Some results on the risk capital allocation rule induced by the Conditional Tail Expectation risk measure
aa r X i v : . [ q -f i n . R M ] F e b Some results on the risk capital allocation rule induced by theConditional Tail Expectation risk measure
Nawaf Mohammed ∗ †
Edward Furman ‡ Jianxi Su § Abstract
Risk capital allocations (RCAs) are an important tool in quantitative risk management, wherethey are utilized to, e.g., gauge the profitability of distinct business units, determine the price ofa new product, and conduct the marginal economic capital analysis. Nevertheless, the notion ofRCA has been living in the shadow of another, closely related notion, of risk measure (RM) in thesense that the latter notion often shapes the fashion in which the former notion is implemented. Infact, as the majority of the RCAs known nowadays are induced by RMs, the popularity of the twoare apparently very much correlated. As a result, it is the RCA that is induced by the ConditionalTail Expectation (CTE) RM that has arguably prevailed in scholarly literature and applications.Admittedly, the CTE RM is a sound mathematical object and an important regulatory RM, butits appropriateness is controversial in, e.g., profitability analysis and pricing. In this paper, weaddress the question as to whether or not the RCA induced by the CTE RM may concur withalternatives that arise from the context of profit maximization. More specifically, we provideexhaustive description of all those probabilistic model settings, in which the mathematical and regulatory
CTE RM may also reflect the risk perception of a profit-maximizing insurer.
Key words and phrases : Conditional tail expectation-based allocation, conditional geometric tailexpectation-based allocation, conditional covariance, size-biased transform, standard simplex.
JEL Classification : C60, C61. ∗ Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada. † Corresponding author; postal address: 4700 Keele St, Toronto, ON M3J 1P3, Canada; email: [email protected] ‡ Department of Mathematics and Statistics, York University, Toronto, ON M3J 1P3, Canada. § Department of Statistics, Purdue University, West Lafayette, IN 47906, U.S.A. Introduction
Consider positive random variables (RVs) X , . . . , X n , n ∈ N , which represent losses due to distinctbusiness units (BUs) of an insurer, and denote by the sets N = { , . . . , n } and X the collections ofthese BUs and losses, respectively. Then, for the aggregate loss RV S X := X + · · · + X n , the map A : X × X → [0 , ∞ ) ∪ { + ∞} , which assigns finite or infinite values to random pairs ( X, S ) ∈ X × X , iscalled a risk capital (RC) allocation rule (e.g., Denault, 2001; Dhaene et al., 2012; Furman and Zitikis,2008b). Additionally, if A ( X, X ) = H ( X ), where the map H : X → [0 , ∞ ) ∪ { + ∞} is called a riskmeasure and assigns finite or infinite values to the random loss X ∈ X , then the allocation rule A issaid to be induced by the risk measure H .RC allocation rules have gained major importance in risk management and insurance applicationsin the context of price determination, profitability assessment, budgeting decision making, to namejust a few (Guo et al., 2020; Venter, 2004). Similarly, the academic significance of - also, interest in -the subject of RC allocations have been strong, as evidenced by the large and growing body of schol-arly literature (e.g., Boonen et al., 2019; Furman et al., 2020a,c; Kim and Kim, 2019; Shushi and Yao,2020, for recent references in the Insurance: Mathematics and Economics journal, alone).Not surprisingly, therefore, numerous RC allocation rules have been proposed and studied, withthe RC allocation rule induced by the conditional tail expectation (CTE) risk measure being arguablythe most popular (Kalkbrener, 2005). More specifically, for q ∈ [0 , s q := VaR q ( S X ) = inf (cid:8) s ∈ [0 , ∞ ) : P ( S X ≤ s ) ≥ q (cid:9) , loss portfolio X = ( X , . . . , X n ) ∈ X n and BU i ∈ N , the CTE-based RCallocation rule, when well-defined and finite, is given byCTE q ( X i , S X ) = E [ X i | S X > s q ] . (1)As it is common in real applications to use RC allocation (1) - also, other RC allocation rules - toattribute the exogenous aggregate risk capital, say κ ∈ R + , to distinct BUs in the set N , and, in orderto guarantee the total additivity of the RC allocation rule, it is beneficial to explore the quantity (e.g.,2haene et al., 2012), κ i = κ × r q,i , where r q,i = CTE q ( X i , S X )CTE q ( S X ) , i ∈ N (2)is the associated proportional RC allocation rule induced by the CTE risk measure CTE q ( X ) =CTE q ( X, X ) for any X ∈ X and q ∈ [0 , r q,i raises a number of concerns:(a) it hinges on a contentious two-step procedure (Chong et al., 2019), and (b) it neglects the riskperception of the insurer and the economic environment in which they operate (Bauer and Zanjani,2016). Admittedly, it is not surprising in any way that regulations, which are driven by the notion of3rudence, and insurers’ targets, which are profit-oriented, diverge. Nevertheless, it is instrumental todetermine whether or not there exist model settings under which the RC allocation rule induced bythe CTE risk measure yields outcomes that address Points (a) and (b) above. This is what we do inthe present paper.We have organized the rest of this papers as follows. In Section 2, we motivate in detail andformulate the problem of interest. We then solve this problem in Sections 3 and 4, which provideample elucidating examples. Some of our analysis and conclusions carry over to a family of riskmeasures that contains the CTE risk measure as a special case, which is demonstrated in Section 5.Section 6 concludes the paper. Note that the CTE-based allocation exercise (2) can be framed within the context of the standard n -dimensional simplex space (Aitchison, 1986): S n = (cid:8) ( r , . . . , r n ) : r i ∈ [0 , , i = 1 , . . . , n and r + · · · + r n = 1 (cid:9) . Specifically, for x , . . . , x n ∈ [0 , ∞ ) and s := x + · · · + x n , as well as for the special map C : [0 , ∞ ) n → S n with C i ( x , . . . , x n ) = x i /s, i = 1 , . . . , n , proportional allocation rule (2) is obtained via setting x i = CTE q ( X i , S X ), and so (Belles-Sampera et al., 2016; Boonen et al., 2019) r q,i = C i (cid:0) CTE q ( X , S X ) , . . . , CTE q ( X n , S X ) (cid:1) = CTE q ( X i , S X ) / CTE q ( S X ) , q ∈ [0 , . We note in passing that a similar reformulation of the RC allocation exercise in the context of the n -dimensional simplex space can be achieved effortlessly for the whole class of weighted RC allo-cation rules (Furman and Zitikis, 2008b), which are induced by the class of weighted risk measures(Furman and Zitikis, 2008a) and of which allocation rule (1) is a special case (Furman et al., 2020b).An alternative way to determine the proportional contribution of the i -th BU of an insurer to the4ggregate risk capital - under the assumption that it is the CTE risk measure that induces the desiredallocation rule - is by considering the ratio RV R i = X i / S X , i = 1 , . . . , n , directly. That is, while, fora fixed q ∈ [0 , r q,i , confined with the help of the normalizing constantCTE q ( S X ) ∈ R + to the unit interval, I = [0 , X i , S X ) ∈ X × X , analternative to r q,i proportional allocation, call it ˜ r q,i , is chosen to operate on random pairs ( R i , S X ) ∈I × X , i = 1 , . . . , n , and so˜ r q,i = CTE q ( C i ( X , . . . , X n ) , S X ) = CTE q ( R i , S X ) , q ∈ [0 , . While various properties of the proportional allocation r q,i have been well-studied, this is not sofor its counterpart, ˜ r q,i . Further, we report a number of important properties of the latter quantity.In this respect, our first proposition shows that the quantity ˜ r q,i agrees with the economic capitalallocation rule proposed recently by Bauer and Zanjani (2016). Namely, while the motivation forthe RC allocation rule r q,i is the central role that the CTE risk measure plays in today’s (insurance)regulation, the proportional allocation ˜ r q,i turns out to be a well-justified choice for a profit maximizinginsurer with risk-averse counterparties in an incomplete market setting with frictional capital costs.Consider the aggregate loss RV S X ∈ X and the Geometric Tail Expectation (GTE) risk measure:GTE q ( S X ) := exp (cid:8) E [log( S X ) | S X > s q ] (cid:9) , q ∈ [0 , . (3)The connection of risk measure (3) to the notion of geometric means (e.g., Hardy et al., 1952) mo-tivates its name; also, risk measure (3) is a tail quasi-linear mean risk measure in the sense ofB¨auerle and Shushi (2020). It is not difficult to see that, for any q ∈ [0 , q ( S X ) ≤ GTE q ( S X ) ≤ CTE q ( S X ) , q ∈ [0 , . Another immediate but worth-mentioning observation is that risk measure (3) is neither coherent5n the sense of Artzner et al. (1999) nor convex in the sense of F¨ollmer and Schied (2002), as it violatestranslation-invariance. Nevertheless, when viewed through the prism of a profit maximizing insurer,risk measure (3) induces the optimal RC allocation outcome, ˜ r q,i ; intuitively, this might be due tothe decreasing marginal effect of the increase in aggregate loss (Bauer and Zanjani, 2016). Our nextstatement about the RC allocation induced by risk measure (3) is formulated as a proposition. Proposition 1.
The GTE-based RC allocation, ˜ r q,i , q ∈ [0 , , is the gradient allocation in thedirection of the loss RV X i ∈ X , i ∈ N induced by risk measure (3) .Proof. Since the GTE risk measure is positively homogenious, by Euler’s theorem we have, for u =( u , . . . , u n ) ∈ I n and S X ( u ) := u X + · · · + u n X n ,GTE q (cid:0) S X ( u ) (cid:1) = n X i =1 u i ∂∂u i GTE q (cid:0) S X ( u ) (cid:1) , with q ∈ [0 , i ∈ N . Therefore, for denoting the n -variate vector of ones, we obtain ∂∂u i GTE q (cid:0) S X ( u ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) u = = (cid:18) GTE q (cid:0) S X ( u ) (cid:1) × E (cid:20) X i S X ( u ) (cid:12)(cid:12)(cid:12)(cid:12) S X ( u ) > VaR q ( S X ( u )) (cid:21)(cid:19) (cid:12)(cid:12)(cid:12)(cid:12) u = = GTE q (cid:0) S X (cid:1) × ˜ r q,i , i ∈ N .Clearly, P ni =1 GTE q (cid:0) S X (cid:1) × ˜ r q,i = GTE q (cid:0) S X (cid:1) × P ni =1 ˜ r q,i = GTE q (cid:0) S X (cid:1) for q ∈ [0 , Proposition 2.
The proportional allocation r q,i is a linear approximation of the proportional allocation ˜ r q,i for i ∈ N .Proof. Consider the function g ( x i , s ) = x i /s for x i , s ∈ R + , i = 1 , . . . , n , and denote its partialderivatives by g i ( x i , s ) = ∂∂x i g ( x i , s ) and g s ( x i , s ) = ∂∂s g ( x i , s ) . g around ( x , s ) = (cid:0) CTE q ( X i , S X ) , CTE q ( S X ) (cid:1) , yields x i / s = g ( x , s ) + g i ( x , s ) ( x i − x ) + g s ( x , s ) ( s − s ) + R ( x i , s ) , where R ( x i , s ) is the reminder term for all x i , s ∈ R + , q ∈ [0 , i = 1 , . . . , n . Consequently, we have˜ r q,i ≈ r q,i + g i ( x , s ) E (cid:2) ( X i − x ) | S X > s q (cid:3) + g s ( x , s ) E (cid:2) ( S X − s ) | S X > s q (cid:3) = r q,i , which establishes the desired approximation and thus completes the proof of the proposition.Finally, our last proposition - which can be considered a follow-up on Proposition 2 - delineates thedifference between the allocations r q,i and ˜ r q,i . The proof is an immediate consequence of the identityCov( R i , S X | S X > s q ) = E [ X i | S X > s q ] − E [ R i | S X > s q ] × E [ S X | S X > s q ] , which holds for all q ∈ [0 ,
1) and ( X i , S X ) ∈ X × X , i = 1 , . . . , n . Proposition 3.
Given that all the quantities below are well-defined and finite, we have r q,i = ˜ r q,i + Cov( R i , S X | S X > s q )CTE q ( S X ) , q ∈ [0 , , i ∈ N . Proposition 3 implies that it is the sign of the covariance between the RVs R i and S X , thatdetermines the order of the allocations r q,i and ˜ r q,i ; note that, as R + · · · + R n = 1 almost surely, wehave P ni =1 Cov( R i , S X | S X > s q ) = 0 (Furman and Zitikis, 2010, for examples, albeit in a differentcontext, of the importance of covariances in insurance and finance). Also, and more importantly,Proposition 3 suggests that the proportional allocation rules induced by the CTE risk measure andthose induced by risk measure (3) coincide if the aforementioned covariance is nil. This motivates thefollowing question that engages us in the rest of this paper. Question 1.
For X i ∈ X , i ∈ N , can we characterize those portfolios of losses X = ( X , . . . , X n ) ∈X n , for which the proportional allocations r q,i and ˜ r q,i agree for every q ∈ [0 , ?
7t the outset, let us note that if the answer to the question above were in the affirmative, thenthis would imply that the regulatory (e.g., Swiss Solvency Test) CTE risk measure induces an optimalRC allocation rule for a profit maximizing insurer; the richer the class of joint CDFs of the RV X ∈X n sought in Question 1, the more common the just-mentioned and apparently desirable agreementbetween the regulatory requirements and the risk perceptions of insurers.Speaking formally, our goal is to characterize the following collection of loss RVs: W = (cid:8) X = ( X , . . . , X n ) ∈ X n : r q,i = e r q,i for all q ∈ [0 ,
1) and i ∈ N (cid:9) . (4)It is to be noted that the set W can not be empty. To see a trivial case in which ˜ r q,i = r q,i for every q ∈ [0 ,
1) and i ∈ N , let the loss RV X = ( X , . . . , X n ) ∈ X n have identically distributed coordinates, X i ∈ X i , and an exchangeable copula function, then we have E [ R i | S X = s ] = 1 /n resulting inCov( R i , S X | S X = s ) ≡
0, and therefore Cov( R i , S X | S X > s q ) = 0 for all q ∈ [0 ,
1) and i = 1 , . . . , n .Consequently, the set of all loss RVs X = ( X , . . . , X n ) ∈ X n , such that the proportional allocationrules r q,i and ˜ r q,i coincide has at least one portfolio of losses in it.We devote the following sections of this paper to studying what other random loss RVs - besidesthe trivial example above - are members of the set W . In the sequel, we routinely work with anatomless and rich probability space (Ω , F , P ), and we let L α and L ∞ denote, respectively, the set ofall RVs that have finite α -th moment, α ∈ [0 , ∞ ), and the set of all essentially bounded RVs on theprobability space (Ω , F , P ). For every X ∈ L , we denote by F X and φ X the CDF and the Laplacetransform of the RV X , and we use to denote the indicator function. In this section, we devise the necessary and sufficient conditions for the equality, ˜ r q,i = r q,i , for all q ∈ [0 ,
1) and i ∈ N . For this, we need a few auxiliary notions first. That is, Definitions 1 and2 below introduce the univariate size-biased transform and its multivariate extension (Arratia et al.,2019; Furman et al., 2020c; Patil and Ord, 1976), both playing major roles in our analysis.8 efinition 1. Let X ∈ L α be a positive loss RV, then the size-biased counterpart of order α ∈ R + ofthe loss RV X , call it X [ α ] , is defined via: P (cid:16) X [ α ] ∈ dx (cid:17) = x α E [ X α ] P ( X ∈ dx ) for all x ∈ R + . (5) When α = 1 , we simply write X ∗ for the size-biased of order one variant of the RV X ∈ L . The RVs X and X [ α ] are independent for all α ∈ R + . Definition 2.
Let X = ( X , . . . , X n ) ∈ X n be a loss RV with positive univariate coordinates X i ∈ L α i , i = 1 , . . . , n , then the multivariate size-biased counterpart of order α = ( α , . . . , α n ) ∈ R n + of theloss RV X , call it X [ α ] , is defined via P (cid:16) X [ α ] ∈ d x (cid:17) = x α × · · · × x α n n E [ X α × · · · × X α n n ] P ( X ∈ d x ) for all x = ( x , . . . , x n ) ∈ R n + . (6) The RVs X and X [ α ] are independent. We next define the partial size-biased transform , which is a useful special case of the one presentedin Definition 2 (e.g., Arratia et al., 2019; Furman et al., 2020c, for a few recent results in which thepartial size-biased transform plays a central role but is not explicitly defined).
Definition 3.
Consider the size-biased RV X [ α ] , α = ( α , . . . , α n ) as per Definition 2. Then, in thespecial case where the i -th coordinate of the vector α is equal to α i = α ∈ R + , i = 1 , . . . , n , whereasall other coordinates of the vector α are equal to zero, we call the implied transform, the i -th partialsize-biased transform of order α , and denote the corresponding RV by X [( α ) i ] . Namely, we have P (cid:16) X [( α ) i ] ∈ d x (cid:17) = x αi E [ X αi ] P ( X ∈ d x ) for all x = ( x , . . . , x n ) ∈ R n + . (7) The RVs X and X [( α ) i ] are independent. For the case α = 1 and X i ∈ L , we simply write, X ( ∗ ) i ,for the partial size-biased counterpart of the RV X . The operation of size-biasing has an important interpretation in the context of actuarial scienceand, more generally, in quantitative risk management, where it is considered loading for model/sample9isk. Indeed, it is not difficult to see that the size-biased loss RVs X [ α ] and X [ α ] (also, X ( α ) i ) dominatestochastically the loss RVs X and X , respectively.Furthermore, the partial size-biased RV, X [( α ) i ] , plays an important role for size-biasing sums ofRVs. Namely, let S ∗ X = ( X + · · · + X n ) ∗ , then the distribution of the RV S ∗ X admits a finite-mixturerepresentation (e.g., Arratia et al., 2019) in terms of the partial size-biased RVs. Indeed, let φ S ∗ X denote the Laplace transform of the RV S ∗ X , then, for p i = E [ X i ] / E [ S X ], we have φ S ∗ X ( t ) = n X i =1 p i × φ S ( ∗ ) iX ( t ) , Re( t ) > , (8)where S ( ∗ ) i X is the sum of the coordinates of the partially size-biased RV X ( ∗ ) i , i = 1 , . . . , n .The following lemma is a variation of Equation (8) that we find useful in this paper. Lemma 1.
Consider the RV X + = ( X , . . . , X n , Y n +1 , . . . , Y n + m ) ∈ X n + m , n, m ∈ N , and let S X = P ni =1 X i , S Y = P n + mi = n +1 Y i , S + = S X + S Y , and S ( ∗ ) X + = S ∗ X + S Y . Then the distribution of theRV S ( ∗ ) X + admits a mixture representation in the sense that we have S ( ∗ ) X + = d S ( ∗ ) K + , where the RV K ∈ { , . . . , n } is such that P ( K = k ) = E [ X i ] / E [ S X ] , k = 1 , . . . , n Proof.
Let φ S ( ∗ ) X + denote the Laplace transform of the RV S ( ∗ ) X + , then, with the help of Equation (7),we have, for p i = E [ X i ] / E [ S X ], φ S ( ∗ ) X + ( t ) = E (cid:2) S X e − tS + (cid:3) E [ S X ] = n X i =1 p i E (cid:2) X i e − tS + (cid:3) E [ X i ] = n X i =1 p i × φ S ( ∗ ) i + ( t ) , Re( t ) > , which establishes the desired result and thus completes the proof.The next assertion spells out the sufficient and necessary conditions for the loss portfolios X =( X , . . . , X n ) ∈ X n to belong to the set W , and hence it answers Question 1. The non-technicalinterpretation of the assertion is that for loss portfolios in the set W and under the paradigm ofloading for model/sample risk, the choice of the load direction as per Definition 3 does not impact thedistribution of the loaded aggregate loss RV. Theorem 1.
Consider the loss RV X = ( X , . . . , X n ) ∈ X n and assume that X i ∈ L , then we have he equality r q,i = ˜ r q,i (= E [ X i ] / E [ S X ]) for all q ∈ [0 , , i ∈ N , if and only if S ( ∗ ) i X = d S ( ∗ ) j X (= d S ∗ X ) , i = j ∈ N .Proof. Assume that r q,i = ˜ r q,i for all q ∈ [0 ,
1) and i = 1 , . . . , n . By Proposition 3, this is equivalentto requesting that Cov( R i , S X | S X > u ) = 0 for all u ≥
0, or, in other words with the notation G i ( s ) = E [ R i | S = s ] − E [ R i ] , i = 1 , . . . , n , that E (cid:2) S X G i ( S X ) | S X > u (cid:3) = E (cid:2) G i ( S X ) | S X > u (cid:3) E [ S X | S X > u ] for all u ≥ , from which we must have G i ( u ) = E [ G i ( S X ) | S X > u ] for all u ≥ . Hence, G i ( u ) ≡ const, which alongside the fact that E [ G i ( S X )] = 0, implies G i ( u ) ≡
0. Furthermore,as we assumed that r q,i = ˜ r q,i for all q ∈ [0 , r ,i = ˜ r ,i , and so E [ R i | S X ] = E [ X i ] E [ S X ]or, equivalently, E [ X i | S X ] = E [ X i ] E [ S X ] S X for i = 1 , . . . , n . Finally, we have the following implication in terms of the Laplace transform of theloss RV S ( ∗ ) i X and for i = 1 , . . . , n , φ S ( ∗ ) iX ( t ) = E [ X i e − tS X ] E [ X i ] = E (cid:2) E [ X i | S X ] e − t S X (cid:3) E [ X i ] = E [ S X e − tS X ] E [ S X ] = φ S ∗ X ( t ) , Re( t ) > . This implies S ( ∗ ) i X = d S ( ∗ ) j X for all 1 ≤ i = j ≤ n and so completes the ‘only if’ direction of the theorem.In order to prove the ‘if’ direction of the theorem, let us assume that the distributional equality S ( ∗ ) i X = d S ( ∗ ) j X (= d S ∗ X ) holds for all i = 1 , . . . , n , which means E [ X i e − tS X ] E [ X i ] = E [ S X e − tS X ] E [ S X ]11r, equivalently, E (cid:20) E [ X i | S X ] E [ X i ] e − tS X (cid:21) = E (cid:20) S X E [ S X ] e − tS X (cid:21) , with the immediate implication E [ R i | S X = u ] = E [ X i ] E [ S X ] for all u ≥ . Therefore, we necessarily have E [ R i ] = E [ X i ] / E [ S X ], i = 1 , . . . , n . Finally, we obtain the followingstring of equations:Cov( R i , S X | S X > u ) = E [ R i S X | S X > u ] − E [ R i | S X > u ] E [ S X | S X > u ]= E (cid:2) E [ R i | S X ] S X (cid:12)(cid:12) S X > u (cid:3) − E (cid:2) E [ R i | S X ] (cid:12)(cid:12) S X > u (cid:3) E [ S X | S X > u ]= E [ X i ] E [ S X ] E [ S X | S X > u ] − E [ X i ] E [ S X ] E [ S X | S X > u ]= 0for all u ≥ i = 1 , . . . , n . The ‘if’ direction of the theorem is then proved by evoking Proposition3. This completes the proof of the theorem.Some properties of the portfolios of losses X ∈ W are studied next. Specifically, it turns out thatthese portfolios are consistent in the sense that the answer to Question 1 must be in affirmative forall their sub-portfolios. This is formulated and proved next. Theorem 2.
Consider the loss RV X + = ( X , . . . , X n , Y n +1 , . . . , Y n + m ) ∈ X n + m and assume that X + ∈ W , then the sub-portfolios ( X , . . . , X n ) and ( Y n +1 , . . . , Y n + m ) also belongs to the set W .Proof. We prove that if X + ∈ W , then X = ( X , . . . , X n ) ∈ W ; the case Y = ( Y n +1 , . . . , Y n + m ) ∈ W follows in the same fashion. Let S X = P ni =1 X i , S Y = P n + mi = n +1 Y i and S + = S X + S Y , as in Lemma 1.Because X + ∈ W and by Theorem 1, we have, for all i = j ∈ { , . . . , n } and Re( t ) > φ S ( ∗ ) i + ( t ) := E (cid:2) exp {− t S ( ∗ ) i + } (cid:3) = E (cid:2) X i e − t ( S X + S Y ) (cid:3) E [ X i ]12 E (cid:2) X j e − t ( S X + S Y ) (cid:3) E [ X j ] = E (cid:2) exp {− t S ( ∗ ) j + } (cid:3) =: φ S ( ∗ ) j + ( t ) . (9)Therefore, we have E " e − t S Y E (cid:20) X i E [ X i ] e − tS X (cid:12)(cid:12)(cid:12) S Y (cid:21) = E " e − t S Y E (cid:20) X j E [ X j ] e − tS X (cid:12)(cid:12)(cid:12) S Y (cid:21) for all Re( t ) > , from which we can conclude E (cid:20) X i E [ X i ] e − tS X (cid:12)(cid:12)(cid:12) S Y (cid:21) = E (cid:20) X j E [ X j ] e − tS X (cid:12)(cid:12)(cid:12) S Y (cid:21) for all Re( t ) > . The assertion of the theorem follows by the law of total expectation and evoking again Theorem 1.Theorem 2 remains true if a split results in more than two loss portfolios and implies that, whenstarting with a loss portfolio in the set W , the split operation yields loss portfolios that are also in theset W .The next result emphasizes that the merge operation - an opposite of split - is more intricate, butthat the merge of loss portfolios belonging to the set W may result in a loss portfolio that also belongsto the set W . The statement of Theorem 3 - with appropriately adjusted Condition (10) - stays validif more than two loss portfolios are merged. Theorem 3.
Consider independent loss portfolios, ( X , . . . , X n ) ∈ W and ( Y n +1 , . . . , Y n + m ) ∈ W ,and denote by S X and S Y the corresponding sums of coordinates. Also, let X + = ( X , . . . , X n , Y n +1 ,. . . , Y n + m ) ∈ X n + m be the merged portfolio. Then, X + ∈ W if and only if, for i ∈ { , . . . , n } and j ∈ { n + 1 , . . . , n + m } , φ S ( ∗ ) iX ( t ) φ S ( ∗ ) jY ( t ) = φ S X ( t ) φ S Y ( t ) , Re( t ) > . (10) Proof.
Let S + = S X + S Y . We need to show that S ( ∗ ) i + = d S ( ∗ ) j + for all i = j ∈ { , . . . , n + m } . First,consider the case in which the indices i, j belong to either one of the sets { , . . . , n } or { n +1 , . . . , n + m } ,say i ∈ { , . . . , n } and j ∈ { n + 1 , . . . , n + m } . Then by Lemma 1 with the addition of the independence13ssumption and since ( X , . . . , X n ) ∈ W and ( Y n +1 , . . . , Y n + m ) ∈ W , we have φ S ( ∗ ) i + ( t ) = φ S ( ∗ ) iX ( t ) × φ S Y ( t ) = φ S ( ∗ ) jY ( t ) × φ S X ( t ) = φ S ( ∗ ) j + ( t )for all Re( t ) > i = j are both in { , . . . , n } or both in { n + 1 , . . . , n + m } follows similarly. Thiscompletes the proof of the theorem.The assertion that concludes this section reveals that an amalgamation - on a BU basis - of acollection of loss portfolios, each of which belongs to the set W , may result in a loss portfolio thatalso belongs to the set W . The statement of Theorem 4 - with appropriately adjusted Condition (11)- stays valid if more than two loss portfolios are amalgamated. Theorem 4.
Consider independent loss portfolios, ( X , . . . , X n ) ∈ W and ( Y , . . . , Y n ) ∈ W . Let S = ( S , . . . , S n ) , where S i = X i + Y i , i = 1 , . . . , n . Then S ∈ W if and only if E [ X i ] E [ X j ] = E [ Y i ] E [ Y j ] for all i = j ∈ N . (11) Proof.
Let S X = X + · · · + X n , S Y = Y + · · · + Y n , and S + = S X + S Y . By Lemma 1 with theaddition of the independence assumption, we have, for i = 1 , . . . , n , φ S ( ∗ ) i + ( t ) = E (cid:2) S i e − t S + (cid:3) E [ S i ] = E [ X i ] E [ S i ] φ S ( ∗ ) iX ( t ) φ S ( ∗ ) Y ( t ) + E [ Y i ] E [ S i ] φ S ( ∗ ) iY ( t ) φ S ( ∗ ) X ( t ) for all Re( t ) > . By Theorem 1, Condition (11) is required so that the equality φ S ( ∗ ) i + ( t ) = φ S ( ∗ ) j + ( t ) holds for allRe( t ) > i = j ∈ N . This completes the proof of the theorem. In this section, we review a few examples of those loss RVs, X ∈ X n , for which the equality r q,i = ˜ r q,i holds for all q ∈ [0 ,
1) and i ∈ N . That is, we now construct a few examples of the loss portfolios X ∈ X n , for which the RC allocations induced by the CTE risk measure reflect the diminishing impactof large losses on the insurers’ perception of risk. 14ur first example is the Liouville distributions (e.g., Gupta and Richards, 1987, for a compre-hensive treatment, and Hua, 2016; McNeil and Neˇslehov´a, 2010, for applications in the context ofdependence modelling). To start with, for γ ∈ R + , denote by Γ( γ ) the complete gamma function, thatis Γ( γ ) = Z ∞ x γ − e − x dx. Also, for γ , . . . , γ n ∈ R + and γ • := γ + · · · + γ n , define the multivariate Beta function as B ( γ , . . . , γ n ) = Q nj =1 Γ( γ j )Γ( γ • ) . Example 1.
The positive and absolutely-continuous RV, X = ( X , . . . , X n ) ∈ X n , is said to bedistributed Inverted-Dirichlet, succinctly X ∼ ID n ( γ , . . . , γ n , β ) with the parameters β, γ , . . . , γ n ∈ R + , if its probability density function (PDF) is: f X ( x , . . . , x n ) = 1 B ( γ , . . . , γ n , β ) n Y j =1 x γ j − j n X j =1 x j − ( γ • + β ) , x , . . . , x n ∈ R + , (e.g., Gupta and Song, 1996; Ignatov and Kaishev, 2004, for a general discussion and applications inactuarial science, respectively).It is not difficult to show that φ S ( ∗ ) i ( t ) = φ S ( ∗ ) j ( t ) , Re( t ) > for all ≤ i = j ≤ n , and hence byTheorem 1, we have r q,i = γ i /γ • = ˜ r q,i for q ∈ [0 , and i ∈ N . An interesting observation that paves the way for a fairly general proposition, which is stated next,is that for X ∼ ID n ( γ , . . . , γ n , β ), we have the stochastic representation X j = Z × Y j , j = 1 , . . . , n ,where the RV Z = P nj =1 X j has a univariate inverted beta distribution, Z ∼ IB ( γ • , β ), with theparameters γ • , β ∈ R + , and the RV Y = ( Y , . . . , Y n ), independent on the RV Z , is distributedmultivariate Dirichlet (Ng et al., 2011).The proof of the following assertion is readily obtained via the routine conditioning and thenevoking Theorem 1 and is thus omitted. 15 roposition 4. Let the RV Y = ( Y , . . . , Y n ) be independent on the RV Z and such that, for aconstant b ∈ R + , the equality, P ni =1 Y i = b , holds almost surely. Further, let the loss portfolio X =( X , . . . , X n ) ∈ X n admit the stochastic representation X j = Y j × Z, j ∈ N , then X ∈ W . Proposition 4 implies that the loss RVs X , . . . , X n that admit the Multiplicative BackgroundRisk Model (MBRM) stochastic representation with the idiosyncratic risk factors (RFs), Y , . . . , Y n distributed Dirichlet with parameters γ , . . . , γ n ∈ R + and the systemic RF Z having the PDF f Z ,such that f Z ( z ) ∝ g ( z ) z γ • − , z ∈ R + , (12)where γ • = P ni =1 γ i , the function z g ( z ) is positive, continuous and integrable in the sense ofGupta and Richards (1987), all belong to the set W . Some examples, in addition to the already-mentioned inverted beta distribution, of the probability distribution of the systemic RF, Z , are: thegamma distribution and the generalized mixture of exponential distributions.The class of multivariate probability distributions that admit the stochastic representation de-scribed in Proposition 4 is called the class of Liouville distributions , and these distributions are oneway to extend the multivariate Dirichlet distribution to the unbounded domain, R n + . Another wayis via the class of mixed-Gamma (MG) distributions, which has recently been presented and studiedin Furman et al. (2020b). Speaking briefly and avoiding unnecessary technicalities - thus consideringthe simplest possible case - the loss RV X = ( X , . . . , X n ) is said to be distributed n -variate MGdistribution if it has the PDF: f X ( x , . . . , x n ) = m X k =1 p k n Y i =1 x γ i,k − i Γ( γ i,k ) θ γ i,k i e − x i /θ i , x , . . . , x n ∈ R + , (13)where γ i,k ∈ R + and θ i ∈ R + are, respectively, the shape and scale parameters, and p k > , k =1 , . . . , m are the mixture weights satisfying P mk =1 p k = 1; succinctly, we write X ∼ MG n ( γ , θ , p ),where γ and θ are the n × m - and m - dimensional vectors of shape and scale parameters, respectively,and p = ( p , . . . , p m ).The class of MG distributions is a generalization of the popular class of multivariate Erlang mix-tures considered in Willmot and Woo (2014), albeit with (a) positive - and not positive and integer -16hape parameters, and (b) possibly distinct - and not all equal - scale parameters (e.g. Lee and Lin,2012; Verbelen et al., 2016). The class of MG distributions is connected to Question 1 in the exampleand proposition that follow. Example 2.
Consider a loss portfolio X ∼ MG n ( γ , θ , p ) with the PDF as per Equation (13), butwith θ i ≡ θ . Then, for i ∈ N , we have φ S ( ∗ ) iX ( t ) = E [exp {− tS ( ∗ ) i X } ] = m X k =1 p ( ∗ ) i k (1 + θt ) − γ • ,k − , Re( t ) > , where p ( ∗ ) i k = γ i,k P mk =1 γ i,k × p k p k , k = 1 , . . . , m, which can be viewed as the i -th partial size-biased transform of the PMF underlying the stochastic shapeparameters. Consequently, for the equality S ( ∗ ) i X = d S ( ∗ ) j X to hold, we must require (due to Theorem 1) γ i, (cid:16)P nj =1 γ j, (cid:17) = · · · = γ i,m (cid:16)P nj =1 γ j,m (cid:17) (14) for all i ∈ N . The observation presented in Example 2 is strengthened in the following proposition, which con-cludes this section.
Proposition 5.
Let X ∼ MG n ( γ , θ , p ) , then we have X ∈ W if and only if both of θ i ≡ θ andEquation (14) hold true.Proof. Example 2 establishes the ‘if’ direction. In order to prove the ‘only if’ direction, we pursueproof by contradiction. To this end, consider X ∼ MG n ( γ , θ , p ) in which the coordinates of thevector of parameters θ = ( θ , . . . , θ n ) are all distinct, and suppose X ∈ W . (If some scale parameterswere equal, then we would introduce the vector of distinct scales, b θ = ( b θ , . . . , b θ n ′ ), n ′ < n as wellas, for d = 1 , . . . , n ′ and T d = (cid:8) i ∈ { , . . . , n } : θ i = b θ d (cid:9) , the corresponding shape parameters b γ d,l = P i ∈ T d γ i,l , l = 1 , . . . , m . We would then proceed with the proof, as it is outlined below.) Further,17ithout loss of generality, assume that the shape parameters are ordered as γ d, ≤ γ d, ≤ · · · ≤ γ d,m , d = 1 , . . . , n .With the above in mind and for any BU, j ∈ N , we have that the Laplace transform of the RV S ( ∗ ) j X is: φ S ( ∗ ) jX ( t ) = m X k =1 p ( ∗ ) j k (cid:0) θ j t (cid:1) − (1+ γ j,k ) n Y d =1 ,d = j (1 + θ d t ) − γ d,k , Re( t ) > . Furthermore, as X ∈ W , we have that Theorem 1 implies, for 1 ≤ i = j ≤ n and all Re( t ) > φ S ( ∗ ) iX ( t ) = φ S ( ∗ ) jX ( t ) . However, this is impossible, which is easily seen by comparing, e.g., the m -th terms of the Laplacetransforms φ S ( ∗ ) iX and φ S ( ∗ ) jX . Hence, we have arrived at a contradiction and the proposition is proved. In this section, we explore the loss portfolios X ∈ X n that have independent constituents. Admittedly,the assumption of independence simplifies the problem postulated in Question 1 considerably, yet norit means that the RV R = ( R , . . . , R n ) has independent coordinates, neither that the RVs R and S X are independent, thus warranting a separate discussion. Theorem 5.
Assume that X = ( X , . . . , X n ) ∈ X n is a portfolio of independent losses, then we havethe equality r q,i = ˜ r q,i (= E [ X i ] / E [ S ]) for all q ∈ [0 , and i ∈ N , if and only if the equality φ X i ( t ) = (cid:0) φ X j ( t ) (cid:1) E [ X i ] / E [ X j ] holds for all i = j ∈ N and Re( t ) > .Proof. By Theorem 1 and assuming that the RVs X , . . . , X n are mutually independent, we have X ∈ W if and only if the Laplace transforms of the RVs X i + X ∗ j and X ∗ i + X j agree for all i = j ∈ N .18hat is, we must have 1 E [ X j ] ddt φ X j ( t ) φ X j ( t ) = 1 E [ X i ] ddt φ X i ( t ) φ X i ( t ) , for all i = j ∈ N and Re( t ) >
0. This, in turn, is equivalent to ddt log φ X j ( t ) ddt log φ X i ( t ) = E [ X j ] E [ X i ] , (15)implying, for all Re( t ) >
0, log φ X j ( t )log φ X i ( t ) = E [ X j ] E [ X i ] (16)by Pinelis’ L’hopital-type rules. The fact that Equation (16) leads to Equation (15) is easy to checkby routine differentiation in the latter equation. This completes the proof of the theorem. Examples of the loss portfolios X ∈ X n that have independent constituents and also belong to theset W are really numerous. For instance, consider losses X i , i ∈ N , that have infinitely divisibledistributions and such that the condition in Theorem 5 holds, then we have ˜ r q,i = r q,i = E [ X i ] / E [ S X ]for any q ∈ [0 , Example 3.
Assume that the portfolio of losses X ∈ X n has independent constituents X , . . . , X n ,then it belongs to the set W given that these constituents have the following probability distributions: • X i ∼ Negative - Binomial( β i , p ) , β i ∈ R + , p ∈ (0 , , with mean E [ X i ] = β i (1 − p ) / p and Laplacetransform φ X i ( t ) = (cid:16) p − (1 − p ) e − t (cid:17) β i , Re( t ) > . X i ∼ Gamma( γ i , β ) , γ i ∈ R + , β ∈ R + , with mean E [ X i ] = γ i β and Laplace transform φ X i ( t ) = (1 + β t ) − γ i , Re( t ) > . • X i ∼ Inverse - Gaussian( µ i , µ i ) , γ i ∈ R + , with mean E [ X i ] = µ i and Laplace transform φ X i ( t ) = exp n µ i (1 − √ − t ) o , Re( t ) > . We mentioned in Section 2 that the CTE risk measure is a member of the class of weighted riskmeasures and that it induces the RC allocation r q,i , q ∈ [0 , i ∈ N . In fact, a more encompassingclass of risk measures - and hence a generalization of the CTE risk measure - can be defined as follows(Furman and Zitikis, 2008a). Let v, w : [0 , ∞ ) → [0 , ∞ ) be two (non-decreasing) functions, then the generalized weighted risk measure is the map H v,w : X → [0 , ∞ ) ∪ { + ∞} , which, when well-definedand finite, is given by H v,w ( X ) = E (cid:2) v ( X ) w ( X ) (cid:3) E (cid:2) w ( X ) (cid:3) , X ∈ X . (17)For w ( x ) = { x ≥ VaR q ( X ) } and v ( x ) = x , where x ∈ [0 , ∞ ) and q ∈ [0 , k ∈ N , set v ( x ) = x k , x ∈ [0 , ∞ ) and keep the weight function x w ( x ) equal theindicator function as before in order to emphasize the tail loss scenarios, then generalized weightedrisk measure (17) yields the k -th order CTE risk measure. Furthermore, extending the notation inSection 2, let, for q ∈ [0 , , k ∈ N and i ∈ N ˜ r kq,i = E [ R ki | S > s q ] and r kq,i = E [ X ki | S > s q ] / E [ S k | S > s q ] . In general, the proportional k -th order CTE-based allocation r kq,i is not fully-additive. Nevertheless,20t is a meaningful quantity in quantitative risk management (e.g., Furman and Landsman, 2006; Kim,2010; Landsman et al., 2016, for elaborations and applications).It is not difficult to see that, for a fixed k ∈ N , the equality ˜ r kq,i = r kq,i holds for all q ∈ [0 ,
1) and i ∈ N , if an only if we have Cov( R ki , S k | S > s q ) ≡
0. Therefore, it is natural to reformulate Question1 as follows.
Question 2.
For loss RVs X i ∈ L k , can we characterize those loss portfolios X = ( X , . . . , X n ) ∈ X n ,for which the equality ˜ r kq,i = r kq,i holds for all q ∈ [0 , , i ∈ N , and a fixed k ∈ N ? Question 2 seeks to characterize the RVs that belong to the set W k = (cid:8) X = ( X , . . . , X n ) ∈ X n : r kq,i = e r kq,i for all q ∈ [0 ,
1) and i ∈ N (cid:9) , (18)which we do next. To start off, note that if the equality r kq,i = e r kq,i holds for all q ∈ [0 , q = 0, implies that for all loss portfolios in the set W k , we must have r kq,i = e r kq,i = E [ X ki ] / E [ S kX ]. Theorem 6.
If the portfolio of losses X = ( X , . . . , X n ) ∈ X n with X i ∈ L k belongs to the set W k and so the equality r kq,i = ˜ r kq,i (= E [ X ki ] / E [ S kX ]) holds for all q ∈ [0 , , i ∈ N , and a fixed k ∈ N , then S ( k ) i X = d S ( k ) j X , ≤ i = j ≤ n . The opposite direction does not hold.Proof. The proof follows the same argumentation as in Theorem 1 with the quantities S X , R i and G i ( s ) = E [ R i | S X = s ] − E [ R i ] replaced with the quantities S kX , R ki and G ki ( s ) = E [ R ki | S X = s ] − E [ R ki ] , s ∈ [0 , ∞ ).To see that the distributional equality, S ( k ) i X = d S ( k ) j X , ≤ i = j ≤ n , does not imply X ∈ W k ,consider the RV ( X , X ) ∈ X that has independent and identically distributed constituents; X i ∼ Uni[0 , , i = 1 ,
2. Clearly, we have S ( k ) X = d S ( k ) X , k ∈ N . Nevertheless, with some algebra we obtain,for i ∈ { , } , E [ X ki | S = s ] = s k k { ≤ s< } + 1 − ( s − k +1 (1 + k )(2 − s ) { ≤ s ≤ } , s ∈ R + , which implies E [ R ki | S X = s ] = const and hence ˜ r kq,i = r kq,i . This completes the proof of the theorem.21ccording to Theorem 6, if the equalities r q,i = ˜ r q,i and r q,i = ˜ r q,i hold for all q ∈ [0 ,
1) and i ∈ N ,then we must have E [ R i | S X = s ] ≡ const and E [ R i | S X = s ] ≡ const, respectively. Therefore, thefact X ∈ W does not imply X ∈ W (also due to the counter example in the proof of Theorem 6).Next example demonstrates that this statement, when formulated in the opposite direction, does nothold either. Example 4.
Consider again the MG distribution as per Example 2, i.e., let X ∼ MG n ( γ , θ , p ) , andset θ i ≡ θ ∈ R + and γ i, ( γ i, + 1) (cid:16)P nj =1 γ j, (cid:17) (cid:16)P nj =1 γ j, + 1 (cid:17) = · · · = γ i,m ( γ i,m + 1) (cid:16)P nj =1 γ j,m (cid:17) (cid:16)P nj =1 γ j,m + 1 (cid:17) for all i ∈ N . Then it is not difficult to check directly that r q,i = ˜ r q,i , i ∈ N , and therefore we have X ∈ W . However, the choice of parameters above does not guarantee Equation (14) , and consequentlyby Theorem 1, we do not necessarily have X ∈ W . We conclude this section by outlining a situation in which the fact, X ∈ W , does imply the fact, X ∈ W , and vice versa; curiously, this connects Questions 1 and 2 to the celebrated Lukacs theorem(Lukacs, 1955). For this, recall that the fact that the loss portfolio X = ( X , . . . , X n ) ∈ X n belongsto the set W k , k ∈ N , or in other words, that the RVs R i , i = 1 , . . . , n , and S X are uncorrelatedconditionally on S X > s for all s ∈ [0 , ∞ ), does not in general imply the fact that the loss RVs R and S X are independent. This statement holds true even if the constituents of the loss portfolio X ∈ X n are independent. The following assertion delineates the cases, in which the RVs R and S X are independent in the context of Questions 1 and 2. Corollary 1.
Assume that the loss RVs X , . . . , X n ∈ X are independent. The loss portfolio X =( X , . . . , X n ) ∈ X n belongs to the sets W and W if and only if the loss RV X i ∈ X is distributedgamma with the shape and scale parameters γ i > and θ > , respectively, i ∈ N . In this case, theRVs R and S X are independent.Proof. In order to prove the ‘if’ direction, we note that by Lukacs’ theorem, the assumption X i ∼ Ga ( γ i , θ ) implies R i ⊥⊥ S X for all i ∈ N , which in turn implies X ∈ W ∩ W .22urther, let us prove the ‘only if’ direction. For this, fix i ∈ N and note that by Theorems 5and 6 - with the assumption of independence made in the latter case - we arrive at the following twoequations, where φ X i ( t ) and φ X j ( t ) denote, respectively, the Laplace transforms of the loss RVs X i and X j , and a := E [ X j ] / E [ X i ] and b := E [ X i ] / E [ X j ] , ≤ i ≤ j ≤ n , for the simplicity of exposition φ X j ( t ) = ( φ X i ( t )) a , Re( t ) > φ X j ( t ) d dt φ X i ( t ) = b φ X i ( t ) d dt φ X j ( t ) , Re( t ) > . (20)Rewriting the equations above in terms of the Laplace transform φ X i only, we obtain, for Re( t ) > φ X i ( t )) a d dt φ X i ( t ) = b φ X i ( t ) " a ( a −
1) ( φ X i ( t )) a − (cid:18) ddt φ X i ( t ) (cid:19) + a ( φ X i ( t )) a − d dt φ X i ( t ) , which, with some algebra and the notation c = a b ( a −
1) (1 − a b ) − , simplifies to d dt φ X i ( t ) ddt φ X i ( t ) = c ddt φ X i ( t ) φ X i ( t ) , Re( t ) > . After some more algebra, we arrive at the following first-order non-linear ODE ddt φ X i ( t ) = − E [ X i ] ( φ X i ( t )) c , Re( t ) > , with the solution φ X i ( t ) = h c − E [ X i ] t i − / ( c − . Finally, substituting the expressions for the first and second moments of the gamma distribution inthe constant c and hence noticing that c − /γ i , we arrive at φ X i ( t ) = (1 + θ t ) − γ i , Re( t ) > , γ i > θ >
0, respectively. Hence, X i ∽ Ga ( γ i , σ ) , i ∈ N . Also, for γ • = γ + · · · + γ n as before,we have S X ∼ Ga ( γ • , σ ), and by Lukacs’ theorem, the RVs R i = X i / S X and S X are independent.This completes the proof of the ‘only if’ direction as well as of the corollary as a whole. The striking majority of the risk capital allocation rules that exist nowadays are induced by riskmeasures, with the CTE risk measure being arguably the most popular. While the CTE risk measure- and the risk capital allocation based on it - are sound mathematical objects that have been studiedextensively in a great variety of contexts and even embedded in some regulatory accords, they are notexplicitly linked to the risk preferences of an insurer. Yet, these risk preferences are of fundamentalimportance and should presumably drive - or at least impact - the allocation exercise.In this paper, we have demonstrated that there are model settings, in which the risk capitalallocation induced by the CTE risk measure does reflect insurer’s risk preferences by accounting forthe decreasing marginal effect of the increase in the aggregate loss. This implies that there exist lossportfolios for which the utilization of the CTE risk measure as a basis for allocating risk capital may bejustified from both the regulatory and profit maximizing perspectives. Moreover, we have developedexhaustive description of the just-mentioned special loss portfolios and elucidated the findings withample specific examples.Our message in this paper is two-fold. On the one hand-side, the feasibility of achieving theapparently desirable agreement between the prudence- and profitability- driven considerations forsome loss portfolios gives hope that the CTE-based risk capital allocation may indeed be appropriate for profitability analysis and price determination. On the other hand-side, the sparsity of the collectionof the loss portfolios for which the just-mentioned agreement is achieved implies that more research isrequired in order to explicitly connect the risk capital allocation exercise to its objectives as well as tothe economic traits of the insurer. 24 eferences
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