ETERPRISE RISK MAAGEMET I ISURACE GROUPS: MEASURIG RISK COCETRATIO AD DEFAULT RISK ADIE GATZERT HATO SCHMEISER STEFA SCHUCKMA WORKIG PAPERS O RISK MAAGEMET AD ISURACE O. 35 EDITED BY HATO SCHMEISER CHAIR FOR RISK MAAGEMET AD ISURACE FEBRUARY 2007
ETERPRISE RISK MAAGEMET I ISURACE GROUPS: MEASURIG RISK COCETRATIO AD DEFAULT RISK adne Gatzert Hato Schmeser Stefan Schuckmann JEL Classfcaton: G22, G28, G32 ABSTRACT In fnancal conglomerates and nsurance groups, enterprse rsk management s becomng ncreasngly mportant n controllng and managng the dfferent ndependent legal enttes n the group. The am of ths paper s to assess and relate rsk concentraton and jont default probabltes of the group s legal enttes n order to acheve a more comprehensve pcture of an nsurance group s rsk stuaton. We further examne the mpact of the type of dependence structure on results by comparng lnear and nonlnear dependences usng dfferent copula concepts under certan dstrbutonal assumptons. Our results show that even f fnancal groups wth dfferent dependence structures do have the same rsk concentraton factor, jont default probabltes of dfferent sets of subsdares can vary tremendously.. ITRODUCTIO Durng the last several years, there has been a trend toward consoldaton (M&A actvtes) n the fnancal sector of many countres (see Amel, Barnes, Panetta, and Salleo, 2004). Regulatory authortes, as well as ratng agences, are concerned wth new types of rsk and rsk concentratons arsng n fnancal conglomerates and wth how to properly assess them n group supervson (for the European Unon, see, e.g., CEIOPS, 2006). In ths context, enterprse rsk management (ERM) has become ncreasngly mportant. ERM takes a comprehensve vew of rsk and helps manage rsks n a holstc and consstent way (CAS, 2003). The am of ths paper s to provde a detaled and more comprehensve The authors are wth the Unversty of St. Gallen, Insttute of Insurance Economcs, Krchlstrasse 2, 900 St. Gallen, Swtzerland, E-Mal: nadne.gatzert@unsg.ch, hato.schmeser@unsg.ch, stefan.schuckmann@unsg.ch.
2 pcture of an nsurance group s rsk stuaton by assessng and relatng the rsk concentraton and jont default probabltes of ts legal enttes. We further examne the mpact of the type of dependence structure on results by comparng lnear and nonlnear dependences usng the concept of copulas. Determnaton of the rsk concentratons of an nsurance group can be based on an analyss of dversfcaton effects at a corporate level snce dversfcaton s the opposte of concentraton. In partcular, rsk concentratons, lke nterdependences or accumulaton, reduce dversfcaton effects. The dversfcaton effect s measured wth the economc captal of an aggregated rsk portfolo, whch mplctly reles on the assumpton that dfferent legal enttes are merged nto one. An essental aspect n aggregatng rsks s modelng the dependence structure usng lnear and nonlnear dependences. Copula theory can be used to model nonlnear dependences n extreme events and to test the fnancal stablty of a conglomerate structure. There has been steady growth n research on the applcaton of copula theory to rsk management. Embrechts, Mcel, and Straumann (2002) ntroduce copulas n fnance theory and analyze the effect of dependence structures on value at rsk. L (2000) apples copulas to the valuaton of credt dervatves. Frey and Mcel (200) model dependent defaults n credt portfolos, wth a specal emphass on tal dependences. An ntroducton to the use of copulas n the actuaral context can be found n Frees and Valdez (998) and Embrechts, Lndskog, and Mcel (2003). A central aspect of ERM s the aggregaton of dfferent types of rsk to calculate the economc captal necessary as a buffer aganst adverse outcomes. Wang (998, 2002) gves an overvew on the theoretcal background of economc captal modelng, rsk aggregaton, and the use of copula theory n enterprse rsk management. Kurtzkes, Schuermann, and Wener (2003) aggregate rsks at dfferent levels of a fnancal holdng company under the assumpton of jont normalty; n an emprcal study, they compute the relatve dversfcaton effect for several conglomerates. Ward and Lee (2002) use a normal copula approach to aggregate the rsks of a dversfed nsurer n a combned analytcal and smulaton model. Dmakos and Aas (2004) apply a smlar method to model total economc captal, and combne rsks by parwse aggregaton; they present a pract-
3 cal approach to estmate the jont loss dstrbuton of a orwegan bank and a orwegan lfe nsurance company. Favre (2003) models the overall loss dstrbuton for a four-lnes-of-busness nsurance company and examnes the nfluence of dfferent types of copulas on the value at rsk and the company s default probablty. Tang and Valdez (2006) smulate the economc captal requrements for a multlne nsurer, takng nto account dfferent types of dstrbutons and dfferent types of copulas; the resultng values for economc captal are used to compute absolute dversfcaton benefts. Rosenberg and Schuermann (2006) relax the jont normalty assumpton and use copula theory to aggregate rsks wth nonnormal margnals; they analyze the nfluence of the busness mx between credt, market, and operatonal rsk on value at rsk and calculate dversfcaton benefts by comparng the value at rsk of the dversfed conglomerate wth the stand-alone value at rsk. However, most of the lterature does not take nto account the specal rsk profle of fnancal conglomerates that arses from the group-holdng structure. Fnancal conglomerates or nsurance groups consst of several ndependent legal enttes, each wth lmted lablty. In the European Unon, for example, combnng bankng and nsurance actvtes n the same legal entty s prohbted (see Artcle 6(b) of the Lfe Insurance Drectve 2002/83/EC and Artcle 8(b) of the on- Lfe Insurance Drectve 73/239/EEC). Artcle 8() of the Lfe Insurance Drectve prohbts the combnaton of lfe and non-lfe busness n the same legal entty. Hence, where such s even lawful, a transfer of funds between dfferent legal enttes n case of an nsolvency of one entty occurs only f a transfer-of-losses contract has been sgned or f the management of the corporate group decdes n favor of cross-subsdzaton (e.g., for reputatonal reasons). Snce the contractng party s usually not the whole group, but a sngle subsdary. In prncple, the structure of an nsurance group s not mportant to those buyng nsurance from the subsdary n respect to the nsurer s default rsk. Thus, generally, for polcyholders and other debt holders, only the default rsk of ndvdual legal enttes and ther ablty to meet outstandng labltes are of relevance when there s no transfer-of-losses contract between members of the group. However, for the executve board of the nsurance group and for shareholders, nformaton on dversfcaton and thus on rsk concentraton and jont default probabltes s mportant when consderng the rsk profle of the
4 conglomerate for ERM. Informaton on rsk concentraton may also be helpful n obtanng a certan ratng level from a ratng agency. From the perspectve of regulatory agences, rsk concentraton nformaton can be valuable n analyzng systemc rsk of nsolvency. The default of a whole fnancal group wll, n general, have a stronger mpact on fnancal markets than the default of a sngle subsdary company. Ths paper extends prevous lterature by analyzng rsk concentratons n an nsurance group and by concurrently reportng jont default probabltes for sets of legal enttes wthn a fnancal conglomerate. otng that jont default probabltes only depend on ndvdual default probabltes and the couplng dependence structure, we further study the nfluence of dfferent dependence structures usng the concept of copulas. In partcular, we consder an nsurance group wth three legal enttes and compare results for Gauss, Gumbel, and Clayton copulas for normal and nonnormal margnal dstrbutons. Our results demonstrate that even f dfferent dependence structures mply the same rsk concentraton factor for the fnancal group, jont default probabltes of dfferent sets of subsdares can vary tremendously wth the dependence structure. The analyss shows that the smultaneous consderaton of rsk concentraton and default probabltes can provde nformaton of substantal value. The remander of the paper s organzed as follows. Secton 2 ntroduces the concept of dversfcaton on economc captal and rsk concentraton of a fnancal conglomerate. Dependence structures are presented n Secton 3, ncludng lnear and nonlnear dependences modeled wth copulas. In the numercal analyss, Secton 4, we compare results for Gauss, Gumbel, and Clayton copulas under dfferent dstrbutonal assumptons. Secton 5 summarzes the fndngs. 2. RISK COCETRATIO AD DEFAULT RISK Ths secton descrbes, frst, a framework for measurng rsk concentratons by calculatng the dversfcaton effect on the economc captal of an nsurance group, assumng that the dfferent legal enttes are merged. The economc captal s the amount necessary to buffer aganst unexpected losses from busness actvtes so as to prevent default at a specfc rsk tolerance level for a fxed tme horzon. Dversfcaton s generally ntended to reduce the overall rsk level n an n-
5 surance group and thus acts to allevate the dangers nherent n rsk concentraton. Calculatng rsk concentraton n an nsurance group can thus be based on an examnaton of dversfcaton effects at the group level. In a second step, the jont default probabltes of legal enttes wthn a conglomerate are ntroduced. We focus on an nsurance group wth a holdng structure and dfferent companes (legal enttes) wth lmted lablty. Generally, such a conglomerate s subject to market rsks, credt rsks, underwrtng rsks, and operatonal rsks. If each member of the group s faced wth dentcal rsks, one would expect the stochastc labltes of the dfferent enttes to be hghly correlated. Ths wll usually be the case f several frms of the same type, for example, fnancal frms, form a group. However, f the corporate group s composed of companes from wdely dfferent ndustres, the labltes between the dfferent legal enttes mght be rather uncorrelated. In what follows, we wll consder an nsurance group consstng of a bank, a lfe nsurer, and a non-lfe nsurer. In our framework, the equty value of each subsdary legal entty s modeled at two ponts n tme (t = 0, ). The value of the assets (labltes) at tme t = of company s defned as A ( L ). Debt and equty captal n t = 0 s nvested n rskless assets, leadng to a determnstc cash flow for the assets n t =, whereas labltes pad out n t = are modeled stochastcally. Stand-alone economc captal The amount of necessary economc captal depends on the specfc rsk tolerance level and on the measure chosen to evaluate corporate rsk. In the followng, we determne the necessary amount of captal usng the default probablty. The default probablty α of each legal entty can be wrtten as ( < ) =. P A L α In the next step, the nvested assets A are dvded nto two parts the expected value of the labltes E( L ) and the economc captal EC : ( ( ) ) ( ) P E L + EC < L = α ( ) α. P L E L > EC = ()
6 Hence, gven a probablty dstrbuton for the labltes and a certan safety level α, the economc captal EC can be derved. The necessary economc captal EC for dfferent legal enttes wthn an nsurance group can be calculated by ( ) ( ) EC = VaR α L E L =,,. (2) For consstency, all companes wthn the conglomerate should have the same safety level α. Therefore, value at rsk (VaR) s defned by { } ( ) ( ) nf ( ) VaRα L = F α = x F x α, where Aggregaton L L F L stands for the dstrbuton functon of the labltes for company. Assumng that the several companes n the nsurance group are merged nto one company (full lablty between legal enttes), the necessary economc captal for the safety level α on an aggregate level for L can be wrtten as aggr = α = = EC VaR L E L =. (3) To calculate the quantle n Equaton (3), nformaton about the cumulatve dstrbuton of the labltes s needed. Closed-form solutons for L = can be derved only for a lmted number of dstrbutons. In the case of a normal dstrbuton, only the varance of the portfolo s needed to determne the aggregate economc captal EC aggr. If no closed-form soluton can be obtaned, the quantle of the dstrbuton of the aggregate labltes L = can be estmated usng ether numercal smulaton technques or analytcal approxmatons (for an overvew, see Daykn, Pentkanen, and Pesonen, 994, pp. 9 ff.). Dversfcaton versus concentraton Gven Equatons (2) and (3), dversfcaton can be measured wth the rato of aggregated economc captal to the sum of stand-alone economc captal (see, e.g., Kurtzkes, Schuermann, and Wener, 2003),
7 d = EC = aggr EC. (4) In the case of lnear dependences, the factor d takes on values between zero and one and can be used to compare the level of rsk concentraton n conglomerates. A value of one corresponds to perfect correlaton, whch means that there would be no dversfcaton benefts f the dfferent legal enttes merged nto one company. When rsk factors are less than perfectly correlated, some of the rsk can be dversfed. Absolute measures for dversfcaton can be found n the lterature (see, e.g., Tang and Valdez, 2006); however, absolute measures do not allow the comparson of companes and conglomerates that are of dfferent szes. Gven a benchmark company, a hgher value of d mples possble rsk concentraton, snce lower values of d mean a hgher dversfcaton and thus a lower rsk concentraton. We henceforth refer to the coeffcent n Equaton (4) as the rsk concentraton factor. To keep the dfferent quanttes n Equaton (4) comparable, t s mportant to use the same rsk measure and the same tme horzon for all legal enttes when calculatng the economc captal. Generally, dversfcaton of the group s of no relevance to debt holders of the group s ndvdual companes (e.g., polcyholders n the case of an nsurance group) snce the whole group s not the contractng party, that s, the contract s between the polcyholder and the nsurance subsdary only (although there mght be transfer-of-loss contracts). However, for management and shareholders of the corporate group, nformaton about rsk concentraton n the dfferent sectors s of hgh mportance. Determnaton of default probabltes Even though calculaton of the dversfcaton factor may enable the detecton of rsk concentratons wthn the conglomerate, the factor s n most cases only a hypothetcal number snce ndvdual legal enttes generally do not (fully) cover the losses of the other enttes. To obtan further nsght about the conglomerate s rsk stuaton, jont default probabltes are approprate and can provde addtonal and valuable nformaton.
8 In contrast to the determnaton of the rsk concentraton factor, whch requres a convoluton over dfferent enttes L =, default probabltes make use of only the jont dstrbuton functon. To determne the jont default probablty of two or more legal enttes, the jont cumulatve dstrbuton functon s needed. For the case of a conglomerate comprsed of three legal enttes, the jont default probabltes of exactly one, two, and three legal enttes are gven by ( ) P = P L A, L A, L A P P, 2 2 3 3 2 3 ( j j k k ) P2 = P L > A, L > A, L A, j k, (,, ) P = P L > A L > A L > A. 3 2 2 3 3 It s assumed that no transfer of losses between companes wll occur. 3. MODELIG THE DEPEDECE STRUCTURE In rsk management, approprate modelng of dependence structures s very mportant. One recommendaton n the lterature s to apply copulas n addton to lnear correlaton to ensure an adequate mappng of dependence (see Embrechts, Mcel, and Straumann, 2002). Copulas allow for the ncluson of features such as fat tals and skewness for nonellptcally dstrbuted rsks. In ths secton, frst, the concept of copulas s presented n regard to modelng nonlnear dependences. Second, lnear correlatons are dscussed as a specal case, whch further allows for a closed-form soluton for economc captal f labltes are normally dstrbuted. 3.. Copulas For contnuous multvarate dstrbuton functons, copulas serve to separate the unvarate margns and the multvarate dependence structure. The copula C represents the dependence structure and couples the margnal dstrbutons to a jont multvarate dstrbuton. Let the random varables X (wth =,..., ) have (contnuous) margnal dstrbuton functons F. From Sklar s theorem t follows that (see elsen, 999, p. 5)
9 ( <,..., < ) = X ( ),..., X,..., = C F ( x ) F ( x ) P X x X x F x x ( X X ),...,. For jont default probabltes, ths result mples that for fxed ndvdual default probabltes ( ) P X < 0 = α, =,...,, one can obtan for the jont default probablty of all enttes ( < 0,..., < 0) = ( 0,...,0) P X X FX,..., X = C( F ( 0 ),..., ( 0 ) X FX ) = C( α,..., α). (5) Hence, the jont default probablty depends on the dependence structure expressed by the copula C and on the margnal default probabltes α. In our case, these quanttes are gven and fxed snce the economc captal for each entty s adjusted for each entty such that the margnal default probabltes reman constant. For example, n the case of the three enttes consdered here, the probablty that legal entty and 2 default, and entty 3 survves s ( ) ( ( ) ( ) ( ) X X X X X ) ( < < > ) = ( ) ( ) = C( α α ) C( α α α ) P X 0, X 0, X 0 C F 0, F 0, C F 0, F 0, F 0 2 3 3 2 2,,,,. 2 2 3 The probablty of default for exactly two legal enttes can thus be calculated by 2 ( 0, 0, 0, j k ) ( α α ) ( α α ) ( α α ) ( α α α ) P = P X < X < X > j k = C,, + C,, + C,, 3 C,,. 2 3 3 2 2 3 (6) Furthermore, the probablty that exactly one company defaults s determned by ( ) P = P X > 0 =,2,3 P P (7) 2 3 where
0 ( ) ( ) P = P X < 0, X < 0, X < 0 = C α, α, α. (8) 3 2 3 2 3 In the analyss, we wll compare several copulas. To obtan boundares, we nclude the case of ndependence and perfect dependence (comonotoncty), represented by the copula (Mcel, Frey, and Embrechts, 2005, p. 89) (,..., ) mn {,..., } M u u = u u. The ndependence copula s gven by (Mcel, Frey, and Embrechts, 2005, p. 89) (,..., ) Π u u = u. = The default probablty for all three enttes s thus the product of the margnal default probabltes. From Equatons (6) (8), t follows that n the case of ndependence the default probabltes of exactly one, two, and three companes are ( α ) ( α ) ( α ) P = P P 2 3 2 3 P = α α + α α + α α 3 α α α 2 2 3 2 3 2 3 P3 = α α2 α3. The two most common Archmedan copulas, Clayton and Gumbel, are used to model the dependence structure between the enttes. These are explct copulas that have closed-form solutons and are not derved from multvarate dstrbuton functons as s the mplct Gaussan copula. In general, an -dmensonal Archmedan copula may be constructed by usng the respectve generator Φ ( t) (,..., ) ( ( )... ( )) C u u =Φ Φ u + +Φ u, whch must have certan propertes (as descrbed n Mcel, Frey, and Embrechts, 2005, p. 222). In the case of the -dmensonal Clayton copula, the generator and ts nverse are gven by (see Wu, Valdez, and Sherrs, 2006, p. 7)
() t ( t θ ) Φ = and () t ( t ) / θ Φ = θ +, θ whch leads to the followng representaton / θ Cl θ C, ( u,..., u ) = u θ +, = where 0 θ <. For θ, one obtans perfect dependence; θ 0 mples ndependence (Mcel, Frey, and Embrechts, 2005, p. 223). For the -dmensonal Gumbel copula, the generator and ts nverse are gven by (Wu, Valdez, and Sherrs, 2006, p. 7) ( ) θ / θ Φ () t = log() t and Φ ( t) = exp( t ), whch mples the expresson / θ Gu θ Cθ, ( u,..., u) = exp ( logu) =, where θ. For θ, one obtans perfect dependence; θ mples ndependence (Mcel, Frey, and Embrechts, 2005, p. 220). Both Clayton and Gumbel copulas exhbt asymmetres n the dependence structure. The Clayton copula s lower tal dependent; the Gumbel copula s upper tal dependent. 3.2. The specal case of lnear dependence Lnear dependence s a specal case of the copula concept. If X s a multvarate Gaussan random vector, then ts copula s a so-called Gauss copula (Mcel, Frey, and Embrechts, 2005, p. 9) ( ) (,..., ) ( ),..., ( ) C u u =Φ Φ u Φ u, Ga R where R s the correlaton matrx, Φ denotes the standard unvarate standard normal dstrbuton functon, and Φ denotes the jont dstrbuton functon of X. The Gauss copula measures the degree of monotonc dependence and has no
2 closed-form soluton, only an ntegral representaton. The copula may be constructed by the nverse method, whch maps lnear dependence n the form of the lnear correlaton of ranks, as descrbed by Iman and Conover (982) and by Embrechts, Mcel, and Straumann (2002). The use of rank correlatons ensures the exstence of a multvarate dstrbuton wth the prescrbed margnals. Gven lnear correlatons ρ ( X, X j), Spearman s rank correlaton matrx for the Gauss copula can be derved from (Mcel, Frey, and Embrechts, 2005, p. 25) ρ ( X, X ) = 2 sn (, j ρ X X S j) π 6. (9) If the jont dstrbuton s a multvarate normal wth standard normal margnals, the economc captal EC for each entty can be calculated by (see, e.g., Hull, 2003, pp. 350 ff.) ( ) EC = σ L z α, (0) where z α denotes the α-quantle of the standard normal dstrbuton and σ stands for the standard devaton. To aggregate the economc captal under the assumpton that all sectors are carred n one company, correlatons between the labltes of the dfferent enttes are needed. The symmetrc correlaton matrx R wth coeffcents ρ j between the labltes of entty and entty j s gven by ρ ρ 2 ρ ρ 2 2 R =. ρ ρ 2 The correspondng entres n the covarance matrx Σ are gven by Σ j = ρj σ ( L ) σ( Lj ) and the standard devaton of the portfolo of the labltes L = L can be calculated by = σ ( L) j. (), j= = Σ
3 The aggregated economc captal, assumng that all sectors n the nsurance group are merged, can be calculated from ( ) EC σ L z aggr α =. (2) Equaton (2) can be reformulated usng Equaton () to obtan the followng formula (see Kurtzkes, Schuermann, and Wener, 2003; Groupe Consultatf, 2005): ( ) α α α ( ) α ( ) EC = σ L z = Σ z = z σ L ρ z σ L aggr j j j, j=, j= = EC ρ EC, j= j j (3) T EC ρ2 ρ EC EC2 ρ2 ρ2 EC2 =. EC ρ ρ EC 2 Equaton (3) llustrates that the effect of dversfcaton on the aggregated economc captal EC aggr depends on the number of legal enttes, the relatve porton of the economc captal of the ndvdual companes EC, and the correlaton between the labltes of the dfferent companes. One way to calculate economc captal for lablty dstrbutons that are not normally dstrbuted s to use analytcal approxmaton methods such as the normalpower concept (see, e.g., Daykn, Pentkanen, and Pesonen, 994, pp. 29 ff.). 4. UMERICAL EXAMPLES In ths secton we present numercal examples n order to examne the nfluence of the dependence structure (nonlnear vs. lnear dependence) and the dstrbutonal assumptons (normal vs. non-normal) on rsk concentraton and default probabltes. Frst, the case of lnear dependence s presented for normally and non-normally dstrbuted labltes wth dfferent szes. Second, nonlnear dependences are examned for normalty and non-normalty.
4 Table sets out the nput parameters that are the bass for the numercal examples analyzed n ths secton. The values and dstrbutons are chosen to llustrate central effects. TABLE Economc captal for ndvdual enttes n an nsurance group for dfferent dstrbutonal assumptons gven a default probablty α = 0.50% and E( L ) = 00, =, 2, 3. Legal entty Dstrbuton type Case (A) Case (B) normal σ ( L ) EC ( ) EC σ L Bank ormal 5.00 38.64 35.00 90.5 Lfe nsurer ormal 5.00 38.64 5.00 2.88 on-lfe nsurer ormal 5.00 38.64 5.00 2.88 Sum 5.9 5.9 non-normal Bank ormal 5.00 38.64 35.00 90.5 Lfe nsurer Lognormal 5.00 45.22 5.00 3.59 on-lfe nsurer Gamma 5.00 42.84 5.00 3.35 Sum 26.70 7.09 Table contans values for two dfferent cases, (A) and (B), for normally and non-normally dstrbuted labltes. The gven default probablty of 0.50% s adapted to Solvency II regulatory requrements, whch are currently beng debated (European Commsson, 2005). For normally dstrbuted labltes, economc captal can be calculated usng Equaton (0) wth a standard normal quantle of z α = 2.5758. The conglomerate under consderaton conssts of a bank, a lfe nsurance company, and a non-lfe nsurer. In case (A), the labltes of all three enttes have the same standard devaton and thus requre the same economc captal. In case (B), the bank has a substantally hgher standard devaton than the nsurance enttes. Accordngly, the resultng economc captal dffers. We next change the dstrbuton assumpton to allow for non-normalty. ow, only the labltes of company (Bank) are normally dstrbuted, whereas the labltes of company 2 (Lfe Insurer) and company 3 (on-lfe Insurer) follow, respectvely, a lognormal and a gamma dstrbuton. To keep the cases compara-
5 ble, the expected value µ and standard devaton σ reman fxed. In the case of lognormal (a, b) dstrbuton, the parameters can be calculated by 2 a = ln ( µ ) b / 2 and b 2 = ln ( + σ 2 / µ 2 ) (Casella and Berger, 2002, p. 09). For 2 2 gamma dstrbuton (α, β), the parameters are gven by α = µ / σ and β = σ 2 / µ (Casella and Berger, 2002, pp. 63 64). The assumpton of non-normal dstrbutons leads to dfferent ndvdual economc captal values n case (A) and case (B) compared to the values under the normalty assumpton. As a result, the sum of the ndvdual economc captal (26.70 n case (A) and 7.09 n case (B)) dffers also (5.9 for both cases under the normalty assumpton). The numercal analyss proceeds as follows. Frst, we calculate the necessary aggregated economc captal based on the value at rsk at the group level for a confdence level α= 0.50% (Equaton (3)). The concentraton factor can then be derved usng the stand-alone economc captal of the legal enttes gven n Table by way of Equaton (4). Subsequently, we calculate the correspondng default probabltes P, P 2, P 3. 4.. umercal results for lnear dependence To calculate the necessary economc captal at the group level, the correlaton matrx for the labltes s needed. Estmaton of dependences can be made on the bass of macroeconomc models. For nstance, Estrella (200) derves correlatons from stock market returns to measure possble dversfcaton effects between the bank and nsurance sectors. To obtan more comprehensve nformaton on the rsk stuaton of conglomerate under consderaton (see Table ), we compare the effect of dstrbutonal assumptons on the concentraton factor and default probabltes. Fgure shows a plot of the default probabltes for dfferent choces of the correlaton matrx wth ncreasng dependency and the correspondng concentraton factors for dfferent dstrbutonal assumptons. In partcular, we compare the cases (A) and (B) gven n Table when labltes follow a normal dstrbuton (normal) and when they are partly nonnormally dstrbuted (non-normal). For ease of exposton we use the same coeffcent of correlaton ρ between the labltes of all enttes,.e.,
6 ( j) ρ L, L = ρ, j. Fgure shows how the concentraton factor and nformaton on default probabltes can complement each other. Part a) llustrates that the jont default probabltes depend on the dependence structure between the legal enttes and ndvdual default probabltes. Hence, for normal and non-normal dstrbutons, the jont default probabltes reman unchanged, whereas the concentraton factor can dffer substantally. In the case of ndependence, jont default probabltes of two and three companes are (approxmately) zero n the example consdered and only ndvdual default occurs wthn the group. Wth ncreasng dependence, the probablty of a sngle default (P ) decreases, whle the probablty of combned defaults (P 2, P 3 ) ncreases. For hgher correlatons, the probablty of a combned default of two enttes (P 2 ) decreases agan. For perfectly correlated labltes, all three enttes default wth probablty 0.50%, whle P = P 2 = 0.
7 FIGURE Default probabltes and rsk concentraton factor for lnear dependence on the bass of Table. a) Jont default probabltes for lnear dependence.60%.40%.20%.00% 0.80% 0.60% 0.40% P P2 P3 0.20% 0.00% 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.97 0.99 rho b) Rsk concentraton factor for lnear dependence 00% 90% 80% 70% normal, case (A) non-normal, case (A) normal, case (B) non-normal, case (B) 60% 50% 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.97 0.99 rho otes: P = probablty that exactly one entty defaults; P 2 = probablty that exactly two enttes default; P 3 = probablty that all three enttes default. Part b) of Fgure llustrates that gven that the labltes have the same standard devatons (case (A)) the dstrbutonal assumpton has only margnal nfluence on the concentraton factor, but that dfferent correlaton factors and frm sze (case (B)) do matter. As an example, consder the case ρ = 0.3, mplyng a correlaton matrx
8 0.3 0.3 R = 0.3 0.3. 0.3 0.3 The correspondng concentraton factor can be derved usng Equaton (3). For normally dstrbuted labltes n case (A), the aggregated economc captal s A EC aggr = 84.65. Ths s lower than the sum of stand-alone economc captal (5.9) due to dversfcaton effects. Hence, the concentraton factor s gven by d = 84.65/5.9 = 73.03%. Changng only the dstrbutonal assumpton to a non-normal dstrbuton leads to a lower value of d = 90.78/26.70 = 7.65%. Hence, the concentraton factor decreases n case of non-normal dstrbuton even though the aggregated economc captal ncreases to 90.78. Ths llustrates that an absolute comparson of aggregated economc captal may be msleadng. The concentraton factor s very smlar to the results for the normal case, wth a dfference of only.38 percentage ponts, whch can be explaned by the calbraton of the dstrbutons. We calculated the parameters of the lognormal and gamma dstrbutons usng the same values for expected value and standard devaton so as to acheve better comparablty between dfferent stuatons. Hence, we can say that n the example gven the choce of a non-normal dstrbuton has very lttle mpact on concentraton factors. A much larger effect can be observed when comparng case (A) wth case (B). For normal dstrbuton, the concentraton factor for case (B) s d = 99.76/5.9 = 86.07%. Hence, ths stuaton leads to a hgher concentraton factor than case (A) (d = 73.03%). Thus, the stuaton gven n case (B) ndcates a possble exstence of rsk concentraton wthn the conglomerate, orgnatng from the bank. The bank s relatvely large rsk contrbuton to total group rsk causes a less effectve dversfcaton of rsks. Losses resultng from bankng actvtes n case (B) are less lkely to be compensated by good results from nsurance actvtes than n case (A). Thus the concentraton factor d s useful for examnng the exstence of rsk concentratons whenever a benchmark company s avalable. Overall, Fgure, part b) demonstrates that the dfference between the concentraton factors of cases (A) and (B) decreases wth ncreasng correlaton. In the
9 case of perfect postve correlaton, ρ =, the dfference vanshes and the concentraton factor takes on ts maxmum of 00%. Even though all four curves mply the same jont default probabltes, they have dfferent rsk concentraton factors. The dfferences n d result from changes n the amount of economc captal needed to retan a constant default probablty. 4.2. umercal results for nonlnear dependence In ths secton, we alter the assumpton for the dependence structure and examne the mpact of nonlnear dependences on rsk concentraton and jont default probabltes usng Clayton and Gumbel copulas as descrbed n Secton 3.. Both copulas are constructed usng Monte Carlo smulaton wth the same 200,000 paths so as to ncrease comparablty. The Clayton and Gumbel copulas are smulated usng the algorthms n Mcel, Frey, and Embrechts (2005, p. 224). The algorthm for the Gumbel copula uses postve stable varates, whch were generated wth a method proposed n olan (2005). umercal results for the Clayton and Gumbel copulas are llustrated n Fgures 2 and 3, respectvely. In both fgures, part a) dsplays default probabltes as a functon of the dependence parameter θ and part b) shows the correspondng concentraton factors. The ndependence copula marked as Π n the fgures serves as a lower boundary, whle the case of comonotoncty (M) represents perfect dependence and s thus an upper bound. At frst glance, both dependence structures n Fgures 2 and 3 appear to lead to smlar results as n the lnear case n Fgure. Overall, the probablty that any company defaults decreases wth ncreasng θ. As before, under perfect comonotoncty, all three enttes always become nsolvent at the same tme wth probablty 0.50%, whle the probablty for one or two defaulted companes s zero (P = P 2 = 0). In fact, n ths case, the concentraton factor exceeds 00% snce the value at rsk s not a subaddtve rsk measure (for a dscusson, see Embrechts, Mcel, and Straumann, 2002, p. 22).
20 FIGURE 2 Default probabltes and rsk concentraton factor for Clayton copula on the bass of Table. a) Jont default probabltes for Clayton copula.60%.40%.20%.00% 0.80% 0.60% 0.40% 0.20% 0.00% Π 0 20 40 60 80 00 20 M theta P P2 P3 b) Rsk concentraton factor for Clayton copula 00% 90% 80% 70% normal, case (A) non-normal, case (A) normal, case (B) non-normal, case (B) 60% 50% Π 0 20 40 60 80 00 20 M theta otes: P = probablty that exactly one entty defaults; P 2 = probablty that exactly two enttes default; P 3 = probablty that all three enttes default. A comparson of Fgures 2 and 3 reveals that the type of tal dependence (upper vs. lower) has a sgnfcant mpact on the partcular characterstcs of the jont default probabltes curves. In case of the upper tal dependent Gumbel copula, companes become nsolvent far more often, and hence the jont default probablty of all three enttes quckly approaches 0.50% n the lmt M. In contrast, the default probabltes of the lower tal dependent Clayton copula converge to 0.50% much more slowly. In fact, even for θ close to 20, the generaton of random numbers from the Clayton copula becomes ncreasngly dffcult, despte the fact that the jont default probablty of all three enttes s only 0.3%.
2 FIGURE 3 Default probabltes and rsk concentraton factor for Gumbel copula on the bass of Table. a) Jont default probabltes for Gumbel copula.60%.40%.20%.00% 0.80% 0.60% 0.40% P P2 P3 0.20% 0.00% Π..3.5 2 2.5 3 3.5 4 M theta b) Rsk concentraton factor for Gumbel copula 00% 90% 80% 70% normal, case (A) non-normal, case (A) normal, case (B) non-normal, case (B) 60% 50% Π..3.5 2 2.5 3 3.5 4 M theta otes: P = probablty that exactly one entty defaults; P 2 = probablty that exactly two enttes default; P 3 = probablty that all three enttes default. Even though results for default probabltes and concentraton factors under the Gauss, Gumbel, and Clayton copulas look very smlar at frst glance, they can dffer tremendously, whch wll be demonstrated n the next subsecton. 4.3. Comparng the mpact of nonlnear and lnear dependences To compare and dentfy the consderable effects of the underlyng dependence structures on default probabltes, we take examples from the Fgures, 2, and 3 that have the same concentraton factor, usng case (A) wth normally dstrbuted margnals so as to make the results comparable.
22 Two examples are presented n Fgure 4 for fxed concentraton factors of 90% n part a) and 99.40% n part b) from the Clayton, Gauss, and Gumbel copulas. The examples n each part of the fgure have the same concentraton factor and thus exhbt the same value at rsk. Although the compared companes have the same rsk, default probabltes dffer substantally wth the dependence structure. FIGURE 4 Comparson of jont default probabltes for one (P ), two (P 2 ), and three (P 3 ) companes for dfferent dependence structures; case (A), normal dstrbutons. a) Rsk concentraton factor d = 90%. Clayton (theta = 20) Gauss (rho = 0.7) P3 P2 P Gumbel (theta =.3) b) Rsk concentraton factor d = 99.40%. 0.00% 0.30% 0.60% 0.90%.20%.50% Clayton (theta = 20) Gauss (rho = 0.95) P3 P2 P Gumbel (theta = 3) 0.00% 0.30% 0.60% 0.90%.20%.50% otes: P = probablty that exactly one entty defaults; P 2 = probablty that exactly two enttes default; P 3 = probablty that all three enttes default. A comparson of parts a) and b) of Fgure 4 shows that the sum of default probabltes (= P + P 2 + P 3 ).e., the probablty that one, two, or three companes default s hgher for the lower concentraton factor d = 90%. Furthermore, for d = 99.40%, the parttonng between the three jont default probabltes (P, P 2, P 3 ) s shfted toward P 3, whle P decreases. Hence, a hgher concentraton factor
23 s accompaned by a lower sum of default probabltes, but nduces a sgnfcantly hgher jont default probablty of all three enttes. Fgure 4 demonstrates the consderable nfluence that the choce between Clayton, Gauss, and Gumbel copulas has on jont default probabltes. The Clayton copula leads to the hghest sum of default probabltes, but has the lowest probablty of default for all three companes (P 3 ). The other extreme occurs under the Gumbel copula, where P 3 s hghest and P takes the lowest value, whle the Gauss copula nduces values between those of the Clayton and Gumbel copulas. Our results show that even f dfferent dependence structures mply the same value at rsk and thus the same rsk concentraton factor, jont default probabltes can dffer tremendously. Furthermore, our analyss demonstrates that the smultaneous reportng of rsk concentraton factors and default probabltes can be of substantal value, especally for the management of the corporate group. By comparng lnear and nonlnear dependences, we found that the effect of msmodelng dependences may not only lead to sgnfcant dfferences n assessng rsk concentraton, but can also lead to msestmatng jont default probabltes. Hence, there s a substantal model rsk nvolved wth respect to dependence structures. 5. SUMMARY Ths paper assessed and related rsk concentratons and jont default probabltes of legal enttes n a conglomerate composed of three enttes, a bank, a lfe nsurance company, and a non-lfe nsurance company. Our procedure provded valuable nsght regardng the group s rsk stuaton, whch s hghly relevant for enterprse rsk management purposes. An nsurance group (a conglomerate) typcally conssts of several legally ndependent enttes, each wth lmted lablty. However, dversfcaton concepts assume that these enttes are fully lable and all together meet all outstandng labltes of each. Even f dversfcaton s of no mportance from a polcyholder perspectve, t s useful n determnng rsk concentraton n an nsurance group because greater dversfcaton generally mples less rsk. To determne default probabltes, we focused on the case of lmted lablty wthout transfer of losses between the dfferent legal enttes wthn the group.
24 Jont default probabltes only depend on ndvdual default probabltes and the couplng dependence structure. Hence, we studed the effect of dfferent dependence structures usng the concept of copulas. In the numercal analyss, we consdered an nsurance group comprsed of three legal enttes and compared results from the Gauss, Gumbel, and Clayton copulas for normal and non-normal margnal dstrbutons. Economc captal was adjusted for each stuaton to satsfy a fxed ndvdual default probablty. In contrast to the rsk concentraton factor, jont default probabltes only depend on ndvdual default probabltes and on the dependence structure, not on dstrbutonal assumptons. For all models, we found that the rsk concentraton factor and the jont default probablty of all three enttes ncrease wth ncreasng dependence between the enttes, whle the probablty of a sngle default decreases. Overall, the sum of default probabltes of one, two, or three enttes decreases wth ncreasng dependence. Furthermore, one entty s large rsk contrbuton, n terms of volatlty, led to a much hgher rsk concentraton factor for the group as a whole. Our fndngs further demonstrated that even f dfferent dependence structures mply the same rsk concentraton factor for the group, jont default probabltes for dfferent sets of subsdares can vary tremendously. In partcular, the lower tal dependent Clayton copula led to the lowest probablty of default for all three enttes, whle the upper tal dependent Gumbel copula exhbted the hghest default probablty. The analyss showed that a smultaneous consderaton of rsk concentraton factor and default probabltes can be of substantal value, especally for the management of the corporate group wth respect to enterprse rsk management.
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