6.3 Review of some principles
6.3.2 Other capital allocation principles based on
There are other examples of proportional capital allocation principles based on partial contributions fitting expression (6.2). Two examples are given here, one from a probabilistic perspective and another one from a game-theoretic perspective.
The covariance allocation principle
This principle is proposed, for instance, inOverbeck[2000]. It takes into account the variance as the risk measure for the whole portfolio: Ω(S)= V(S). The partial contribution of theith agentXi is the covariance ofXi with respect toS, soΩ(Xi |S)=Cov(Xi,S). Therefore, this principle is expressed as
Ki=K·Cov(Xi,S)
V(S) , 8i=1,...,n. (6.15) Note thatΩ(S)=Pn
j=1Ω(Xi|S)because of the (bi)linearity of the covari-ance:
From the perspective of the Euler’s Theorem on homogeneous functions, this principle can be understood in two different (but related) ways. The first interpretation considers as the risk measureΩthe variance in expres-sion (6.4), in order to interpret the covariance principle similarly to a gradi-ent principle. The resulting functionf~X=V±sis not an homogeneous func-tion of degreer=1but anhomogeneous function of degreer =2, because the variance is not a positively homogeneous risk measure but satisfies the following relationship: for all∏2Rand for allX2°,V(∏·X)=∏2·V(X). From Theorem6.1this means that expression
2·V
holds or, in other words, that
"
Let us check this last equivalence:
If last expression is evaluated at~u =(1,1,...,1)then the desired result is found.
The second interpretation allows to understand the covariance allocation principle as a pure gradient allocation principle as explained in Section6.3.1.
The key is to consider as the risk measureΩin (6.4) the covariance of a ran-dom variable with respect to the sumSof the components of~Xinstead of the variance. So the function fX~ is taken as f~X =Cov(·,S)±s. As long as Cov(∏·X,S)=∏·Cov(X,S)for all∏2Rand for allX2°, f~X is an homo-geneous function of degreer =1and Theorem6.1may be applied in this case as in Proposition6.1.
Example 6.3. Let us apply the covariance principle to obtain an allocation linked to the portfolio of Example6.1.
The first step is to recall the expression for the variance of(s(~u))and to de-rive the expression for the homogeneous (covariance) function:
V(s(~u))=u21+u1·u2+u22,
=4.165. In relative terms, the covariance alloca-tion principle is assigning a45.84%of the risk (measured by the variance) to the first asset and a54.16%to the second one.
Finally, some comments on strengths and weaknesses of the covariance prin-ciple may be pointed out. As a strength in front of other gradient allocation principles, estimators of bothV(S)andCov(Xi,S)for alli =1,...,ncan be found satisfying that the sum of the estimated covariances add up to the estimated variance of the overall portfolio, whatever the set of random vari-ables{Xi}i=1,...,nis. Hence, the covariance principle overcomes the second drawback commented at the end of the previous section. As a weakness, the allocation only takes care of linear dependence structures between random variablesXi,i=1,...,n, and may lead to negative allocated capitalsKi. The article ofWang[2014] is inspired by the covariance allocation principle and the tail variance risk measure presented inFurman and Landsman[2006].
In this work the author define the capital allocation principles based on the Tail Covariance Premium Adjusted and tackles a possible non linear depen-dence between business lines.
The Shapley value principle and one of its simplifications
Another proportional allocation principle based on partial contributions can be derived from game theory. The capital allocation problem can be
understood as a cooperative game in which capitalKhas to befairlyshared by the agents, taking into account that the cost of a coalition is linked to the risk that this coalition assumes. The key concept to find such a fair alloca-tion is the Shapley value (sometimes also called Bondareva-Shapley value).
Let us use the following notations: N ={1,...,n}, AµN denotes a subset ofNwith cardinalitya= |A|andR(A)=Ω°P
k2AXk¢
. A capital allocation principle based on the Shapley value is of the form (6.2), where
Ω(Xi|S)= X
AµN‡{i}
a!·(n°a°1)!
n! ·[R(A[{i})°R(A)] . (6.17) Note thatR(N)=Ω(S). Additionally, it can be proved thatΩ(S)=Pn
i=1Ω(Xi
|S)using the properties of the Shapley value. The contribution ofith agent to the overall risk is, basically, a weighted average of all the marginal contri-butions thatith agent makes on the risk of each of the coalitions that can be obtained withoutith agent. This principle can require a high computa-tional demand for obtaining eachΩ(Xi|S)ifnis large.
In order to avoid this drawback, some authors propose an alternative ap-proach, that is a simplification of this principle. InBalog[2010] this alter-native is calledincremental principle. It is built by reducing the number of terms added up in expression (6.17) only to the one linked to the setN‡{i}. In other words, the incremental principle is of the form (6.1) where
fi(Xi)=R(N)°R(N‡{i})=Ω(S)°Ω
√X
j6=i
Xj
!
,8i=1,...,n.
This alternative principle assigns as partial contribution ofith agent the dif-ference between the overall risk and the risk quantified in absence of theith agent. To some extent, this principle can be considered as a hybrid between a proportional principle based on partial contributions and a stand-alone proportional principle. This principle cannot be considered a proportional principle based on partial contributions, becausePnj=1fj(Xj)6=Ω(S). But, at the same time, some relationship betweenith agent and the rest of partic-ipants is taken into account by fi, so it cannot be considered a stand-alone proportional principle. The loss of information is the price that must be paid to reduce the computational cost of the Shapley value for largen.
Example 6.4. Let us illustrate the Shapley value principle. We will consider three random variables, two of which are identical. They can only take four possible values. So, for the following states (!1,!2,!3,!4), random variables
X1,X2andX3are defined as follows: the other partial sums can also be computed.
≠ P X1 X2 X3 X1+Xi=2,3 X2+X3 S
!1 1/10 60 3 3 63 6 66
!2 1/10 0 30 30 30 60 60
!3 2/5 30 °7.5 °7.5 22.5 °15 15
!4 2/5 °15 15 15 0 30 15 The reader can easily compute the survival function for each of the random variables. Let us show how to compute the probability distribution function and the survival function forS, namely the sum of the three initial random variables.
The question is now to illustrate how to perform the allocation of a risk measure based on the Shapley principle. Let us consider theTVaR85%(S). Then, the following measures are needed: TVaR85%(X1), TVaR85%(X2), TVaR85%(X3), TVaR85%(X1+Xi=2,3), TVaR85%(X2+X3)and, obviously, TVaR85%(S).
Recall (see, Chapter1, eq. (1.4)) that an expression for theTVaRÆ(Z), for a random variableZ, is:
TVaRÆ(Z)=VaRÆ(Z)+ 1
1°Æ·ESÆ(Z). (6.18) Then since,VaR85%(S)=60and
ES85%(S)= X4 j=1
(sj°VaR85%(S))+·pj=(66°60)· 1 10= 6
10, (6.19) it follows that
TVaR85%(S)=60+100 15 · 6
10=60+60
15=64. (6.20) Similarly,TVaR85%(X1)=50andTVaR85%(X2)=TVaR85%(X3)=25, while TVaR85%(X1+Xi=2,3)=52andTVaR85%(X2+X3)=50.
Since all the necessary values of the risk measureΩ=TVaR85%are ready, then two tables are constructed in order to obtainΩ(X1|S),Ω(X2|S)and Ω(X3|S). Let us first concentrate onX1.
A P
k2AXk a a! (n°n!a°1)! X1+ P
k2AXk R(A)R(A[{1})
? ? 0 1 2/6 X1 0 50
{2} X2 1 1 1/6 X1+X2 25 52
{3} X3 1 1 1/6 X1+X3 25 52
{2,3} X2+X3 2 2 1/6 S 50 64
Using expression (6.17), it follows that:
Ω(X1|S)=2
6·(50°0)+1
6·(52°25)+1
6·(52°25)+2
6·(64°50)
=30+1 3
Using a similar procedure, it follows thatΩ(X2|S)=Ω(X3|S)=16+5/6. Therefore,Ω(S)=P3
i=1Ω(Xi|S)=64.