Banking regulation and prompt corrective action *

Transcription

1 Bankng regulaton and prompt correctve acton * Xaver Frexas Department of Economcs and Busness, and CEPR, UK Unverstat Pompeu Fabra, Barcelona, Span xaver.frexas@econ.upf.es Bruno M. Parg Department of Economcs Unversty of Padova, Italy, and CESfo, Germany brunomara.parg@unpd.t Ths verson May 29, 2007 Abstract We explore the ratonale for regulatory rules that prohbt banks from developng some of ther natural actvtes when ther captal level s low, as eptomzed by the US Prompt Correctve Acton (PCA). Ths paper s bult on two nsghts. Frst, n a moral hazard settng, captal requrement regulaton may force banks to hold a large fracton of safe assets whch, n turn, may lower ther ncentves to montor rsky assets. Second, agency problems may be more severe n certan asset classes than n others. Taken together, these two deas explan why, surprsngly, captal regulaton, whch may cope wth rsk and adverse selecton, s not enough. Hence, nstead of forcng banks to hold a large fracton of safe assets, prohbtng some types of nvestment and allowng ample scope of nvestment on others may be the only way to preserve ncentves and guarantee fundng. In partcular, provdng ncentves to montor nvestments n the most opaque asset classes may prove to be excessvely costly n terms of the requred captal and thus neffcent. We show that the optmal captal regulaton conssts of a rule that a) allows well captalzed banks to nvest any amount n any rsky assets, b) prohbts banks wth ntermedate levels of captal to nvest n the most opaque rsky assets, and c) prohbts undercaptalzed banks to nvest n any rsky assets. Keywords: bankng, prudental regulaton, moral hazard JEL Classfcatons: E58, G21 * We acknowledge fundng from the Italan Mnstry of Unversty and Research, and ontly from the Italan and Spansh Mnstres of Unversty and Research through a grant Azon Integrate/Accon Integradas We are grateful to Hans Degryse, Jean-Charles Rochet, and the audences at the Unversty of Mlan, and at the CEsfo Appled Mcroeconomcs Conference Munch 2007 for useful comments. The usual dsclamer apples. Correspondng author: Bruno M. Parg, Department of Economcs, Va del Santo 33, Padova, Italy; e-mal: brunomara.parg@unpd.t; tel ; fax

2 1. Introducton The am of ths paper s to understand the logc behnd the US bankng regulaton called Prompt Correctve Acton (PCA). PCA was ntroduced n 1991 by the Federal Depost Insurance Corporaton Improvement Act (FDICIA) n response to the bankng crses of the 1980s to ntegrate captal regulaton wth the man goal to preclude supervsory forbearance (Calem and Rob 1999). The defnton of banks permtted range of actvtes s tradtonally part of bank regulaton. What s orgnal about PCA s that t places mandatory restrctons on bank s actvtes dependng on captal ratos. 1 Banks are classfed n 5 categores dependng on (varous measures of) captal ratos: for example, well captalzed, wth captal rato (total rsk-based captal) 10%; Adequately captalzed 8%; Undercaptalzed < 8%; Sgnfcantly Undercaptalzed < 6%; Crtcally Undercaptalzed 2% of tangble equty. Well captalzed and adequately captalzed banks face no restrctons; banks n the three bottom categores face restrctons whch become more and more severe the lower ther captal ratos. Examples of the restrctons placed on banks actons are: lmts to dvdends payments and compensaton to senor managers; ncreased montorng; restrctons to asset growth; restrctons to nterafflate transactons; requred authorzaton for acqustons and new busness lnes; requred authorzaton to rase addtonal captal; lmts to credt for hghly leveraged transactons; and n the most extreme cases, recevershp. A key aspect of PCA s that t specfes a mx of dscretonary and mandatory provsons for nsttutons n each category rather than relyng only on regulatory dscreton (Benston and Kaufman 1997). In a context where the scope of bankng actvtes has been expandng, wth the repeal of the Depresson-era bankng legslatons (as the Glass-Steagall act) both n the US and n some European countres, banks engagng n a wder range of actvtes, and bankng operatons growng 1 Here we ust sketch the man features of PCA. For a detaled descrpton and dscusson of ts functonng we refer the reader to e.g. Jones and Kng (1995) and Benston and Kaufman (1997). 2

3 n complexty due to fundamental changes n the fnancal ndustry, the ssue of the benefts of a PCA-type of legslaton n countres outsde the US s both tmely and relevant. Introducng regulaton smlar to PCA s complex, as t requres an mportant amendment both to the bankruptcy code and to the law delegatng powers to the regulator. Wth the exstng law ts mplementaton s mpossble n Europe and n the maorty of countres, and t s not part of Basle s Core Prncples for Effectve Bankng Supervson. Yet, the benefts of PCA seem to be mpressve, as ndvdual bank crses are replaced by low cost open bank resoluton. Thus, understandng the cost-beneft analyss of PCA and assessng whether ths pece of regulaton should be exported to other countres has been a key motvaton of our research. Snce there s no model of the mpact of PCA, the obectve of ths paper s to develop a model that allows us to dentfy and measure the effects of PCA. Although ntutvely the benefts of PCA are obvous, ts modellng may be challengng. Indeed, t may seem reasonable, prma face, to restrct a bank s rsky actvtes as ts captal s depleted. But n order to do that the standard captal regulaton wth approprate rsk weghts should suffce. So, the ssue s more nvolved and requres a careful dstncton between rsk and asymmetrc nformaton. The novelty of our analyss stems from the observaton that the complexty of many bank s nvestments per se, ndependently from rsk, s a source of agency problems that cannot be addressed solely by means of quanttatve captal regulaton; nstead a combnaton of qualtatve and quanttatve restrctons succeeds n provdng effcent regulaton. Our approach goes beyond the tradtonal vew of captal regulaton as a buffer aganst losses and hence falure (Dewatrpont and Trole 1994). The tradtonal vew mantans that captal lowers rsk-takng ncentves, and algns ncentves of bank owners wth depostors and other credtors. But other studes argue that captal requrements may have the unntended effect of ncreasng rsk-takng behavor because of the loss of franchse value (Helmann, Murdock, Stgltz 3

4 2000), and the compoundng of moral hazard n effort and rsk choce 2 (Besanko and Kanatas 1996). Calem and Rob (1999) fnd that the amount of rsk a bank takes depends on ts level of captal, wth the most severely undercaptalzed banks takng on maxmal rsk. As captal ncreases banks take less rsk, and as captal rses further banks take more rsk. Calem and Rob (1999) argue that ths U-shaped relatonshp between captal poston and rsk-takng shows that there are lmtatons to the usefulness of captal regulaton to lower nsolvency rsk, and that nstead t provdes a ratonal for the PCA provson of FDICIA. A recent strand of theoretcal and emprcal lterature on the role of captal n banks makes two ponts that are qute mportant for our analyss: frst, market forces exert a promnent nfluence on bank leverage decsons; second, ths lterature challenges the assumpton that banks holds as lttle captal as possble (Flannery and Rangan 2004). In partcular two studes have noted that captal ratos are mportant strategc varables n bank competton. Km et al. (2005) usng a sample of Norwegan banks show that banks can use captal ratos to dfferentate ther servces and soften competton. Allen et al. (2005) have lnked captal regulaton to the competton banks face n the credt market. When banks compete for proects because there s an excess supply of funds relatve to nvestment opportuntes, market dscplne may be so strong to mpose a level of captal hgher than that mposed by the regulator. Flannery and Rangan (2004) document that book captal ratos at the 100 largest U.S. Bank Holdng Companes (BHC) have rased substantally n the perod between 1986 and The average bank has always exceeded the mnmum requred captal rato, and the percentage of constraned BHC dropped to the pont that captal restrctons became effectvely non-bndng for the 100 largest U.S. BHC after They fnd support to the hypothess that large banks captal growth has been a delberate response to market changes makng bank counterpartes more senstve to the default rsk of banks. As a result banks ncreased ther captal ratos to lower ther fundng costs. 2 Although Gordon and Wnton (2003) queston both that the very noton that moral hazard n banks nduces excessve rsk-takng, and the resultng ratonale for captal regulaton. 4

5 The lterature on PCA s manly emprcal. Snce the ntroducton of PCA there have been several attempts to assess ts functonng. It s generally beleved that t has worked well: n partcular PCA has had a sgnfcant mpact both n terms of rasng captal ratos and reducng rsk for banks (e.g. Benston and Kaufmann 1997, Aggarwal and Jaques 2001, Elzalde and Repullo 2006). However, Barth et al. (2004) n a study of bank regulaton and supervson n more than 100 countres rase doubts about government polces that rely excessvely on drect government regulaton and supervson of banks. Two theoretcal papers on PCA are relevant for our work. Both focus on the optmal closure polcy of banks ht by shocks and not on the benefts of restrctng bankng actvtes dependng on ther captal level. In Shm (2004) the banker can dvert profts and affect rsk and return. In a dynamc game between the depost nsurance regulator and the banker t s shown that t s optmal to base the regulaton on the level of captal and to use a stochastc termnaton threat to nduce the banker to exert effort and report ncome truthfully. In Kocherlakota and Shm (2005) loans repayment are enforced only through rsky collateral. The optmal contract among frms, depostors, and taxpayers has bank termnaton or forbearance dependng on the ex ante probablty that taxpayers money s nvolved. 3 All these papers enhance our understandng of optmal captal regulaton of banks, but they do not ustfy the exstence of restrctons to bank actons whch s the man feature of PCA. The obect of our research s precsely to construct a model of bankng regulaton that combnes two types of restrctons on bank nvestments. The frst one corresponds to forcng the bank to hold a mnmum fracton of safe assets (whch s very smlar to captal requrements). The second one mposes restrctons to the types of rsky assets the bank s allowed to nvest n. 3 Modelng optmal captal regulaton n a dynamc context s related to the study of dynamc optmal captal structure. Bas et al. (2006) consder a model of corporate fnance where the manager can dvert ncome, and show that the nfnte repetton of the game between fnancers and manager, where the frm s downszed when t runs out of cash, provdes approprate ncentves to the manager. 5

6 Our paper s bult on two nsghts. Frst, n a moral hazard settng, forcng banks to hold a large fracton of safe assets may lower ther ncentves to exert effort to montor rsky assets;.e. t s necessary to guarantee the banks enough profts to recover montorng costs. Hence nstead of forcng the bank to hold a large fracton of safe assets, prohbtng certan types of nvestment and allowng ample scope of nvestment on others may be the only way to preserve ncentves and guarantee fundng. Second, t s well known that bank assets are generally opaque precsely because banks specalze n lendng to nformaton senstve customers; however, opacty and hence agency problems may be more severe n certan asset classes than n others. Provdng ncentves to montor nvestments n the most opaque asset classes may thus be more costly n the sense that these nvestments can pay out less cash flow and requre more captal. Prohbtng some actvtes could thus allow the bank to have better access to fundng t would have been deprved of because of moral hazard. Our man fndng s that a regulator amng to maxmze the expected value produced by the bankng ndustry should restrct the composton of a bank portfolo accordng to the followng captal rato rule: the regulator should a) allow well captalzed banks to nvest any amount n any rsky assets, b) prohbt banks wth ntermedate levels of captal to nvest n the most opaque rsky assets, and c) prohbt undercaptalzed banks to nvest n any rsky assets. Ths rule captures n a stylzed fashon one of the key aspects of the PCA regulaton adopted n the US, namely the noton that restrctons to bank assets becomes more strngent the less captalzed banks are. Notce that these restrctons to bank choces could not have been replcated wth standard rsk-adusted captal regulaton only, except f, by some concdence, the rskyness of loans and the moral hazard possbltes are perfectly algned. Fnally we use our basc framework to analyze the adustment of the scale of bank operatons to address regulatory concerns. We show that for a gven level of captal the regulator can explot a trade-off between the scope of bank operatons (the types of allowed nvestments) and the scale of bank assets. 6

7 The rest of the paper s organzed as follows. In Secton 2 we set up the basc model of nvestment under moral hazard wth fxed bank sze. In Secton 3 we analyze the functonng of the unregulated bankng ndustry. In Secton 4 we ntroduce PCA and determne the optmal regulaton. In Secton 5 we extend the man result to the varable bank sze, and n Secton 6 we conclude. 2. Model set up We consder a statc model of bank nvestments wth a rsk-neutral bank owner-manager. Bank asset sze s fxed and normalzed to 1. In Secton 5 we consder the general case wth varable bank sze. Assets are funded by bank captal, K < 1 and by unnsured labltes n the form of a loan,1 K from a rsk-neutral perfectly compettve lendng market wth opportunty cost of the funds equal to 1+, where r 0 ndcates the rskless net return. As we wll llustrate below, by rf f focusng on unnsured labltes we allow market dscplne to work most effectvely. We denote wth D the break-even repayment promsed to the lender on a 1 K loan. For modelng purposes we wll consder a newly-created bank that operates wth an exogenous level of captal. The economc ntuton we wll produce wll allow us to analyze the case of an ongong bank wth a standng portfolo of assets and labltes where changes n the level of captal can be thought of as the result of prevous perod s random cash flows. The bank can nvest n several rsky assets and n a rskless asset. The rskless asset returns1+. There are n classes of rsky assets ndexed by= 1,..., n. Unobservable effort n the rf fxed amount e > 0 may be devoted to montor nvestments n any rsky asset class. Montorng effort can be devoted to more than one rsky asset class, but as we wll show later, f the bank montors t wll chose to montor only one rsky asset. Only the bank has the sklls to montor rsky nvestments, whch ustfes fnancal ntermedaton. 7

8 We consder a moral hazard problem smlar to Holmström and Trole (1997). Wth montorng effort e the probablty of success of nvestment n asset class s p, as opposed to p Δ > 0 absent montorng effort, wth Δ > 0. The return X per unt of nvestment n asset class s X > 0 n case of success, and 0 n case of falure for all asset classes. Asset returns are statstcally ndependent. The key feature of ths set up s that some asset classes ental hgher agency problems so that the cash flows that can be pad to outsders are lower. Snce, for reasons that wll be clear later, we do not want that the market can condton fundng on the class of rsky assets the bank chooses, we assume that the returns X are observable but not verfable by the market. Banks can also nvest n the rsk-free asset that requres no montorng. Unversal rsk neutralty allows us to abstract from loan portfolo dversfcaton to concentrate on the basc problem of moral hazard n asset choces. We make a number of assumptons about parameters values. Assumpton 1 (postve expected value): only when montorng takes place the expected return of the nvestments n a rsky assets net of montorng cost exceeds the return from the rsk-free asset;.e. ( ) p X e > 1 + r > p Δ X,. (2.1) f Assumpton 2 (assets rankng by expected value): rsky assets wth hgher ndex have a hgher expected value wth and wthout montorng;.e. a) p X e p X e,..., p X e ( Δ ) ( Δ ) ( Δ ) b) p X p X,..., p X n n n n n (2.2) Furthermore we assume: 8

9 Assumpton 3: (assets rankng by rsk): asset classes wth hgher expected value have a lower probablty of success, wth and wthout montorng;.e. a) p <,..., < p < p, n 2 1 bp ) Δ <,..., < p Δ < p Δ. n n (2.3) The nformaton structure of the fnancng and nvestment game s the followng. The regulator, but not the market, observes both the asset class and the proporton α of rsky assets n bank s portfolo, and can set and enforce restrctons on both. The market nfers both the equlbrum asset choce and the bank s effort decson, and prces debt accordngly. Montorng effort s unobservable to ether the market or the regulator. The tmng of the model s as follows: at t=0 the level of bank captal s exogenously determned and made publc; at t=1 the regulator determnes both the maxmum fracton, and the class of rsky assets allowed for a certan level of captal; at t=2 the market sets promsed repayments, and provdes funds; at t=3 the bank decdes both the class of rsky assets t nvests n, and the montorng effort; at t=4 returns are realzed, and repayment made. 3. Unregulated bankng To begn wth, t s crucal to nvestgate under what condtons a bank can operate wthout any regulatory restrcton. In our model, ths wll be possble provded the bank holds suffcent captal so that t has ncentves to montor the nvestments n rsky assets. As we wll see later on n ths secton only n ths case the market provdes funds to the bank. For the sake of exposton t s convenent to assume that the bank chooses to nvest n, and montor only one asset. Later n ths Secton we wll prove that ths s ndeed the optmal choce. In the absence of regulaton bank s nvestment choce depends only upon the level of captal. Recall that the market nfers both the equlbrum asset choce and the bank s effort 9

10 decson, and prces debt accordngly. Thus market dscplne mples that for each rsky asset class the break-even condton for the perfectly compettve credt market s ( 1 K)( 1 r ) pd ( 1 p )( 1 r )( 1 α ) + = + + (3.1) f f where D s the repayment promsed to the lender n case of success when the fracton of rsky nvestment n asset class s α and the rest s nvested n the rsk-free asset. Observe that from (3.1) D ( 1 K)( 1+ r ) ( 1 p )( 1+ r )( 1 α ) f f = (3.2) p D and, that debt s rsk-senstve, as > 0. α For the moment let us hold fxed the rsky asset class to nvest n. Usng (3.2) the bank s obectve functon wth montorng becomes ( α ( 1 α)( 1 ) ) p X + + r D e (3.3) f f ( 1 )(( 1 α) ( 1 )) p αx + + r K e (3.4) whch s ncreasng nα snce, by assumpton 1, p X > 1+ r. Hence absent regulaton the bank f wll never nvest n the safe asset,.e. the bank would chooseα = 1 for all. Let us now return to the ssue of how many rsky assets the bank wll montor, f t montors, and establsh the followng result. Lemma 1. The bank wll devote effort to montor only one rsky asset, the one wth the hghest expected value. Proof. See the appendx. 10

11 The ntuton s straghtforward. Dsregardng montorng costs, gven rsk neutralty and ndependence of assets returns, the obectve functon of the bank s convex, and a corner soluton s optmal: the bank wll nvest only n the asset wth the hghest expected value. Furthermore, montorng costs ncrease f the bank nvests n more assets. Next, we turn to the exploraton of the banks ncentves to montor ther rsky nvestment. Notce that, regardless of the level of captal, unless the bank has ncentve to montor the rsky nvestments t wll not be funded. Indeed wthout montorng the sum of the expected returns of the bank and of the market s less than the opportunty cost of the nvestment by assumpton 1;.e. ( p )( X D ) ( p ) D 1 r,. Δ + Δ < + (3.5) f Absent regulaton the bank s ncentve to montor an nvestment n asset class s ( ) ( )( ), p X D e p Δ X D (3.6) e or, equvalently, X D. Havng establshed that absent regulaton the bank wll never nvest Δ n the safe asset, and snce for Lemma 1 the bank wll montor only one rsk asset, we can now derve the montorng ncentve constrant as a functon of bank captal. Usng the market clearng condton (3.2) forα = 1equaton (3.6) becomes K ( px ( 1 rf )) pe Δ + 1+ r f (3.7) whch dentfes the montorng captal constrant for an nvestment n rsky assets. Equaton (3.7) mposes a mnmum level of captal as a necessary condton for outsde fundng n asset. Notce that Lemma 1 mples that the bank could not satsfy the montorng ncentve constrant combnng any two rsky assets. Equaton (3.7) can also be expressed as e p X K + r ( 1 )( 1 ) f Δ (3.8) 11

12 e whch has the usual nterpretaton that the expected cash flow payable to outsders p X, Δ should not be smaller than the opportunty cost of outsde funds (Trole 2005). From (3.8) snce, e K<1 t follows that p X > 0,. Δ We now turn to the queston whether unregulated market fnance may arse. For the market not to collapse, a necessary condton s that n equlbrum there s no shrkng. Lemma 1 tells us that, f there s no shrkng, then nvestment and montorng wll occur n asset n. Consequently we have to determne whether the bank has any ncentve to devate from montorng asset n, and shrk n the rsky asset that gves the hghest expected payoff when the promsed repayment s D n. From part b) of assumptons 2 and 3 t follows that the best alternatve to montor asset n s to shrk n asset n. Therefore necessary condton for unregulated market fnance s Recallng that from (3.2) pndn ( 1 K)( 1 rf ) ( p )( X D ) p ( X D ) e. Δ (3.9) n n n n n n n D = + and observng that n < 0, then (3.9) s volated K for hgh e and low K. If we defne K n the value of K such that equaton (3.9) s satsfed wth equalty, then, f a bank has a level of captal K such that for K< K n the market collapses and we cannot have unregulated market fnance. 4. Regulaton wth moral hazard Havng dentfed the necessary condton for unregulated market fnance we now nvestgate how the regulator can mprove welfare f the market collapses. In partcular we ask whether the regulator can aval two regulatory tools, the restrctons on the maxmum amount of nvestment n rsky assets, and the composton of the portfolo of rsky assets,.e. prohbt nvestments n certan rsky assets to ncrease the expected value of the bankng ndustry. Indeed one dmenson of bankng regulaton s the power to nspect banks thus obtanng nformaton that markets do not 12

13 have, and to grant and revoke lcences on the bass of ths nformaton (see e.g. Bhattacharya et al. 2002). Although another mportant dmenson of modern bankng regulaton s depost nsurance here we allow only for unnsured labltes. Our results would not change f we were to consder nsured labltes. If labltes are nsured by a farly-prced depost nsurance scheme ther cost s dentcal to that mposed by a rsk-neutral perfectly compettve lendng market. Under a flat depost nsurance scheme nstead, rents on deposts would be maxmzed by choosng the rskest asset, as shown n Lemma 1 where the nterest rate charged to the bank by the market does not depend on the chosen proect. In the context of our assumptons of fxed sze and fxed captal, settng a captal requrement s equvalent to mposng an nvestment n a safe asset. Snce for Lemma 1 the bank wll not nvest n more than one rsky asset, then the requrement to nvest at least 1 α n the safe asset s logcally dentcal to a standard captal regulaton mandatng a mnmum captal rato. In fact usng the break-even condton (3.2) the bank obectve functon (3.3) becomes (3.4) where t s mmedate to observe that the fractons nvested n the safe asset and n captal are perfect substtute. Gven α, for each asset class the banker s ncentve to montor an nvestment n the rsky asset class s ( α ( 1 α)( 1 ) ) ( ) α ( 1 α)( 1 ) ( ) p X + + r D e p Δ X + + r D (4.1) f f whch becomes e αx + ( 1 α)( 1 + rf ) + D. Δ (4.2) Snce for Lemma 1 the bank wll montor only one rsky asset we can now derve the captal constrant when the bank may face restrctons on the fractonα of a rsky asset. Usng the break even condton (3.2) for the perfectly-compettve credt market, equaton (4.2) becomes 13

14 αx ( 1 α)( 1 rf ) Δ e ( 1 K)( 1+ r ) ( 1 p )( 1+ r )( 1 α ) f f p (4.3) or K K pe Δ ( px ( 1 rf )) α + 1+ r f. (4.4) The RHS of (4.4), K, ndcates the mnmum level of captal, for each valueα 0, that a bank must own n order to have the ncentve to montor an nvestment n asset class. Part b) of assumptons 2 and 3 guarantee that f (4.4) s satsfed the bank has no ncentve to shrk n an asset wth an ndex <. To guarantee that such a mnmum level of captal s postve, or, n other words, that t s not possble to fnance a proect exclusvely through external debt, we make the followng assumpton. Assumpton 4 (postve captal): For every asset, the expected cash flow that can be pad to e outsders p X Δ s smaller than the opportunty cost of funds, ( 1 rf ) + ;.e. e 1 + rf > p X,. Δ Ths leads us to our frst result. Proposton 1. For any rsky assets =1, n-1 f a bank has a level of captal such that K< K n (.e. unregulated market fnance s mpossble) then ether the parameters are such that equaton (4.4) s satsfed for α = 1, that s there are no restrctons on the fracton of assets that can be devoted to rsky nvestments, or there s no α 0 that can satsfy (4.4). In the latter case, the only way to 14

15 guarantee that the market funds the bank s for the regulator to setα = 0, that s to prohbt any nvestment n asset class. The proof s obvous gven assumptons 1 and 4 that guarantee that K s a decreasng functon of α. The ntuton for Proposton 1 s that by forcng the bank to nvest a large fracton n the safe asset the regulator lowers the bank s expected proft and thus ts ncentve to montor the rsky asset. Snce the lower the level of captal the hgher s the promsed debt repayment, then there are parameter constellatons such that t s mpossble to provde ncentve to montor the nvestment n rsky asset gven the amount of captal. In such cases, captal regulaton s powerless and prohbton of any nvestment n that asset class s the only alternatve to have the bank funded. 4 It s well known (Trole 2005) that n ths type of models there are two sources of agency problems: the lkelhood rato Δ and the effort level e. The two sources of agency problems yeld p a maxmum level of expected cash flow from nvestment n rsky asset that can be pad to e outsders equal to p X Δ. Our paper makes the pont that agency problems may be more severe n certan asset classes than n other so that t may be more costly to provde ncentves to montor nvestments n those asset classes. More nnovatve nvestments, for example n dervatves, brdge loans for M&A, propretary equty tradng, hedge fund fnancng, may be more opaque, and therefore leave more scope for manageral dscreton, than more tradtonal credt operatons, thus requrng more costly ncentves. To reflect ths dea n a straghtforward way, we make the followng assumpton that lnks the expected value from nvestng n a rsky asset to the cash flow that can be pad to outsders once we account for the cost of provdng manageral ncentves. 4 Proposton 1 s related to the credt ratonng results of Aghon and Bolton (1997) and Pketty (1997) n growth models wth moral hazard where ndvduals have heterogeneous wealth endowments. 15

16 Assumpton 5 (negatve correlaton between expected values and expected pledgeable cash flows): asset classes wth hgher expected value have lower expected pledgeable cash flow;.e. e e e p1 X1 > p2 X2 >,..., > p n X n. Δ1 Δ2 Δn (4.5) Thus asset class 1 leaves more expected cash flow payable to outsders than asset class 2, and so on. Fgure 1 provdes an example of the decreasng relatonshp between expected pay-offs and pledgeable cash-flows. [Insert Fgure 1 about here] Assumpton 5 deserves some comments. Frst, ts motvaton stems from the desre to focus on a smplfed case where a maor tenson exsts between net present value maxmzaton and moral hazard. Alternatve assumptons, whle leadng to the same qualtatve results regardng the benefts of regulatory restrctons to complement captal ratos, would make the analyss more cumbersome and would ntroduce addtonal complextes that would make the ntuton of our result less clear. Ths tenson between net present value and moral hazard s requred n order to pont out why prohbtng some actvty could be effcent. Second, notce that the rankng s a characterstc of the assets n the whole economy and not of the asset portfolo of a partcular bank. Before ntroducng our man result on optmal bank regulaton we have to establsh the bank optmal choce of an asset class assumng the bank always montors. Ths wll be the one wth the hghest expected value among those that satsfy the montorng constrant. Lemma 2. Assume there s a subset of asset classes {1,,} n whch montorng takes place,.e. equaton (4.4) s satsfed for α = 1. The bank prefers to any proect k such that k<. Proof. See the appendx. 16

17 Recall that K n (4.4) defnes the montorng threshold for nvestment n asset class and that, for α=1, the negatve correlaton assumpton mples an orderng of the montorng captal threshold such that K 1 <,,<K <,,< K n and Assumpton 4 on mnmum captal mples K 1 >0. We are now n a poston to establsh our man result about optmal bank captal regulaton. Proposton 2. The optmal captal regulaton s characterzed as follows. 1) For banks wth a level of captal K such that K Kn t s optmal to set no restrctons ether on the type of rsky nvestment that the bank s allowed to undertake or on the percentage of nvestment n rsky asset classes, that s t s optmal to set α = 1. 2) For banks wth a level of captal K such that K + 1 > K K for = 1,,n-1 t s optmal to prohbt nvestments n all rsky asset classes wth ndex > and allow the bank to nvest all( α = 1) n rsky asset classes wth ndex (see Fgure 2). 3) For banks wth a level of captal K such that 0 < K < K1 t s optmal to prohbt nvestments n all rsky asset classes and allow nvestment n the rsk-free asset only, that s t s optmal to setα = 0. Proof. See the appendx. Allowed asset classes Prohbted asset classes 1 +1 n Fgure 2. Optmal regulaton for banks wth captal K +1 >K K, =1,,n-1 17

18 The frst part of Proposton 2 says that f a bank has a suffcently hgh level of captal such that the ncentve to montor s preserved even for the nvestments n the most opaque asset classes, then regulaton should not restrct ts choces and the market wll fund t n any way. The second part of Proposton 2 says that for banks wth an ntermedate level of captal the only way to guarantee that the bank has ncentves to montor, and thus t s funded by the market, s to prohbt nvestments n the most opaque asset classes even f those nvestments have the hghest expected value. The thrd part of Proposton 2 says that for banks wth low level of captal the only way to guarantee that the market funds the bank s to prohbt nvestments n rsky assets altogether. Several comments are n order. Frst, ths result captures n a stylzed way one of the man messages of the PCA regulaton, namely that the lower the bank captal rato the fewer are the types of nvestments that the bank s allowed to undertake. Notce that these are both qualtatve and quanttatve restrctons on bank s actons that acheve results (bank fundng) that could not have been acheved wth tradtonal captal regulaton alone as shown n Proposton 1. 5 Second, snce the prohbton of certan types of nvestments allows a reducton of the level of captal to satsfy the montorng constrant, then ths set of rules effectvely economzes on captal for a gven bank sze or, alternatvely, t allows ncreasng the sze of the bankng sector gven the captal. In the followng Secton we wll consder the case where bank sze s varable and we wll show that sze can be ncreased through the prohbton of the nvestment n some rsky assets. Thrd, as mentoned, the role of the negatve correlaton assumpton (assumpton 5) n proposton 2 s merely to smplfy the analyss and provde a clear cut case. A qualtatve smlar, but less clear cut, result would arse also f the expected values and the expected cash flow payable to outsders were not negatvely related. One would need to construct a set of rsky assets such that the ones preferred by the bank are not necessarly those wth the hghest cash flow payable to outsders. 5 The result that qualtatve restrctons on the set of allowed nvestments may be optmal s smlar to the prohbton of certan dffcult-to-observe tasks n the mult-task prncpal-agent model of Holmström and Mlgrom (1991). 18

19 As noted, one of the ratonales of the PCA regulaton was to mandate some nterventons for undercaptalzed banks rather than relyng on regulatory dscreton only. Although n ths paper we do not address the ssue of the costs and benefts of havng rules rather than relyng on regulatory dscreton, the rule we have derved n Proposton 2 can be nterpreted as the optmal contract between the regulator and the bank. The bank would fnd t optmal to subscrbe to that contract because havng ts hands ted,.e. beng n the mpossblty to nvest n some moral hazard senstve proects would allow t to have access to fnance. In fact n ths model commtment to that contract would domnate an ad hoc method of bank crss resoluton; absent unforeseen contngences that mght make dscreton preferable wth respect to rgd rules, dscreton would only lead to forbearance wth some postve probablty; antcpatng ths possble lack of dscplne the market would (weakly) provde less funds. It s mportant to nterpret our results n the context of the lterature on captal regulaton. Frst, notce that n ths model we do not exclude a pror tradtonal captal regulaton. The fact that the optmal value of α s 1 s the result of the need to satsfy an ncentve constrant whch could not be done wth quanttatve restrctons on captal alone. Second, observe that n ths model the tradtonal role of captal regulaton mantanng stablty and solvng depostors collectve acton problems s not operatonal because lablty holders subect fnancal ntermedares to perfect market dscplne as loans are farly prced. However, because of lack of nformaton, lablty holders are not capable to wrte contngent contracts wth bankers as to the asset choces. Hence PCA regulaton wth qualtatve asset prohbtons s needed to ntegrate quanttatve captal regulaton when the opacty of bank assets s an ssue. 5. Varable bank sze Up to now we have consdered a fxed captal and fxed sze framework, so that captal regulaton would mply that banks had to nvest a fracton of ther assets n the rskless asset. In fact, the bank could comply wth captal regulaton by contractng the scale of ther operatons. But scale would 19

20 be observed by the market and therefore the bank mght be subect to market dscplne. Consequently, consderng the possblty of a varable bank sze s qute relevant for our analyss, as t allows examnng the possble trade-offs between scale and scope of banks. For ths reason we now turn to the case of varable asset sze and the bank has to decde the scale of ts overall actvtes as well as the composton of the portfolo of rsky assets. Assume that asset sze I s such that I 0, I, I beng the maxmum capacty. Ths ntermedate assumpton between fxed asset sze and completely varable sze captures the dea that after a certan sze returns are sharply decreasng. Asset sze I s funded by captal K and by farly prced loans, I-K. Producton has constant returns to scale: wth success the return s IX for all, wth falure 0. Effort s assumed proportonal to asset sze so that ei s the effort to montor any rsky asset of sze I. Probablty of success wth and wthout montorng are as n the fxed sze case. Snce bank sze s determned together wth funds request and thus, as n the prevous sectons, t s gven at the stage of asset choce, then Lemma 1 apples,.e. t wll montor only one rsky asset f t montors. The tmng and the remanng assumptons are as n the fxed sze case. The break-even constrant for the rsk-neutral compettve lender becomes and the bank obectve functon s Usng (4.6) equaton (4.7) becomes ( )( 1 ) ( 1 )( 1 )( 1 α ) I K + r = pd + p + r I (4.6) f f ( α ( α)( ) ) p XI r I D ei. (4.7) f ( α ( 1 α)( 1 )) ( 1 )( ) f f I p X + + r ei + r I K (4.8) 20

21 whch s ncreasng n α snce p X > 1+ r. Thus agan gven the amount borrowed I-K the bank f wll not nvest n the safe asset unless forced to do so. The montorng ncentve constrant becomes 6 ( px ( 1 rf )) pe α + Δ 1 K I = I 1+ rf m (4.9) where the multpler m s m 1+ rf pe α + Δ ( px ( 1 rf )). (4.10) Notce that once bank sze s determned, banker s decson follows the same logc of the m fxed asset sze. For all asset classes, > 0, because of assumpton 1. Therefore, gven asset α sze and asset class, ether the montorng constrant (4.9) s satsfed for α = 1 or there s no value α > 0 that can satsfy t, a result smlar to Proposton 1. Thus consderng the value of (4.10) whenα = 1, the nvestment multpler becomes: m ( 1+ rf ) > 1 e ( 1+ rf ) p X Δ (4.11) e because of the mantaned assumpton 4, that s1 + rf > p X,. Δ Notce that there are two ways to satsfy the montorng constrant: ncrease the multpler or lower bank sze. Under the negatve correlaton assumpton (assumpton 5) t follows that m 1 >,,> m n whch mples that the multpler can be ncreased by prohbtng nvestments n the most opaque asset classes. Defne the varable I Km, as the maxmum sze of asset gven the level of captal K, compatble wth montorng ncentves. Absent regulaton, unless the bank has enough 6 The dervaton of the expresson (4.9) follows the same logc of the equvalent constrant n Secton 4. For a very smlar expresson of the borrowng capacty see Trole (2005) p

22 captal such that I n I, then n order to get funded t wll lower ts sze to the pont that the montorng constrant s satsfed. Thus for the banks wth captal such that In < I a regulaton prohbtng nvestment n the most opaque assets allows to ncrease bank sze. Ths leads us to the followng result. Proposton 3: 1) If the level of captal s such that I n I then placng restrctons on bank asset choce would not ncrease fundng; 2) If nstead the level of captal s such that I n < I a regulaton prohbtng nvestment n the most opaque asset classes can ncrease the multpler and thus fundng. The proof follows n a straghtforward way from the assumpton that I 0, I, and the defnton of I. In case 1, the bank s suffcently captalzed so that even the nvestment n the most opaque asset n satsfes the montorng constrant for the relevant range of nvestment opportuntes; no restrcton on asset choces would mprove welfare. The low margnal returns after reachng the maxmum sze (or decreasng returns zone) are suffcent for the bank to choose a sze for whch t has ncentves to montor. In case 2, despte the market provdng some fundng, sze s lmted by the scarcty of captal and by the opacty of bank assets, so that, absent regulaton, only partal fundng to explot nvestment opportuntes can be obtaned. Case 2 llustrates the fact that the constrant (4.9) can be satsfed n two ways: ether ncreasng the multpler (.e. prohbtng nvestment n certan asset classes) or by lowerng bank sze. Thus the regulator faces a trade-off between allowng nvestments n many asset classes (low multpler) and allowng banks to pursue large levels of overall nvestment. 22

23 The man mplcaton of case 2 s that snce the welfare functon s the net aggregate expected present value of bankng nvestment, the regulator has to consder both the expected return and the sze of the nvestment. Sze depends on the moral hazard dmenson va the multpler m. Thus regulaton affectng the multpler m effectvely determnes the sze of the overall bank nvestments. Under our assumpton of negatve correlaton the hgher the ndex of the asset class the lower the maxmum sze of the nvestment for a gven captal compatble wth montorng ncentves. Thus as we move from asset 1 to n, we go from a combnaton of hgh expected returnlow sze to a combnaton of low expected return-hgh sze. The optmal choce depends on the characterstcs of the rsky assets and n partcular on the relatonshp between expected value and expected pledgeable cash flow. The man dfference wth respect to the fxed-sze result of Proposton 2 s that banks wth lttle captal can stll nvest n opaque assets albet at a small scale. Furthermore, the noton that the ncentve constrant can be satsfed also by allowng the bank to nvest n opaque assets thus effectvely lowerng the overall bank sze can be nterpreted as downszng a bank to force t to satsfy a certan captal requrement. Fnally, the ncentve constrant (4.9) can also be seen as a way to mnmze the amount of captal needed to satsfy the montorng constrant to nvest n rsky assets of gven sze. Hence our framework can be appled to study the ssue of recaptalzaton of undercaptalzed banks wth a gven portfolo of loans. 6. Concluson We have developed a smple framework to study nvestment choces under moral hazard that can be appled to a varety of general corporate fnance decson processes. What makes t partcularly sutable for the analyss of bank captal regulaton s the regulator s advantage, from ts supervsory and lcensng role, n gatherng nformaton about bank portfolos and enforcng portfolo restrctons. Our model shows that the logc behnd PCA regulaton s well rooted n the mcroeconomc analyss of banks ncentves. However, our paper has not attempted to provde any analyss for one 23

24 of the other ratonales behnd the adopton of PCA n the US, namely the desre to avod forbearance by tghtenng the regulator s hands and requrng some mandatory actons as a functon of captal ratos. In other countres, n partcular n the European Monetary Unon, bank captal regulaton at the natonal level often follows both the tradtonal quanttatve captal regulaton based on rsk-weghted captal ratos, and a mx of moral suason and ad hoc resoluton of bank crses. A polcy mplcaton of our paper s that a pece of regulaton smlar to PCA should also be adopted outsde the US, partcularly n countres where a dscretonary approach to bank crss resoluton may lead to regulatory capture by the ndustry. Ths s partcularly true n countres wth weak nsttutonal envronments where, as Bart. et al. (2006) argue, gvng strong dscretonary powers to bank supervsors may actually make matters worse. 24

25 Appendx Proof of Lemma 1. We wll frst show that the bank wll choose to mplement only one proect, usng a convexty argument, and then we wll prove that the selected proect s the one wth the hghest net present value. Recall that at t=2 the market sets a repayment D as a functon of the captal and the equlbrum behavor of the banker. Let λ be a nx1 vector whose components λ, 0 λ 1, are the n proportons of nvestment n rsky assets, wth λ = 1. Denote wth λ * the equlbrum vector of proportons chosen by the bank. We ntroduce the followng defnton: proect s sad to belong to the set S f ts realzed return X s such that X=X ; proect s sad to belong to the set S f ts realzed return X s such that X=0, where S s the complement set of S. Let us denote wth Z ( S ) the set of all subsets of S (ncludng the empty set). Snce the returns of the nvestments n the rsky assets are statstcally ndependent the expected value of bank profts gross of montorng cost can be expressed as At t=3 the bank wll choose λ to maxmze where 1( λ 0) Φ S Z S S S ( λ) p ( 1 p) max λx D;0. (6.1) n ( λ) e 1( λ 0) Φ > (6.2) > s a vector whose -th component takes value 1 when λ > 0, and 0 otherwse. Notce that D s a functon of the equlbrum vector λ * and not of λ. We wll show that Φ ( λ ) s convex, that s, that for every vector So t must be the case that 1 λ and 2 λ t must be true that ( αλ 1 ( α) λ 2 ) α ( λ 1 ) ( α) ( λ 2 ) α [ ] Φ + 1 Φ + 1 Φ 0,1. (6.3) 25

26 S Z S 1 2 p ( 1 p) max ( αλ + ( 1 α ) λ ) X D;0 S S ( ) ( ) 1 2 p 1 p αmax λx D;0 + 1 α max λ X D;0. S Z S S S S (6.4) 1 2 ( α λ ) + X D then the LHS of (6.4) s equal to 0 and (6.4) S ( α λ ) X > D then S Two cases are possble: f αλ ( 1 ) s satsfed. If αλ ( 1 ) S Z S Z S S 1 2 p ( 1 p) max ( αλ + ( 1 α ) λ ) X D;0 = S S 1 2 p ( 1 p) ( αλ + ( 1 α ) λ ) X D = S S p p α λ X + α λ X D ( 1 ) ( 1 ) 1 2 S Z S S S S S Z 1 2 p ( 1 p) αmax λx D ( α) λ S S Ths shows that Φ ( λ ) s convex. ;0 + 1 max X D;0. S S (6.5) n It s standard to show that the maxmzaton of the convex functon Φ ( λ ) over the hyperplane λ = 1 yelds a corner soluton. From ( αλ ( 1 α) λ ) α ( λ ) ( 1 α) ( λ ) Φ + Φ + Φ { ( ) ( )} ( ) ( ) ( ) { } { ( ) ( )} α max Φ λ, Φ λ + 1 α max Φ λ, Φ λ max Φ λ, Φ λ (6.6) by quasconvexty. Recall that we denote wth λ * the vector that maxmzes (6.2). Assume that * ( ) λ 0,0,...,1,...,0 δk (6.7) 26

27 where δk s the canoncal base where the k-th element s 1, so that * n λ = 1 = αδ and let { { }} J = = 1, 2,..., n, α > 0 follows that * Snce λ s optmal then * So that Φ ( λ ) = Φ ( δ ) max. k J. Because the δ k are n the feasble set, then by quasconvexty t k * ( λ ) k J ( δk) Φ max Φ. (6.8) * ( λ ) k J ( δk) Φ max Φ. (6.9) Snce the ntroducton of effort s cost benefts a lower number of rsky asset classes montored, the optmal choce condtonal on montorng s to choose only one rsky asset. We now show that ths wll be the n-th asset. Ths s the case snce D s ndependent of k. Notce frst that Φ ( δ ) = p ( X D) for any D such that p D ( r )( K) n n n * = 1+ f 1 where * s the asset chosen by the bank. For any such that Φ ( δ ) = 0asset n s obvously preferred. For such that ( δ ) 0 Φ >, combnng assumpton 1 and 3 we have pnxn px and pnd pd. Consequently ( δ ) ( δ ), 1,2,..., Φ n Φ = n, wth equalty only for =n. Proof of Lemma 2. Recall that by assumpton 2 we have that for k<, p X pkxk, and by assumpton 3 p k > p. Hence and the result follows. ( ) ( ) p X D e> p X D e (6.10) k k 27

28 Proof of Proposton 2. Proposton 1 allows us to set α = 1 and focus on the allowed rsky nvestments. Recall that the negatve correlaton assumpton (Assumpton 5) mples an orderng of the montorng captal threshold, such that K 1 <,,< K <,,< K n and that Assumpton 4 on mnmum captal mples that K 1 >0. To prove part 1. For a level of captal K K n equaton (3.9) on market dscplne s satsfed. Because of the negatve correlaton assumpton we have K > K, =1, n-1. Hence no shrkng wll occur. Thus bank s choce s among n montored nvestments. Lemma 2 states that asset class n s chosen. To prove part 2. Because of the negatve correlaton assumpton equaton (4.4) s not satsfed for asset classes +1,,n whle t s satsfed for asset classes 1,,. Part b) of assumptons 2 and 3 guarantee that f (4.4) s satsfed the bank has no ncentve to shrk n asset wth an ndex <. Thus the bank wll shrk for all asset classes +1 to n and wll montor nvestments n asset classes 1,,. Usng Lemma 2 we know that s the bank s preferred choce among montored assets. Recall that from assumpton 1 all non-montored nvestments n rsky assets have negatve expected value. Thus the best acton for the regulator s to prohbt all assets +1,,n. To prove part 3. For 0 < K <K 1, the set of montored nvestments n rsky asset classes s empty. Thus the best acton for the regulator s to force the bank to nvest n the safe asset only. Ths mples forbddng all rsky assets and settng α = 0. 28

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32 Expected value p n X n -e p 1 X 1 -e p n (X n -e/δ n ) p 1 (X 1 -e/δ 1 ) Expected pledgeable cash flow Fgure 1: Expected values and expected pledgeable cash flows: an example. 32