Applications of axiomatic capital allocation and generalized weighted allocation

Transcription

1 ( ) Journal of East China Normal University (Natural Science) No. 6 Nov Article ID: (2010) Applications of axiomatic capital allocation and generalized weighted allocation HU Feng-xia, YAO Ding-jun (School of Finance and Statistics, East China Normal University, Shanghai , China) Abstract: Applications of two kinds of capital allocation principles were studied: one was axiomatic allocation which had been known, and the other was generalized weighted allocation. Firstly, several known risk measures were listed. Based on a axiomatic method, allocation formulae with respect to these risk measures were given. Then in view of a specific numerical model, its allocation results were calculated. Secondly, generalized weighted allocation, which was an extension of weighted allocation, was considered. A method for comparing the amounts of generalized weighted capital allocations under two different aggregate losses, which stems from diversifying property, was mainly discussed. And some special examples were given. Key words: capital allocation; risk measure; directional derivative; conditional expectation CLC number: O211 Document code: A, (, ) : :,.,,.,.,,.. : ; ; ; : : ( ); (973 )(2007CB814904); (NCET ) :,,,. fengxia.hu@gmail.com. :,,,. yaodingjun@yahoo.com.cn.

2 6, : Introduction Capital allocation can be considered as a method of measuring the performance of a portfolio in terms of the diversification that it contributes to the whole company. There are many kinds of methods about capital allocation in portfolio management and risk-based performance measurement, and different methods usually lead to different strategies. Cummins [1] provided an overview of several methods for capital allocation in the insurance industry, and built a relation between capital allocation and decision making tools. Denault [2] discussed capital allocations based on game theory, where a risk measure was used as cost function. Kalkbrener [3] proposed an axiomatic allocation system. Furman and Zitikis [4] introduced weighted risk capital allocations, which stemmed from weighted premium calculation principles. Dhaene et al. [5] considered CTE-based capital allocation. Moreover, we can refer to Dhaene, Goovaerts and Kaas [6], and Fischer [7] for other allocation principles. In this paper, our discussions are based on two kinds of capital allocation principles one is the axiomatic allocation introduced by Kalkbrener [3], the other is generalized weighted allocation. And we study some of their applications respectively. The rest of this paper is organized as follows. Section 1 recalls the axiomatic allocation of Kalkbrener [3], and considers some applications of this method. In Section 2, we propose generalized weighted capital allocation, and study its application. Then we discuss a method for comparing the amounts of generalized weighted capital allocations under two different aggregate losses. We conclude the paper in Section 3. 1 Application of an axiom capital allocation Kalkbrener [3] proposed an axiomatic method for capital allocation. Based on this axiomatic method, we will study its application. To begin with, let us introduce the axiom system Introduction of the axiom model Let (Ω, F, P) be a probability space, and L 0 be the space of all equivalent classes of real valued random variables on Ω. V is a subspace of the vector space L 0, representing the set of portfolios of a organization (an insurance company or rating agency, say). V is chosen depending on the specific problem at hand, and different problems may choose different V. For any portfolio X V, X(ω) specifies the loss of X at a future date in state ω Ω. Suppose that X consists of subportfolios X 1,, X m V, i.e., X = X 1 + +X m. Then define a risk measure ρ : V R, ρ(x) identifying the risk capital associated with X. Indeed, ρ(x) is the level of safely invested capital that the owner of X has to hold in order to make the future worth X ρ(x) acceptable to the regulators (note that the future worth is a little different from that of Artzner et al. [8], where X means net worth, so its future worth is X + ρ(x)). The axiom system is based on the assumption that the amount of risk capital allocated to subportfolio X i only depends on X i and X, but not on the decomposition of the rest X X i = X j. j i

3 148 ( ) 2010 In view of above assumptions, a risk capital allocation problem is to allocate the risk capital ρ(x) among the subportfolios, and an allocation principle can be defined. Definition 1.1 Let ρ : V R be a risk measure. Suppose that X V consists of subportfolios X 1,, X m V, i.e., X = X X m. A capital allocation is a function : V V R that maps each (X i, X) to a unique real number k i, i,e, (X i, X) = k i representing the amounts of capital allocated to X i, and also satisfies n k i = ρ(x). i=1 Moreover, if for every X V, (X, X) = ρ(x), then (, ) is called a capital allocation with respect to ρ. Definition 1.2 Let (, ) be a capital allocation, (1) If for each a, b R, and each X, Y, Z V, (ax + by, Z) = a (X, Z) + b (Y, Z), then (, ) is called linear. (2) If for each X, Y V, (X, Y ) (X, X), then (, ) is called diversifying. (3) Let Y V, if for each X V, lim (X, Y + ɛx) = (X, Y ), then (, ) is called continuous at Y. All notations mentioned above and with properties of linear, diversifying, and continuous compose the axiom system. Remark 1.1 Kalkbrener [3] had demonstrated the completeness of the axiom system and existence of the relevant capital allocations. Assumed ρ( ) was positively homogeneous and subadditive, and let V be the set of all real valued and linear functions on V. Put H ρ = {h V h(x) ρ(x), X V }. For every Y V, he proved that there existed h Y H ρ with h Y (Y ) = ρ(y ). Then for every X, Y V, he defined ρ (X, Y ) = h Y (X). The most important results of Kalkbrener [3] are Theorem 3.1, Theorem 4.2, and Theorem 4.3. We will rewrite them as the following three propositions: Proposition 1.1 Let (, ) be a linear, diversifying capital allocation with respect to ρ. If (, ) is continuous at Y V, then for all X V, Proposition 1.2 ρ(y + ɛx) ρ(y ) (X, Y ) = lim. ɛ (a) If there exists a linear, diversifying capital allocation (, ) with respect to ρ, then ρ is positively homogeneous and subadditive. (b) If ρ is positively homogeneous and subadditive, then ρ is a linear, diversifying capital allocation with respect to ρ. Proposition 1.3 Let ρ( ) be a positively homogeneous and subadditive risk measure, and Y V. Then the following three conditions are equivalent. (a) ρ is continuous at Y, i.e., for all X V, lim ρ(x, Y + ɛx) = ρ (X, Y ).

4 6, : 149 (b) The directional derivative exists for every X V. ρ(y + ɛx) ρ(y ) lim ɛ (c) There exists a unique h H ρ with h(y ) = ρ(y ). If these conditions are satisfied, then for all X V Applications of the axiom model ρ (X, Y ) = lim ρ(y + ɛx) ρ(y ) ɛ In this section, we list three important risk measures: Dutch principle proposed by Van Heerwaarden and Kaas [9], semi-deviation risk measures, and p-mean value risk measure. With the axiom system mentioned above, the allocation formulae with respect to these risk measures are obtained respectively. measures: Definition 1.3 Let c 0 and p 1 be real numbers. We list the following three risk Dutch principle: ρ 1 (X) = EX + ce(x EX) +. Semi-deviation principle: ρ 2 (X) = EX + c{e[(x EX) + ] 2 } 1 2. p-mean value principle: ρ 3 (X) = [E(X p )] 1 p. Theorem 1.1 Proof ρ 1 ( ), ρ 2 ( ), ρ 3 ( ) are positively homogeneous and subadditive. Positively homogeneous properties of the three risk measures are easy to prove, we omit them here. And only subadditivity properties are verified. (a) Let X, Y V, ρ 1 (X + Y ) = E(X + Y ) + ce(x EX + Y EY ) + E(X + Y ) + ce(x EX) + + ce(y EY ) + = ρ 1 (X) + ρ 1 (Y ). (b) Let X, Y V, ρ 2 (X + Y ) = E(X + Y ) + c{e[(x EX + Y EY ) + ] 2 } 1 2 E(X + Y ) + c{e[(x EX) + + (Y EY ) + ] 2 } 1 2 E(X + Y ) + c{e[(x EX) + ] 2 } c{e[(y EY )+ ] 2 } 1 2 = ρ 2 (X) + ρ 2 (Y ), where the second inequality follows from Minkowski inequality. (c) Let X, Y V, and assume X 0, Y 0. By Minkowski inequality, ρ 3 (X + Y ) = [E(X + Y ) p ] 1 p [E(X) p ] 1 p + [E(Y ) p ] 1 p = ρ3 (X) + ρ 3 (Y ). By Proposition 1.2, there exist three linear, diversifying capital allocations with respect to ρ 1 ( ), ρ 2 ( ), and ρ 3 ( ), respectively. The following contents provide the exact allocation

5 150 ( ) 2010 formulae, and we will also give detailed verifications. These allocation formulae could have an extensive applications. Definition 1.4 Let c 0 and p 1 be real numbers. We define three capital allocations: 1 (X, Y ) = ce[(x EX)1 {Y EY } ] + EX; E[(X EX)(Y EY )+] c + EX, if E(Y EY ) 2 (X, Y ) = {E[(Y EY ) +] 2 } 1 + > 0, 2 EX, if E(Y EY ) + = 0; E[X(Y p 1 )], if E(Y p ) 0, 3 (X, Y ) = [E(Y p )] 1 p 1 0, if E(Y p ) = 0. Theorem 1.2 Let c 0 be a real number, V = L 1 be the set of all random variables in L 0 such that X is integrable. Let p 1 be a real number, V = L p + be the set of all non-negative random variables in L 0 such that X p is integrable. Then (1) 1 (, ) is a linear, diversifying and continuous capital allocation with respect to ρ 1, and 1 (X, Y ) = lim ρ 1 (Y + ɛx) ρ 1 (Y ) ɛ (1.1) for every X V. (2) 2 (, ) is a linear, diversifying capital allocation with respect to ρ 2. If E(Y EY ) + > 0, then 2 (, ) is continuous at Y, and 2 (X, Y ) = lim ρ 2 (Y + ɛx) ρ 2 (Y ) ɛ (1.2) for every X V. (3) 3 (, ) is a linear, diversifying capital allocation with respect to ρ 3. If EY p 0, then 3 (, ) is continuous at Y, and 3 (X, Y ) = lim ρ 3 (Y + ɛx) ρ 3 (Y ) ɛ (1.3) for every X V. Proof (1) Obviously, 1 (X, X) = ρ 1 (X), and 1 (, ) is linear. Moreover, E[(X EX)1 {Y EY } ] = E[(X EX)1 {Y EY } 1 {X EX} ] + E[(X EX)1 {Y EY } 1 {X<EX} ] E[(X EX)1 {Y EY } 1 {X EX} ] E[(X EX)1 {X EX} ], so 1 (X, X) 1 (X, Y ), i.e., 1 (, ) is diversifying. Since lim 1 {Y +ɛx EY +ɛex} = 1 {Y EY }, so 1 (, ) is continuous at every Y V. By Proposition 1.1, Equality (1.1) follows.

6 6, : 151 (2) It s easy to prove 2 (X, X) = ρ 2 (X), and 2 (, ) is linear. If E(X EX) + > 0, E(Y EY ) + > 0, then 2 (X, X) = c{e[(x EX) + ] 2 } 1 2 Cauchy-Schwarz inequality, + EX. By so 2 (X, Y ) 2 (X, X). E[(X EX)(Y EY ) + ] E[(X EX) + (Y EY ) + ] {E[(X EX) + ] 2 } 1 2 {E[(Y EY )+ ] 2 } 1 2, If E(X EX) + > 0, E(Y EY ) + = 0, then 2 (X, Y ) = EX 2 (X, X). If E(X EX) + = 0, E(Y EY ) + > 0, then X EX, hence, 2 (X, Y ) EX = 2 (X, X). If E(X EX) + = 0, E(Y EY ) + = 0, then 2 (X, Y ) = EX = 2 (X, X). Therefore, 2 (, ) is diversifying. When E(Y EY ) + > 0, following from lim(y + ɛx EY ɛex) = Y EY, lim {E[(Y + ɛx EY ɛex) +] 2 } 1 2 = lim {E[(Y EY ) + ] 2 } 1 2, we obtain its continuous property at Y. Finally, by Proposition 1.1, Equality (1.2) holds. (3) 3 (X, X) = [E(X p )] 1 p = ρ3 (X), and linear property is easy to verify. Since X 0, Y 0, when E(X p ) > 0, E(Y p ) > 0, by H older inequality, E(XY p 1 ) [E(X p )] 1 p [E(Y p 1 ) p p 1 p 1 ] p = [E(X p )] 1 p [E(Y p ] 1 1 p, so 3 (X, Y ) 3 (X, X). In other cases, we can verify diversifying property by classifying just as step (2) did, we omit them here. When EY p 0, lim E[X(Y + ɛx) p 1 ] 3(X, Y + ɛx) = lim = [E(Y + ɛx) p ] (X, Y ). p Therefore, the conclusion is proved. And Equality (1.3) follows from Proposition 1.1. In order to have a direct recognition about application of the axiomatic allocation, a specific numerical example is given to conclude this section. Example 1.1 Let P (λ) represent Poisson distribution with parameter λ. Suppose X 1 P (1), X 2 P (2), and they are independent. Put X = X 1 + X 2, then X P (3) following from additivity of Poisson distributions for independent random variables. And it is easy to know, EX 1 = 1, EX 2 = 2, and EX = 3. In view of the axiom system, we define a subspace V, such that {X 1, X 2, X} V. ρ 1 ( ) is used to measure risk, by Theorem 1.2, 1 (, ) is the relevant allocation. Therefore, 1 (X 1, X) = ce[(x 1 1)1 {X 3} ] + 1 = ce(x 1 1) ce[(x 1 1)1 {X=0,1,2} ] + 1 = c{e[(x 1 1)1 {X=0} ] + E[(X 1 1)1 {X=1} ] + E[(X 1 1)1 {X=2} ]} + 1 def = c(i 1 + I 2 + I 3 ) + 1,

7 152 ( ) 2010 Note that I 1 = E[(X 1 1)1 {X=0} ] = E[(X 1 1)1 {X1=0}1 {X2=0}] = E[(X 1 1)1 {X1=0}]P(X 2 = 0) = e 1 e 2 = e 3. Similarly, I 2 = 2e 3, I 3 = 3 2 e 3. Hence, 1 (X 1, X) = 9c 2 e By the same way, 1 (X 2, X) = 9ce 3 + 2, and note that 1 (X 2, X) = 2 1 (X 1, X). 2 Application of generalized weighted capital allocation Furman and Zitikis [4] considered weighted risk capital allocation. In this section, we will extend their model to a more general one: generalized weighted allocation. Then we give an example for its application, and propose a comparing method with the associated numerical example. Let X 0 be a risk random variable with cumulative distribution function F X ( ), and let χ denote the set of all such random variables, which are defined on probability space (Ω, F, P). Heilmann [10] and Kamps [11] stated weighted premium, which was called weighted functional in Furman and Zitikis [4]. H w : χ [0, + ] is defined by H w (X) = E[Xw(X)] E[w(X)], where w : [0, + ) [0, + ) is a deterministic and Borel-measurable function, called weight function, and it is usually chosen by the decision maker depending on the problem at hand. Furman and Zitikis [12] gave some special cases of this premium. Definition 2.1 Let X, Y χ, and w : [0, + ) [0, + ) is a weight function. The functional w : χ χ [0, + ] is called weighted allocation induced by the weighted premium principle H w ( ), if it is defined by Remark 2.1 w (X, Y ) = E[Xw(Y )] E[w(Y )]. (1) If Y represents the aggregate loss faced by an insurance company, and X represents the loss faced by one of its business units, w (X, Y ) means the amount of capital allocated to X. A number of well known allocation rules are special cases of w (, ), such as those in [13-15]. (2) Especially, if w(y) = 1 {y F 1 Y (α)} (0 α 1), w (X, Y ) = E[X Y F 1 Y (α)]. This CTE allocation is a linear and diversifying capital allocation, which was proved in [4, Example 3.1]. Now, we extend the definitions of weighted premium and weighted allocation to more general cases in the following. Definition 2.2 weighted premium, if it is defined by Let X χ, the functional H v,w : χ [0, + ] is called generalized H v,w (X) = E[v(X)w(X)], E[w(X)]

8 6, : 153 where v : [0, + ) [0, + ), w : [0, + ) [0, + ) are weight functions. Definition 2.3 Let X, Y χ, the functional v,w : χ χ [0, + ] is called generalized weighted allocation induced by the generalized weighted premium H v,w ( ), if it is defined by v,w (X, Y ) = where v( ) and w( ) are weight functions. E[v(X)w(Y )], E[w(Y )] Here, we make some statements about this definition. Compared to w (, ), v,w (, ) is more general. We can assign different weights to X by varying v( ). Especially, We may think of v( ) as a utility function, which is usually nondecreasing and concave. Remark 2.2 (1) When v(x) = x, H v,w ( ) reduces to the classical weighted premium. On the other hand, when Y = X, we get v,w (X, X) = H v,w (X), that s why we say that v,w (X, Y ) is induced by H v,w (X). (2) Define p(y) = E[v(X) Y = y], and a weighted cumulative distribution function F w,y (y) = E[1 {Y y}w(y )]. E[w(Y )] Then v,w (X, Y ) can be viewed as the mean p(y)df w,y (y) with respect to F w,y (y). And we can refer to [16] for a detailed study about weighted distribution function. In view of the statements mentioned above, we now consider an example about generalized weighted allocation, which mainly verifies diversifying property. Example 2.1 For generalized weighted allocation v,w (, ), we assume v( ) is nondecreasing, and w(y) = 1 {y F 1 Y (α)} (0 α 1). If both F X( ) and F Y ( ) are continuous, then v,w (X, Y ) = E[v(X)1 {y F 1 (α)}] Y P(y F 1 Y (α)) is diversifying. Proof Let 0 α 1. Denote x α = F 1 X (α), y α = F 1 Y (α), so v,w (X, Y ) = E[v(X) Y y α ]. Both F X ( ) and F Y ( ) are continuous, and hence both P(X x α ) and P(Y y α ) are equal to 1 α. Then the diversifying property v,w (X, Y ) H v,w (X) is equivalent to E[v(X)d(X, Y )] 0, (2.1) where d(x, Y ) = 1 {X xα} 1 {Y yα}. So we just need to verify Inequality (2.1): E[v(X)d(X, Y )] = E[v(X)d(X, Y )1 {X xα}] + E[v(X)d(X, Y )1 {X<xα}] E[v(x α )d(x, Y )1 {X xα}] + E[v(x α )d(x, Y )1 {X<xα}] = v(x α )E[d(X, Y )] = 0, where the inequality follows from that d(x, Y )1 {X xα} 0 and d(x, Y )1 {X<xα} 0, and v(x) is non-decreasing. Therefore, we get the conclusion.

9 154 ( ) 2010 Note 2.1 For generalized weighted allocation v,w (, ), its verification of diversifying property is not easy. Here, we will give a general comparing method, not only restricted to verification of diversifying property. Just as Note 2.1 in [4] stated, the weighted function w( ) might depend on F Y ( ). So in precise way, we may write w FY ( ) or w Y ( ), and v,w (, ) is written to be v,wy (, ). But when it does not cause a confusion, we prefer the simpler way w( ) and v,w (, ). Theorem 2.1 Let X, Y and Z be elements of χ, and w Y, w Z be two weight functions, possibly depending on F Y ( ) and F Z ( ), respectively. Then one of the following three relations holds. depending on whether the function v,wy (X, Y ) = v,wz (X, Z), r w (x) = E[w Z(Z) X = x] E[w Y (Y ) X = x] is, respectively, non-decreasing, constant, or non-increasing. Proof By property of conditional expectation, we know v,wy (X, Y ) = E[v(X)h Y (X)] E[h Y (X)] and v,wz (X, Z) = E[v(X)h Z(X)], E[h Z (X)] where h Y (x) = E[w Y (Y ) X = x] and h Z (x) = E[w Z (Z) X = x]. Obviously, they are functions with respect to x, not depending on Y or Z. Denote r w (x) = h Z(x) h Y (x). Now, we consider h Y (x) and h Z (x) as new weight functions, and denote them by w (x) and w (x) respectively. Following from Inequality (4.3) in Section 4 of Furman and Zitikis [12], if r w (x) is non-decreasing, then v,wy (X, Y ) v,wz (X, Z), which completes the first relation s proof. Similarly, proofs of the next two relations follow. Finally, a special numerical example of this method is given for application. Example 2.2 function. By Theorem 2.1, if Assume Z = X, and w is not depending on any cumulative distribution r w (x) = w(x) E[w(Y ) X = x] (2.2) is non-decreasing, we have v,w (X, Y ) v,w (X, X). (2.3) Put ξ = Y X, and assume that X is independent of other subportfolios. formula (2.2) reduces to r w (x) = w(x) E[w(x + ξ)]. Therefore, the

10 6, : 155 In order to check that (2.2) is non-deceasing, we put w(x) = x 2, and then Since its derivative r w (x) = x 2 E(x + ξ) 2. (2.4) r w(x) = 2x{E(x + ξ)2 E[x(x + ξ)]} [E(x + ξ) 2 ] 2 is non-negative, following that x 0 and ξ 0 (recall that w : [0, + ) [0, + ), and ξ χ). Hence, (2.4) is non-decreasing, Inequality (2.3) follows. In addition to w(x) = x 2, other many weight function can been chosen, we can refer to the table in Section 2 in [12]. 3 Conclusions In this paper, allocation formulae with respect to several important risk measures under the axiom system are obtained, and allocation results of a specific model are given. Then we consider generalized weighted allocation, and study an important example of this allocation. At last, just as Section 2 states, a method for comparing generalized weighted capital allocations under two different aggregate losses is given, and some special cases are also given. [ References ] [ 1 ] CUMMINS J D. Allocation of capital in the insurance industry [J]. Risk Management and Insurance Review, 2000, 3(1): [ 2 ] DENAULT M. Coherent allocation of risk capital [J]. Journal of Risk, 2001, 4(1): [ 3 ] KALKBRENER M. An axiomatic approach to capital allocation [J]. Mathematical Finance, 2005, 15(3): [ 4 ] FURMAN E, ZITIKIS R. Weighted risk capital allocations [J]. Insurance: Mathematics and Economics, 2008, 43: [ 5 ] DHAENE J, HENRARD L, LANDSMAN Z, et al. Some results on the CTE-based capital allocation rule [J]. Insurance: Mathematics and Economics, 2008, 42: [ 6 ] DHAENE J, GOOVAERTS M J, KAAS R. Economic capital allocation derived from risk measures [J]. North American Actuarial Journal, 2003, 7(2): [ 7 ] FISCHER T. Risk capital allocation by coherent risk measures based on one-sided moments [J]. Insurance: Mathematics and Economics, 2003, 32: [ 8 ] ARTZNER P, DELBAEN F, EBER J M, et al. Coherent measures of risk [J]. Mathematical Finance, 1999, 9(3): [ 9 ] VAN HEERWAARDEN A E, KAAS R. The Dutch premium principle [J]. Insurance: Mathematics and Economics, 1992(11): [10] HEILMANN W R. Decision theoretic foundations of credibility theory [J]. Insurance: Mathematics and Economics, 1989(8): [11] KAMPS U. On a class of premium principles including the Esscher premium [J]. Scandinavian Actuarial Journal, 1998(1): [12] FURMAN E, ZITIKIS R. Weighted premium calculation principles [J]. Insurance: Mathematics and Economics, 2008, 42: [13] GOOVAERTS M J, LAEVEN R J A. Actuarial risk measures for financial derivative pricing [J]. Insurance: Mathematics and Economics, 2008, 42: [14] SCHECHTMAN E, SHELEF A, YITZHAKI S, et al. Testing hypotheses about absolute concentration curves and marginal conditional stochastic dominance [J]. Econometric Theory, 2008, 24: [15] TSANAKAS A. Risk measurement in the presence of background risk [J]. Insurance: Mathematics and Economics, 2008, 42: [16] PATIL G P, RAO C R, RATNAPARKHI M V. On discrete weighted distributions and their use in model choice for observed data [J]. Communications in Statistics, A: Theory and Methods, 1986, 15: