6.3 Review of some principles
6.3.3 The excess based allocation principle
The last principle explained in this chapter is the principle proposed invan Gulicket al.[2012]. The reason is twofold. On the one hand, because of its
originality and, on the other hand, because of its non-proportionality. Taken the authors’ own words [cf. page 29].
“The allocation rule that we propose determines the allocation that lexicograph-ically minimizes the portfolio’s excesses among a set of allocations that satis-fies two basic properties. First, no portfolio is allocated more risk capital than the amount of risk capital that it would need to withhold if it were on its own.
Second, a portfolio is not allocated less than the minimum loss it can incur”.
To better understand this principle, the following definition fromvan Gulick et al.[2012] must be presented.
Definition 6.5 (Lexicographical ordering). Form2Nand any two vectors
~x,~y2Rn,~xis lexicographically strictly smaller than~y, denoted as~x<lex~y, if there exists ani…msuch thatxi<yi, and for allj<iit holds thatxj= yj. Moreover,~xis lexicographically smaller than~y, denoted by~x…lex~y, if
~x=~yor~x<lex~y.
The authors considers that the capitalK to be shared among the agents is, in fact, equal toΩ(S), whereΩshould be a coherent risk measure. They use notationN ={1,...,n}. Once these preliminaries are established, the idea of the excess based allocation principle may be outlined in four steps:
(i) Consider any capital allocation principleK~such thatPnj=1Kj=Ω(S) and such that the following boundary conditions are satisfied for all i2N: max{0,min!2≠Xi(!)}…Ki…Ω(Xi). The set of all the prin-ciples satisfying these conditions is called theset of feasible principles, denoted asF.
(ii) Compute, for each feasible principle, the vector of dimension2n con-sisting ine(¯K~)=≥
Eh°P
j2A(Xj°Kj)¢
+
i¥
AµN. So, there is a compo-nent for each subsetAµN. This component is equal to the mathe-matical expectation of the random variable that represents the non-negative excess of capital that principleK~assigns to coalitionA. (iii) For each feasible principle~K, the components ofe(¯~K)are ordered in
a decreasing manner. The ordered resulting vector inRnis denoted by µ[ ¯e(K~)].
(iv) The excess based allocation principle, denoted byK~EB A, is the feasi-ble principle which lexicographically minimizes theµ[ ¯e(K~)]. In other words,K~EB Ais chosen among all feasible principles as the principle
associated to the first position in the set of orderedµ[ ¯e(K~)], suppos-ing that this order is similar to the one provided by a librarian who has been increasingly ordering vectorsµ[ ¯e(K~)]alphabetically.
Obviously, last comment on step (iv) is not formal. A more precise way to present the excess based allocation principle is by
K~EB A=©K~2F| µ[ ¯e(K~)]…lexµ[ ¯e(C~)], 8C~2F™, (6.21) taking into account that the set at the right-hand side of expression (6.21) is a single value set [as proved in van Gulicket al.,2012] and that, therefore, there is a simplification of notation when identifying a set consisting on a single element with that element.
Although this perspective on the allocation procedure is very interesting, as in the case of the capital allocation principle based on the Shapley value, this principle can involve a significant computational cost whennis large.
In addition, as long as the coherence property is required to the risk measure Ωto be used in the EBA principle, it has to be noted that VaR could not be used asΩin that allocation procedure.
Example 6.5. The same random variables that were defined in the previous example (6.4) are used here to illustrate the excess based allocation princi-ple, usingTVaR85%.
The following steps are needed:
a) Conditions to be fulfilled by each feasible principle.
b) Calculation of vectorse(¯K~).
c) Ordering the components ofe(¯K~), to obtainµ[ ¯e(K~)]. d) Finding the excess based allocation.
a) A feasible principleK~ =(K1,K2,K3)has to fulfill the following condi-tions:
Condition 1. K =K1+K2+K3. SinceK =TVaR85%(S)=64, thenK1+ K2+K3=64.
Condition 2. K2=K3, because the excess based allocation principle satis-fies the symmetry property.
Condition 3. Conditionmax{0,min!2≠Xi(!)}…Ki…Ω(Xi)in this case correspond to0…K1…50and0…K2…25(similarly,0…K3…25).
Since by Condition 1K2 =K3 =(64°K1)/2it follows that(64° K1)/2=K2…25then14…K1.
Moreover, no sub-portfolio should be allocated more capital than its own risk. Therefore,K1+K2…Ω(K1+K2)=52and then, sinceK2=
b) Vectore(¯K~)has eight components, corresponding respectively to?,{1}, {2},{3},{1, 2},{1, 3},{2, 3}and{1, 2, 3}. It can be written as follows:
Note that the vector can be parametrized in terms ofK1.
c) Ordering depends on the values ofK1. Therefore, for instance ifK1= 14, then the decreasing order is, in terms of the subsets ofN,{1}>{1,2}= {1,3}>{2,3}>{2}={3}>N>?.
The ordering is different for each of the following intervals ofK1, namely [14,23+1/3],[23+1/3,32],[32,33],[33,260/7]and[260/7,40]. Due to linearity, we only need to concentrate on the extremes of the previous inter-vals in order to find the candidates for the excess based allocation principle.
d) To finalize the computation of the allocation principle, we need to choose the principle that minimizes lexicographically theµ[ ¯e(K~)]. We will first choose the principle in each interval and then find the overall choice. So,
In the intervalK12[14,23+1/3], the minimum is found for{1}and it is given byK1=23+1/3, soK~[14,23+1/3]=(23+1/3, 20+1/3, 20+1/3), because K2 =(64°K1)/2. Moreover,e(¯K~[14,23+1/3])=(6.˙3, 1.9˙3, 1.9˙3, 1.9˙3, 0.9˙6, 0.9˙6, 0.2, 0).
In the intervalK12[23+1/3,32], the minimum is at{1}and it is given byK1=32, soK~[23+1/3,32]=(32,16,16). Heree(¯K~[23+1/3,32])=(2.8, 2.8, 1.5, 1.5, 1.4, 1.4, 0.2, 0), corresponding respectively to{1}>{2, 3}>{1, 2}={1, 3}>{2}={3}>N>?.
In the intervalK12[32,33], the minimum is now at{2, 3}and it is given byK1=32, so againK~[32,33]=(32, 16, 16)ande(¯K~[32,33])=(2.8, 2.8, 1.5, 1.5, 1.4, 1.4, 0.2, 0), corresponding respectively to{2, 3}>{1}>
{1, 2}={1, 3}>{2}={3}>N>?.
In the intervalK12[33,260/7], the minimum is at{2, 3}and it is given by K1=33, so againK~[33,260/7]=(33, 15.5, 15.5)ande(¯K~[33,260/7])= (2.9, 2.7, 1.45, 1.45, 1.45, 1.45, 0.2, 0), corresponding respectively to{2, 3}>{1}>{2}={3}>{1, 2}={1, 3}>N>?.
In the intervalK12[260/7,40], the minimum is at{2, 3}and it is given by K1=260/6, so againK~[260/7,40]=(37.14, 13.43, 13.43)and
e(¯~K[260/7,40])=(4.57, 2.286, 2.286, 2.286, 1.2426, 1.2426, 0.2, 0), corresponding respectively to{2, 3}>{2}={3}>{1}>{1, 2}={1, 3}
>N>?.
In the previous allocations, we seek the lexicographical minimum, which corresponds toe(¯~K[23+1/3,32])=e(¯K~[32,33])=(2.8, 2.8, 1.5, 1.5, 1.4, 1.4, 0.2, 0). As a result, the excess based allocation principle is:
K~=(K1,K2,K3)=(32, 16, 16).
Note that invan Gulicket al.[2012], the authors recommend an optimiza-tion procedure, which has not been implemented in the previous example.