The systemic importance of financial institutions 1

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1 Nkola Tarashev Claudo Boro Kostas Tsatsarons The systemc mportance of fnancal nsttutons 1 Prudental tools that target fnancal stablty need to be calbrated at the level of the fnancal system but mplemented at the level of each regulated nsttuton. They requre a methodology for the allocaton of system-wde rsk to the ndvdual nsttuton n lne wth ts systemc mportance. Ths artcle proposes a general and flexble allocaton methodology and uses t to dentfy and quantfy the drvers of systemc mportance. It then llustrates how the methodology could be employed n practce, based on a sample of large nternatonally actve nsttutons. JEL Classfcaton: C15, C71, G20, G28. On 16 September 2008 the US authortes announced that they would take the unprecedented step of offerng emergency fnancal support to AIG, a large nsurance conglomerate. The decson was rooted n concerns about the repercussons of the falure of ths nsttuton on the economy at large, e about ts systemc mportance. 2 Smlar far-reachng and urgent decsons were taken by authortes n other jursdctons. By contrast, n 1995, the Bank of England had allowed merchant bank Barngs to fal because t consdered ths would have no materal mpact on other banks (whch was subsequently confrmed). More generally, the events of the past two years serve as a stark remnder that systemc fnancal dsruptons can have large macroeconomc effects. As a result, the objectve of strengthenng the macroprudental orentaton of fnancal stablty frameworks has rsen to the top of the nternatonal agenda. 3 The man dstncton between the macro- and mcroprudental perspectves s that the former focuses on the fnancal system as a whole, whereas the latter focuses on ndvdual nsttutons. 4 1 The authors thank Marek Hlavacek for excellent research assstance, and Stephen Cecchett, Robert McCauley and Chrstan Upper for helpful comments. The vews expressed n ths artcle are those of the authors and not necessarly those of the BIS. 2 The press release from the Federal Reserve explaned: The Board determned that, n current crcumstances, the dsorderly falure of AIG could add to already sgnfcant levels of fnancal market fraglty and lead to substantally hgher borrowng costs, reduced household wealth, and materally weaker economc performance. 3 See G20 (2009) and de Larosere (2009) for reports on ths nternatonal consensus. 4 See Crockett (2000), Knght (2006) and Boro (2009) for an elaboraton of the macroprudental approach and progress n ts mplementaton. BIS Quarterly Revew, September

2 By necessty, however, the tools of fnancal sector supervson and key polcy nterventons are appled to ndvdual nsttutons, even when decsons are motvated by systemc consderatons. Thus, polcymakers need analytcal tools to help them assess the systemc mportance of ndvdual nsttutons. In tmes of crss, these tools can help to gauge the lkely mpact of dstress at a gven fnancal frm on the stablty of the overall fnancal system. In perods of calm, they can help to calbrate prudental nstruments, such as captal requrements and nsurance premums, accordng to the relatve contrbuton of dfferent nsttutons to systemc rsk. Ths artcle presents a methodology that takes as nputs measures of system-wde rsk and allocates them to ndvdual nsttutons. The methodology s derved drectly from a game-theoretc concept, the Shapley value, whch descrbes a way of allocatng the collectve beneft created by a group to the ndvdual contrbutors. The Shapley value approach satsfes a number of ntutve crtera and s qute general, beng applcable to a wde spectrum of measures of system-wde rsk. The methodology makes t straghtforward to quantfy the mpact of the varous drvers of an nsttuton s systemc mportance. These nclude ther rskness on a standalone bass, ther exposure to common rsk factors and the degree of sze concentraton n the system. A key result s that the contrbuton of an nsttuton to system-wde rsk generally ncreases more than proportonately wth ts sze. We apply the methodology to real-world data on a sample of 20 large nternatonally actve fnancal nsttutons. The results hghlght the nteracton among the varous drvers of systemc mportance. In our sample, none of them, taken n solaton, s a fully satsfactory proxy for systemc mportance. The artcle s organsed n four sectons. The frst secton descrbes the allocaton procedure and ts propertes. The second secton apples the procedure to a specfc measure of systemc rsk n hypothetcal and hghly stylsed fnancal systems n order to analyse the mpact of dfferent drvers of systemc mportance. The thrd secton dscusses how the methodology could be used n practce as a tool to mtgate systemc rsk and apples t to realworld data. The last secton concludes. The allocaton procedure: measurng systemc mportance The problem of allocatng system-wde rsk to ndvdual nsttutons s analogous to that of a rsk controller n an nvestment frm seekng to attrbute the use of the frm s rsk captal to ndvdual desk traders. The fact that the sum of the rsks ncurred by each desk n solaton does not equal the total rsk for the frm complcates the controller s problem. Smple summaton gnores that the nteractons among ndvdual postons could reduce or compound overall rsk. They would reduce t when postons across desks partally cancel each other out; they would compound t when losses n one sde of the busness are ncurred smultaneously wth, or trgger, losses n another. Game theorsts have tackled smlar problems n the context of cooperatve games. These are general settngs where a group of players A general methodology to attrbute systemc rsk BIS Quarterly Revew, September 2009

3 ... draws on game theory... engage n a collectve effort n order to generate a shared beneft 5 (called value ) for the group. The theoretcal problem of allocatng ths value among ndvdual players n a way that satsfes certan fundamental crtera s conceptually dentcal to that of rsk attrbuton descrbed above. Lloyd Shapley proposed a methodology that dstrbutes the overall value among players on the bass of ther ndvdual contrbutons (Shapley (1953)). The dea behnd the allocaton methodology s qute smple. Addng up what ndvdual players can acheve by themselves (the equvalent of summng up the standalone rsk of each tradng desk n the nvestment frm) s unlkely to reflect ther contrbutons to the productvty of others. Smlarly, calculatng the margnal contrbuton of a sngle player as the dfference between what the entre group can acheve wth and wthout the specfc ndvdual gves only a partal pcture of the ndvdual s contrbuton to the work of others. The reason s that ths method also gnores the complextes of blateral relatonshps. By contrast, the Shapley methodology accounts fully for the degree to whch such relatonshps affect the overall outcome. It accomplshes ths by ascrbng to ndvdual players the average margnal contrbuton each makes to each Box 1: Shapley value allocaton methodology: a specfc example Ths box llustrates the Shapley value allocaton methodology by reference to a specfc numercal example where three partes (A, B and C) can cooperate to generate a measurable outcome. If nobody partcpates nothng s produced, and each partcpant alone can produce 4 unts. The output of each possble groupng of the three partcpants s detaled n the left-hand column of the table below. Subgroup Subgroup output Margnal contrbuton of A Margnal contrbuton of B Margnal contrbuton of C A B C A, B A, C B, C A, B, C Shapley value Table A The margnal contrbuton of a player to a subgroup s calculated as the output of the subgroup mnus the output of the same subgroup excludng the ndvdual partcpant. For nstance, the margnal contrbuton of A to the output of the overall group (A, B, C) s equal to the dfference between 15, whch s the overall group s output, and 11, whch s the output of B and C together. The Shapley value of each player s the average of ts margnal contrbutons across all dfferently szed subgroups. For example, the value of B s equal to 5 (see bottom row). It s calculated as the average of 4, whch s ts ndvdual output, 6, whch s the mean contrbuton t makes to subgroups of sze two, and 5, whch s ts margnal contrbuton to the overall group. The calculaton can also be motvated as the expected margnal contrbuton of an ndvdual partcpant n groups that are formed randomly by sequentally selectng players (see Mas-Colell et al (1995)). 5 Ths s a very general concept that could be thought of as wealth, or collectve output. BIS Quarterly Revew, September

4 possble subgroup n whch they partcpate (see Box 1 on the prevous page for a detaled exposton of the methodology and a numercal example). In addton to ts smplcty, the Shapley value has a number of ntutvely appealng features. 6 It ensures that the gans from cooperaton between any two players are dvded equally between them; n other words, t s far n the sense that t does not lead to based outcomes that favour or penalse partcular players. It dstrbutes exactly the total beneft to all players, wthout resultng n any surplus or defct. It s symmetrc, n the sense that two players wth the same characterstcs receve the same share of the overall value. And t assgns no payoff to a player who makes no contrbuton to any subgroup. An applcaton of the Shapley value methodology to the measurement of nsttutons systemc mportance smply transposes the problem of dstrbutng a collectve value among ndvdual players to that of attrbutng overall rsk to ndvdual nsttutons. It requres as an nput a quanttatve measure of rsk for all groupngs of nsttutons. These range from the largest group comprsng all nsttutons to the smallest, whch consst of sngle nsttutons. The methodology then attrbutes the overall (system-wde) rsk to each nsttuton on the bass of ts average contrbuton to the rsk of all the groups n whch t partcpates. The degree of systemc mportance of nsttutons s therefore captured by the share of systemc rsk that s attrbuted to each of them. Insttutons wth hgher systemc mportance wll have a hgher Shapley value than others. A major strength of the Shapley value methodology s ts generalty. It accommodates any systemc rsk measure that treats the system as a portfolo of nsttutons and dentfes rsk wth the uncertanty about the returns (losses) on ths portfolo. In addton, exstng allocaton procedures are specfc applcatons of the Shapley value methodology. Ths s the case, for nstance, of the procedure recently proposed by Acharya and Rchardson (2009) for the calbraton of nsttuton-specfc premums for nsurance aganst systemc dstress. Tarashev et al (2009) dscuss these ponts at some length. Another strength of the Shapley value methodology s that t allows measures of systemc mportance to account for model and parameter uncertanty. Such uncertanty may make t natural to measure systemc rsk under alternatve models and parameter estmates. Ths would lead to alternatve measures of systemc mportance for each nsttuton. Beng lnear, the Shapley value mples that the weghted average of alternatve measures (a lnear combnaton) can be used as a sngle robust measure of systemc mportance.... and has many appealng features: t measures ndvdual contrbutons to rsk; s general and flexble; and s robust to uncertanty 6 For a fuller dscusson of the techncal propertes of the Shapley value, see Mas-Colell et al (1995). Tarashev et al (2009) provde a more detaled descrpton of how to mplement a Shapley value decomposton n the context of the attrbuton of system-wde rsk to ndvdual nsttutons. 78 BIS Quarterly Revew, September 2009

5 Drvers of systemc mportance: stylsed examples Drvers of systemc rsk Systemc mportance ncreases wth... In ths secton we study three drvers of systemc rsk and, hence, of the systemc mportance of ndvdual nsttutons. One s the rskness of ndvdual frms, as captured by ther probabltes of default (PDs). 7 Another s the degree of sze concentraton, or lumpness, of the system, whch ncreases as the number of nsttutons decreases or as ther relatve szes become more dsparate. The fnal drver s the nsttutons exposure to common (or systematc) rsk factors, whch arses ether because fnancal nsttutons are smlar to each other (eg lend to the same sectors) or because they are nterconnected. Importantly, whle the probablty of default (or nsolvency) can be constructed on the bass of nsttuton-specfc characterstcs alone, the other two drvers relate to characterstcs of the system as a whole. As a concrete measure of systemc rsk, we use expected shortfall, whch equals the expected (average) sze of losses n a systemc event (see the appendx on page 86 for detal). In general, a systemc event s defned as one that generates losses deemed large enough to cause dsruptons to the functonng of the system. In ths artcle, a systemc event s defned as the occurrence of extreme aggregate losses that materalse wth a gven small probablty, e losses that exceed a certan threshold. 8 The mpact of the three drvers on systemc rsk s qute ntutve. Keepng everythng else constant, an ncrease n nsttutons PDs leads to a hgher level of systemc rsk. Even f the PDs reman unchanged, greater lumpness of the system reduces dversfcaton benefts, rasng the lkelhood of extreme losses and, wth t, expected shortfall. Smlarly, greater exposure to common rsk factors ncreases the lkelhood of jont falures and hence also the lkelhood of extreme losses n the system. To explore the mpact of the same three drvers on the systemc mportance of ndvdual nsttutons, we resort to numercal exercses. For these exercses, we allocate system-wde expected shortfall to ndvdual nsttutons ( banks ) on the bass of the Shapley value methodology. The results, based on hghly stylsed hypothetcal systems, yeld four key messages. Frst, a rse n an nsttuton s exposure to a common rsk factor ncreases ts systemc mportance. Ths s llustrated n Table 1, whch compares a number of bankng systems, each comprsng 20 banks. In every system there are two homogeneous groups, A and B, whch dffer only wth respect to banks exposures to the common factor. Keepng the strength of exposures to the common factor n group B constant but ncreasng t for group A (across columns to the rght, n each panel) results n an ncrease n these banks share n systemc rsk. In the specfc example of a strongly captalsed system, the 7 Strctly speakng, an nsttuton s standalone rsk depends both on ts PD and on ts loss-gvendefault (LGD). Ths artcle abstracts from LGD by assumng that t s constant and equal for all fnancal nsttutons. Relaxng ths assumpton n order to account for certan emprcal propertes of LGD would not alter any of the qualtatve conclusons derved below. 8 A smlar settng has been used n the context of fnancal stablty by Kurtzkes et al (2005), who measure the expected loss to the depost nsurance fund usng smlar concepts. BIS Quarterly Revew, September

6 Common exposures, systemc rsk and systemc mportance Strongly captalsed system (all PDs = 0.1%) Exposure to the systematc rsk factor (banks n group A) Weakly captalsed system (all PDs = 0.3%) Exposure to the systematc rsk factor (banks n group A) ρ = 0.30 ρ = 0.40 ρ = 0.50 ρ = 0.60 ρ = 0.70 ρ = 0.30 ρ = 0.40 ρ = 0.50 ρ = 0.60 ρ = 0.70 Group A (share) 44.0% 46.2% 50.0% 54.4% 60.4% 41.7% 45.4% 50.0% 56.2% 63.2% Group B (share) 56.0% 53.8% 50.0% 45.6% 39.6% 58.3% 54.6% 50.0% 43.8% 36.8% Total ES Total expected shortfall (ES) equals the expected loss n the 0.2% rght-hand tal of the dstrbuton of portfolo losses; per unt of overall system sze, n percentage ponts. The frst two rows report the share of the two groups (each comprsng 10 banks) n total ES. The exposure of each of the 10 banks n group A to the systematc rsk factor s as gven n the row headngs. The exposure of each of the 10 banks n group B to the systematc rsk factor corresponds to ρ = See the techncal appendx for a defnton of ρ. The probablty of defaut (PD) of each bank s as specfed n the panel headng. Loss-gven-default s set to 55%. All banks are of equal sze, each one accountng for 5% of the overall sze of the system. Table 1 combned contrbuton of group A banks rses from 44% to roughly 60%. The result s smlar for a weakly captalsed system. The reason for ths result s that hgher exposures to the common factor result n a hgher probablty of jont falures n the system. In turn, a hgher probablty of jont falures translates nto hgher average losses n the systemc event, whch leads to a hgher level of systemc rsk, as measured by expected shortfall. Qute ntutvely, the rse n the level of systemc rsk s attrbuted manly to the banks that contrbute most to ths rse, e those that experence an ncrease n ther exposure to the common factor (group A banks n Table 1). Second, the nteracton between dfferent drvers may renforce ther mpact on systemc mportance. A concrete example s provded by Graph 1 (left-hand panel) on the bass of a system n whch banks dffer only n terms of sze. As the strength of exposures to the common factor ncreases unformly across all banks n ths system, the porton of the expected shortfall... the strength of common rsk factors... ndvdual rskness... Systemc rsk: nteracton of dfferent drvers 1 When banks dffer n sze 2 When banks dffer n PD 3 Total systemc rsk 5 bg banks 10 small banks 15 Total systemc rsk 8 hgh-rsk banks 8 low-rsk banks Exposure to the systematc factor Exposure to the systematc factor 4 1 All numbers are n percentage ponts. Total systemc rsk equals the expected loss n the 0.2% rght-hand tal of the dstrbuton of portfolo losses; per unt of overall system sze. The contrbutons of the two groups of banks to the total are plotted as shaded areas. Each group accounts for half of the overall system sze. Loss-gven-default s assumed to be 55%. 2 Each bank s probablty of default (PD) equals 0.3%. 3 The PD of a hgh-rsk bank s 0.3% and that of a low-rsk bank s 0.1%. 4 See the techncal appendx for a defnton. Graph 1 80 BIS Quarterly Revew, September 2009

7 ... and nsttutons relatve sze attrbutable to larger banks ncreases by a greater amount than that attrbutable to smaller banks. In other words, bank sze renforces the mpact of common factor exposures on systemc mportance. The rght-hand panel of Graph 1 llustrates a smlar pont n the context of a system comprsng banks that dffer only wth respect to ther ndvdual PDs. If all of these banks experence the same rse n ther exposures to the common factor, the ncrease n the contrbutons to systemc rsk s greater for rsker banks. Here, ndvdual rskness renforces the mpact of common factor exposures on systemc mportance. Thrd, changng the lumpness of a system affects the systemc contrbutons of banks of dfferent szes dfferently. Ths s reported n Table 2, whch consders hypothetcal bankng systems where all banks feature the same PDs and exposures to the common factor but dffer n sze. There are three bg banks of equal sze, together accountng for 40% of the overall system, and a group of small banks, makng up the rest. As the number (but not the share) of small banks ncreases (across columns to the rght, n each panel), dversfcaton benefts reduce overall systemc rsk. 9 Ths reducton s assocated wth a declne n the systemc mportance of small banks and a rse n that of large banks (the frst two rows n each panel). Moreover, the rse n bg banks systemc mportance reflects not only a rse n the share but also n the amount of systemc rsk that these banks account for. Consderng the example of a strongly captalsed system (left-hand panel), a rse n the number of small banks from fve to 25 results n a drop of systemc rsk from 9.8 to 9.3 cents on the dollar. At the same tme, the amount of ths rsk that bg banks account for rses from 4.3 (or 42.8% of 9.8) to 6.3 (or 68.1% of 9.3) cents on the dollar. 10 System lumpness, systemc rsk and systemc mportance Strongly captalsed system (all PDs = 0.1%) Number of small banks Weakly captalsed system (all PDs = 0.3%) Number of small banks n s = 5 n s = 10 n s = 15 n s = 20 n s = 25 n s = 5 n s = 10 n s = 15 n s = 20 n s = 25 Three bg banks (share) 42.8% 56.8% 62.6% 66.0% 68.1% 41.6% 52.3% 56.5% 59.3% 60.7% n s small banks (share) 57.2% 43.2% 37.4% 34.0% 31.9% 58.4% 47.7% 43.5% 40.7% 39.3% Total ES Total expected shortfall (ES) equals the expected loss n the 0.2% rght-hand tal of the dstrbuton of portfolo losses; per unt of overall system sze, n percentage ponts. The frst two rows report the share of the two groups of banks n total ES. The group of bg banks accounts for 40% of the overall sze of the system and the group of small banks accounts for 60%. The probablty of default (PD) of each bank s as specfed n the panel headng. Loss-gven-default s set to 55%. All banks are assumed to have the same senstvty to common rsk factors, mplyng a common asset return correlaton of 42%. Table 2 9 The declne n systemc rsk s rather subdued because the assumed hgh exposure of banks to the common rsk factor restrcts the dversfcaton benefts obtaned from ncreasng ther number. Ths general result s studed n detal n Tarashev (2009). 10 The effect s even stronger n the case of a weakly captalsed system (rght-hand panel). BIS Quarterly Revew, September

8 Sze and systemc mportance Systemc mportance Sze 1 All numbers are n percentage ponts. The system comprses 10 nsttutons, each represented by a dot. Systemc mportance s measured as the share of each nsttuton n the expected shortfall of the system, defned as the expected loss n the 0.2% rght-hand tal of the dstrbuton of system-wde losses. Sze s measured as a share n the aggregate sze of all nsttutons n the system. Each bank s loss-gven-default and probablty of default equal 55% and 0.1% respectvely. The loadngs on the common factor (see the techncal appendx) are constant across banks and equal 0.6. Graph 2 Fnally, and qute generally, systemc mportance ncreases more than proportonately wth (relatve) sze. Ths relatonshp s a consequence of the fact that larger nsttutons play a dsproportonate role n systemc events. The frst column of Table 2, for example, relates to a system n whch a bg bank s roughly 10% larger than a small one but s assgned a 25% greater share n systemc rsk. 11 Ths effect ncreases as banks szes become more dsparate. For example, the ffth column of the table, whch relates to a system where the szes of bg and small banks are roughly 5:1, reports that the respectve shares n systemc rsk are roughly 18:1. Graph 2 presents further evdence of ths non-lnear relatonshp between sze and systemc mportance. It plots the contrbutons to system-wde rsk of nsttutons that are all dentcal except for ther sze. In the partcular example, the largest nsttuton s about 5 tmes as large as the smallest one, but ts relatve systemc mportance s nearly 10 tmes as hgh. Even though the above examples have been cast n stylsed settngs, they llustrate robust results and pont to concrete polcy lessons. In partcular, all else equal, they suggest that any systemc captal charge appled to ndvdual nsttutons should ncrease more than proportonately wth relatve sze. In other words, there s a clear ratonale for havng tghter prudental standards for larger nsttutons. In addton, the charge should ncrease wth the degree to whch an nsttuton s exposed to sources of systematc rsk. Ths means that hgher captal charges would be appled to nsttutons that are more smlar to the typcal (or average ) nsttuton: f they fal, they are more lkely to fal n a systemc event. The above examples also touch, albet ndrectly, on the noton of dversfcaton from a systemc vewpont. There s a potental trade-off between dversfcaton n the portfolo of an ndvdual nsttuton and dversfcaton for Implcatons for the calbraton of prudental nstruments 11 More precsely, the rato of small and bg bank szes equals (0.4/3)/(0.6/5) = The correspondng rato of systemc mportance measures s (42.8%/3)/(57.2%/5) = BIS Quarterly Revew, September 2009

9 the system as a whole. Ths s because, by dversfyng ther own nvestment portfolos, nsttutons affect systemc rsk n two ways. Frst, greater dversfcaton of each portfolo s lkely to reduce the rskness of ndvdual nsttutons. Second, t s also lkely to result n more smlar portfolos and, thus, n nsttutons beng more exposed to common rsk factors. The net outcome depends on how the frst effect, whch lowers systemc rsk, compares to the second, whch rases t. Implementng the tool: beyond stylsed examples Operatonalsng the methodology An applcaton to real-world data... The prevous analyss provdes a structured framework for examnng what factors are relevant n assessng the systemc mportance of nsttutons. But what steps are needed to apply the Shapley value methodology n practce? What choces do polcymakers have to make? In makng ths general approach operatonal, a number of ssues need to be addressed. Beyond choosng a specfc measure of systemc rsk, these nclude: the defnton of the relevant system ; the defnton of the sze of nsttutons; the choce of nputs; the uncertanty about the correct specfcaton of the rsk model and the true parameter values; and computatonal burden. Except for the last, all of these ssues are related to the measure of systemc rsk, rather than to the Shapley value methodology as such. Box 2 provdes a dscusson of the trade-offs and ptfalls nvolved and outlnes the consderatons that mght gude polcymakers choces. Once these choces are made, the applcaton s straghtforward. To llustrate how the methodology can be appled to real-world data, consder the followng example. The chosen measure of system-wde rsk s expected shortfall, as n the stylsed examples of the prevous secton. We defne the relevant system as comprsng 20 large nternatonally actve fnancal nsttutons and assume that a loss s ncurred when one or more of them fal. We measure an nsttuton s sze as the book value of ts labltes, dvded by A system of large nternatonally actve nsttutons Systemc mportance Systemc mportance Systemc mportance Sze Probablty of default Exposure to the common factor 1 All numbers are n percentage ponts. Systemc mportance s measured as the share of each nsttuton n the expected shortfall of the system, whch s defned as the expected loss n the 0.2% rght-hand tal of the dstrbuton of portfolo losses. The sze of an nsttuton equals the book value of ts labltes, expressed as a share n the sum of the labltes of all nsttutons n the system. The probablty of default s the one-year EDF provded by Moody s KMV for end Exposures to the common factor are derved on the bass of Moody s KMV GCorr estmates of nsttutons asset-return correlatons for end Sources: Moody s KMV. Graph 3 0 BIS Quarterly Revew, September

10 Box 2: Applyng the method n a polcy context: choces and trade-offs Ths box addresses the polcy choces and practcal ssues that have to be confronted when mplementng the methodology as an element n a macroprudental approach to regulaton and supervson. The defnton of the approprate system, as a precondton for calbraton, s not straghtforward. Ths s less of an ssue n current regulatory arrangements whch focus on ndvdual nsttutons but becomes crtcal when the prudental framework focuses on systemc rsk. At least two aspects need to be addressed. The frst relates to the nsttutonal coverage of regulaton ts so-called permeter. A systemc approach would need to take account of the rsks generated by all fnancal nsttutons that are capable, on ther own and as a group, of causng materal system-wde damage. Ths s so regardless of ther legal form. The second aspect relates to the geographcal coverage of regulaton. Should the approach be appled at a domestc level or at a more global level, say to nternatonally actve nsttutons? And f the answer s to both, how would the adjustments be reconcled? Clearly, a large dose of pragmatsm s necessary. And the precse answers wll also depend on the extent of cooperaton across regulatory jursdctons. The defnton of the sze of the nsttutons also merts attenton, and partly overlaps wth that of the system. One queston s whether to nclude only domestc exposures or both domestc and nternatonal ones. Another queston s whether the approprate measure refers to the assets (presumably ncludng off-balance sheet tems) or to the labltes (excludng equty) of the nsttutons. Total assets better reflect the potental overall losses ncurred by all the clamants on the nsttuton; labltes are a better measure of the drect losses lnked to ts falure. Havng defned the system and the sze of the nsttutons, the next practcal queston s how to estmate the addtonal parameters, notably the probabltes of default and the factor loadngs on the systematc rsk factors. The sources of nformaton range from market nputs, at one end, to supervsory nputs, at the other. Combnatons of the two are also possble. Market nputs have a number of attractve features but also lmtatons. On the plus sde: they summarse the consdered opnon of market partcpants based on the nformaton at ther dsposal; they should reflect market partcpants vews of all potental sources of rsk, regardless of ther orgn (eg poor asset qualty, bank runs, counterparty lnkages); and they are easly avalable on a tmely bass. On the mnus sde: they may not be avalable for all nsttutons (eg equty prces for savngs banks); they requre the use of models to ether flter out extraneous nformaton (eg rsk prema, expectatons of balouts) or complete the nformaton they contan (eg to derve probabltes of default from equty prces), gvng rse to model uncertanty; and they may contan systematc bases: for example, t s well known that market prces tend to be especally buoyant as fnancal vulnerabltes buld up durng booms (Boro and Drehmann (2009)). Supervsory estmates have ther own strengths and weaknesses. On the plus sde, they can be based on more granular and prvate nformaton, to whch market partcpants do not have access; on the mnus sde, they may smply not be avalable, or may be hard to construct for certan nputs. For example, supervsors have a long tradton n producng measures of the soundness of ndvdual fnancal nsttutons, such as ratng systems. However, they have as yet not developed tools to derve measures of exposures to systematc rsk factors and correlatons across nsttutons based on balance sheet data. The avalable technques are n ther early stages of development. All ths suggests that, n practce, t mght be helpful to rely on a combnaton of sources and to mnmse ther ndvdual lmtatons. For example, currently market prces appear to be especally suted for the estmaton of exposures to common factors. And long-term averages of such prces would help to address the bases n the tme dmenson. Ths would be especally approprate f the tool s used to calculate relatve contrbutons of nsttutons to systemc rsk and to avod procyclcalty (Boro (2009)). These dffcultes hghlght the need to deal wth the margn of error that wll nevtably surround the estmates of systemc rsk and hence, by mplcaton, of nsttutons contrbutons to t (Tarashev (2009)). Fortunately, as noted above, the lnearty property of the allocaton procedure makes t possble to address ths ssue n a formal, smple and transparent way. Ths property allows one to combne alternatve estmates, weghtng them by the degree of confdence that one attaches to them (Tarashev et al (2009)). In addton, t may be advsable for polcymakers not to rely too heavly on the resultng pont estmates. One possblty would be to allocate nsttutons nto a few buckets, each of them comprsng an nterval of pont estmates akn to a ratng system. Ths groupng has the added advantage of reducng the computatonal burden of assessng rsk at the level of subgroups of nsttutons. 84 BIS Quarterly Revew, September 2009

11 ... llustrates that there s no sngle proxy for systemc mportance the sum of the labltes of all nsttutons n the system. In addton, we measure an nsttuton s standalone rskness as the Moody s KMV estmate of ts one-year probablty of default and assume that loss-gven-default s constant at 55%. We also mpose a sngle-common-factor structure on the Moody s KMV estmate of the 20 nsttutons asset-return correlatons n order to derve the strength of exposures to systematc rsk. Both sets of estmates are based on market prces of equty and relate to end Fnally, we abstract (for smplcty) from model and estmaton uncertanty. Gven these assumptons, we then derve the expected shortfall of the system and each nsttuton s contrbuton to t. The results are shown n Graph 3, whch plots each nsttuton s contrbuton to system-wde rsk aganst three of ts drvers, namely the nsttuton s sze, probablty of default and exposure to the common factor. The results ndcate qute clearly that the nteracton of the varous factors plays a key role. None of them, n solaton, provdes a fully satsfactory proxy for systemc mportance. For example, the largest nsttuton n the system llustrated n Graph 3 s also the one wth the bggest contrbuton (red dot). However, owng to ts comparatvely hgh probablty of default, the nsttuton wth the fourth largest contrbuton s also one of the smallest and the least exposed to the common rsk factor (blue dot). Ths hghlghts an mportant strength of the Shapley value methodology, namely that t allows for a straghtforward quantfcaton of the nteractons of the varous drvers. Concluson Ths paper has presented a very general methodology to quantfy the contrbuton of ndvdual nsttutons to systemc rsk. For a gven measure of systemc rsk, ths s equvalent to calculatng ther systemc mportance. The methodology can be appled to a wde varety of measures of systemc rsk, and s very ntutve and flexble. As shown elsewhere, t subsumes other much more restrctve procedures as specal cases (Tarashev et al (2009)). The methodology s very helpful n structurng an analyss of the drvers of systemc mportance and n quantfyng ther relatve mpact. In practce, any measure of ndvdual nsttutons systemc mportance wll necessarly be based on a specfc measure (or measures) of systemc rsk. The constructon of such measures faces a number of tough challenges. These largely reflect the need to defne what the relevant system s and to estmate the approprate parameters. In the specfc settng used here, these parameters nclude the probablty of default and loss-gven-default of ndvdual nsttutons, exposures to common rsk factors and the sze dstrbuton of the system. We have dscussed how some of these challenges can be met and llustrated ths wth a concrete but smplfed example usng real-world data. In future, tools such as ths one wll nevtably be part of the arsenal of weapons needed to mplement a fnancal polcy framework wth a macroprudental orentaton, as called for by the nternatonal polcy communty. BIS Quarterly Revew, September

12 Techncal appendx: expected shortfall Expected shortfall, also known as expected tal loss, s the measure of systemc rsk we use n all numercal examples. It s defned as the expectaton of default-related losses n the system, condtonal on a systemc event. Ths event occurs when system-wde losses equal or exceed some (n ths artcle, the 98th) percentle of ther probablty dstrbuton. equal s We specfy ths probablty dstrbuton as follows. System-wde losses N = 1 LGD I, where s s the sze of the labltes of nsttuton, LGD s that s lost f that nsttuton defaults, and (loss-gven-default) s the share of I s an ndcator varable that equals 1 f nsttuton defaults and 0 otherwse. Wthout loss of generalty, the overall sze of the system s set to unty, s = 1, = 1 and, for smplcty, t s assumed that LGD = 55% for all nsttutons. Fnally, n lne wth structural credt rsk models, nsttuton s assumed to default when ts assets V fall below a partcular threshold. Specfcally, ths happens when 2 1 V = ρ M + 1 ρ Z < Φ PD, where the value of assets s drven by one rsk ( ) factor that s common to all nsttutons, M, and another rsk factor that s specfc to nsttuton, Z, and both factors are standard normal varables. In addton, PD denotes the uncondtonal probablty of default of nsttuton and 1 Φ s the nverse of the standard normal CDF. Fnally, the loadngs on the common (or systematc) factor, ρ [ 0,1 ] for { 1L,,N} correlaton of defaults wthn the system., determne the We quantfy expected shortfall usng Monte Carlo smulatons that take as nputs the followng parameters for each nsttuton : s, LGD, PD, ρ. N 86 BIS Quarterly Revew, September 2009

13 References Acharya, V and M Rchardson, eds (2009): Restorng fnancal stablty: how to repar a faled system, John Wley & Sons. Boro, C (2009): Implementng the macroprudental approach to fnancal regulaton and supervson, Banque de France Fnancal Stablty Revew, September. Boro, C and M Drehmann (2009): Towards an operatonal framework for fnancal stablty: fuzzy measurement and ts consequences, BIS Workng Papers, no 284, June. Crockett, A (2000): Marryng the mcro- and macro-prudental dmensons of fnancal stablty, BIS Speeches, 21 September. De Larosère, J (2009): The hgh-level group report on fnancal supervson n the EU. Group of 20 (2009): G20 workng group 1: enhancng sound regulaton and strengthenng transparency, 25 March. Knght, M (2006): Marryng the mcro- and macroprudental dmensons of fnancal stablty: sx years on, BIS Speeches, 5 October. Kurtzkes, A, T Schuermann and S Wener (2005): Depost nsurance and rsk management of the US bankng system: what s the loss dstrbuton faced by the FDIC?, Journal of Fnancal Servces Research, no 27:3, pp Mas-Colell, A, A Whnston and J Green (1995): Mcroeconomc theory, Oxford Unversty Press. Shapley, L (1953): A value for n-person games, n H Kuhn and A Tucker (eds), Contrbutons to the theory of games, volume II, Annals of Mathematcal Studes, v 28, pp , Prnceton Unversty Press. Tarashev, N (2009): Measurng portfolo credt rsk correctly: why parameter uncertanty matters, BIS Workng Papers, no 280. Tarashev, N, C Boro and K Tsatsarons (2009): Allocatng systemc rsk to ndvdual nsttutons: methodology and polcy applcatons, BIS Workng Papers, forthcomng. BIS Quarterly Revew, September