A model of bank capital, lending and the macroeconomy: Basel I versus Basel II

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1 A model of bank capial, lending and he macroeconomy: Basel I versus Basel II Lea Zicchino Working Paper no. 270 Bank of England, Financial Indusry and Regulaion Division. lea.zicchino@bankofengland.co.uk This paper represens he views and analysis of he auhor and should no be hough o represen hose of he Bank of England. I would like o hank Ralph Chami, Charles Goodhar, Glenn Hoggarh, Erlend Nier, Hyun Song Shin, Dimirios Tsomocos, Geoffrey Wood, seminar paricipans a he Bank of England, and an anonymous referee for helpful commens. I am, of course, responsible of any remaining errors. This paper was finalisedon15june2005. Copies of working papers may be obained from Publicaions Group, Bank of England, Threadneedle Sree, London, EC2R 8AH; elephone , fax , mapublicaions@bankofengland.co.uk Working papers are also available a The Bank of England s working paper series is exernally refereed. cbank of England 2005 ISSN

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3 Conens Absrac 5 Summary 7 1 Inroducion 9 2 Chami-Cosimano (2001) 12 3 Exension o he model and resuls 17 4 Concluding remarks 30 Appendix 32 References 43 3

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5 Absrac The revised framework for capial regulaion of inernaionally acive banks (known as Basel II) inroduces risk-based capial requiremens. This paper analyses he relaionship beween bank capial, lending and macroeconomic aciviy under he new capial adequacy regime. I exends a model of he bank-capial channel of moneary policy - developed by Chami and Cosimano - by inroducing capial consrains àlabasel II. The resuls sugges ha bank capial is likely o be less variable under he new capial adequacy regime han under he curren one, which is characerised by invarian asse risk-weighs. However, bank lending is likely o be more responsive o macroeconomic shocks. Key words: Capial adequacy, regulaion, bank lending, procyclicaliy. JEL classificaion: E5, G2. 5

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7 Summary The process of reforming he 1988 Basel Accord, ha sared in 1999, has been moivaed by he goal of more closely maching regulaory capial o he risk profile of banks asse porfolios. The raionale for minimum capial requiremens is ha hey miigae financial insiuions moral hazard. Regulaors are imposing a cos on bank owners o encourage hem o avoid cosly defaul. However, he limied number of risk caegories in he curren framework has creaed opporuniies for banks o increase he risk o which hey are exposed wihou increasing he amoun of regulaory capial. ThenewBaselAccordiswidelyrecognisedasamuch needed effor o deal wih he shorcomings of he curren sysem. By realigning capial adequacy rules wih banks incenives i aims a resoring he link beween risk and capial holding. Noneheless, a number of quesions have been raised by cenral bankers, regulaors and praciioners regarding he impac of a more risk-sensiive regulaory framework on macroeconomic sabiliy. Among hem, here is he issue of he poenial procyclical effecs of he new capial adequacy requiremens, ie he possibiliy ha during periods of weak economic growh, a fall in capial raios and an increase in regulaory requiremens implied by a deerioraion in he risk profile of banks asses migh increase he likelihood of credi conracion and, herefore, a furher weakening of growh. This paper analyses he relaionship beween banks capial holdings, banks loans and macroeconomic aciviy under risk-sensiive capial adequacy requiremens. In paricular, i compares he impac of macroeconomic shocks on banks choices of capial srucure and loan supply under he old and new capial adequacy regimes. I does so by exending a model ha invesigaes he impac of moneary policy on lending in an economy where banks operae in an oligopolisic marke and are subjec o minimum capial requiremens. In order o analyse banks reacion o changes in macroeconomic condiions under he new capial adequacy regime, I exend he model by assuming a link beween loan risk-weighs and borrowers crediworhiness. In paricular, I inroduce asse risk-weighs ha vary wih macroeconomic performance, which is a major deerminan of credi risk. The firs resul of he paper is ha he response of banks o shocks ha affec loan demand differs when he minimum capial requiremens are calculaed wih asse risk-weighs ha are sensiive o 7

8 macroeconomic condiions. In paricular, bank capial is less volaile han under capial requiremens wih consan risk-weighs. The inuiion behind his resul can be undersood by considering, for example, a posiive shock o macroeconomic condiions ha increases boh curren and fuure loan demand. If he capial consrain is binding, banks may no be able o expand loan supply in he curren period and hey may need o raise capial o increase supply in he fuure. Therefore, if capial requiremens do no change wih borrowers risk, capial increases in response o posiive macroeconomic shocks and decreases afer negaive shocks. Bu when asse risk-weighs depend on macroeconomic condiions, bank capial migh no need o increase for banks o be able o expand heir credi supply. In fac, following a posiive macroeconomic shock he risk-weighs decrease and he capial consrain hus become looser. This insigh has an imporan policy implicaion. On he one hand banks will end o operae above he minimum regulaory capial o avoid he capial consrain becoming binding in fuure periods. On he oher hand banks may no volunarily accumulae capial in imes of good macroeconomic condiions because i is during hese imes ha he capial consrain becomes looser. This means ha if banks are affeced by an adverse shock during a period of credi expansion, hey migh be forced o raise capial a a ime when marke condiions are unfavourable. A second and relaed resul of he paper concerns he effec of macroeconomic shocks on loan supply. Since capial is more difficul o accumulae in a recession, and easier o accumulae when he economy experiences a posiiveshock,bankcrediislikelyobemoreprocyclical under he new Accord han under he curren one. 8

9 1 Inroducion The aim of his paper is o analyse he relaionship beween macroeconomic condiions, bank capial and lending when banks are subjec o risk-sensiive capial adequacy requiremens as envisaged in he new Basel proposals. (1) The process of reforming he Basel Accord, begun in 1999, has been moivaed by he goal of more closely maching regulaory capial o he risk profile of banks asse porfolios. The raionale for minimum capial requiremens is ha hey miigae financial insiuions moral hazard. Regulaors are imposing a cos on bank owners o encourage hem o avoid cosly defaul. Bu he limied number of risk caegories in he curren framework has creaed opporuniies for banks o increase he risk o which hey are exposed wihou increasing he amoun of regulaory capial. There are a number of ways in which banks are able o engage in capial arbirage. For example, banks can sell or securiise hose asses for which he regulaory capial charge is believed o be higher han he one markes would impose while keeping on he books poorer qualiy asses for which he regulaory capial charge is relaively low. ThenewBaselAccordiswidelyrecognisedasamuch needed effor o deal wih he shorcomings of he curren sysem. By realigning capial adequacy rules wih banks incenives i aims a resoring he link beween risk and capial holding. Noneheless, a number of quesions have been raised by cenral bankers, regulaors and praciioners regarding he impac of a more risk-sensiive regulaory framework on macroeconomic sabiliy. Among hem, here is he issue of he poenial procyclical effecs of he new capial adequacy requiremens, ie he possibiliy ha during periods of weak economic growh, he rise in regulaory requiremens implied by a deerioraion in he risk profile of banks asses migh lead o a reducion of credi supply and hus reinforce he weakening of macroeconomic condiions. Empirical evidence shows some suppor for he idea ha he credi crunch experienced in he Unied Saes during he 1990 recession migh have been caused, or a leas accenuaed, by he inroducion of capial requiremens. Bernanke and Lown (1991) use US sae-level daa on individual banks o argue ha, alhough demand facors migh have caused much of he observed slowdown in bank lending during he period, a shorage of equiy capial limied banks abiliy o (1) See Basel Commiee on Banking Supervision (2004) for a descripion of he Basel II Accord. 9

10 exend loans, paricularly in he norheasern par of he counry. Lower levels of bank capial seem o have been he resul of an increase in loan defauls and wrie-offs. (2) If he weighs used o se capial requiremens are sensiive o changes in risk, as proposed under he new Accord, required regulaory capial may increase a he same ime ha acual levels of bank capial are decreasing. Risk-weighs would increase during recessions because of a higher probabiliy of defaul and poenially also because of a higher loss given defaul on loans. So in order o mainain a given raio of capial o risk-weighed asses, banks will need o hold more regulaory capial. This rise in capial requiremens would coincide wih a period during which banks profis, and poenially acual capial, may be falling as a consequence of bad loans being charged off. In a recen paper, Caarineu-Rabell, Jackson and Tsomocos (2003) sugges ha many banks have moved owards poin-in-ime raing mehods when deermining credi risk. Under an inernal raings based (IRB) approach o he implemenaion of he new capial adequacy regulaion, banks would herefore deermine he risk-weighs based on a Meron-ype model where curren informaion on borrowers equiy price and book liabiliies is used o obain esimaes of borrowers probabiliies of defaul. Since cyclical effecs in asse valuaions would be refleced in he defaul probabiliies, he Meron-model approach delivers risk-weighs ha are highly sensiive o curren economic condiions. Apar from Caarineu-Rabell e al, here are few papers ha analyse he impac of capial regulaion on banks behaviour. However, over he pas few years a relaed lieraure has emerged which analyses he link beween bank capial and lending in models of moneary policy. (3) In addiion o he more convenional mechanisms of propagaion and amplificaion of moneary policy shocks, a new ype of credi channel (a so-called bank capial channel ) has been idenified. (4) According o he bank capial channel view, a change in ineres raes can affec lending hrough banks capial. This ransmission mechanism is operaive when banks lending is consrained by a capial adequacyrequiremen.anincreaseinofficial ineres raes, for example, will raise he cos of banks exernal funding, reduce heir profis and possibly heir capial. (2) For more evidence on he impac of capial requiremens on bank lending see Jackson e al (1999), Hancock, Laing and Wilcox (1995), Haubrich and Wachel (1993), Thakor (1996) and Wagser (1999). (3) See, among ohers, Bliss and Kaufman (2002), Chami and Cosimano (2001), Tsomocos (2003), and Van den Heuvel (2001). (4) See Van den Heuvel (2002) for a brief descripion of he various channels of moneary policy ransmission. 10

11 If he capial consrain becomes binding, banks will need o decrease heir supply of credi. (5) I is imporan, however, o noice ha he bank capial channel may operae in response o facors oher han changes in moneary policy. Changes in regulaory policy or simply shocks o macroeconomic condiions may also shif banks lending by affecing heir regulaory capial consrain. This las mechanism will be he focus of he analysis in his paper. The impac of macroeconomic condiions on bank capial and loans is examined by using he framework developed by Chami and Cosimano (2001). In heir paper, Chami and Cosimano (C-C) analyse he effec of moneary policy in an economy characerised by banks operaing under capial adequacy consrains in an imperfecly compeiive loan marke. The model assumes ha minimum capial requiremens are consan over ime àlabasel I (6) and ha bank loans are risk-free. In order o analyse banks incenives and he impac of macroeconomic condiions on banks choices under he new Accord, I inroduce wo key exensions. Firs, i is assumed ha borrowers can defaul and hus ha banks make credi losses. Loan wrie-offs are modelled as being dependen on macroeconomic condiions, consisen wih empirical evidence ha banks losses on loan porfolios are correlaed wih he business cycle, under any capial adequacy regime. Second, asse risk-weighsareassumedovaryoverhebusiness cycle. This assumpion inends o capure he link beween loan risk-weighs and borrowers crediworhiness esablished in he new Basel rules. In order o compare he impac of he curren and revised capial adequacy sandards (ie Basel I and Basel II) on banks behaviour, I inroduce he wo exensions sequenially. Firs, I presen he resuls of he model under he assumpion ha banks make losses on loans bu ha risk-weighs are consan, as under Basel I. Then, I will show he consequences of asse risk-weighs ha vary wih macroeconomic condiions, as under Basel II. The res of he paper is organised as follows: Secion 2 provides a brief summary of he C-C (5) In Chen (2001) a bank s ne worh can impac on lending wihou a capial adequacy consrain. Bank capial has a role in overcoming he moral hazard problem arising from asymmeric informaion beween banks, deposiors and borrowers. (6) The firs Capial Accord, published in July 1988 and implemened by he end of 1992, prescribes minimum capial o asse raios. All asses are assigned o one of four buckes, which classify he riskiness of he respecive conrac (eg loans o OECD governmens, loans o OECD banks and oher OECD public secor eniies, residenial morgage loans, loans o he privae secor). The same consan risk-weigh is hen associaed o he asses in he same risk bucke. 11

12 paper, Secion 3 presens he exension o he C-C model and he main resuls, Secion 4 concludes and discusses furher work. 2 Chami-Cosimano (2001) The C-C paper analyses he bank capial channel of moneary policy by using a dynamic model in which banks maximise he presen value of fuure profis, subjec o a minimum capial-asse raio. In anicipaing ha he capial consrain may bind in he fuure, banks choose an opimal level of dividend payous (or, equivalenly, capial) each period o minimise his possibiliy. Moneary policy affecs he supply of loans by affecing he value of holding bank capial. For example, a ighening of moneary policy, which causes an increase in curren and fuure deposi raes, reduces banks curren and fuure loan supply. A lower loan supply gives banks he incenive o hold less capial. Bu a lower level of capial will make he consrain on fuure lending more resricive. In sum, conracionary moneary policy reduces fuure credi supply no only because i increases he marginal cos of loans bu also because i causes a reducion in banks capial. Before briefly presening he model, I will discuss banks capial consrain and banks cash flows as presened by C-C. The oal capial consrain requires ha he sum of Tier 1 and Tier 2 capial (7) be no less han a given raio of risk-adjused asses: θ L q 1 s + b (1) where q 1 s (Tier 1 capial) is he previous period s marke value of equiy (price q a ime 1 imesnumberofsharess), L are one-period loans, b are one-period bonds (subordinaed deb in he Basel Accord) issued in he previous period, and θ is he regulaory raio of Tier 1 plus Tier 2 capial o risk-adjused asses. To mainain he book value feaure of regulaory capial, i is assumed ha las period s marke value of equiy and bonds deermines he capial consrain for he presen period. (7) Tier 1 capial is he book value of a bank s sock plus reained earnings. Tier 2 capial is he sum of loan-loss reserves and subordinaed deb. 12

13 The following ideniy describes he sources and uses of cash flows n generaed by he bank: n π = d s r b b q (s +1 s ) b +1 (2) where π are profis, d s are dividend paymens and r b is he ineres rae on one-period bonds. (8) The banking indusry is assumed o operae in an oligopolisic marke for loans: here are N banks ha charge a loan rae consisen wih monopoly power. An individual bank precommis o a quaniy of loans hrough is capial holding (since he curren choice of capial resrics he supply of loans nex period), and loan rae compeiion follows in he nex period. To characerise he equilibrium, C-C analyse he behaviour of a bank operaing as a monopolis. (9) The bank aracs deposis D a a fixed marginal cos c D. The bank holds αd as reserves agains wihdrawals of money. Assume hese wihdrawals x come from a probabiliy disribuion f (x). If a any ime x exceeds he amoun of markeable securiies T (held in addiion o reserves), hen he bank has o liquidae asses a a penaly rae r p. The bank faces he following loan demand: L = l 0 l 1 r L + l 2 M + ε L, (3) where r L is he loan rae, M is a variable represening he level of macroeconomic aciviy and (8) Bank profis π are used o mee obligaions: dividend paymens on equiy, and ineres and principal paymens on bonds, such ha reained earning are described by re = π d s (1 + r b)b. Banks should also inves in plan and equipmen each period. For simpliciy we can assume ha here is no invesmen in new physical capial and ha he depreciaion rae of banks capial sock is ψ. Banks finance he depreciaion ψk hrough reained earnings, new equiy or new bonds: ψk = re + q (s +1 s ) such ha he ne cash flow is given by n π ψk = d s + (1 + r b )b q (s +1 s ) b +1. Wihou loss of generaliy, we can se ψk = 0. Thus, if for example (1 + r b)b = b +1, ie he subordinaed deb is rolled over each period, hen he equaion above becomes: π d s = q (s +1 s ). If π > d s, hen q (s +1 s )<0. This implies ha banks do no keep reained earnings on heir balance shees as cash. If no compleely paid ou as dividends, and if deb is rolled over, earnings are used o decrease liabiliies (unless banks decide o hold more capial o increase loan supply, as we will see laer). (9) See he appendix or Chami-Cosimano (2001) for a discussion of he condiion under which he game played by oligopolisic banks has a co-operaive soluion. 13

14 ε L, represens a random exogenous shock o he demand for loans, which is defined over he inerval L, L. (10) The bank maximises is value by choosing he loan rae, is level of deposis, is invesmen in reasury securiies, and is capial, subjec o he cash-flow consrain (5), he loan demand (6),he financing consrain (7) and he balance-shee ideniy (8): Max v(q 1 s + b, x ) = {π + λ q 1 s + b θ L τ q s +1 + b +1 + E m,1 v(q s +1 + b +1, x +1 ) } (4) subjec o: π = r L L + r T T r D D c L L c D D r p 2 D T (5) 2 D L = l 0 l 1 r L + l 2 M + ε L, (6) q s +1 + b +1 = (d + q ) s r b b π (7) T = (1 α) D L + q 1 s + b (8) (10) As observed by an anonymous referee, some models propose ha firms demand for exernal funds (and hus for bank loans) may increase following a fall in firms inernal funds, ie when macroeconomic condiions deeriorae. Here, I follow C-C in assuming a Hicksian loan demand ha is an increasing funcion of GDP and a decreasing funcion of he ineres rae. 14

15 In he value funcion, λ is he Lagrange muliplier for he capial consrain, τ is he deadweigh cos of capial (11) and m,1 is a sochasic discoun facor. In he profi funcion, c L is he consan marginal cos of loans (represening he cos of monioring and screening), c D is a consan componen of he cos of deposis (represening he cos of cheque clearing and bookkeeping), and 2 D T is he cos of deposi wihdrawals. (12) r p 2D The level of macroeconomic aciviy M, and he ineres rae on deposis r D are assumed o follow a firs-order auoregressive sochasic process: (13) M +1 = ρ M M + ε M,+1 (9) and r D +1 = ρ r Dr D + ε r D,+1 (10) where ε M,+1, and ε r D,+1 are zero-mean sochasic shocks o economic aciviy and deposi rae, respecively. One may hink of ε M,+1 as a emporary shock o GDP, eiher a demand shock, for example a emporary shif in governmen purchases or in consumer confidence, or a supply shock, for example an oil price shock. When solving his problem, las period s choice of capial is aken as given and deermines wheher or no he capial consrain is binding his period. Furher, i urns ou ha he soluion of he problem separaes he bank s desired capial from is oher choices. Therefore, decisions on loan ineres rae, deposis and markeable securiies can be analysed as in a saic problem. When he capial consrain is non-binding, he opimal level of loans occurs where he marginal revenue (11) In he model here is no difference beween he relaive cos of raising subordinaed deb versus equiy. (12) The cos of deposi wihdrawals is obained as follows: C (T ) = r p (x T ) f (x) dx = r p 2 D T T 2D where x is assumed o have a uniform disribuion wih suppor D, D. (13) The fac ha shocks o macroeconomic condiions M and o deposi rae r D (which approximae moneary policy) are no correlaed means ha he model does no have a well-defined moneary policy rule. Oherwise, we would expec shocks in M o affec r D. I owe his poin o an anonymous referee. 15

16 from loans is equal o he marginal cos of loans (poin A in Figure 1 below). If here is an increase in loan demand (an upward shif of he demand curve), boh he quaniy and he price of loans (he ineres rae r L ) increase. If he marginal cos of loans decreases, he quaniy of loans will increase, while he ineres rae will decrease. Figure 1: Loan marke equilibrium in he C-C model Loan rae r L * A Marginal Cos Marginal Revenue Loan Demand L* Loans However, if he capial consrain is binding and loans are a heir ceiling L,henanincreasein he loan demand or a decrease in he marginal cos of loans do no affec he supply of credi bu only he loan rae, r L. In deciding he amoun of capial o hold, he bank compares he marginal deadweigh cos of capial, τ, wih he expeced marginal benefi. The laer has wo componens. Firs, addiional equiy (or subordinaed deb) reduces he need for raising deposis in he nex period. Second, hereisheexpecedmarginalbenefi from he fac ha he capial consrain is less likely o be binding in he nex period, λ +1. (14) The resuling opimal capial held by banks depends on a number of facors: (1) i is a negaive (14) The opimal bank capial is he soluion o he firs-order condiion ha equaes τ, he cos of capial, o he expeced marginal cos of deposis, which is he alernaive source of financing for he bank, plus λ +1,whichishe shadow value of he capial consrain (ie i represens he marginal increase in he opimal value of he objecive funcion of he bank if he capial consrain is relaxed). 16

17 funcion of he expeced marginal cos of exernal funds (which is equal o he difference beween he marginal deadweigh cos of capial τ and he expeced marginal benefi of capial, as jus described); (2) a posiive funcion of he expeced demand for loans (which is, in urn, a funcion of curren economic condiions, due o he auoregressive naure of he sochasic process characerising M); (3) a negaive funcion of he expeced marginal cos of loans ( r D +c D + c 1 α L), and wihin ha of he deposi rae r D ; (4) a posiive funcion of he volailiy of loan demand; (5) a negaive funcion of he elasiciy of he loan demand (bigger elasiciy implies less monopoly power); and (6) a posiive funcion - under a parameric assumpion - of he regulaory capial raio θ. An increase in economic aciviy has a posiive impac on he level of capial held by he bank. An improvemen in macroeconomic condiions increases he expeced loan demand nex period and he probabiliy ha he capial consrain will become binding. The response of banks is o increase capial o keep profis a heir maximum. The capial held by banks is herefore procyclical, in he sense ha i follows he level of expeced loan demand, which is assumed o depend on economic condiions. 3 Exension o he model and resuls Two key feaures - and limiaions - of he C-C model are ha loans, which have a one-period mauriy, are always repaid in full and ha he capialconsrainisindependen of macroeconomic condiions. Bu wha is ineresing is he link beween macroeconomic aciviy, loan defauls (wrie-offs) and asse risk-weighs. I herefore inroduce wo exensions. Firs, I consider he possibiliy ha bank borrowers can defaul on heir principal paymen. Moreover, he proporion of defauled deb is assumed o be a negaive funcion of curren macroeconomic condiions, ie wrie-offs are higher when economic condiions are bad han when hey are good. Bank loan defauls ranslae ino an exra cos erm in he banks profi funcion. As will become clear, unlike in C-C, he marginal cos curve will now shif downwards (upwards) as macroeconomic condiions improve (worsen). In oher words, economic condiions affec loan supply as well as demand. The second assumpion I inroduce is ha he risk-weigh assigned o loans in he capial adequacy raio is no consan, as in C-C, bu is a decreasing funcion of curren macroeconomic condiions, as implied by Basel II. 17

18 As regarding he firs assumpion, here is evidence o suppor he noion ha loan wrie-offs are higher during recessions. Weak economic condiions are likely o be associaed wih a deerioraion in asse qualiy as borrowers financial healh weakens. Char 1 shows he aggregae annual ne wrie-off o loan raio, consruced from daa published on he public accouns of he major UK banks, and he (invered) annual UK oupu gap (15) over he period The wo series seem o follow he same paern, wih wrie-offs following he oupu gap wih a lag. Char 1: Wrie-off/loan raio and UK (invered) oupu gap Per cen 6 (Invered) Oupu gap (LHS) Wrie-off/loan raio (RHS) Per cen _ Sources: Bank of England; OECD Economic Oulook. Char 2 shows he annual aggregae ne wrie-off o loan raio ogeher wih (invered) UK GDP growh rae from 1988 o The char confirms he inuiion of a counercyclical movemen in loan losses for he Unied Kingdom. Moreover, he negaive correlaion beween bank losses and he business cycle does no seem o be a unique feaure of he UK economy. Borio, Furfine and Lowe (2001) show evidence of a cyclical paern of bank provisions for loan losses in many OECD counries, wih a paricularly srong link beween he provision o asse raio and he oupu gap in Ausralia, Sweden, Norway, Japan and Spain. The assumpion made in his paper is ha wrie-offs reac o he curren value of M, which can be hough of as a proxy for GDP. However, hey depend on pas macroeconomic condiions as well, (15) As calculaed by he OECD. 18

19 given ha M is described by an auoregressive process. Char 2: Wrie-off/loan raio and UK GDP growh rae Per cen 2 Wrie-off/loan raio (RHS) (Invered) GDP growh rae (LHS) + _ Per cen Source: Bank of England. The second assumpion inroduced here is ha risk-weighs vary wih macroeconomic condiions. In he revised Basel Accord banks are given he choice beween wo mehodologies for calculaing heir minimum capial requiremens. Under he so-called sandardised approach, exposures included in banks reail porfolios would mainain a consan risk-weigh (16) while exernal credi assessmens, such as credi scores by raing agencies, would be used o consruc he risk-weighs for claims on sovereigns, banks, and corporaes. In assessing a borrower s crediworhiness, he major raing agencies aim a mainaining a sable raing hrough he business cycle. There is, however, evidence ha credi raings show a cyclical paern wih more downgrades han upgrades during recessions (see, for example, Nickell, Perraudin and Varoo (2000)). Under he alernaive mehod (he inernal raings based approach) some banks will use heir own credi raings o deermine capial requiremens. This approach migh induce more cyclicaliy in risk-weighs. Mos inernal raing sysems use a shor-erm horizon o measure risk. In paricular, borrowers probabiliy of defaul, which is one of he erms in he formula for risk-weighs, is deermined over (16) Claims in he reail porfolio, which includes small businesses, credi cards, personal loans and leases, may be risk-weighed a 75%, wih he excepion of loans secured by morgages on residenial propery and pas due loans. Claims secured by residenial propery will be risk-weighed a 35%, while pas due loans, which will have a risk-weigh of eiher 100% or 150%, depending on wha percenange of he ousanding amoun of he loan is covered by specific provisions. 19

20 a one-year period and borrowers are assigned o raing grades using models such as Moody s KMV. (17) A number of papers, including Caarineu-Rabell e al (2003) and Kashyap and Sein (2004), have shown ha his inroduces considerable cyclicaliy in risk-weighed asses. In is hird Consulaive Paper (CP3), (18) he Basel Commiee has proposed flaer risk-weigh curves for small and medium enerprises (19) as a way o address he concern ha procyclicaliy will increase under he IRB approach. This iniiaive will cerainly dampen he volailiy in capial raios bu probably no eliminae i. Indeed, evidence presened in Kashyap and Sein (2004) among ohers is based on he revised curves for corporae and SME credi. Our assumpion reflecs he idea ha, under he new Capial Accord, risk-weighs migh become more closely relaed o curren macroeconomic condiions. 3.1 Bank loan quaniy and price responseochangingmacroeconomiccondiions Case I: banks loan losses depend on macroeconomic condiions and asse risk-weighs are consan (Basel I) In wha follows I presen he resul of he model under he assumpion ha bank loans may no be repaid, ie ha borrowers can defaul. The proporion of defauled deb is a negaive funcion of curren macroeconomic condiions, ie wrie-offs are higher when economic condiions are bad han when hey are good. The risk-weigh on loans is consan, as in Basel I. In he model banks operae in an oligopolisic marke for (risky) loans and inves in a safe asse (reasury securiies) for liquidiy purposes. I solve he profi maximisaion problem for a monopolis bank and, in he appendix, derive he condiion under which banks co-operae and equally share he profis of he indusry. (20) The capial consrain ha a bank faces when maximising is profis is he following: (17) See he discussion paper by he Basel Commiee on Banking Supervision (2000) for a survey on banks Inernal Raings Sysems. (18) See BCBS (2003). (19) For any given ype of asse, hese curves describe he relaionship beween he capial charge and he probabiliy of defaul. (20) The condiion is idenical o he one obained in C-C. 20

21 θ q 1s + b w L L + w T T (11) where he risk-weigh on reasuries (w T ) is assumed o be zero and he one on loans (w L ) is se equal o 1. The bank faces a downward sloping demand curve and equaes marginal revenue o marginal cos o find he opimal quaniy of loans L and he opimal loan rae r L, in each period. Thebank sprofis are given by he following expression: π = r L L + r T T c L L r D D c D D r p 2D (D T )2 δ (M ) L (12) where δ (M ) L represens defauled loans. The defaul rae δ (21) varies beween 0 and 1 and increases when macroeconomic condiions deeriorae, ie 0 δ (M ) 1and δ(m ) M < 0. The objecive is o analyse he impac on banks decisions of macroeconomic condiions when losses depend on macroeconomic condiions and a risk-insensiive capial adequacy consrain, described in (11), is in place. In order o achieve his, I need o solve for he opimal bank allocaion of asses beween loans and reasuries, and liabiliies beween deposis and oal capial (equiy plus subordinaed deb). This is shown in he appendix. I is possible hen o analyse how bank capial and he quaniy and price of loans are affeced by changes in macroeconomic condiions. Proposiion 1 Under he assumpion of posiive loan losses and consan asse risk-weighs, banks raise (decrease) heir capial holdings in response o a posiive (negaive) shock in he curren level of economic aciviy. The proof of his proposiion is given in he appendix. This resul is similar o he one obained in C-C bu here bank capial is more procyclical. This is because lending is more procyclical in a model wih loan losses, as will be shown in he nex proposiions. (21) δ corresponds o he loss given defaul (LDG). 21

22 We proceed now o esablish he impac of changes in M on loans and loan raes. We presen firs he resuls under a non-binding capial consrain and hen under a binding consrain. When λ = 0, ie he consrain does no bind, hen: L + j = 1 M 2 l 2ρ j M ρ j M l 1 δ M + j (13) 2 M + j r+ L j = 1 l 2 ρ j M M 2l + ρ j M δ M + j (14) 1 2 M + j Proposiion 2 When he capial consrain is non-binding, bank loans increase as a resul of an improvemen in macroeconomic aciviy, while he loan rae can eiher increase or decrease depending on he parameers characerising he loan demand funcion and he loan defaul rae. The proof of his proposiion follows direcly from (13), (14) and he condiion δ (M + j) M + j < 0. When macroeconomic condiions improve, loan demand shifs o he righ and, because defaul raes decrease, he marginal cos curve shifs downwards. (22) The marginal cos curve moves downwards (from MC 1 o MC 2 in Figure 2) and banks find i profiable o increase he supply of loans. The rae charged on loans, however, can eiher increase or decrease depending on he parameers characerising he loan demand and on he loan defaul rae. I is imporan o noice ha, when accouning for he possibiliy of loan defauls, he loan supply increases by more han in he C-C model - o L 2 raher han L 2,CC in Figure 2 - in response o beer macroeconomic condiions, because he marginal cos curve shifs downwards a he same ime he demand (and he marginal revenue) curve shifs upwards. (23) (22) The derivaive of he marginal cos curve wih respec o M is described by he following expression: MC( + j) δ( + j) = ρ j M M M < 0 (15) + j (23) I should be noed ha he impac of a change in macroeconomic condiions dies ou over ime, since ρ j M < 1, ie he effec of he curren value of M on fuure macroeconomic condiions declines over ime. 22

23 Figure 2: Loan marke equilibrium when he capial consrain is non-binding Loan rae r L 2,CC r L 2 r L 1 MC 1 MC 2 MR 1 MR 2 D2 D 1 L 1 L 2,CC L 2 Bank Loans I consider now he case in which he capial consrain binds, ie λ > 0. In his case loan supply is deermined by he capial adequacy consrain and is equal o L = q 1s + b θ (16) while he loan rae response o changes in macroeconomic aciviy is: r L M = l 2 l 1 (17) Proposiion 3 When he capial consrain is binding, curren bank lending does no change in response o an improvemen in macroeconomic aciviy. Theloanraehoweverincreasesasa consequence of a higher loan demand. The proof of his proposiion follows immediaely from (16) and (17). 23

24 An increase in M causes an upward movemen of he loan demand and a downward movemen of he marginal cos curve (see Figure 3). Banks are willing o increase heir loan supply bu hey are no are able o do so because hey are capial-consrained. The loan rae increases up o r L. Figure 3: Loan marke equilibrium wih a binding capial consrain Loan rae r L CC=r L A r L* MC MC 1 MR 1 D 1 L*=L MR D Bank Loans Case II: banks loan losses and asse risk-weighs depend on macroeconomic condiions (Basel II) In his secion analogous proposiions o he ones in he previous secion will be derived for a model where boh loan losses and loan risk-weighsareafuncionofmacroeconomic condiions M. The capial consrain is now described by he following expression: θ q 1 s + b w (M ) L + w T T (18) where he risk-weigh on reasuries (w T ) is again assumed o be zero and he one on loans is a 24

25 decreasing funcion of curren macroeconomic condiions, ie w(m ) M < 0. Proposiion 4 Under he assumpion of posiive loan losses and a risk-sensiive capial consrain, banks can eiher raise or lower heir capial holdings in response o a posiive shock in he curren level of economic aciviy. Their choice depends on he effec ha he change in M has on he likelihood of he capial consrain binding nex period. The proof of his proposiion is given in he appendix. This resul is in conras o he Basel I model and provides an ineresing implicaion for he policy debae abou he risk ha he new capial sandards may exacerbae he cyclicaliy of bank lending. If he criical shock o loan demand, ε L,+1 (24) decreases when M increases, ie if he likelihood hahecapialconsrainwill bind nex period increases, hen banks will choose o hold more capial o be able o gran more loans nex period. However, i is possible ha ε L,+1 M > 0, and ha bank capial decreases (see he appendix). The reason for his possibiliy is ha, in he Basel II model, beer macroeconomic condiions have wo couneracing effecs on bank capial. On he one hand, since banks expec a higher loan demand in he fuure, hey wan o be able o supply more loans. In order o be able o increase heir lending, hey need o hold more capial. A he same ime hough, banks expec lower risk-weighs and, herefore, a higher capial o asse raio, which makes i less likely ha he capial consrain will bind. Since holding capial enails a deadweigh cos for banks, hey have an incenive o reduce he amoun of capial. If he second effec dominaes, banks will choose o hold less capial. To summarise, once allowance ismadeforvaryingrisk-weighsinhecapial raio, he impac of he curren macroeconomic condiions on banks capial holdings is ambiguous. In conras o he model presened in he previous paragraph, a posiive shock o M does no necessarily induce banks o increase capial. I urn now o he effec of a shock o M on bank loans and he loan rae, and analyse boh cases: when he capial consrain is slack and when i is binding. When λ = 0, ie he consrain does no bind, hen: (24) In he appendix ε L,+1 is derived as he criical shock o loan demand a ime + 1 such ha he capial consrain jus binds. 25

26 L + j = 1 M 2 l 2ρ j M ρ j M l 1 δ M + j (19) 2 M + j r+ L j = 1 l 2 ρ j M M 2l + ρ j M δ M + j (20) 1 2 M + j Proposiion 5 Under he same assumpions of Proposiion 4, and when he capial consrain is non-binding, bank loans increase as a resul of an improvemen in macroeconomic aciviy. The loan rae can eiher increase or decrease depending on he parameers characerising he loan defaul rae and he loan demand funcion. The proof of his proposiion follows direcly from (19), (20) and he condiion δ (M + j) M + j < 0. Clearly, when he capial consrain is non-binding, he models wih consan and non-consan risk-weighs yield he same resuls as regards exension of loans. The impac on he quaniy and price of loans of a posiive shock o M can be seen in Figure 2 above. I consider now he case in which he capial consrain binds, ie λ > 0. By aking he firs derivaive of (A-19) in he appendix wih respec o M, I obain: L M = q 1s + b θ[w (M )] 2 w(m ) M (21) while he loan rae response o changes in macroeconomic aciviy is: r L M = l 2 l 1 + q 1s + b θl 1 [w (M )] 2 w(m ) M (22) Proposiion 6 Under he same assumpions of Proposiion 4, and when he capial consrain is binding, curren bank lending increases inresponseoanimprovemen in macroeconomic aciviy. The loan rae can eiher increase or decrease depending on he parameers characerising he loan demand and he risk-weighs. 26

27 The proof of his proposiion follows from (21), (22) and he assumpion w (M + j) M + j < 0. An increase in M induces boh a slackening of he capial consrain and an upward shif of he loan demand (see Figure 4). Banks are willing o increase heir loan supply and are able o do so because heir capial raio has increased. The direcion of he change in he loan rae depends on he elasiciy of loan demand wih respec o he loan rae, and he sensiiviy of loan demand and he risk-weigh o macroeconomic condiions. The higher he sensiiviy of loan demand o M (ie he bigger he parameer l 2 ), he more he loan rae ends o increase. The higher he elasiciy of loan demand and he sensiiviy of risk-weighs o M (ie he bigger l 1 and w(m ) M ),hemorehe loan rae ends o decrease. A he limi, if he posiive shock o macroeconomic condiions makes he capial consrain slack, loan supply goes from L o L and he loan rae from r L o r L in Figure 4. If he capial consrain sill binds, he equilibrium loan quaniy will be somewhere beween L and L and he loan rae beween r L CC and r L. This is in conras o he Basel I model, where loans would remain a L hroughou his exercise and he loan rae would respond by moving from r L o r L CC. Figure 4: Loan marke equilibrium when he bank capial consrain is binding Loan rae r L CC A r L r L* B MC MC 1 MR D 1 L* MR L D Bank Loans Finally, I analyse he effec of a curren shock o macroeconomic condiions on bank loans and 27

28 loan raes in fuure periods. This impac is analogous o he one on curren loans and loan raes, ie i depends on wheher he capial consrain binds or no in fuure periods. If he capial consrain is slack, he response of loans and loan raes in period j 1 o a curren macroeconomic shock is described by Proposiion 4 bu he impac of he shock dies ou over ime, since ρ j M < 1. If he capial consrain binds in fuure periods, he response of loans and loan raes o a change in M is described by: L + j M = ρ j 1 M θw q 1+ j s + j + b + j ρ M M + j M 1+ j q 1+ j s + j + b + j w w M + j M + j M + j (23) and r L + j M = l 2ρ j M ρ j 1 M w + l 2 ρ M + j M q 1+ j s + j + b + j 1 θl 1 w M+ j M + j q 1+ j s + j + b + j M + j 1 (24) Therefore, fuure bank loans can eiher increase or decrease in response o a posiive shock o curren macroeconomic condiions depending on he relaive magniudes of he erms in he righ-hand side of equaion (23). Recall ha Proposiion 4 ells us ha under Basel II capial does no necessarily increase in response o a posiive shock o M. If capial decreases in response o a posiive shock o macroeconomic condiions, loans can decrease as well, provided ha he firs erm in he curly brackes in (23) is bigger han he second erm, ie ha he effec of he shock on capial dominaes he effec of he shock on loan risk-weighs. Finally, he sign of he change in loan raes is undeermined, as i can be seen from equaion (24). As a summary of he resuls of he paper, Table A below shows he effec of a posiive shock o macroeconomic condiions on banks lending under he wo capial adequacy regimes (Basel I and Basel II). 28

29 Table A: Responses o a posiive shock o macroeconomic condiions Loans Loan rae Bank capial Basel I, non-binding capial consrain Basel I, binding capial consrain Basel II, non-binding capial consrain Basel II, binding capial consrain = Undeermined, lower han in he Basel I model Undeermined, lower han in he Basel I model Undeermined, lower han in he Basel I model In boh versions of he model presened here, here are wo cases: wih he curren capial consrain binding, and wih i non-binding. In he Basel I model, boh bank loans and loan rae increase in response o a posiive shock o M, when he capial consrain is no binding. If insead he capial consrain binds, an upward shif of he loan demand will induce an increase in he loan rae, which will be higher han in he unconsrained case, bu leave he loan supply unchanged. Moreover, since a posiive shock o macroeconomic condiions increases he likelihood ha he capial will be binding in he fuure, banks increase he amoun of capial hey hold. In he Basel II model, a shock o curren macroeconomic condiions affecs no only he loan demand bu also he risk-weighs in banks capial o asse raios. If he capial consrain is slack, bank loans increase by he same amoun han in he Basel I model. If he capial consrain is binding, hen, unlike under Basel I, banks can sill expand heir credi supply, even hough by less han if he capial is non-binding. They areableodosobecauseaposiiveshocko macroeconomic condiions induces lower risk-weighs and herefore a slackening of he capial consrain. The loan rae can eiher increase or decrease, depending on he relaive size of he change in loan demand and capial raio. Analogously, a negaive shock o macroeconomic condiions resuls in a possibly greaer reducion of credi han in he Basel I model because of boh a downward shif of he loan demand and a ighening of he capial consrain. 29

30 Moreover, under Basel II, he sign of he change in banks capial holding is undeermined as a posiive shock o M has wo couneracing effecs on he equilibrium value of bank capial. On he one hand, a posiive macroeconomic shock has a persisen posiive effec on loan demand and herefore raises he likelihood of he capial consrain binding in he fuure. A he same ime, however, he capial raio increases so ha he likelihood ha he capial consrain will bie becomes lower. 4 Concluding remarks The model by Chami and Cosimano analyses, among oher hings, he effec of macroeconomic shocks on bank capial holding. Their conclusion is ha capial - he numeraor in he capial adequacy raio - is procyclical. An improvemen in macroeconomic condiions induces an increase in he loan demand and, as a consequence, an increase in he probabiliy ha he capial consrain will become binding. The response of banks is o increase capial o maximise profis. Thus, a bank s capial is procyclical because i changes according o expeced loan demand, which is assumed o be a funcion of he curren level of macroeconomic aciviy. By exending he model o include loan defaul and loan risk-weighs as variables depending on macroeconomic aciviy, I inroduce wo oher channels hrough which economic condiions affec banks capial and lending. The firs channel is acive under any capial adequacy regime: empirical evidence suppors he assumpion ha beer macroeconomic condiions reduce he loan defaul rae and hus he loan marginal cos. This implies a greaer variabiliy in loan supply han in he C-C economy, where loans are defaul free. Under Basel II, a macroeconomic shock will also affec he loan risk-weighs in he capial o asse raio. As a resul, he capial consrain migh become eiher igher, if he shock is negaive, or looser, if he shock is posiive. Therefore, if banks face a binding capial consrain, hey will be able o increase heir loan supply in response o beer macroeconomic condiions bu hey migh be forced o reduce supply if he shock is negaive. By conras, in he model wih consan risk-weighs, a shock affecs only he loan rae while leaving he loan supply unchanged when he capial consrain binds. The oher implicaion of risk-sensiive weighs àlabasel II is ha banks do no necessarily 30

31 increase heir capial in response o a posiive macroeconomic shock. Similarly, a deerioraion of macroeconomic condiions does no necessarily induce banks o decrease heir capial. This happens because macroeconomic condiions no only affec loan demand bu also banks capial consrain. Since a posiive shock induces a decrease in he asse risk-weighs, and hus an increase in capial raios, banks do no necessarily need o raise new capial o expand heir loan supply. Analogously, since a negaive shock ighens he capial consrain, a reducion in loan demand does no necessarily induce banks o decrease heir capial. Said differenly, under Basel II banks migh no have he necessiy o mainain he same level of capial during periods of high economic aciviy as under Basel I. For his reason banks migh be more vulnerable o unexpeced negaive shocks. If he economy falls ino a recession or experiences a weakening in is growh, i will be more likely for banks capial consrain o be binding and hus for credi o be raioned. These findings may have implicaions for public policy. Risk-sensiive weighs may lead o a greaer reducion of credi following a negaive macroeconomic shock. No only will loan demand fall during an economic downurn bu banks may be forced o reduce loan supply o saisfy igher capial requiremens. In order o avoid such an evenualiy, supervisors may herefore wan o encourage banks o mainain a capial buffer (during he periods of srong economic growh) above he one banks would choose volunarily. Indeed, such an approach has been favoured by he Basel Commiee and is recommended under Pillar 2 of he new Accord (he Supervisory Review Process). Under he Supervisory Review Process, banks may in fac be required o hold more capial han he regulaory minimum as a proecion agains risks, including he poenial impac of an economic downurn, which are no fully capured by Pillar 1. Alernaively, cyclicaliy in capial requiremens could be addressed under Pillar 1, eg by adjusing capial requiremens in response o changes in macroeconomic condiions, as proposed by Kashyap and Sein (2004) and Gordy and Howells (2004). 31

32 Appendix The bank s opimisaion problem Here I presen he model wih loan losses and asse risk-weighs ha depend on macroeconomic condiions. Banks balance shee ideniy is: L + T = (1 α)d + q 1 s + b (A-1) while banks cash flows are given by (12). The bank maximisaion problem is he following: (25) r L max,d,(q s +1 +b +1 ) v(q 1 s + b, x ) ={π + λ q 1 s + b θw(m )L τ q s +1 + b +1 + E m,1 v(q s +1 + b +1, x +1 ) } (A-2) subjec o: π = r L L + r T T c L L r D (25) Given ha he uses of cash flow are: π = d s D c D D r p 2D (D T )2 δ (M ) L 1 + r b b q s + q s +1 + b +1 (A-3) where 1 + r b b + q s is given a ime, choosing q s +1 + b +1 or dividends d is equivalen. 32

33 L = l 0 l 1 r L + l 2 M + ε L, (A-4) q s +1 + b +1 = (d + q ) s r b b π (A-5) T = (1 α) D + q 1 s + b L (A-6) The firs-order condiions of he maximisaion problem (A-2) are: r L : π L 1 + τ E r L L +λ θw(m ) l 1 = 0 V m,1 df ε L,+1 (q s +1 + b +1 ) (A-7) D : π D L V 1 + τ E m,1 (q s +1 + b +1 ) L df ε L,+1 = 0 (A-8) λ : θw(m )L q 1 s + b (A-9) L V q s +1 + b +1 : τ + E m,1 df ε L,+1 = 0 (q s +1 + b +1 ) L (A-10) When he capial consrain does no bind (ie λ = 0), I can subsiue (A-10) ino (A-7) and (A-8), and rewrie he firs-order condiions as follows: L l 1 r L r T c L r p D T δ (M ) = 0 (A-11) D 33

34 r T (1 α) r D c D + (1 α) r p D T = 0 D (A-12) θw(m )L q 1 s + b < 0 (A-13) From he above condiions and he loan demand (A-4) I obain he opimal bank loan supply and loan rae: L = 1 2 l0 + l 2 M + ε L, l 1 2 c L + r D + c D 1 α + δ (M ) (A-14) r L = 1 1 l0 + l 2 M + ε L, + c L + r D + c D 2l α + δ (M ) (A-15) From (A-11), I obain he following expression for he reasury bonds: T = D + D r p r T r D + c D 1 α (A-16) From he balance-shee ideniy (A-1), I ge he deposis: D = 1 D + L (q 1 s + b ) + 1 α D r p (1 α) r T r D + c D 1 α (A-17) And I finally obain bank profis: 34

35 π = 1 4l 1 l0 2 + l 2 M + ε L, (l1 ) 2 c L + r D + c D c L + r D 2 1 α + δ (M ) + c D 1 1 α + δ (M l 1 ) l0 + l 2 M + ε L, c L + r D 2 2 r T r D 2 + c D + D r T r D + r D + c D 1 α (q 1s + b ) + D 2r p 1 α + c D 1 α + δ (M ) + c D (A-18) 1 α When he oal capial consrain is binding, ie λ > 0, he opimal bank loans are given by he following expression: L = q 1s + b θw(m ) (A-19) By equaing he loan supply and demand, I ge he equilibrium loan rae: r L = 1 l 1 l0 + l 2 M + ε L, q 1 s + b l 1 θw(m ) (A-20) The amoun of reasuries is unaffeced by he binding capial consrain: T = D + D r p r T r D + c D 1 α (A-21) while deposis are now equal o: D = 1 D + L (q 1 s + b ) + 1 α D r p (1 α) r T r D + c D 1 α (A-22) Finally, bank profis are now described by he expression below: 35