Implementing Monetary Policy Without Reserve Requirements

Transcription

1 Implementng Monetary Polcy Wthout Reserve Requrements Cornela Holthausen Cyrl Monnet Flemmng Würtz Aprl 2, 2008 Abstract We propose a new framework to mplement monetary polcy n a corrdor system, whch does not rely on mposng reserve requrements. A man lendng faclty s ntroduced, where banks can borrow a lmted amount of cash at a rate that we suppose s n the mddle of the corrdor. There s no mantenance perod, so that short term rates are not a ected by an end of perod e ect. We formally show that ths framework wll reduce the volatlty of the overnght rate relatve to systems that use reserve averagng. Also, parametrzng the framework, we show that the varablty of the short term rate can be practcally elmnated wthout a ectng tradng n the nterbank market. Introducton Most central banks now mplement monetary polcy by targetng some level for short term money market rates, normally the overnght rate. It s therefore mportant for the credblty of monetary polcy that money market rates are e ectvely steered to the target set by the central bank. To adjust the overnght rate and control ts varablty, central banks use manly three nstruments, wth whch they steer the margnal value of lqudty. Frst, usng open market operatons, central banks steer aggregate lqudty n the system. Second, central banks can o er a lendng and a depost faclty, where commercal banks can borrow or depost reserves at some xed nterest rates. These rates then form a natural corrdor for the overnght money market rate, whch lmts the sze of ts uctuatons. Generally, central banks use a combnaton of both tools. For nstance, the Federal Reserve conducts reverse repos every day and although t does not o er a depost faclty wth a rate d erent from We are grateful to Nuno Cassola and Jens Tapkng for comments and dscussons. The vews do not re ect those of the ECB or of the Eurosystem.

2 zero, t mantans a lendng faclty, the dscount wndow. Smlarly, the European Central Bank o ers a depost and a lendng faclty and conducts weekly re nancng operatons and occasonal ne tunng operatons. Fnally, central banks can also requre commercal banks to hold a fracton of ther deposts as reserves. When banks are requred to hold an average amount of reserves over a certan perod, the mantenance perod (usually one month), these reserve requrements can be used to bu er aganst ther lqudty shocks. The bu er role of average reserves functons n a smple way. Whenever a bank receves an (unexpected) lqudty shock, t can ether adjust ts reserves holdngs for that day or, f stll possble, borrow funds on the nterbank market. Adjustng reserves balance for a day s an opton as banks need to ful ll requrements on average: If a bank taps n ts reserves today, then t wll have to adjust ts reserves by the correspondng amount later n the perod. As t also helps to smooth dosyncratc shocks, the bu erng functon of reserve requrements s an mportant tool to mantan a low level of volatlty of the short term money market rates. Moreover, reserve requrements s a tool for reducng the number of open market operatons, as t s helpful n absorbng uctuatons n aggregate lqudty. However, reserve averagng creates ssues of ts own. Frst, from a practtoner pont of vew, t s rather complcated for banks to dentfy ther optmal reserve requrements ful llment path. Second, the bu er functon of average reserves s nexstent n the last day of the mantenance perod and, as s the case n the euro money market, the varablty of rates wll ncrease as the end of the mantenance perod approaches. 2 The volatlty on the last day of a mantenance perod s bascally a consequence of banks beng eager to borrow (or lend) the requred reserves and dodge the penalty rate assocated wth a recourse to the facltes. To some extent, ths volatlty splls over to days farther away from the end of the mantenance perod. Thrd, there s evdence that the overnght rate does not satsfy the martngale property accordng to whch the prevalng rate s the one expected at the end of the mantenance perod. 3 Perez-Quros and Rodrguez-Mendzabal (2006) argue that rskneutral banks back-load ther reserve requrements to use the bu er functon of requred reserves to ts full extent, thus puttng upward pressure on rates durng the last days of the mantenance perod. Smlarly one could also argue that rsk-averse banks front-load ther reserve requrements so as to lmt the recourse to the borrowng faclty on the last See Cassola (2007) for a recent study on banks requred reserves ful llment path n the euro area. 2 See for nstance Würtz (2003) or Hamlton (996). 3 Prat et. al. (2002) nd that the euro overnght rate drops towards the end of the mantenance perod. Also, Perez-Quros and Mendzabal (2006) nd evdence that the rate ncreases on the last tradng day of the mantenance perod. 2

3 day of the mantenance perod, thus puttng downward pressure on rates. Whether there s an upward or a downward pressure on rates, the fact that the overnght rate s not a martngale s problematc, as t s ether a symptom that there s some complcatons n the money market, or that the central bank s unable to acheve the rate t targets. 4 As a matter of facts, t seems that the ECB resorted on several occasons to open market operatons on the last day of the mantenance perod n order to counter the deleterous e ects of reserve averagng. Another drawback of requred reserves s that they ncrease the overall sze of a central bank s balance sheet, whch s nconsstent wth the lean balance sheet prncple. As a matter of prncple, a central bank should am to reduce the amount of nancal resources t absorbs nto ts balance sheet. Even though reserve requrements are sometmes fully remunerated they can stll be regarded as a consumpton of nancal resources, tyng up collateral of banks that could potentally be used more pro tably. 5 Aganst ths background, we propose an mplementaton framework whch does not use reserve requrements, and therefore elmnates the problems assocated wth dealng wth mantenance perods. The bu er role of reserve requrements s nstead replaced by a Man Lendng Faclty (MLF). In ths system, commercal banks hold remunerated current accounts wth the central bank. In case ther balance s negatve, banks borrow overnght at the MLF, at xed rate and aganst proper collateral, any amount of reserves up to a lmt. In the proposed framework, the central bank stll operates the lendng and depost facltes and one may suppose that the MLF rate s the target rate of the central bank. Therefore, f banks need lqudty (have negatve balance on ther current account) that they cannot obtan on the nterbank market, they wll automatcally be drected to the MLF and then to a resdual lendng faclty. Snce the amount borrowed at the MLF s capped, banks wll stll have recourse to the resdual lendng faclty f they receve a lqudty shock that s larger than the borrowng lmt at the MLF. In ths framework, the central bank carres out re nancng operatons, calbrated wth the vew to steer the average expected draw from the MLF to be one half of the aggregate lmt. In ths way, the central bank ensures equal possblty to absorb lqudty dranng and lqudty absorbng shocks. The frequency and average volume of these operatons need to be made consstent wth the MLF lmt. We show that the lower the sze of the lmt, the more precsely the expected draw from the MLF should be calbrated n order to avod uctuatons n the overnght rate. A more frequent calbraton of the draw from the MLF n turn necesstates a larger average volume 4 On the contrary, Cassola (2007) argues the Euro area overnght rate (the EONIA) sats es the martngale property. 5 See Papada and Würtz (2007) for a detaled analyss of the lean balance sheet prncple. 3

4 n, and a hgher frequency of, re nancng operatons. At the extreme the MLF lmt could be set su cently large to absorb all uctuatons n the temporary component of the lablty sde of a central bank s balance sheet. To llustrate the e ect of ntroducng the MLF, the two tables below show a smpl ed verson of a central bank balance sheet under the two frameworks. The rst table shows the balance sheet wth reserve averagng. Assets Labltes Re nancng operatons ( week & 3 months) 300 Banknotes 400 Net nancal assets 200 Reserve requrements 00 Total assets 500 Total labltes 500 The followng table presents the smpl ed balance sheet of a central bank that would adopt the MLF framework. Assets Labltes Man lendng faclty 00 Banknotes 400 Re nancng operatons ( week & 3 months) 00 Reserve requrements 0 Net nancal assets 200 Total assets 400 Total labltes 400 Note that buldng the bu er functon of the MLF on the asset sde of the balance sheet, rather than on the lablty sde as s the case of requred reserves, reduces the sze of the balances sheet and s therefore consstent wth the lean balance sheet prncple. We propose a smple model of the MLF framework, as a rst step to analyse how the MLF framework fares relatve to a setup wth reserve requrements. The result of the stylsed analyss s that a reasonable lmt on the MLF reduces the varablty of the nterbank rate sgn cantly, wthout a ectng much nterbank market actvty. The reason s that, wth an aggregate lqudty de ct, the nterbank market rate s set at the MLF rate so that banks trade ther dosyncratc shocks away. Also, we show that the MLF just takes over the role of reserves n an average reserve requrement system. In the Appendx, we use the model of the nterbank market rate determnaton wth requred reserves proposed by Gaspar et. al. (2007) to compare the varablty of short term rates, and the actvty on the nterbank market for both mplementaton frameworks. The paper s structured as follows. In Secton 2, we present the theoretcal envronment for the new framework. In Secton 3 we solve for an equlbrum and present some basc 4

5 results. In Secton 4 we study how MLF systems compare wth reserve averagng systems. Secton 5 extends the basc analyss to consder the e ects of transacton costs on tradng actvty. We consder d erent ways to set up the MLF lmts n Secton 6 and we conclude n Secton 7. 2 Envronment Ths envronment s partally based on Poole (968) and Gaspar, Perez-Quros, Rodrguez- Mendzabal, (GPR, 2007). There are n commercal banks and a central bank. Commercal banks mantan deposts wth the central bank, called current accounts, to ful l payments oblgatons. We wll call balances the amounts on these current accounts. At the start of the day, all current accounts are cleared n the sense that banks have a zero balance on ther current account. A typcal day for a commercal bank can be decomposed n three stages. In the rst stage, banks receve an early lqudty shock caused by autonomous factors and central banks operatons. Gven balances on ther current account, banks can trade balances n the nterbank market. In a second stage, banks receve a (late) lqudty shock. Gven the sze of ther current account balance after ths shock, banks are drected to one of the three standng facltes, the man lendng faclty (MLF) where they can borrow up to a lmt B for each bank at rate b, the resdual lendng faclty (RLF) where they can borrow any amount at a rate ` b and nally, postve account balances are swept nto the depost faclty, where the remuneraton rate s d. In the last stage, and dependng on ther nancal clams contracted n the prevous two stages, banks receve nterest rates on ther current account and pay nterest rates from ther current account. Banks wll seek to maxmse(mnmse) the nterest payments they receve(make) from ther operatons. We abstract from default ssues, so that the central bank does not requre any collateral. 5

6 Banks are not requred to hold reserve, so there s no mantenance perod. Instead, banks use all pro ts accurng n the last stage, so that they start each cycle/day wth a zero account balance. As a result, there s no dynamcs nvolved, and we can assume that there s only one day wthout loss of generalty. Followng the early shock bank starts the day wth an amount of balances s 2 R. 6 The aggregate amount of balances s then P n = s = S. We assume that there s a structural lqudty de ct due to, for nstance, banknotes n crculaton, whch s partally o set by the central bank s conduct of lqudty provdng operatons. Hence, n general S s negatve. We do not model central bank s operatons here. We descrbe n detals below the functonng of the nterbank market and of the three standng facltes. Interbank market When the nterbank market opens, banks balances are heterogeneous so that they have d erent expected lqudty needs. As a consequence, they have an ncentve to trade n the nterbank market f there s a wedge between the lendng rate and the depost rate. When bank accesses the nterbank market, we wll denote ts trade by y. If y < 0 then bank lends balances, whle f y > 0; bank borrows balances. The nterbank market rate s set so as to clear the market. Standng facltes and end of day shocks At the end of the tradng sesson, gven banks ntal balances and ther trades on the nterbank market, bank has balances s + y. Each bank then receves a late shock v F ( ; ) whch dstrbuton s common knowledge. Ths shock can be bank spec c, and s not necessarly ndependent across banks. The bank must have enough balances to cover ths lqudty shock. Otherwse, the bank s automatcally drected toward the standng facltes provded by the central bank to borrow balances. If a bank ends the day wth a postve account balance, t s automatcally swept to the depost faclty. There are three standng facltes, the man lendng faclty (MLF), the resdual lendng faclty (RLF) and the depost faclty (DF). Any bank that ends the day wth a negatve account balance s rst drected to the MLF. The amount borrowed from the MLF s lmted and cannot be more than B 0 for bank. All banks pay the MLF rate b on any amount borrowed there. Once the MLF cap s reached, bank s automatcally drected to the RLF and the central bank charges ` b on any amount borrowed there. Fnally, postve account balances are remunerated at the DF rate d. We wll assume that shocks are evenly 6 s also denotes the sze of the early lqudty shocks. We therefore do not model explctly ths shock. 6

7 dstrbuted across banks, so that a postve shock for bank corresponds to a negatve shock for some bank j and there s no aggregate lqudty creaton. Settlement After the lqudty shock s realsed, banks repay loans and redeem deposts. Therefore, the amount of balances of bank at ths stage s s 0 = s y ( + m ) b ( + b ) ` ( + `) + d ( + d ) where b s the amount borrowed from the MLF, ` s the amount borrowed at the RLF and d s the amount deposted at the DF. We assume that, at any stage, bank seeks to maxmze s 0. 3 Equlbrum We now solve for the equlbrum n the nterbank market,.e. the nterbank market clearng nterest rate, rst ndng an expresson for banks payo as a functon of ther tradng actvtes on the nterbank market. Lqudty shocks and access to standng facltes We let V (s ; y ) be the expected payo of bank when t exts the nterbank market wth balances s, of whch y has been borrowed from the nterbank market. Each bank then receves a late shock v F ( ; ). If the bank has enough balances to cover ths shock (s ), then t stll has a postve balance on ts current account s, whch s remunerated at the depost faclty rate d. Otherwse, f s <, bank s current account s automatcally credted wth the amount s > 0, so that t holds a zero current account balance. If s + B, bank s current account s credted wth the b = s and s charged the MLF rate b. Bank repayment to the central bank s then ( s ) b for all such that s < s + B. Otherwse, f > s + B, then bank s current account s credted wth B from the MLF, and wth the remanng mssng amount (s + B ) from the lendng faclty. In ths case, bank s repayment from shock to the central bank s [ (s + B )] ` + B b = ( s ) ` + B ( b `). Therefore, bank s automtc recourse to the facltes can be summarsed as follows: s, depost s at the DF s < s + B, borrow s from MLF s + B <, borrow B from MLF, borrow (s + B ) at RLF 7

8 Therefore, bank s expected payo from borrowng y on the nterbank market V (s ; y ) has the followng expresson V (s ; y ) = ( + m ) y + Z+ Z s ( + d ) (s ) f () d s Z +B s +B [( + b ) B + ( + `) ( (s + B ))] f () d: s ( + b ) ( s ) f () d The expected value of leavng the nterbank market wth balance s, of whch y has been borrowed on the nterbank market s consttuted of four parts: rst, the cost to repay the loan contracted on the nterbank market. Second, the expected gans from depostng reserves at the DF when the lqudty shock s not large. Thrd, the expected cost of borrowng at the MLF f the lqudty needs are lower than the cap. Fnally, the expected cost of borrowng up to the cap at the MLF and even more at the RLF when the lqudty needs are greater than the cap. The bank wll seek to borrow lqudty on the nterbank market to maxmze ths expected value. Interbank Market In the nterbank market, a bank solves Z (s) = max V (s + y; y) y Note that we have omtted the dependency of ths value functon on the aggregate state S. Implctly, we are therefore assumng that banks are too small to be able to a ect the equlbrum rate on ther own. 7 The rst order condton gves V y + V s = 0. From the de nton of V, we have (see the Appendx for detals), V y (s + y ; y ) = ( + m ) V s (s + y ; y ) = ( + `) ( b d ) F (s + y ) (` b ) F (s + y + B ) = ( + d ) F (s + y ) + ( + b ) [F (s + y + B ) F (s + y )] + ( + `) [ F (s + y + B )] In words, the value of an ncremental ncrease n borrowng on the nterbank market s the cost to pay back ths loans, ( + m ). Also, the margnal value of bank s account balance s the expected gan from havng a postve account balance at the end of the day. If the shock s not severe, the extra balance can be deposted at the DF and earn nterest rate 7 See Ewerhart et. al. (2006) for a framework where ths s not the case. 8

9 d. If the schock takes some ntermedate values (between s + y and s + y + B ) then the ncremental balance s used to cover the shock, and therefore saves on the borrowng cost at the MLF, + b. Fnally, f the shock s severe enough, the ncremental balance s used nstead of havng recourse to the RLF, whch saves + `. In equlbrum, the margnal bene t of borrowng on the nterbank market s the margnal bene t of ncreasng one s account balance V s, whle the margnal cost s the nterbank market rate, + m. Hence, y solves d F (s + y ) + b [F (s + y + B ) F (s + y )] + ` [ F (s + y + B )] = m () Fgure llustrates the demand curve for banks on the nterbank market for d erent level of caps, where borrowng (lendng f negatve) s on the x-axs. Note that the cap has a non-lnear e ect on the demand schedule. In partcular, as the sze of the cap becomes relatvely large compared wth the uncertanty regardng the sze of the lqudty shock, banks demand schedule become very elastc, up to the pont where t reaches ts cap. The reason s that when the nterbank market rate s hgher than the MLF rate, a bank arbtrages both rates, by lendng on the nterbank market, as t can borrow from the MLF at a cheaper rate. As the borrowng lmt ncreases, the lendng actvty for all nterbank rates above the MLF rate also ncreases. Importantly, snce the demand s very elastc for rates n a neghborhood of the MLF rate, a large varaton n the uncertan component of the demand for lqudty can be accomodated by a relatvely small change n the nterbank market rate. Therefore, the equlbrum rate volatlty s decreasng when the lmt on the MLF becomes larger. Note that when banks are perfectly homogeneous, so that s = s j = s, and B = B j = B for any 6= j, the only equlbrum s when y = y j = 0, so that the equlbrum rate sats es, d F (s) + b [F (s + B) F (s)] + ` [ F (s + B)] = m : In ths case, Fgure shows, the nterbank market rate wll only be n a regon near b f the structural lqudty de ct - de ned as P n = s =n = s < 0 - s su cently large. Below we show rgorously that ths ntuton s correct. Fnally, Fgure also shows that a relatvely small cap sze wll not a ect the varablty of the nterbank market rate much relatve to a standard daly reserve requrement system. We can now de ne an equlbrum for ths economy. De nton Gven polcy rates (`; b ; d ), and aggregate balances S, an equlbrum s an nterbank market rate m and allocaton fy g n =, such that, gven m, y solves () for all and P y = 0. 9

10 Fgure : Demand functon of a bank on the nterbank market, gven shocks are dstrbuted accordng to a N(0; 66 ), where d = 3, ` = 5 and b = 4, for d erent levels of caps. Proposton An equlbrum exsts and s unque. Proof. Let us rst de ne the functon (x) = ( b d ) F (x) + (` b ) F (x + B ). Note that s a functon of B and therefore can d er across banks. (x) s de ned over the nterval de nng F () (whch we wll take to be [ ; +]). Hence ( ) = 0 and () = ` d. Also, s strctly ncreasng over ths nterval. Therefore, ts nverse s well de ned. Then an equlbrum s de ned by m and y such that (s + y ) = ` m for all and P y = 0. Hence, we have s + y = (` m ). Usng the market clearng condton we obtan that m solves X (` m ) = X s Snce s contnuous and monotone for all and (0) = and (` d ) =, there exsts by the ntermedate value theorem a unque m 2 [ d ; `] such that the above equalty holds. Note that f B = B for all, then we can derve a smple expresson for m as X m = ` ( b d ) F s n (` X b ) F s n + B : (2) 0

11 In ths case, we also have that y = X s n s : (3) An mportant mplcaton of ths equaton s that the sze of the lmt on the MLF does not a ect tradng on the nterbank market, but only a ects the equlbrum nterbank market rate. Also, note that f B = 0 for all, m = ` ( F (s + y)) + d F (s + y), whch s the standard equlbrum equaton n an envronment wth daly requred reserves and no averagng. 8 In words, settng the MLF lmts to zero, or equvalently elmnatng the MLF, s smlar to ntroducng a regme wth daly requred reserves and no averagng. Then n ths case, y k s gven by m F (s + y) = ` : ` d Also f B = +, so that n nte recourse to the MLF s possble, then y s gven by m F (s + y) = b : b d Corollary 2 For all S, there s B such that f B > B for all, then b m. Proof. Set B to a su cently large amount that F (s + y + B ) =. Then equaton (??) becomes m = b ( b d ) F (s + y ) < b. By contnuty, ether m < b for all B 0, or there s B such that f B > B for all, then b m. Fnally, under some mld assumptons, we show that there s an average reserve de ct S=n so that m = b = (` + d ) =2. Lemma 3 Suppose the densty functon of the shock dstrbuton s centered around ts mean, B = B for all and b = (` + d ) =2. If S=n = B=2, then m = (` + d ) =2. Proof. We showed that f B = B for all then m s m = ` ( b d ) F (S=n) (` b ) F (S=n + B) : Replacng m = b = (` + d ) =2, we obtan that S=n must solve F (S=n) = F (S=n + B) : However, snce F s centered around ts mean, we have that F ( x) = F (x) for all x. Hence, S=n = B=2 solves ths equaton. Therefore, n order to ht the mddle range of the corrdor mpled by the depost and lendng facltes rates, t s su cent for the central bank to target an average lqudty de ct 8 See for nstance Gaspar et. al (2007).

12 that equals half the cap n the MLF. When ths s the case, note that we can decompose each s n two parts: s = B=2 + A, where A = d + a can be tself decomposed between the uncertan porton of the aggregate lqudty de ct d and bank dosyncratc early shock, a. When n s large, the law of large number mples that d =n 0, so that P n = A =n P n = a =n,.e. the number of banks mpacts the extent to whch the uncertanty about the aggregate lqudty shock s a ectng ndvdual banks relatve to ther dosyncratc shock. As n become large, the nterbank market s used n order to trade away only bank s dosyncratc shock, whle the aggregate lqudty de ct s covered by a sure recourse to the MLF. As larger caps are mposed, banks can also cover some of ther ndvdual schock at the MLF, whch then reduces the varablty of the nterbank market rate. 4 How do MLF systems compare wth Reserve Averagng systems? In ths secton, we show that a system relyng on the MLF can replcate the outcome of a reserve averagng system n the rst days of the mantenance perod, that s, n a reserve averagng system when the money market rate s the least volatle. We show that ths s the case when we restrct the mantenance perod to two days. 9 It s relatvely straghtforward to derve the soluton of the problem of a bank n the rst day of a two-days mantenance perod. 0 When bank starts ths day wth balance s ; on ts current account, and faces a daly reserve requrement of R (or a reserve requrement of 2R over the whole mantenance perod), bank chooses to borrow y ; on the nterbank market, where y ; sats es d F (s ; + y ; 2R ) + ` [ F (s ; + y ; )] +E [ m;2 ] [F (s ; + y ; ) F (s ; + y ; 2R )] = m; (4) Intutvely, a bank chooses y so as to equate ts margnal cost m;, and ts margnal bene t. The margnal bene t from borrowng s composed of three parts. Frst, when the bank current account balance s ; + y ; s nsu cent to cover the lqudty shock, the addtonal unt borrowed allows the bank to borrow an addtonal unt less at the lendng faclty, whch has a cost `. Ths event has probablty F (s ; + y ; ). Snce the bank 9 It s possble to extend the results to an n-day mantenance perod. 0 See for nstance Whtesell (2006b). 2

13 s requred to hold a daly average of R, t does not have to hold R and therefore t can use all of ts account balance to bu er ts shock. Second, when the bank current account s large enough to cover both ts lqudty shock and ts reserve requrement for the whole mantenance perod 2R, then t can depost the rest at the depost faclty and earn return d on the addtonal unt borrowed. Ths event has probablty F (s ; + y ; 2R ). Fnally, for moderate shocks, the bank wll nether access the lendng nor the depost faclty, and the addtonal unt borrowed can be used on the nterbank market n the last day of the mantenance perod, whch has an expected return of E [ m;2 ] (where m;2 s the money market rate n the second perod of the mantenance perod). Ths event has probablty F (s ; + y ; ) F (s ; + y ; 2R ). Now, n the last day of the mantenance perod, the bank wll set y 2 so that m = ` [ F (s ;2 + y ;2 R ;2 )] + d F (s ;2 + y ;2 R ;2 ) (5) where s 2 and R 2, are respectvely the account balance and the reserve de cency of the bank at the start of the last day of the mantenance perod. We can now state the man result of ths secton. Proposton 4 Any equlbrum fy g n = n an average reserve requrement system s an equlbrum allocaton n a MLF system. However, there are equlbrum allocatons n a MLF system that are not mplemented wth an average reserve requrement system. Proof. To replcate the allocaton of an averagng system, t s enough to show that approprately chosen non-stochastc component of the account balance s, borrowng lmts B at the MLF and the MLF rate b, can generate the exact same borrowng level for bank, for all. If ths s the case then the equlbrum rate wll be dentcal across the two systems. Snce the averagng system s dynamc, the MLF system wll only be able to replcate t f we allow the borrowng lmt to be tme dependent. Now, gven s ;, R, and E [ m;2 ] for bank n the average reserve requrement system, set s, B ; and b such that: s = s ; 2R B ; = 2R b = E [ m;2 ] Then t s obvous that equaton () de nng y MLF n the MLF system s equvalent to (4) de nng y RA n the reserve averagng system. Snce there s a unque equlbrum n the RA and MLF systems, we obtan y MLF = y RA for all and MLF m = m;. Now, gven s ;2 and R ;2 for bank n the average reserve requrement system, set s, B ; and b such that: s = s ;2 R ;2 and B = 0. Then t s clear that equaton () de nng y MLF s equvalent 3

14 to (5) de nng y RA ;2. To show the second part of the proposton, t s enough to set B > 0 and b 2 ( d ; `) for all perod. Indeed, n such a case, snce there s no bu er functon from reserves n the last day of the mantenance perod, t wll be the case that y RA some. 6= y MLF, for Another way to state the above proposton s that the varablty of the overnght rate wll be smaller n the MLF system. The reason s that on the last day of a mantenance perod, requred reserves do not play ther bu er functon anymore, whle the MLF can always act as a bu er as long as B > 0. Therefore, gven B s set to replcate the allocaton n the rst days of a mantenance perod (when the volatlty of the overnght rate s the smallest), the rate under a reserve averagng system wll always be more volatle than under the MLF scheme, as the end of the mantenance perod gets closer. 5 Transacton costs In ths secton we extend our theoretcal framework to ntroduce transacton costs. When banks decde to be actve on the nterbank market, they have to pay a xed nomnal cost > 0 whch s ndependent of the sze of ther transactons. Therefore, when bank accesses the nterbank market, we wll denote ts trade by y. If y < 0 then bank lends balances, whle f y > 0; bank borrows balances. However, ts balance at the closng of the nterbank market s ether s + y f bank was actve on the nterbank market, or s, otherwse. What d erentate the xed cost from tradng s that there s no nterest beng pad/receve on the xed cost. The decson of bank s now whether to become actve or nactve on the nterbank market and f so, whch level of actvty to choose. We wll denote by z (s ; m ) 2 f0; g, the decson of bank to become actve (z = ) or nactve (z = 0) on the nterbank market gven ts ntal account balance s and the market rate m. Therefore, a bank now solves, Z (s) = max V (s + z (s; m) (y ) ; z (s; S) y) y;z(s; m)2f0;g Hence, bank wll access the nterbank market whenever V (s + y ; y ) V (s ; 0). We therefore obtan the followng result, whch proof has been relegated to the Appendx. Lemma 5 Gven m and B, bank does not access the nterbank market f s 2 (s ; s ) and otherwse, sets y = y such that ( b d ) F (s + y ) + (` b ) F (s + y + B ) = ` m (6) 4

15 The above gure shows the demand functon for a typcal bank when t faces transacton costs = 0. The dotted lne shows the same demand functon when there s no transacton costs. Under the chosen parameters, a bank wll not access the nterbank market (y = 0) for rates that are close to the MLF rate. The money market wll therefore only be used to accomodate large shocks to the bankng sector, whle small shocks wll be accomodated by the MLF. When there are transacton costs, the use of the MLF s welfare enhancng as t reduces the costs of tradng relatve to a reserve averagng system. Indeed, on the last day of the mantenance perod of a reserve averagng system, banks wll have to cover ther early shock by accessng the market, snce the bu er role of requred reserves s not any more present. However, n ths case, they have to bear transacton costs. It s also nterestng to know how tradng actvty n the presence of tradng costs s a ected by a change n the borrowng lmt at the MLF. However, whle t s possble (and actually easy) to derve the e ect of a change n B on the demand schedule of a gven bank (the no-trade regon as depcted above decreases), t s more nvolved to derve the general equlbrum e ects of an ncrease n MLF borrowng lmts, snce t wll also a ect the equlbrum nterest rate. We therefore leave ths ssue for future work. 5

16 6 Settng the MLF lmts To mplement the proposed scheme, a central bank wll have to decde how t sets the lmts on the MLF. In ths secton, we propose three possbltes to set lmts on the MLF for each bank. Average reserve requrements As shown n Secton 4, settng the MLF lmts to the level of requred reserves over an hypothetcal mantenance perod would surely acheve low nterest rate varablty, as long as the central bank targets a su cently large lqudty de ct. The mplementaton of the MLF lmts would be very smlar to the functonng of the reserve averagng system: based on data from ther balance sheets, the central bank would calculate banks MLF lmts vald for, say, the forthcomng month. Each month, a new value for the MLF lmts would be calculated. Asde from the calculaton methods of the MLF lmts, all other ngredents of the average reserve scheme would be removed from the mplementaton framework. Auctonng lmts The prevous methods of calculaton only reles on a rule mposed by the central bank. However, there s lttle theoretcal groundng for the optmal level of requred reserves. In fact, many central banks now do not mpose requred reserves any more, but rather trust bank n holdng the approprate amount of voluntary reserves. Also, the MLF lmts as calculated on the bass of requred reserves could be nterpreted as beng a subsdy to banks elgble to access the central bank facltes. Rather the central bank could decde on an aggregate amount of borrowng at the MLF and aucton ths amount to elgble banks. Detals of the aucton ( xed or varable rate tender, etc.) should be carefully studed. The aucton would ntroduce an nterestng element as those banks n needs for larger lmts would prce ther needs themselves. Also, ths would allow the central bank to control the overall maxmum amount to be borrowed from the MLF, and as such would facltate the targetng of the lqudty de ct. Buyng lmts (Whtesell) Fnally, an alternatve to auctonng lmts, whch may brng problems of ts own, s for the central bank to sell, on demand, MLF lmts for a fee. The fee could just be a couple of bass ponts or a more complex prcng functon. The man d erence wth auctonng lmts s that the central bank would then not be able to set the aggregate amount avalable at the MLF. 6

17 7 Concluson In ths paper, we analyzed the behavour of nterbank market rates, when the mplementaton framework does not use reserve averagng over a mantenance perod, but o er a bu er through lendng at a man lendng faclty. Ths faclty o ers banks the possblty to borrow at a rate whch s the average of the depost and lendng facltes rates. However, the central bank can mpose a lmt on the amount borrowed at the man lendng faclty. Wthn ths framework, we showed that rates are less volatle than under an mplementaton framework that uses reserve averagng, and ths wthout a ectng the functonnng of the nterbank market. These results are promsng, but further analyss should now be devoted to explore the optmal lmt sze of the man lendng faclty. 8 Techncal Appendx 8. Dervaton of V s and V y From the de nton of V we have V (s ; y ) = Z s + + [ ( + m ) y + ( + d ) (s )] f () d sz +B [ ( + b ) ( s ) ( + m ) y ] f () d Z+ s +B [ ( + b ) B ( + m ) y ( + `) ( (s + B ))] f () d: Ths can be easly smpl ed to V (s ; y ) = ( + m ) y + ( + d ) s F (s ) ( + d ) Z s f () d sz +B + ( + b ) s [F (s + B ) F (s )] ( + b ) f () d [( b `) B ( + `) s ] [ F (s + B )] ( + `) s Z+ f () d: s +B 7

18 or V (s ; y ) = ( + m ) y d f () d Z s + [( + d ) s ( + b ) s ] F (s ) b sz +B s f () d ` Z+ s +B f () d + [( + b ) s + ( b `) B ( + `) s ] F (s + B ) ( b `) B + ( + `) s whch s equvalent to V (s ; y ) = ( + m ) y ( b `) B + ( + `) s ( b d ) s F (s ) (` b ) (s + B ) F (s + B ) d Z s f () d Therefore we obtan, usng Lebnz Rule: b sz +B s f () d ` Z+ s +B f () d V y (s ; y ) = ( + m ) V s (s ; y ) = ( + `) ( b d ) F (s ) ( b d ) s f (s ) (` b ) F (s + B ) (` b ) (s + B ) f (s + B ) d s f (s ) + b s f (s ) b (s + B ) f (s + B ) + ` (s + B ) f (s + B ) = ( + `) ( b d ) F (s ) (` b ) F (s + B ) 8.2 Transacton costs Gven bank choses to access the nterbank market, t solves the same problem as before, wth rst order condton gves V y + V s = 0. From the de nton of V, we have (see the Appendx for detals), V y (s + y ; y ) = ( + m ) V s (s + y ; y ) = ( + `) ( b d ) F (s + y ) (` b ) F (s + y + B ) = ( + d ) F (s + y ) + ( + b ) [F (s + y + B ) F (s + y )] + ( + `) [ F (s + y + B )] so that bank tradng actvty s gven by ( b d ) F (s + y ) + (` b ) F (s + y + B ) = ` m 8

19 Bank expected payo n ths case s V (s + y ; y ) = ( + m ) (s ) ( b `) B [ F (s + y + B )] d s Z +y f () d b s +y Z +B s +y f () d ` Z+ s +y +B f () d: Note that n ths case, gven B and m, bank exsts the nterbank market wth the same level of account balance s + y = A, ndependent of s. However, f bank choses to reman nactve, ts expected payo s V (s ) = ( + `) s (` b ) s F (s + B ) ( b d ) s F (s ) ( b `) B [ F (s + B )] d Z s f () d b sz +B s f () d ` Z+ s +B Hence, bank wll access the nterbank market whenever f () d V (s + y ; y ) V (s ; 0) ; or ( m `) s + ( b d ) s F (s ) + (` b ) s F (s + B ) ( b `) B [F (s + B ) F (s + y + B )] 2 s Z+y Z s s +y Z +B sz +B ( + m ) + d 4 f () d f () d b 4 f () d ` 4 Z+ s +y +B f () d Z+ s +B 3 7 f () d5 s +y f () d s Snce, gven m and B, the account balance level after trade on the nterbank market does not depend on s (the tradng actvty y, wll!), we can rewrte ths nequalty as (` m ) s + ( b d ) s F (s ) + (` b ) s F (s + B ) ( b `) B [F (s + B ) F (A + B )] (7) 2 Z A Z s AZ +B sz +B 4 f () d f () d5 b 4 f () d f () d5 d ` 2 4 Z+ A f () d Z+ s 3 f () d5 ( + m ) A +B s +B Proposton 6 Gven B and m, there are two unque levels of account balance s and s, such that z (s ; m ) = 0 f s 2 (s ; s ), and z (s ; m ) =, otherwse. 9

20 Proof. When s = A, the left hand sde of the nequalty s zero. So the nequalty does not hold and bank wll not pay the xed cost to enter the nterbank market. When s! +, the left hand sde tends to ( m d ) s ( b `) B [ F (A + B )] d 4 whch tself tends to + snce m > d. When s! (` m ) s +( b `) B F (A + B ) d Z A 2 Z A f () d f () d 3 AZ +B 5 b f () d ` A, the left hand sde tends to 2 b AZ +B Z+ Z+ A +B f () d f () d 4 ` f () d 5 A A +B whch tself tends to +, snce ` > m. Therefore, by contnuty, there s s and s, such that z (s ; m ) = 0 f s 2 (s ; s ), and z (s ; m ) =, otherwse. Now we prove that s and s are unque for each. To show ths, t su ces to show that the left sde of nequalty (7) s monotone. Takng ts dervatve of the left hand sde of (7) and smplfyng, we obtan (` m ) + ( b d ) F (s ) + (` b ) F (s + B ) (8) Ths depcts the margnal gans from trade, gven ntal account balance s. Gven the de nton of A, (8) s obvously zero at s = A. Also, (8), s strctly postve for s > A, and (8) s strctly negatve for s < A. Ths proves that s and s are unque for each Smulatons We smulate the model usng smlar assumpton as n GPR (2007). In ths way, ther results can be used as a benchmark for our analyss. Therefore, there wll be a very closed lnk between ths secton and the same secton n GMP. We smulate an economy where the lendng rate at the resdual lendng faclty s ` = 5 percent and the depost rate s d = 3 percent. Ths leaves a 200 bass pont corrdor, as n the Eurosystem. The man lendng faclty rate s set to b = 4 percent, the mddle of the corrdor. We assume that the cap s the same for all banks and we smualte the economy for d erent levels of cap B = 0; 00; 230; 500; 000 and 500. For each economy, the equlbrum outcome can be characterzed by a sngle nterest rate each day and a dstrbuton of quanttes traded. Ths dstrbuton and the nterest rate depends on the heterogenety of banks, that s the level of reserves they start day t wth, s (t), after the early shock ht. We model the early shock e as a random draw from a normal dstrbuton wth mean B=2 and standard devaton = 20. All banks face the same structure of early shocks. 20

21 To solve for the equlbrum prces and quanttes we smulate Z = 5000 days dentcal to the one descrbed at the begnnng of the prevous secton. We construct late shocks as follows. We assume that represents the uncertanty about the changes n the current account by the end of the day because of clercal errors, etc. As n GPR (2007), we de ne (k) as the random transfer of funds between bank and bank k. Ths transfer s assumed to be dstrbuted accordng to a normal dstrbuton wth mean 0 and standard devaton = 20. If the shock s postve, funds are moved from bank k to bank and nversely. We assume () = 0. The shock s therefore = nx (k) : k= Hence, follows a normal dstrbuton wth zero mean and standard devaton = p 2. Also, snce late shocks sum up to zero across banks, no balances enter or leave the system. Moreover these shocks have co-varances and correlatons equal to E ( k ) = E (k) 2 = 2 and ( k ) = E ( k ) = k 2 : In our smulaton, the standard devaton of ndvdual transfers s = 20 whch mples that banks are subject to shocks whch are jontly normal wth standard devaton of j = (2 ) =2 = 66:33 and a correlaton between shocks of ( ; k ) = 0:09. Even n ths smple case wth no shocks that change the overall supply of reserves, GPR (2007) show that the statstcal propertes of prces and quanttes traded of ther model follow the pattern found n the data for the EONIA. Table reports descrptve statstcs for the state of the system de ned by the ntal levels of reserves s, and trade y. for n = 2. The column mean (s) correspond to the followng statstcs mean (s) = Z ZX z= n nx s (z) where s (z) s the ntal reserves for bank after the early shock s realsed n the z th smulaton. The column (s) s the standard devaton of the varable s across smulatons and s computed as v u (s) = t Z ZX z= n = 2 nx [s (z) mean (s)]! : = The computaton of the statstcs for y s smlar, although we restrcted our attenton to borrowng (.e. to the observatons such that y > 0). For convenence we also report the statstcs computed n GPR (2007) for the same varable. 2

22 e s N ( B=2; 20) Gaspar et. al. (2007) B mean(s) (s) mean (y) (y) mean (y) (y) 0 0: 9:2 303:7 69: t = 2 44: 82: :9 9:2 302:6 67:8 t = 3 687:4 50: :4 9:2 302:6 69: :2 304:9 69: :3 8:8 304:2 68: :3 304:3 70: Table : Average and standard devaton of banks level of early reserves and borrowng levels. Note that ntroducng the MLF does not reduce market actvty when lmts ncrease. Ths should not come as a surprse, as equaton (3) shows that the level of borrowng does not depend on the sze of the lmt at the MLF. That banks behavour on the nterbank market s not a ected by the lqudty de ct, s explaned by the fact that banks trade away ther dosyncratc shock on the nterbank market, as when there s no aggregate lqudty de ct, and access the MLF to cover the aggregate lqudty de ct. However, market actvtes reman relatvely more subdued that n a framework wth reserves. The reason s that along the mantenance perod, heterogenety across banks ncreases, gven more ground for trade. It s also mportant to note that n our framework, although the sze of the MLF lmt ncreases, the actvty on the nterbank market remans as mportant as when there s no MLF (B = 0) - that s when we would mpose a daly reserve requrement and no reserves averagng. Table 2 below shows that the equlbrum nterest rate decreases as the lmt on the MLF loosens. The downward pressure on the nterest rate comes from the MLF lmt tself, snce the hgher ths lmt, the less lkely banks are to access the resdual lendng faclty. e s N ( B=2; 20) Gaspar et. al. (2007) B mean ( m ) d ( m ) d( m ) 0 3:99 0:099 t = 2 0: :99 0:059 t = 3 0: : Table 2: Average nterbank rate and dsperson meaures. Table 2 reports some statstcs on the nterbank market rate. We compute the average 22

23 nterest rate for ths economy. Then we compute a measure of the dsperson of the nterbank market rate as the square of the change between the aggregate nterest rate n day t and ts value at t, as n GPR (2007). The average of that seres s shown n the column labelled d ( m ), whle we report the standard devaton of m n the column ( m ). Both numbers gve an ndcaton of how volatle the aggregate nterest rate s n the tme seres dmenson. The nterbank market rate volatlty s decreasng wth the lmt on the MLF, and when B 500, the varablty dsappears contrary to what GPR (2007) nd. Interbank rate varablty remans n a reserve averagng framework, as banks also take nto account future shocks aganst whch they may want to nsure. Ths dynamc aspect creates some dsperson n nterbank rates. As our smulaton shows, we obtan approxmately the same varablty as n GPR (2007), when B = ther daly reserve requrements for a 2 days mantenance perod. probablty to access the three standng facltes n Table 3 below. 230, whch s approxmately twce the sze of B Pr RLF RLF Pr MLF MLF Pr DF DF 0 0:5 27:5 N an N an 0:5 33: 00 0:23 9:4 26:9 0:005 9: :05 :6 02:8 0 6: :9 0 0: : GPR (2007) - - t= t= t= Table 3: Probablty and expected amount at each standng faclty. Fnally, we report the Tables 3 presents the use of the standng facltes for d erent levels of borrowng lmts at the MLF. GPR (2007) computes the probabltes n the followng way: for each realsaton for the Monte Carlo smulaton, the probablty of gong to a faclty s computed for each bank gven the dstrbuton of shocks and then those probabltes are averaged over banks and smulatons. In our (repeated, but statc) framework, the probabltes are entrely de ned by B and mean (s), for each bank, snce s + y = mean(s) and we do not need to average across all banks. Table 3 ncludes the expected use that one bank wll GPR (2007) e ectvely study a two-days reserve averagng system, where banks receve an early shock only n the mornng of the rst day and no shock otherwse. 23

24 make of the facltes (multply by n to get the aggregate expected recourse to a faclty). As one would expect, n our framework wthout reserves averagng, banks cannot postpone the access to the standng facltes by usng the dynamcs nherent to a mantenance perod. However, ntroducng the MLF greatly reduces the lkelhood of accessng the resdual lendng faclty. Also, note that although there may be more systematc recourse to the facltes, the expected amount borrowed from these facltes s comparable to GPR (2007), and are actually lower when we compare our results to the last day of ther mantenance perod. 9 Bblography Cassola, N. (2007). On the Reserve Ful lment Path of Euro Area Commercal Banks: Theory and Emprcal Testng Usng Panel Data. Mmeo, European Central Bank. Ewerhart, C., N. Cassola, S. Ejerskov, N. Valla (2006). Overnght Rate. ECB WP 378. Lqudty, Informaton and the Gaspar, V., G. Perez-Quros and H. Rodrguez-Mendzabal (2007). Interest rate dsperson and volatlty n the market for daly funds. European Economc Revew, In Press. Hamlton, J. D. (996). The Daly Market for Federal Funds. The Journal of Poltcal Economy. Vol. 04, No.. Papada, F. and F. Würtz (2006). Central Bank Balance Sheets: Comparsons, Trends and Some Thoughts, n Central Bank Reserve Management, New Trends, from Lqudty to Return, Age F. P. Bakker, Edtor. Edward Elgar Publshng, Perez-Quros, G. and H. Rodrguez-Mendzabal (2006). The Daly Market for Funds n Europe: What Has Changed wth the EMU. Journal of Money Credt and Bankng, Vol. 38, No.. Poole, W. (968). Commercal Bank Reserve Management n a Stochastc Model: Implcatons for Monetary Polcy. Journal of Fnance 23, Prat, A., L. Bartoln and G. Bertola (2002). The overnght nterbank market: evdence from the G7 and the euro zone. Centre for Economc Polcy Research. 24

25 Whtesell, W. (2006a). Interest rate corrdors and reserves. Journal of Monetary Economcs 53, Whtesell, W. (2006b). Monetary Polcy Implementaton Wthout Averagng or Rate Corrdors. Fnance and Economcs Dscusson Seres, Workng Paper Würtz, F. (2003). A comprehensve model on the euro overnght rate, ECB WP