Macroeconomic Implications of Maturity Transformation

Transcription

1 Macroeconomic Implicaions of Mauriy Transformaion Xiaobei He Sep 213 Absrac In his paper I develop a DSGE model in which he banking secor engages in boh monioring and mauriy ransformaion. Mauriy mismach on he bank balance shee, namely, long-erm asses financed wih shor-erm liabiliies, has been widely documened empirically, bu has so far been ignored in macroeconomic models. Macroeconomic models ypically assume ineres raes on loans are rese each period, bu in realiy many financial conracs are long-erm conracs wih fixed ineres raes. My simulaion resuls show ha he incorporaion of mauriy mismach alers he ransmission channels beween he financial and real secors. Unlike much of he exising lieraure, in his model he bank ne worh funcions as a shock absorber and dampens he business cycles. Specifically, he responses of he real secor o shocks are aenuaed, while he responses of he financial secor are magnified. These resuls sugges ha by omiing long erm loans previous sudies overesimae he amplificaion effecs of financial fricions and he effeciveness of moneary policy, and herefore he implicaions of financial fricions for business cycles and moneary policy should be revisied. Keywords: mauriy mismach, DSGE, financial fricions, balance shee channel, ransmission mechanism JEL codes: E22, E32, E44, G21 Goehe Universiy Frankfur ( Xiaobei.He@hof.uni-frankfur.de). I would like o hank Marin Andreasen, Eser Faia, Ansgar Rannenberg, Mirko Wiederhol, James Yeman for discussion and helpful commens. I would like o give special hanks o Cirad Slavík for his deailed discussions and valuable commens. I also would like o hank Séphane Moyen for his encouragemen and guidance. I am deeply graeful o Thomas Laubach, my supervisor, for his consan suppor, grea paience, and invaluable advice. All errors are mine.

2 2 1 Inroducion Recen macroeconomics developmens have underlined he imporance of credi marke imperfecions in business cycle dynamics. Financial fricions are considered o be one of he prime deerminans of macroeconomic flucuaions, and more imporanly, he criical propagaors and amplifiers of shocks. The fricions in he credi marke, which ofen include he asymmeric informaion and moral hazard problems in financial conracs, have been widely employed in dynamic general equilibrium models o reproduce he disrupion in financial markes and large business cycle flucuaions as observed in he daa. Alhough a wide range of financial facors have been accouned for in he exising sudies, such as he financial acceleraor channel and he credi supply channel, lieraure has remained inconclusive on he dynamic effecs of financial fricions on he real aciviies. In his paper I will argue ha a financial facor ha has been omied in macroeconomic sudies, i.e., mauriy mismach on banks balance shees, also plays an influenial role in shaping he business cycles hrough an alernaive ransmission channel. To complemen he previous sudies on he macroeconomic consequences of financial fricions, I develop a real business cycle model accouning for he mauriy mismach and assess he quaniaive imporance of his feaure in business cycle dynamics. Figure 1: Average Mauriy of Newly-iniiaed Loans Figure 1 shows he average mauriy of newly-iniiaed loans for corporae purposes 1. Empirical evidences 2 indicae ha he average mauriy of oal bank asses are four imes he mauriy of heir liabiliies. Mos macroeconomic models, however, are buil under he conjecure ha boh loans and deposis have a one-period mauriy (one period ofen refers o one quarer). The assumpion ha all ineres raes of loans are rese each period fails o agree wih he fac ha in pracice many loan conracs are fixed rae conracs. Figure Thomson Reuers LPC s DealScan. 2 Documened by GSünner (21) 3 Flow of Funds and NIPA.

3 3 plos he average ineres expenses of non-financial corporae secor agains he spo ineres raes, i.e., he prime rae (loans) and he BAA corporae bond yield (bonds). If all he ineres raes on loans were rese each period as DSGE models assume, he adjusmens of average ineres expenses would rack he movemens of he prime rae and he BAA yield closely. However, in realiy he average borrowing cos, represened by he average ineres expenses of he non-financial corporae secor, follows a much smooher paern. Have misinerpreed he average borrowing cos, macroeconomic models wih only one-period loans end o overesimae he flucuaions of he real secor s borrowing cos. As will be shown in his paper, a model accouning for he sluggish borrowing cos predics a disinc ransmission channel ha offses he effecs of financial fricions. Unlike he exising DSGE models which emphasize he role of he financial secor as a shock amplifier, his model also highlighs he role of he bank ne worh as a shock absorber. To his end, he model in his paper is specifically ailored o capure he feaure of mauriy mismach in he banking secor. Figure 2: Fixed-rae Loans Mauriy mismach is desirable because more profiable projecs frequenly require a longer invesmen horizon bu invesors liquidiy needs are uncerain. The bank finances he longer erm invesmen by pooling he demand deposis, hereby providing "ransformaion" service. As an imporan feaure of he banking secor, mauriy mismach has no obained enough aenion from macroeconomiss. In his paper I will show he imporance of mauriy mismach on economic dynamics by comparing a full model wih mauriy mismach o a baseline model wihou i. The baseline model shares many similariies wih much of he available lieraure; he full model builds on op of he baseline model wih he incorporaion of a non-rivial srucure of mauriy mismach. The srucure of he baseline model is oulined as follows. The credi demand of he real secor is modeled in a very similar manner of Bernanke, Gerler and Gilchris (1999) (henceforh BGG). The banker and he enrepreneur

4 4 engage in a financial conrac, in which he conracual loan rae is ied o he enrepreneur s financial condiion (i.e., he leverage raio); his is he so-called financial acceleraor channel. In addiion o his, a (consrained) opimizing banking secor is embedded so as o model he bank lending and pricing behavior. The role of bank ne worh is moivaed by considering a moral hazard problem beween bankers and deposiors, borrowed from Gerler and Karadi (21) (henceforh GK). In he presence of he moral hazard problem he size of he bank balance shee is endogenously consrained by he level of bank ne worh. The bank balance shee channel and he perceived defaul risk joinly deermine he loan rae and he credi supply. The baseline model oulined feaures he inerplay beween he credi demand and he credi supply. As has been widely documened in his srand of lieraure, credi marke fricions affec real aciviies by magnifying he impac of shocks including moneary policy shocks. For example, Chrisensen e al. (27) show ha financial fricions boos he response of oupu afer an increase in policy raes by abou a hird. Bernanke e al. (1999) and Gerler and Karadi (21), among many ohers, also have similar predicions on he quaniaive effec of he credi channel on he ransmission of shocks. Alhough he credi channel heory grealy improves he abiliy of DSGE models o mach empirical evidences, I will argue wih he following model ha his resul is subsanially weakened in a seing in which banks are engaging in mauriy ransformaion. The demand for long-erm loans is moivaed by he sochasic duraion of invesmen projecs. A projec faces a consan probabiliy o erminae in each period, and upon erminaion, he enrepreneur refinances by engaging in a new loan conrac. The consan expeced loan mauriy across enrepreneurs and across ime grealy simplifies he aggregaion. Given he presence of long-erm loans, banks face mauriy mismach on he balance shees: on he asse side of he balance shee are muli-period loans wih fixed ineres raes, while on he liabiliy side here are he demand deposis. 4 The sluggish adjusmens of loan raes consiue a ransmission channel beween he enrepreneurial and bank balance shees ha has been under-explored in lieraure. To show business cycle implicaions I simulae he model o various shocks. Impulse response funcions sugges ha he presence of mauriy mismach on he bank balance shee aenuaes he responses of he real secor in face of shocks a he cos of amplifying he flucuaions of financial variables, such as he bank leverage raio and reained earnings. Unlike much of he exising lieraure, he bank ne worh in his model acs as a shock absorber and dampens he business cycle flucuaions. The inuiion behind hese resuls is simple: he real secor is parially immunized from he flucuaions of ineres raes due o fixed-rae loans, while he banking secor is exposed o ineres rae risks because of he saggered ineres rae adjusmens. Therefore bank ne worh funcions as a buffer and couneracs he amplificaion effec caused by financial fricions. 4 I is worh noing ha in principle long-erm loans don necessarily imply mauriy mismach; in his paper he underlying assumpion of deposis being one-period allows me o use long-erm loans and mauriy mismach inerchangeably.

5 5 These resuls sugges ha previous sudies wih exclusively one-period loans overesimae he amplificaion effec of financial fricions. In realiy, in he presence of mauriy mismach he amplificaion mechanism of financial fricions are largely mued. An exercise comparing he modeled economy wih he efficien economy indicaes ha a cerain degree of mauriy mismach on banks balance shees help he economy realize is full poenial. This paper is relaed o he banking lieraure on mauriy mismach. In fac, mauriy mismach as a core aciviy of financial inermediaries has been sudied exensively in microeconomic lieraure, beginning wih he seminal work of Diamond and Dybvig (1983). Allen and Gale (27), Roche and Vives (24) and Holmsrom and Tirole (1995, 1994), among many ohers, have conribued o his srand of lieraure. However, all he models menioned above are parial equilibrium models wih a wo or hree period horizon, and hence hey do no allow for sudying he role of mauriy mismach in business cycle flucuaions. This paper is closely relaed o he rapidly growing lieraure on DSGE models wih financial fricions. The seminal work of Bernanke Gerler and Gilchris (1999) sudies he role of he borrower s ne worh in a New Keynesian framework. The model generaes a counercyclical exernal finance premium, which is he so-called financial acceleraor channel. The banking secor in his model serves as a monior and diversifies invesmen risks for he invesors. Gerler and Kiyoaki (21) and Gerler and Karadi (21) insead focus on he credi supply. They inroduce a moral hazard problem beween he financial secor and households, and consequenly he credi supply is endogenously consrained by he banks ne worh. Rannenberg (211) incorporaes BGG framework ino he GK model in order o sudy he ineracion beween he credi demand and he credi supply. He finds ha wih boh a financial acceleraor and he bank balance shee channel he model ouperforms BGG in erms of maching he daa. Meh and Moran (21) shows ha he bank ne worh propagaes and magnifies he shocks in a model wih double moral hazard problems. However, he models menioned above consider one-period loans exclusively and only idenify he amplificaion channel hrough which financial fricions magnify shocks. Specifically, Gerler and Karadi (21) and Meh and Moran (21) boh highligh he role of he bank ne worh as a shock amplifier. My model complemens he exising sudies and and underlines he role of he bank ne worh as a buffer and shock aenuaor, which cass some doubs on he quaniaive resuls in he available lieraure. Andreasen, Ferman and Zabczyk (212) (henceforh AFZ), o my knowledge, is he firs aemp o incorporae he mauriy mismach in a DSGE framework. In heir model he demand for long-erm loans is moivaed by firms infrequen capial adjusmen. The model predics an aenuaing effec on all he aggregae variables such as oupu and invesmen, bu lile implicaion for he banking secor can be drawn in heir model because of he lack of imporan financial fricions. My model shares many similariies wih AFZ, bu by furher exploiing he credi channel my model arrives a a raher differen resul and

6 6 delivers new messages for policymakers. Insead of an ad hoc long-erm loan conrac like in AFZ, he long-erm financial conrac beween enrepreneurs and bankers in my model is buil on credi fricions, aking he defaul risks of enrepreneurs ino accoun. By allowing for he enrepreneurs defaul, my model no only capures he ransmission of sicky loan raes on he bank balance shee, bu also he feedback effec driven by he perceived defaul risk. This feaure has been proven o be very imporan in deermining he banks lending behaviors, and hence he main finds of my model differ from AFZ s in a key respec: AFZ predic a credi aenuaor effec of mauriy mismach on he economic flucuaions; My model, while providing similar predicions on he flucuaions of real aciviies, suggess ha financial flucuaions are magnified in he presence of mauriy mismach. The paper is srucured as follows: Secion 1 is an inroducion and moivaion; Secion 2 discusses he model in deail; Secion 3 provides he simulaion resuls and inspecs he mechanism; finally Secion 4 concludes. 2 An RBC Model wih Mauriy Mismach 2.1 Workers There is a coninuum of idenical households in he economy. Wihin each household here are hree ypes of agens: workers, enrepreneurs and bankers. Workers make consumpion plans and supply labor. They reurn heir wages o he respecive households and deposi funds in compeiive banks. Enrepreneurs inves in long-erm projecs. The acquisiion of working capial for he projec is parially financed by aking loans from banks. Bankers finance he loans wih heir own ne worh and deposis. Over ime he enrepreneur/banker exis he indusry and ransfers his ne worh back o he respecive household. A corresponding number of workers randomly become enrepreneurs and bankers, keeping he number of each occupaion consan. There is perfec consumpion insurance wihin a household. Le C be he consumpion of a represenaive household, and H be worker s labor supply. The worker is risk averse, and he or she derives uiliy from consumpion and disabiliy from labor supply max E β i [ln(c +i hc +i 1 ) χ 1 + φ H1+φ +i ] i= wih < β < 1, < h < 1 and χ > φ >. β is he discoun facor. To faciliae comparison o many DSGE models I also allow for habi formaion h. Demand deposis D are he only saving vehicle, which offers a risk-free reurn R in a given period. W is he real wage and Π denoes he ne ransfers from enrepreneurs and bankers o he household. The worker s

7 7 consumpion decision has o saisfy he household s budge consrain, C = W H + R D D +1 + Π Le ϱ denoe he marginal uiliy of consumpion. consumpion and labor supply are given as follows The household s opimal choices for 1 βh = ϱ (1) C hc 1 C +1 hc ϱ W = χh ϕ (2) E R ρ +1 ρ = 1 (3) 2.2 Enrepreneurs Physical Seup Enrepreneurs are in perfec compeiion, and each enrepreneur invess in a long-erm projec a a ime. Enrepreneurs are risk neural and aim o maximize he erminal wealh accumulaed hrough reained earnings. The producion echnology is given by he convenional Cobb-Douglas producion funcion Y = A K ɛ H 1 ɛ where A is he oal facor produciviy. K and H are he facors of producion, physical capial and labor, respecively. The facor marke is perfecly compeiive, and he real wage is given by he marginal produc of labor W = (1 ɛ) Y H The acquisiion of physical capial is financed by he enrepreneur s ne worh as well as exernal funds from he banking secor. Enrepreneurs have finie business horizons. Specifically, each period a consan fracion 1 θ e of enrepreneurs reire and exi he indusry, implying an expeced business horizon of 1 1 θ e. This assumpion ensures ha he real secor doesn ulimaely accumulae sufficien wealh o say independen of exernal financing. To moivae he demand for long-erm loans, in his model he invesmen projec has a sochasic duraion. In paricular, condiioning on reaching a given period each projec faces a consan probabiliy 1 α o erminae in ha period. The assumpion of sochasic duraion of he projec is inended o capure he fac ha projecs frequenly require muliperiod commimens, and herefore here is demand for long-erm financial conrac in order o secure sable funding. The conrac ensures ha he ineres rae of he loan is fixed hroughou he lifeime of he projec. Upon projec erminaion, he enrepreneur engages

8 8 in a new loan conrac o finance he nex projec. An alernaive way o inroduce fixed ineres rae would be o assume ha he invesmen is long-lived while he enrepreneur has sochasic opporuniy o refinance. Unil he enrepreneur is "apped by he Calvo fairy" he enrepreneur is locked in he conraced fixed ineres rae. The random renewal of he financial conrac can be inerpreed as a refinancing risk faced by he banking secor. Given ha he wo assumpions are echnically idenical, I sick o he firs one for is real-life relevance. The sochasic naure of he financial conrac implies an average loan mauriy 1 of 1 α, and he parameer α governs he degree of mauriy mismach on he bank balance shee: The greaer he α is, he greaer he degree of mauriy mismach. Enrepreneurs refinancing a ime j are denoed as vinage j. There is an infinie number of vinages in he economy, and he size of each vinage is geomerically declining overime. Le Q be he price of capial and δ denoe he depreciaion rae of capial 5. The aggregae reurn on capial across all enrepreneurs is given by R k +1 = ɛ Y +1 K + Q +1 (1 δ) Q (4) The realized reurn for an individual projec in each period, however, is subjec o boh aggregae and idiosyncraic risk. Individual enrepreneur i s reurn is given by ω i R k, wih ω i being he idiosyncraic produciviy shock. ω i is i.i.d across enrepreneurs and ime, following a log-normal disribuion wih he probabiliy disribuion funcion of f(ω) and cumulaive disribuion funcion of F (ω). The idiosyncraic shock has a mean value E{ω} = 1 and he sandard deviaion σ. The idiosyncraic risks and he realized capial reurn are privaely observed by he enrepreneur. Bankers need o pay an audiing cos o verify he realized reurn ω i R k. In he beginning of period enrepreneur i has available ne worh N i. To finance he difference beween his expendiures on capial goods and his ne worh he akes ou loans of L i. The balance shee of enrepreneur i is given by Q K i = N i + L i (5) An enrepreneur akes up a loan a a fixed ineres rae hroughou he enire life-ime of he invesmen, bu he volume of he loan varies in he meanime depending on he opimal capial acquisiion in ha period. The banker moniors in each period independen of wheher he loan maures or no. I is worh noing ha unlike audiing, monioring is cosless for bankers. If an enrepreneur s cash flow is no sufficien o cover he deb paymen (ineress plus he principal), he enrepreneur declares defaul and exis he indusry. Once he projec erminaes he enrepreneur refinances by aking up a loan a a new ineres rae. 5 Q will be derived laer

9 9 Financial Conrac The enrepreneur makes he invesmen decision, aking he price of capial goods and he expeced reurn o capial as given. Afer he opimal demand for capial is deermined, he capial prices and he reurns are endogenized as par of a general equilibrium soluion. The financial srucure of his model draws exensively from Bernanke, Gerler and Gilchris (1999), he core of which is a "cosly sae verificaion" (CSV) problem in he manner of Townsend (1979). The idiosyncraic produciviy shocks are privaely observed by he enrepreneur. Bankers mus pay a fixed audiing cos in order o verify he realized reurns. The audiing cos is inerpreable as he cos of bankrupcy and equal o a proporion µ of he realized gross payoff o he capial, i.e., µω i R k Q 1 K i 1. The financial conrac is characerized by he enrepreneur s leverage raio φ e,i (alernaively, demand for capial K) i and he conracual loan rae R L,i. I is worh noing ha he conracual loan rae R L,i is fixed in he conrac, in conras o he sae coningen loan rae in BGG. This feaure makes he conrac more relevan for he real world in ha bankers share he defaul risks wih enrepreneurs. Therefore he ex pos reurn on loans earned by banks (adjused for defaul cos) generally differs from he conracual loan rae. I is assume ha bankers monior in each period, and enrepreneurs have o remain solven in each period regardless of wheher he projec erminaes or no. Each period he enrepreneur has o repor o he banker ha he enrepreneur has sufficien cash flow o cover deb paymen (ineress plus he principal), oherwise he banker audis. This is a raher sark assumpion 6, bu i grealy simplifies he conracing problem and doesn affec he resuls qualiaively. The solvency condiion is characerized by a hreshold value of he idiosyncraic shock ω i +1 : for values of he idiosyncraic shock lower han ωi +1, he enrepreneur is no able o honor he loan a he conracual loan rae R L,i ω i +1R k +1Q K i = R L,i L i (6) When ω i ω i, enrepreneur is able o pay off he loan R L,i L i and keeps he difference, equal o ω i R+1 k Q +1K+1 i RL,i L i. Under he solvency condiion, whenever ω i < ω i he enrepreneur declares defaul and he financial conrac erminaes auomaically. Upon his he banker pays he audiing cos and ge o keep he remaining, i.e., (1 µ)ω i +1 Rk +1 Q K i. A defauling enrepreneur earns zero profi and exis he indusry. The financial conrac is a dynamic version of he conracing problem in BGG. Γ( ω i +1 ) denoes he expeced gross share of profis going o he banks Γ( ω i +1) = ω i +1 ω i +1 ω i +1 f(ω)dω + ωf(ω)dω 6 In principal, given a muli-period conrac he lender doesn monior each period, and he borrower should be able o accumulae profis and only pay he principle when he loan falls due.

10 1 The firs erm on he righ hand side measures he expeced reurn of loans earned from he enrepreneurs who honor he deb a he conracual loan rae. The second erm measures he remaining oucome seized by bankers when he enrepreneur declares defaul. Le µg( ω i ) denoe he expeced audiing cos as a fracion of he oal oucome ω i µg( ω +1) i +1 = µ ωf(ω)dω The expeced ne share of profis earned by banks is he gross profis less he deadweigh loss [Γ( ω i +1 ) µg( ωi +1 )]. The expeced share of profis earned by enrepreneurs is herefore given by 1 Γ( ω +1 i ). Enrepreneur s value funcion Ω is he expeced discouned profis, aking ino accoun he coninuaion value of he conrac (weighed by he coninuaion probabiliy α). Enrepreneur s opimizaion problem is given as following Ω = φ e,i max E {R+1[1 K Γ( ω +1)]φ i e,i,e ω +1 N e,i + βα[1 F ( ω i +1)]Ω +1} The enrepreneur chooses he opimal leverage raio φ e,i and he expeced defaul hreshold value E ω +1 i o maximize he discouned profis. The enrepreneur may equally choose he conracual loan rae R L,i as he alernaive o he expeced defaul hreshold value. In principle, Ω is an infinie sum of profis of all fuure periods, and Ω +1 also depends on he choices in period. To avoid he complicaions of a fully dynamic conracing problem while preserving he essence of a long-erm conrac, I simplify he conracing problem by assuming ha all non-refinancing enrepreneurs have idenical leverage raio in he beginning of he period. Deails of he assumpion will be made clear below. Under his assumpion Ω +1 has a slighly differen form han Ω. 7 The enrepreneur akes ino accoun he fac ha he individual leverage raio will be he aggregae raio φ e +1 in he beginning of he nex period if he financial conrac coninues o run, and herefore Ω +1 is a funcion of aggregae leverage raio and expeced aggregae echnology Ω +1 = {R K +2[1 Γ( ω i +2)]φ e +1N e,i +1 + βα[1 F ( ω i +2)]Ω +2}} The enrepreneur s opimal choices are subjeced o bankers paricipaion consrain. Each banker holds an marke porfolio of loans, and hence he paricipaion consrain is idenical for each banker. Bankers ake he expeced defaul risk ino accoun, and he expeced reurn 7 Appendix.

11 11 of loans is given by E {(Q K N e ) R }{{} +1} D = E {R K Q K [Γ( ω +1 ) µg( ω +1 )]} }{{} ousanding loan ne share of profis earned by banks (7) E R+1 D can be inerpreed as he expeced reurn ha bankers require for a given amoun of loans L issued. The ex pos reurn R+1 D is adjused for defaul and hence lower han he conracual loan rae R L. E R+1 D is joinly deermined by he demand and supply of credi, and he enrepreneur akes i as given when enering he conrac. Afer solving he opimizaion problem by choosing he opimal leverage raio and expeced defaul hreshold value, he conracing problem yields he following demand equaion ( ) E{R+1} k N e,i = s Q K i R+1 D (8) wih s ( ) < Deails of he conracing problem are derived in he echnical appendix. Equaion (8) gives a relaionship beween capial demand and borrowing cos. The proporionaliy facor s( ) is increasing in mauriy parameer α, which reflecs he erm premium. A rise in E { Rk +1 } reduces he expeced defaul probabiliy, herefore he enrepreneur akes on more R+1 D loan and expand he invesmen scale. The assumpion of consan reurns o scale implies ha relaionship (8) doesn depend on any enrepreneur-specific facor, which ensures ha he leverage raios are idenical wihin he opimizing enrepreneurs. By he same oken, he refinancing enrepreneurs share he same expeced defaul hreshold value. Therefore equaion (8) can be easily aggregaed over refinancing enrepreneurs and expressed in erms of he leverage raio φ e = Ψ(E { Rk +1 R+1 D }) (9) wih Ψ(1) = 1, Ψ ( ) >. Equaion 9 gives he opimal leverage raio for he refinancing vinage. To avoid abusing noaion le ( ) denoe he variable associaed wih he refinancing enrepreneurs, o be disinguished from he aggregae variables. Aggregaion of he Real Aciviies The following char summarizes he evens ha could happen o an enrepreneur wih boh exogenous and endogenous probabiliies

12 12 1 θ e Reired and Exi Enrepreneur 1 α New Financial Conrac θ e 1 F ( ω) Refinance? α Solven? F ( ω) Incumben Defaul and Exi Among he hree probabiliies ha rigger ransiions, 1 θ e, F ( ω) and 1 α, only he defaul probabiliy F ( ω) is endogenous. The defaul probabiliy of an enrepreneur depends on boh he conracual ineres rae and he leverage raio. The former is fixed bu he laer varies over ime. Specifically, he leverage raio of a enrepreneur depends on he whole hisory of idiosyncraic and aggregae shocks, and because of his mainaining racabiliy of he leverage raios as well as he defaul probabiliies is very hard. To circumven his problem I make he following assumpion o minimize he degrees of heerogeneiy. Before refinancing enrepreneurs obain new loans, all enrepreneurs engage in an insurance conrac, which saes ha a he end of his period (afer he realizaion of shocks) low-leveraged enrepreneurs ransfer ne worh o highly-leveraged enrepreneurs up o he poin where he leverage raios across enrepreneurs are equal. This conrac works as a parial insurance scheme, since highly-leveraged enrepreneurs face higher defaul risk han low-leveraged ones and herefore ex ane enrepreneurs have he incenive o share he risk. These ransfers reshuffle he ne worh across incumben enrepreneurs while keeping each exising loan o he firm i was iniiaed o, as well as he associaed ineres rae. Therefore he insurance conrac doesn imply full risk-sharing in he sense ha he ineres raes of loans peraining o he firms say unchanged and he associaed risk for each firm is carried over. By making his simplifying assumpion he only heerogeneiy across incumben firms is he fixed ineres rae of loans, and he defaul cos is a reflecion of he dispersion of loan raes insead of he leverage raios. Given ha loan raes are vinage-specific, he defaul probabiliies are vinage-specific as well. However, he loan rae of a cerain vinage is fixed over ime (unil refinancing) while he defaul probabiliy of a cerain vinage is a dynamic process. The evoluions of defaul probabiliies will be discussed a he end of his secion. Afer he ransfer of he ne worh, he common leverage raio of incumben enrepreneurs is deermined by he condiion α(1 δ)q K 1 = αθ e [1 F ( ω )]N e e,φ e

13 13 The lef hand side is he capial sock (ne of depreciaion) held by incumben enrepreneurs. The leverage raio of he incumben enrepreneurs φ e is pinned down by heir capial holding and ne worh. Therefore here are only wo prevailing leverage raios among all firms a a ime, φ e peraining o incumben enrepreneurs and φ e peraining o refinancing enrepreneurs. The aggregae enrepreneur ne worh N e is comprised of he ne worh of exising enrepreneurs, N e e, and he ne worh brough by newly-enered enrepreneurs N e n, N e = θ e [1 F ( ω )]N e e, + N e n, (1) The evoluion of he ne worh of he exising enrepreneur in period, N e e, = (1 α)r k N e 1 φ e 1[1 Γ( ω 1 )] (11) + α(1 α)r k N e 1φ e 1[1 Γ( ω 2 )] + α 2 (1 α)r k N e 1φ e 1[1 Γ( ω 3 )] α τ 1 (1 α)r k N e 1φ e 1[1 Γ( ω τ )] +... Incoming enrepreneurs ener he indusry wih sarup funds from heir respecive households. The amoun of sarup funds is proporional o he value of he oal capial sock, wih he implicaion ha he household considers how much an enrepreneur needs o sar a business. N e n, = ξ e Q K (12) The refinancing enrepreneurs capial demand K (as menioned above, variable refers o he refinancing enrepreneurs) is comprised of he demand of incumben enrepreneurs and he demand of newly-enered enrepreneurs (1 α)q K = { (1 α)θ e [1 F ( ω )]Ne, e }{{} + ξ e Q K }{{} } φ e incumben enrepreneurs whose loans fall due newly-enered enrepreneurs φ e is he common leverage raio of all he refinancing enrepreneurs. Newly-iniiaed loans L acquired by he refinancing enrepreneurs and he exising loans add up o oal ousanding loans L = (1 α) L + αθ e [1 F ( ω )]L 1 (13) R L can be inerpreed as he spo ineres rae on newly-iniiaed loans, which differs from he aggregae conracual loan rae R L since he laer consiss of all he loan raes fixed in he previous periods. To be specific, he average borrowing cos R L is a weighed average of

14 14 he spo rae R L and ineres raes of all vinages R L L = (1 α) R L L + αθ e [1 F ( ω )]R L 1L 1 (14) Le F ( ω τ ) be he defaul rae of vinage τ. The aggregae defaul rae is he weighed average of defaul raes across all vinages, expressed in he following way F ( ω) = (1 α)f ( ω 1 ) + α(1 α)f ( ω 2 ) + α 2 (1 α)f ( ω 3 )... (15) Le L τ and R D,τ be he ousanding loans and ex pos loan rae of vinage τ respecively. The aggregae reurn of loans R D can be obained by adding up reurns of all vinages R D L 1 = (1 α) R D, 1 L 1 + α(1 α)r D, 2 L 2 + α 2 (1 α)r D, 3 L (16) Equaions 31, 16 and 15 conain indefinie erms, which generaes an ineresing echnical challenge since hey canno be calculaed wihou keeping rack of he defaul raes of all vinages. This problem is solved in he following way. Given he assumpion of he insurance conrac ha ensures a common leverage raio across incumben enrepreneurs, he defaul rae of a cerain vinage τ a ime (τ 2) depends on he aggregae echnology R k and balance shee φ, and he vinage-specific loan rae R L τ. However, since he only vinagespecific facor, he loan rae, is fixed over ime, he evoluion of he defaul rae only depends on he evoluions of he wo aggregae variables. In paricular, he defaul hreshold value of vinage τ can be expressed in a recursive form ω τ +1 = RL τ (φ 1) R k +1 φ = ω τ +1 = ω τ R k R k +1 φ 1 φ 1 φ 1 1 φ log linearized = ˆ ω τ +1 = ˆ ω τ + ˆR k ˆR k +1 + ˆφ ˆφ 1 φ 1 The las equaion is in log-linearized fashion, wih ˆx being he percenage deviaion, and x being he seady sae. Wih ω τ being in a recursive form, he evoluions, in erms of percenage deviaions, of aggregae defaul probabiliy F ( ω ) and he enrepreneur s share of profis [1 Γ( ω 1 )], boh of which are linear funcions of ˆ ω τ, can be expressed in recursive form. Therefore he evoluions of aggregae enrepreneur ne worh, aggregae defaul rae and aggregae ex pos reurn of loans can be raced over ime in he log-linearized fashion.

15 Banking Secor Bankers are risk neural, and each one manages a bank. In each period a banker reires wih a consan probabiliy 1 θ b. Upon reiremen he banker becomes a worker and reurns his ne worh o he respecive family. A corresponding number of workers randomly become bankers o keep he number of each occupaion consan. The newly-enered bankers receive sarup funds from he families o ake over he long-erm loans from he reiring bankers. Each bank holds a marke porfolio of loans. The banker j aims o maximize his expeced erminal wealh, denoed by V j V j = max E = max E τ τ β τ (1 θ b )(θ b ) τ N b,j +τ+1 β τ (1 θ b )(θ b ) τ [R D +τ+1l j +τ R +τ (L j +τ N b,j +τ )] N b,j is he ne worh of bank j. Alernaively, V j fashion can be expressed in he forward-looking V j = ν L j + η N b,j (17) where he respecive shadow value of L j and N b,j are given by he following, ν = E {(1 θ b )β(r D +1 R ) + βθ b x,+1 ν +1 } η = E {(1 θ b )βr + βθ b z,+1 η +1 } wih x,+1 L j +1 /Lj, z,+1 N b,j b,j +1 /N. x,+1 is he gross growh rae of loans, and z,+1 is he gross growh rae of bank ne worh. ν can be inerpreed as he expeced discouned marginal gain of iniiaing one more uni of loan, holding ne worh consan, while η is he expeced discouned value of building up anoher uni of ne worh, holding he balance shee size consan. In a fricionless capial marke he banker will build up his asses indefiniely by borrowing from workers unil he ne reurn of asses is zero. To place a limi on arbirage, I follow he seup of GK and inroduce a moral hazard problem beween bankers and workers, in order o preven excess credi supply ha wipes ou he ineres spread beween loans and deposis. I is assumed ha he banker has an incenive o renege on debs (deposis) and diver a fracion λ of his asses (loans) for his own consumpion. If his happens he bank is forced ino bankrupcy and he workers recover he remaining (1 λ) of asses. To preven he banker from divering funds, workers refrain from lending excessively o bankers. In paricular, he amoun of lending will be consrained by V j λl j (18)

16 16 The lef-hand side is he erminal wealh ha he banker would lose if he divers he asses; he righ-hand side is his gain from doing so. The incenive consrain can be equivalenly expressed in a forward-looking way ν L j + η N b,j λl j (19) The opimal decision for he banker is o borrow up o he poin when he gain of divering asses is exacly balanced by he cos. Equaion (19) is binding only if < ν < λ: if ν > λ, i is no worh divering he funds since he franchise value of he bank is higher. Under reasonable parameerizaions he borrowing consrain is always binding around he seady saes. The binding borrowing consrain gives rise o an endogenous bank leverage raio L j = η N b,j λ ν = φ b N b,j where φ b denoes he leverage raio of bank j. Since φ b doesn depend on any bank-specific facor, he leverage raio is idenical across all banks. The ne worh of bank j is evolving according o N b,j = θ b [(R D R 1 )φ b 1 + R 1 ]N b,j 1 I follows ha he gross growh raes of bank ne worh and loans are no bank-specific eiher z,+1 N b,j b,j +1 /N = (R+1 D R )φ b + R (2) x,+1 L j +1 /Lj = (φb +1/φ b )(N b,j b,j +1 /N ) = (φ b +1/φ b )z,+1 (21) Due o he common growh raes and leverage raios, he evoluion of oal bank ne worh can be aggregaed over individual banks N b e, = θ b [(R D R 1 )φ b 1 + R 1 ]N b 1 (22) where Ne, b on he lef hand side denoes he ne worh of banks surviving o period + 1. The aggregae bank ne worh N b consiss of he former and he sarup funds brough in by incoming bankers N b = Ne, b + Nn, b (23) wih Nn, b being he sarup funds of incoming bankers, aken from he corresponding families. The amoun of sarup funds is made sufficien for new banks o ake up he nonmauring loans from reiring bankers. The amoun of sarup funds is a fracion ξ b of he oal value of loans, wih ξ b calibraed o ensure he akeover of he ousanding long-erm loans of reiring banks. An alernaive way, as suggesed by AFZ, o guaranee he inheriance of old loans is o inroduce an insurance agency financed by a proporional ax on banks profis. In each

17 17 period he insurance agency endows he new banker wih funds ou of he ax revenue. The wo alernaive assumpions resul in differen calibraed values for λ, θ b and ξ b, bu he seady sae values are no affeced by he choice of assumpions. Here I sick o he firs assumpion for is breviy. Nn, b = ξ b L (24) Combining equaions (22) and (24) yields he law of moion of aggregae bank ne worh N b = θ b [(R D R 1 )φ b 1 + R 1 ]N b 1 + ξr D L (25) Given a common leverage raio across banks, he aggregae credi supply is obained by L = φ b N b (26) 2.4 Capial Producing Firms Capial goods producers are compeiive and owned by households. Each period hey buy capial from enrepreneurs, refurbish depreciaed capial and produce new capial. The cos of repairing worn ou capial is uniy. In he beginning of each period capial producing firms package boh he new and repaired capial and re-sell hem o enrepreneurs a a uni price Q. Following Chrisiano, Eichenbaum and Evans (25), I allow for adjusmen coss of invesmen flow f( ). Capial producer s objecive is o maximize profis by choosing I max E I τ β τ ϱ +1+τ ϱ +τ {Q +τ I +τ [1 + f ( I+τ I +τ 1 ) ]I +τ } The adjusmen echnology has he following properies: f(1) = f (1) =, and f (1) >. These properies are sufficien o jusify ha here is no adjusmen cos in he seady sae. The opimal invesmen decision yields he price of capial Q = 1 + f ( I I 1 ) + I ( ) f I ϱ +1 E I 1 I 1 ϱ ( I+1 I ) 2 ( ) f I+1 I 2.5 Marke Clearing Condiions The aggregae demand for capial is he sum of he demand by refinancing enrepreneurs and he exising capial sock of ongoing projecs,ne of depreciaion. K = (1 α) K + α(1 δ)k 1 (27)

18 18 Table 1: Convenional Parameers Parameer Descripion Value β Household discoun facor.995 ɛ Facor share of capial.33 ϕ Inverse Frisch elasiciy.27 h Exernal habi formaion.7 η i Invesmen adjusmen cos 1.5 δ Depreciaion rae.25 The economy-wide resource consrain is given by Y = C + I + R k Q 1 K 1 µγ( ω ) + [1 + f( I I 1 )]I (28) 2.6 Macroeconomic Implicaions Parameers The convenional parameer values are se o be consisen wih much of he available lieraure, as repored in Table Table repors he parameers specific o financial fricions. The parameers peraining o he enrepreneurial secor are mosly se o be in line wih BGG: he seady sae quarerly defaul rae is.75; he bankrupcy cos µ is.12. The sandard deviaion of he idiosyncraic produciviy shock σ is calibraed o achieve he argeed enrepreneurial leverage raio of 1.8, which maches he average leverage raio of he nonfinancial corporae business secor in he Flow of Funds. The defaul rae, he bankrupcy cos and sandard deviaions joinly pin down he seady sae ineres spread R k R D, which equals o 4 basis poins, and he seady sae ineres spread of R D R is se o 2 basis poins 8. θ e and ξ e are pinned down by he balance shee of non-refinancing enrepreneurs and refinancing enrepreneurs respecively. λ, θ b are calibraed o achieve he argeed bank leverage raio of 8, consisen wih Gerler and Kiyoaki (212). ξ b is calibraed o ensure ha incoming bankers have sufficien ne worh o effecively ake up he ousanding long-erm conracs. In he following experimens, he parameer α ha governs he average mauriy of loans are se o various values in order o examine is implicaions for business cycles Seady Sae Analysis As shown by Figure 3, he sensiiviy of ineres spread R K R D o he enrepreneurial leverage raio φ e, measured by χ, is increasing in he degree of mauriy mismach α. The 8 Daa source: Bankscope

19 19 Table 2: Parameers Peraining o Financial Fricions Parameer Descripion Value µ Bankrupcy cos.12 σ 2 Variance of idiosyncraic shocks.3 F ( ω) Seady sae defaul rae.75% θ e Survival probabiliy of enrepreneur.968 φ e Firm leverage raio 1.8 λ Fracion ha banker can diver.33 θ b Survival probabiliy of banker.975 φ b Bank leverage raio 8 R K R Spread 6 BPS posiive relaionship beween χ and α reflecs he erm premium, which implies ha he fuure defaul risk has been priced ino he loan rae. Since χ is one of he measures of he srengh of he credi channel, i can be inferred ha he greaer he degree of mauriy mismach, he greaer he magniude of he credi channel in he ransmission of shocks. However, i is worh noing ha he seady sae values, including he ineres spread R K R D, don depend on he value of α. Pu differenly, he degree of mauriy mismach affecs he economy dynamics, bu doesn have an impac on resource allocaion in he seady sae Impulse Responses Analysis Technology shock The properies of he model are shown by impulse response funcions. To illusrae he ransmission mechanism of shocks, I sar wih he analysis on he baseline mode, which is shown by he black dashed line. In he baseline model all loans are singleperiod (α = ), and he financial conrac collapses o he one-period conrac as in BGG. The echnology shock considered here is characerized by a one percen innovaion in echnology wih a quarerly auoregressive facor of.9. On impac, he aggregae reurn on capial declines and more enrepreneurs fall ino bankrupcy. Aggregae enrepreneurial ne worh falls sharply due o he lower reurn and he higher defaul rae, and he leverage raio rises. Perceiving persisenly low produciviy, invesmen falls and drags down he price of capial. The reurn on capial shoos up in he second period because he capial price bounces back righ afer he shock. Due o he surge of capial reurn soon afer he shock, enrepreneurs profis sar o pick up. The defaul rae says high persisenly and banks require a higher expeced reurn on loans o compensae for he defaul cos. This is observed in he hike of boh he conracual loan rae and he ex pos loan rae. The banking secor builds up ne worh since he rise in he loan rae is only parially passed o deposiors and he ineres spread increases. Deposiors are less willing o lend o banks, and i can be seen as banks undergo deleveraging. The conracion of he bank balance shee means a decline in loans

20 2 iniiaed. Boh higher demand for credi and lower credi supply conribue o he surge of ineres spreads. R L R D represens he magniude of he deadweigh loss arising from defaul, and R D R is he banking secor s profi margin. The spread beween he reurn on capial and he deposi rae R k R measures he overall financial (in)efficiency. The macroeconomic implicaions of mauriy mismach can be analyzed by varying α. Solid blue lines show he scenario when α =.75, i.e., average loan mauriy of four quarers. The degree of mauriy mismach has been se o 4 because empirical evidences sugges ha he observed average duraion of bank asses is abou four imes he duraion of heir liabiliies, as documened by Jochen Guenner (21). In wha follows, α =.75 will be mos of he ime used o calibrae he economy wih mauriy mismach. As shown by he solid blue lines, he responses of oupu, invesmen and capial price in an economy wih mauriy mismach are milder han in he baseline model represened by he dashed black lines. Since hree quarers of he enrepreneurs are locked in he fixed ineres raes, hey are immune o he surge in conracual loan rae. Therefore he decline in aggregae enrepreneur ne worh is relaively mued. On he oher hand, he responses of financial secor are more pronounced in he presence of long-erm loans. Due o sluggish adjusmens of loan raes, bank ne worh and leverage raio reach he peak a a laer period in comparison o he baseline model. Ineres spread R D R rises more sharply, suggesing ha banks are exposed o more volaile movemens of ne ineres margin. This explains he amplified flucuaions in bank ne worh and leverage raio. Given ha he rise in reained earnings ouweighs he deleveraging effec, he conracion in credi supply is relaively moderae, and herefore he declines in oal invesmen and oupu are miigaed. Financial shocks Financial shocks have received much aenion in recen sudies since hey are considered o be good candidaes for explaining many sylized facs. Financial shocks, in general, are defined as he shocks affecing he abiliy of borrowing. In his model boh enrepreneurs and bankers are borrowers wihin paricular financial relaionships, and hence financial shocks may emerge from eiher he real secor or he financial secor, boh of which deserving of aenion. Firsly, I consider a one-ime negaive shock (5%) o he enrepreneurial ne worh. The negaive wealh shock direcly affecs enrepreneurs capial buffer and he cos of borrowing. Alhough all enrepreneurs ne worh is hi by he negaive shock, hree quarers of he enrepreneurs are immune o he surge of high borrowing cos due o he fixed ineres raes. Defaul probabiliy is much lower in he scenario wih fixed ineres raes, and so is he defaul cos. The banks require a higher expeced loan rae o lend ou loans in response o he shock. This is shown by a more volaile ineres margin R D R and he reained earnings for banks. Bank ne worh funcions as a buffer in he propagaion of his shock, and dampens he impac of he shock on he real secor. Nex I consider wo financial shocks originaing from he financial secor: a negaive shock

21 21 o bank ne worh and a reducion of he deposiors confidence in bankers. Given ha banker s abiliy o borrow is consrained by he bank ne worh, a 5% negaive wealh shock reduces he amoun of deposis ha banks can ake up and hence he amoun of loans iniiaed. Since he shock emerges from he financial secor and affecs all banks in he same way, he ineres spreads of R D R in boh baseline model and he model wih mauriy mismach rack closely, and so do he financial variables. Bu in he presence of mauriy mismach, he conracual loan rae and he ex pos loan rae have milder adjusmen pahs and enrepreneurs are less exposed o defaul risk. The aenuaion effec on he real secor implied from he previous experimens carries over o his scenario. A reducion in deposiors confidence direcly affecs he banking secor s leverage raio allowed by he deposiors. This is done by giving a persisen (auoregressive facor of.9) negaive shock of one percen o Λ, he fracion of asses ha bankers can diver. The confidence shock is ransmied form he liabiliy side of he bank balance shee o he asse side, opposie o shocks originaing from he real secor. To arac deposis he bankers have o offer a higher real ineres rae, bu again, due o he sicky loan raes he banking secor canno fully pass he higher borrowing cos o enrepreneurs. The banking secor akes he ineres rae risks and parially shields he real secor from flucuaions of ineres raes. In general, he conclusion ha he banking secor wih mauriy mismach is sabilizing he economy over he business cycles is robus o various shocks considered, independen of where he shocks originae. The inroducion of mauriy mismach offses he acceleraor effecs caused by financial fricions, and he bank ne worh funcions as a buffer by dampening business cycle flucuaions. Opimaliy of mauriy ransformaion The nex quesion ha comes naurally is wha degree of mauriy mismach on he bank balance shee is desirable from a macroeconomic poin of view. As an aemp o answer his quesion, Figure 9 provides a comparison beween economies wih various degrees of mauriy mismach o is efficien level, i.e., he economy absen of any financial fricions. This exercise is conduced by simulaing he modeled economies o a negaive echnology shock of one percen in size. The dash doed (red) line represens he IRFs of he simple real business cycle model wihou any financial fricions, i.e., in he absence of asymmeric informaion problem and moral hazard problem. The incorporaion of he asymmeric informaion problem beween enrepreneurs and banker as well as he moral hazard problem beween bankers and deposiors grealy amplifies he shocks, as shown by he baseline model (he dash black line). The doed green line sands for he economy wih average loan mauriy of en quarers (α =.9), which predics a sronger aenuaion effec han α =.75. Bu in he scenario of α =.9 he economy is driven furher apar from he efficien level, since he real secor is consrained and doesn fully respond o he disurbance in an opimal manner. Poenially here exiss an opimal degree of mauriy

22 22 mismach ha offses he amplificaion effec of financial fricions and helps he economy o realize is full poenial. This exercise provides insigh ino he financial regulaion from a macroeconomic perspecive. Exising financial regulaion policies primarily aim o limi he disress of individual insiuions; he macroprudenial policies, however, can improve he overall welfare of he economy by designing proper regulaions and ools. As his exercise suggess, an opimal argeed degree of mauriy mismach on banks balance shees can help he economy realize is full poenial. 3 A Model wih Nominal Rigidiies Allowing for a role of moneary policy, in his secion I exend he model wih nominal rigidiies. While he major framework of he RBC model is preserved, a few modificaions are made on he seup of he real secor and he moneary auhoriy. 3.1 Inermediae Goods Firms Following he convenional seup of New Keynesian models, he real secor is comprised of inermediae goods firms and reail firms. Enrepreneurs, as presened in he previous secion, produce inermediae goods and hus will be labeled as inermediae goods firms in his secion. Inermediae goods are evenually sold o reail firms a he relaive price of mc (in he period ). The real wage and he reurn on capial herefore follow W = (1 ϖ) Y H mc R k +1 = mc +1ϖ Y +1 K + Q +1 (1 δ) Q (29) The remaining seup and he evoluion of he real variables are he same as in Secion Reail Firms Final oupu Y is a CES (consan elasiciy of subsiuion) composie of a coninuum of mass uniy of differeniaed reail firms. Y = [ 1 Y f ε 1 ε df] ε ε 1 (3)

23 23 is he produc of reail firm f. Cos mini- where ε is he elasiciy of subsiuion, and Y f mizaion implies he sandard demand funcion Y f = ( P f P ) ε Y (31) where P f is he price of he reail good of firm f. The aggregae price level is hus given by P [ 1 P f 1 ε df] 1/1 ε (32) Reailers re-package inermediae oupu. I akes one uni of inermediae oupu o make a uni of reail oupu. The real marginal cos is hus he relaive inermediae oupu price mc. Reailers se nominal prices in a saggered fashion à la Calvo (1983). Each period he reailer is able o freely adjus price wih probabiliy 1 ς, and he opimal price se in period is denoed as P. The remaining fracion ς of reailers simply keep heir prices consan P f = P f 1. The reailer s pricing problem is o choose he opimal price P o maximize he expeced profis max P E i= ς i β i ρ +i ( P mc +i )Y f +i (33) ρ P +i subjec o he individual demand funcion (31). From he law of large number, he evoluion of he price level is given by P = [(1 ς)p 1 1 ε + ςp 1/1 ε 1 ] 1 ε (34) 3.3 Moneary Policy Moneary policy is characerized by a simple Taylor rule wih ineres-rae smoohing. Le R n be he seady sae nominal rae, Ȳ he seady sae level of oupu and Π he seady sae inflaion rae. [ R n = ρ R nr 1 n + (1 ρ R n) ( ) R n Π + φ Π log + φ Y log( Y ] Π Ȳ ) + ɛ Rn (35) where he smoohing parameer ρ R n lies beween zero and uniy, and ɛ is an exogenous shock o moneary policy. The relaionship beween nominal and real ineres raes is given by he Fisher equaion R n = R Π +1 = R P +1 P (36)