Monetary Shocks and Bank Balance Sheets

Transcription

1 Moneary Shocks and Bank Balance Shees Sebasian Di Tella and Pablo Kurla Sanford Universiy PRELIMINARY June 215 Absrac We propose a model o explain why banks balances shees are exposed o ineres rae risk despie he exisence of markes where ha risk can be hedged. A rise in nominal ineres raes raises he opporuniy cos of holding currency; since bank liabiliies are close subsiues of currency, demand for bank liabiliies rises and banks earn higher spreads. If risk aversion is higher han 1, he opimal dynamic hedging sraegy is o susain capial losses when nominal ineres raes rise and, conversely, capial gains when hey fall. A radiional bank balance shee wih long duraion nominal asses achieves ha. Keywords: Moneary shocks, bank deposis, ineres rae risk JEL codes: E41, E43, E44, E51 pkurla@sanford.edu and sdiella@sanford.edu. We hank V.V. Chari, Arvind Krishnamurhy, Ben Moll and Ed Nosal for helpful commens. 1

2 1 Inroducion Moneary policy can have major redisribuive effecs. One of he channels of redisribuion is hrough banking sysem. Typically, banks hold long duraion nominal asses such as fixed-rae morgages and herefore susain capial losses in mark-o-marke erms) when nominal ineres raes rise. One could conjecure ha a mauriy-mismached balance shee is inheren o he banking business and he resuling ineres rae risk is an ineviable side effec. However, here exis deep and liquid markes for ineres rae derivaives where banks could hedge agains ineres rae changes if hey waned. Furhermore, Begenau e al. 213) show ha, if anyhing, banks end o use ineres rae derivaives o increase raher han reduce heir exposure o ineres rae risk. Why, hen, do hey choose his exposure? We argue ha banks choose o bear ineres rae risk as par of opimal dynamic hedging. We model a flexible price, complee markes, moneary economy, wih hree key ingrediens. Firs, he economy consiss of banks and households, who are idenical excep ha banks can issue deposis which are close subsiues o currency, up o a leverage limi. Second, here are indeed moneary shocks which move nominal ineres raes. Third, risk aversion is high, wih a CRRA coefficien greaer han 1. In his economy, banks opimally choose o be exposed o ineres rae risk. The mechanism works as follows. Because deposis provide liquidiy services, banks earn he spread beween he nominal ineres rae on bonds and he lower ineres rae on deposis. If nominal ineres raes rise, he opporuniy cos of holding currency rises so, given ha currency and deposis are subsiues, demand for deposis rises. This drives up he spread beween he nominal ineres rae and he ineres rae on deposis, increasing banks reurn on wealh. This has boh income and subsiuion effecs. Because risk aversion is higher han 1, he income effec dominaes and banks wan o ransfer wealh from saes of he world wih high-reurn-on-wealh o saes of he world wih low-reurn-on-wealh. They are willing o ake capial losses when ineres raes rise because spreads going forward will be high, and wan o make gains when ineres raes fall because spreads going forward will be low. Choosing a porfolio of long-duraion nominal asses is a way o achieve his exposure and hey do no wan o undo i even if complee markes allow hem o do so. The fac ha bank deposi raes move less han one-for-one wih marke ineres raes has been observed before. Hannan and Berger 1991) and Driscoll and Judson 213) aribue i o a form of price sickines; Drechsler e al. 214) aribue i o imperfec compeiion among bank branches. Nagel 214) makes a relaed observaion: he premium on oher near-money asses besides banks deposis) also co-moves wih ineres raes. He aribues 2

3 his, as we do, o he subsiuabiliy beween money and oher liquid asses. Krishnamurhy e al. 215) documen a negaive correlaion beween he supply of publicly issued liquid asses and he supply of liquid bank liabiliies, also consisen wih heir being subsiues. Relaive o his lieraure, he conribuion of our work is o derive he implicaions for equilibrium risk managemen in a model where he underlying risk in modeled explicily. Landier e al. 213) shows cross-secional evidence ha exposure o ineres rae risk has consequences for bank lending. 2 The Model Preferences and echnology. Time is coninuous. There is a fixed capial sock k which can be used o produce a flow of consumpion goods wih a linear echnology y = ak. There are wo ypes of agens in he economy: households and bankers, a coninuum of each. Boh have idenical Epsein-Zin preferences wih ineremporal elasiciy of subsiuion equal o 1, risk aversion and discoun rae ρ: wih U = E f x, U) = ρ 1 ) U ˆ f x s, U s ) ds log x) 1 ) log 1 ) U) 1 x is a Cobb-Douglas composie of consumpion c and liquidiy services from money holdings m: x c, m) = c β m 1 β 1) Money iself is a CES composie of real currency holdings h and real bank deposis d, wih elasiciy of subsiuion ɛ: 1 m h, d) = α 1 ɛ 1 ɛ h ɛ ) + 1 α) 1 ɛ ɛ 1 ɛ 1 ɛ d ɛ 1 Throughou, uppercase leers denoe nominal variables and heir corresponding lowercase leer are real variables. Hence h H p and d D p where p is he price of consumpion goods in erms of currency, which we ake as he numeraire. 2) 3

4 Currency supply. The governmen issues a nominal amoun of currency H. We ake moneary policy as exogenously given by he following sochasic process dh H = µ H, d + σ H, db where B is a sandard Brownian moion. The process B drives equilibrium dynamics. The governmen disribues or wihdraws currency o and from agens hrough lump-sum ransfers or axes. Markes. There are complee markes. We denoe he real price of capial by q, he nominal ineres rae by i, he real ineres rae by r, and he price of risk by π so an asse wih exposure σ o he process B will pay an excess reurn σπ). All hese processes are coningen on he hisory of shocks B. The oal real wealh of privae agens in he economy includes he value of he capial sock qk, he real value of ousanding currency h and he ne presen value of fuure governmen ransfers and axes, which we denoe by g. Toal wealh is denoed by ω: ω = qk + h + g Toal household wealh is denoed by w and oal bankers wealh is denoed by n, so n + w = ω 3) Noice ha wih complee markes i is no necessary o specify who receives governmen ransfers when he supply of currency changes: all hose ransfers are priced in and included in he definiion of wealh. We denoe by z n he share of he aggregae wealh ha is ω owned by bankers. The only difference beween households and bankers is ha bankers may issue deposis. These pay a nominal ineres rae i d and also ener he uiliy funcion according o equaion 2). 2 The amoun of deposis bankers can issue is subjec o a leverage limi. A banker whose 2 Noe ha even hough deposi conracs are specified in nominal erms, nohing prevens a banker and a deposi holder from also rading securiies such as ineres rae or inflaion swaps o choose any exposure o nominal variables. 4

5 individual wealh is n can issue deposis d S up o d S φn 4) where φ is an exogenous parameer. Consrain 4) may be inerpreed as eiher a regulaory consrain or a level of capializaion required for deposis o acually have he liquidiy properies implied by 1). Moneary policy. As is sandard, moneary policy can be described in erms of he supply of currency or in erms of he nominal ineres rae. We assume ha he governmen chooses a pah for H such ha i follows he Cox e al. 1985) sochasic process: 3 di = λ i ī) d + σ i db 5) Shocks o B are our represenaion of moneary shocks. There is more han one sochasic process H ha will resul in 5). Le dp p = µ p, d + σ p, db be he sochasic process for he price level which is endogenous). We assume ha he governmen implemens he unique process H such ha in equilibrium 5) holds and σ p, =. Informally, his means ha moneary shocks affec he rae of inflaion µ p bu he price level moves smoohly. 3 Equilibrium Households problem. solves a sandard porfolio problem: Saring wih some iniial nominal wealh W, each household max U x) W,x,c,h,d,σ W 3 This is a square roo process. I is always nonnegaive and if 2λī σ 2 hen i is sricly posiive almos surely and has a saionary disribuion. 5

6 subjec o he budge consrain: dw = i + σ W, π ĉ W ĥi ˆd ) ) i i d d + σ W, db W 6) and equaions 1) and2). A ha denoes he variable is normalized by wealh, i.e. ĉ = pc W = c w. The household obains a nominal reurn i on is wealh. I incurs an opporuniy cos i on is holdings of currency. I also incurs an opporuniy cos i i d ) on is holdings of deposis. Le s = i i d denoe he spread beween he deposi rae and he marke ineres rae. Furhermore, he household chooses is exposure σ W and obains he risk premium πσ W in reurn. Consrain 6) can be rewrien in real erms as o he moneary shock dw = r + σ w, π ĉ w ĥi ˆd ) ) i i d d + σ w, db 7) where r = i µ p. is he real ineres rae. Bankers problem. Bankers are like households, excep ha hey can issue deposis denoed d S ) up o he leverage limi and earn he spread s on hese. The banker s problem, expressed in real erms, is: subjec o: and equaions 1) and2). max U x) n,x,c,h,d,d S,σ n dn = r + σ n, π ĉ n ĥi + ˆdS ˆd ) ) s d + σ n, db ˆd S φ n 8) Equilibrium definiion Given an iniial disribuion of wealh beween households and bankers z and an ineres rae process i, a compeiive equilibrium is 1. a process for he supply of currency H 2. processes for prices p, i d, q, g, r,π 3. a plan for he household: w, x h, c h, m h, h h, d h, σ w 6

7 4. a plan for he banker: n, x b, c b, m b, h b, d b, d S, σ n such ha 1. Households and bankers opimize aking prices as given and w = 1 z ) q k + h + g ) and n = z q k + h + g ) 2. The goods, deposi and currency markes clear: c h + c b = ak d h + d b = d S h h + h b = h 3. Wealh holdings add up o oal wealh: w + n = q k + h + g 4. Capial and governmen ransfers and nominal claims are priced by arbirage: [ ˆ q = E Q a [ˆ g = E Q exp exp ˆ s ˆ s ) ] r u du ds ) ] dhs r u du p s 9) 1) where Q is he equivalen maringale measure implied by r and π. 5. Moneary policy is consisen i = r + µ p, σ p, = 4 Equilibrium Characerizaion Hamilon-Jacobi-Bellman equaions and FOCs. We sudy he banker s problem firs. I can be separaed ino a saic problem choosing c, m, h and d given x) and a dynamic problem choosing x and σ n ). 7

8 Consider he saic problem firs. Given he form of he aggregaors 1) and 2), we immediaely ge ha he minimized cos of one uni of liquidiy services is given by ι: ιi, s) = αi 1 ɛ + 1 α) s 1 ɛ) 1 1 ɛ 11) and he minimized cos of one uni of he aggregaor x is given by χ: and he saic choices of c, m, h and d are given by: Turn now o he dynamic problem. ) 1 β ι χ i, s) = β β 12) 1 β c = βχ 13) x m x = 1 β) χ 14) ι h ι ) ɛ m = α 15) i d ι ) ɛ m = 1 α) 16) s In equilibrium i will be he case ha i d < i so bankers leverage consrain will always bind. This means ha 8) reduces o dn = r + σ n, π χ i i, s ) ˆx + φs ) n }{{} d + σ n,db 17) µ n, Given he homoheiciy of preferences and he lineariy of budge consrains he problem of he banker has he value funcion: V b n) = ξ n) 1 1 ξ capures he value of he banker s invesmen opporuniies, i.e. his abiliy o conver unis of wealh ino unis of lifeime uiliy, and follows he law of moion dξ ξ = µ ξ, d + σ ξ, db where µ ξ, and σ ξ, are equilibrium objecs. 8

9 The associaed Hamilon-Jacobi-Bellman equaion is = max f ) [ ] x, V b + E dv b x,σ n,µ n Using Io s lemma and simplifying, we obain: = max ρ 1 ) ξ n ) 1 [ log ˆxn ) 1 ˆx,σ n,µ n 1 1 log ξ n ) 1 ) ] + ξ 1 n 1 µ n + µ ξ 2 σ2n ) 2 σ2ξ + 1 )σ ξ σ n s..µ n = r + σ n π + φs ˆxχ The household s problem is similar. The only difference is ha he erm φs is absen from he budge consrain. The value funcion has he form where and he HJB equaion is V h w) = ζ w) 1 1 dζ ζ = µ ζ, d + σ ζ, db = max ρ 1 ) ζ w ) 1 [ log ˆxw ) 1 ˆx,σ w,µ w 1 1 log ζ w ) 1 ) ] + ζ 1 w 1 µ w + µ ζ 2 σ2w ) 2 σ2ζ + 1 )σ ζ σ w s..µ w = r + σ w π ˆxχ Aggregae sae variables. We look for a recursive equilibrium aking he saic opimizaion choosing c, m, h and d given x) as given. There are wo sae variables: he ineres rae i which is exogenous) and he bankers share of aggregae wealh z which is endogenous). Using he definiion of z = n, we obain a law of moion for z from Io s n+w 9

10 lemma and he budge consrains dz = 1 z ) σ n, σ w, )π + φs ˆx b ˆx h )χ + σ w, σ w, σ n, ) ) z ) σz, 2 d z 1 z }{{ } µ z, 18) + 1 z ) σ n, σ w, ) db } {{ } σ z, Equilibrum objecs are hen funcions of i and z, e.g. ξ = ξi, z ). We can use Io s lemma o compue heir laws of moion, e.g. µ ξ, = ξ i ξ λ ī i ) + ξ z ξ µ z,z + 1 ξii 2 ξ i σ ξ iz i σσ z, z + ξ ) zz ξ ξ σ2 z,z 2 σ ξ, = ξ i ξ σ i + ξ z ξ σ z,z Definiion 1. A recursive equilibrium is a se of funcions of i and z: value funcions ξ and ζ, policy funcions ˆx b, σ n, µ n ) and ˆx h, σ w, µ w ) ; prices q, g, h, r, π, i d ; and funcions µ z and σ z ha define a law of moion for z: dz = µ z z d + σ z z db such ha 1. ξ and ζ, and he corresponding policy funcions solve he HJB equaions of bankers and households respecively. 2. Markes clear: a) for goods: [ˆx h 1 z) + ˆx b z ] qk + h + g) βχ = ak b) For deposis: [ˆx h 1 z) + ˆx b z ] qk + h + g) 1 α)1 β) χ ι ι s ) ɛ = φz qk + h + g) c) For currency: [ˆx h 1 z) + ˆx b z ] qk + h + g) α1 β) χ ι ι i ) ɛ = h 1

11 3. Arbirage pricing: a) For capial: a q + µ q r = πσ q b) For governmen ransfers µ h + i r) h + µ g rg = σ h h + σ g )π 4. The law of moion of z saisfies 18) The goods marke clearing condiion is obained by using 13), n = z qk + h + g) and w = 1 z) qk + h + g). The deposi marke clearing condiion is obained similarly, using 14) and 16) and he fac ha deposi supply is φn. The currency marke clearing condiion is obained similarly, using 14) and 15). The arbirage pricing condiions are jus he differenial form of 9) and 1). Toal Wealh, spreads and currency holdings. banker and household problem are boh given by: The firs order condiions for ˆx in he ˆx = ρ χ 19) Since he ineremporal elasiciy of subsiuion is 1, boh he banker and he household spend heir wealh a a consan rae ρ independen of prices. Using 19) and he goods marke clearing condiion we can solve for oal wealh: ω = ak βρ 2) Hence in his economy oal wealh will be consan. This is because he Cobb-Douglas form of he x aggregaor implies ha consumpion is a consan share of spending he res is liquidiy services), he rae of spending ou of wealh is consan and oal consumpion is consan and equal o ak. Using 19), he deposi marke clearing condiion simplifies o: ρ1 α)1 β)ι ɛ 1 s ɛ = φz 21) 11

12 Solving 21) for s implicily defines bank spreads s i, z) as a funcion of i and z. Replacing 11) ino 21) and using he implici funcion heorem: s i, z) i s i, z) z α 1 ɛ) i ɛ = 1 α) s i, z) ɛ + αi 1 ɛ ɛs i, z) 1 22) φ αi 1 ɛ + 1 α) s i, z) 1 ɛ) 2 s i, z) ɛ = ρ1 α)1 β) [ 1 α) s i, z) ɛ + αi 1 ɛ ɛs i, z) 1] 23) By equaion 22), he spread is increasing in i as long as ɛ > 1. If currency and deposis are close subsiues, an increase in i, which increases he opporuniy cos of holding currency, increases he demand for deposis, so he spread mus rise o clear he deposi marke. By equaion 23), he spread is always decreasing in z. If bankers have a larger fracion of oal wealh, hey can supply more deposis so he spread mus fall o clear he deposi marke. Finally, using 19) and 2), he currency marke clearing condiion simplifies o: ak β α1 β)ιɛ 1 i ɛ = h 24) Having solved for s, 24) immediaely defines currency holdings h i, z) as a funcion of i and z. Aggregae risk sharing. household s choice of σ w are respecively: The firs order condiions for he banker s choice of σ n and he σ n, = π + 1 σ ξ, 25) σ w, = π + 1 σ ζ, 26) The firs erm in each of 25) and 26) relaes exposure o B o he risk premium π; his is he myopic moive for choosing risk exposure. The second erm capures he dynamic hedging moive, which depends on an income and a subsiuion effec. If he agen is sufficienly risk averse > 1), hen he income effec dominaes. The agen will wan o have more wealh when his invesmen opporuniies capured by ξ and ζ respecively) are worse. From 25) and 26) we obain he following expression for σ z : σ z, = 1 z ) 1 12 σ ξ, σ ζ, ) 27)

13 The objec σ z measures how he bankers share of wealh responds o he aggregae shock. The erm σ ξ, σ ζ, in 27) capures he relaive sensiiviy of bankers and households invesmen opporuniies o he aggregae shock. How his differenial sensiiviy feeds ino changes in he wealh share depends on income and subsiuion effecs. If agens are no very risk averse < 1), hen he subsiuion effec dominaes and in equilibrium hey will shif aggregae wealh o bankers afer aggregae shocks ha improve heir invesmen opporuniies relaive o households, i.e. when ξ goes up. In conras, if agens are highly ζ risk averse > 1) hey will shif aggregae wealh o bankers afer shocks ha worsen heir invesmen opporuniies relaive o households, i.e. ξ goes down. In he quaniaive ζ secion we focus on his second, more empirically relevan, case. We can use Io s lemma o obain an expression for σ ξ σ ζ : σ ξ σ ζ = ξz ξ ζ ) z ξi σ z z + ζ z ξ ζ ) i σ i 28) ζ Noice ha σ z eners he expression for σ ξ σ ζ : he response of relaive invesmen opporuniies o aggregae shocks depends in par on aggregae risk sharing decisions as capured by σ z. This is because in equilibrium invesmen opporuniies depend on he disribuion of wealh z, so we mus look for a fixd poin. Replacing 28) ino 27) and solving for σ z : Implemenaion. σ z = 1 z) 1 ξ i 1 z1 z) 1 ζ i ξ ζ ξ z ξ ) ζz ζ )σ i 29) Wih complee markes, here is more han one way o aain he exposure dicaed by equaions 25) and 26). We are ineresed in seeing wheher one possible way o do his is for banks o have a radiional balance shee: long-erm nominal asses, deposis as he only liabiliy and no derivaives. To be concree, imagine ha a banker wih wealh n issues deposis φn in order o buy 1 + φ) n worh of nominal zero-coupon bonds ha maure in T years. I s easy o show ha wih he ineres process 5), his balance shee produces he following exposure: σ n = 1 e λt ) 1 + φ λ σ i < 3) Expression 3) has a sandard inerpreaion: asses will be more sensiive o changes in ineres raes if hey have longer mauriy high T ) or if ineres changes are more persisen low λ). Furhermore, a more highly leverage bank will have greaer exposure, oher hings 13

14 being equal. Noice ha his implemenaion only works if he desired σ n is negaive, i.e. if bankers wan o lose wealh when ineres raes rise. Conversely, if he banker wans exposure σ n, invering 3) ells us he mauriy of he nominal asses he needs o hold: 5 Numerical Resuls T = 1 λ log 1 + λ 1 + φ We solve for he recursive equilibrium by mapping i ino a sysem of parial differenial equaions for he equilibrium objecs. We solve hem numerically using a finie difference scheme. In order o obain a saionary wealh disribuion we add ax on bankers wealh a a rae τ ha is redisribued o households as a wealh subsidy. Appendix A explains he numerical procedure in deail. Parameer values. σ n σ i ) 31) We choose parameer values so ha he model economy maches some key feaures of he US economy. Our choice of parameers is shown on Table 1. TO BE COMPLETED Parameer Value a 1 k 1 15 ī.4 λ.1 σ.3 ρ.8 α.95 β.75 φ 8 τ.19 ɛ 8 Table 1: Parameer values Aggregae risk sharing. Figure 1 shows aggregae risk sharing. The op panels show bankers exposure o ineres rae risk. If he nominal ineres rae rises by 1 basis poins, 14

15 T T bankers ne worh changes by σn σ i %. I is always negaive, so banks face large financial losses afer an increase in nominal ineres raes. This means i can be implemened wih a radiional banking srucure as explained above. Expression 31) gives us he mauriy of nominal asses bankers need o hold in order o implemen he desired exposure o ineres rae risk. This is shown in he lower panels of Figure 1. < n / < i z =.1 z =.7 z = i < n / < i i =.1 i =.4 i = z z =.1 z =.7 z = i =.1 i =.4 i = i z Figure 1: Aggregae risk sharing as capured by σn σ i of nominal asses T lower panels). upper panels) and he implied mauriy Exploring he mechanism. To undersand he mechanism, i is useful o spli i ino several pars. Firs, an increase in he nominal ineres rae i makes holding currency more cosly for agens. Since currency and deposis are subsiues, his increases he demand for deposis, oher hings being equal. We can see his in he expression for oal demand for deposis ρω1 α)1 β) αi 1 ɛ + 1 α) s 1 ɛ) 1 s ɛ which is increasing in i. Since he supply of deposis φzω is fixed, he spread on deposis s mus go up o clear he marke for deposis. Inuiively, banks are able o charge a higher spread for he liquidiy services hey provide when holding currency becomes more cosly. In addiion, if bankers share of aggregae wealh z goes down, his reduces he supply of 15

16 9/1 9/1 s s deposis and also drives he spread s up. This is shown in he upper panels of Figure 2. Since banks earn he spread s, and households don, bankers relaive invesmen opporuniies are beer when he ineres rae i is high and heir share of aggregae wealh z is low. This is capured by he raio ξ, shown in he lower panels of Figure 2. As a resul, bankers ζ relaive invesmen opporuniies are beer afer a moneary shock ha raises he ineres rae. Since > 1, his means ha he righ hand side of 27) is negaive: bankers share of aggregae wealh z goes down afer ineres raes go up. Since bankers benefi from a moneary shock ha raises ineres raes via higher spreads) i makes sense ha hey are willing o susain financial losses relaive o households in ha sae. These financial losses in urn reduce he supply of deposis and drive he spread s up, furher improving bankers invesmen opporuniies and amplifying heir incenives o ake on ineres rae risk ex-ane..8.6 z =.1 z =.7 z = i =.1 i =.4 i = i z z =.1 z =.7 z =.12 2 i =.1 i =.4 i = i z Figure 2: Spread on deposis s upper panels) and bankers relaive invesmen opporuniies capured by ξ lower panels). ζ Dynamics. Agens endogenous exposure o ineres rae risk leads o ineresing equilibrium dynamics, shown in Figure 3. The upper panels show he drif of bankers share of aggregae wealh z. The volailiy σ z is negaive hroughou, so afer a posiive shock db ha increases nominal ineres raes, z goes down. The drif of z is posiive for small z and negaive for high z. The drif is also higher when he ineres rae i is high. This dynamic 16

17 behavior is driven primarily by he spread s, which is higher when i is high and z low, and leads o a saionary disribuion. This is shown in Figure 7 z z =.1 z =.7 z = i 7 z i =.1 i =.4 i = z z =.1 z =.7 z = i =.1 i =.4 i =.7 < z -.6 < z i z Figure 3: The drif of z, µ z upper panels) and is volailiy σ z lower panels). References Achdou, Y., Lasry, J.-M., Lions, P.-L. and Moll, B.: 214, Online appendix: Numerical mehods for heerogeneous agen models in coninuous ime, Unpublished manuscrip, Princeon Univ., Princeon, NJ. Begenau, J., Piazzesi, M. and Schneider, M.: 213, Banks risk exposures, Sanford Universiy Working Paper. Cox, J. C., Ingersoll, Jonahan E., J. and Ross, S. A.: 1985, An ineremporal general equilibrium model of asse prices, Economerica 532), Drechsler, I., Savov, A. and Schnabl, P.: 214, The deposis channel of moneary policy, New York Universiy Working Paper. Driscoll, J. C. and Judson, R.: 213, Sicky deposi raes. 17

18 Seady sae pdf Figure 4: Saionary disribuion over i, z). Hannan, T. H. and Berger, A. N.: 1991, The rigidiy of prices: Evidence from he banking indusry, The American Economic Review pp Krishnamurhy, A., Vissing-Jorgensen, A. e al.: 215, The impac of reasury supply on financial secor lending and sabiliy, Technical repor. Landier, A., Sraer, D. and Thesmar, D.: 213, Banks exposure o ineres rae risk and he ransmission of moneary policy, Technical repor, Naional Bureau of Economic Research. Nagel, S.: 214, The liquidiy premium of near-money asses, Technical repor, Naional Bureau of Economic Research. Appendix A: Soluion Mehod NOT UP TO DATE) Reiremen In order o have a saionary disribuion for z we assume ha bankers reire randomly wih Poisson inensiy θ. Upon reiremen, hey keep heir wealh bu lose heir 18

19 abiliy o issue deposis, effecively becoming households. The HJB equaion hen becomes: 4 ρ ξ ñ) 1 1 = max x,σ n,µ n ˆx 1 1 ñ1 + ξ 1 ñ 1 µ n + µ ξ, 2 σ2 n 2 σ2 ξ, + 1 )σ ξ, σ n + θ )) ζ ñ) 1 1 ξ ñ) 1 1 s.. µ n = r + σ n π + φ i i d ) ˆxχ Replacing he firs order condiions 19) and??) which are unaffeced), we obain: ρ + θ 1 = 1 1 ξ 1 χ + r + 2 σ2 n, + φ ) i i d + µξ, 2 σ2 ξ, + θ ζ ξ ) ) Similarly, replacing??) and 26) in??) we obain he following HJB equaion for households: ρ 1 = 1 1 ζ Overview of he soluion procedure 1 χ + r + 2 σ2 w + µ ζ, 2 σ2 ζ, 33) The soluion mehod finds endogenous objecs as funcions of sae variables. We ll divide he equilibrium objecs ino wo groups. Denoe he firs group of variables by X = { ξ i, z), ζ i, z), q i, z), h i, z), g i, z), i d i, z) }. We ll express hese as a sysem of differenial equaions and solve i backwards. Denoe he second group of variables by Y = {ˆx b, ˆx h, σ z, σ n, σ w, π, r }. These variables con be solved saically for every possible value of X. Solving for Y given X Suppose we had found all he variables in X as funcions of i, z). By Io s Lemma i follows ha he law of moion of any of hese variables X is: dx i, z) = µ X i, z) d + σ X i, z) db 34) 4 Inroducing reiremen implies ha here is a disincion beween he ne worh of an individual banker and he collecive ne worh of all bankers, since he group of individuals who are bankers keeps shrinking. We reain he noaion n o refer o he collecive ne worh and denoe he ne worh of an individual banker by ñ. 19

20 where he drif and volailiy are µ X i, z) = X z i, z) µ z i, z) + X i i, z) µ i i) + 1 [ Xzz i, z) σz 2 i, z) z 2 + X ii i, z) σi 2 i) + 2X zi i, z) σ i i) zσ z i, z) ] 2 σ X = X z i, z) σ z i, z) z + X i i, z) σ i i) or, in geomeric form: where he drif and volailiy are dx i, z) X i, z) = µ X i, z) d + σ X i, z) db 35) µ X i, z) = X z i, z) X i, z) µ z i, z) + X i i, z) X i, z) µ i i) + 1 [ Xzz i, z) 2 X i, z) σ2 z i, z) z 2 + X ii i, z) X i, z) σ2 i i) + 2 X ] zi i, z) X i, z) σ i i) zσ z i, z) σ X = X z i, z) X i, z) σ z i, z) z + X i i, z) X i, z) σ i i) Hence if we know he funcions X and heir derivaives, we know heir drifs and volailiies a every poin of he sae space. Numerically, we approximae he derivaives wih finiedifference marices such for any se of values of X on a grid, he values of he derivaives on he grid are: X i D i X X z XD z X ii D ii X X zz XD zz X iz D i XD z The variables in Y can be found as follows. ι i, z) and χ i, z) are immediae from 11) and 12). ˆx b i, z) and ˆx h i, z) follow from he firs order condiions 19) and??). σ z i, z) follows from 29). By definiion, z = n qk + h + g 2

21 which implies: 5 σ z = σ n qkσ q + hσ h + σ g qk + h + g Knowing g, q, h and heir volailiies, as well as σ z, σ n i, z) can be obained from 36). π i, z) can hen be obained from he FOC??). σ w i, z) follows from he FOC 26). r i, z) follows from 33). 36) Solving for X The remaining equilibrium condiions are: [ˆx h 1 z) + ˆx b z ] βχ η k = a 37) qk + h + g [ˆx h 1 z) + ˆx b z ] χ ) ) η s ι 1 α)1 β) = φz 38) ι i i d [ˆx h 1 z) + ˆx b z ] χ ) η ι ) s h α1 β) = 39) ι i qk + h + g 1 ξ 1 1 χ + r + 2 σ2 n + φ ) i i d + µξ 2 σ2 ξ + θ ζ ξ ) 1 1 = ρ + θ 4) 1 a q + µ q r = πσ q 41) µ h + µ p ) h + µ g rg = σ h h + σ g )π 42) Equaion 37) is he marke clearing condiion for he goods marke; 38) is a marke clearing condiion for he deposis marke; 39) is a marke clearing condiion for he currency marke; 4) is he banker s HJB equaion; 41) is an arbirage-pricing condiion for capial and 42) is an arbirage-pricing condiion for governmen ransfers. We find he funcions X by differeniaing equaions 37)-42) wih respec o ime and finding X such ha he ime derivaives are equal o zero. Differeniaing yields he following sysem of differenial equaions: ξ ζ q A ḣ ġ i d = B 43) 5 g is expressed in absolue erms using 34) bu n, q, h are expressed in geomeric erms using 35) 21

22 where A is a 6 6 marix wih enries: 1 ξ χ 1 a 11 = βχ η z 1 a 12 = βχ η 1 z) 1 k 2 a 13 = a qk + h + g) 2 k a 14 = a qk + h + g) 2 k a 15 = a qk + h + g) 2 [ a 16 = zξ 1 1 ζ χ z) ζ 1 ) 1 1+ χ βχ η + [ˆx h 1 z) + ˆx b z ] ] βηχ η 1 a χ χ ) η ι a 21 = 1 α)1 β) ι χ a 22 = 1 α)1 β) ι a 23 = a 24 = a 25 = ) s z 1 i i d ) s 1 z) 1 ) η ι i i d 1 ξ χ 1 1 ζ χ 1 a 26 = [ˆx h 1 z) + ˆx b z ] 1 α)1 β) i i d) [ s ι s η χ η ηχ 1 a χ + s η) ι 1 a ι + s i i d) ] 1 ] [ζ z) + ξ 1 1+ z χ χ ) ) η s ι 1 α)1 β) a ι i i d χ 22

23 χ ) η ι a 31 = α1 β) ι i χ ) η ι a 32 = α1 β) ι i hk a 33 = qk + h + g) 2 qk + g a 34 = qk + h + g) 2 h a 35 = qk + h + g) 2 ) s z 1 1 ξ χ 1 ) s 1 1 z) 1 ζ χ 1 a 36 = [ˆx h 1 z) + ˆx b z ] α)1 β)ι s η χ η i [ ] s ηχ 1 a χ + s η) ι 1 a ι + ] [ζ z) + ξ 1 1+ z χ χ ) η ι ) s α)1 β) aχ ι i a 41 = 1 ξ a 42 = 1 ζ a 43 = a 44 = a 45 = a 46 = a 51 = a 52 = 1 ζ a 53 = 1 q a 54 = a 55 = a 56 = 23

24 a 61 = a 62 = g + h ζ a 63 = a 64 = 1 a 65 = 1 a 66 = and B is a 6 1 vecor wih enries b 1 = b 2 = b 3 = b 4 = 1 ξ 1 ζ ξ χ 1 + ρ ) σ2 ξ + θ 1 ρ + θ 1 b 5 = a q + µ q ρ 1 + b 6 = µ h + i) h + µ g 1 ζ 1 1 ζ 1 1 χ ρ 1 1 ζ 1 where for any variable X, µ X is defined as The algorihm for finding X is as follows. χ 1 2 σ2 w µ ζ + 2 σ2 ζ + 2 σ2 n + φ i i d ) + µξ + 2 σ2 w + µ ζ 2 σ2 ζ πσ q 1 χ µ X µ X Ẋ X 2 σ2 w µ ζ + 2 σ2 ζ 1. Guess values for X a every poin in he sae space ) g + h) σ h h + σ g )π 2. Compue he derivaives wih respec o i and z by a finie difference approximaion 3. Compue Y a every poin in he sae space given he guess for X. 4. Compue Ẋ a evey poin in he sae space using 43) 5. Take a ime-sep backwards o define a new guess for X 24

25 6. Repea seps 1-5 unil Ẋ. The condiion Ẋ = is equivalen o saying ha equilibrium condiions 37)-42) hold. Finding he seady sae Once we solve for he equilibrium, his defines drifs and volailiies for he wo sae variables: µ i i, z), σ i i, z), µ z i, z), σ i, z). The densiy f i, z) of he seady sae disribuion is he soluion o he saionary Kolmogorov Forward Equaion: = i [µ i i, z) f i, z)] z [µ z i, z) f i, z)] 44) [ σi i, z) 2 f i, z) ] + 2 [ σz i, z) 2 f i, z) ] ) i 2 z 2 i z [σ i i, z) σ z i, z) f i, z)] We solve his equaion by rewriing i in marix form. 6 The firs sep is o discreize he sae space ino a grid of N i N z poins and hen conver i o a N i N z 1 vecor. Le vec ) be he operaor ha does his conversion. We hen conver he differeniaion marices so ha hey are properly applied o vecors: D vec i I Ni D i D vec ii I Ni D ii D vec z M I Nz D z ) M D vec zz M I Nz D zz ) M D vec iz Di vec where denoes he Kreonecker produc and M is he vecorized ranspose marix such ha Mvec A) = vec A ). Now rewrie 44): D vec z [ D vec i D vec ii diag vec µ z )) vec f)) ] diag vec σi 2 )) vec f)) + Dzz vec diag vec σz)) 2 vec f)) diag vec σ i )) diag vec σ z )) vec f)) diag vec µ i )) vec f)) D vec z +2D vec iz = and herefore Avec f) = 45) 6 See Achdou e al. 214) for deails on his procedure. 25

26 where A = Di vec + 1 [ D vec ii 2 diag vec µ i )) Dz vec diag vec µ z )) diag vec )) σi 2 + D vec zz diag vec σz 2 )) + 2D vec iz diag vec σ i )) diag vec σ z ))) ] Equaion 45) defines an eigenvalue problem. We solve i by imposing he addiional condiion ha f inegraes o 1. 26