Group cohesion under individual regulatory constraints
aa r X i v : . [ q -f i n . M F ] O c t GROUP COHESION UNDER INDIVIDUAL REGULATORYCONSTRAINTS
DELIA COCULESCU AND FREDDY DELBAEN
Abstract.
We consider a group consisting of N business units. We suppose there areregulatory constraints for each unit, more precisely, the net worth of each business unit isrequired to belong to a set of acceptable risks, assumed to be a convex cone. Because of theserequirements, there are less incentives to operate under a group structure, as creating onesingle business unit, or altering the liability repartition among units, may allow to reduce therequired capital. We analyse the possibilities for the group to benefit from a diversificationeffect and economise on the cost of capital. We define and study the risk measures that allowfor any group to achieve the minimal capital, as if it were a single unit, without alteringthe liability of business units, and despite the individual admissibility constraints. We callthese risk measures cohesive risk measures. Introduction
We consider an insurance group structured in N ≥ i has someexogenous liability, modelled as a random variable X i ≥ , F , P ).We suppose that the net worth of each unit is subject to constraints, e.g. that are set bya regulator, namely it needs to belong to a certain set A of “acceptable positions”. Weconsider that A is a convex cone so that the functional ρ : L (Ω , F , P ) → R : ρ ( ξ ) = inf { m : ξ + m ∈ A} is a coherent risk measure (see [1]).Whenever the aggregation of the units’ liabilities is possible, the group is only requiredto hold the capital ρ (cid:16) − P Ni =1 X i (cid:17) . By convexity of ρ we have that ρ (cid:16) − P Ni =1 X i (cid:17) ≤ P Ni =1 ρ ( − X i ), which reflects the fact that the group would achieve a lower required cap-ital as compared with N separated entities with the same liabilities. In this situation, riskaggregation is beneficial because it reduces the capital.In this paper we are assuming that there are legal or geographical limitations that preventrisk transfers or aggregation of liability to take place with the aim of reducing the regulatorycapital. Each business unit must face the regulatory requirements individually. At the grouplevel nevertheless, it is possible to manage the available capital and make certain monetarytransfers to compensate for losses occurring at time 1 within the different business units.The set of all such possible monetary compensations will be called admissible payoffs. Forinstance, when at the group level the available capital is m >
0, then the set of admissiblepayoffs will be denoted by A X ( m ); it contains the nonnegative, N dimensional random Date : October 6, 2020.The first author thanks the participants at the seminar Finance and Insurance at the University of Zurich,in particular Pablo Koch Medina for helpful interactions. ectors Y = ( Y , ..., Y N ) such that P i Y i = m , and fulfilling some additional rules specifyingthe payment priority of the different units (as given in Definition 4 below).In this framework, unit i receives a payoff Y i so that its net worth is Y i − X i . Thelowest overall capital that the group needs to hold when it has liability given by the vector X = ( X , ..., X N ) is: K ( X ) := inf { m ≥ | ∃ Y ∈ A X ( m ) , ∀ i, Y i − X i ∈ A} . (1.1)In general, when the business units are facing such individual admissibility constraints, itis the case that the cost of capital K ( X ) is higher than the minimal cost obtained withaggregating the risks, ρ (cid:16) − P Ni =1 X i (cid:17) . Hence, the existence of individual constraints reducesthe benefit of being a group. In such circumstances the incentives are to organise the businessdifferently, as a unique entity, or form some optimised subgroups, depending on the liabilityvector.The topic of this paper, is to characterise the acceptability sets A that satisfy the property K ( X ) = ρ − N X i =1 X i ! , ∀ X ∈ ( L ∞ ) N . (1.2)We will show that the relation (1.2) is rather restrictive. We call the corresponding riskmeasures cohesive, as any group requires the same amount of capital as if it were a singleentity, despite the impossibility to aggregate liability to take advantage of the convexity of theregulatory constraint. When the risk measure is cohesive, the group benefits of the maximaldiversification gain, as if it were a single entity, even with individual capital constraints forthe group members.On the way of characterising the cohesive risk measures, we will show how admissiblepayoffs can be designed in order to offset the liability for each business unit and achieve anacceptable net worth.Our problem formulation is connected to the topic of optimal risk transfers based onconvex risk measures. Optimal risk transfers within a group is a topic that has been studiedin a substantial body of literature. We refer to Heath and Ku [10], Barrieu and El Karoui[2], [3], Jouini et al. [11], Filipovi´c and Kupper [9], Burgert and R¨uschendorf [4], Embrechtset al. [8]. In these papers, the problem is formulated generally asinf ξ N X i =1 ρ i ( − ξ i )over all vectors ξ = ( ξ , · · · , ξ N ) satisfying P Ni =1 ξ i = η for some η fixed. Note that eachbusiness unit may use a specific risk measure in this framework. The main difference withthe optimal risk transfer literature is to introduce individual risk admissibility constraintsfor every unit. At the same time, we consider that the liability at the level of each unit is nottransferable among units and we introduce solvability constraints, namely that paymentscan only be made within the limits of the available capital. Further, when the group isinsolvent, we introduce fixed rules for how the payments are to be made. This framework issimilar to the one we have introduced in [5], where the question addressed was the fairness ofinsurance contracts in presence of default risk. With similar rules for payments in bankruptcyand admissibility conditions for the payments, we have shown that it is not possible in eneral to perfectly offset the default risk exposure of the insured agents by proposing thema benefit participation. The question of when such offsetting payments can take place wasnot addressed there and the current paper also brings clarifications in that context.2. Setup and main definitions
Let us introduce the mathematical setup more clearly. We work in a two date model: time0 where everything is known and time 1, where randomness is present. Possible outcomesat time 1 are modelled as random variables on a probability space (Ω , F , P ), consideredto be atomless. Unless otherwise specified, all equalities and inequalities involving randomvariables are to be considered in an P a.s. sense. The space of risks occurring at time 1is considered to be L ∞ (Ω , F , P ), simply denoted L ∞ , i.e., the collection of all essentiallybounded random variables.At time 0, a regulator measures risks by means of a convex functional ρ fulfilling theproperties detailed below. Definition 1.
A mapping ρ : L ∞ → R is called a coherent risk measure if the followingproperties hold:(1) if ξ ≥ ρ ( ξ ) ≤ ρ is convex: for all ξ, η ∈ L ∞ , 0 ≤ λ ≤ ρ ( λξ +(1 − λ ) η ) ≤ λρ ( ξ )+(1 − λ ) ρ ( η );(3) for a ∈ R and ξ ∈ L ∞ , ρ ( ξ + a ) = ρ ( ξ ) − a ;(4) for all 0 ≤ λ ∈ R , ρ ( λξ ) = λρ ( ξ );(5) (the Fatou property) for any sequence ξ n ↓ ξ (with ξ n ∈ L ∞ ) we have ρ ( ξ n ) ↑ ρ ( ξ ).We refer to [1], [6], [7] for an interpretation of these mathematical properties and howthey apply to the framework of risk regulation. The main idea is that the regulator onlyaccepts risks ξ that satisfy ρ ( ξ ) ≤
0, hence we say that a random variable ξ is acceptablewhenever ρ ( ξ ) ≤
0. Remark that ξ + ρ ( ξ ) is always acceptable, so that ρ ( ξ ) is interpretedas the capital required for the risk ξ . If ρ is coherent then the acceptability set A := { ξ | ρ ( ξ ) ≤ } is a convex cone.The Fatou property allows to apply convex duality theory and establishes a one-to-onecorrespondence between a coherent risk measure and a convex closed set S consisting ofprobabilities which are absolutely continuous with respect to P (the so called scenario set of ρ ): Theorem 1. If ρ is coherent, there exists a convex closed set S ⊂ L , consisting of probabilitymeasures, absolutely continuous with respect to P , such that for all ξ ∈ L ∞ : ρ ( ξ ) = sup Q ∈S E Q [ − ξ ] . Conversely each such a set S defines a coherent utility function. We shall use the assumption that S is weakly compact, so that we will be able to replacethe sup by a max. Indeed, as a direct application of James’s theorem, weak compactnessis equivalent to the nonemptyness of the subgradient of ρ at any point: for every ξ ∈ L ∞ , ∇ ρ ( ξ ) = ∅ , that is, there is a Q ∈ S with ρ ( ξ ) = E Q [ − ξ ].An additional assumption that we will use is that the risk measure ρ used by the regulatoris commonotonic. efinition 2. We say that two random variables ξ, η are commonotonic if there exist arandom variable ζ as well as two non-decreasing functions f, g : R → R such that ξ = f ( ζ )and η = g ( ζ ). Definition 3.
We say that ρ : L ∞ → R is commonotonic if for each couple ( ξ, η ) of com-monotonic random variables we have ρ ( ξ + η ) = ρ ( ξ ) + ρ ( η ). Remark . Loosely speaking, two risks are commonotonic if they are bets on the same event.Indeed, ξ and η being nondecreasing functions of ζ , neither of them is a hedge against theother. The commonotonicity of ρ can therefore be seen as a translation of the rule: if thereis no diversification, there is also no gain in putting these claims together. Remark . If ρ is commonotonic then for nonnegative random variables f, g ∈ L ∞ satisfying P [ f > , g >
0] = 0 we have that ρ ( f + g ) = ρ ( f ) + ρ ( g ). In particular for ξ ∈ L ∞ : ρ ( ξ ) = ρ ( ξ + ) + ρ ( − ξ − ). We also have that for Q ∈ S satisfying ρ ( ξ ) = E Q [ − ξ ], necessarilyalso ρ ( ξ + ) = E Q [ − ξ + ] and ρ ( − ξ − ) = E Q [ ξ − ]. This easily follows from the subadditivity of ρ .All notions above are standard in the theory of risk measures. We now introduce somedefinitions that are specific to the framework of this paper, that is the one of a groupconsisting of N distinct units under regulatory supervision. Definition 4.
We denote by X the space of N dimensional random variables which arepositive and bounded. We consider a liability vector X = ( X , · · · , X N ) ∈ X and consider m ∈ R + . The class of admissible payoffs from a total capital m , corresponding to the liability X is defined as: A X ( m ) = ( Y ∈ X (cid:12)(cid:12)(cid:12)(cid:12) N X i =1 Y i = m ; ∀ k ∈ { , ..., N } : if P Ni =1 X i > m then Y k = X k P Ni =1 X i m if P Ni =1 X i ≤ m then Y k ≥ X k ) . Admissible payoffs respect some rules as follows. If the capital m is less than the aggregateliability P i X i , the group defaults. In this case, all liabilities have the same priority ofpayment, regardless the unit to which they are corresponding. Hence, in default, all capital m is distributed towards the units proportionally to their liability size. Whenever the groupis solvent at the aggregate level ( P i X i ≤ m ), every unit must be solvent as well, hence thecentral unit distributes for each unit i a payment that should cover the liability X i . As thereis a surplus in this case, some units will get more than their liability as a payoff. Example . Let us consider that each business unit receives some constant proportion of thesurplus ( m − P i X i ) + . The corresponding admissible payoffs (that we shall call standardpayoffs) are given as follows: Y k = " X k + α k m − X i X i ! { P i X i ≤ m } + X k (cid:18) m P i X i (cid:19) { P i X i >m } , i = 1 , . . . , N (2.1)where each α k is a nonnegative constant and P Ni =1 α i = 1. Definition 5.
Given a level of capital m , an offsetting payoff corresponding to the liability X ∈ X , is a vector of random variables Y ∈ A X ( m ) satisfying Y i − X i ∈ A for all i ∈ { , ..., N } . (2.2)that is, the net worth of any unit is acceptable. ffsetting payoffs cannot be achieved when there is not sufficient overall capital m . Theanalysis in the next section will reveal the fact that the offsetting payoffs can never beachieved when the capital m is less than K := ρ ( − P i X i ), that is the minimal capital re-quired for the aggregated liability. Also, in general, holding the capital K does not guaranteethe existence of these payoffs, so that the group may be required to hold more capital. Thisjustifies to introduce the following additional definition: Definition 6.
A coherent risk measure ρ is called cohesive if for any risk vector X ∈ X there exists Y ∈ A X ( ρ ( − P Ni =1 X i )) such that Y i − X i ∈ A for all i ∈ { , ..., N } . (2.3)3. Properties of offsetting payoffs with minimal capital
We use the setup and notation from the previous section; in particular ρ is a coherent riskmeasure that is commonotonic, with a corresponding scenario set S assumed to be weaklycompact. Also, we shall consider a fixed liability vector X ∈ X and denote the aggregatedgroup liability by S X := N X i =1 X i . Also we denote K := ρ − N X i =1 X i ! that is the minimum capital for the aggregated liability. Under individual regulatory con-straints for the business units, the minimum regulatory capital for the group is denoted K ( X ) and its expression was introduced in (1.1).First observation is that the group needs to hold at least a capital of K . Lemma 1.
The minimal capital for the group K ( X ) satisfies K ( X ) ≥ K .Proof. We assume Y ∈ A X ( m ) offsetting, that is Y i − X i ∈ A , or ρ ( Y i − X i ) ≤ i ∈ N . From this and the sub-additivity of ρ , we get:0 ≥ N X i =1 ρ ( Y i − X i ) ≥ ρ N X i =0 ( Y i − X i ) ! = ρ m − N X i =0 X i ! = K − m. (3.1) (cid:3) We now investigate what happens if the company holds a capital of K . Is it possible tosplit the capital K in a vector of admissible payoffs Y , so that the net worth Y i − X i of eachunit i is acceptable? The existence of such payoffs is not granted. Below we show that thiscondition is rather restrictive and we characterise the situations where the answer to thequestion is positive.A first remark is that the existence of offsetting payoffs with capital K requires thatno further improvement of the net worth of the business units can be reached by furtherdiversification. This is what the next lemma says. emma 2. If the payoff vector Y ∈ A X ( K ) is offsetting the liability X , then: N X i =0 ρ ( Y i − X i ) = ρ N X i =0 ( Y i − X i ) ! . (3.2) Proof.
The proof is similar to the one of Lemma 1. It suffices to take m = K in (3.1),therefore we need to have only equalities. We remark that the admissibility of Y does playa role in establishing this result only through the condition P i Y i = K (cid:3) Proposition 1.
Consider a payoff vector Y ∈ A X ( K ) . The following are equivalent: (i) The payoff vector Y is offsetting the liability X . (ii) If Q ∗ ∈ ∇ S X , then for all i ∈ { , ..., N } : ρ ( Y i − X i ) = E Q ∗ [ X i − Y i ] = 0 . (3.3)(iii) Relation (3.2) holds and for all i ∈ { , ..., N } : E Q ∗ [ X i − Y i ] = 0 . (3.4) for some Q ∗ ∈ ∇ S X . (iv) The following hold: the minimal group capital K ( X ) satisfies K = K ( X ) and Y is a solution of inf ξ ∈ A X ( K ) N X i =0 ρ ( ξ i − X i ) . Proof. (i) ⇒ (ii). We can apply Proposition 2 below, taking E = { , . . . , N } and ξ i := Y i − X i .Indeed, by Lemma 2, the condition in Proposition 2 (1) is fulfilled and it is equivalent to (3).(ii) ⇒ (i). Obvious.(ii) ⇒ (iii). The relation (3.3) implies that relation (3.2) holds as it can easily checked; also(3.3) implies (3.4) obviously.(iii) ⇒ (ii). If relation (3.2) holds, we can apply Proposition 2 below to deduce that ρ ( Y i − X i ) = E Q ∗ [ X i − Y i ], for all Q ∗ ∈ ∇ S X and these expressions are null, again by (iii).(i) ⇔ (iv). In general, P Ni =0 ρ ( ξ i − X i ) ≥ ρ (cid:16)P Ni =0 ( ξ i − X i ) (cid:17) so thatinf ξ ∈ A X ( K ) N X i =0 ρ ( ξ i − X i ) ≥ ρ ( K − S X ) = 0 . The condition K = K ( X ) means that there are offsetting payoff vectors with a capital K , while from the above inequality we see that whenever there exist such offsetting payoffvectors, they are solving the minimisation problem (as ρ ( Y i − X i ) = 0 whenever Y offsetting X ). Hence the proof is complete. (cid:3) Proposition 2.
We consider some random variables ( ξ i ) i ∈ E , with E some countable set andlet Q E ∈ ∇ ρ (cid:0)P i ∈ E ξ i (cid:1) . The following are equivalent: (1) ρ X i ∈ E ξ i ! = X i ∈ E ρ ( ξ i ) . (3.5) For all λ i ≥ , i ∈ E ρ X i ∈ E λ i ξ i ! = X i ∈ E λ i ρ ( ξ i ) . (3.6)(3) For all i ∈ E ρ ( ξ i ) = E Q E [ − ξ i ] . (3.7) Proof.
We show (1) ⇒ (3). For any i ∈ E , let Q { i } ∈ S be such that ρ ( ξ i ) = E Q { i } [ − ξ i ] . Then: X i ∈ E ρ ( ξ i ) = X i ∈ E E Q { i } [ − ξ i ] ≥ X i ∈ E E Q E [ − ξ i ] = ρ X i ∈ E ξ i ! . From (3.5), it follows that we must have only equalities above, hence: X i ∈ E (cid:0) E Q { i } [ − ξ i ] − E Q E [ − ξ i ] (cid:1) = 0 , which implies (as all terms in the above sum are nonnegative) that ρ ( ξ i ) = E Q { i } [ − ξ i ] = E Q E [ − ξ i ].Now, we show (3) ⇒ (2). Using the linearity of the expectation and (3.7) we obtain: ρ X i ∈ E λ i ξ i ! ≥ E Q E " − X i ∈ E λ i ξ i = X i ∈ S λ i ρ ( ξ i ) . (3.8)On the other hand, ρ being convex we also have for all λ i ≥ ρ X i ∈ E λ i ξ i ! ≤ X i ∈ E λ i ρ ( ξ i ) (3.9)Combining (3.8) and (3.9) we get the equality (3.6).The implication (2) ⇒ (1) is trivial, hence the proof is complete. (cid:3) Proposition 1 (ii) emphasizes that a payoff vector Y satisfying E Q ∗ [ X i − Y i ] = 0 with Q ∗ ∈ ∇ ρ ( − S X ) is a potential candidate to be an offsetting payoff (necessary condition).It follows that when payoffs are standard, i.e., as in (2.1), the proportions ( α i ) can beidentified via these equalities. Their expressions are given below in Proposition 3, togetherwith another necessary and sufficient condition for these to be indeed offsetting vectors. Tobe noted that this time we will use commonotonicity of the risk measure ρ to obtain thiscondition, the previous results remaining true also when ρ not commonotonic. Proposition 3.
The minimal capital for the group K ( X ) satisfies K ( X ) = K if and only if ρ (cid:16) − (cid:0) K − S X (cid:1) − (cid:17) = N X i =1 ρ (cid:18) − X i S X ( K − S X ) − (cid:19) . (3.10) If this condition is satisfied, then Y is offsetting X , where Y is a standard payoff (see (2.1))with ∀ i : α i : = E Q ∗ (cid:2) X i S X ( K − S X ) − (cid:3) E Q ∗ [( K − S X ) + ] = E Q ∗ (cid:2) X i S X ( S X − K ) + (cid:3) E Q ∗ [( S X − K ) + ] here Q ∗ ∈ ∇ ρ ( − S X ) .Proof. By Proposition 1 (iii), e Y ∈ A X ( K ) is ofsetting X if and only if the condition (3.2) issatisfied together with E Q ∗ [ e Y i − X i ] = 0 for all i .Using the commonotonicity of ρ we obtain that an equivalent expression for (3.2) is (ap-plied to e Y ): ρ N X i =1 e Y i − X i ! + − N X i =1 ρ (cid:16) ( e Y i − X i ) + (cid:17) + ρ − N X i =1 e Y i − X i ! − − N X i =1 ρ (cid:16) − ( e Y i − X i ) − (cid:17) = 0and because of the subadditivity property of ρ each of the two expressions in the brackets issmaller or equal to 0. It follows the equivalent expression of (3.2) is: ρ N X i =1 e Y i − X i ! + = N X i =1 ρ (cid:16) ( e Y i − X i ) + (cid:17) (3.11)and ρ − N X i =1 e Y i − X i ! − = N X i =1 ρ (cid:16) − ( e Y i − X i ) − (cid:17) . (3.12)We notice that because e Y is admissible, the expression (3.12) equals (3.10), so that thecondition (3.10) is necessary for the existence of ofsetting payoffs. We now show that it asufficient condition. Indeed, we can always choose e Y = Y with Y standard and as stated inthe proposition, we have that the random variables ( Y i − X i ) + = α i ( K − S X ) + , ∀ i = 1 , ..., N .We then use the commonotonicity property of ρ to conclude that (3.11) is verified. Theparticular proportions α i are found through the equality E Q ∗ [ Y i − X i ] = 0 Proposition 1 (ii).Hence, once (3.10) is verified, the vector Y fulfils the necessary and sufficient conditions tobe offsetting. (cid:3) A class of cohesive risk measures
We consider a random variable H ≥ ≤ E P [ H ] < ∞ and introduce therisk measure: ρ H ( ξ ) := sup Q ∈S H E Q [ − ξ ] (4.1)with a scenario set: S H := (cid:26) Q | ≤ d Q d P ≤ H P a.s. (cid:27) (4.2)The main result in this subsection is that all cohesive risk measures that are commonotonichave this representation. A generalisation of these risk measures will follow afterwards.Before we prove this result, we give some alternative characterisations of ρ H : roposition 4. (1) Let us denote E P [ H ] = h and introduce the probability measure H ≪ P as: d H d P := Hh .
For any ξ ∈ L ∞ we have: ρ H ( ξ ) = AV @ R ( H , /h ) ( ξ ) , i.e., the average value at risk for ξ , at the level /h and under the probability H . Werecall AV @ R ( P ,λ ) ( ξ ) is defined as: AV @ R ( P ,λ ) ( ξ ) = max Q ∈S λ E Q [ − ξ ] , where S λ is the set of all probability measures Q ≪ P whose density d Q /d P is P a.s. bounded by /λ . (2) For any ξ ∈ L ∞ we have: ρ H ( ξ ) = E Q ξ [ − ξ ] for a probability Q ξ satisfying: d Q ξ d P = H on { ξ < q } ch H on { ξ = q } on { ξ > q } , (4.3) with q = inf { x : E P [ H ξ ≤ x ] ≥ } and: c = ( if E P [ H ξ = q ] = 0 − E P [ H ξ The risk measure ρ H is commonotonic. We are now ready to prove the main result of this section. Proposition 5. A commonotonic risk measure ρ satisfying the weak compactness propertyis cohesive if and only if it has the representation (4.1) for some random variable H ∈ L ( P ) .Proof. We first proof that ρ H is cohesive. The risk measure ρ H is commonotonic (Corrollary1) and clearly satisfies the weak compactness property. It is cohesive if for any given liabilityvector X offsetting payoffs exist. For X ∈ X and S X = P i X i , let Q ∗ be such that K = ρ H ( − S X ) = E Q ∗ [ S X ]. It is sufficient to show that for arbitrary X ∈ X and for Y as in roposition 3, the conditions Y i − X i ∈ A , ∀ i ∈ { , ..., N } are satisfied. By construction thecondition E Q ∗ ( X i − Y i ) = 0 is satisfied for all i ≥ ρ H ( Y i − X i ) = E Q ∗ ( X i − Y i ) . (4.4)We recall that Q ∗ is defined as in (4.3) where h = E P [ H ]. With a slight change in notation,let q be such that P [ S X ≥ q ] ≥ /h ≥ P [ S X > q ]. Then Q ∗ [ { S X < q } ] = 0 and Q ∗ [ S X >K ] = E P (cid:2) H { S X >K } (cid:3) (since obviously K ≥ q ).For i ≥ Q ∈ S we have: E Q [ X i − Y i ] = − Z S X 1) (4.7)We consider a partition of A , τ ( A ) = { A , A } ⊂ F , the risks X := A , X := A andtheir sum S X = X + X = A . We let K := ρ − X i X i ! = ρ ( − A ) , and K ∈ (0 , 1) due to (4.7). This in turn leads to A = { S X > K } and A c = { S X < K } .There is Q A ∈ ∇ ρ ( − A ) with ρ ( − A ) = Q A [ A ]. Hence ρ ( A c ) = ρ (1 − A ) = − ρ ( − A ) = − Q A [ A c ]. As ρ is cohesive, there exist offsetting payoffs for the risk X = ( X , X , , .., e now consider Y ∈ A X ( K ) standard payoffs offsetting X , with Y i = 0 for i > 2. Theresidual risks of the units i = 1 , X i − Y i = − α i K A c + (1 − K ) A i . By commonotonicity of ρ and of the random variables ( Y i − X i ) + and − ( Y i − X i ) − thefollowing hold (for i = 1 , ρ ( Y i − X i ) = ρ (cid:0) ( Y i − X i ) + (cid:1) + ρ (cid:0) − ( Y i − X i ) − (cid:1) = α i Kρ ( A c ) + (1 − K ) ρ ( − A i )= − α i K Q A [ A c ] + (1 − K ) ρ ( − A i )while by Proposition 1 we also know that for Q A ∈ ∇ ρ ( − A ): ρ ( Y i − X i ) = − α i K Q A ( A c ) + (1 − K ) Q A ( A i ) = 0 . (4.8)Therefore (remember that 0 < K < ρ ( − A i ) = Q A ( A i ) , ∀ A i ∈ τ ( A ) . We emphasize that the probability Q A is chosen independently of the partition of the set A , τ ( A ). That means: if Q A ∈ ∇ ρ ( − A ) then: ρ ( − B ) = Q A ( B ) , ∀ B ⊂ A. (4.9)From (4.9) we deduce that for all probabilities measures Q ∈ S d Q A d P A ≥ d Q d P A P a.s. , that is, d Q A d P A ≥ H A , and because Q A ∈ S we must get equality. That means that if (4.7)holds we have d Q A d P = H on the set A . In other words ρ ( − A ) = E [ H A ] . For general sets B ∈ F , we distinguish between several cases.(a) If B ∈ F is such that ρ ( − B ) ∈ (0 , ρ ( − B ) = E [ H B ], as was proved above.(b) If B ∈ F satisfies ρ ( − B ) = 0, then, B is a null set for all probabilities in S ,consequently H B = 0 P a.s. and in a trivial way ρ ( − B ) = E P [ H B ].(c) If B ∈ F satisfies ρ ( − B ) = 1 we will use the weak compactness and the property thatthe probability space is atomless. There is a nondecreasing family of sets A t ; 0 ≤ t ≤ P [ A t ] = t P [ B ] and A = B . The Lebesgue property (weak compactness)then shows that the function t → ρ ( − A t ) is continuous. It is nondecreasing, startsat 0 and ends at 1. Therefore there is a unique number s ≤ t < s , ρ ( − A t ) < t ≥ s , ρ ( − A t ) = 1. For t < s we have ρ ( − A t ) = E [ H A t ] andby continuity we get 1 = ρ ( − A s ) = E [ H A s ]. Since A s ⊂ B we have 1 = ρ ( − B ) = E [ H B ] ∧ H ∈ L . This is rather obvious since ρ ( − { H>n } ) = E [ H { H>n } ] ∧ n → ∞ . Hence eventually E [ H { H>n } ] < < ∞ and H ∈ L . (cid:3) orollary 2. Suppose that the risk measure ρ is commonotonic, satisfies the weak compact-ness property, is cohesive and law determined (rearrangement invariant). Then ρ is a tailexpectation, i.e. there is a level α with ρ = AV @ R ( P ,α ) . Indeed if ρ ( ξ ) is determined by the distribution of ξ , then the set S must be rearrangementinvariant, i.e. if f ∈ S and g has the same distribution as f , then also g ∈ S . It is easy tosee (for an atomless space) that this implies that the function H constructed above, mustbe a constant. This is nothing else than the characterisation of AV@R. Remark . For a scenario set S we can introduce a space of random variables. We define E = { ξ | for all Q ∈ S : E Q [ | ξ | ] < ∞} . For elements ξ ∈ E we have that also ρ ( −| ξ | ) = sup Q ∈S E Q [ | ξ | ] < ∞ and this expressiondefines a norm for which E becomes a Banach space. Examples show that in general L ∞ isnot dense in this space. The risk measure ρ has a natural extension to E . Indeed we candefine ρ ( ξ ) = sup Q ∈S E Q [ − ξ ]. In case 0 < H ∈ L ( P ) , E [ H ] ≥ 1, the scenario set S H alsodefines such a space which in this case is easy to describe. As for tail expectation we havethe following inequalities Z | ξ | d H ≤ ρ ( −| ξ | ) ≤ h Z | ξ | d H where h = E [ H ] and d H = Hh d P . It follows that the space E is nothing else but the space L ( H ), with an equivalent norm.5. Cohesion for fixed aggregated liability Above, we analysed group cohesion when the class of possible liability vectors is X . Wenow suppose that the overall (or aggregated) group liability is fixed and equals Z ∈ L ∞ so that the class of all possible liability vectors becomes: { X ∈ X | P Ni =1 X i = Z } . Wegeneralise the class of risk measures from the previous section as follows. We define: ρ L,H ( ξ ) := sup Q ∈S L,H E Q [ − ξ ] , (5.1)with a scenario set: S L,H := (cid:26) Q : L ≤ d Q d P ≤ H P a.s. (cid:27) , (5.2)and where L, H are nonnegative random variables that satisfy 0 ≤ L ≤ H , with E [ H ] < ∞ and E [ L ] > 0. From the calculations below it will turn out that this risk measure is alsocommonotonic. We shall denote ℓ := E [ L ] and h := E [ H ]. We suppose ℓ < h > S L,H = P . We introduce the following probability measures: d H d P := H − Lh − ℓ and d L d P := Lℓ { ℓ> } + { ℓ =0 } . (5.3)For a given random variable ξ , we define a probability measure Q ξ as follows: d Q ξ d P := H on { ξ < q ( ξ ) } c ( ξ ) H + (1 − c ( ξ )) L on { ξ = q ( ξ ) } L on { ξ > q ( ξ ) } (5.4) ith q ( ξ ) and c ( ξ ) being constants (derived from the distribution of ξ ) to ensure that E P h d Q ξ d P i = 1. These constants can be computed as follows. Let us denote: F ( ξ, x ) := E P [ H ξ ≤ x + L ξ>x ] = ℓ + E P [( H − L ) ξ ≤ x ] . The function F ( ξ, · ) is increasing, right continuous, and satisfies lim x →−∞ F ( ξ, x ) = ℓ ≥ x →∞ F ( ξ, x ) = h ≥ 1. We denote: q ( ξ ) := inf { x : F ( ξ, x ) ≥ } (5.5)and define c ( ξ ) as satisfying c ( ξ ) F ( q ( ξ )) + (1 − c ( ξ )) F ( q ( ξ ) − ) = 1, that is: c ( ξ ) = ( F is continuous at q ( ξ ) − F ( q ( ξ ) − ) F ( q ( ξ )) − F ( q ( ξ ) − ) otherwise. (5.6)We observe that indeed E P h d Q ξ d P i = c ( ξ ) F ( q ( ξ )) + (1 − c ( ξ )) F ( q ( ξ ) − ) = 1 as required for Q ξ to be a probability measure. Proposition 6. For any ξ ∈ L ∞ we have the following alternative representations for ρ L,H : ρ L,H ( ξ ) = ℓ E L [ − ξ ] + (1 − ℓ ) AV @ R ( H ,γ ) ( ξ ) . for γ = (1 − ℓ ) / ( h − ℓ )) < and ρ L,H ( ξ ) = E Q ξ [ − ξ ] , where Q ξ defined in (5.4).Proof. We have by definition: ρ L,H ( ξ ) = sup {− E P [ ϕξ ] | L ≤ ϕ ≤ H , E P [ ϕ ] = 1 } (5.7)Using the transformation ψ := ( ϕ − L )( h − ℓ )( H − L )(1 − ℓ ) we obtain that { L ≤ ϕ ≤ H , E P [ ϕ ] = 1 } = { ≤ ψ ≤ h − ℓ − ℓ , E H [ ψ ] = 1 } and: E P [ ϕξ ] = E P [ Lξ ] + E P [( ϕ − L ) ξ ]= ℓ E P (cid:20) ξ d L d P (cid:21) + (1 − ℓ ) E P (cid:20) ψξ d H d P (cid:21) = ℓ E L [ ξ ] + (1 − ℓ ) E H [ ψξ ] . Therefore, an equivalent expression for ρ L,H is: ρ L,H ( ξ ) = ℓ E L [ − ξ ] + (1 − ℓ ) sup (cid:26) − E H [ ϕξ ] | ≤ ϕ ≤ h − ℓ − ℓ , E H [ ϕ ] = 1 (cid:27) = ℓ E L [ ξ ] + (1 − ℓ ) AV @ R ( H ,γ ) ( ξ ) , where AV @ R ( H ,γ ) ( − ξ ) is the average value at risk for − ξ , at the level γ = (1 − ℓ ) / ( h − ℓ ))and under the probability H .We notice that γ < h > 1. The optimiser probability for AV @ R is known to be Q ξ : d Q ξ d H = 1 γ ( ξ Consider the regulator’s risk measure is ρ = ρ L,H . Let Z ∈ L ∞ + with q ( − Z ) the corresponding constant, as defined in (5.5). Suppose further that ρ L,H ( − Z ) ≥ − q ( − Z ) , (for this inequality to hold it is sufficient that AV @ R ( H ,γ ) ( − Z ) ≤ E L [ Z ] ). Then, for all riskvectors X satisfying P Ni =1 X i = Z , the following equality is satisfied K ( X ) = ρ ( − Z ) . Proof. Let us denote K = ρ L,H ( − Z ) and Q ∗ be the probability in (5.4) with ξ = − Z . If K ≥ − q ( − Z ), then { Z > K } ⊂ { Z > − q ( − Z ) } = {− Z < q ( − Z ) } = { d Q ∗ d P = H } .For any X ∈ X satisfying P Ni =1 X i = Z and for any corresponding vector Y ∈ A X ( K ), wehave: ∀ i { X i > Y i } ⊂ { Z > K } . Therefore, we obtain: ρ ( − ( Y i − X i ) − ) = E Q ∗ (cid:2) − ( Y i − X i ) − (cid:3) = E P (cid:2) − ( Y i − X i ) − H (cid:3) . It follows that the condition (3.10) is satisfied and, as ρ L,H is commonotonic, the equality K ( X ) = ρ ( − Z ) holds as an application of Proposition 3.It remains to prove the claim that if AV @ R ( H ,γ ) ( − Z ) ≤ E L [ Z ] then ρ L,H ( − Z ) ≥ − q ( − Z ).We have that − q ( − Z ) ≤ AV @ R H ,γ ( − Z ) (this is always true). Then, the condition AV @ R ( H ,γ ) ( − Z ) ≤ E L [ Z ] implies AV @ R ( H ,γ ) ( − Z ) ≤ ρ ( − Z ), and hence the claim is proved. (cid:3) References [1] Artzner, Ph., F. Delbaen, J.-M. Eber, and D. Heath: Coherent Measures of Risk , Mathematical Finance (3), 203–228, (1999).[2] Barrieu, P. and N. 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