ABSTRACT 1. INTRODUCTION

Transcription

1 MULI-LEVEL RISK AGGREGAION BY DAMIR FILIPOVIC * ABSRAC In this paper we compare the current Solvenc II standard and a genuine bottom-up approach to risk aggregation. his is understood to be essential for developing a deeper insight into the possible differences between the diversification assumptions between the standard approach and internal models. 1. INRODUCION here are various approaches to model diversification benefits using linear correlation at solo level. In the current Solvenc II standard two-level approach, there is a base correlation matrix within each risk class (market, life, non-life, health, default) and a top level correlation matrix between these risk classes. Internal models tend to follow a genuine bottom-up approach which uses a correlation matrix that combines all risk tpes. See e.g. the recent CEIOPS documents [1,2] and the CRO Forum QIS3 Benchmarking stud [6]. In this paper we compare the two approaches. In particular, we discuss the interpla between the top level correlation between risk classes and the base correlation matrix between the risk tpes. his is understood to be essential for developing a deeper insight into the possible differences between the diversification assumptions between the standard approach and internal models. In Section 2, we show that in general onl correlation parameters set at the base level lead to unequivocall comparable solvenc capital requirements across the industr (heorem 2.2). his also supports the findings in the technical paper [7] of the Groupe Consultatif. In Section 3, we then consider portfolio dependent base correlations. hese implied correlations are not unique in general. We provide at least three possible specifications and give sufficient conditions such that the actuall qualif as positive semi-definite correlation matrices. We then show that there exists a unique minimal base correlation matrix (heorem 3.2). his distinguished correlation matrix ma serve as a benchmark for comparison of standard and internal correlation specifications. In Section 4, * I am grateful to Isaac Alfon and an anonmous referee for helpful comments. Financial support from WWF (Vienna Science and echnolog Fund) is gratefull acknowledged. Astin Bulletin 39(2), doi: /AS b Astin Bulletin. All rights reserved.

2 566 D. FILIPOVIC we compute the three base correlation matrices for a life and non-life insurance portfolio which reflect the EEA average from the QIS3 Benchmarking Stud of the CRO Forum [6]. hese results ma be used as benchmark for a possible standard specification of base correlations between market, life and non-life risk tpes. Section 5 concludes. 2. CONSAN BASE CORRELAION For the sake of illustration, we consider two risk classes composed of m and n risk tpes with stand alone solvenc capital requirements (i.e. 99.5%-quantiles) given b the vectors J x N K 1 O x = h K x O! m + and = L m P J N K 1 O h K O! + L n P Here and subsequentl, with a vector we mean a column vector. It is straightforward to extend the following to more than two risk classes. he current Solvenc II standard model is based on a two-level correlation aggregation. First, some m m and n n base correlation matrices A for x and B for ield the solvenc capital requirements per risk class X = x $ A $ x and Y = $ B $, (1) n. respectivel. Second, some top level correlation factor R between X and Y ields the total solvenc capital requirement 2 2 SCR = X + 2RXY + Y. (2) he standard model provides base and top level correlation parameters A, B and R. Input from the insurance compan are the portfolio specific data x,. A genuine bottom-up model, in contrast, uses a full (m +n) (m + n) base correlation matrix M = A C e o (3) C B that aggregates all risk tpes, across risk classes, in one go: SCR = `x, j $ M$ d x n. (4)

3 Equalling (2) and (4) implies x C = R x $ A $ x $ B $. (5) It is understood that the full base correlation matrix M is fundamental part of the risk model. It reflects the underling nature of the risks, which is generic and compan independent. Compan specific, in contrast, is the individual exposure to the risks. hus, M should simultaneousl appl to all companies. In that sense, M should not depend on the compan specific portfolio x,, while the implied top level correlation x $ C $ R = R(x,) = x $ A $ x $ B $ then does. Conversel: Definition 2.1. For A, B, R given, we call C(x, ) a base correlation matrix for x, ifmin(3) is a correlation matrix (i.e. positive semi-definite) and (5) is satisfied. Now suppose the standard model (1)-(2) specified b A, B, R is applied b N companies with portfolio data x(i), (i), i =1,,N. We then shall sa that the resulting solvenc capital requirements SCR(i) are unequivocall comparable if there exists a common base correlation matrix C(x(i), (i)) / C for all i =1,,N. In other words, the solvenc capital requirements are unequivocall comparable if the are based on a common underling risk model. Checking for unequivocal comparabilit is an inverse problem, which is difficult to solve in general. Indeed, (5) can be seen as linear sstem of N equations for the mn-vector (C ij ): m n!! x (k) j (k)c ij = z(k), k =1,,N, (6) i = 1 j = 1 MULI-LEVEL RISK AGGREGAION 567 i with z(k) :=R x ( k) $ A $ x( k) ( k ) $ B $ ( k ). Since the set of solutions C to (6) equals the (possibl empt) preimage of the N-vector z(k) for the N (mn)-matrix (x i (k) j (k)), a necessar and sufficient condition for the existence of a solution C of (6) is that the N-vector (z(k)) lies in the image of the N (mn)-matrix (x i (k) j (k)). For a generic vector z(k) in N + this essentiall requires that N # mn, see e.g. [5, Paragraph.2.3]. But the number of tested companies will tpicall be greater than mn (there are currentl six risk tpes in market and seven in life, that is mn = 42). Moreover, even if a solution C of (6) exists, it et has to satisf that M in (3) is positive semi-definite. he following example shows that a solution ma not exist if N > mn: let m =2, n =1, A = 1 c 1 m, B =(1), R =.4, and N = 3. he portfolio data are

4 568 D. FILIPOVIC x(1) = c 3 4 m, x(2) = c 1 1 m, x(3) = c 1 2 m, (1) = (2) = (3)=1 ( actuall cancels out in (5) and (6), respectivel). hen x(1), x(2) alread uniquel determine the 26 solution C = c, 3, mof (6). But a straightforward inspection shows that (5) does not hold for x(3). he next result shows that the above inverse problem is in fact genericall ill-posed. Indeed, if we assume infinitel man companies with a continuum of portfolio data (x, )! m+n +, then a common base correlation matrix C(x,) / C exists for all (x,)! m+n + if onl if the risk tpes are either uncorrelated or full correlated. Denote b J m n the m n-matrix with all entries equal 1. heorem 2.2. Suppose A, B, R are given as above. hen there exists a common base correlation matrix C(x,) / C for all (x,)! + m+n if and onl if C = RJ m n and either R =or A = J m m and B = J n n. PROOF. Sufficienc of the statement is clear. For necessit, we insert x = e i and = e k (the standard basis vectors in m or n, respectivel) in (5) and obtain C = RJ m n.ifr = we are done. Otherwise, we divide (5) b R and square on both sides to obtain m n!! x x j k l =!! A B kl x i x j k l. i i, j = 1 kl, = 1 m n ij i, j = 1 kl, = 1 Matching coefficients ields A ij B kl = 1 and, since A ii = B kk = 1, thus the claim. Based on heorem 2.2, we ma state that in general onl correlation parameters set at the base level lead to unequivocall comparable solvenc capital requirements across the industr. 3. MINIMAL BASE CORRELAION heorem 2.2 showed that it is genericall impossible to find a common base correlation matrix C(x,) / C for all (x,)! m+n +. In this section, we relax this assumption and find base correlation matrices C(x,) which ma depend on the given portfolio data (x,)! m+n +. Such solutions exist, but are not unique, in general. he next lemma provides some examples. For two vectors u! m, v! n we write u v for the m n-matrix (u i v j ). We denote b D = tr ( D $ D ) the Euclidian norm of a matrix D. 2 2 Hence, in particular, u = u 1 + g + u m for an m-vector u. It follows b inspection that the following matrices satisf (5): A $ x $ B C(x, ) =R $, x $ A $ x $ B $ (7)

5 MULI-LEVEL RISK AGGREGAION 569 x $ A $ x $ B $ C(x, ) =R x, 2 2 $ (8) x x $ A $ x $ B $ C(x, ) =R J m # n. (9) x $ J $ m # n We next provide sufficient conditions on A and B such that examples (7)-(9) qualif as base correlation matrix. Lemma 3.1. (i) (7) is a base correlation matrix for x,. (ii) If there exists p, q $ with pq = R 2 such that both matrices A - x $ A $ x p 4 x $ x and B - q x $ B $ 4 $ (1) are positive semi-definite, then (8) is a base correlation matrix for x,. (iii) If there exists p, q $ with pq = R 2 such that both matrices x $ A $ x A - p x $ J $ x m # m $ B $ Jm # m and B - q $ Jn# n $ J (11) n# n are positive semi-definite, then (9) is a base correlation matrix for x,. PROOF. It remains to show that M in (3) is positive semi-definite, that is, u A u +2u C(x,) v + v B v $, 6(u,v)! m+n. his is equivalent to (u A u)(v B v) $ (u C(x,) v) 2, 6(u,v)! m+n. (12) For (7), propert (12) follows b the Cauch Schwarz inequalit (u A x) 2 # (u A u)(x A x) and analogousl for B. his proves (i). For (8), propert (12) holds if and onl if (u A u)(v B v) $ R 2 x $ A $ x $ B $ u $ x $ x $ u u $ $ $ u x x for all (u,v)! m+n. his proves (ii). Part (iii) follows similarl.

6 57 D. FILIPOVIC We now show that (8) is distinguished among all base correlation matrices. heorem 3.2. Suppose C * = C(x,) in (8) is a base correlation. hen it is minimal in the following sense: C * = { C C is base correlation matrix for x, }. PROOF. We consider the scalar product GC, DH =tr(c D ) on m n. hen C = CC,, and the left hand side of (5) is just the scalar product: x C = GC, x H. Hence ever C that satisfies (5) can uniquel be decomposed into C x $ A $ x $ B $ = R x N 2 2 # ` $ + j x where N is orthogonal to x, i.e. Gx, NH =. Hence `x $ A $ xj ` $ B $ j C C C x 2 * * 2 = + R N 4 4 $ with equalit if and onl if N =. B the ver definition, for given A, B, R and x,, ever base correlation matrix for x, ields the same total solvenc capital requirement SCR. Hence there is no wa in distinguishing a base correlation matrix for x, b its diversification effect for the portfolio (x, ). Nonetheless, from heorem 3.2 we ma infer the following universal minimalit propert of C * = C(x, ) in (8). Suppose C is an other base correlation matrix for x,. hen heorem 3.2, combined with the Cauch Schwarz inequalit (see e.g. [5, Paragraph.6.3]), sas that the maximal weighted sum of the entries of C * is strictl smaller than the respective maximal sum for C in the sense that m n m n sup!! Dij C * ij = * = sup!! D = 1 i = 1 j = 1 D = 1 i = 1 j = 1 C < C D C. ij ij In this sense, (8) allocates the prescribed top level correlation R among the base risk tpes in a uniquel minimal wa, as illustrated in the next section. In terms of diversification effects, this can be expressed as follows. Suppose A, B and some base correlation matrix C = C(x,) calibrated to the portfolio (x,) are going to be used as benchmark risk model for other portfolios. Moreover, suppose we measure the diversification effect for an portfolio

7 MULI-LEVEL RISK AGGREGAION 571 (z, j)! + m+n as difference between the squared total solvenc capital requirement and the squared solvenc capital requirement with zero top level correlation: D(z, j,c) =z R Az +2z R Cj + j R Bj (z R Az + j R Bj) =2z R Cj. hat is, the less D(z, j,c ), the higher the diversification effect. It then follows as above that sup z j # x D(z, j,c * )=2x R C * # sup z j # x D(z, j,c ). In words, the lowest diversification effect among all portfolios (z, j)! + m+n with z j # x for C * is higher than the respective lowest diversification effect for an other base correlation matrix C = C(x,). 4. APPLICAION O QIS3 DAA Figures 1 and 2 in the appendix show the EEA-average solvenc capital requirements per risk tpe for a life and non-life insurer, respectivel, taken from the QIS3 Benchmarking Stud 2 of the CRO Forum [6]. Figures 3-6 show the top and base level correlation matrices according to the QIS3 standard model [1]. One then checks numericall b computing the eigenvalues that the two matrices in (1) (for p =.1 and q =.625) and in (11) (for p = q =.25) are positive semi-definite, both for the life and non-life portfolio. B Lemma 3.1 it follows that all examples (7)-(9) are base correlations for the given capital requirements in Figures 1 and 2. he resulting base correlations between market and life and non-life risk tpes for the life and non-life insurer, respectivel, are shown in Figures 7-9 and Cells with correlations greater than.1 are indicated. It becomes obvious that the minimal base correlation matrix (8) assigns less correlation to risk tpes than the other two examples (7) and (9). 5. CONCLUSION In this paper, we rigorousl demonstrated the fact that onl correlation parameters set at the base level lead to unequivocall comparable solvenc capital requirements across the industr. 2 hese figures are derived from the proportion splits of QIS3 capital charges as shown on pages 29, 31, 55 and 39, 41, 43 in the document [6]. he capital requirements are thus normalized such that the undiversified Basic SCR results in 1 1. he risk class default is negligible, both for life and non-life insurers, and therefore is omitted.

8 572 D. FILIPOVIC Relaxing the assumptions, we then found portfolio dependent base correlation matrices that correspond to a prescribed top level correlation. Narrowing further the choice we arrived at a unique minimal solution, which we then explicitl computed for QIS3 data from [6]. I suggest that further empirical comparison of standard and internal correlation specifications is carried out with this minimal solution as a benchmark. However, I also stress the fact that Value-at- Risk and correlation aggregation does not appropriatel capture tails and tail dependence of risks in the insurance business. In that regard, I encourage the additional use of risk and dependence modeling beond correlation such as indicated in e.g. [4, 3]. REFERENCES [1] COMMIEE OF EUROPEAN INSURANCE AND OCCUPAIONAL PENSIONS SUPERVISORS (27) QIS3 echnical Specifications. PAR I: INSRUCIONS, URL: 118/124. [2] COMMIEE OF EUROPEAN INSURANCE AND OCCUPAIONAL PENSIONS SUPERVISORS (27) QIS4 echnical Specifications, URL: solvenc/qis4/technical_specifications_en.pdf [3] FILIPOVIC, D. and KUNZ, A. (27) Realizable Group Diversification Effects, Life & Pensions, Ma 28. URL: [4] FILIPOVIC, D. and KUPPER, M. (27) On the Group Level Swiss Solvenc est, Bulletin of the Swiss Association of Actuaries 1, [5] HORN, R.A. and JOHNSON C.R. (1985) Matrix Analsis, New York: Cambridge Universit Press. [6] HE CHIEF RISK OFFICER FORUM (27) A benchmarking stud of the CRO forum on the QIS III calibration, URL: [7] GROUPE CONSULAIF ACUARIEL EUROPÉEN (25) Diversification, echnical paper, URL: DAMIR FILIPOVIC Vienna Institute of Finance Universit of Vienna and Vienna Universit of Economics and Business Administration

9 MULI-LEVEL RISK AGGREGAION 573 APPENDIX Results Mkt int 1536 eq 2624 prop 512 sp 148 conc 64 fx 256 Life mort 14 long 119 dis 245 lapse 7 exp 385 rev CA 84 FIGURE 1: EEA-average solvenc capital requirements per risk tpe for a life insurer. Source: [6]. Mkt int 572 eq 258 prop 396 sp 264 conc 572 fx 132 NL pr 4187 CA 1113 FIGURE 2: EEA-average solvenc capital requirements per risk tpe for a non-life insurer. Source [6]. BSCR mkt def life health nl mkt 1,25,25,25,25 def,25 1,25,25,5 life,25,25 1,25 health,25,25,25 1 nl,25,5 1 FIGURE 3: op level correlation matrix between risk classes. Source: [1]. Mkt int eq prop sp conc fx int 1,5,25,25 eq 1,75,25,25 prop,5,75 1,25,25 sp,25,25,25 1,25 conc 1 fx,25,25,25,25 1 FIGURE 4: Base level correlation matrix between market risk tpes. Source: [1].

10 574 D. FILIPOVIC Life mort long dis lapse exp rev CA mort 1,5,25 long 1,25,25,25 dis,5 1,5 lapse,25 1,5 exp,25,25,5,5 1,25 rev,25,25 1 CA 1 FIGURE 5: Base level correlation matrix between life risk tpes. Source: [1]. NL pr CA pr 1 CA 1 FIGURE 6: Base level correlation matrix between non-life risk tpes. Source: [1]. mort long dis lapse exp rev CA int,2,1,3,8,8,3 eq,4,15,5,12,12,4 prop,4,16,5,13,13,4 sp,3,11,4,3 conc,,,,,, fx,2,8,3,6,6,2,5,8,7,,4 FIGURE 7: Base level correlation matrix (7) between market and life risk tpes. mort long dis lapse exp rev CA int,1,12,2,7,4, eq,2,2,4,12,7, prop,,4,1,2,1, sp,1,11,2,6,4, conc,,,,,, fx,,2,,1,1,,8,14,3,8,,1 FIGURE 8: Minimal base level correlation matrix (8) between market and life risk tpes. mort long dis lapse exp rev CA int eq prop sp conc fx FIGURE 9: Uniform base level correlation matrix (8) between market and life risk tpes.

11 MULI-LEVEL RISK AGGREGAION 575 pr CA int,7,2 eq,23,6 prop,21,6 sp,2 conc,4,1 fx,8,2 FIGURE 1: Base level correlation matrix (7) between market and non-life risk tpes. pr CA int,6,2 eq,26,7 prop,4,1 sp,3,1 conc,6,2 fx,1, FIGURE 11: Minimal base level correlation matrix (8) between market and non-life risk tpes. pr CA int,14,14 eq,14,14 prop,14,14 sp,14,14 conc,14,14 fx,14,14 FIGURE 12: Uniform base level correlation matrix (8) between market and non-life risk tpes.