Top-Down versus Bottom-Up Approaches in Risk Management

Transcription

1 To-Down vesus Bottom-U Aoaches in isk Management PETE GUNDKE 1 Univesity of Osnabück, Chai of Banking and Finance Kathainenstaße 7, Osnabück, Gemany hone: ++49 (0) fax: ++49 (0) ete.gundke@uni-osnabueck.de Abstact: Banks and othe financial institutions face the necessity to mege the economic caital fo cedit isk, maket isk, oeational isk and othe isk tyes to one oveall economic caital numbe to assess thei caital adequacy in elation to thei isk ofile. Beside just adding the economic caital numbes o assuming multivaiate nomality, the to-down and the bottom-u aoach have been emeged ecently as moe sohisticated methods fo solving this oblem. In the to-down aoach, coula functions ae emloyed fo linking the maginal distibutions of ofit and losses esulting fom diffeent isks. In contast, in the bottom-u aoach, diffeent isk tyes ae modeled and measued simultaneously in one common famewok. Thus, thee is no need fo a late aggegation of isksecific economic caital numbes. In this ae, these two aoaches ae comaed with esect to thei ability to edict loss distibutions coectly. We find that the to-down aoach can undeestimate the tue isk measues fo lowe investment gade issues. The accuacy of the maginal loss distibutions, the emloyed coula function, and the loss definitions have an imact on the efomance of the to-down aoach. Unfotunately, given limited access to times seies data of maket and cedit isk loss etuns, it is athe difficult to decide which coula function an adequate modelling aoach fo eality is. JEL classification: G 21, G 32 Key wods: banking book, bottom-u aoach, coula function, cedit isk, goodness-of-fit test, integated isk management, maket isk, to-down aoach 1 Fo stimulating discussions, I would like to thank aticiants of the annual meetings of the Geman Association of Pofessos fo Business Administation (Veband de Hochschullehe fü Betiebswitschaftslehe, Padebon, 2007), the Euoean Financial Management Association (Vienna, 2007) and the Financial Management Association (Olando, 2007). Fo helful comments, I also thank two anonymous efeees.

2 1 1. Intoduction Banks ae exosed to many diffeent isk tyes due to thei business activities, such as cedit isk, maket isk, o oeational isk. The task of the isk management division is to measue all these isks and to detemine the necessay amount of economic caital which is needed as a buffe to absob unexected losses associated with each of these isks. Pedominantly, the necessay amount of economic caital is detemined fo each isk tye seaately. That is why the oblem aises how to combine these vaious amounts of caital to one oveall caital numbe. Consideing divesification effects equies to model the multivaiate deendence between the vaious isk tyes. In actice, some kind of heuistics, based on stong assumtions, ae often used to mege the economic caital numbes fo the vaious isk tyes into one oveall caital numbe. 2 Fo examle, fequently, it is assumed that the loss distibutions esulting fom the diffeent isk tyes ae multivaiate nomally distibuted. Howeve, this is cetainly not tue fo cedit o oeational losses. Two theoetically moe sound aoaches consist in the so-called to-down and bottom-u aoaches. Both aoaches ae a ste towads an enteise-wide isk management famewok, which can suot management decisions on an enteise-wide basis by integating all elevant isk comonents. Within the to-down aoach, the seaately detemined maginal distibutions of losses esulting fom diffeent isk tyes ae linked by coula functions. The difficulty is to choose the coect coula function, esecially given the limited access to adequate data. Futhemoe, thee ae comlex inteactions between vaious isk tyes, fo examle between maket and cedit isk, in bank otfolios. Changes in maket isk vaiables, such as isk-fee inteest ates, can have an influence on the default obabilities of obligos, o the develoment of the undelying maket isk facto detemines the exosue at default of OTC-deivatives with counteaty isk. It might suggest itself that coula functions ae likely to only insufficiently eesent this comlex inteaction because all 2 Fo an oveview on isk aggegation methods used in actice, see Joint Foum (2003), Bank of Jaan (2005), and osenbeg and Schuemann (2006).

3 2 inteaction has to be essed into some aametes of the (aametically asimoniously chosen) coula and the functional fom of the coula itself. In contast, bottom-u aoaches model the comlex inteactions descibed above aleady on the level of the individual instuments and isk factos which should make them moe exact. These aoaches allow to detemine simultaneously, in one common famewok, the necessay amount of economic caital needed fo diffeent isk tyes (tyically cedit and maket isk), wheeby ossible stochastic deendencies between isk factos can diectly be taken into account. Thus, thee is no need fo a late aggegation of the isk-secific loss distibutions by coulas. In this ae, we deal with the question how lage the diffeence between economic caital comutations based on to-down and bottom-u aoaches is fo the maket and cedit isk of banking book instuments. In ode to focus on the diffeences caused by the diffeent methodological aoaches, we geneate maket and cedit loss data with a simle examle of a bottom-u aoach fo maket and cedit isk. Aftewads, the to-down aoach is estimated and imlemented based on the geneated data and the esulting integated loss distibution is comaed with that one of the bottom-u aoach. Thus, it is assumed that the bottom-u aoach eesents the eal-wold data geneating ocess and we evaluate the efomance of the to-down aoach elative to the bottomu aoach. Doing this, we can ensue that the obseved diffeences between the loss distibutions ae not ovelaid by estimation and model isk fo the bottom-u aoach, but ae only due to inaccuacies of the to-down aoach. The ae is stuctued as follows: in section 2, elevant liteatue with esect to the to-down and the bottom-u aoach is eviewed. Aftewads, in sections 3 and 4, the model set u and the methodology of the comaison ae exlained. In section 5, the esults ae esented, and finally, in section 6, the main conclusions ae summaized.

4 3 2. Liteatue eview Sound aoaches fo isk aggegation can oughly be classified accoding to the two gous aleady mentioned in section 1. Let us stat with biefly eviewing bottom-u aoaches. Paes of this gou exclusively deal with a combined teatment of the two isk tyes maket isk and cedit isk. Kiesel et al. (2003) analyze the consequences fom adding ating-secific cedit sead isk to the CeditMetics model fo a otfolio of defaultable zeo couon bonds. The ating tansitions and the cedit seads ae assumed to be indeendent. Futhemoe, the isk-fee inteest ates ae nonstochastic as in the oiginal CeditMetics model. Howeve, Kijima and Muomachi (2000) integate inteest ate isk into an intensity-based cedit otfolio model. Gundke (2005) esents a modified CeditMetics model with coelated inteest ate and cedit sead isk. He also analyzes to which extent the influence of additionally integated maket isk factos deends on the model s aameteization and secification. Jobst and Zenios (2001) emloy a simila model as Kijima and Muomachi (2000), but additionally intoduce indeendent ating migations. Beside the comutation of the futue distibution of the cedit otfolio value, Jobst and Zenios (2001) study the intetemoal ice sensitivity of couon bonds to changes in inteest ates, default obabilities and so on, and they deal with the tacking of cooate bond indices. This latte asect is also the main focus of Jobst and Zenios (2003). Dynamic asset and liability management modelling unde cedit isk is studied by Jobst et al. (2006). Bath (2000) comutes by Monte Calo simulations vaious wost-case isk measues fo a otfolio of inteest ate swas with counteaty isk. Avanitis et al. (1998) and osen and Sidelnikova (2002) also account fo stochastic exosues when comuting the economic caital of a swa otfolio with counteaty isk. The most extensive study with egad to the numbe of simulated isk factos is fom Banhill and Maxwell (2002). They simulate the isk-fee tem stuctue, cedit seads, a foeign exchange ate, and equity maket indices, which ae all assumed to be coelated. Anothe extensive study with esect to the modeling of the bank s whole balance sheet (assets, liabilities, and off-balance sheet items) has ecently been esented by Dehmann et al. (2006). They assess the imact of cedit and

5 4 inteest ate isk and thei combined effect on the bank s economic value as well on its futue eanings and thei caital adequacy. Thee ae also fist attemts to build integated maket and cedit isk otfolio models fo commecial alications, such as the softwae Algo Cedit develoed and sold by the isk management fim Algoithmics (see Iscoe et al. (1999)). Examles of the to-down aoach ae fom Wad and Lee (2002), Dimakos and Aas (2004), and osenbeg and Schuemann (2006). Dimakos and Aas (2004) aly the coula aoach togethe with some secific (in)deendence assumtions fo the aggegation of maket, cedit and oeational isk. 3 osenbeg and Schuemann (2006) deal with the aggegation of maket, cedit and oeational isk of a tyical lage, intenationally active bank. They analyze the sensitivity of the aggegate Va and exected shotfall estimates with esect to the chosen inte-isk coelations and coula functions as well as the given business mix. Futhemoe, they comae the aggegate isk estimates esulting fom an alication of the coula aoach with those comuted with heuistics used in actice. Kuitzkes et al. (2003) discuss and emiically examine isk divesification issues esulting fom isk aggegation within financial conglomeates, wheeby they also conside the egulato s oint of view. Finally, using a nomal coula, Wad and Lee (2002) aly the coula aoach fo isk aggegation in an insuance comany. An aoach, which does not fit entiely neithe into the to-down aoach no into the bottom-u aoach (as undestood in this ae), is fom Alexande and Pezie (2003). They suggest to exlain the ofit and loss distibution of each business unit by a linea egession model whee changes in vaious isk factos (e.g., isk-fee inteest ates, cedit seads, equity indices, o imlied volatilities) until the desied isk hoizon ae the exlaining factos. Fom these linea egession models, the standad deviation of the aggegate ofit and loss is comuted and finally multilied with a scaling 3 Late, this wok has been significantly extended by Aas et al. (2007), whee ideas of to-down and bottomu aoaches ae mixed.

6 5 facto to tansfom this standad deviation into an economic caital estimate. Howeve, this scaling facto has to be detemined by Monte Calo simulations. The main contibution of this ae to the liteatue is that it is the fist which diectly comaes the economic caital equiements based on the bottom-u and the to-down aoach. Fo this, we estict ouselves to two isk tyes, maket isk (inteest ate and cedit sead isk) and cedit isk (tansition and default isk), and we estict ouselves to the isk measuement of banking book instuments only. Obviously, it would be efeable to conside futhe isk tyes, such as oeational isk. Howeve, it would be athe difficult to integate oeational isk into a bottom-u aoach (actually, the autho is not awae of any such an aoach) so that a comaison between the bottomu and the to-down aoach would not be ossible. Futhemoe, it would be efeable to extend the analysis to tading book instuments. Howeve, measuing diffeent isk tyes of banking book and tading book instuments simultaneously in a bottom-u aoach, would make in necessay to emloy a dynamic vesion of a bottom-u aoach because only in dynamic vesion, changes in the tading book comosition, as a bank s eaction to advese maket movements, could be consideed fo measuing the total isk of both books at the common isk hoizon. As this extension would intoduce much moe comlexity, we estict ouselves to the banking book. 4 Nevetheless, measuing the maket and cedit isks of the banking book ecisely would aleady be a significant ste fowad because the volume of the banking book of univesal banks is tyically lage comaed to the tading book. The ofit and loss distibution of the banking book comuted by a otentially moe exact bottom-u aoach could then ente into a to-down aoach which integates all bank isks. Being able to assess the maket and the cedit isk of the banking book and the inteaction of these isk tyes is also of cucial imotance fo fulfilling the equiements of the second illa of the New Basle Accod (see Basle Committee of Banking Suevision (2005)). The second illa equies that banks have a ocess fo assessing thei oveall caital adequacy in elation to thei isk ofile. Duing the caital assessment ocess, all mateial isks faced by the bank should be addessed, including fo 4 osenbeg and Schuemann (2006) link the cedit isk of the banking book and the maket isk of the tading book togethe with the oeational isk. Howeve, they only conside a to-down aoach.

7 6 examle inteest ate isk in the banking book. 5 Howeve, fo identifying the bank s isk ofile, it is imotant to coectly conside the intelay between the vaious isk tyes. 3. Model Setu 3.1 Potfolio Comosition Fo the uose of the comaison between to-down and bottom-u aoaches, a stylized banking book comosition is emloyed. It is assumed that the banking book is exclusively comosed of assets and liabilities with fixed inteest ates and that the bank usues a tyical stategy of ositive tem tansfomation (see figue 1). The cedits n {1,, N} on the asset side ae defaultable and mainly exhibit matuity dates T n of seven to ten yeas. 6 All cedits ae stuctued as zeo couon bonds with a face value of one Euo and ae issued by a diffeent cooate. The liabilities m {1,, M} of the bank ae also stuctued as zeo couon bonds, but they ae assumed to be non-defaultable. inset figue 1 about hee 3.2 Bottom-U Aoach As a bottom-u aoach fo measuing the cedit and maket isk of banking book instuments simultaneously, an extended CeditMetics model is emloyed. This extension exhibits coelated inteest ate and cedit sead isk. 7 The isk hoizon of the cedit otfolio model is denoted by H. P denotes the eal wold obability measue. The numbe of ossible cedit qualities at the isk hoizon 5 Anyway, with the New Basle Accod, the egulatoy authoities ay moe attention to the inteest ate isks in the banking book. Banks will have to eot the economic effect of a standadized inteest ate shock alied to thei banking book. If the loss in the banking book as a consequence of the standadized inteest ate shock amounts moe than 20% of the tie 1 and tie 2 caital, the bank is qualified as an outlie bank. The consequence is that the egulatoy authoities analyze the inteest ate isk of this bank moe thooughly. At the end, they can even equie that the bank educes its inteest ate exosue o inceases its egulatoy caital. 6 7 Such long cedit eiods can tyically be obseved in Gemany. A simila model has been used by Gundke (2005).

8 7 is K : one denotes the best ating and K the wost ating, the default state. The conditional obability of migating fom ating class i {1,..., K 1} to k {1,, K} within the isk hoizon H is assumed to be: n n ( η η ) P = k = i, Z = z, X = x : = f ( z, x ) H 0 i, k i 2 i 2 k ρ ρx, z ρx, x k+ 1 ρ ρx, z ρx, x =Φ Φ 1 ρ 1 ρ (1) n whee η 0 and hoizon t η, esectively, denotes the ating of obligo n {1,, N} in t = 0 and at the isk n H = H, esectively, and Φ () is the cumulative density function of the standad nomal distibution. Given an initial ating i, the conditional migation obabilities ae not assumed to be obligo-secific. The thesholds ae deived fom a tansition matix Q= ( qik ) 1 i K 1,1 k K, whose i k elements q ik secify the (unconditional) obability that an obligo migates fom the ating gade i to the ating gade k within the isk hoizon. 8 The above secification of the conditional migation obabilities coesonds to defining a twofacto-model fo exlaining the etun n on fim n s assets in the CeditMetics model: = Z + X + ( n {1,, N} ) (2) 2 n ρ ρx,, 1 ρx ρε n whee Z, X, and ε 1, ε ae mutually indeendent, standad nomally distibuted stochastic, N vaiables. The stochastic vaiables Z and X eesent systematic cedit isk, by which all fims ae affected, wheeas ε n stands fo idiosyncatic cedit isk. An obligo n with cuent ating i is assumed to be in ating class k at the isk hoizon when the ealization of n lies between the two i thesholds k + 1 and i i, with k+ 1 < k. The secification (2) ensues that the coelation i k Co( n, m) (n m) between the asset etuns of two diffeent obligos is equal to ρ. The coelation Co( n, X ) between the asset etuns and the facto X is ρ X,. As X is also the 8 Fo details concening this ocedue, see Guton et al. (1997,. 85).

9 8 andom vaiable which dives the tem stuctue of isk-fee inteest ates (see (4) in the following), ρ is the coelation between the asset etuns and the isk-fee inteest ates. X, Fo simlicity, the stochastic evolution of the tem stuctue of isk-fee inteest ates is modeled by the aoach of Vasicek (1977). 9 Thus, the isk-fee shot ate is modeled as a mean-eveting Onstein-Uhlenbeck ocess: d() t = κθ ( ()) t dt + σdw () t (3) whee κθσ,, + ae constants, and W () t is a standad Bownian motion unde P. The solution of the stochastic diffeential equation (3) is: 2 κ σ 2κ t () = θ + ( t ( 1) θ) e + ( 1 e ) X 2κ P = E [ ( t)] (4) whee X N(0,1). As the andom vaiable X also entes the definition (1) of the conditional tansition obabilities, tansition isk and inteest ate isk ae deendent in this model. The ice of a defaultable zeo couon bond at the isk hoizon H, whose issue n has not aleady n defaulted until H and exhibits the ating η {1,, K 1}, is given by: H ( ( n ) ( ) η ) H ( X, S, H, T ) = ex ( X, H, T ) + S ( H, T ) T H. (5) n η n n n n H Hee, ( X, H, T ) denotes the stochastic isk-fee sot yield fo the time inteval [ H, T ] calculated n fom the Vasicek (1977) model (see de Munnik (1996,. 71), Vasicek (1977,. 185)). In the Vasicek n model, the stochastic isk-fee sot yields ae linea functions of the single isk facto X, which n dives the evolution of the whole tem stuctue of inteest ates. S n ( H, T n ) ( η {1,, 1} η H K ) is the H 9 It is well-known that the Vasicek model can oduce negative inteest ates. Howeve, fo emiically estimated aametes, the obability fo negative inteest ates is usually vey small. Actually, the CeditMetics model could be combined with any othe tem stuctue model.

10 9 stochastic cedit sead of ating class η fo the time inteval [ H, T ]. 10 The ating-secific cedit n H seads ae assumed to be multivaiate nomally distibuted andom vaiables. This is what Kiesel et al. (2003) found fo the joint distibution of cedit sead changes, at least fo longe time hoizons such as one yea, which ae usually emloyed in the context of isk management fo banking book n instuments. 11 Futhemoe, it is assumed that the inteest ate facto X is coelated with the cedit seads. Fo the sake of simlicity, this coelation aamete is set equal to a constant ρ X, S, egadless of the ating gade o the emaining time to matuity. Besides, it is assumed that the idiosyncatic cedit isk factos ε n ( n {1,, N} ) ae indeendent of the ating-secific cedit seads n S n ( H, T n ) ( η {1,, 1} η H K ) fo all consideed matuity dates T n. The ice ( X, H, Tn) of a H default isk-fee zeo couon bond is comuted by discounting the standadized nominal value only with the stochastic isk-fee sot yield ( X, H, T ). n n If the issue n of a zeo couon bond has aleady defaulted ( η = K ) until the isk hoizon H, the value of the bond is set equal to the minimum of a beta-distibuted faction δ n of the value ( X, H, T ) of a isk-fee, but othewise identical, zeo couon bond and the value of the bond n without any ating tansition of the obligo: H 10 n Actually, S n ( H, T n ) is the stochastic aveage cedit sead of all obligos in the ating class η η H. The gas H between the fim-secific cedit seads and the aveage cedit sead of obligos with the same ating ae not modeled, but all issues ae teated as if the cedit sead aoiate fo them equals the aveage cedit sead of the esective ating gade. This assumtion also imlies indeendence between the cedit seads and the idiosyncatic isk factos. Without this assumtion, the ealized asset etun of each obligo would have to be linked to a fim-secific cedit sead by a Meton-style fim value model which has to be calibated fo each obligo. This seems not to be adequate fo actical uoses. 11 Obviously, the assumtion of multivaiate nomally distibuted cedit seads imlies the ossibility of negative ealizations. Howeve, as fo the isk-fee inteest ates, this haens only with a vey small obability.

11 10 { n η } ( X, η = K, δ, HT, ) = min δ X (, HT, ); X (, S, HT, ). (6) n H n n n n n 0 This is a modified vesion of the so-called ecovey-of-teasuy assumtion, which ensues that the ecovey ayment is neve lage than the value of the defaultable bond without a default. The ecovey ate is assumed to be indeendent acoss issues and indeendent fom all othe stochastic vaiables of the model. Fo icing the liabilities lx (, S, HT, m) of the bank, it is assumed that the bank cannot default, but η bank 0 bank emains in its initial ating gade η 0 = Aa. Thus, only the obability distibutions of the isk-fee inteest ates and the Aa cedit seads ae elevant fo the icing of the bank s liabilities. 12 Finally, the value Π ( H ) of the entie banking book at the isk hoizon H, comising the effects of maket and cedit isks as measued within the bottom-u aoach descibed above, is: N Π ( H ) = X (, S, HT, ) lx (, S, HT, ) n bank η n H η 0 = Aa m n= 1 m= 1 M. (7) Accodingly, the absolute loss L() t of the banking book within a eiod ( t 1, t] is defined as L() t ( t 1) () t =Π Π, and the log-loss etun is BU () t ln ( ( t 1) () t ) L = Π Π. 3.3 To-Down Aoach Accoding to Skla s Theoem, any joint distibution function F, ( xy, ) can be witten in tems of a coula function Cuv (, ) and the maginal distibution functions F ( x ) and F ( y ): 13 XY X Y 12 It does not ose any methodological oblems to intoduce a vaying ating of the bank, which deends on the ealized etun on the bank s assets. Additional simulations show that, as exected, the necessay amount of economic caital deceases due to this modification because a bad efomance of the cedit otfolio causes a ating downgade of the bank and, hence, a eduction of the maket value of the bank s liabilities. 13 Standad efeences fo coulas ae Joe (1997) and Nelsen (1999). Fo a discussion of financial alications of coulas, see, e.g., Cheubini et al. (2004).

12 11 FXY, ( xy, ) = CF ( X( x), FY( y)). (8) The coesonding density eesentation is: fxy, ( x, y) = fx( x) fy( y) c( FX( x), FY( y)) (9) whee cuv 2 (, ) ( / u vcuv ) (, ) = is the coula density function, and f ( x ) and f ( y ) ae the X Y maginal density functions. Fo ecoveing the coula function of a multivaiate distibution FXY, ( xy,, ) the method of invesion can be alied: Cuv = F F u F v (10) 1 1 (, ) XY, ( X ( ), Y ( )) whee F 1 X ( x) and F 1 ( y) Y ae the invese maginal distibution functions. In the context of ou todown aoach, FX ( x ) and FY ( y ) ae the maginal distibutions of the maket and cedit isk loss etuns of the banking book, measued on a stand-alone basis. Two coula functions fequently used fo isk management and valuation uoses ae the nomal coula and the t-coula, which ae also emloyed in this ae. These two coula functions ae also tyically used when a to-down aoach is imlemented in actice. 4. Methodology In the following, the thee-ste-ocedue of geneating time seies of maket and cedit isk loss etuns by the bottom-u aoach, estimating the maginal distibutions and the coula aametes and comaing the esulting loss distibutions of the bottom-u and the to-down aoach ae descibed in detail. Futhemoe, the emloyed aameteization of the model is exlained. 4.1 Paametes The isk hoizon H is set equal to one yea. The simulations ae done fo homogeneous initial atings η0 {Aa, Baa}. As tyical aametes fo the Vasicek tem stuctue model, κ = 0.4 and σ = 0.01 ae chosen. The mean level θ and the initial shot ate (0) ae set equal to As maket ice of

13 12 inteest ate isk λ a value of 0.5 is taken. 14 As the mean and the standad deviation of the betadistibuted ecovey ate, µ δ = and σ δ = ae emloyed. These ae tyical values obseved fo senio unsecued bonds by ating agencies. The one-yea tansition matix Q equals the aveage tansition obabilities of all cooates ated by Moody s in the eiod (see Hamilton et al. (2006,. 25)). The means and standad deviations of the multivaiate nomally distibuted ating-secific cedit seads S ( H, T ) ( k {1,, K 1} ) as well as thei coelation aametes ae taken fom Kiesel et k n al. (2003). They use fo estimation daily Bloombeg sead data coveing the eiod Ail 1991 to Ail Unfotunately, they only estimate these aametes fo times to matuity of two and five yeas. The coelations between cedit seads S ( H, T ) and S ( H, T ) with diffeent times to matuity Tn Tn ae also not estimated by them. Thus, fo the uose of this simulation study, it is assumed that the cedit seads with diffeent times to matuity ae efectly coelated and that the k n k n cedit sead distibutions ae identical fo all times to matuity. 15 This unique cedit sead distibution is based on the aveage aamete values of the distibutions detemined by Kiesel et al. (2003) fo times of matuity of two and five yeas. The value of the coelation aamete ρ between the asset etuns is chosen as 0.1. This value is nea the lowe bounday of the egulatoy inteval [0.08;0.24] defined fo cooates in the New Basle Accod and, hence, close to those values obseved in emiical studies about asset etun coelations 14 Fo examle, Banhill and Maxwell (2002) estimate a shot ate volatility of 0.007, wheeas Lehbass (1997) finds σ = 0.029, and Huang and Huang (2003) even wok with σ = With egad to the mean evesion aamete and the maket ice of inteest ate isk, Lehbass finds κ = and absolute values of 0.59, and fo the aamete λ, wheeas Huang and Huang choose κ = and an absolute value of fo λ. 15 In the Vasicek model, the sot yields fo diffeent times to matuity ae also efectly coelated.

14 13 (see, e.g., Düllmann and Scheule (2003), Hahnenstein (2004), o Dietsch and Petey (2004)). As a obustness check, also the exteme asset etun coelation ρ = 0.4 is tested. The coelation aamete ρ X, S between the cedit seads and the isk-fee inteest ates is set equal to 0.2. The emiical evidence hints at a negative elationshi between changes in isk-fee inteest ates and changes in cedit seads (see, e.g., Duffee (1998), Düllmann et al. (2000), Kiesel et al. (2002); oosite esults ae found by Neal et al. (2000)). This obsevation is in line with theoetical icing models fo cedit isks (see, e.g., Longstaff and Schwatz (1995)). The stength of the coelation deends on the ating gade: the absolute value is lage the lowe the ating is. Howeve, fo simlicity, this effect is neglected in this simulation study. With esect to the aamete ρ X,, which detemines the coelation between the fims asset etuns and the tem stuctue of isk-fee inteest ates, emiical studies about the ability of fim value models to coectly ice cedit isks usually assume a negative coelation between asset etuns and isk-fee inteest ates (see, e.g., Lyden and Saaniti (2000,. 38) o Eom et al. (2004,. 505)). Howeve, Kiesel et al. (2002) find that in yeas with negative inteest ate changes, less ating ugades take lace and Va values ae inceased. Thei esult hints at a ositive sign fo ρ X,, which would also be comatible with ρ, < 0. Due to this uncetainty, vaious aametes X S ρ X, { 0.2,0,0.2} ae tested as a obustness check. 4.2 Data Geneation The samle data matix D= { ( t), ( t )} T = of cedit and maket isk loss etuns of the banking book L1 L2 t 1 is simulated by means of the bottom-u aoach. The ealization of the cedit isk loss etun L1 0 cedit ( η ) () t = ln Π Π () t at time t is geneated by the cedit otfolio model without consideing maket isk, but only the isk of tansitions between the ating classes. In this case, the futue ayments ae discounted with those isk-fee discount factos that coesond to the initial mean level of the

15 14 shot ate and with those default-isky discount factos which coesond to the exected cedit sead discount factos. Thus, fluctuations in the tem stuctue of isk-fee inteest ates o stochastic cedit seads ae not consideed fo comuting the losses. Π η 0 denotes the value of the banking book when all obligos ae in thei initial ating class η 0 and no maket isk is consideed fo discounting. In contast, the ealization of the maket isk loss etun () ln ( maket L t η () t ) = Π Π at time t is 2 0 geneated by only consideing maket isk, but no tansition isk. In this case, it is assumed that all obligos stay in thei initial ating class within the time eiod ( t 1, t]. The futue ayments ae discounted with the isk-fee sot yields and the cedit seads of the esective ating gades, obseved in t (fo the distibutional assumtions of the isk-fee shot ate and the cedit seads, see section 3.2). Fo the simulation of the loss data, it is assumed that at the beginning of each eiod ( t 1, t], the cash flow stuctue is as esented in figue 1 and that all obligos ae in thei initial ating class η 0. Thus, the cash flow of the evious eiod and maybe additional caital ae used to ecove the cash flow stuctue and, in aticula, to comensate cedit losses. In aticula, a dynamic deteioation of the cedit quality of the banking book is not consideed. Futhemoe, losses due to a deceasing time to matuity ae not consideed. The time eiod between each samle oint t of the data matix D= { ( t), ( t )} T = is chosen as one L1 L2 t 1 yea. Oveall, the data of 60 bank yeas is simulated. Altenatively, quately data could be simulated by adjusting the tansition matix and the cedit sead distibution oely. One quate is the fequency with which Geman banks have to measue, fo examle, the inteest ate isk of thei banking book accoding to the tie 2 equiements of the New Basle Accod (see BaFin (2005)). The data of the simulated 60 bank yeas would coesond to quately data gatheed by the bank since 15 yeas. Howeve, comaed to what one finds cuently in actice, even 15 yeas of cedit loss histoy would aleady be a vey long time eiod.

16 Estimation of the Maginal Distibutions and the Coula Paametes Fo the to-down aoach, we need the maginal distibutions of the cedit and maket isk loss etuns L 1 and L 2. Thee ae seveal ossibilities how obtain these maginal distibutions. Fist, based on the simulated samle data matix D= { ( t), ( t )} T =, the maginal distibutions can be estimated L1 L2 t 1 aametically. Howeve, this aametic aoach suffes fom the oblem that we have to choose a distibution family à ioi and that missecified maginal distibutions can cause a missecification of the deendence stuctue exessed by the coula (see Femanian and Scaillet (2005)). Fo modeling the etun of maket isk ositions, a t-distibution o a nomal mixtue distibution ae often used because they eflect the fat tails usually obseved fo maket isk etuns. Fo modeling the loss etun of cedit isk ositions, the beta distibution, the lognomal distibution, the Weibull distibution o the Vasicek distibution have been oosed. Second, emiical maginal distibutions fo loss etuns can be deived fom single-isk-tye models which exist in most banks. 16 In this study, we use both aoaches fo estimating the maginal distibutions of the andom vaiables L 1 and L 2. Fo the second aoach, a lage numbe of bank yeas (200,000 to 2,000,000) ae simulated using the data geneating model descibed in section 3.2. Fo the aametic estimation of the maginal maket isk loss etun distibution, a nomal distibution is chosen. Fo the aametic estimation of the maginal cedit isk loss etun distibution, a beta distibution β, ( ) with ab L1 aametes a and b is taken. As the suot of the (standad) beta distibution is the unit inteval [0,1], negative ealizations of the loss etun L 1 cannot be modeled by this choice of the maginal distibution. Howeve, these ae ossible, in aticula fo lowe initial ating gades, due to the makto-maket aoach of the data geneating ocess fo the cedit losses. Fo the late simulation of the cedit isk loss etuns within the to-down aoach, the lowe bounday of zeo fo the ossible ealizations of the cedit isk loss etun does not ose any oblems because fo comuting isk 16 A thid aoach, which is not usued in this ae, would be to deive aametic estimates of the maginal loss etun distibutions based on single-isk tye models.

17 16 measues only the ight tail of the loss distibution is elevant. Futhemoe, due to the usage of the beta distibution, loss etuns lage than 100% cannot be simulated, too. 17 Loss etuns lage than 100% ae ossible because the etuns ae defined as log-etuns. Thus, in these cases, the to-down aoach with the maginal cedit isk loss etun aameteized as a beta distibution is likely to undeestimate the isk measues. The aamete ˆρ of the bivaiate nomal coula and the aametes ( nˆ, ˆ ρ ) of the bivaiate t-coula can be comuted, fo examle, by maximum likelihood estimation. 18 Basically, the aametes of the maginal distibutions and the aametes of the coula, combined in the aamete vecto θ, can be estimated simultaneously by the maximum likelihood method. Taking into account the density eesentation (9), the log-likelihood function is: 19 T l(θ) = ln f (), ()( ( ), ( );θ) L t 1 L t L t 2 1 L t 2 t= 1 T 2 Cuv (, ;θ) = ln ln ( f ( ( );θ) ( ( );θ) L L t f 1 2 ) 1 L L t +. (11) 2 t = 1 u v ( uv, ) = ( F ( L ( t);θ), F ( ( );θ)) L1 1 L L t Fo estimating the aametes of the coula functions, in those cases in which the ealization of the loss etun basis, L 1 is negative, which imlies an incease in the value of the cedit otfolio on a mak-to-maket L 1 is set equal to a small ositive numbe (e.g., ). Wheneve L 1 is lage than one, L 1 is set equal to Besides, it is ossible that negative otfolio values (with as well as without integated maket and cedit isk) ae geneated, in aticula fo low initial atings and lage asset etun coelations. As comuting a log-etun is not ossible in these cases, negative otfolio values ae not consideed in the simulated times seies, on which the calibation of the to-down aoach deends, and duing the simulations of the otfolio value with the bottom-u aoach. 18 osenbeg and Schuemann (2006) do not estimate the aametes of the coula functions, but instead, fo ˆρ, they emloy the esults of othe studies and exet inteviews and the degee of feedom ˆn of the t- coula is chosen ad hoc. 19 It is assumed that the coula function does not vay within the data eiod.

18 17 As usual, the maximum likelihood estimato (MLE) can be obtained by maximizing the log-likelihood function (11): ˆθ MLE = ag max l(θ) (12) θ Θ whee Θ is the aamete sace. Of couse, to aly (11) and (12), an à ioi choice of the tye of the maginal distibution and the coula (with unknown aametes) is necessay. To educe the comutational costs fo solving the otimization oblem (12), which esults fom the necessity to estimate jointly the aametes of the maginal distibutions and the coula, the method of infeence functions fo magins (IFM) can be alied (see Joe (1997,. 299), Cheubini et al. (2004,. 156)). The IFM method is a two-ste-method whee, fist, the aametes θ 1 of the maginal distibutions ae estimated and, second, given the aametes of the maginal distibutions, the aametes θ 2 of the coula ae detemined. The IFM ocedue is used in this ae, wheeby the aametes of the maginal distibutions ae comuted by the method of moments. A futhe ossibility fo estimating the coula function is the canonical maximum likelihood (CML) estimation. Fo this method, thee is no need to secify the aametic fom of the maginal distibutions because these ae elaced by the emiical maginal distibutions. Thus, only the aametes of the coula function have to be estimated by MLE (see Cheubini et al. (2004,. 160)). In the following, this aoach is emloyed when the maginal distibutions ae estimated non-aametically based on single-isk-tye models. 4.4 Comaison of the Loss Distibutions Next, the loss etun distibutions BU L and TD L, esectively, of the banking book ove a isk hoizon of one yea ae comuted. Fo this, on one hand, the bottom-u aoach, as descibed in section 3.2, is emloyed and, on the othe hand, the to-down aoach with a nomal and a t-coula is used. Fo comuting the loss etun TD L within the to-down-aoach, the simulated cedit isk loss etun L 1

19 18 and the simulated maket isk loss etun L 2 ae aggegated in the following way to a total loss etun: 20 Π η 1 = ln = ln. (13) Πη ( Π e ) ( ( e )) e e 1 0 η Π 0 η Π 0 η Π 0 η Π 0 η + 0 losses due to cedit isks losses due to maket isks TD 0 L L 1 L 2 L 1 L2 The numbe of simulations D vaies fom 200,000 to 2,000,000; the exact numbe is indicated below each table. To comute, fo examle, the 1% -ecentile of L, the geneated ealizations of these andom vaiables ae soted with a Quicksot-algoithm in ascending ode and the (0.01 D) th of these soted values is taken as an estimate of the 1% -ecentile. As isk measues, the Value-at-isk (Va) and the exected shotfall (ES) coesonding to a confidence level of {99%,99.9%,99.97%} ae comuted: ( ) P > E[ ] + Va( ) = 1, (14) L L ES( ) = E P L L > E[ L] + Va( ). (15) Futhemoe, 95%-confidence intevals ae comuted fo these isk measues (see Glasseman (2004,. 491) and Maniste and Hancock (2005)). To take into account the uncetainty in the aamete estimates of the maginal distibutions of L 1 and L 2 as well as the coula aametes caused by the shot time seies, the thee stes descibed in the thee evious sections (1. data geneation, 2. estimation of the maginal distibutions and the coula 20 It might be temting to define = +. Howeve, this definition would cause a systematic TD L L1 L2 undeestimation of the isk measues oduced by the to-down aoach. This undeestimation would be lage the lage the absolute losses ae. Fo examle, assume that the initial otfolio value is Π = 100, η0 cedit the simulated otfolio value consideing only cedit isks is Π = 70, and the simulated otfolio value maket consideing only maket isks is Π = 80. Then, the above additive definition yields a total loss etun 2 + ln ( ) ln ( ) = ln 100 (80 70) = Howeve, the coect total loss etun L1 L2 = + ( ) = ln 100 ( ) = TD defined accoding to (13) is ( ) L

20 19 aametes, 3. comaison of the loss etun distibutions esulting fom the to-down and the bottomu aoach) ae eeated 200 times fo the to-down aoach. Doing this, an emiical obability distibution fo the aametes and the isk measues of the to-down aoach can be comuted and comaed with the isk measues oduced by the bottom-u aoach esults 5.1 Base Case Table 1 shows the fist fou moments of the emiical maginal distibutions fo the maket and cedit isk loss etuns. These ae based on 2,000,000 bank yeas. Fo the maket isk loss etun, the skewness and excess kutosis do not contadict the assumed nomal distibution. The cedit isk loss etun is clealy non-nomally distibuted. Skewness and excess kutosis ae lage the lowe the initial ating and the lage the asset etun coelation is. - inset table 1 about hee - Table 2 shows the isk measues fo the banking book loss etun comuted, on one hand, with the bottom-u aoach and, on the othe hand, with the to-down aoach. Fo the to-down aoach, a nomal coula is used, and the dislayed isk measues coesond to the mean of the esective numbes ove 200 eetitions. Fo the initial ating Aa, the fit between the isk measues is quite good: the Va and ES ae slightly undeestimated by the to-down aoach, but the isk measues of both aoaches have the same ode of magnitude. Howeve, fo the initial ating Baa, this is not tue any moe: hee, we can obseve a significant undeestimation of the isk measues by the to-down aoach, which is lage the highe the confidence level of the isk measue is. Unfotunately, 21 Altenatively, estimation isk could be consideed based on the asymtotic joint maximum likelihood distibution of the aametes of the maginal distibutions and the coula function (see Hamele et al. (2005) and Hamele and ösch (2005)). Howeve, due to the shot time seies, the asymtotic maximum likelihood distibution of the aametes might diffe significantly fom the eal distibution of the aametes.

21 20 esecially in cedit isk management, isk measues coesonding to high confidence levels (usually lage than 99.9%) ae needed. - inset table 2 about hee - One eason fo the undeestimation of the isk measues by the to-down aoach in the case of an initial ating of Baa might be the fact that the beta distibution cannot oduce cedit isk loss etuns lage than one. This is checked by emloying the emiical maginal loss distibutions based on 200,000 bank yeas instead of assuming secific distibutions fo the maket and cedit isk loss etuns and fitting them to the data. As table 3 shows, using this non-aametic aoach fo the maginal distibutions imoves the fit between the isk measues oduced by the bottom-u and the to-down aoach significantly. Futhemoe, the uncetainty of the isk measues of the to-down aoach is educed because the uncetainty about the maginal loss distibutions is educed. - inset table 3 about hee - Table 4 shows the esults when the numbe of simulations on which the emiical maginal distibutions and the comutation of the isk measues ely is inceased to 2,000,000. Fo table 4, the estimation of the coelation aamete of the nomal coula is also based on 2,000,000 bank yeas (instead of a eeated estimation based on 60 bank yeas). Futhemoe, the asset etun coelation and the coelation aamete ρ X, between the asset etuns and the isk-fee inteest ates ae vaied. As can be seen, fo the initial ating Baa, the fit between the isk measues oduced by the bottom-u and the to-down aoach wosens with inceasing coelation between the asset etuns and the iskfee inteest ates. - inset table 4 about hee - ρ Table 4 also eveals in which way the estimated coelation aamete CML ˆnomal ρ of the nomal coula deends on the cedit quality of the otfolio and the vaious inut coelation aametes. The lagest absolute values fo ˆ CML nomal 43.67% ρ esult fom a good cedit quality Aa, a low asset etun

22 21 coelation ρ = 0.1 and lage absolute values of the coelation ρ X, between the asset etuns and the isk-fee inteest ates. Positive values fo ρ ae only oduced by ρ, < 0. Fo comaison, CML ˆnomal X osenbeg and Schuemann (2006) assume a benchmak coelation between maket and cedit isk of 50%. This is the midoint value eoted by othe studies (see osenbeg and Schuemann (2006, table 7,. 595)). 22 Howeve, the inte-isk coelation of 50% is intended to be the coelation between the cedit loss etuns of the banking book and the maket isk loss etuns of the tading book. In contast, in this study, instuments only. CML ˆnomal ρ is the coelation between the cedit and maket isk of banking book 5.2 obustness Checks In the following, vaious obustness checks ae caied out and futhe asects ae analyzed Estimation isk fo the Bottom-U Aoach U to now, we have assumed that the bottom-u model coesonds the eal wold data geneating ocess and that we know its aametes with cetainty. In contast, fo the to-down aoach, we have estimation isk fo the coelation aamete of the nomal coula and the maginal distibution functions. The advantage of this assumtion is that we can ensue that the obseved diffeences between the loss distibutions oduced by the to-down and the bottom-u aoach ae only due to methodological diffeences, but not due to estimation and/o model isk of the bottom-u aoach. Howeve, in eality, we neithe know the data geneating ocess no its aametes, but fo imlementation, the bottom-u aoach also has to be estimated. That is why it has to be analyzed whethe the sensitivity with esect to estimation and/o model isk is diffeent in both aoaches. As isk measuement techniques fo each seaate isk tye ae moe (fo maket isk) o less (fo cedit isk) well develoed, we focus in the following on the estimation isk fo the coelation 22 This midoint value is consideably uwad biased due to a lage coelation aamete of 80% found in exet inteviews. The maket and cedit isk coelation eoted by the two othe studies is aound 30%.

23 22 aamete ρ X,, which govens the intelay between inteest ate and tansition isk in the bottom-u aoach. With esect to infomations on distibutions and aametes that ae only elevant fo each seaate isk tye, we assume that these ae exact. Fo each simulated time seies of 60 bank yeas, the aamete ρ X, is estimated by maximum likelihood. Fo this, the numbe of tansitions (ugades and downgades) away fom the initial ating ae counted e eiod. Then, ˆ ρ X, is given by the solution of the following otimization oblem (see, similaly, Gody and Heitfeld (2002), Hamele et al. (2003), van Landschoot (2005)): 60 N max ln f ( z, x ) (1 f ( z, x )) φ( z ) φ( x ) dzdx t X,, t = 1 ρ ρ ρ (16) t ( Nt tt ) tt η0, η0 η0, η0 t whee t t is the numbe of tansitions away fom the initial ating η 0 in eiod t, N t is the numbe of obligos in eiod t, fη, η (, z x ) is the conditional obability to stay in the initial ating gade within 0 0 one eiod, and φ() is the density function of the standad nomal distibution. Fo comuting the above integals, the Gauss-Legende integation ule with n = 96 gid oints is alied, and the integation intevals ae tansfomed to the unit inteval [0,1]. Fo the esults of table 5, 200 time seies of 60 bank yeas ae geneated and fo each time seies, the coelation aametes ˆ ρ X, and CML ˆnomal ρ of the bottom-u and the to-down aoach, esectively, ae estimated. Based on these estimates, the Va and ES isk measues ae comuted by Monte Calo simulation with the bottom-u and the to-down aoach fo each simulation un. Fo the to-down aoach, the emiical maginal distibutions based on 200,000 bank yeas ae emloyed. Fo both aoaches, the comuted isk measues ae influenced by the Monte Calo simulation eo and the estimation eo simultaneously. As table 5 shows, the estimato ˆ ρ X, is biased due to a small samle eo. Fo the initial ating Aa, the uncetainty of the isk measue estimates, measued by thei standad deviation and the width of the [5%,95%-ecentile] inteval, is comaable. Howeve, fo the initial ating Baa, the influence of the estimation isk is moe onounced fo the to-down aoach than fo the bottom-u aoach. - inset table 5 about hee -

24 t-coulas fo the To-Down Aoach Table 6 shows the isk measues when a t-coula and the emiical maginal distibutions ae emloyed fo the to-down aoach. Due to its tail deendence, it is exected that in the case of a t- coula, we have bette fit between the isk measues oduced by both aoaches. Indeed, as can be seen in table 6, choosing the degee of feedom n of the t-coula small enough can oduce isk measues that ae even lage than those of the bottom-u aoach. In aticula, the choice n = 15 yields a quite good fit. - inset table 6 about hee - Howeve, fo the esults of table 6, we just assumed that a t-coula with a secific degee of feedom is the coect coula fo descibing the deendence between the maket and cedit isk loss etuns. The imotant question is whethe this assumtion is also suoted by the simulated data. This, howeve, seems not to be the case. Based on time seies of maket and cedit isk loss etuns with length of 50,000 bank yeas, the aametes ˆtCML ρ and n of an assumed t-coula have been estimated fo vaious scenaios ( η 0 {Aa, Baa}, ρ {0.1,0.4}, and ρ X, { 0.2,0,0.2} ). Fo most scenaios, the estimated degee of feedom is lage than 300 and in no case it is smalle than 100 (without table). This indicates that the t-coula is not an adequate modeling aoach because it is not backed by the data Goodness-of-Fit Test fo the Coula Function of the To-Down Aoach Based on osenblatt s Tansfomation The esults of the evious section show that a coect identification of the coula function is of essential imotance fo the isk measues oduced by the to-down aoach. In actical alications, the coula function itself is just assumed to be the coect one and often even the aametes of this assumed coula function ae not estimated on time seies data, but ae based on socalled exet views (an excetion is fo examle Cech and Fotin (2006)). Howeve, doing this, nealy any desied isk measue can be oduced. Thus, an imotant question is whethe it is ossible to

25 24 infe fom given time seies data of loss etuns the coect coula function o, at least, to eject the null hyothesis of secific coula assumtions. This means that we have to do goodness-of-fit (GoF) tests fo the coula function emloyed by the to-down aoach. GoF tests ae not extensively discussed in the liteatue, but ecently some contibutions emeged (see, e.g., Malevegne and Sonette (2003), Beymann et al. (2003), Chen et al. (2004), Dobić and Schmid (2005), Femanian (2005), Genest et al. (2006), Kole et al. (2007)). Fom the oosed GoF tests, we choose a test based on osenblatt s tansfomation (see fo the following Dobić and Schmid (2007)). The osenblatt tansfomation S( X, Y ) of two andom vaiables X and Y is defined as (see osenblatt (1952), Dobić and Schmid (2007)): 2 ( ) ( ( )) 2 X Y X 1 1 S( X, Y) = Φ F ( X) + Φ C F ( Y) F ( X) (17) whee C( F ( Y) F ( X) ) C( u, v) u Y X ( uv, ) = ( F ( X ), F ( Y )) = is the conditional distibution function of X Y v= F ( Y) given u = F ( X) and Cuv (, ) is the coula function descibing the deendence between Y X the andom vaiables X and Y (see (10)). As the andom vaiables Z 1 = U = F ( X) and ( ) ( Y X ) Z2 = CVU = C F( Y) F( X) ae indeendent and unifomly distibuted on [0,1], the osenblatt tansfomation S( X, Y ) is chi squaed distibuted with two degees of feedom. Thus, the validity of the null hyothesis of inteest H 0 : ( X, Y ) has coula Cuv (, ) imlies the validity of the null hyothesis * H 0 : S( X, Y ) is 2 χ2 -distibuted and H 0 can be ejected if * H 0 is ejected. Fo testing the X null hyothesis * H 0, Dobić and Schmid (2007) oose to emloy the Andeson Daling (AD) test statistic due to its vey good owe oeties: j= 1 ( 2 1) ln ( ( ( j) )) ln ( 1 ( ( N j+ 1) )) N 1 AD = N j G S + G S (18) N whee ( x1, y 1),, ( xn, y N ) is a andom samle fom ( X, Y ) and S(1) S( N ) ae the inceasingly odeed osenblatt tansfomations Sx (, y ) ( j {1,, N} ) of this andom samle. G is the j j distibution function of a chi squaed distibuted andom vaiable with two degees of feedom. The test statistic AD could then be comaed against the citical values of its theoetical distibution.