denoted by f (Xk,S). H3: The d risks have the same penalty fnction g k = g, k {1,..., d}. The first assmption is verified when the indicator is strictly convex, this is particlarly tre when for at least one k {1,..., d}, g k is strictly convex; and the joint density f (Xk,S) spport contains [, ] 2 (see [1]) Coherence properties. In his article [11], Denalt introdced the notion of a coherent allocation, fixing for axioms that mst be verified by a principle of capital allocation, driven by coherent nivariate risk measres, according to the criteria defined by Artzner et al (1999) [1], in order to be qalified as coherent. Or optimal capital allocation is not directly derived from a nivariate risk measre, even if it is obtained by minimizing a mltivariate risk indicator, we reformlate coherence axioms in or context Coherence. We follow Denalt s idea to define a coherent capital allocation. Definition 1.2 (Coherence). A capital allocation ( 1,..., d ) R +d of an initial capital R + is coherent if it satisfies the following properties: 1. Fll allocation: All of the capital R + mst be allocated between the branches: i =. i=1 2. Symmetry: If the joint distribtion of the vector (X 1,..., X d ) is nchanged by permtation of the risks X i and X j, then the allocation remains also nchanged by this permtation, and the i th and j th bsiness lines both make the same contribtion to the risk capital: (X 1,..., X i 1, X i, X i+1..., X j 1, X j, X j+1,..., X d ) L = (X 1,..., X i 1, X j, X i+1..., X j 1, X i, X j+1,..., X d ), then i = j. 3. Riskless allocation: For a deterministic risk X = c, where the constant c R + : A X,X1,...,X d () = (c, A X1,...,X d ( c)). This property means that the optimal allocation method relates only risky branches, the presence of a deterministic risk has no impact on the share allocated to the risky branches. 4. Sb-additivity: M {1,..., d}, let (, 1,..., r) = A i M X i,x j {1,...,d}\M (), where r = d card(m) and ( 1,..., d ) = A X1,...,X d (): i. i M This property mains that the optimal allocation takes into accont the diversification gain. It is related to the no nderct property defined by Denalt, which has no sense in or context. 4

6 5. Comonotonic additivity: For r d comonotonic risks, A Xii {1,...,d}\CR, k CR X k () = ( i i {1,...,d}\CR, where ( 1,..., d ) = A X1,...,X d () is the optimal allocation of on the d risks (X 1,..., X d ) and CR denote the set of the r comonotonic risk indexes. The concept of comonotonic random variables is related to the stdies of Hoeffding (194) [16] and Fréchet (1951) [14]. Here we se the definition of comonotonic risks as it was first mentioned in the actarial literatre in Borch (1962) [5]. k CR k ), Other desirable properties. We define also some desirable properties that an optimal allocation mst natrally satisfy. These properties are based on the ideas presented by Artzner et al (1999) [1] for coherent risk measres and on the axiomatic characterization of coherent capital allocations given by Kalkbrener (29) [17]. Definition 1.3 (Positive homogeneity). An optimal allocation is positively homogeneos, if for any α R +, it satisfies: A αx1,...,αx d (α) = αa X1,...,X d (). In other words, a capital allocation method is positively homogeneos, if it is insensitive to the cash changes. Definition 1.4 (Translation invariance). An optimal allocation is invariant by translation, if for all (a 1,..., a d ) R d, it satisfies: ) A X1 a 1,...,X d a d () = A X1,...,X d ( + a k (a 1,..., a d ). The translation invariance property shows that the impact of an increase (decrease) of a risk by a constant amont of its share of allocation of the capital, boils down to an increase (decrease) of its share in the allocation of sch capital decreased (increased) by the same amont. Definition 1.5 (Continity). An optimal allocation is continos, if for all i {1,..., d}: lim A X1,...,(1+ɛ)X ɛ i,...,x d () = A X1,...,X i,...,x d (). This property reflects the fact that a small change to the risk of a bsiness line, have only limited effect on the capital part that we attribte to it. Let s recall the definition of the order stochastic dominance, as it is presented in Shaked and Shanthikmar (27)[21]. For random variables X and Y, Y first-order stochastically dominates X if and only if: F X (x) F Y (x), x R +, and in this case we denote: X st Y. This definition is also eqivalent to the following one: X st Y E[(X)] E[(Y )], for all increasing fnction Definition 1.6 (Monotonicity). An optimal allocation satisfies the monotonicity property, if for (i, j) {1,..., d} 2 : X i st X j i j. 5

7 The monotonicity is a natral reqirement, it reflects the fact that if a branch X j is riskier than branch X i. Then, it is natral to allocate more capital to the risk X j. The RORAC compatibility property defined by Dirk Tasche [24] loses its meaning in absence of the risk measre sed in the constrction of the allocation method Optimality conditions. In this section, we focs on the optimality condition for the indicators I and J. For an initial capital, and an optimal allocation minimizing the mltivariate risk indicator I, we seek R d + sch that: I ( ) = inf I (v), v v 1 + +v d = Rd +. Under assmption H2, the risk indicators I and J are differentiable, and in this case, we can calclate the following gradients: ( I(v)) i = and, ( J(v)) i = + + v k g k (v k x)f Xk,S(x, )dx + E[g i(v i X i ) 1 {Xi >v i } 1 {S } ] v k g k (v k x)f Xk,S(x, )dx + E[g i(v i X i ) 1 {Xi >v i } 1 {S } ]. Under H1 and H2, sing the Lagrange mltipliers method, we obtain an optimality condition verified by the niqe soltion to this optimization problem: (1.1) E[g i( i X i ) 1 {Xi > i } 1 {S } ] = E[g i( j X j ) 1 {Xj > j } 1 {S } ], j {1,..., d} 2. A natral choice for penalty fnctions is the rin severity: g k (x) = x. In that case, and if the joint density f (Xk,S) spport contains [, ] 2, for at least one k {1,..., d}, or optimization problem has a niqe soltion. We may write the indicators as follows: ( ) I ( 1,..., d ) = E R k 1 {R k <} 1 d { i=1 Ri } and, = J( 1,..., d ) = = ( ) E (X k k ) 1 {Xk > k } 1 d { = i=1 X i } ( ) E R k 1 {R k <} 1 d { i=1 Ri } ( ) E (X k k ) 1 {Xk > k } 1 d { = i=1 X i } E ( ) (X k k ) + 1 {S }, E ( ) (X k k ) + 1 {S }. The respective components of the gradient of these indicators are of the form: K I P X 1 > 1, X j,..., K I P X d > d, X j, and, j=1 K J P X 1 > 1, X j,..., K J P X d > d, X j, j=1 j=1 6 j=1

8 where, + K I = K J = (x k )f Xk,S(x, )dx. k Using the Lagrange mltipliers to solve or convex optimization problem nder the only constraint d =, the following optimality conditions are obtained from 1.1 in the special case where g k (x) = x : (1.2) P (X i > i, S ) = P (X j > j, S ), (i, j) {1, 2,..., d} 2. For the J indicator, this condition can be written: (1.3) P (X i > i, S ) = P (X j > j, S ), (i, j) {1, 2,..., d} 2. Some explicit and semi-explicit formlas for the optimal allocation can be obtained with this optimality condition. Or problem redces to the stdy of this allocation depending on the natre of the distribtions of the risk X k and on the form of dependence between them Main theoretical reslts. In section 2, we shall prove the following theorem. Theorem 1.7. In the case of penalty fnctions g k (x) = x k {1,..., d}, and for continos random vector (X 1,..., X d ), sch that the joint density f (Xk,S) spport contains [, ] 2, for at least one k {1,..., d}, the optimal allocation by minimization of the indicators I and J is a symmetric riskless fll allocation. It satisfies the properties of comonotonic additivity, positive homogeneity, translation invariance, monotonicity, and continity. The main theoretical reslt of section 3 is Theorem 3.7, from which the asymptotic (as + ) behavior of the optimal allocation may be derived for large classes of independent distribtions. 2. Properties of the optimal allocation method In what follows, we show that the capital allocation minimizing the indicator I, satisfies the coherence axioms of Definition 1.2, except the sb-additivity. We show also that it satisfies other desirable properties in the second sbsection. The same holds for the indicator J Coherence. Firstly, the fll allocation axiom is verified by constrction, since any optimal allocation satisfies the eqality: i =. Proposition 2.1 shows that the optimal allocation satisfies the symmetry property. i=1 Proposition 2.1 (Symmetry). Under H1, if for (i, j) {1, 2,..., d} 2, i j, the coples (X i, S i ) and (X j, S j ) are identically distribted and the penalty fnctions g i and g j are the same g i = g j, then: i = j. Proof. Let (i j) {1, 2,..., d} 2 be sch that (X i, S i ) and (X j, S j ) have the same distribtion and the same penalty fnction g i = g j = g. If i j, we may assme i < j, and denote: then, ( 1,..., i,..., j,..., d ) = A X1,...,X i,...,x j,...,x d (), I( 1,..., i,..., j,..., d ) = inf I (v) = inf v U d v U d 7 E ( ) g k (v k X k ) 1 {Xk >v k } 1 {S }.

9 On the other hand, and since g i = g j = g and (X i, S i ) (X j, S j ), then: I( 1,..., i 1, j, i+1,..., j 1, i, j+1,..., d ) =,k i,k j E ( g k ( k X k ) 1 {Xk > k } 1 {S } ) + E ( g( i X i ) 1 {Xi > i } 1 {S } ) + E ( g( j X j ) 1 {Xj > j } 1 {S } ) = I( 1,..., i,..., j,..., d ). From H1, the indicator I admits an niqe minimm in U d, we dedce that: ( 1,..., i,..., j,..., d ) = ( 1,..., i 1, j, i+1,..., j 1, i, j+1,..., d ). We conclde that i = j. Corollary 2.2. Under Assmptions H1 and H3, if (X 1,..., X d ) is an exchangeable random vector, then the allocation by minimizing I and J indicators is the same and is given by: ( A X1,...,X d () = d, d,..., ). d The following proposition shows that the optimal allocation verifies the Riskless allocation axiom. Proposition 2.3 (Riskless Allocation). Under Assmptions H1 and H3, and for 1-homogeneos penalty fnctions, for any c R: A c,x1,...,x d () = (c, A X1,...,X d ( c)), where (c, A X1,...,X d ( c)) is the concatenated vector of c and the vector A X1,...,X d ( c). Proof. The presence of a discrete distribtion makes the indicator I not differentiable, so we cannot se neither the gradient, nor the optimality condition obtained in the case of existence of joined densities. Let, (, 1,, d) = A c,x1,...,x d () and ( 1,, d ) = A X1,...,X d ( c). We denote S = d i=1 X i, and the common penalty fnction g = g k, k {1,..., d}, the fnction g is convex on R and g() =, we dedce that g is also positively homogeneos. We distingish between three possibilities: Case 1: < c In this case, I(, 1,..., d) = inf I (v) = inf E ( ) g(v k X k ) 1 {Xk >v k } 1 {S c} v U d+1 v U d+1 k= = E ( ) g( c) 1 {S c} + E ( ) g( k X k ) 1 {Xk > k } 1 {S c}, for all k {1,..., d} we pt, for example, α k = α = c <, and since the fnction g is d convex and g() =, it satisfies for all real < β < 1, g(βx) βg(x), x R. Then: ( ( I(, 1,..., ) ) c d) E d g 1 {S c} + E ( ) g( k X k ) 1 {Xk > d k } 1 {S c} = E ( ) d g ( ( α) + ) 1 {S c} + E ( ) g( (X k k) + ) 1 {S c} = E ( ) [g( (X k k) + ) + g( ( α k ) + )] 1 {S c}, 8

10 x g( (x) + ) is also a 1-homogeneos convex fnction, then: I(, 1,..., d) E ( ) g( (X k ( k + α k )) + ) 1 {S c}, we remark that d ( k + α k ) = c, then ( 1 + α,..., d + α) U d c. So, then, I(, 1,..., d) E ( ) g(( k + α k ) X k )) 1 {Xk > k +α k} 1 {S c} inf E ( ) g(v k X k ) 1 {Xk v U c d >v k } 1 {S c} = E ( ) g( k X k ) + 1 {S c} = I(c, 1,..., d ), I(, 1,..., d) I(c, 1,..., d ). That is contradictory with the niqeness of the minimm on the set U d+1. Case 2: > c We have : and, I(, 1,..., d) = E ( ) g( k X k ) 1 {Xk > k } 1 {S c}, I(c, 1,..., d ) = E ( ) g( k X k ) 1 {Xk > k } 1 {S c}. Let α = c >, we remark that,( d 1+α,..., d+α) U c, d and that the penalty fnction g is decreasing on R becase g (x), x R and g ( + ) =. Then, I(c, 1,..., d ) = E ( ) g( (X k k ) + ) 1 {S c} = inf E ( ) g(v k X k ) 1 {Xk v U c d >v k } 1 {S c} E ( ) g( (X k ( k + α)) + ) 1 {S c} < E ( ) g( (X k k)) + 1 {S c} = E ( ) g( k X k ) 1 {Xk > k } 1 {S c} = I(, 1,..., d). That is contradictory with the fact that I(, 1,..., d) = We dedce that the only possible case is the third one = c. 9 inf v U d+1 I (v).

11 Case 3: = c The niqeness of the minimm implies that: (, 1,..., d) = (c, 1,..., d) = arg min U d+1 Finally, we have proven that: = arg min U c d = (c, 1,..., d ). (, 1,..., d) = (c, 1,..., d ). E[g( k X k ) 1 {Xk > k } 1 {S c} ] E[g( k X k ) 1 {Xk > k } 1 {S c} ] Lemma 2.4 is related to the sb-additivity property. It will be sed in the proof of the comonotonic additivity property. Lemma 2.4. Under Assmptions H1,H2 and H3, and For all (i, j) {1,..., d} 2, and where, x.e i is the dot prodct of the vector x R d and the i th component of the canonical basis of R d. if A X1,...,X i 1,X i +X j,x i+1,...,x j 1,X j+1,...,x d ().e i < A X1,...,X d ().(e i + e j ), then: k {1,..., d} \ i, j, A X1,...,X i 1,X i +X j,x i+1,...,x j 1,X j+1,...,x d ().e k > A X1,...,X d ().e k, if A X1,...,X i 1,X i +X j,x i+1,...,x j 1,X j+1,...,x d ().e i > A X1,...,X d ().(e i + e j ), then: k {1,..., d} \ i, j, A X1,...,X i 1,X i +X j,x i+1,...,x j 1,X j+1,...,x d ().e k < A X1,...,X d ().e k, if A X1,...,X i 1,X i +X j,x i+1,...,x j 1,X j+1,...,x d ().e i = A X1,...,X d ().(e i + e j ), then: k {1,..., d} \ i, j, A X1,...,X i 1,X i +X j,x i+1,...,x j 1,X j+1,...,x d ().e k = A X1,...,X d ().e k. Proof. In order to simplify the notation, and withot loss of generality, we assme i = d 1 and j = d. We pt,( 1,..., d 1, d ) = A X1,...,X d () and ( 1,..., d 2, d 1) = A X1,...,X d 2,X d 1 +X d (). The optimality condition for ( 1,..., d 1, d ) is given (i, j) {1,..., d} 2 by eqation 1.1: E[g i( i X i ) 1 {Xi > i } 1 {S } ] = E[g i( j X j ) 1 {Xj > j } 1 {S } ] = λ, and for ( 1,..., d 2, d 1) is i {1,..., d 2} E[g i( i X i ) 1 {Xi > i } 1 {S } ] = E[g i( d 1 (X d 1 + X d )) 1 {Xd +X d 1 > d 1 } 1 {S } ] = λ. Now, we sppose that d 1 > d + d 1. In this case there exists k {1,..., d 2} sch that k < k, and since the fnction x g ( (x) + ) is decreasing on R +, then: E[g i( k X k ) 1 {Xk > k } 1 {S } ] = λ < E[g i( k X k ) 1 {Xk > k } 1 {S } ] = λ we dedce from this that for all k 1,..., d 2 : k < k. The proof is the same if we sppose that d 1 < d + d 1, and the additive case is a corollary of the two previos ones. Proposition 2.5 (Comonotonic additivity). Under Assmption H2, and for g k (x) = x, for all k {1,..., d}, if r d risks X ii CR are comonotonics, then: A Xii {1,...,d}\CR, k CR X () = ( k i i {1,...,d}\CR, k ), where ( 1,..., d ) = A X1,...,X d () is the optimal allocation of on the d risks (X 1,..., X d ), A Xii {1,...,d}\CR, k CR X k () is the optimal allocation of on the n d+1 risks (X i i {1,...,d}\CR, k CR X k ), and CR denote the set of the r comonotonic risk indexes. 1 k CR

12 Proof. For (i, j) {1,..., d} 2, if X i and X j are comonotonic risks, then, there exists an increasing non negative fnction h sch that X i = h(x j ), and we remark that h is strictly increasing nder Assmption H2. Let f be the fnction x f(x) = x + h(x), so that X i + X j = f(x j ). We denote ( 1,..., d ) = A X1,...,X d () and ( 1,..., d 1) = A Xi {1,...,d}\{i,j},Xi +X j (), then, A Xi {1,...,d}\{i,j},Xi +X j ().e d 1 = d 1 and A X1,...,X d ().(e i + e j ) = i + j. From the optimality condition for the allocation A X1,...,X d (), given in Eqation 1.2: P(X i i, S ) = P(X j j, S ), we dedce that i = h( j ) and that i + j = f( j ). If there exists k {1,..., d} \ {i, j}, sch that k < k, then k {1,..., d} \ {i, j}: so, P(X k k, S ) > P(X k k, S ), P(X i + X j d 1, S ) = P(X j f 1 ( d 1), S ) = P(X k k, S ) > P(X k k, S ) = P(X j j, S ), finally, we dedce that: f 1 ( d 1) < j, then d 1 < f( j ) = i + j and, k {1,...,d}\{i,j} k < k {1,...,d}\{i,j} k which is absrd. In the same way, the case k < k for k {1,..., d} \ {i, j} leads to the contradiction. Using Lemma 2.4, and nder Assmption H3, we dedce the optimal allocation for the other risks X k, k {1,..., d} \ {i, j}. The additivity property for two comonotonic risks is trivially generalizable to several comonotonic risks. Concerning the sb-additivity property, we have not yet managed to bild a demonstration for this property. However, the simlations sing the optimization algorithm presented in Cénac et al (212) [1], seem to confirm the sb-additivity of the allocation by minimizing the indicators I and J, even for classic examples of non sb-additivity of the risk measre VaR. Remark 2.6. The previos properties have been demonstrated for the optimal allocation by minimizing the risk indicator I, they can be demonstrated with the same argments for the optimal allocation by minimization of the indicator J Other desirable properties. In this section, we show that the optimal allocation by minimization of the indicators I and J satisfies some desirable properties. We consider the allocation by minimizing the mltivariate risk indicator I, the demonstrations are almost the same in the case of the indicator J. The following proposition shows that the optimal allocation by minimization indicators I and J satisfies the property of positive homogeneity. Proposition 2.7 (Positive homogeneity). Under Assmption H1, and for 1-homogeneos penalty fnctions g k, k {1,..., d}, for any α R + : A αx1,...,αx d (α) = αa X1,...,X d (). 11

13 Proof. Since the penalty fnctions are convex and 1-homogeneos, then, for any α R + : A αx1,...,αx d (α) = arg min E[g k ( ( 1,..., d ) U k αx k ) 1 {αxk > α d k } 1 {αs α} ] ( ) = arg min αe[g k k ( 1,..., d ) U α d α X k 1 1 {Xk > k {S } ] α } ( ) = arg min E[g k k ( 1,..., d ) U α d α X k 1 1 {Xk > k {S } ] α } = α arg min E[g k ( k X k ) 1 {Xk > k } 1 {S } ] ( 1,..., d ) U d = αa X1,...,X d (). Proposition 2.8 (Translation invariance). Under Assmptions H1, H2 and for all (a 1,..., a d ) R d, sch that the joint density f(x k, S) spport contains [, + d a k ] 2, for all k {1,..., d}: ) A X1 a 1,...,X d a d () = A X1,...,X d ( + a k (a 1,..., a d ). Proof. We denote by ( 1,...(, d) the optimal allocation A X1 a 1,...,X d a d (), and by ( 1,..., d ) the ) optimal allocation A X1,...,X d + d a k. Using the optimality condition (1.1), ( 1,..., d) is the niqe soltion in U d of the following eqation system: E[g i( i (X i a i )) 1 {Xi a i > i } 1 {S a } ] = E[g j( j (X j a j )) 1 {Xj a j > j } 1 {S a } ], j {1,..., d}, where a = d a k. Then, ( 1,..., d) satisfies also: E[g i( i + a i X i ) 1 {Xi > i +a i} 1 {S +a} ] = E[g i( i + a j X j ) 1 {Xj > j +a i} 1 {S +a} ], j {1,..., d}. Since, ( 1 + a 1,..., d + a d ) U+a, d and from the soltion niqeness of the optimality condition (1.1) for the allocation A X1,...,X d ( + a), we dedce that: k+a k = k for all k {1,..., d}. Proposition 2.9 (Continity). Under Assmptions H1 and H2, and if k {1,..., d}, ɛ > sch that: ɛ, ɛ < ɛ, E[ sp g k(v (1 + ɛ)x k ) ] < +, v [,] then, if (X 1,..., X d ) is a continos random vector, for all i {1,..., d}: lim ɛ A X1,...,(1+ɛ)X i,...,x d () = A X1,...,X i,...,x d (). Proof. Let ( 1,..., d ) be the optimal allocation of on the d risks (X 1,..., X d ): ( 1,..., d ) = A X1,...,X i,...,x d (), then ( 1,..., d ) is the niqe soltion in U d of Eqation system (1.1): E[g i( i X i ) 1 {Xi > i } 1 {S } ] = E[g i( j X j ) 1 {Xj > j } 1 {S } ], j {1,..., d}. For ɛ R, let ( ɛ 1,..., ɛ d) be the optimal allocation of on the d risks (X 1,..., X i 1, (1 + ɛ)x i, X i+1,..., X d ): ( ɛ 1,..., ɛ d) = A X1,...,X i 1,(1+ɛ)X i,x i+1,...,x d (), 12

14 then ( ɛ 1,..., ɛ d) is the niqe soltion in U d of the following eqation system: E[g i( ɛ i (1 + ɛ)x i ) 1 {(1+ɛ)Xi > ɛ i } 1 {S+ɛXi }] = E[g i( ɛ j X j ) 1 {Xj > ɛ j } 1 {S+ɛXi }], j {1,..., d}. Since U d is a compact on (R + ) d, we may consider a convergent sbseqence ( ɛ k 1,..., ɛ k d ) of ( ɛ 1,..., ɛ d). Since the penalties fnctions satisfy: ɛ >, ɛ, ɛ < ɛ, E[ sp g k(v (1 + ɛ)x k ) ] < +, v [,] we se Lebesge s dominated convergence Theorem to get: E[g i(lim ɛ k i X i ) 1 ɛ {Xi > lim ɛ k i ɛ } 1 {S } ] = E[g i(lim ɛ k j X j ) 1 ɛ {Xj > lim ɛ k j } 1 {S } ], j {1,..., d}, ɛ thereby (lim ɛ k 1,..., lim ɛ k ɛ ɛ (lim ɛ k 1,..., lim ɛ k ɛ ɛ d ) U. d From the soltion niqeness of (1.1) in U, d we dedce that: lim ɛ k i k ɛ k ɛ l d ) is a soltion of Eqation (1.1), becase d lim = lim ɛ d ɛ l =, = i for all i {1,..., d}. For all convergent sbseqence of ( ɛ 1,..., ɛ d) the limit point is ( 1,..., d ), we dedce that: lim( ɛ ɛ 1,..., ɛ d) = ( 1,..., d ). Proposition 2.1 (Monotonicity). Under Assmption H2, and for (i, j) {1,..., d} 2, sch that g i = g j = g: X i st X j i j. Proof. Let ( 1,..., j ) be the optimal allocation A X1,...,X i,...,x d (), nder Assmption H2, the optimality condition (1.1) is written as follows: E[g ( i X i ) 1 {Xi > i } 1 {S } ] = E[g ( j X j ) 1 {Xj > j } 1 {S } ]. Now if X i st X j, and since, x g ( ( i x) + ) 1 {S } is an increasing fnction on R +, then: We dedce that: E[g ( i X j ) 1 {Xj > j } 1 {S } ] E[g ( i X i ) 1 {Xi > i } 1 {S } ]. E[g ( i X j ) 1 {Xj > j } 1 {S } ] E[g ( j X j ) 1 {Xj > j } 1 {S } ], and since, g j i. is an increasing fnction, and the distribtions are all continos, that implies: By combining all the properties demonstrated in this section, we show Theorem 1.7. These properties are therefore desirable from an economic point of view, the fact that they are satisfied by the proposed optimal allocation implies that this allocation method may well be sed for the economic capital allocation between the different branches of a grop, in terms of their actal participation in the overall risk, taking into accont both their marginal distribtions and their dependency strctres with the remaining branches. 13

15 3. Some general reslts in independence case In this section we generalize the reslts presented in dimension 2 by Cénac et al. in the first section of their paper [9] to higher dimension. The reslts presented here give explicit forms to the optimal allocation for some specific distribtions. This cold be sed as a benchmark to test optimization algorithms convergence. We also get some asymptotic reslts, when the capital goes to infinity. We stdy both the exponential and the sb-exponential cases, and we determine the difference between their asymptotic behavior for exponential and Pareto distribtions cases. We consider from now on that the penalty fnctions are identical and eqal to the severity of local rin g k (x) = g(x) = x, k {1, 2,..., d}. The optimality conditions for minimizing the mltivariate risk indicators I and J are given respectively by Eqations (1.2) and (1.3). In this section we focs on the independence case. Remark that α i = i [, 1], then when +, we may consider convergent sbseqences in the proofs below. By abse of notation, we consider lim α i. In fact, we consider a convergent + sbseqence and get the existence of the limit by obtaining the niqeness of the limit point Independent exponentials. Assme X 1, X 2,..., X d are independent exponential random variables with respective parameters < β 1 < β 2 < < β d. Remark that in the particlar case where β 1 = β 2 = = β d, the optimal allocation is 1 = = d = /d. Proposition 3.1 (The optimal allocation for the indicator I). The allocation minimizing the risk indicator I is the niqe soltion in U, d of the following eqation system : (3.1) h(β i α i ) h(β j α j ) A l h(β l )[h(α i (β i β l )) h(α j (β j β l ))] =, (i, j) {1, 2,..., d} 2, where h is the fnction defined by h(x) = exp( x), and A l denotes the constants A l = d β j, l = 1,..., d. β j β l j=1,j l Proof. If X i E(β i ) are independent exponential random variables, then S i = j=1;j i generalized Erlang distribtion with parameters (β 1, β 2,..., β i 1, β i+1,..., β d ), so we write: P X i > i, X j = P (X i > i ) P X i > i, X j > j=1 = F Xi ( i ) F Xi () i j=1 FS i( s)f Xi (s)ds = h(β i α i ) h(β i ) A l h(β l )h(α i (β i β l )) + A l h(β i ) = h(β i α i ) A l h(β l )h(α i (β i β l )), 14 X j have a

16 d becase, FXi (x) = e βix β j, FS i(x) = ( )e βlx and A l = 1.,l i j=1,j l,j i β j β l The srvival fnction of the generalized Erlang distribtion with parameters (β 1, β 2,..., β d ) is given by ([18]): d β j F X (x) = ( )e βlx = A l e βlx. β j β l j=1,j l The optimal allocation is the niqe soltion in U, d of the following eqation system : ( ) P X i > i, X k = P X j > j, X k, (i, j) {1, 2,..., d} 2, which leads to (3.1). The reslting system is a system of nonlinear eqations, which can be solved nmerically. Proposition 3.2 (The asymptotic optimal allocation for the indicator I). When the capital goes to infinity, the asymptotic optimal allocation satisfies: 1 ( 1 lim α = lim, 2,..., ) d β = i. 1 Proof. Eqation system (3.1) is eqivalent to: j=k j=1 β j i=1,2,...,d (3.2) (i, j) {1, 2,..., d} 2, h(β i α i )[1 A l h((1 α i )β l )] = h(β j α j )[1 A l h((1 α j )β l )]. Firstly, remark that for all i {1, 2,..., d}, lim sp i = lim then taking if necessary i sch that lim i and Eqation system (1.2) cannot be satisfied. Eqation system (3.2) is eqivalent to: < 1 becase, if this reslt was not satisfied j α i = 1. Then for all j i, lim (i, j) {1, 2,..., d} 2, h(β i α i β j α j ) = 1 d A l h((1 α j )β l ) 1 d A l h((1 α i )β l ), = lim α j =, the right side of the last system eqations tends to 1 when tends to, therefore, we dedce that lim h(β i α i β j α j ) = 1 and conseqently: then, for all i {1, 2,..., d}: (i, j) {1, 2,..., d} 2, lim α i = β j β lim α j, i lim α i = 1 β i dj=1 1 β j. Remark 3.3. Based on the above reslt, we can conclde that asymptotically: if β i < β j then α i > α j, this means that we allocate more capital to the most risky bsiness line. 15

17 α i is a decreasing fnction of β i. This observation is consistent with the previos conclsion. α j for j i is an increasing fnction of β i. Proposition 3.4 (The optimal allocation for the indicator J). The allocation minimizing the risk indicator J is the niqe soltion in U d, of the following eqation system : (3.3) (i, j) {1, 2,..., d} 2, A l h(β l )[h(α i (β i β l )) h(α j (β j β l ))] =. Proof. The proof is similar to that of Proposition 3.1. Proposition 3.5 (The asymptotic optimal allocation for the indicator J). When the capital goes to infinity, the optimal allocation minimizing the risk indicator J is the following: 1 lim j = 1 and lim = j {2, 3,..., d}. Proof. Eqation system (3.3) is eqivalent to the following one: (i, j) {1, 2,..., d} 2, A l h((1 α i ) (β l β i )) = A l h(α j (β j β l ) + β l β i ). If lim 1 < 1, as goes to + in the eqations of the previos system for (i = 1) we get: j {2, 3,..., d}, lim A l h(α j (β j β l ) + β l β 1 ) = A 1. The first terms of these eqations can be decomposed into three parts as follows: lim A l h(α j (β j β l ) + β l β 1 ) = lim A 1 h(α j (β j β 1 )) j + lim A l h(α j (β j β l ) + β l β 1 ) l=2 + lim A l h(α j (β j β l ) + β l β 1 ), l=j+1 j and since, for all, j > 1, lim A l h(α j (β j β l ) + β l β 1 ) =, becase β l β 1 > and l=2 β j β l for l {2, 3,..., j}. Moreover, lim A l h(α j (β j β l ) + β l β 1 ) =, becase l=j+1 for all l {j + 1, j + 2,..., d}, α j (β j β l ) + β l β 1 = (β l β j )(1 α j ) + β j β 1 >, then lim h(α j (β j β 1 )) = 1. We dedce that j {2, 3,..., d}; lim α j = o( 1 ). This contradicts the necessary condition: lim d α j = Some distribtions of the sb-exponential family. In most cases of risk distribtions, we cannot give explicit or semi-explicit optimal allocation formlas, the difficlty comes from the lack of a simple form of the risks sm and its joint distribtion with each risk. In this section, we present explicit formlas for some distribtions of the sb-exponential family. By this way, we generalize reslts of [9] to higher dimension. 16

18 We recall the sb-exponential distribtions family definition, consisting in distribtions of positive spport, with an distribtion fnction that satisfies: F 2 (x) F (x) x + 2, where F 2 is the convoltion of F. In Asmssen (2) [2], it is proven that the sb-exponential distribtions satisfy also the following relation, for all d N : F d (x) F (x) x + d, where F d is the d th convoltion of F. We shall se the following theorem proved in Cénac and al. Theorem 3.6 (Sb-exponential distribtions [9]). Let X be a random variable with sb-exponential distribtion F X, Y a random variable with spport R +, independent of X, and (, v) (R + ) 2, sch that: Then, there exists < κ 1 < κ 2 < 1 sch that for large enogh, κ 1 v κ 2, x + = O(1), if y = Θ(x) 1. F X (y) F X (x) P(X v, X + Y ) lim F X () The asymptotic behavior. Propositions 3.8 and 3.1 concern the asymptotic behavior of the optimal allocation by minimizing the indicator I, in the cases of some sb-exponential distribtions. In what follows, ( 1,..., d ) denotes the optimal allocation of associated to the risk indicator I. Theorem 3.7. Let (X 1, X 2,..., X d ) be continos positive and independent random variables. Assme that (i, j) {1, 2,..., d} 2 : (1) F Xi (x) x + = Θ( F Xj (x)), (2) F Xi (s) s + = o( F Xi (t)), if t = o(s), then, there exist κ 1 > and κ 2 < 1 sch that, = 1. (3.4) κ 1 l κ 2 l {1, 2,..., d}. Note that the first condition of Theorem 3.7 is not satisfied for exponential distribtions. However, Pareto distribtions verify the hypothesis of Theorem 3.7. Proof. We sppose that, i {1,..., d} sch that: i + 1 or i +, the first case implies that foll all j i, j +, then, it is sfficient to prove that the existence of an i {1,..., d} sch that i + is impossible. We assme the existence of i {1,..., d}, sch that: i +. 1 For (x, y) R 2+, we shall denote y = Θ(x) if there exist < C 1 C 2 <, sch that for x large enogh, C 1 y x C 2 17

19 Then, j {1,..., d}\i sch that lim j ], 1], therefore, + j Using Assmptions (1) and (2), we dedce that: F Xj ( j ) + (3.5). F Xi ( i ) + + and i + = o( j ). The optimality condition ((1.2)) can also be written for all j i as follows: F Xi ( i ) P (X i > i, S > ) = F Xj ( j ) P (X i > j, S > ). (3.6) That presents a trivial contradiction if i remains bonded. k=d Now, assme that i +. Recall that S i = X k, then: We have:,k i P (X i > i, S > ) = P ( X i > i, S i >, S > ) + P ( X i > i, S i <, S > ). P ( X i > i, S i >, S > ) P (X i > i ) P ( S i > ) + = o( F Xi ( i )). Using assmption (2) and since i = o(), We dedce that: (3.7) We remark also that: (3.8) Eqation (3.6) leads to: P ( X i > i, S i <, S > ) F Xi ( ) + = o( F Xi ( i )). P (X i > i, S > ) + = o( F Xi ( i )). P (X j > j, S > ) + = O( F Xj ( j )) + = o( F Xi ( i )). 1 P (X i > i, S > ) F Xi ( i ) }{{} T 1 = F Xj ( j ) F Xi ( i ) }{{} T 2 P (X j > j, S > ). F Xi ( i ) }{{} T 3 Now, relations: (3.7), (3.5), and (3.8), imply that T 1, T 2, and T 3, go to zero, and this is a contradiction. Proposition 3.8. Let X 1,..., X d be continos, positive and independent random variables sch that the spport of the density of (X i, S) is (R + ) 2. Let ( 1,..., d ) be the optimal allocation of associated to the risk indicator I. We assme: (1) there exist < κ 1 < κ 2 < 1 sch that for all i = 1,..., d and for all R +, κ 1 i κ 2, (2) for all i = 1,..., d, if y = y(x) is sch that y < lim inf x x then Then, for all i, j = 1,..., d, F Xi (x) F Xi (y) F Xi ( i ) F Xj ( j ) lim sp x 18 x. 1. y x < 1

20 Assmptions of Proposition 3.8 are satisfied for distribtions of exponential type (see remark (3.9) below). Its application gives another proof to Proposition 3.2. In contrast, Proposition 3.8 cannot be sed for Pareto distribtions. Proof. Following the lines of the proof of Theorem 3.7, take < γ < 1 κ 2, As before, P(X i > i, S > ) = P(X i > i, S i > γ, S > ) + P(X i > i, S i > γ, S > ). On the other hand, P(X i > i, S i > γ, S > ) P(X i > i )P(S i > γ) + = o(f Xi ( i )). P(X i > i, S i > γ, S > ) P(X i > γ) = F Xi ((1 γ)) + = o(f Xi ( i )) becase < κ 1 1 γ i (1 γ) κ 2 1 γ < 1. So that, P(X i > i, S > ) + = o(f Xi ( i )) and the same comptation gives P(X j > j, S > ) + = o(f Xj ( j )). Now, i and j satisfy Eqation (1) and ths, This implies that F X j ( j ) F Xi ( i ) 1 + o(1) + = F X j ( j ) F Xi ( i ) + o(1)f X j ( j ) F Xi ( i ). is bonded from above and ths F Xj ( j ) F Xi ( i ) + = 1 + o(1). Remark 3.9. We remark that the hypothesis of Proposition 3.8 are satisfied for distribtion of exponential type, that is distribtions verifying: F Xi (x) = Θ(e µ ix ). Indeed, in this case, we have < lim i lim i < 1, i {1,..., d}. In fact, if i then, j {1,..., d}\{i} sch that j κ ], 1], so + i = o( j ). + Since µ i, µ j >, µ i i = o(µ j j ). And as in the proof of Theorem 3.7, we get: and, From the optimality condition, F Xi ( i ) e µ i i P (X i > i, S > ) e µ i i }{{} P (X i > i, S > ) + = o( F Xi ( i )) P (X j > j, S > ) + = o( F Xj ( j )). T 1 = which is absrd becase as + : F Xi ( i ) = Θ(e µ i i ), T 1 = o(1), T 2 = F Xj ( j ) e µ j j e µ j j +µ i i, since µ i i = o(µ j j ), 19 F Xj ( j ) P (X j > j, S > ) e µ i i }{{} e µ i i }{{} T 2 T 3, + = o(),

21 and T 3 = o(1)e µ j j +µ i i. Proposition 3.1 (The asymptotic optimal allocation for the indicator I). Under the same conditions of Theorem 3.7, and if, for all i {1,..., d}, F Xi is a sb-exponential distribtion, that verifies: F Xi (y) x + = O(1), for < κ 1 y F Xi (x) x κ 2 < 1. Then, by minimizing the I indicator, i and j satisfy: (3.9) FXi ( i ) F Xi () + = F Xj ( j ) F Xj () + o( F Xi ()). Proposition 3.1 is applicable in the case of Pareto distribtions. So, we will se it for determining the optimal asymptotic allocation, for independent risks of Pareto distribtions in the next sbsection. Proof. The proof of this theorem is a direct application of Theorems 3.6 and 3.7. Now, we focs on the asymptotic optimal allocation by minimizing the risk indicator J, and we stdy the case of sb-exponential distribtion family. Proposition 3.11 (The asymptotic optimal allocation for the indicator J). Let (X 1, X 2,..., X d ) be continos positive and independent random variables, sch that, there exist i {1,..., d} with a sb-exponential distribtion, the optimal capital allocation by minimizing the J indicator ( 1,..., d ) verifies, for all j i: P(X j j, S ) lim F Xi () = 1. Proof. The soltion to (1.3) satisfies : j {1, 2,..., d}, P (X i > i, S ) P (X i > ) = P (X j > j, S )). P (X i > ) When goes to +, and sing Theorem 3.6, we obtain Proposition application to Pareto independent distribtions. We consider d independent random variables {X 1, X 2,..., X d } of Pareto distribtions, with parameters (a, b i ) {i=1,2,...,d} respectively, sch that b 1 > b 2 > > b d >. Therefore, these distribtions will be characterized by densities and srvival fnctions of the following forms: f Xi (x) = a b i (1 + x b i ) a 1, and F Xi (x) = ( 1 + x b i ) a. Proposition 3.12 (The asymptotic optimal allocation minimizing I). Asymptotically, the niqe soltion to (1.2) satisfies: (i, j) {1, 2,..., d} 2, lim α a i lim α a ( ) ( ) j 1 a a 1 =. bi b i b j b j Proof. Follows from Proposition

22 Proposition 3.13 (The asymptotic optimal allocation minimizing J). The niqe soltion to (1.3) satisfies: lim α 1 = 1 et lim α i =, i {2, 3,..., d}. Proof. We sppose that, {1,..., d} sch that < lim α j < 1. From Theorem 3.6 : P(X j j, S ) lim = 1. F Xj () On the other hand, and applying Proposition 3.11 in the Pareto distribtions case, we get for i {1,..., d} \ {j}: P(X j j, S ) F Xj () then, for i {1,..., d} \ {j}: = P(X j j, S ) F Xi () P(X j j, S ) lim F Xj () That is absrd. We dedce that i {1, 2,..., d}: F Xi () F Xj () = lim α i {, 1}, ( bj b i + ) a 1. F Xi () F Xj () = 1 + b i 1 + b j and since d i=1 α i = 1, then, there is a niqe i sch that lim α i = 1 and for all j i lim α j =. Now, and sing the stochastic dominance order, we have X d st st X 2 st X 1, becase b 1 > b 2 > > b d. Finally, since the optimal capital allocation satisfies the monotonicity property, we dedce that: max { lim α i} = lim i {1,2,...,d} α 1 = Comparison between the asymptotic behaviors in exponential and sb-exponential cases. Let s consider the following risk indicator: I loc ( 1,..., d ) = E ( ) (X k k ) 1 {Xk > k } = E ( (X k k ) +). With this indicator, the impact of each branch on the grop is not taken into accont. This indicator ths takes only local effects into accont. If, for any k = 1,..., d, the random vector (X k, S) admits a density whose spport contains [, ] 2, then the optimality condition associated to I loc gives: P(X i > i ) = P(X j > j ). We remark that this corresponds to the asymptotic reslt for indicator I of Proposition 3.8, where F Xj ( j ) + = 1 + o(1). In other words, in the case of exponential type and independent risks, F Xi ( i ) the grop effect is negligible. This behavior is also clear in Proposition 3.2, where we fond the asymptotic capital allocation for I for independent exponential distribtions as: 1 a, as for I loc. i = β i dj=1 1 β j, for all i = 1,..., d, 21

23 Proposition 3.1 shows that for independent sb-exponential distribtions the asymptotic behavior is different. Indeed, in Eqation (3.9), the terms F Xi () and F Xj () lead to take into accont the grop effect. In the Pareto distribtions case as example, recall that Proposition 3.12 gives that the asymptotic behavior of the optimal allocation for I is described by : lim α a i lim α a ( ) ( ) j 1 a a 1 =, bi b i b j whereas the optimal allocation for I loc for independent Pareto distribtions is given by α i = for all i {1,..., d}. b j b i d, b l For optimal allocation by minimizing the J indicator, the asymptotic behavior is identical for the two families of distribtions, the riskiest branch is considered as first responsible for the overall rin, and ths, the optimal soltion is to allocate the entire capital to this bsiness line. 4. The impact of the dependence strctre In this section, we focs on the impact of the dependence strctre on the optimal allocation. We stdy at first the impact of mixtre exponential-gamma to constrct a Pareto distribtion, compared to the independence case presented in the previos section, then we analyze the optimal allocation composition in the case of comonotonic risks. The last sb-section is devoted to the stdy of the impact of the dependence natre on the optimal allocation, sing some bivariate models with coplas Correlated Pareto. Let (X 1,..., X d ) be a mixtre of exponential distribtions sch that for all i {1, 2,..., d}, X i E(β i θ), with (β 1 < β 2 < β d ), and θ Γ(a, b). Therefore, X i have srvival fnctions of the form: ( F Xi (x) = F Xi Θ=θf Θ (θ)dθ = e βiθx f Θ (θ)dθ = 1 + β ) a ix, b conseqently, X i have Pareto distribtion of parameters ( a, b β i ). They are conditionally independents. So, the idea is conditioning on the random variable θ and then integrate the formlas fond for the case of independent exponential distribtions. This model has been stdied in e.g.[2, 26]. Proposition 4.1 (The optimal allocation for the indicator I). The optimal allocation minimizing the mltivariate risk indicator I is the niqe soltion in U d, of the following eqation system : (i, j) {1, 2,..., d} 2, (4.1) s(β i α i ) s(β j α j ) A l [s(α i β i + (1 α i )β l )) s(α j β j + (1 α j )β l ))] =, where s is the fnction defined by s(x) = (1 + x b ) a and α i = i for all i {1,..., d}. Proof. It sffices to integrate Eqation system (3.1), mltiplied by the density fnction of θ. Proposition 4.2 (The asymptotic optimal allocation for the indicator I). When the capital goes to infinity, the optimal allocation by minimization of the risk indicator I is the niqe soltion in U d of the following eqation system: (i, j) {1, 2,..., d} 2, 22

24 (4.2) (β i α i ) a (β j α j ) a A l [(α i β i + (1 α i )β l ) a (α j β j + (1 α j )β l ) a ] =. Proof. We divide Eqation system (4.1) by s(1), and let go to + to get Eqation system (4.2). Proposition 4.2 shows the impact of the dependence related to the mixtre. Indeed, in the case of independent Pareto distribtions, of parameters ( a, b β i, the asymptotic optimal allocation )i=1,...,d for the indicator I is given by Proposition 3.12 as the soltion of the eqation system: (i, j) {1, 2,..., d} 2, (β i α i ) a (β j α j ) a = (β i ) a (β j ) a. Each eqation in this system depends only on two risks, nlike the mixtre case, where the eqations of Eqation system (4.2), depend on all the risks. Proposition 4.3 (The optimal allocation for the indicator J). The optimal allocation minimizing the mltivariate risk indicator J is the niqe soltion in U d, of the following eqation system : (4.3) (i, j) {1, 2,..., d} 2, A l [s(α i β i + (1 α i )β l )) s(α j β j + (1 α j )β l ))] =. Proof. It sffices to integrate Eqation system (3.3), mltiplied by the density fnction of θ. Proposition 4.4 (The asymptotic optimal allocation for the indicator J). When the capital goes to infinity, the optimal allocation by minimization of the risk indicator J, is the niqe soltion in U d of the following eqation system: (4.4) (i, j) {1, 2,..., d} 2, A l [(α i β i + (1 α i )β l ) a (α j β j + (1 α j )β l ) a ] =. Proof. we divide Eqation system (4.3) by s(1), and we let fo to + to get Eqation system (4.4). Proposition 4.4 shows that for the indicator J, the asymptotic behavior of the optimal capital allocation takes into accont the mixtre effect. In fact, for independent Pareto distribtions, we have proved in Proposition 3.13, that we allocate the entire capital to the riskiest branch X 1, while the asymptotic optimal allocation in the correlated Pareto distribtions case is the soltion of Eqation system (4.4) Comonotonic risks. A vector of random variables (X 1, X 2,..., X n ) is comonotonic if and only if there exists a random variable Y and non decreasing fnctions ϕ 1,..., ϕ n sch that: (X 1,..., X n ) = d (ϕ 1 (Y ),..., ϕ n (Y )). Note that the joint distribtion fnction of comonotonic random variables corresponds to the pper Fréchet bond. In the case where the risks X 1,...,X d are comonotonic, we can give explicit formlas for the optimal allocation minimizing the mltivariate risk indicators I and J, and for some risk models. For that, we se the existence of an niform random variable U sch that: X i = FX 1 i (U) for all i {1,..., d}, and S = d i=1 FX 1 i (U) = ϕ(u), where ϕ(t) = d i=1 FX 1 i (t), ϕ is a non decreasing fnction. The main reslt of this section is given below. 23