Article Bank credit risk management and rating migration analysis on the business cycle

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1 econstor er Oen-Access-Publkatonsserver der ZBW Lebnz-Informatonszentrum Wrtschaft The Oen Access Publcaton Server of the ZBW Lebnz Informaton Centre for Economcs Gavalas, mtrs; Syrooulos, Theodore Artcle Bank credt rsk management and ratng mgraton analyss on the busness cycle Internatonal Journal of Fnancal Studes Provded n Cooeraton wth: Multdsclnary gtal Publshng Insttute (MPI), Basel Suggested Ctaton: Gavalas, mtrs; Syrooulos, Theodore (204) : Bank credt rsk management and ratng mgraton analyss on the busness cycle, Internatonal Journal of Fnancal Studes, ISSN , MPI, Basel, Vol. 2, Iss., , htt:// dx.do.org/0.3390/jfs20022 Ths Verson s avalable at: htt://hdl.handle.net/049/03572 Nutzungsbedngungen: e ZBW räumt Ihnen als Nutzern/Nutzer das unentgeltlche, räumlch unbeschränkte und zetlch auf de auer des Schutzrechts beschränkte enfache Recht en, das ausgewählte Werk m Rahmen der unter htt:// nachzulesenden vollständgen Nutzungsbedngungen zu vervelfältgen, mt denen de Nutzern/der Nutzer sch durch de erste Nutzung enverstanden erklärt. Terms of use: The ZBW grants you, the user, the non-exclusve rght to use the selected work free of charge, terrtorally unrestrcted and wthn the tme lmt of the term of the roerty rghts accordng to the terms secfed at htt:// By the frst use of the selected work the user agrees and declares to comly wth these terms of use. zbw Lebnz-Informatonszentrum Wrtschaft Lebnz Informaton Centre for Economcs

2 Int. J. Fnancal Stud. 204, 2, 22 43; do:0.3390/jfs20022 Artcle OPEN ACCESS Internatonal Journal of Fnancal Studes ISSN Bank Credt Rsk Management and Ratng Mgraton Analyss on the Busness Cycle mtrs Gavalas and Theodore Syrooulos,2, * 2 Shng, Trade & Transort eartment, Busness School, Unversty of Aegean, Chos 8200, Greece; E-Mal: [email protected] Audenca Nantes School of Management, Nantes Cedex , France * Author to whom corresondence should be addressed; E-Mal: [email protected]; Tel.: ; Fax: Receved: 6 November 203; n revsed form: February 204 / Acceted: 5 February 204 / Publshed: 3 March 204 Abstract: Credt rsk measurement remans a crtcal feld of to rorty n bankng fnance, drectly mlcated n the recent global fnancal crss. Ths aer examnes the dynamc lnkages between credt rsk mgraton due to ratng shfts and revalng macroeconomc condtons, reflected n alternatve busness cycle states. An nnovatve emrcal methodology ales to bank nternal ratng data, under dfferent economc scenaros and nvestgates the mlcatons of credt rsk qualty shfts for rsk ratng transton matrces. The emrcal fndngs are useful and crtcal for banks to algn to Basel gudelnes n relaton to core catal requrements and rsk-weghted assets n the underlyng loan ortfolo. Keywords: credt ratng mgraton; busness cycles; stress testng; Basel gudelnes JEL Classfcatons: C5; C58; E02; E32; G2. Introducton Credt rsk s defned as the otental that a bank borrower or counterarty wll fal to meet ts oblgatons n accordance wth agreed terms by the Basel Commttee on Bankng Suervson []. Credt rsk s usually conceved n loan exosures and nterest-generatng securtes whch are core revenue sources for fnancal nsttutons. In an envronment of global fnancal crss and economc

3 Int. J. Fnancal Stud. 204, 2 23 recesson, banks are now ncreasngly exosed to credt rsk from a dversty of fnancal nstruments other than loans, ncludng nterbank transactons, trade fnancng, foregn exchange transactons, fnancal futures, swas, bonds, equtes and otons, nter ala. Accordng to Basel II and III gudelnes, banks are allowed to ncororate ther own estmates of dfferent rsk exosures, adotng nternal rsk ratng systems, n order to realstcally assess ther regulatory catal requrements, known as the nternal-ratngs-based (IRB) aroach. Only banks meetng certan mnmum condtons, dsclosure requrements and aroval from ther natonal suervsor are allowed to aly ths aroach [2]. Furthermore, the noton of ncremental rsk charge (IRC) requres banks to calculate a one-year 99.9% VaR (value-at-rsk) for losses assocated wth credt senstve roducts n the tradng book [3]. In ths dynamc rocess, banks need to montor credt ratng shfts and otental defaults n a volatle economc envronment [4]. In a broader context, the assgnment rocess of credt rsk ratng takes nto account both qualtatve and quanttatve elements ([5 7], nter ala). An mortant feature of external agency ratngs s that, assumng tme-homogenety, they are assgned tycally wth a through-the-cycle (TTC; also, stressed ratngs ) aroach, based on an undefned long-term ersectve on credt rsk but of a more stable ersectve [8]. Ths mles an evaluaton of the borrower based on a downsde scenaro (e.g., the worst hase n the macroeconomc cycle) rather than on current market condtons, as n the so-called ont-n-tme (PIT) aroach [9,0]. TTC ratngs, therefore, tend to be more stable than PIT ratngs, as ratng transtons (mgratons).e., movements from one ratng class to another are less frequent. Also, hstorcal default rates for each ratng class may be more unstable, snce n bad years the default rate may ncrease wthout comanes beng moved to a dfferent ratng class, as occurs under the PIT aroach. Snce PIT ratngs are senstve to cyclcal changes n the borrower s credt rsk, rsk levels and corresondng catal requrements tend to ncrease durng recesson erods and to declne durng economc exansons. Because of ncreasng (regulatory) catal requrements durng downturns, banks tend to restrct lendng actvtes or to ncrease rce margns, both reducng fundng oortuntes. The economc downturn may be exacerbated by reduced fundng due to ncreasng catal requrements from PIT ratng systems and ths henomenon s referred to as rocyclcalty. Emrcal evdence suggests already that banks, for a number of reasons, reduce lendng more than the reducton of economc actvty durng downturn. Therefore, rocyclcalty has been a crtcal ssue n Basel II and II catal gudelne rules that clearly advse TTC ratngs. Ratng stablty, on the other hand, may be n conflct and not reflect recent changes n default rsk, as ratngs are adated slowly to these changes. Many of the nternal ratng systems bult by fnancal nsttutons are PIT systems measurng credt rsk takng nto account the current condtons and stuaton of a borrower [6,7]. To ths end, credt ratngs facltate bank evaluaton of borrower s credt qualty and worthness. Credt ratng (or scorng) transton, n secfc, s the mgraton of a debt nstrument from one ratng to another ratng over a erod of tme. Ths mgraton s the movement ether as an ugrade or a downgrade from an exstng ratng and ndcates the change n the credt qualty of an entre loan ortfolo as well as the otental for sgnfcant fnancal stress and loan default []. In relaton to that, transton matrces reresent a key tool when assessng the rskness of a credt exosure. A transton matrx reorts the dstrbuton of ssuers/borrowers based on ther ntal ratng class (on rows) and on ther fnal status (n columns: fnal ratng class, wthdrawn ratng or default) at the end of a artcular

4 Int. J. Fnancal Stud. 204, 2 24 tme nterval (e.g., one-quarter or one-year). It s now wdely recognzed that the mlementaton of a transton analyss aroach would suort banks to accurately model credt ratng mgraton robabltes as an ntegral art of modern credt rsk management [2]. Indeed, the recent turmol n global catal markets and the adverse revalng economc condtons have resulted n a wdesread shft towards negatve credt ratngs, sgnallng severe deteroraton n credt qualty. Credt rsk transtons are seen to exhbt dfferent behavoural atterns over dfferent macroeconomc hases and busness cycle states (booms contractons). To take nto account the mlcaton of busness cycles and real economc actvty on credt rsk and ratng mgraton behavour, a number of studes dstngush between dfferent economc hases (regmes), treatng booms searately from contractons ([3 7], nter ala). The core objectve of the aer contrbutes to the emrcal nvestgaton of credt rsk transton atterns n dfferent busness cycle states. More secfcally, the study attemts to model cororate ratng mgraton robabltes on bank loan ortfolos, ncororatng the state of real economc actvty as well. The emrcal aroach assumes an IRB bank nternal ratng framework. The aer roceeds as follows. Secton 2 revews ast emrcal lterature on crtcal factors that arguably nfluence bank loan ortfolo montorng as well as the assessment of borrowers credtworthness. Secton 3 rooses and ales a dscrete-tme maxmum lkelhood (TML) model [8 20] and ncororates key macroeconomc factors to reflect dfferent states of the real economy. Secton 4 evaluates the emrcal fndngs on the bass of a bank s nternal ratng system and, fnally, Secton 5 concludes. 2. Lterature Revew As BCBS ostulates borrowers and facltes must have ther ratngs refreshed at least on an annual bass; certan credts, esecally hgher rsk borrowers or roblem exosures, must be subject to more frequent revew. In addton, banks must ntate a new ratng f materal nformaton on the borrower or faclty comes to lght [2]. BCBS, furthermore, ndcates that montorng shfts n new ratng atterns can be a useful nut n antcatng future default sgnals, when scrutnsng oblgors hstorcal track records. Such montorng can be founded uon the noton that hases of ncreased default rsk are seen to follow earler hases of hghly frequent new ratngs. Nevertheless, a bank s feasblty to contnuously montor a rsky borrower and to effcently assess the level of credtworthness remans a crtcal ssue. Past relevant studes have dealt wth credt ratng mgraton analyss takng nto consderaton the followng ssues. In case a sequence follows a frst-order Markov rocess, the mgraton robablty matrx can be crtcal n revealng transton roertes. Ths outcome, however, may be based towards the deendence of mmedately successve events and does not contan all ror events [22,23]. The tye on deendence s often known as the memory of the rocess, whch n ths case has a length of one-ste (dstance between adjacent events). Furthermore, whenever one-ste transton robabltes, + do not deend on tme, so that n n j = j ( n), they are assumed to be statonary and the Markov Whereas credt ratng mgraton robabltes characterze the robablty of a credt ratng beng ugraded, downgraded or remanng unchanged wthn a secfc tme erod, credt mgraton matrces characterze the evoluton of credt qualty for ssuers wth the same aroxmate lkelhood of default; they are constructed by mang the ratng hstory of the enttes nto transton robabltes.

5 Int. J. Fnancal Stud. 204, 2 25 chan s also statonary or tme-homogeneous [24]. Fnally, key macroeconomc factors are argued to be crtcal nuts to be ncororated n the modelng rocess, n order to roxy busness cycles and states of the economy. 2.. Frst-Order Markov Process Transton matrces are at the centre of modern credt rsk management. A standard secfcaton for ratng transton robabltes s the frst-order, tme-homogeneous Markov model, whch s based on the assumtons that (a) the robablty of mgratng from one ratng class to another deends on the current ratng only and (b) the robablty of changng from one ratng class at tme t to another class at tme t + n does not deend on t. Past emrcal evdence, nevertheless, ndcates that transton robabltes are nfluenced by ratng hstory (see: [25 27]). Consecutve ratng changes n the same drecton are seen to be more frequent than n the ooste drecton. Snce the effect s stronger n the case of downgrades, ths henomenon s referred to as downward momentum [28]. A number of studes ostulate that credt ratngs under the frst-order Markov assumton rovde a reasonable ractcal aroxmaton, as long as dfferent transton matrces are consdered n booms from contracton states [5,26,29,30]. Markovan models are seen to roduce nformaton on the robabltes of rare transtons (comared wth a tycal multnomal or cohort aroach), even n case these transtons are not actually observed n the data [27]. The state sace of the ratng rocess can exand to render the Markov assumton reasonable. Furthermore, a Markovan model can be convenently secfed for frms that are downgraded and enter, subsequently, an excted state for a stochastc tme-horzon [27,3]. A comany s ratng s termed as excted ( non-excted ) n case the last ratng change was a downgrade (otherwse) Tme-Homogenety The cross-sectonal and temoral homogenety of transton ntenstes s crtcal for dervng the modelng aroach to transton matrces. A number of ast studes nvestgate transton robabltes functons of dstance onts n tme between dfferent rates and study whether the same transton matrx can be used for each-ont-n tme (PIT) ndeendently of the date onts n tme. Some studes conclude that credt ratng transtons can be adequately modeled as a Markov chan for tycal forecast horzons (one or two years), based on n-samle datasets [32,33]. Furthermore, heterogenety n default robablty, measured by a credt ratng scorng for nstance, s shown to be of crtcal mortance n affectng the shae of the loss dstrbuton [34,35]. Several studes secfy non-homogeneous models that ncororate the noton of cyclcalty and treat credt ratng mgratons as a non-memory-less rocess [35 37]. Nckell et al. [38], for nstance, nvestgate the deendence of ratngs transton robabltes on ndustry, country and stage of the busness cycle and argue that the busness cycle dmenson s crtcal n exlanng varatons n transton robabltes. Banga et al. [5] dstngush between transton matrces deendent on the busness cycle and fnd that the loss dstrbuton and economc catal of a synthetc bond ortfolo can vary consderably n dfferent economc envronments. Jafry and Schuermann [39] roose a moblty-metrc (a measure of overall moblty for a rated ssuer), estmated as the average of the sngular values of the moblty matrx for the ssuer s rofle. Smlarly, Truck and Rachev [40]

6 Int. J. Fnancal Stud. 204, 2 26 conclude that mgraton matrces estmated at dfferent busness cycle onts roduce sgnfcantly dfferent VaR measures for an underlyng ortfolo. When modelng credt ratng mgraton over a secfc tme-horzon (not n nfnty), tme-homogenety s a key ssue. Modelng ratng transton as a Markov chan rocess mles that default s consdered to be an absorbng state, that s, n the long-term all assets are n default. Past studes based on external ratng data and u to two-year forecasts, ostulate that credt ratng dynamcs can be adequately modeled as a Markov chan [32,4]. Furthermore, the ncororaton of dscrete-tme aroaches rather than of contnuous-tme duraton models wth tme-homogenety s argued to be ambvalent n terms of loss effcency n cohort data analyss. fferent estmaton aroaches can result to large dfferences n ortfolo rsk rofle and catal requrements. Estmatons based on contnuous-tme duraton settngs are found to be more effcent relatve to multnomal cohort based aroaches [42,43] Macroeconomc Factors A number of studes ncororate crtcal macroeconomc exlanatory factors and busness cycle states n modelng ratng transton robabltes, n order to establsh lnkages between underlyng economc condtons and credt rsk mgraton. Transton robabltes are seen to deend on ndustry, comany, locaton and busness cycle states [5,38,44,45]. Average tme-to-default s found to decrease when economc actvty decreases [46 49]. The tme-varyng cyclcal nature of default rates over long hstorcal erods has been also nvestgated [50 52]. Jafry and Schuermann [39] develo a metrc that ales to mgraton matrces, n order to comare dfferent estmaton methods of mgraton robabltes. Smlarly, Truck and Rachev [53] roose a class of drected dfference ndces to aly on transton matrces and these measures exhbt hgh correlaton wth dfferences n credt VaRs. To better understand the economc structure of tme deendence, Parnes [35] examnes shfts n key macroeconomc busness condtons, ncororatng a scaled GP varable, contracton exanson states and the CBOE volatlty ndex (VIX). A number of studes test models of ratng transtons n a calendar tme framework over dscrete tme slots [36,54,55]. GP s seen to be a key macroeconomc varable [56 57]. Recent studes ncororate alternatve models n a dynamc framework, such as generalzed lnear mxed models (GLMM) and Markov chan Monte Carlo (MCMC) technques [36,37] or conclude asymmetry n credt cycle dynamcs [58]. Macroeconomc factors are found to affect the ersstence of a low default regme more than a hgh default regme, suortng correlatons of credt and busness cycles. On the other hand, credt ortfolo models condtonal only uon busness cycle roxes may mss out on a sgnfcant art of systematc ortfolo credt rsk. 3. The screte-tme Maxmum Lkelhood Framework Α model of dscrete-tme maxmum lkelhood (TML) s now dscussed (quarterly ntervals); t ncororates a arametrc aroach to account for the state of the economy by ncludng key macroeconomc factors. The framework aled here s sutable for roducng maxmum lkelhood (ML) estmates, whenever the exact tme of movement between ratng classes and the class occuancy

7 Int. J. Fnancal Stud. 204, 2 27 between observaton tmes are unknown; t also takes nto account that the nter-examnaton erods (tme-sans between successve ratngs) may vary for dfferent frms or ndvdual borrowers. 3.. Tme-Homogenety n Transton Probabltes Intally, certan crtcal dfferences between dscrete and contnuous-tme frameworks are dscussed. For that, a frst-order tme-homogenous Markov chan X (t) s ncororated to ndcate credt ratng categores at tme t (t 0). Assumng dscrete data and dentcal observaton tmes equally saced for each borrower n the Markov chan ( t o < t < t j < K < t jn ), such that s j = t j t j = s, for all j = K,, j montorng stes, the ML estmator of the statonary transton robabltes drawn by n Equaton () s the cohort estmator: Nll ' ˆ l l ( s) = l N ' = l ll' l,l =,,l where, 2,, l are fnte numbers of the Markov chan states (, 2,, l corresond to decreasng levels of credtworthness and l ndcates default); P(s) denotes the l l transton robablty matrx wth ts (ll ) th element, namely (s), corresondng to mgraton robablty from state l to l wthn a s-length tme nterval; l l j N ll = n j = N ll s the total number of verfed transtons from l to l. In case that observaton tme onts are not dentcal and equally saced, the cohort estmator remans oor as t does not corresond to the ML estmator of l l (s) any longer, due to the fact that only the observatons wth dentcal and equal montorng stes are consdered. In other words, the cohort aroach estmates the mgraton robabltes as observatons runnng from the begnnng to the end of a tme erod under study (any observatons wthn ths erod nterval are gnored). On the contrary, the TML framework rovdes an alternatve aroach that contans ths latter flaw. In the contnuous-tme maxmum lkelhood (CTML) framework, the Markov chan ales transton ntenstes q l l ( t, s) = q ll, whereby, gven that state l enters n tme t and s stll occued at t + s, then the transton out of l s determned by a set of l transton ntenstes q > 0. (The ntensty, q j (t), from state to state j can be erceved as the rate of change of the robablty P j n a very short tme nterval, t ). Gven a tme-homogenous Markov chan, the relatonsh between an l l ) ntensty matrx 2, Q, and the transton matrx, P (s), s gven by Equaton (2): ( ll () P (s) = e Qs (2) 2 Smlar to a robablty transton matrx, an ntensty matrx, Q, can be constructed, contanng all ossble ntenstes between varous states. An outcome contanng K states, for nstance, would have the followng ntensty matrx, q( t ) q2( t ) L qk ( t) Q: Q (t) = q2( t) q22( t) K q2k ( t ). The followng constrants aly on the row entres, q j (t), of ntensty matrces: M M O M q q L q K( t) K 2( t ) KK ( t ) (a) the off-dagonals must be non-negatve,.e., q j ( t) 0 for j; (b) the rows must sum to zero.e., q j ( t) = 0 [59]. K j=

8 Int. J. Fnancal Stud. 204, 2 28 In ths case, the ML estmator of the transton ntensty s gven by Equaton (3): where N ll denotes the number of l l Q ll = n N = l l n T l = transtons made by an oblgor and T s the overall tme that oblgor has sent n class l [26]. Ths s the duraton estmator whch counts every ratng drft n a gven erod dvded by the total tme sent n each ratng class. Wth no nformaton avalable about the tmng of events between observaton tmes or about the exact transton tme, nether the numerator nor the denomnator of Equaton (3) can be calculated [30]. Assumng dscrete data (observatons corresondng to a sequence of dscrete tme onts), the duraton ntenstes cannot be used to evaluate the transton ntenstes; moreover, the observaton tmes must be dentcal and equally saced, n order for the cohort estmator (Equaton ()) to rovde ML estmatons. Based on earler emrcal studes [30,60 62] rooses a model to roduce ML estmates of the ntensty matrx, Q. Observaton tme onts are assumed to be arbtrary and the exact tmes of ratng class transtons and the class occuancy between observaton tmes are assumed to be unknown. Transton robabltes,, are exressed n terms of ntenstes; hence, t s vector θ of ntenstes ll to be estmated. Incororatng a canoncal decomoston and assumng that Q (θ ) has dstnct egenvalues, λ, K,λ l, Equaton (2) develos nto Equaton (4): where A 0 s a P (s) = A 0 dag λs λ ( e,, e s l ) A0 l (3) q ll L (4) l l matrx whose l th column s the rght egenvector for λ l. The ndvdual contrbuton to the total lkelhood functon that s to be maxmzed s dvded nto observatons of the frms defaultng durng the observed tme t 0 =0, t, K, tj, wth the ratng of frm as x 0, x, K, xj, resectvely x { l, K, l }] and those that are censored at the end of the observaton tme erod. [ Assumng that the exact tme of default s known but the ratng class before the default s not known, the censored observatons (n the set R {, K, l } ) contrbutes to the lkelhood functon by: (θ ) = j= o j j+ + j # L = [ x x ( tj+ tj ) x ( tj tj )] (5) j where for l =,, l tj+ tj ) = lr( tj + t ) (6) # x ( j r R j The non-censored observatons (frms enterng default durng the observed erod t j ) contrbute to the lkelhood functon by: where, for l =,,l j 2 # L = [ x x ( tj+ tj ) x l ( tj tj )] (θ) j= o j j+ + # (7) ( t) l ll = l = ll ( t j ) q (8) l l

9 Int. J. Fnancal Stud. 204, 2 29 The total lkelhood functon s the roduct of the lkelhood contrbutons over all n frms, condtonal uon the dstrbuton of frms among the states at t 0 [30]: n L (θ) = = L (θ ) (9) Consderng the dscrete-tme framework (quarterly ntervals), by transformng Equaton (2): P (s) = e Qs n Q = 4 e = Q 4 ( e ) n = n P quarter (0) where n =,, m quarters and P quarter s the one-quarter transton matrx. The total lkelhood functon s the outcome of the lkelhood contrbuton over frm, as n Equaton (): where = ( l, l ) ll L ( P quarter j = 0 ( tj + tj ) ll j ) = () P quarter. Equaton () can be rewrtten as n Equaton (2): L ( P quarter ) = n = j j = 0 ( tj + tj ) ll Thus, the log-lkelhood functon s gven n Equaton (3) as: ln L ( P quarter ) n j = = j= 0 (2) ( t j + tj ) ln (3) ll Equaton (3) ndcates that the quarterly transton matrx s evaluated wthn every observaton avalable, where only the exact tme of default s known but the revous ratng class (ror to default) s unknown. Hence, an attractve feature of TML aganst CTML relates to the ssue that the former aroach does not requre teratve egenvalue calculatons to roduce an outcome. In order to obtan a ML outcome n the general case of l 2, a Quas-Newton algorthm wth fnte dfferences can aly to roduce numercal dervatves aroxmatons [59]. Snce the algorthm s an teratve rocess, a startng ont can be the cohort estmator [30] Busness Cycles n Transton Probabltes The ncororaton of busness cycles and the state of the economy nto a tme-homogenous Markov chan n a TML framework s now dscussed. As ast studes ostulate, the aroach to estmate matrces condtoned on the busness cycle remans a crtcal ssue [5,22,33,63]. In bref, a mgraton matrx descrbes all ossble transton robabltes gven a ratng scale [64], as n Equaton (4): P (t) = M, 2, L, 0 M.2 2,2 L,2 0 L K O L L M. L 2, L L, L (4) where each, j reresents the transton robablty from state to state j, f j at tme erod t; the rows reresent current ratngs of oblgors whereas the columns reresent future ratngs. The last row, L, reresents the absorbng state of default,.e., the robablty of leavng the default state equals zero.

10 Int. J. Fnancal Stud. 204, 2 30 Wth the hghest ratng n the frst row, the elements below the dagonal are the robabltes for ugrades and the elements above the dagonal are the robabltes for downgrades. The uer art of the matrx also ncludes the L th column whch gves default robabltes for dfferent ratngs. The dagonal elements reresent the robabltes for the ratngs to be reserved n erod t. Two searate tme-homogenous ratng mgraton matrces can be estmated, condtoned on the boom and contracton state of the economy, resectvely. The total transton matrx (estmated on all avalable observatons) s used to construct the average transton robabltes. The transton matrx s antcated to shft between uward or downward market hases, dected by the exlanatory macroeconomc varables (such as GP er cata; GP growth; nflaton rate; external debt/exorts, fscal/external balance; [65]). In order to dect the statstcal correlaton between macroeconomc varables and transton matrces on the state of economy, Equaton (5) s set as: P quarter (γ ) = Paverage + ( α + β * γ )( Pboom Pcontracto n ) (5) where γ s a key macroeconomc factor and α and β reflect the boom and contracton coeffcents, resectvely. In accordance wth Equaton (2), the log-lkelhood functon can be set as n Equaton (6): where ) ) ( j ( t t j + ll j γ s the l l th element of: L [( P quarter ( γ )] = n = j ( tj + tj ) = 0 ( ) j ll γ (6) j tj + tj P quarter (γ j ) = [ P + ( α + β * γ )( P average where γ s the macroeconomc varable at tmet j j. boom P contracton In case of ncluson of economc varables, Equaton (5) can be transformed nto Equaton (7): )] t j + t j P quarter(, γ, L γ ) = Paverage + ( α j + β j * γ j )( P j= boom P contracton ) (7) A number of ast studes examne the ssue of busness cycles and states of the economy n the context of credt rsk [5,7,8,22,33,38,39,42]. Some aroaches classfy the states of the economy as fnte regmes and assume transton matrces change over tme wth the states of the economy. Accordng to Xng et al. [66], ths gnores the fact that the states of the economy at dfferent erods mght be dfferent even f they are n the same regme. From ths ersectve, t would be better not to restrct the number of regmes. In ther study, they assume the state of the economy to be contnuous, and model the structural changes n the economy as the shfts of the state of the economy n a contnuous sace. Ths mles that the generators of the ratng transton matrces are constant between two adjacent structural changes n the economy. Xng et al. [66] show that the generator and transton matrces of ratng transtons are ndeed changng over tme and the estmated structural breaks are not only statstcally sgnfcant but economcally meanngful as well. They also demonstrate that the generator or transton matrces n dfferent ndustry categores have dfferent behavors and, secfcally, ndustry sectors related to fnance servces are more suscetble to economc changes than other sectors.

11 Int. J. Fnancal Stud. 204, Statstcal Measurements Past studes ncororate a varety of statstcal aroaches, models and measurements to deal wth the emrcal analyss of ratng mgraton matrces. The next sectons dscuss brefly the sngular value decomoston metrc and bootstrang aroaches. The latter are frequently ncororated n transton matrx analyss, as they arguably bear attractve roertes and flexblty [42] Sngular Value ecomoston Metrc The sngular value decomoston (SV) s a generalzaton of the egen-decomoston whch can be used to analyze rectangular matrces. The man dea of the SV s to decomose a rectangular matrx nto three smle matrces: two orthogonal matrces and one dagonal matrx. Because t gves a least square estmate of a gven matrx by lower rank matrx of same dmensons, the SV s equvalent to rncal comonent analyss (PCA) and metrc multdmensonal scalng (MS) and s, therefore, an essental tool for multvarate analyss [67]. In the context of transton matrces, Jafry and Schuermann [39] roose a new metrc for comarng these matrces a moblty ndex termed the the sngular value of decomoston metrc, M SV, whch has an ntutvely-aealng sze related to the average robablty of mgraton of the orgnal matrx. Ths metrc for a transton robablty matrx, P, of dmenson l l s gven n Equaton (8) as: M SV (P) = l = λ ([ I P] [ I P] ) where the dentty matrx of the same I and λ (G) j denotes the j th egenvalue of a l l matrx G. By subtractng the dentty matrx I from the mgraton matrx, only the dynamc art of the orgnal matrx remans, reflectng the magntude of P n terms of mled moblty. The value of M SV s shown to ndcate the average amount of mgraton contaned n P; the larger the M SV value, the hgher the average robablty of ratng change. The dstance metrc between the mgraton matrces P I and P II, s gven as n Equaton (9): Bootstra Method l (8) M P, P ) = M P ) M ( P ) (9) SV( I II SV( I SV II The bootstra rocedure nvolves choosng random samles wth relacement from a data set and analyzng each samle n the same way, wth each observaton selected searately n random from the orgnal dataset [68]. A artcular data ont from the orgnal data set could aear multle tmes n a gven bootstra samle. The number of elements n each bootstra samle equals the number of elements n the orgnal data set. An mortant advantage of bootstra aganst other methods s that the researcher does not need to have a reconceved noton of the orgnal samle dstrbuton [42]. A number of ast studes [39,69] ndcate that 200 and 000 bootstra relcatons are suffcent for obtanng standard errors of bootstraed statstcs and confdence ntervals, resectvely. On the bass of data nut constrants and n lne wth standard earler ractce, ths study assumes the number of B = 000 relcatons to be suffcent.

12 Int. J. Fnancal Stud. 204, 2 32 The bootstra aroach aled n ths study nvolves the followng stes: frst, selecton of a number of frms n random out of the samle under study along wth ther ratng hstores; subsequent relacement untl the number of frms s the same as n the orgnal samle; second, ncororaton of a cohort and a TML estmaton aroach; thrd, reetton (B-) tmes of the two revous stes, wth comutaton of the comaratve metrc M SV for every estmaton, along wth the confdence nterval of α magntude; α s obtaned by sortng the comaratve metrcs n descendng order and examnng the breakonts of to and bottom ( α)/2 ercentles. The standard devaton of the transton robabltes s calculated as: 4. Emrcal Alcaton 2 σ jj = n B = x = 2 B x B The dataset of the study s based on the entre loan ortfolo and nternal ratng system of a major Austran bank (anonymty reserved for confdentalty reasons). An ntal emrcal ste relates to defnng the segmentaton of busness sectors; ths s based on the resectve Standard & Poor s (S&P) segregaton, ncludng: fnancal nsttutons; nsurance; consumer/servces; energy and natonal resources; real estate; lesure tme/meda; and transortaton. Crtcal ssues nclude also the secfcaton of a ratng system wth credt qualty classes, grades and robabltes of mgraton from one class to another over the credt rsk horzon. Ratng agences, such as Moody s [70], Ftch Ratngs [7] and Standard & Poor s [72], for nstance, ncororate a 2-, 20-, and a 7-grade scale, resectvely, for cororate borrowers. Past studes on nternal ratng systems [73], conclude a 5-grade ratng system for the medan bank out of the 50 largest US bankng organzatons. Other studes [30,56] use a 6-grade and a 7-grade ratng system, resectvely. A number of banks are nterested n ncreasng the number of nternal grades, ether through the addton of ± modfers or by slttng rsker grades. For the uroses of ths study, a smle 5-grade ratng system s constructed, wth grade denotng the worst (lowest) credt grade and grade 5 the best (hghest) credt grade, resectvely. A crtcal ste relates to the secfcaton of the rsk horzon. The study samle ncludes both large and small and medum-szed enterrse (SME) oblgors wth an annual turnover of at least 2.5 mllon and frms orgnatng from a dversty of busness sectors and countres of orgn. These borrowers are classfed accordng to ther ratng hstory over a 0-year horzon. The ratng hstory of all borrowers (not defaulted by that year-end) n the reference (data-rovder) bank s loan ortfolo s obtaned, as of 3 ecember In addton, ratng nformaton for all borrowers that defaulted durng s collected. Legal bankrutcy roceedngs and loan loss rovsons are used as roxes for default. The observaton erod s eventually restrcted to , n order to avod survvorsh bas and to overcome nformaton constrants. The dataset contans ntally 20,000 observatons on 5,000 borrowers, at quarterly ntervals [7,58]. Subsequent to a careful data cleansng rocess, the dataset fnally comes down to 85,000 observatons on 48,000 frms. A thorough evaluaton s normally conducted by the bank once a year (on a year-on-year bass as to the ntal oblgor s ratng). (20)

13 Int. J. Fnancal Stud. 204, Macroeconomcs and States of the Economy Followng the ntal dscusson of Secton 2.3, a crtcal ssue relates to the selecton of the most arorate macroeconomc varable n order to dect the state of the economy, n the TML framework. Past studes roose a varety of relevant varables correlated wth default rsk, such as, ndcatvely, real GP growth and unemloyment rate [74]; S&P500 returns [75]; and ndustral caacty utlzaton (CUI ndex, that s rato of actual outut level to sustanable maxmum outut level [76]. For the uroses of ths aer, the CUI ndex s selected as a convenent ndcator, justfed on several grounds [77]. Frst, t s avalable quarterly (contrary to annual varables). Second, t can be ncororated nto a well-secfed multvarate model wth a mnor mact on redcton error bas. Thrd, t s more stable over tme relatve to other wdely used cyclcal ndcators. Snce the samle under study conssts of heterogeneous frms that cannot be treated as a homogenous grou, a sngle macroeconomc varable for the samle as a whole s rendered narorate. As a result, a smaller samle, consstng only of domestc (Austran) frms/oblgors, s fnally undertaken; ths results n 22,000 observatons of 5000 frms (76.5% of the sub-samle exhbts an nter-examnaton frequency of one year; 6% of less than one year; and, 7.5% of more than one year). Ths samle s subsequently further dvded nto two sub-samles that corresond nto a boom and a contracton economc state, resectvely, n lne wth ast relevant studes [7,27]. A number of studes roose dfferent aroaches n order to assess threshold values for macroeconomc varables, such as, for nstance, calculaton of average or medan values of the factor under study; the best erformng art s assumed to reflect the boom state and the worst erformng art the contracton state, resectvely ([54,78], nter ala). Ths study chooses to set the threshold value to the medan. Based on that, the sub-samles n the boom and contracton states consst of aroxmately 5000 frms wth 2,000 observatons and 4000 frms wth 0,000 observatons, resectvely Transton Probabltes n Tme-Homogenety Average transton matrces n tme-homogenety for boom and contracton hases are now estmated. The estmaton of a one-quarter transton matrx roceeds on the bass of the full samle of the bank loan ortfolo under study, consstng of 85,000 observaton ars and targetng to maxmze the log-lkelhood functon as stated n Equaton (3). Table summarzes the one-year cohort and the TML estmaton results. The value of M quantfes the noton of average amount of mgraton contaned n the SV transton robablty matrx, P. The hgher the value of ratng changes. In ths study, MSV, the hgher the average robablty of s estmated at (at % statstcal sgnfcance; standard MSV error of less than 0.00). Past studes reort transton matrces (estmated wth a arametrc tme-homogeneous duraton method for boom and contracton states) to exhbt a dfference of n M SV [39]. Furthermore, comarsons of cohort and CTML estmatons ndcate the observed value of the dstance metrc to be equal to (at 5% statstcal sgnfcance), [64]. Other studes estmate the M dfference between the two matrces at 0.042, [79]. SV

14 Int. J. Fnancal Stud. 204, 2 34 The cohort transton matrx s seen to be sgnfcantly more dagonal-domnant (meanng that most of the robablty mass resdes along the dagonal) comared wth the ML transton matrx, ndcatng that the latter matrx catures moblty (shfts) more effcently. The two methods generate dfferent robabltes of default. The one-year cohort estmator ncororates only yearly noted mgratons and, n general, t s seen to tycally underestmate default rsk along all ratng grades. Broadly, ast studes conclude mxed emrcal fndngs on the cohort method, ndcatng over- or under-estmated Ps [7,39,80]. The ML aroach, on the other hand, ncororates events that occurred durng the year as well as at year-end. Basel gudelnes set a lower bound of 0.03% on P estmates whch may be used to comute regulatory catal [2]. Emrcal evdence ndcates that only the ML transton matrx s margnally below the crtcal bound of 0.03% (shaded transton robablty cell; Table ). Table. Transton robabltes n tme-homogenety: Full samle. From To efault Panel A: The one-year cohort transton matrx (0.0073) (0.0052) ( ) ( ) ( ) ( ) ( ) (0.0028) (0.0092) (0.0008) ( ) (0.0002) ( ) (0.0002) ( ) ( ) ( ) ( ) ( ) (0.000) ( ) ( ) ( ) ( ) (0.0002) ( ) ( ) ( ) ( ) (0.0056) efault Panel B: The one-year dscrete-tme ML transton matrx ( ) ( ) (0.0008) ( ) (0.0009) ( ) (0.0088) ( ) ( ) (0.0093) ( ) (0.0033) (0.0007) ( ) (0.0052) (0.002) ( ) (0.0005) ( ) ( ) ( ) ( ) ( ) (0.0008) ( ) ( ) ( ) ( ) ( ) ( ) efault Both transton matrces are estmated for The TML transton matrx s based on all observaton ars wth an nter-examnaton tme of less than or equal to fve years. Standard errors (n arentheses) are based on a non-arametrc bootstra aroach wth B =,000 relcatons. The 99% confdence nterval s calculated at by non-arametrc bootstrang. The dagonal entres are n bold for convenence.

15 Int. J. Fnancal Stud. 204, Transton Probabltes n Tme-eendence Pror to nvestgatng transton matrces n tme-deendence, average transton matrces n tme-homogenety are ntally estmated, on the bass of the bank loan ortfolo under study, focusng only on the sub-samle of the domestc oblgors. The one-year average cohort and the TML estmatons are summarzed n Table 2. The M SV s estmated at (at % statstcal sgnfcance; standard error of 0.006). Ths transton matrx dslays smlar characterstcs as the earler transton matrx estmated on the full samle. Table 2. Transton robabltes n tme-homogenety: Sub-samle domestc oblgors only. From To efault Panel A: The one-year cohort transton matrx ( ) (0.0233) (0.078) ( ) ( ) ( ) (0.0099) ( ) (0.0044) (0.0023) ( ) ( ) (0.0002) ( ) ( ) ( ) (0.0007) ( ) ( ) (0.0002) (0.0062) (0.0027) (0.0088) ( ) ( ) ( ) efault Panel B: The one-year dscrete-tme ML transton matrx ( ) (0.024) ( ) ( ) (0.0009) (0.0000) (0.0098) ( ) (0.0055) ( ) (0.0075) ( ) ( ) (0.0052) ( ) ( ) ( ) ( ) (0.0002) (0.0003) ( ) ( ) ( ) ( ) ( ) (0.0006) (0.0040) (0.0662) (0.0442) ( ) efault Both transton matrces are estmated for The TML transton matrx s based on all observaton ars wth an nter-examnaton tme of less than or equal to fve years. Standard errors (n arentheses) are based on a non-arametrc bootstra aroach wth B = 000 relcatons. The 99% confdence nterval s calculated at , obtaned by non-arametrc bootstrang. The dagonal entres are n bold for convenence. Table 3 summarzes the results on tme-homogenous one-year transton matrces n boom and contracton hases, based on the TML aroach. The M SV s estmated at (at 6% statstcal sgnfcance; standard error of 0.007). As ndcated (shaded transton robablty cells), four transton robabltes n each of the two ML-estmated transton matrces (boom and contracton hases)

16 Int. J. Fnancal Stud. 204, 2 36 reman below the 0.03% bound, whch satsfes relevant Basel gudelnes. The robablty of remanng at the lowest rsk grade s sgnfcantly larger n the boom than n the contracton transton matrx (by ont). Furthermore, a hgher robablty of mrovement n the exansonary hase s seen n grade 4 (4, 4 2 and 4 3). The other transton robabltes exhbt a waverng behavor, deendng uon the busness cycle state and the rsk grade they corresond to. Most of transton robabltes (0 out of 5) that reflect rsk-grade mrovement (cells on the left of the dagonal) are hgher durng booms, as antcated. Sgnfcant dfferences are also seen n default robabltes (all of whch are found to be lower n booms). These fndngs suort the robustness of CUI as a key macroeconomc varable to roduce transton matrces n dfferent states of the economy n the context of ths study. Table 3. Transton robabltes n tme-homogenety: screte-tme maxmum lkelhood (TML) results on busness cycle states. From To efault Panel A: The one-year ML boom transton matrx ( ) ( ) (0.0042) (0.0729) ( ) (0.0000) (0.0043) (0.009) ( ) ( ) (0.0065) ( ) ( ) (0.0044) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.0006) (0.0664) (0.0773) (0.0052) efault Panel B: The one-year ML contracton transton matrx ( ) (0.0326) ( ) (0.0929) ( ) (0.0000) (0.0022) ( ) ( ) ( ) (0.0082) ( ) ( ) (0.0077) ( ) ( ) (0.0042) ( ) ( ) (0.0005) ( ) ( ) ( ) (0.006) ( ) ( ) ( ) (0.0662) (0.0734) (0.009) efault Both transton matrces are estmated for Both TML transton matrces are based on all observaton ars wth an nter-examnaton tme of less than or equal to fve years. Standard errors (n arentheses) are based on a non-arametrc bootstra aroach wth B = 000 relcatons. The 99% confdence nterval s calculated at , obtaned by non-arametrc bootstrang. The dagonal entres are n bold for convenence.

17 Int. J. Fnancal Stud. 204, 2 37 The log-lkelhood functon (Equaton (6)) s now mlemented to determne the tme-deendent one-quarter transton matrx (as n Equaton (7)). All avalable observaton ars are ncororated to estmate α and β by maxmzng Equaton (6); ths results nto α = 5.62 and β = Convenent restrctons are then set on γ, n order to ensure that that P(γ) fulflls transton matrx requrements. Thus, the range of uer and lower values γ s allowed to take runs from ercent; these values corresond to mnmum and maxmum values that γ has recorder hstorcally over the study erod ( ). Hence, Equaton (5) can be re-wrtten as n Equaton (2): P = P * γ )( P P ) (2) quarter(γ) average ( exanson recesson Table 4 summarzes the emrcal fndngs from the estmaton of one-year transton matrces on dfferent busness cycle states of the economy. The γ coeffcent can obtan four dfferent values (one for every quarter of the year) that reresent dfferent economc scenaros, on the bass of quarterly observatons (as n anels A to ; Table 4). Panels A and B corresond to hases of boom and contracton, resectvely; anel C corresonds to transton states n the economy, namely a slum or recovery hase (n-between recesson and boom hases); fnally, anel corresonds to mxed economc states, ncororatng all elgble γ values and demonstratng the course of the economy on a busness cycle (from boom to contracton). The largest s n M SV are obtaned n a contracton scenaro (as of Equaton (9); at % statstcal sgnfcance). The default robabltes for rsk-grades, 2, and 3 are seen to be lower n the average state anel than n the mxed state anel. More frequently, the largest art of data for defaults corresonds to rsk grade 5 surassng any other rsk-grade. Ths mles that default transton comarsons could be restrcted to the transton from rsk-grade 5 to default. When comarng mgraton robabltes from rsk-grade 5 nto default n all four matrces on dfferent busness cycle scenaros, Ps ncrease from the lowest n the boom state, followed by the average and mxed states and endng, fnally, nto the hghest n the contracton state. As antcated, the worse (more unstable) the state of the economy s, the hgher the resectve Ps are. Past studes argue that Ps should be referably overestmated n boom states rather than underestmated n contracton states [39]. efault rsk durng contracton (boom) s statstcally and economcally overestmated (underestmated) by the nave cyclcal aroach relatve to the mxture of Markov chans aroach and more so for longer redcton horzons [7]. The comarson of tme-homogenous and tme-deendent functons n ths study, durng the boom state for nstance, reveals that the estmated ratng mgraton matrx n the context of the latter functon dslays a hgher robablty from rsk-grade 5 to default than of the former functon ( and , resectvely). Ths ndcates that the tme-deendent functon overestmates Ps durng boom hases. A reason for ths may be that the Ps of the tme-deendent contracton transton matrx (Panel B; Table 4) are hgher than the tme-homogenous average contracton transton matrx (Panel B; Table 3).

18 Int. J. Fnancal Stud. 204, 2 38 Table 4. Transton robabltes n tme-deendence: TML results on dfferent busness cycle scenaros. From To efault Panel A: The one-year ML boom transton matrx efault γ = 89.8 ( =, 2, 3, 4) Panel B: The one-year ML contracton transton matrx efault γ = ( =, 2, 3, 4) Panel C: The one-year ML average transton matrx efault γ = ( =, 2, 3, 4) Panel : The one-year ML mxed transton matrx efault γ = 89.8; γ 2 = 74.33; γ 3 = 74.33; γ 4 = The TML transton matrces are estmated for and are based on all observaton ars wth an nter-examnaton tme of less than or equal to fve years. The γ s values refer to the quarter of the year ( =, 2, 3, 4) and corresond to dfferent economc scenaros and combnatons of busness cycle states for each quarter of the year. Panels A and B corresond to boom and contracton hases, resectvely; anel C corresonds to transton states n the economy, namely a slum or recovery hase; anel corresonds to mxed economc states, ncororatng all elgble γ values and demonstratng the course of the economy on a busness cycle (from boom to contracton). The dagonal entres are n bold for convenence. Μ SV (Boom Contracton) = ; Μ SV (Average Boom) = ; Μ SV (Average Contracton) = 0.072; Μ SV (Mxed Boom) = ; Μ SV (Mxed Contracton) = ; Μ SV (Mxed Average) =

19 Int. J. Fnancal Stud. 204, 2 39 To sum u, ths study contrbutes certan nterestng and useful emrcal fndngs to the relevant emrcal lterature. Frst, comarng dfferent estmaton methods on tme-homogenous average transton robabltes, the ML transton matrx s found to have a better cature ratng transton moblty relatve to the more dagonal-domnant cohort estmated transton matrx. Second, the alcaton of the TML method to busness cycle states (n order to examne tme-homogenous average transton robabltes) ndcates that most transton robabltes reflectng rsk-grade mrovement are, concevably, hgher durng the boom state. Thrd, there aears to be sgnfcant dfferences n default robabltes but all of them are seen to be lower n the boom state. Forth, emrcal evdence suorts the robustness of CUI ndex as a key macroeconomc varable that can adequately contrbute to transton matrces that ncororate dfferent states n the economy. Fnally, comarng robablty mgratons from rsk-grade 5 to default n dfferent busness cycle scenaros, the TML aroach ndcates that the Ps ncrease from the lowest n the boom state, followed by the average and mxed states and endng n the hghest n the contracton state, justfyng that the worse the state of the economy s, the hgher the resectve Ps are. The dscrete-tme (versus contnuous-tme) framework s seen to be convenent and effcent n studyng credt ratng mgraton settngs. 5. Conclusons A focal ssue n Basel drectves emhaszes on the ntensfed senstvty banks should ay to catal requrements n corresondence to the rsk exosures of the bank s assets. In other words, the level of catal a bank should hold s to be drectly related to the rskness of ts underlyng assets ortfolo. Nevertheless, busness cycle fluctuatons can exert an mact on bank catal adequacy requrements. Under Basel gudelnes, banks are allowed to ncororate own estmates to crtcal rsk arameters n order to calculate regulatory catal. Gven that regulatory measures of fnancal robustness (such as the Ter- catal rato) refer to core bank catal and rsk-weghted assets, the emrcal estmaton of convenent credt rsk mgraton matrces remans a crtcal exercse for bank catal requrements aganst unexected losses n the loan ortfolo as well as for effcent rsk control. Ths aer enrches and exands ast emrcal lterature by examnng the lnkages between credt rsk management and macroeconomc states. Ths aears to be one of the frst studes to emrcally estmate and comare the TML and CTML aroaches and then aly the TML aroach to four dfferent busness cycle scenaros (boom, contracton, average and mxed economc states) to roduce rsk transton matrces. Further emrcal research n ths feld remans useful and tmely. Author Contrbutons mtrs Gavalas and Theodore Syrooulos have wrtten the aer jontly contrbutng to ts theoretcal foundatons and emrcal alcatons. Ths aer relates to ntal outut from. Gavalas doctoral research, suervsed by T. Syrooulos. Conflcts of Interest The authors declare no conflct of nterest.

20 Int. J. Fnancal Stud. 204, 2 40 References. Basel Commttee on Bankng Suervson (BCBS). Prncles for the Management of Credt Rsk; Bank for Internatonal Settlements: Basel, Swtzerland, Basel Commttee on Bankng Suervson (BCBS). Internatonal Convergence of Catal Measurement and Catal Standards: A Revsed Framework; Bank for Internatonal Settlements and Basel Commttee on Bankng Suervson Publcatons: Basel, Swtzerland, Basel Commttee on Bankng Suervson (BCBS). Gudelnes for Comutng Catal for Incremental Rsk n the Tradng Book; Bank for Internatonal Settlements: Basel, Swtzerland, Hull, J. Rsk Management and Fnancal Insttutons, 3rd ed.; Wley Fnance: Hoboken, NJ, USA, Gordy, M.B. A rsk-factor model foundaton for ratngs-based bank catal rules. J. Fnanc. Intermed. 2003, 2, Van Gestel, T.; Baesens, B. Credt Rsk Management: Basc Concets: Fnancal Rsk Comonents, Ratng Analyss, Models, Economc and Regulatory Catal; Oxford Unversty Press: Oxford, UK, Saunders, A.; Allen, L. Credt Rsk Measurement In and Out of the Fnancal Crss: New Aroaches to Value at Rsk and Other Paradgms, 3rd ed.; Wley Fnance: Hoboken, NJ, USA, Andersson, A; Vann, P. Credt Mgraton Rsk Modellng. Natonal Centre of Cometence n Research Fnancal Valuaton and Rsk Management, Workng Paer No. 539, 9 June e Servgny, A.; Renault, O. Measurng and Managng Credt Rsk; McGraw Hll: Columbus, OH, USA, Rkkers, F.; Thbeault, A.E. The otmal ratng hlosohy for the ratng of SMEs. Vlerck Leuven Gent Management School, 2007, do:0.239/ssrn Altman, E.I.; Saunders, A. Credt rsk measurement: eveloments over the last 20 years. J. Bank. Fnanc. 998, 2, Tsag, Y.; Levy, A.; Wang, Y. Analysng the mact of credt mgraton n a ortfolo settng. J. Bank. Fnanc. 20, 35, Belkn, B.; Suchover, S.; Forest, L. A one-arameter reresentaton of credt rsk and transton matrces. Credt Metr. Mont. 998,, Km, J. Condtonng the transton matrx. Credt Rsk 999, October, Banga, A.; ebold, F.X.; Kronmus, A.; Schagen, C.; Schuermann, T. Ratngs mgraton and the busness cycle, wth alcaton to credt ortfolo stress testng. J. Bank. Fnanc. 2002, 26, We, J. A mult-factor, credt mgraton model for soveregn and cororate debts. J. Int. Money Fnanc. 2003, 22, Fe, F.; Fuertes, A.M.; Kalotychou, E. Credt ratng mgraton rsk and busness cycles. J. Bus. Fnanc. Account. 202, 39, Aït-Sahala, Y. Maxmum lkelhood estmaton of dscretely samled dffusons: A closed-form aroxmaton aroach. Econometrca 2002, 70, Rabe-Hesketh, S.; Skrondalb, A.; Pckles, A. Maxmum lkelhood estmaton of lmted and dscrete deendent varable models wth nested random effects. J. Econom. 2005, 28,

21 Int. J. Fnancal Stud. 204, Chambers, R.L.; Steel,.G.; Wang, S.; Welsh, A.H. Maxmum Lkelhood Estmaton for Samle Surveys; Taylor and Francs Grou: Boca Raton, FL, USA, Basel Commttee on Bankng Suervson (BCBS). Internatonal Convergence of Catal Measurement and Catal Standards: A Revsed Framework; Bank for Internatonal Settlements: Basel, Swtzerland, Truck, S. Forecastng credt mgraton matrces wth busness cycle effects: A model comarson. Eur. J. Fnanc. 2008, 4, Truck, S.; Rachev, T. Ratng Based Modellng. of Credt Rsk: Theory and Alcaton of Mgraton Matrces; Elsever Academc Press: Amsterdam, The Netherlands, Regna, E.; Rodrgues, J.; Achcar, A. Alcatons of screte-tme Markov Chans and Posson Processes to Ar Polluton Modellng and Studes; Srnger: New York, NY, USA, Altman, E.I.; Kao,.L. The mlcatons of cororate bond ratngs drft. J. Fnanc. Anal. 992, 48, Lando,.; Skodeberg, T.M. Analysng ratng transtons and ratng drft wth contnuous observatons. J. Bank. Fnanc. 2002, 26, Chrstensen, J.; Hansen, E.; Lando,. Confdence sets for contnuous-tme ratng transton robabltes. J. Bank. Fnanc. 2004, 28, Guttler, A.; Rauach, P. The mact of downward ratng momentum. J. Fnanc. Serv. Res. 200, 37, Arvants, A.; Gregory, J.; Laurent, J.P. Buldng models for credt sreads. J. erv. 999, 6, Mahlmann, T. Estmaton of ratng class transton robabltes wth ncomlete data. J. Bank. Fnanc. 2006, 30, Hamlton,.T.; Cantor, R. Ratng Transton and efault Rates Condtoned on Outlooks. J. Fxed Income. 2004, 4, Kefer, N.M.; Larson, C.E. A smulaton estmator for testng the tme-homogenety of credt ratng transtons. J. Emr. Fnanc. 2007, 4, Frydman, H.; Schuermann, T. Credt ratng dynamcs and Markov mxture models. J. Bank. Fnanc. 2008, 32, Hanson, S.; Pesaran, H.; Schuermann, T. Frm Heterogenety and Credt Rsk versfcaton; Workng Paer; Wharton Fnancal Insttutons Centre: Phladelha, PA, USA, Parnes,. Tme seres atterns n credt ratngs. Fnanc. Res. Lett. 2007, 4, McNel, A.; Wendn, J. Bayesan nference for generalzed lnear mxed models of ortfolo credt rsk. J. Emr. Fnanc. 2007, 4, Czado, C.; Pfluger, C. Modellng deendences between ratng categores and ther effects on redcton n a credt rsk ortfolo. Al. Stoch. Models Bus. Ind. 2008, 24, Nckell, P.; Perraudn, W.; Varotto, S. Stablty of ratng transtons. J. Bank. Fnanc. 2000, 24, Jafry, Y.; Schuermann, T. Measurement, estmaton and comarson of credt mgraton matrces. J. Bank. Fnanc. 2004, 28, Truck, S.; Rachev, T. Credt ortfolo rsk and robablty of default confdence sets through the busness cycle. J. Credt Rsk 2005a,, 6 88.

22 Int. J. Fnancal Stud. 204, Wessbach, R.; ette, H. Kolmogorov Smrnov-tye testng for the artal homogenety of Markov rocesses wth alcaton to credt rsk. Al. Stoch. Models Bus. Ind. 2007, 23, Hanson, S.; Schuermann, T. Confdence ntervals for robabltes of default. J. Bank. Fnanc. 2006, 30, Kadam, K.; Lenk, P. Bayesan nference for ssuer heterogenety n credt ratngs mgraton. J. Bank. Fnanc. 2008, 32, Wlson, T. Portfolo credt rsk. Econ. Polcy Rev. 998, 4, ebold, F.X.; Marano, R.S. Comarng redctve accuracy. J. Bus. Econ. Stat. 2002, 20, Kavvathas,. Estmatng Credt Ratng Transton Probabltes for Cororate Bonds; Workng Paer; AFA New Orleans Meetng: Chcago, IL, USA, Carlng, K.; Jacobson, T.; Lnde, J.; Roszbach, K. Catal Charges under Basel II: Cororate Credt Rsk Modellng and the Macroeconomy; Workng Paer, No. 42; Sverges Rksbank: Stockholm, Sweden, Couderc, F.; Renault, O. Tme-to-efault: Lfe Cycle, Global and Industry Cycle Imacts; Workng Paer, No. 42; Internatonal Center for Fnancal Asset Management and Engneerng, Unversty of Geneva: Geneva, Swtzerland, uffe,.; Sata, L.; Wang, K. Mult-erod cororate default redcton wth stochastc covarates. J. Fnanc. Econ. 2007, 83, Fledelus, P.; Lando,.; Nelsen, J.P. Non-arametrc analyss of ratng transton and default data. J. Invest. Manag. 2004, 2, Kooman, S.J.; Lucas, A. Busness and default cycles for credt rsk. J. Al. Econom. 2005, 20, Kooman, S.J.; Lucas, A.; Klaassen, P. Emrcal credt cycles and catal buffer formaton. J. Bank. Fnanc. 2005, 29, Truck, S.; Rachev, T. Changes n Mgraton Matrces and Credt VAR. A New Class. of fference Indces; Workng Paer; Unversty of Karlsruhe: Karlsruhe, Germany, 2005b. 54. Gaglardn, P.; Goureroux, C. Mgraton correlaton: efnton and effcent estmaton. J. Bank. Fnanc. 2005, 29, Kooman, S.J.; Lucas, A. A non-gaussan anel tme seres model for estmatng and decomosng default rsk. J. Bus. Econ. Stat. 2007, 26, Kooman, S.J.; Lucas, A.; Montero, A. The mult-state latent factor ntensty model for credt ratng transtons. J. Econom. 2008, 42, Feng,.; Goureroux, C.; Jasak, J. The ordered qualtatve model for ratng transtons. J. Emr. Fnanc. 2008, 5, Banachewcz, K.; Lucas, A.; van der Vaart, A. Modellng ortfolo defaults usng hdden Markov models wth covarates. Econom. J. 2008,, Rolsk, T.; Schmdl, H.; Schmdt, V.; Teugels, J. Stochastc Processes for Insurance and Fnance; John Wley & Sons: Chchester, UK, Cox,.R.; Mller, H.. The Theory of Stochastc Processes; Chaman & Hall: London, UK, Kalbflesch, J..; Lawless, J.F. The analyss of anel data under a Markov assumton. J. Am. Stat. Assoc. 985, 80,

23 Int. J. Fnancal Stud. 204, Kalbflesch, J..; Lawless, J.F.; Vollmer, W.M. Estmaton n Markov models from aggregate data. Bometrcs 983, 39, a, Q.; Sngleton, K.G.; Yang, W. Regme shfts n a dynamc term structure. Rev. Fnanc. Stud. 2007, 20, Blochlnger, A. Lnkng the TTC and PIT efault Probabltes; Workng Paer; Kantonalbank: Zürcher, Swtzerland, Bennell, J.A.; Crabbe,.; Thomas, S.; Gwlym, O. Modellng soveregn credt ratngs: Neural networks versus ordered robt. Exert Syst. Al. 2006, 30, Xng, H.; Sun, N.; Chen, Y. Credt ratng dynamcs n the resence of unknown structural breaks. J. Bank. Fnanc. 202, 36, Abd, H. The Sngular Value ecomoston. In Encycloeda of Measurement and Statstcs; Salknd, N., Ed.; Sage: Thousand Oaks, CA, USA, 2007; Efron, B.; Tbshran, R.J. An. Introducton to the Bootstra; Chaman & Hall: New York, NY, USA, Andrews,.W.K.; Buchnsky, M. On the number of bootstra reettons for BCa confdence ntervals. Econom. Theory 2002, 8, Moody s. Ratngs Symbols and efntons; Moody s Investors Servce: New York, NY, USA, Ftch Ratngs. efntons of Ratngs and Other Forms of Onon; Ratngs efntons, Ratngs Research, Ftch Ratngs: New York, NY, USA, Standard & Poor s. What Are Credt Ratngs and How o They Work? Gude to Credt Ratng Essentals, Verson.4; Standard & Poor s: New York, NY, USA, Treacy, W.F.; Carey, M. Credt rsk ratng systems at large US banks. J. Bank. Fnance. 2000, 24, Fglewsk, S.; Frydman, H.; Lang, W. Modellng the effect of macroeconomc factors on cororate default and credt ratng transtons. Int. Rev. Econ. Fnanc. 202, 2, Stefanescu, C.; Tunaru, R.; Turnbull, S. The credt ratng rocess and estmaton of transton robabltes: A Bayesan aroach. J. Emr. Fnanc. 2009, 6, Corrado, C.; Mattey, J. Caacty utlzaton. J. Econ. Persect. 997,, Baumohl, B. The Secrets of Economc Indcators: Hdden Clues to Future Economc Trends and Investment Oortuntes, 3rd ed.; Wharton School Publshng: Uer Saddle Rver, NJ, USA, elaneds, G.; Geske, R. Credt Rsk and Rsk Neutral efault Probabltes: Informaton about Ratng Mgratons and efaults; Workng Paer; The Anderson School, UCLA: Los Angeles, CA, USA, Lvngston, M.; Naranjo, A.; Zhou, L. Slt bond ratngs and ratng mgraton. J. Bank. Fnanc. 2008, 32, Van eventer,.; Ima, K. Credt Rsk Models and the Basel Accords; Wley Fnance: New York, NY, USA, by the authors; lcensee MPI, Basel, Swtzerland. Ths artcle s an oen access artcle dstrbuted under the terms and condtons of the Creatve Commons Attrbuton lcense (htt://creatvecommons.org/lcenses/by/3.0/).