Mathieu Bargès

Hélène Cossette

Etienne Marceau

Étienne Marceau

TVaR-based capital allocation with copulas

Keywords: Capital allocation, Tail value at risk, Dependence models, Copulas, Discretization methods

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Introduction

In recent years, a lot of research has focused on insurance capital allocation. Indeed the European Solvency II project and the recent events encourage insurance companies to consolidate their financial reserves and investments. Risk measures are well-known tools to determine the capital amount that has to be allocated to a risk portfolio. [START_REF] Artzner | Coherent measures of risk[END_REF] proposed an axiomal definition of a coherent risk measure that can be used for allocation issues. This coherence property has also been discussed in [START_REF] Wang | A set of new methods and tools for enterprise risk capital management and portfolio optimisation[END_REF]. Using their definition, [START_REF] Artzner | Coherent measures of risk[END_REF] proposed the tail value at risk (TVaR), also called expected shortfall (ES), as a coherent alternative to the non-coherent risk measure value at risk (VaR). Applied to continuous random variables, the TVaR can identically be defined as the conditional tail expectation (CTE). But these two risk measures differ in discrete contexts where the CTE is no longer coherent. The differences between these definitions and properties have been highlighted in [START_REF] Acerbi | Expected shortfall as a tool for financial risk management[END_REF] and [START_REF] Acerbi | On the coherence of expected shortfall[END_REF].

In the literature on capital allocation, continuous situations are widely studied in contrast with discrete cases. That is why most of the references speak in terms of CTE. The capital 1 allocation principle has first been introduced by [START_REF] Tasche | Risk contributions and performance measurement[END_REF] where the capital allocated to each risk is expressed in terms of the CTE of the aggregate risk. This top down allocation method has then been used to provide several closed formulae and approximations of the CTE and the CTE-based allocations for different types of multivariate continuous distributions. The first multivariate top down model was considered by [START_REF] Panjer | Measurement of risk, solvency requirements and allocation of capital within financial conglomerates[END_REF] where the risks have a multivariate normal distribution. This work has been extended to a multivariate elliptical distribution in [START_REF] Landsman | Tail conditional expectations for elliptical distributions[END_REF] and in [START_REF] Dhaene | Some results on the cte-based capital allocation rule[END_REF]. A multivariate gamma distribution for risks has been studied in [START_REF] Furman | Risk capital decomposition for a multivariate dependent gamma portfolio[END_REF] as well as a multivariate Tweedie distribution in [START_REF] Furman | Economic capital allocations for non-negative portfolios of dependent risks[END_REF]. In these papers, explicit expressions for the CTE and the CTE-based allocation are derived. Other closed form expressions for the CTE of the sum of multivariate phase-type distributed risks and the contribution of one risk to the portfolio have been given in [START_REF] Cai | Conditional tail expectations for multivariate phase-type distributions[END_REF]. More recently, [START_REF] Chiragiev | Multivariate pareto portfolios: Tce-based capital allocation and divided differences[END_REF] found a CTE and CTE-based allocation for multivariate Pareto risks. Further information on the CTE-based allocation of risk capital can be found in [START_REF] Kim | Estimation and allocation of insurance risk capital[END_REF].

In most papers mentioned above, the dependence between the different lines of business of the insurance company is due to the construction of a multivariate distribution. In the present paper, we propose introducing dependence with a copula. Copulas are currently seen as effective and flexible tools to represent dependence between random variables. Furthermore, in order to have the risk measure coherence property in every continuous and discrete situation, we propose using the TVaR as defined in [START_REF] Acerbi | Expected shortfall as a tool for financial risk management[END_REF] and [START_REF] Acerbi | On the coherence of expected shortfall[END_REF] to develop a top down approach of the capital allocation. Indeed, the Committee of European Insurance and Occupational Pensions Supervisors (CEIOPS) advises in the Solvency II context the use of the TVaR for the evaluation of the Solvency Capital Requirement (SCR); see CEIOPS (2006) and CEIOPS (2007).

First, closed form expressions for the TVaR and then the TVaR-based contribution of one risk over the aggregation of all risks are obtained when the Farlie-Gumbel-Morgenstern copula describes the dependence between the risk marginals. With most copulas introducing dependence between different risks however, we are not able to reach closed form expressions. Consequently, we also present approximation methods to evaluate the TVaR and the TVaR-based allocation by the use of different discretization methods of continuous random variables which are applicable with any copula and any marginals.

In the first section, we give the general definitions for the tail value at risk of the aggregate risk and the contribution of one of the risks. The second section deals with the application of the TVaR-based allocation rule using the FGM copula and exponential distributed risks. We first consider two lines of business and then pursue to a multivariate context. We widen our results to risks that are distributed as mixture of exponentials in section 3. For these two last sections, we are able to have closed form expressions for both the TVaR and the individual risk contribution based on it. Then, we expose approximation methods for the TVaR and TVaR-based allocation when the dependence structure is defined by any copula. The results are illustrated with numerical applications.

Definition of the TVaR and the TVaR-based allocation

In this section, we define the tail value at risk (TVaR) for the aggregate risk and the TVaR-based allocation rule. We consider the aggregate claim amount (or loss) S of a portfolio of n risks. The claim amount (or loss) for risk i is denoted by X i . Thus we have S = X 1 + ... + X n where all X i 's are non-negative random variables.

The value at risk at level κ, 0 < κ < 1, of S is defined by

V aR κ (S) = inf (x ∈ R, F S (x) ≥ κ).
It is well known that the VaR is a risk measure that is not coherent. Thus we choose to work with the tail value at risk of S as introduced in Acerbi and [START_REF] Acerbi | On the coherence of expected shortfall[END_REF], [START_REF] Schmock | Allocation of risk capital and performance measurement[END_REF] and [START_REF] Schmock | Modelling dependent credit risks with extensions of creditrisk+ and application to operational risk[END_REF] at level κ, for κ ∈ (0, 1). Its definition is

T V aR κ (S) = 1 1 -κ ∫ 1 κ V aR u (S)du = E [ S1 {S>V aRκ(S)} ] + V aR κ (S) ( P r (S ≤ V aR κ (S)) -κ ) 1 -κ ,
which is a coherent risk measure. When S is continuous, P r(S ≤ V aR κ (S)) = κ which implies that the TVaR is exactly the conditional tail expectation (CTE) meaning

T V aR κ (S) = E[S|S > V aR κ (S)] = CT E(S).
In financial risk management, the TVaR is called the expected shortfall.

The additivity of the expectation allows the decomposition of the TVaR (CTE) into the sum of TVaR contributions as follows

T V aR κ (S) = n ∑ i=1 T V aR κ (X i ; S),
where the TVaR contribution of the ith risk to the total risk represents the part of the capital that is allocated to risk i. For κ ∈ (0, 1), it can be expressed as

T V aR κ (X i ; S) = E [ X i × 1 {S>V aRκ(S)} ] + β S E [ X i × 1 {S=V aRκ(S)} ] 1 -κ , with β S = { P r(S≤V aRκ(S))-κ P r(S=V aRκ(S)) , if P r (S = V aR κ (S)) > 0, 0, otherwise. 3 For continuous distributions, we have T V aR κ (X i ; S) = 1 1 -κ E [ X i × 1 {S>V aRκ(S)} ] = E[X i | S > V aR κ (S)] = CT E κ (X i |S).
That means that the TVaR-based contribution of one risk is equal to the CTE-based contribution of the same risk when all the marginals are continuous.

Note that

E[X i | S = s] = E[X i | X 1 + ... + X n = s] = ∫ s 0 xf X i |S (x | S = s)dx = ∫ s 0 x f X i ,S (x, s) f S (s) dx.
Then, the CTE-based contribution can be expressed as

CT E κ (X i |S) = ∫ ∞ V aRκ(S) E[X i | S = s]f S|S>V aRκ(S) (s)ds = 1 P r(S > V aR κ (S)) ∫ ∞ V aRκ(S) E[X i | S = s]f S (s)ds = 1 1 -κ ∫ ∞ V aRκ(S) ∫ s 0 xf X i ,S (x, s)dxds.
(1)

3 TVaR and the TVaR-based allocation with exponential marginals and the FGM copula

In this section, we derive the expression for the TVaR and the TVaR-based allocation for two exponentially distributed risks joined by a FGM copula. We also extend the results obtained with this bivariate model to a multivariate model. The exponential distribution is a classical distribution for the risk random variables. Its convenient and practical mathematic properties permit to develop explicit results. We are aware that the FGM copula introduces only light dependence. However, it admits positive as well as negative dependence between a set of random variables. As said in [START_REF] Yeo | Claim dependence with common effects in credibility models[END_REF] where the FGM copula is used to link claim variables in a credibility model, even if it can model only weak dependence, the FGM copula permits to assign a unique dependence parameter for each pair or group of risks and allows a more complex dependence structure than most of the copulas which use only one or few parameters. Furthermore, its handy form allows explicit calculus and thus exact results. This copula was also used to describe different correlation relations on the financial markets in [START_REF] Gatfaoui | How does systematic risk impact us credit spreads ? a copula study[END_REF] and [START_REF] Gatfaoui | Credit default swap spreads and u.s. financial market: Investigating some dependence[END_REF].

The bivariate case

Let X 1 and X 2 be two exponentially distributed random variables representing the claim amounts of two insurance risks. Their cumulative distribution functions (cdf) and probability density functions (pdf) are given by

F X i (x i ) = 1 -e -λ i x i , f X i (x i ) = λ i e -λ i x i , for i = 1, 2.
In order to simplify our presentation, we restrain our study to the constraints

λ 1 ̸ = λ 2 , λ 1 ̸ = 2λ 2 , λ 2 ̸ = 2λ 1 .
It is possible to find adjusted results without these constraints by applying a similar method as the one exposed below.

A dependence structure for (X 1 , X 2 ) based on the bivariate FGM copula is introduced. The FGM copula is defined by

C F GM θ (u 1 , u 2 ) = u 1 u 2 + θu 1 u 2 (1 -u 1 )(1 -u 2 ) for u i ∈ [0, 1], i = 1, 2, and dependence parameter θ ∈ [-1, 1].
The density of the bivariate FGM copula is

c F GM θ (u 1 , u 2 ) = ∂ 2 C F GM θ (u 1 , u 2 ) ∂u 1 ∂u 2 = 1 + θ(1 -2u 1 )(1 -2u 2 ) = 1 + θ(2u 1 -1)(2u 2 -1),
where

u i = 1 -u i , i = 1, 2.
The TVaR of the aggregate risk S = X 1 + X 2 is given in the following proposition.

Proposition 1 Let X 1 and X 2 be two exponentially distributed random variables with joint cdf defined by a bivariate FGM copula as follows

F X 1 ,X 2 (x 1 , x 2 ) = C F GM θ (F X 1 (x 1 ), F X 2 (x 2 )), with θ ∈ [-1, 1]. Then, the TVaR of the aggregate risk S = X 1 + X 2 at level κ, 0 < κ < 1, is T V aR κ (S) = 1 1 -κ [ (1 + θ)ζ(V aR κ (S); λ 1 ; λ 2 ) -θζ(V aR κ (S); 2λ 1 ; λ 2 ) -θζ(V aR κ (S); λ 1 ; 2λ 2 ) + θζ(V aR κ (S); 2λ 1 ; 2λ 2 ) ] ,
(2)

where ζ(x; γ 1 , γ 2 ) = γ 2 γ 2 -γ 1 e -γ 1 x ( x + 1 γ 1 ) + γ 1 γ 1 -γ 2 e -γ 2 x ( x + 1 γ 2
) .

Proof. The joint pdf of (X 1 , X 2 ) is given by

f X 1 ,X 2 (x 1 , x 2 ) = c F GM θ (F X 1 (x 1 ), F X 2 (x 2 ))f X 1 (x 1 )f X 2 (x 2 ) = f X 1 (x 1 )f X 2 (x 2 ) + θf X 1 (x 1 )f X 2 (x 2 )(1 -2F X 1 (x 1 ))(1 -2F X 2 (x 2 )) = (1 + θ)λ 1 e -λ 1 x 1 λ 2 e -λ 2 x 2 -θ2λ 1 e -2λ 1 x 1 λ 2 e -λ 2 x 2 -θλ 1 e -λ 1 x 1 2λ 2 e -2λ 2 x 2 + θ2λ 1 e -2λ 1 x 1 2λ 2 e -2λ 2 x 2 .
Let h(x, λ 1 , λ 2 ) be the distribution function of a generalized Erlang random variable

X h(x, λ 1 , λ 2 ) = λ 1 λ 2 λ 2 -λ 1 e -λ 1 x + λ 1 λ 2 λ 1 -λ 2 e -λ 2 x .
Then, the pdf of S can be expressed as a combination of generalized Erlang pdf's

f S (s) = ∫ s 0 f X 1 ,S (x, s)dx = ∫ s 0 f X 1 ,X 2 (x, s -x)dx = (1 + θ)h(s; λ 1 ; λ 2 ) -θh(s; 2λ 1 ; λ 2 ) -θh(s; λ 1 ; 2λ 2 ) + θh(s; 2λ 1 ; 2λ 2 ), (3)
where the first component of f S (s) is

∫ s 0 (1 + θ)λ 1 e -λ 1 x λ 2 e -λ 2 (s-x) dx = (1 + θ)λ 1 λ 2 e -λ 2 s ∫ s 0 e x(λ 2 -λ 1 ) dx = (1 + θ) λ 1 λ 2 λ 2 -λ 1 e -λ 1 s + (1 + θ) λ 1 λ 2 λ 1 -λ 2 e -λ 2 s = (1 + θ)h(s; λ 1 ; λ 2 ).
The three other components can be found similarly.

Thus, the TVaR of S takes the form

T V aR κ (S) = E[S | S > V aR κ (S)] = ∫ ∞ V aRκ s f S (s) P r(S > V aR κ (S)) ds = 1 1 -F S (V aR κ (S)) ∫ ∞ V aRκ(S) sf S (s)ds = 1 1 -κ ∫ ∞ V aRκ(S) s [ (1 + θ)h(s; λ 1 ; λ 2 ) -θh(s; 2λ 1 ; λ 2 ) -θh(s; λ 1 ; 2λ 2 ) + θh(s; 2λ 1 ; 2λ 2 ) ] ds. (4) Define ∫ ∞ V aRκ(S) sh(s; λ 1 ; λ 2 )ds = ∫ ∞ V aRκ(S) s ( λ 1 λ 2 λ 2 -λ 1 e -λ 1 s + λ 1 λ 2 λ 1 -λ 2 e -λ 2 s ) ds = λ 1 λ 2 λ 2 -λ 1 { [ -s e -λ 1 s λ 1 ] ∞ V aRκ(S) + ∫ ∞ V aRκ(S) e -λ 1 s λ 1 ds } + λ 1 λ 2 λ 1 -λ 2 { [ -s e -λ 2 s λ 2 ] ∞ V aRκ(S) + ∫ ∞ V aRκ(S) e -λ 2 s λ 2 ds } = λ 1 λ 2 λ 2 -λ 1 ( V aR κ (S) e -λ 1 V aRκ(S) λ 1 + e -λ 1 V aRκ(S) λ 2 1 ) + λ 1 λ 2 λ 1 -λ 2 ( V aR κ (S) e -λ 2 V aRκ(S) λ 2 + e -λ 2 V aRκ(S) λ 2 2 ) = λ 2 λ 2 -λ 1 e -λ 1 V aRκ(S) ( V aR κ (S) + 1 λ 1 ) + λ 1 λ 1 -λ 2 e -λ 2 V aRκ(S) ( V aR κ (S) + 1 λ 2 ) = ζ(V aR κ (S); λ 1 , λ 2 ). ( 5
)
Inserting ( 5) in ( 4), we obtain

T V aR κ (S) = 1 1 -κ [ (1 + θ)ζ(V aR κ (S); λ 1 ; λ 2 ) -θζ(V aR κ (S); 2λ 1 ; λ 2 ) -θζ(V aR κ (S); λ 1 ; 2λ 2 ) + θζ(V aR κ (S); 2λ 1 ; 2λ 2 ) ] , with ζ(x; γ 1 , γ 2 ) = γ 2 γ 2 -γ 1 e -γ 1 x ( x + 1 γ 1 ) + γ 1 γ 1 -γ 2 e -γ 2 x ( x + 1 γ 2
) .

A closed form expression for the TVaR-based capital attributed to risk i, i = 1, 2, is given in the next proposition.

Proposition 2 Let X 1 and X 2 be two exponentially distributed random variables with joint cdf defined by a bivariate FGM copula. Then, the TVaR-based contribution of risk i

, i = 1, 2, to the aggregate risk S = X 1 + X 2 at level κ, 0 < κ < 1, is T V aR κ (X i ; S) = 1 1 -κ [ (1 + θ)ξ (V aR κ (S); λ i ; λ j ) -θξ (V aR κ (S); 2λ i ; λ j ) -θξ (V aR κ (S); λ i ; 2λ j ) + θξ (V aR κ (S); 2λ i ; 2λ j ) ] , ( 6
)
where ξ (x;

γ i ; γ j ) = γ j e -γ i x (x+ 1 γ i ) γ j -γ i - γ j e -γ i x -γ i e -γ j x (γ j -γ i ) 2
and i ̸ = j.

Proof. Let i = 1 and j = 2. Recall that for continuous random variables the TVaR-based allocation is equal to the CTE-based allocation. From (1), we have

T V aR κ (X 1 ; S) = E[X 1 | S > V aR κ (S)] = 1 P r(S > V aR κ (S)) ∫ ∞ V aRκ(S) ∫ s 0 xf X 1 ,S (x, s)dxds, where ∫ s 0 xf X 1 ,S (x, s)dx = ∫ s 0 xf X 1 ,X 2 (x, s -x)dx = ∫ s 0 x ( (1 + θ)λ 1 e -λ 1 x λ 2 e -λ 2 (s-x) -θ2λ 1 e -λ 1 x λ 2 e -λ 2 (s-x) -θλ 1 e -λ 1 x 2λ 2 e -λ 2 (s-x) + θ2λ 1 e -λ 1 x 2λ 2 e -λ 2 (s-x)
) dx

= (1 + θ)λ 1 λ 2 ( se -λ 1 s λ 2 -λ 1 - e -λ 1 s -e -λ 2 s (λ 2 -λ 1 ) 2 ) -θ2λ 1 λ 2 ( se -2λ 1 s λ 2 -2λ 1 - e -2λ 1 s -e -λ 2 s (λ 2 -2λ 1 ) 2 ) -θλ 1 2λ 2 ( se -λ 1 s 2λ 2 -λ 1 - e -λ 1 s -e -2λ 2 s (2λ 2 -λ 1 ) 2 ) +θ2λ 1 2λ 2 ( se -2λ 1 s 2λ 2 -2λ 1 - e -2λ 1 s -e -2λ 2 s (2λ 2 -2λ 1 ) 2
) .

The first component on the right-hand side of the last equality is given by

∫ s 0 x(1 + θ)λ 1 e -λ 1 x λ 2 e -λ 2 (s-x) dx = (1 + θ)λ 1 λ 2 e -λ 2 s ∫ s 0 xe x(λ 2 -λ 1 ) dx = (1 + θ)λ 1 λ 2 e -λ 2 s ([ x e x(λ 2 -λ 1 ) λ 2 -λ 1 ] s 0 - ∫ s 0 e x(λ 2 -λ 1 ) λ 2 -λ 1 dx ) = (1 + θ)λ 1 λ 2 e -λ 2 s ( s e s(λ 2 -λ 1 ) λ 2 -λ 1 - [ e x(λ 2 -λ 1 ) (λ 2 -λ 1 ) 2 ] s 0 ) = (1 + θ)λ 1 λ 2 e -λ 2 s ( s e sλ 2 e -sλ 1 λ 2 -λ 1 - e sλ 2 e -sλ 1 -1 (λ 2 -λ 1 ) 2 ) = (1 + θ)λ 1 λ 2 ( se -λ 1 s λ 2 -λ 1 - e -λ 1 s -e -λ 2 s (λ 2 -λ 1 ) 2
) .

Denoting V = V aR κ (S), we have

T V aR κ (X 1 ; S) = 1 1 -F S (V ) ∫ ∞ V { (1 + θ)λ 1 λ 2 ( se -λ 1 s λ 2 -λ 1 - e -λ 1 s -e -λ 2 s (λ 2 -λ 1 ) 2 ) -θ2λ 1 λ 2 ( se -2λ 1 s λ 2 -2λ 1 - e -2λ 1 s -e -λ 2 s (λ 2 -2λ 1 ) 2 ) -θλ 1 2λ 2 ( se -λ 1 s 2λ 2 -λ 1 - e -λ 1 s -e -2λ 2 s (2λ 2 -λ 1 ) 2 ) +θ2λ 1 2λ 2 ( se -2λ 1 s 2λ 2 -2λ 1 - e -2λ 1 s -e -2λ 2 s (2λ 2 -2λ 1 ) 2 ) } ds = 1 1 -F S (V ) [ (1 + θ) ( λ 2 e -λ 1 V (V + 1 λ 1 ) λ 2 -λ 1 - λ 2 e -λ 1 V -λ 1 e -λ 2 V (λ 2 -λ 1 ) 2 ) -θ ( λ 2 e -2λ 1 V (V + 1 2λ 1 ) λ 2 -2λ 1 - λ 2 e -2λ 1 V -2λ 1 e -λ 2 V (λ 2 -2λ 1 ) 2 ) -θ ( 2λ 2 e -λ 1 V (V + 1 λ 1 ) 2λ 2 -λ 1 - 2λ 2 e -λ 1 V -λ 1 e -2λ 2 V (2λ 2 -λ 1 ) 2 ) +θ ( 2λ 2 e -2λ 1 V (V + 1 2λ 1 ) 2λ 2 -2λ 1 - 2λ 2 e -2λ 1 V -2λ 1 e -2λ 2 V (2λ 2 -2λ 1 ) 2 ) ]
.

We finally obtain

T V aR κ (X 1 ; S) = 1 1 -κ [ (1 + θ)ξ (V aR κ (S); λ 1 ; λ 2 ) -θξ (V aR κ (S); 2λ 1 ; λ 2 ) -θξ (V aR κ (S); λ 1 ; 2λ 2 ) + θξ (V aR κ (S); 2λ 1 ; 2λ 2 ) ] , where ξ (x; γ 1 ; γ 2 ) = γ 2 e -γ 1 x (x + 1 γ 1 ) γ 2 -γ 1 - γ 2 e -γ 1 x -γ 1 e -γ 2 x (γ 2 -γ 1 ) 2 .
The TVaR-based allocation for the second risk is symmetrically given by

T V aR κ (X 2 ; S) = 1 1 -κ [ (1 + θ)ξ (V aR κ (S); λ 2 ; λ 1 ) -θξ (V aR κ (S); 2λ 2 ; λ 1 ) -θξ (V aR κ (S); λ 2 ; 2λ 1 ) + θξ (V aR κ (S); 2λ 2 ; 2λ 1 ) ] , where ξ (x; γ 2 ; γ 1 ) = γ 1 e -γ 2 x (x + 1 γ 2 ) γ 1 -γ 2 - γ 1 e -γ 2 x -γ 2 e -γ 1 x (γ 1 -γ 2 ) 2 .
Remark 3 It can be verified that the TVaR of S is the sum of the risk contributions

T V aR κ (S) = 2 ∑ i=1 T V aR κ (X i ; S).
Explicit expressions for the TVaR and the one risk TVaR-based contribution cannot only be obtained in the bivariate case but even for an undefined number of risks.

The multivariate case

Suppose now that there are n different exponential risks joined by a multivariate FGM copula characterized by

C (u 1 , u 2 , ..., u n ) = u 1 u 2 ...u n   1 + n ∑ k=2 ∑ 1≤j 1 <...<j k ≤n θ j 1 j 2 ...j k u j 1 u j 2 ...u j k   ,
where u (.) = 1 -u (.) (see [START_REF] Nelsen | An introduction to copulas[END_REF] p.108). We have here 2 n -n -1 copula parameters to describe the dependence between each pair or group of risks. Its density can be written as

c (u 1 , u 2 , ..., u n ) = 1 + n ∑ k=2 ∑ 1≤j 1 <...<j k ≤n θ j 1 j 2 ...j k (2u j 1 -1)(2u j 2 -1)...(2u j k -1).
As in the bivariate case, we suppose that the parameters of the n exponential risks satisfy the conditions below

λ i ̸ = λ j and λ i ̸ = 2λ j for i ̸ = j. (7)
Then, the following proposition holds.

Proposition 4 Let S n = X 1 +X 2 +...+X n be the sum of n dependent exponential random variables with joint cdf defined by a multivariate FGM copula as follows

F X 1 ,...,Xn (x 1 , ..., x n ) = C F GM θ (F X 1 (x 1 ), ..., F Xn (x n )) with θ ∈ [-1, 1]. Then, the TVaR of S n at level κ, 0 < κ < 1, is T V aR κ (S n ) = 1 1 -κ × [ 1 + n ∑ k=2 ∑ 1≤j 1 <...<j k ≤n θ j 1 j 2 ...j k ×   k ∑ l=0 ∑ (a 1 ,...,a k )∈A l,k (-1) l ζ ( V aR κ (S n ); 2 i 1 λ 1 , ..., 2 i k λ k , λ i k+1 , ...λ in )   ] , (8) where ζ(x; γ 1 , ..., γ n ) = ∑ n i=1 ( ∏ n j=1,j̸ =i γ j γ j -γ i ) e -γ i x ( x + 1 γ i
) , i k+1 , ..., i n are the missing indexes of j 1 , ..., j k to complete 1, ..., n and A l,k are the sets of k-tuples composed of l zeros and (k -l) ones, for l = 0, 1, ..., k and k = 2, ..., n.

In fact, the A l,k are defined by

A 0,k = { (1, 1, ..., 1) 1×k } , A 1,k = { (1, 1, ..., 0) 1×k , ..., (0, 1, ..., 1) 1×k } , A 2,k = { (1, 1, ..., 0, 0) 1×k , ..., (0, 0, ..., 1) 1×k } , ... , A k,k = { (0, 0, ..., 0) 1×k } . Proof. The joint pdf of (X 1 , X 2 , ..., X n ) is f X 1 ,X 2 ,...,Xn (x 1 , x 2 , ..., x n ) = c(F X 1 (x 1 ), F X 2 (x 2 ), ..., F Xn (x n ))f X 1 (x 1 )f X 2 (x 2 )...f Xn (x n ) = f X 1 (x 1 )f X 2 (x 2 )...f Xn (x n ) × [ 1 + n ∑ k=2 ∑ 1≤j 1 <...<j k ≤n θ j 1 j 2 ...j k (1 -F X j 1 (x j 1 ))(1 -F X j 2 (x j 2 ))...(1 -F X j k (x j k ))
] .

As the n risk random variables are exponentially distributed with parameters λ i , i = 1, ..., n with constraint (7), we have

f X 1 ,X 2 ,...,Xn (x 1 , x 2 , ..., x n ) = λ 1 e -λ 1 x 1 λ 2 e -λ 2 x 2 ...λ n e -λnxn × [ 1 + n ∑ k=2 ∑ 1≤j 1 <...<j k ≤n θ j 1 j 2 ...j k (2e -λ j 1 x j 1 -1)(2e -λ j 2 x j 2 -1)...(2e -λ jn x jn -1) ] = ν(x 1 , x 2 , ..., x n ; λ 1 , λ 2 , ..., λ n ) + n ∑ k=2 ∑ 1≤j 1 <...<j k ≤n θ j 1 j 2 ...j k × [ k ∑ l=0 ∑ (a 1 ,...,a k )∈A l,k (-1) l ν(x j 1 , x j 2 , ..., x j k , x i k+1 , ..., x in ; 2 a 1 λ j 1 , 2 a 2 λ j 2 , ..., 2 a k λ j k , λ i k+1 , ..., λ in ) ] , ( 9
)
where i k+1 , ..., i n and A l,k are defined as in the proposition and

ν(x 1 , ..., x n ; γ 1 , ..., γ n ) = γ 1 e -γ 1 x 1 × γ 2 e -γ 2 x 2 × ... × γ n e -γnxn .
Given that

f Sn (s) = ∫ s 0 ∫ s-x 1 0 ... ∫ s-x 1 -...-x n-1 0 f X 1 ,X 2 ,...,X n-1 ,Xn (x 1 , x 2 , ..., x n-1 , s -x 1 -x 2 -... -x n-1 )dx 1 dx 2 ...dx n-1 = ∫ s 0 ∫ s-x 1 0 ... ∫ s-x 1 -...-x n-2 0 f X 1 ,X 2 ,...,X n-2 ,X n-1 +Xn (x 1 , x 2 , ..., x n-2 , s -x 1 -x 2 -... -x n-2 )dx 1 dx 2 ...dx n-2 = ... = ∫ s 0 f X 1 ,X 2 +...+Xn (x 1 , s -x 1 )dx 1 and that ∫ s 0 ∫ s-x 1 0 ... ∫ s-x 1 -...-x n-1 0 ν(x 1 , x 2 , ..., x n ; γ 1 , γ 2 , ..., γ n )dx 1 dx 2 ...dx n-1 = n ∑ i=1   n ∏ j=1,j̸ =i γ j γ j -γ i   γ i e -γ i s = h(s; γ 1 , γ 2 , ...γ n ), one can write f Sn (s) = h (s; λ 1 , ..., λ n ) + n ∑ k=2 ∑ 1≤j 1 <...<j k ≤n θ j 1 j 2 ...j k   k ∑ l=0 ∑ (a 1 ,...,a k )∈A l,k (-1) l h ( s; 2 a 1 λ j 1 , 2 a 2 λ j 2 , ..., 2 a k λ j k , λ i k+1 , ...λ in )   .
Then, the TVaR of S n for n ≥ 2 and 0 < κ < 1 is

T V aR κ (S n ) = E[S n | S n > V aR κ (S n )] = ∫ ∞ V aRκ s f Sn (s) P r(S n > V aR κ (S n )) ds = 1 1 -κ ∫ ∞ V aRκ(Sn) sf Sn (s)ds = 1 1 -κ × [ 1 + n ∑ k=2 ∑ 1≤j 1 <...<j k ≤n θ j 1 j 2 ...j k ×   k ∑ l=0 ∑ (a 1 ,...,a k )∈A l,k (-1) l ζ ( V aR κ (S n ); 2 i 1 λ 1 , ..., 2 i k λ k , λ i k+1 , ...λ in )   ] ,
where

ζ(x; γ 1 , ..., γ n ) = n ∑ i=1   n ∏ j=1,j̸ =i γ j γ j -γ i   e -γ i x ( x + 1 γ i ) .
As in the bivariate case, the capital allocation for risk i can also be explicitly given. In order to find that expression, we first need to introduce the nth order divided difference of a function f as in [START_REF] Chiragiev | Multivariate pareto portfolios: Tce-based capital allocation and divided differences[END_REF]. Consider x 1 , x 2 , ..., x n , x n+1 arbitrary points such that x i ̸ = x j for i ̸ = j. The nth order divided difference of f on x 1 , x 2 , ..., x n , x n+1 is defined by

f (x 1 , x 2 , ..., x n , x n+1 ) = n+1 ∑ i=1 f (x i ) ∏ j̸ =i (x i -x j )
.

The following proposition gives the expression of the TVaR-based allocation.

Proposition 5 Let X 1 , ..., X n be n exponentially distributed random variables with joint cdf defined by a multivariate FGM copula. Then, the TVaR-based contribution of risk i, i = 1, ..., n, to the sum

S n = X 1 + ... + X n at level κ, 0 < κ < 1, is T V aR κ (X i ; S n ) = (-1) n-1 Λ 1 -κ [ H i (V aR κ (S n ); λ 1 ; ...; λ n ) + n ∑ k=2 ∑ 1≤j 1 <...<j k ≤n ∩ i∈{j 1 ,...,j k } θ j 1 ...j k × ( k-1 ∑ l=0 ∑ (a1,...,a k ) -i ∈A l,k-1 { (-1) l+1 2 k-1-l H i (V aR κ (S n ); 2 a 1 λ j 1 ; ...; 2 a k λ j k ; λ i k+1 ; ...λ in ) +(-1) l 2 k-1-l G i (V aR κ (S n ); 2 a 1 λ j 1 ; ...; 2 a k λ j k ; λ i k+1 ; ...λ in ) } ) + n-1 ∑ k=2 ∑ 1≤j 1 <...<j k ≤n ∩ i / ∈{j 1 ,...,j k } θ j 1 ...j k × ( k ∑ l=0 ∑ (a1,...,a k )∈A l,k (-1) l 2 k-l H i (V aR κ (S n ); 2 a 1 λ j 1 ; ...; 2 a k λ j k ; λ i k+1 ; ...λ in ) )] , ( 10
)
where

H i (x; γ 1 , ..., γ i-1 , γ i+1 , ..., γ n ) and G i (x; γ 1 , ..., γ i-1 , γ i+1 , ..., γ n ) are the (n -2)-th order di- vided differences of respectively H i (x; γ) = xe -γ i x γ i (γ i -γ) + e -γ i x γ 2 i (γ i -γ) + γe -γ i x -γ i e -γx γγ i (γ i -γ) 2 and G i (x; γ) = 2 ( xe -2γ i x 2γ i (2γ i -γ) + e -2γ i x (2γ i ) 2 (2γ i -γ) + γe -2γ i x -2γ i e -γx γ2γ i (2γ i -γ) 2
) , Λ = λ 1 × ... × λ n is the product of the parameters of exponential distributions, i k+1 , ..., i n are the missing indexes of j 1 , ..., j k to complete 1, ..., n and A l,k are the sets of k-tuples composed of l zeros and (k -l) ones, for l = 0, 1, ..., k and k = 2, ..., n.

Proof. The capital attributed to the continuous distributed risk i can be expressed as

T V aR κ (X i ; S n ) = CT E κ (X i |S n ) = 1 F Sn (V aR κ (S n )) ∫ ∞ V aRκ(Sn) ∫ s 0 x i f X i ,Sn (x i , s)dxds. (11) A recursive formula for f X i ,Sn (x i , s) = f X i ,Sn-X i (x i , s -x i
) is needed to evaluate this expression. Given that the risk random variables here are not independent, we cannot directly separate

f X i ,Sn-X i (x i , s -x i ) into the product of f X i (x i ) and f S n-1 (s -x i ).
First, we have

f X i ,Sn-X i (x i , s -x i ) = f X i ,X 1 +X 2 +...+X i-1 +X i+1 +...+Xn (x i , s -x i ) = ∫ s-x i 0 ∫ s-x i -x 1 0 ∫ s-x i -x 1 -x 2 0 ... ∫ s-x 1 -...-x n-1 0 f X 1 ,X 2 ,...,X n-1 ,Xn (x 1 , x 2 , ..., x n-1 , s -x 1 -... -x n-1 ) × dx 1 dx 2 ...dx i-1 dx i+1 ...dx n-1
with f X 1 ,X 2 ,...,Xn (x 1 , x 2 , ..., x n ) defined as in ( 9) and which can be extended to

f X 1 ,X 2 ,...,Xn (x 1 , x 2 , ..., x n ) = f X i (x i ) ( ν -i (x 1 ; x 2 ; ...; x n ; λ 1 ; λ 2 ; ...; λ n ) + n ∑ k=2 ∑ 1≤j 1 <...<j k ≤n ∩ i∈{j 1 ,...,j k } θ j 1 j 2 ...j k (1 -2F X i (x i )) × [ k ∑ l=0 ∑ (a 1 ,...,a k ) -i ∈A l,k-1
(-1) l ν -i (x j 1 ; x j 2 ; ...; x j k ; x i k+1 ; ...; x in ; 2 a 1 λ j 1 ; 2 a 2 λ j 2 ; ...;

2 a k λ j k ; λ i k+1 ; ...; λ in ) ] + n ∑ k=2 ∑ 1≤j 1 <...<j k ≤n ∩ i / ∈{j 1 ,...,j k } θ j 1 j 2 ...j k × [ k ∑ l=0 ∑ (a 1 ,...,a k ) -i ∈A l,k-1
(-1) l ν -i (x j 1 , x j 2 ; ...; x j k ; x i k+1 ; ...; x in ; 2 a 1 λ j 1 ; 2 a 2 λ j 2 ; ...;

2 a k λ j k ; λ i k+1 ; ...; λ in ) ] ,
where i k+1 , ..., i n are the missing indexes of j 1 , ..., j k to complete 1, ..., n, A 0

,k = { (1, 1, .., 1) 1×k } , A 1,k = { (1, 1, .., 0) 1×k , ..., (0, 1, .., 1) 1×k } , A 2,k = { (1, 1, .., 0, 0) 1×k , ..., (0, 0, .., 1) 1×k } , ... , A k,k = { (0, 0, .., 0) 1×k } and
ν -i (x 1 ; x 2 ; ...; x n ; γ 1 ; γ 2 ; ...; γ n ) = γ 1 e -γ 1 x 1 ×γ 2 e -γ 2 x 2 ×...×γ i-1 e -γ i-1 x i-1 ×γ i+1 e -γ i+1 x i+1 ×...×γ n e -γnxn .
As in the proof of the Proposition 4, we use the fact that

∫ s-x i 0 ∫ s-x i -x 1 0 ∫ s-x i -x 1 -x 2 0 ... ∫ s-x 1 -...-x n-1 0 ν -i (x 1 ; x 2 ; ...; x n ; γ 1 ; γ 2 ; ...; γ n )dx 1 dx 2 ...dx i-1 dx i+1 ...dx n-1 = n ∑ j=1 j̸ =i    n ∏ k=1,k̸ =j k̸ =i γ k γ k -γ j    γ j e -γ j (s-x i ) = h -i (s -x i ; γ 1 , γ 2 , ...γ n ),
and we obtain

f X i ,Sn-X i (x i , s -x i ) = f X i (x i ) [ h -i (s -x i ; λ 1 , ..., λ i-1 , λ i+1 , ..., λ n ) + n ∑ k=2 ∑ 1≤j 1 <...<j k ≤n ∩ i∈{j 1 ,...,j k } θ j 1 j 2 ...j k (1 -2F X i (x)) ×   k ∑ l=0 ∑ (a 1 ,...,a k ) -i ∈A l,k-1 (-1) l h -i (s -x i ; 2 a 1 λ j 1 , ..., 2 a k λ j k , λ i k+1 , ...λ in )   + n ∑ k=2 ∑ 1≤j 1 <...<j k ≤n ∩ i / ∈{j 1 ,...,j k } θ j 1 j 2 ...j k ×   k ∑ l=0 ∑ (a 1 ,...,a k )∈A l,k (-1) l h -i (s -x i ; 2 a 1 λ j 1 , ..., 2 a k λ j k , λ i k+1 , ...λ in )   ] . ( 12
)
Using the divided difference as in [START_REF] Chiragiev | Multivariate pareto portfolios: Tce-based capital allocation and divided differences[END_REF], notice that

h -i (x; γ 1 , ..., γ i-1 , γ i+1 , ..., γ n ) = (-1) n-2 × γ 1 × ... × γ i-1 × γ i+1 × ...γ n × F (x; γ 1 ; ...; γ i-1 ; γ i+1 ; ...; γ n ),
where

F (x; γ 1 ; ...; γ i-1 ; γ i+1 ; ...; γ n ) is the (n -2)-th order divided difference of F (x; γ) = e -γx .
Then, we have

∫ s 0 x i f X i (x i )h -i (s -x i ; λ 1 , ..., λ i-1 , λ i+1 , ..., λ n )dx i = ∫ s 0 x i f X i (x i )(-1) n-2 Λ -i F (x i ; λ 1 ; ...; λ i-1 ; λ i+1 ; ...; λ n )dx i = (-1) n-1 ΛH i (s; λ 1 ; ...; λ i-1 ; λ i+1 ; ...; λ n ), ( 13
)
where

Λ -i = λ 1 × ... × λ i-1 × λ i+1 × ... × λ n , Λ = λ 1 × λ 2 × ... × λ n and H i (s; λ) = - ∫ s 0 x i e -λ i x i F (s -x i ; λ)dx i .
Let also

G i (s; λ) = - ∫ s 0 2x i e -2λ i x i F (s -x i ; λ)dx i .
Given that the risks here are exponentially distributed, H i and G i take the form

H i (s; λ) = - ∫ s 0 x i e -λ i x i F (s -x i ; λ)dx i = -e -λs ∫ s 0 x i e -(λ i -λ)x i ds = se -λ i s λ i -λ + e -λ i s (λ i -λ) 2 - e -λs (λ i -λ) 2 and G i (s; λ) = - ∫ s 0 2x i e -2λ i x i F (s -x i ; λ)dx i = -2e -λs ∫ s 0 x i e -(2λ i -λ)x i ds = 2 ( se -2λ i s 2λ i -λ + e -2λ i s (2λ i -λ) 2 - e -λs (2λ i -λ) 2
) .

Using ( 13), we can express

∫ s 0 x i f X i ,Sn (x i , s)dx i as ∫ s 0 xf X i ,Sn (x, s)dx = (-1) n-1 Λ [ H i (s; λ 1 ; ...; λ n ) + n ∑ k=2 ∑ 1≤j 1 <...<j k ≤n ∩ i∈{j 1 ,...,j k } θ j 1 ...j k × { k-1 ∑ l=0 ∑ (a1,...,a k ) -i ∈A l,k-1 ( (-1) l+1 2 k-1-l H i (s; 2 a 1 λ j 1 ; ...; 2 a k λ j k ; λ i k+1 ; ...λ in ) +(-1) l 2 k-1-l G i (s; 2 a 1 λ j 1 ; ...; 2 a k λ j k ; λ i k+1 ; ...λ in ) ) } + n-1 ∑ k=2 ∑ 1≤j 1 <...<j k ≤n ∩ i / ∈{j 1 ,...,j k } θ j 1 ...j k × { k ∑ l=0 ∑ (a1,...,a k )∈A l,k (-1) l 2 k-l H i (s; 2 a 1 λ j 1 ; ...; 2 a k λ j k ; λ i k+1 ; ...λ in ) } ] . ( 14
)
To calculate the risk contribution as in ( 11), the H i (s) and G i (s) terms in ( 14) must be integrated on s as follows

H i (V ; λ) = ∫ ∞ V H i (s; λ)ds = ∫ ∞ V ( se -λ i s λ i -λ + e -λ i s (λ i -λ) 2 - e -λs (λ i -λ) 2 ) ds = V e -λ i V λ i (λ i -λ) + e -λ i V λ 2 i (λ i -λ) + λe -λ i V -λ i e -λV λλ i (λ i -λ) 2
and

G i (V ; λ) = ∫ ∞ V G i (s; λ)ds = ∫ ∞ V 2 ( se -2λ i s 2λ i -λ + e -2λ i s (2λ i -λ) 2 - e -λs (2λ i -λ) 2 ) ds = 2 ( V e -2λ i V 2λ i (2λ i -λ) + e -2λ i V (2λ i ) 2 (2λ i -λ) + λe -2λ i V -2λ i e -λV λ2λ i (2λ i -λ) 2
) .

Finally, expression (11

) for T V aR κ (X i ; S n ) is obtained T V aR κ (X i ; S n ) = (-1) n-1 Λ 1 -κ [ H i (V aR κ (S n ); λ 1 ; ...; λ n ) + n ∑ k=2 ∑ 1≤j 1 <...<j k ≤n ∩ i∈{j 1 ,...,j k } θ j 1 ...j k × ( k-1 ∑ l=0 ∑ (a1,...,a k ) -i ∈A l,k-1 { (-1) l+1 2 k-1-l H i (V aR κ (S n ); 2 a 1 λ j 1 ; ...; 2 a k λ j k ; λ i k+1 ; ...λ in ) +(-1) l 2 k-1-l G i (V aR κ (S n ); 2 a 1 λ j 1 ; ...; 2 a k λ j k ; λ i k+1 ; ...λ in ) } ) + n-1 ∑ k=2 ∑ 1≤j 1 <...<j k ≤n ∩ i / ∈{j 1 ,...,j k } θ j 1 ...j k × ( k ∑ l=0 ∑ (a1,...,a k )∈A l,k (-1) l 2 k-l H i (V aR κ (S n ); 2 a 1 λ j 1 ; ...; 2 a k λ j k ; λ i k+1 ; ...λ in ) ) ]
.

Remark 6 To obtain the TVaR-based contribution ( 6) in the bivariate case from ( 10), we just have to use the equalities

H 1 (x; λ 2 ) = - 1 λ 1 λ 2 ξ(x; λ 1 ; λ 2 ), H 1 (x; 2λ 2 ) = - 1 λ 1 2λ 2 ξ(x; λ 1 ; 2λ 2 ), G 1 (x; λ 2 ) = -2 1 2λ 1 λ 2 ξ(x; 2λ 1 ; λ 2 ), G 1 (x; 2λ 2 ) = -2 1 2λ 1 2λ 2 ξ(x; 2λ 1 ; 2λ 2 ).

Numerical application

We illustrate here our results with a numerical example for the bivariate exponential case. Suppose that the parameters of the distributions of X 1 and X 2 are respectively λ 1 = 1/2 and λ 2 = 1/3. Let us calculate the VaR, TVaR and TVaR-based allocations for X 1 and X 2 for different risk levels κ and different FGM copula parameters θ. We write bellow the cumulative distribution function of S which can be expressed in the current case as a combination of generalized Erlang cdf's

F S (s) = (1 + θ)H(s; λ 1 ; λ 2 ) -θH(s; 2λ 1 ; λ 2 ) -θH(s; λ 1 ; 2λ 2 ) + θH(s; 2λ 1 ; 2λ 2 ), where H(s; λ 1 ; λ 2 ) = λ 2 λ 2 -λ 1 ( 1 -e -λ 1 s ) + λ 1 λ 1 -λ 2 ( 1 -e -λ 2 s )
is the cdf of a 2-generalized Erlang distribution with parameters (λ 1 , λ 2 ). The numerical results for the VaR, TVaR and TVaR-based allocations are displayed in Tables 1, 2 and3

. θ = -1 κ = 0.5 κ = 0.75 κ = 0.95 κ = 0.99 κ = 0.995 V aR κ (S)

TVaR and the TVaR-based allocation with mixtures of exponential marginals and the FGM copula

Let us consider now that we have two risks X 1 and X 2 which are distributed as a mixture of exponentials. Their cdf's and pdf's can be written as

F X 1 (x) = α 11 (1 -e -λ 11 x ) + α 12 (1 -e -λ 12 x ) F X 2 (x) = α 21 (1 -e -λ 21 x ) + α 22 (1 -e -λ 22 x ) f X 1 (x) = α 11 λ 11 e -λ 11 x + α 12 λ 12 e -λ 12 x f X 2 (x) = α 21 λ 21 e -λ 21 x + α 22 λ 22 e -λ 22 x ,
where we restrict our model to

λ 1i ̸ = λ 2j , λ 1i ̸ = 2λ 2j , λ 11 + λ 12 ̸ = λ 2j , λ 11 + λ 12 ̸ = 2λ 2j and λ 11 +λ 12 ̸ = λ 21 +λ 22 .
As for the exponential distribution case, the calculations can be done without these constraints but the results are not presented here. Mixtures of exponential distributions, also called hyper-exponential distributions, belong to the phase-type distribution family, see [START_REF] Neuts | Matrix-geometric solutions in stochastic models[END_REF] and [START_REF] Asmussen | Matrix-analytic models and their analysis[END_REF]. They can be used to approximate light-or heavy-tailed distributions with completely monotone pdf's and decreasing failure rates as shown in Feldmann and Whitt (1998), [START_REF] Keatinge | Modeling losses with the mixed exponential distribution[END_REF] and [START_REF] Khayari | Fitting world-wide web request traces with the em-algorithm[END_REF]. The practical form of the mixture of exponential distributions also permits explicit results.

The following two propositions can be proven similarly as Proposition 1 and Proposition 2 in the previous section given that a mixture of exponentials is just an extension of the exponential distribution.

Proposition 7 Let X 1 and X 2 be two random variables with mixture of exponential distributions and a joint cdf defined by a bivariate FGM copula as follows

F X 1 ,X 2 (x 1 , x 2 ) = C F GM θ (F X 1 (x 1 ), F X 2 (x 2 )) with θ ∈ [-1, 1]. Then, the TVaR of the aggregate risk S = X 1 + X 2 at level κ, 0 < κ < 1, is T V aR κ (S) = (I + J + K) × 1 1 -κ ,
where

I = 2 ∑ i=1 2 ∑ j=1 { [ α 1i α 2j + θ ( α 1i α 2j -2α 2 1i α 2j -2α 11 α 12 α 2j -2α 1i α 2 2j -2α 1i α 21 α 22 +4α 2 1i α 2 2j + 4α
S = X 1 + X 2 at level κ, 0 < κ < 1, is T V aR κ (X i ; S) = (L + M + N ) × 1 1 -κ , where L = 2 ∑ i=1 2 ∑ j=1 { [ α 1i α 2j + θ(α 1i α 2j -2α 2 1i α 2j -2α 11 α 12 α 2j -2α 1i α 2 2j -2α 1i α 21 α 22 +4α 2 1i α 2 2j + 4α 2 1i α 21 α 22 + 4α 11 α 12 α 2 2j + 4α 11 α 12 α 21 α 22 ) ] ξ(V aR κ (S); λ 1i ; λ 2j ) } , M = θ 2 ∑ i=1 2 ∑ j=1 { [ α 2 1i α 2j -2α 2 1i α 2 2j -2α 2 1i α 21 α 22 ] ζ(V

Approximation methods for TVaR-based allocation

We have seen in the previous sections that it is possible to have an exact expression for the TVaR of a sum of several dependent random variables and the contribution of each random variable to the aggregate TVaR for some specific situations, in particular when using the FGM copula. For most copulas, it is more complicated to directly calculate this risk measure. [START_REF] Embrechts | Fast computation of the distribution of the sum of two dependent random variables[END_REF] proposed an algorithm to compute numerically the cdf of the sum of two random variables joined by a copula. They used an approximation of the set {(x 1 , x 2 ) ∈ [0, +∞) 2 : x 1 + x 2 ≤ s} by a countable union of disjoint rectangles to obtain an evaluation of F S (s) with S = X 1 + X 2 .

In the present paper, we expose a simple alternative to approximate this cumulative distribution function with the use of common discretization methods that can be found in [START_REF] Klugman | Loss models: From data to decisions[END_REF]. Then we evaluate the TVaR and its contributions when the random variables are linked by any copula. The method is here exposed for two random variables but can be expanded to more random variables as shown in the numerical applications.

Discretization methods

We use three discretization methods in our study that are defined just below. For these three methods, we suppose that X is a continuous random variable with cdf F X and that h is the discretization span.

Definition 9 (Lower method)

The lower method provides a probability mass function of the discretized random variable X given by { f X (0) = 0 f X (jh) = F X (jh) -F X ((j -1)h), for j = 1, 2, ... .

Definition 10 (Upper method)

The upper method provides a probability mass function of the discretized random variable X given by { f X (0) = F X (h) f X (jh) = F X ((j + 1)h) -F X (jh), for j = 1, 2, ... .

Definition 11 (Mean preserving method)

The mean preserving method provides a probability mass function of the discretized random variable X given by {

f X (0) = 1 -E[X∧h] h f X (jh) = 2E[X∧jh]-E[X∧(j-1)h]-E[X∧(j+1)h] h , for j = 1, 2, ... .
This method ensures that the mean of the discretized distribution is the same as the original distribution.

Remark 12 It is shown e.g. in [START_REF] Müller | Comparison methods for stochastic models and risks[END_REF] and [START_REF] Denuit | Actuarial Theory for Dependent Risks: Measures, Orders and Models[END_REF] that X ≤ sd X under the lower method, X ≤ sd X under the upper method, and X ≤ icx X under the mean preserving method, where ≤ sd and ≤ icx designate the stochastic dominance order and the increasing convex order respectively (see the same references for the definitions).

The bivariate case

Suppose that we have two continuous distributed risks X 1 and X 2 . The joint cdf F X 1 ,X 2 is defined by a fixed copula C which introduces a dependence structure between the risks. We discretize X 1 and X 2 with one of the three methods described before. We denote by X1 , X2 the discretized random variables obtained and keep the same dependence relation between these two new random variables with the copula C. Then, we define S = X1 + X2 that we use two approximate S = X 1 +X 2 .

For a constant discretization span h, the cdf of ( X1 , X2 ) for k ≥ 0 and l ≥ 0 is

F X1 , X2 (kh, lh) = k ∑ i=0 l ∑ j=0 P r( X1 = ih, X2 = jh).
The cdf of S for j ≥ 0 is given by

F S (jh) = j ∑ i=0 P r( S = ih),
where the probability mass function (pmf) of S is P r( S = 0) = P r( X1 = 0; X2 = 0), and

P r( S = jh) = j ∑ i=0 P r( X1 = ih; X2 = (j -i)h), for j = 1, 2, ... .
The joint pmf of ( X1 , X2 ) is obtained with the copula as follows

P r( X1 = 0, X2 = 0) = C ( F X1 (0), F X2 (0) ) , P r( X1 = 0, X2 = jh) = C ( F X1 (0), F X2 (jh) ) -C ( F X1 (0), F X2 ((j -1)h) ) , P r( X1 = ih, X2 = 0) = C ( F X1 (ih), F X2 (0) ) -C ( F X1 ((i -1)h), F X2 (0) ) , P r( X1 = ih, X2 = jh) = C ( F X1 (ih), F X2 (jh) ) -C ( F X1 ((i -1)h), F X2 (jh) ) -C ( F X1 (ih), F X2 ((j -1)h) ) + C ( F X1 ((i -1)h), F X2 ((j -1)h)
) .

Then, the TVaR of S can be approximated by the TVaR of S which is given by

T V aR κ ( S) = E[ S × 1 { S>V aRκ( S)} ] + V aR κ ( S) ( P r( S ≤ V aR κ ( S)) -κ ) 1 -κ = E[ S × 1 { S>k 0 h} ] + k 0 h ( P r( S ≤ k 0 h) -κ ) 1 -κ , where V aR κ ( S) = k 0 h.
The TVaR-based allocation of risk X i over the global risk S can be approximated by the TVaRbased allocation of risk Xi over the global discretized risk S where

T V aR κ ( Xi ; S) = E [ Xi × 1 { S>V aRκ( S)} ] + β S E [ Xi × 1 { S=V aRκ( S)} ] 1 -κ = E [ Xi × 1 { S>k 0 h} ] + β S E [ Xi × 1 { S=k 0 h} ] 1 -κ , where β S = { P r( S≤k 0 h)-κ P r( S=k 0 h) , if P r ( S = k 0 h ) > 0, 0, otherwise.
Remark 13 In corollary 4.6 of [START_REF] Müller | Stochastic comparison of random vectors with a common copula[END_REF], it is shown that if (X 1 , X 2 ) and ( X 1 , X 2

) are random vectors with a common conditionally increasing copula and if X i ≤ cx X i for i = 1, 2, then for all non-negative scalars a 1 and a 2 we have

a 1 X 1 + a 2 X 2 ≤ cx a 1 X 1 + a 2 X 2 .
This result also holds for the increasing convex order since it is implied by the convex order. From [START_REF] Bäuerle | Stochastic orders and risk measures: consistency and bounds[END_REF], it follows that

T V aR κ (a 1 X 1 + a 2 X 2 ) ≤ T V aR κ ( a 1 X 1 + a 2 X 2 )

Conclusion

This paper introduces the use of copulas in TVaR-based capital allocation. We obtain explicit expressions for the TVaR and TVaR-based allocation for risks that have exponential and mixture of exponentials distributions linked by a FGM copula. The handy form of this copula permits a direct calculation of the coherent risk measure and its decomposition when we suppose only two different risks. In the multivariate situation, we use divided differences as in [START_REF] Chiragiev | Multivariate pareto portfolios: Tce-based capital allocation and divided differences[END_REF]. For other copulas, we present approximations for the TVaR and the TVaR-based allocation using three discretization methods for continuous distributions.

Figure 2 :

 2 Figure 1: Discretized cdf's vs exact cdf for 2 risks

Figure 3 :Figure 4 :

 34 Figure 3: TVaR of S with different copulas and correlation coefficients

Table 1 :

 1 Bivariate exponential example with θ = -1.

		4.3188	6.5053	11.0436	15.5235	17.4860
	T V aR κ (S)	7.3270	9.3394	13.8369	18.3810	20.3716
	T V aR κ (X 1 ; S) 2.7244	3.1489	3.5085	3.2649	3.0613
	T V aR κ (X 2 ; S) 4.6026	6.1905	10.3283	15.1161	17.3103
	θ = 0	κ = 0.5 κ = 0.75 κ = 0.95 κ = 0.99 κ = 0.995
	V aR κ (S)	4.1589	6.7187	11.9994	16.9914	19.1073
	T V aR κ (S)	7.6589	9.9967	15.0984	20.0310	22.1324
	T V aR κ (X 1 ; S) 2.9206	3.5756	4.6115	5.2234	5.4002
	T V aR κ (X 2 ; S) 4.7383	6.4211	10.4869	14.8075	16.7323

Table 2 :

 2 Bivariate exponential example with θ = 0.

	θ = 1	κ = 0.5 κ = 0.75 κ = 0.95 κ = 0.99 κ = 0.995
	V aR κ (S)	3.9328	6.9975	12.8673	18.0635	20.2236
	T V aR κ (S)	7.9817	10.6369	16.0906	21.1529	23.2818
	T V aR κ (X 1 ; S) 3.1066	3.9947	5.4022	6.2662	6.5272
	T V aR κ (X 2 ; S) 4.8750	6.6422	10.6883	14.8867	16.7546

Table 3 :

 3 Bivariate exponential example with θ = 1.

  2 1i α 21 α 22 + 4α 11 α 12 α 2 2j + 4α 11 α 12 α 21 α 22 12 α 2j -4α 11 α 12 α 2 2j -4α 11 α 12 α 21 α 22 (S); λ 11 + λ 12 ; +2λ 2j ) +4α 11 α 12 α 21 α 22 λ 1i λ 2j (λ 11 + λ 12 )(λ 21 + λ 22 ) ζ(V aR κ (S); λ 11 + λ 12 ; λ 21 + λ 22 ) Let X 1 and X 2 be two mixture of exponentials distributed random variables with joint cdf defined by a bivariate FGM copula. Then, the TVaR-based contribution of risk i, i = 1, 2, to the aggregate risk

				∑	2 ∑	{	[	α 2 1i α 2j -2α 2 1i α 2 2j -2α 2 1i α 21 α 22	]	ζ(V aR κ (S); 2λ 1i ; λ 2j )
			i=1 + [ 2α 11 α ] j=1	λ 1i λ 11 + λ 12	ζ(V aR κ (S); λ 11 + λ 12 ; λ 2j ) }
	and
	K = θ	2 ∑	2 ∑	{	[	α 1i α 2 2j -2α 2 1i α 2 2j -2α 11 α 12 α 2 2j	]	ζ(V aR κ (S); λ 1i ; 2λ 2j )
		i=1	j=1
	+α 2 1i α 2 2j ζ(V aR κ (S); 2λ 1i ; 2λ 2j )
	+	[	2α 1i α 21 α 22 -4α 2 1i α 21 α 22 -4α 11 α 12 α 21 α 22	]	λ 2j λ 21 + λ 22	ζ(V aR κ (S); λ 1i ; λ 21 + λ 22 )
	+2α 2 1i α 21 α 22	λ 2j λ 21 + λ 22	ζ(V aR κ (S); 2λ 11 ; λ 21 + λ 22 ) + 2α 11 α 12 α 2 2j	λ 1i λ 11 + λ 12 ζ(V aR κ }
									.
	Proposition 8
									) ]	} ζ(V aR κ (S); λ 1i ; λ 2j ) ,

  12 α 21 α 22 λ 1i λ 2j (λ 11 + λ 12 )(λ 21 + λ 22 ) ξ(V aR κ (S); λ 11 + λ 12 ; λ 21 + λ 22 )

	∑	2 ∑	{	[	α 1i α 2 2j -2α 2 1i α 2 2j -2α 11 α 12 α 2 2j	]	ξ(V aR κ (S); λ 1i ; 2λ 2j )
	i=1	j=1
	+α 2 1i α 2 2j ξ(V aR κ (S); 2λ 1i ; 2λ 2j )
	+	[	2α 1i α 21 α 22 -4α 2 1i α 21 α 22 -4α 11 α 12 α 21 α 22	]	λ 2j λ 21 + λ 22	ξ(V aR κ (S); λ 1i ; λ 21 + λ 22 )
	+2α 2 1i α 21 α 22	λ 2j λ 21 + λ 22	ξ(s; 2λ 1i ; λ 21 + λ 22 ) + 2α 11 α 12 α 2 2j	λ 1i λ 11 + λ 12	ξ(V aR κ (S); λ 11 + λ 12 ; +2λ 2j )
	+4α 11 α
								aR κ (S); 2λ 1i ; λ 2j )
		+	[ 2α 11 α 12 α 2j -4α 11 α 12 α 2 2j -4α 11 α 12 α 21 α 22	]	λ 1i λ 11 + λ 12	} ζ(V aR κ (S); λ 11 + λ 12 ; λ 2j ) ,

}

.

Acknowledgements

The authors acknowledge the Natural Sciences and Engineering Research Council of Canada and the Chaire d'actuariat de l'Universit Laval for their support. This work has also been partially supported by the French Research National Agency (ANR) under the reference ANR-08-BLAN-0314-01. The authors would like to thank an anonymous referee for her/his comments.

for κ ∈ (0, 1) .

Numerical applications

Bivariate case

Suppose that X 1 and X 2 are exponentially distributed with parameters λ 1 = 1/2 and λ 2 = 1/3 respectively. The dependence between these two risks is defined by the bivariate FGM copula with parameter θ F GM = 0.8. This value of θ F GM implicates a correlation of 0.2 between X 1 and X 2 . Figure 1 in the appendix illustrates the accuracy of the approximations of the cdf of S 2 when using the three discretization methods. Indeed, the decrease of the discretization span h implicates a convergence of the discretized cdf to the real one.

The discretization methods allow an approximation of the cdf of S 2 with any copula. We illustrate the use of the FGM, the Clayton, the Frank and the Gumbel copula. The copulas' parameters are chosen such that we have the same coefficient of correlation between X 1 and X 2 . Figure 2 in the appendix shows the impact of the copulas on the cdf of S 2 which is discretized with the mean preserving method when the correlation coefficient is fixed at 0.2. The first graph displays the drawing of the approximated cdf's of S 2 . We then do the difference between a dependent cdf and the independent cdf and trace this difference for the FGM, the Frank, the Clayton and the Gumbel copula in the second graph of Figure 2. These two graphs highlight the fact that the Clayton copula introduces dependence in lower values and that the Gumbel copula permits dependence in the upper queue.

In Figure 3 in the appendix, we trace the differences between the TVaR of S 2 using dependent risks with one of the copulas discussed before and the TVaR of S 2 using the independent copula against the risk level κ. This is done for three increasing values of correlation between X 1 and X 2 . Given that the FGM copula only allows weak dependence it just appears on the first graph. The graphs confirm the fact that the Gumbel copula introduces dependence in high values.

Tables 4 and5 expose the numerical results for the TVaR and the TVaR-based allocation for X 1 and X 2 with the four copulas discussed above and confidence levels equal to 0.99 and 0.995. The calculations are done with the mean preserving discretization method with span h = 0.05 for the four copulas and also with the exact expression for the FGM copula. The tables attest the good precision of the approximation method and confirm the high dependence values inserted by the Gumbel copula.

Trivariate case

We illustrate here the difference between the exact and the approximated methods with three risk variables dependent through a trivariate FGM copula. Suppose that X 1 , X 2 and X 3 are exponentially distributed with parameters λ 1 = 1/2, λ 2 = 1/3 and λ 3 = 1/5 respectively. The copula parameters θ F GM i , θ F GM ij and θ F GM ijk , for i = 1, 2, 3, j = 1, 2, 3, k = 1, 2, 3, j ̸ = i, k ̸ = i


Atif Mashkoor

Abderrahman Matoussi

Towards validation of requirements models

HAL is a

Introduction

The use of formal methods for software development is escalating over the period of time. Most of the formal methods refine the initial mathematical model up to an extent where final refinement contains enough details for an implementation. The input to this formal specification phase is often the documents obtained during the requirements analysis activity which are either textual or semi-formal. Now there is a traceability gap between analysis and specification phases as verification of the semi-formal analysis model is difficult because of poor understandability of lower level of formalism of verification tools and validation of the formal specification is difficult for customers due to their inability to understand formal models.

Our objective is to bridge this gap by a gradual introduction of formalism into the requirement model in order to facilitate its validation. We analyse our requirements with KAOS (Knowledge Acquisition in autOmated Specification) [START_REF] Van Lamsweerde | Requirements Engineering: From System Goals to UML Models to Software Specifications[END_REF] which is a goaloriented methodology for requirements modeling, then we translate the KAOS goal model, following our derived precise semantics, into an Event-B [START_REF] Abrial | Modeling in Event-B: System and Software Engineering[END_REF] formal specification with the help of the platform RODIN 3 , and finally we rigourously animate our specification with the help of the animator Brama [START_REF] Servat | BRAMA: A New Graphic Animation Tool for B Models[END_REF], incorporated into platform RODIN, in order to validate the conformance of the specification to original requirements. With this approach we aim to reap benefits at two levels: customers can be involved into the development right form the start and consequently the requirement errors can be detected right on the spot.

KAOS and Event-B

To analyze our requirements, we use KAOS which builds a data model in UML-like notation. The main KAOS goal defines an objective the system should meet, usually through the cooperation of multiple agents such as devices or humans, followed by several sub goals. Contrary to other requirements methods, such as i* [START_REF] Yu | Towards Modeling and Reasoning Support for Early-Phase Requirements Engineering[END_REF], KAOS is well suited for our purpose because it can be extended with an extra step of formality which can fill in the gap between requirements and the later phases of development. The choice of Event-B as a formal specification language is due to its similarity and complementarity with KAOS. Firstly, Event-B is based on set theory with the ability to use standard first-order predicate logic facilitating the integration with the KAOS requirements model that is based on first-order temporal logic. Secondly, both Event-B and KAOS have the notion of refinement (constructive approach). Finally, KAOS and Event-B have the ability to model both the system and its environment.

Discussion

Fig. 1. The rigorous requirements validation process

There are two main steps of our approach. First we translate our goal model into an Event-B specification with the help of our Event-B semantics, and later we animate the specification in order to validate that captured requirements are in accordance with original customer requirements. The whole process of validation is summed up by figure 1. Following is the brief elaboration of our approach:

The semantics step

The first step of our approach aims to express the KAOS goal model with Event-B by staying at the same level of abstraction which allows us to give this expression precise semantics. To achieve this objective, we use Event-B to formalize the KAOS refinement patterns that analysts use to generate a KAOS goal hierarchy. We primarily focus on most frequently used "Goal Patterns": the Achieve goals. The assertions in Achieve goals are expressed as following: G-Guard ⇒ ⋄G-PostCond, where G-Guard and G-PostCond are predicates. Symbol ⇒ denotes the classical logical implication. Symbol ⋄ (the open diamond) represents the temporal operator "eventually" which ensures that a predicate must occur "at some time in future". Hence, such assertions demonstrate that from a state in which G-Guard holds, we can reach sooner or later to another state in which G-PostCond holds.

If we refer to the concepts of guard and postcondition that exist in Event-B, a KAOS goal can be considered as a postcondition of the system, since it means that a property must be established. The crux of our formalization is to express each KAOS goal as a B event, where the action represents the achievement of the goal. Then, we will use the Event-B refinement relation and additional custom-built proof obligations to derive all the subgoals of the system by means of B events. One may wonder whether formalization of KAOS target predicates (i.e. the predicate after the diamond symbol) as B post conditions is adequate, since the execution of B events is not mandatory. At this very high level of abstraction, there is only one event for representing the parent goal. In accordance with the Event-B semantics, if the guard of the event is true, then the event necessarily occurs. For the new events built by refinement and associated to the subgoals, we guarantee by construction that no event prevent the post conditions to be established. For that, we have proposed Event-B semantics for each KAOS refinement pattern by constructing set-theoretic mathematical models. This process continues until the complete specification of KAOS goal model into Event-B. A detailed discussion on this step can be found in [START_REF] Matoussi | Expressing KAOS Goal Models with Event-B[END_REF]. Formalization of the goal patterns other than Achieve goals is a work in progress.

The animation step

Following the precise semantics discussed in previous section, we derive an initial Event-B specification of the KAOS goal model. The aim of this animation step is to validate this derived specification. Our approach to address this issue is based on following hypothesis: we presume if the animation of the specification reveals the same behavior that we intended while writing our goal model, then the Event-B specification would be considered as a valid formal representation of the customers' requirements. In order to achieve this, we execute our specification to check its behavior with the approach defined in [START_REF] Mashkoor | Transformation Heuristics for Formal Requirements Validation by Animation[END_REF]. We rigourously animate the specification at each refinement step. It not only indicates any deviation from original requirements right on the spot but also helps fixing the specification errors. If any deviation from the intended behavior is discovered, we go back to the source and rectify the error. The process continues until the specification fully adheres to the requirements.

Conclusion and future work

We present an approach to validate a semi-formal requirement model by the animation of its formal counterpart. We express a KAOS goal model capturing users' requirements into an Event-B specification language for a stepwise requirement validation process.

At theoretical level our approach seems promising as we have obtained some initial results at its both steps independently. However, we hope that our proposed combined approach of analysis, specification and validation is also feasible collectively. We aim to target at transportation domain [START_REF] Mashkoor | B Événementiel pour la Modélisation du Domaine: Application au Transport[END_REF] to test our hypothesis.

http://rodin-b-sharp.sourceforge.net

Acknowledgements

This work has been partially supported by the ANR project TACOS (Ref: ANR-06-SETI-017). We would also like to thank Jean-Pierre Jacquot and Régine Laleau for their valuable comments.