Real-Time Gross Settlement and Hybrid Payment Systems: A Comparison. Preliminary Please Do Not Quote

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1 Rel-Tme Gross Settlement nd Hybrd Pyment Systems: A Comprson Mtthew Wllson* Prelmnry Plese Do Not Quote Abstrct Ths pper contrsts Rel-Tme Gross Settlement nd Hybrd pyment systems tht re bsed on pyment offset, usng two perod, mult-bnk model. The comprson s performed ccordng to two crter: lqudty needs nd speed of settlement. We ssume tht the exstence of pyment s common knowledge but tht the specfc degree of tme-crtclty of pyment s the prvte nformton of the bnk sendng the pyment. Hybrd pyment systems re shown to out perform Rel-Tme Gross Settlement when pyments re offset n the frst perod nd when they re offset n both perods. Ths suggests tht n Hybrd system, the offsettng fclty should be n operton ll dy, or, t the very lest, for some tme fter the system opens n the mornng. A system n whch the offsettng fclty ws only swtched on lte n the dy would not necessrly be preferred to Rel-Tme Gross Settlement. These results re shown to be robust to chnges n the trnsprency of the centrl queue of pyments wtng offset. However, ths robustness my not hold wth dfferent forms of nformton symmetry. * Mrket nfrstructure Dvson, Fnncl Stblty, Bnk of Englnd, Thredneedle Street, London, ECR 8AH. E-ml: mtthew.wllson@bnkofenglnd.co.uk The vews expressed n ths pper re those of the uthor, nd not necessrly reflect those of the Bnk of Englnd. grtefully cknowledge the help nd nsghtful comments receved from Pul Bedford, Stephen Mllrd, Mrco Polengh nd Hyun Song Shn. would lso lke to thnk semnr prtcpnts t the Bnk of Englnd nd Smon Buckle, wth whom begn ths work. All remnng errors nd dmssons re the sole responsblty of the uthor.

2 ntroducton Rel-Tme Gross Settlement (RTGS) hs become the foremost system for the settlement of hgh-vlue pyments n developed economes. (See BS Report, 997.) The rtonle behnd the trend towrds RTGS hs been the perceved need to reduce the rsk potentlly found n Deferred Net Settlement (DNS), the predomnnt system for settlng hgh-vlue pyments prevously. RTGS entls pyments beng settled on gross bss n rel tme. As consequence credt rsk nd settlement rsk between settlement bnks (herefter, bnks) s elmnted. But RTGS does not domnte DNS n ll respects. Wth pyments settled on gross bss, bnks lqudty needs under RTGS re greter thn those under DNS. Ths mples burden for the wder economy prmrly becuse t s drn of resources but lso becuse, n n ttempt to sve lqudty, bnks dely pyment ctvty n order to wt for ncomng pyments tht cn then be used s lqudty. Centrl bnks (CBs) hve mtgted these problems to n extent by provdng ntr-dy lqudty. Of course, lendng lqudty for free genertes credt rsk for the CB. Thus, ths lqudty s collterlsed to remove ths rsk. Yet, the benefts from reducng the rsk one ssoctes wth DNS re consdered to exceed the costs of greter lqudty needs nd hence, the number of RTGS systems hs grown. The debte surroundng the optml bss for settlng pyments hs shfted of lte wth the dvent of so-clled hybrd pyment systems. A hybrd, s the nme suggests, combnes fetures of both RTGS nd DNS. (For survey of ths ctegory of pyment system desgn see McAndrews nd Trundle, 00.) More precsely, hybrd typclly tkes one of the desgns nd ugments t wth fetures ssocted wth the other desgn. Gven tht RTGS s the prme pyment system desgn, recent debte hs mnly concentrted on the benefts of complementng RTGS wth lqudty-svng feture clled pyment offset. A pyment s offset when t s settled smultneously wth set of other pyments rther thn beng settled ndvdully lke n RTGS. When pyments re settled smultneously, the pyments re self-collterlsng, to the extent tht ther vlues re lke. Bnks only need lqudty equl to the net vlue Although lendng s collterlsed CB stll fces some mrket rsk, the rsk tht the vlue of collterl declnes drmtclly dong the dy. To counterct ths, CB wll typclly requre hrcut;.e., the vlue of the collterl when t s posted hs to exceed the vlue of the lqudty tht s borrowed by some mrgn.

3 of ts pyments n the set to settle ts pyments. The Germn lrge-vlue pyment system, Plus RTGS s one RTGS system tht ncorportes pyment offset. However, there do exst hybrd desgns wth DNS bss; e.g., NewCHPS n the Unted Sttes. An mportnt desgn feture of ny hybrd s tht pyments cn be plced n centrl queue. Whlst n ths queue, the system opertor serches for offsettng pyments Otherwse, pyments cn stll be settled by RTGS or DNS wthout necessrly enterng the centrl queue. The beneft of complementng RTGS wth pyment offset fclty s tht lqudty needs cn fll nd the ncentve to dely plcng pyments nto the system s reduced, reltve to RTGS, whle not necessrly rentroducng the rsk present under DNS. The reltve merts of RTGS nd hybrd pyment systems hve been nlysed n number of ppers wth the focus on vrous effects. Roberds (999) compres systems wth regrd to ther mpct on bnks portfolo choces. He fnds tht system wth pyment offset cn produce results mmckng those found wth RTGS nd DNS. Other ppers n the lterture use smulton technques to ssess the dvntges of hybrd pyment systems. These nclude Lenonen nd Sormäk (999) nd Johnson et l. (00). However, mor drwbck wth these smultons s tht bnk behvour s tken to be exogenous. For exmple, the lqudty svngs tht hybrds cn generte re clculted usng RTGS dt. Yet we would expect certn ctons tken by bnks, such s the tmes t whch pyments re submtted to the system, to depend on the pyment system desgn n plce. n ths pper we llow the tmng of pyment ctvty to be endogenous nd hence, enble t to vry cross system desgns. n ths pper, we exmne the ssue of optml pyment system desgn. We compre the performnce of n RTGS system gnst sx hybrd systems bsed on pyment offset. We ssume tht, when pyments re offset, they re consdered leglly to be fnl nd rrevocble. So, the hybrd systems ntroduce no credt rsk reltve to the RTGS benchmrk. We compre the system desgns bsed on two crter: ther lqudty demnds nd the speed wth whch pyments re settled. The reson for usng ths second crteron s tht t cptures the potentl mpct of opertonl rsk,

4 snce ny opertonl event wll hve lrger effect the more pyments tht stll remn to be settled when the event occurs. The crter cpture n mportnt feture of RTGS. There s trde-off between lqudty effcency nd exposure to opertonl rsk. Ths trde-off ensures tht the frst-best s not ttnble under RTGS. For exmple, f bnks ncrese the mount of pyments settled erly, they cnnot explot ncomng pyments s source of lqudty s hevly nd hence, must post more lqudty. Menwhle, postng less lqudty shfts pyment ctvty towrds lter n the dy. We ssume tht bnks re concerned bout exposure to opertonl rsks becuse ther customers re;.e., some pyments re tme-crtcl. Hence, f bnk delys pyment settlement t fces cost. Ech bnk fces trde-off between the costs of obtnng lqudty from the CB nd the costs of delyng pyments when choosng how mny pyments to settle n certn perods of the dy. The result s tht under RTGS bnks wll dely some pyments n order to wt for ncomng pyments. Ths grees wth the fndngs of Angeln (998), Bech nd Grrtt (003), Buckle nd Cmpbell (003), nd Kobykw (997). n our model we ssume tht bnks fce prtculr form of nformton symmetry. Wheres we ssume tht the exstence of pyments s common knowledge, the tmecrtclty of pyments s not publcly known. Only bnk sendng pyment knows the tme-crtclty of ths pyment. The mplcton s tht lthough bnk my know how mny pyments other bnks re ech settlng t does not know how mny re drected to t untl the pyments hve rrved. Snce ncomng pyments re substtute for (costly) borrowng from the CB, bnk s uncertn bout the mpct of other bnks ctons on ts pyoff. Of course, dscussng ths knd of nformton symmetry/uncertnty s only sensble f there re more thn two bnks n the pyment system. Even when there re more thn two bnks, the uncertnty wll dspper when bnks settle ll or none of ther pyments. We ssume ths prtculr knd of nformton symmetry s t enbles us to focus on the problems bnks fce coordntng ther usge of the centrl queue nd how certn fetures of hybrd systems could potentlly ffect ther blty to overcome these dffcultes. Pyments cn only be offset f offsettng pyments re n the centrl queue t the sme tme. Thus, one mght suspect tht the trnsprency of the centrl queue wll nfluence the

5 lkelhood of ths by mprovng the nformton tht bnks hve t ther dsposl. (Trnsprency determnes whether nd to wht degree bnk cn observe the pyments tht other bnks hve submtted to the centrl queue.) We nlyse the effects tht the frequency t whch pyments re offset cn hve. When pyments cn only be offset lte n the dy, hybrd wll not offer mprovements on RTGS ccordng to both crter. However, when offset occurs erly or ll dy the frst-best s obtned. Wht s surprsng s tht ths result holds whether or not the centrl queue s vsble to ll bnks. Becuse the tme-crtclty of pyments s prvte nformton, bnk s unsure bout wht pyments other bnks wll smultneously submt to the centrl queue. Ths mens tht ech of bnk s pyments cn be offset wth postve probblty f ech other bnk submts t lest one pyment. Hence, bnks wll fully explot the offset fclty, by submttng ll of ther pyments t the frst opportunty they hve. Thus, the lrgest possble set of pyments s offset, erly n the dy. ntroducng further uncertnty wll not necessrly eoprdse the result unless we ssume tht usng the centrl queue s costly for bnks. Movng n the other drecton, by ssumng perfect nformton, there s the possblty of other equlbr when pyments re offset erly or ll dy. Secton ntroduces the model. Equlbr under RTGS nd hybrds re presented n sectons 3 nd 4, respectvely. Secton 5 concludes. The Model. Bnks We present the model for the benchmrk pyment system desgn, Rel-Tme Gross Settlement (RTGS). There re n, rsk neutrl, bnks. Ech bnk hs ccess to the pyment system nd the fxed cost of ccess s equl to zero. Assume tht n s fnte nd exceeds two. Ech bnk hs sngle pyment of one unt to send to ech other bnk through the pyment system. n other words, ech bnk hs n- pyments to The ssumpton tht ll pyments hve the sme vlue mens tht the model s more trctble whle not hnderng the blty of the model to cpture the slent fetures of RTGS. f pyments dd hve dfferent vlues the only (qulttve) effect would be to complcte the rules for prortsng pyments

6 send. Ths pttern of pyments s common knowledge. Denote by pyment from bnk to bnk. The pyment system (n RTGS) s modelled s two stge gme. At ech stge bnks cn settle pyments. The two stges tken together s clled dy wth the frst (second) stge referred to s perod () or the mornng (fternoon). The tme-lne for the gme s llustrted n chrt (). [nsert Chrt ()] When bnk fls to settle one of ts pyments t s sd to hve cncelled ths pyment. Of course, n the rel world such pyment could be settled on subsequent dy but here we restrct ttenton to ust sngle dy so the term to cncel s not npproprte. Cncellton of pyment results n cost λ for bnk. Ths could be drect compenston for the customer requestng the pyment or some expected loss of future busness. Smlrly, f bnk delys pyment t ncurs cost. 3 Becuse of the two stge ssumpton, pyment dely occurs when pyment s not settled n the mornng. Unlke the cncellton cost, the cost from delyng pyment dffers cross pyments. Tht s, pyments vry n ther tme-crtclty. We denote the dely cost for pyment by ζ( ) nd ssume tht ζ( ) cn tke n- possble vlues. We ssume tht ζ( )= ζ( k) f nd only f =k; tht s, bnk hs no two pyments wth the sme dely cost. Thus, ech bnk hs sngle pyment ssocted wth ech of the n possble dely costs. A bnk receves ll requests to send pyments t the begnnng of the dy. t orders ts pyments n n nternl queue Q. The orderng of n nternl queue s functon but would not otherwse lter bnk behvour. But conformty n pyment vlues mens tht when we nlyse hybrd pyment systems we wll not consder pyment splttng nor mperfect pyment offset. (mperfect pyment offset mens tht the ech bnk s pyments n the set tht re smultneously settled cn hve dfferent combned vlues.) 3 f there s lso cost of delyng pyment, the true cncellton cost s the sum of the dely cost nd λ. However, n the two stge gme, bnk cn only decde to cncel pyment hvng lredy decded to dely t. Thus, when decdng whether to cncel pyment the dely cost s sunk. The only relevnt cost to the decson s λ. Ths s why we dsentngle the two components of the true cncellton cost nd cll them, respectvely, dely nd cncellton costs.

7 of dely costs. f the dely cost ssocted wth pyment exceeds tht ssocted wth nother pyment k t wll st closer to the front of Q. Thus, the most tme-crtcl pyment s t the hed of the nternl queue nd the lest s t the end of the queue. Degrees of tme-crtclty re llocted to pyments by move by nture t the strt of the dy. A bnk cn perfectly observe the tme-crtclty of ech of ts own pyments but not those of other bnks pyments. Thus, we ssume tht bnk does not know the orderng of other bnks nternl queues. Ths mples tht even when bnk s certn bout how mny pyments nother bnk s gong to settle n prtculr perod cnnot be certn bout whether t wll receve n ncomng pyment from n ths perod. Ths s unless s settlng ll or none of ts pyments n ts nternl queue n ths perod. Before proceedng to dscuss bnks cton sets n ech perod, we ntroduce mthemtcl tool tht we use n the nlyss. Ths s the coverng relton more detls see Dvey nd Prestley, 00.) The coverng relton s used to. (For compre members of n ordered set. Consder n ordered set K, where elements n K re ordered ccordng to n order. Tke two elements x y when x > y nd there exsts no element z such tht x, y K. We cn wrte x > z y, unless z = y. A bnk s cton set n the mornng s the set of pyments t chooses to settle t ths stge,. t s chosen from the mornng cton set tht s shown n equton (). A {,{},{,},..., Q \ { n Q A = }, } () mens tht no pyments re settled (.e., bnk chooses the empty set). {} s the pyment t the hed of the nternl queue, {,} s the frst two pyments n the queue nd so on. Q \ { n } s ll the pyments n the nternl queue br the pyment t the very bck of the nternl queue, {n-}. A does not nclude ll of the fesble combntons of pyments bnk could settle n the mornng. Ths follows snce t does not mke sense for bnk to settle pyment n certn poston n the nternl

8 queue f t does not lso settle pyments hed of ths prtculr pyment. After ll, f there s common cost ssocted wth settlng pyment, bnk should show preference for settlng the pyments tht generte the gretest benefts;.e., those pyments tht generte the lrgest svngs n dely cost. We cn represent dely costs s functon of mornng ctons. The cost of delyng pyment, wth poston ε n the nternl queue, s β ( ) = ζ ( ), where = {,..., ε}. n the fternoon bnk chooses n cton from the cton set shown n equton (). A A = ( Q \ ) () ( Q \ ) s the power set of wht remns of the nternl queue fter the mornng cton hs been tken, Q. The power set contns ll possble combntons of pyments n Q s well s. Tht s, we re not restrctng n the wy tht we restrct A \ \. The explnton for ths s tht there s no reson why pyment n the poston ɛ n nternl queue, whch s not settled n the mornng, should not be settled n the fternoon when the pyment n the poston ɛ- s cncelled. Ths s A becuse the cost of ncludng ny pyment n t the expense of nother pyment s λ. The more pyments there re n the fewer members there re of. The extreme cse s when = becuse = Q. A A To be cpble of settlng pyment n gven perod bnk must hve suffcent lqudty t the begnnng of ths perod to do so. Let be the lqudty vlble to bnk n perod t. Lqudty cnnot be wthdrwn from the system durng the dy. All bnks begn wth zero lqudty. ncomng pyments n the mornng cn only be used s lqudty n the fternoon whle ntr-dy credt provded by the CB s vlble t the begnnng of both the mornng nd the fternoon. n order to borrow t L

9 unt of lqudty from the CB, bnk must post one unt of collterl (We ssume tht the CB does not requre hrcut.) Thus, we refer to borrowed lqudty s collterl. Bnks repy the CB nd redeem ther collterl only t the end of the dy. Ech bnk hs suffcent elgble ssets for t not to be credt constrned. Postng unt of collterl n the mornng genertes cost γ. When unt of collterl s posted n the fternoon bnk ncurs cost γ. Assume tht γ γ > 0. Ths mples tht bnk wll not hve n ncentve to post collterl n the mornng tht t ntends not to use untl the fternoon. 4 Three further ssumptons re mde bout collterl costs: λ > γ ; β ({ }) > γ ; nd β ( Q ) γ γ. These ssumptons mply tht: () The cost of cncellng pyment s greter thn the cost of cqurng the necessry lqudty to mke the pyment n the fternoon; () The cost of delyng the pyment t the hed of bnk s nternl queue exceeds the cost of cqurng the necessry unt of lqudty for the pyment to be settled n the mornng; nd (3) the cost of delyng the pyment t the end of bnk s nternl queue s less thn the cost of obtnng the lqudty requred for t to be settled n the mornng. We denote the level of collterl chosen by bnk n perod t by. Collterl choces re functons of the ctons tken by ll bnks n both perods nd must be such tht bnk hs non-negtve lqudty t ll ponts n the dy. We derve two condtons tht collterl choces hve to stsfy for lqudty to never drop below zero, one condton for ech perod. n RTGS, wht we cll the mornng collterl condton s for =,...,n, t c L c = = ( ) (3) Where ( )=(0) when pyment s (not) settled n the mornng nd s the vlue of pyments n the set. As ncomng pyments cnnot be used s lqudty γ 4 t s not cler whch wy the nequlty between nd should necessrly go. The drecton of the nequlty tht we hve ssumed could be ustfed by the de tht postng ssets s collterl tes those ssets down for the rest of the dy nd the resultng restrcton to bnks freedom mples cost. Alterntvely, one mght beleve t s hrder to obtn elgble ssets durng the dy thn t s t the strt of the dy. n whch cse, the cost dfference should be n the opposte drecton. nvestgtng the cse where γ γ > 0 s somethng we ntend to pursue n future work. γ

10 untl the fternoon, collterl must smply be equl to or greter thn the vlue of outgong pyments n the mornng. The fternoon collterl condton s functon of the lqudty vlble nd the pyments bnk wnts to settle n ths perod. Once gn, bnk cnnot use ncomng pyments tht rrve n the fternoon s source of lqudty to settle pyments n ths perod. Becuse the orderng of nternl queues s prvte nformton, bnk does not know wth certnty the mount of ncomng pyments t the begnnng of the dy when collterl s chosen. The level of fternoon lqudty s shown n equton (4). L = c + c ( ) + ( ) (4) The fternoon collterl condton s c + c ( ) ( ) (5) Snce collterl crres cost, bnks wll seek to mnmse the mount of collterl tht they ech hold n ether perod whle stsfyng both collterl condtons. Tht s, equtons (3) nd (5) wll both hold wth equlty or s close to equlty s s possble. t wll lwys be optml for bnk to use collterl posted n the mornng for settlement purposes n ths perod;.e, there s no ncentve to post collterl n the mornng for fternoon purposes. Therefore, equton (3) holds wth equlty. As the mornng collterl condton bnds we cn rewrte the fternoon collterl condton. The level of collterl tht bnk posts t the begnnng of the fternoon s shown n equton (6). c = mx 0, ( ) ( ) (6) Equton (5) wll hold wth equlty f c > 0 but not necessrly f = 0. At the end of the dy bnk must repy the CB the lqudty tht t borrows to cover ny shortfll c

11 of lqudty t expects to hve durng the dy. A bnk hs end of the dy lqudty equl to c + c ( ) + ( ) For bnk to hve suffcent lqudty to repy the CB t the end of the dy t must be tht ( ) + ( ) c c + c c + (7) Equton (7) s referred to s the sustnblty condton. For the pyment system to be sustnble ech bnk must be ble to borrow the lqudty t needs for settlement purposes whle beng cpble of repyng t t the end of the dy. We cn see from equton (7) tht the pyment system s sustnble, when ech bnk begns wth zero lqudty, f nd only f for ech bnk the vlue of ts ncomng pyments over the dy s equl to or greter thn the vlue of ts outgong pyments over the dy. But equton (7) holds for every bnk f nd only f for ech bnk, the vlue of ts ncomng pyments equls the vlue of ts outgong pyments. Otherwse, nter-bnk lendng s requred for ll bnks to hve enough lqudty to repy the CB. 5. Comprng dfferent pyment system desgns Dfferent pyment system desgns re compred wth reference two crter. The frst s lqudty demnds, n prtculr, the lqudty demnds under gven pyment system reltve to the theoretcl mnmum mount requred. As shown n Roberds (993), the theoretcl mnmum s equl to lqudty demnds under end of dy, multlterl DNS. n ths pper, where ech bnk hs one pyment to send to ech other bnk nd ll pyments hve common vlue, the theoretcl mnmum lqudty demnd for ech bnk equls zero. 5 We re grteful to the nternl referee for pontng out the mportnce of nter-bnk lendng n ensurng tht ll bnks cn repy the CB t the end of the dy.

12 The second crteron s the speed of settlement. Ths cptures the exposure of the pyment system to opertonl rsk. We ssume tht the occurrence of n opertonl ncdent s exogenous to the system nd tht ncdents cn occur n both perods. The speed of settlement ffects the mpct of n opertonl ncdent on the system becuse the slower t s (.e., the fewer pyments tht re settled n the mornng) the more pyments could be cncelled f n opertonl ncdent occurs n the fternoon. f n ncdent occurs n the mornng, there s the possblty tht the pyment system could be bck up nd runnng come the fternoon, llowng pyments to be settled. For smplcty we do not model explctly the lkelhood of such ncdents nor ther ctul effects. Rther, we ssume tht bnks fl to tke nto ccount ll of the negtve effects rsng from opertonl dsrupton, whle the system opertor s concern bout these effects s reflected n smple preference for pyment beng settled n the mornng. The frst-best outcome ccordng to these crter s for ll pyments to be settled n the mornng nd for no bnk to post ny collterl. We hve now set out the bsc buldng blocks of the model we use to nlyse RTGS nd how we ssess dfferent pyment system desgns. Ech bnk chooses one cton n ether perod of the dy. The ctons re the sets of pyments t wnts to settle n ech of the perods. From these decsons we bck out the collterl tht ech bnk must post to ensure tht t hs the necessry lqudty t ll tmes n the dy. We now proceed to descrbe equlbrum ctons under RTGS. 3 Rel-Tme Gross Settlement n ths secton we chrcterse equlbr under RTGS. We wll show tht ny equlbrum wll nvolve the postng of collterl nd the delyng of some pyments. We solve the gme bckwrds nd restrct ttenton to equlbr n pure strteges. The subgme tht mterlses n the fternoon depends on the hstory of ply up to ths pont. Denote by h the hstory of ply where h = {,..., }. Select subgme nd consder n ndvdul bnk s optml fternoon cton. Bnk hs n lqudty L vlble for settlement purposes n the fternoon. As λ > 0, t s not

13 optml for ny bnk to choose ts fternoon cton to be such tht t does not use up ll of ts lqudty tht s provded by ncomng pyments tht hve rrved n the mornng when dong so mens tht some pyments re cncelled. t s lso not optml for bnk to choose n fternoon cton such tht pyments re cncelled when the lqudty from ncomng pyments n the mornng hs vlue less thn tht of bnk s remnng pyments. Ths s becuse by ssumpton the beneft of cncellng one less pyment λ exceeds the cost of obtnng the necessry lqudty from the CB, fternoon, Q γ \. Thus, the optml cton s the set of remnng pyments n the. Ths s clerly bnk s strctly domnnt cton n the fternoon. The result s stted formlly n lemm. \ Lemm : = Q strctly domntes ll other for ll, gven ny h. A Gven tht cncellton costs re suffcently hgh we hve rrved t the unsurprsng result tht no bnk chooses to cncel ny of ts pyments. Ths mples tht every bnk both sends nd receves n- pyments durng the dy. Substtutng the vlues of outgong nd ncomng pyments nto the left-hnd sde of the sustnblty condton n equton (7) we see tht the condton holds wth equlty. Thus, the pyment system s sustnble nd there s no nter-bnk lendng t the end of the dy. A further mplcton of Lemm s tht the mount of collterl tht bnk posts n the fternoon s c = mx 0, (n ) ( ) (8) A bnk s pyoff functon n the mornng when t chooses n cton ' s shown n equton (9). t s bnk s expected cost over the entre dy from the perspectve of the mornng. Hence, bnk chooses ts mornng cton n order to mnmse expected cost. Let = {,...,, +,..., } ;.e., the ctons tken by ll bnks other thn n bnk n the mornng. The frst prt of expected cost s the dely cost ncurred. The

14 second s the mornng collterl cost whle the thrd s the expected fternoon collterl cost. 6 ( + + = ' ' ) 0,( mx ' ) ( ) ', ( n E c γ γ β ) (9) The expected vlue of fternoon collterl s shown n equton (0). ( ) ( ) ( ) n n n E ' ) ( Pr ' ) ( ' ) ( (0) When bnk chooses to settle more pyments n the mornng there re three effects on ts expected cost. Frst, dely cost declnes. Second, mornng collterl cost rses. Thrd, t reduces expected fternoon collterl cost. Ths s becuse n ncrese n reduces the condtonl expectton nd lso reduces the probblty tht bnk hs to post collterl n the fternoon;.e., wth fewer pyments left to settle n the fternoon, t s more lkely tht ncomng pyments n the mornng provde suffcent lqudty to settle these remnng pyments. When decdng to settle fewer pyments n the mornng the three effects work n the opposte drectons. [nsert Chrt ()] The costs fced by bnk re shown n Chrt (). On the horzontl xs s the vlue of pyments settled n the mornng, '. (One must remember tht gven vlue s ssocted wth prtculr member of the mornng cton set.) The mornng collterl cost equls A ' γ. The dely cost s decresng functon of '. The wy tht n nternl queue s ordered, wth pyments poston n the queue nversely relted to ts dely cost, genertes the shpe of the dely cost curve;.e., the mrgnl 6 Obvously the only source of uncertnty s the vlue of ncomng pyments rrvng n the mornng.

15 chnge n dely cost s bnk settles more pyments s declnng n the vlue of pyments settled. From these costs we derve totl cost excludng the expected fternoon collterl cost. The ssumpton tht the reducton n the dely cost from settlng the pyment t the hed of the nternl compred wth no pyments, β ({ }), s greter thn γ s clerly necessry nd suffcent for the cton mnmsng ths cost to be non-empty. Hence, we hve the U-shped curve llustrted n Chrt (). The cost ncludng the expected fternoon collterl cost les (wekly) bove ths curve for ll '. Snce the expected fternoon collterl cost s decresng n ' the gp between the two curves s lso decresng n '. (The curves concde when bnk settles ll of ts pyments n the mornng.) Thus, gven the mornng ctons of other bnks, the optml mornng cton for bnk s tht whch mnmses the totl cost curve. We cn refer to ths cton s the best reply of bnk to the ctons of others, BR ( ). Denote n equlbrum mornng cton by equlbrum fternoon cton s Q \ *. (Lemm mples tht the *.) The equlbrum s chrctersed by * ( *) () BR Prtculr rrys of mornng (nd ccompnyng) fternoon ctons cnnot be equlbr for ny number of bnks. Any equlbrum s such tht ll bnks settle some pyments n the mornng nd t lest some bnks settle pyments n the fternoon. Proposton : Under RTGS, postve vlue of pyments wll be settled n the mornng by ll bnks nd n the fternoon by some bnks for ny n. The proof of proposton hs two prts. The frst demonstrtes tht no bnk wll choose to settle zero pyments n the mornng gven the ssumptons we hve mde bout dely nd collterl costs. Defne n cton ~. Ths s the cton such tht

16 β ( γ ~ ). 7 Clerly, ~ s unquely defned. For ll ctons to the left of t n A (.e., ~ < ), devtng from ths cton, by settlng more pyments, wll lwys reduce bnk s totl cost regrdless of the other bnks ctons. n other words, gnorng ts mpct on expected fternoon collterl cost, the ct of settlng ddtonl pyments s optml for bnk. (Of course, gven tht expected fternoon collterl cost s declnng n the number of pyments settled, f devton s worthwhle wthout tkng nto ccount ths cost reducton t wll lso be f we dd consder t.) Ths s so for ny ctons where the reducton n dely cost exceeds the rse n ~ mornng collterl cost. Tht s, ny cton to the left of n. Hence, no bnk wll ever choose such n cton. The ssumpton tht β ({ }) > γ choosng mornng. 8 A mples tht ~ entls the settlement of t lest one pyment by ech bnk n the The second prt shows tht t s not n equlbrum for ll bnks to settle ll of ther, respectve, pyments n the mornng. Suppose bnks choose the ctons { Q,..., Q n } nd consder the ncentve of n ndvdul bnk to devte from ts cton Q. f ny devton strctly reduces s costs { Q,..., Q } cnnot be n equlbrum. As ech n other bnk s settlng ts entre nternl queue of pyments n the mornng, s not uncertn bout the vlue of ts ncomng pyments n the mornng. t knows for sure tht t wll receve n- pyments n the mornng. Ths mens tht for ny mornng cton t chooses, wll hve suffcent lqudty n the fternoon to meet ts needs wthout hvng to borrow from the CB for second tme. Settlng one less pyment n the mornng genertes beneft γ, the reducton n mornng collterl cost. t lso results n cost β ( Q ), whch s the ncrese n dely cost. However, the expected fternoon collterl cost s unchnged. The ssumpton tht β ( Q ) γ γ mens tht ths devton strctly reduces the costs of bnk. Therefore, {,..., Q n } cnnot be n equlbrum. The ntuton behnd ths s tht when ll other bnks settle ll of Q 7 We defne ths reltonshp by pplyng the coverng relton to the ordered set β ( Q ),..., γ,..., β ({});. 8 As ll bnks hve the sme cost structure, ~ hs the sme vlue for ll bnks.

17 ther pyments n the mornng, bnk optmlly shfts settlement ctvty towrds the fternoon so s to explot the hgh (nd certn) vlue of ncomng pyments n the mornng gven tht the cost of delyng some pyments s not too hgh. n equlbrum, under RTGS, pyments re settled n both perods. Thus, the system wll be exposed to opertonl rsk n the fternoon. The fct tht pyments re settled n the mornng mples tht collterl posted n the mornng hs strctly postve vlue. n the fternoon, collterl posted cn lso be strctly postve or equl to zero. The bsc structure of RTGS precludes the frst-best beng ttnble. To speed up settlement requres bnks to post more collterl n the mornng (wth less thn fully compenstng reducton n expected fternoon collterl cost). To reduce the mount of collterl posted n the mornng, the system s exposure to opertonl rsk hs to rse. t s ths trde-off tht mples tht the frst-best s smply not possble. 4 Hybrd Pyment Systems The debte bout the reltve merts of dfferent pyment systems hs often been centred on two opposng clsses of system desgns. These re RTGS nd DNS. A preference for one of the two depends on the perceved optml mx of credt rsk nd lqudty effcency. DNS requres low levels of lqudty but genertes credt rsk. RTGS needs more lqudty but there s no rsk. However, there exsts set of lterntves to RTGS nd DNS known s hybrd pyment systems. As the nme suggests, hybrd ncorportes elements of both RTGS nd DNS. But to be more precse, hybrd envelops one of RTGS nd DNS but ugments ths desgn wth rules nd procedures one would normlly ssocte wth the other. The rgument n fvour of hybrd desgns s tht they wll mplement levels of socl welfre tht re not ttnble wth ether RTGS or DNS. As we hve tken RTGS to be our benchmrk pyment system desgn, we wll ssume tht ny hybrd desgn consdered here ncorportes t rther thn DNS. The stndrd fetures of RTGS re ugmented by pyment offset fclty.

18 A pyment s deemed to be offset when t s prt of set of pyments tht re smultneously settled. The settlement tkes plce on gross settlement bss nd cn nvolve pyments from two or more bnks. When t s only two, offset s sd to be on blterl bss. Wth more thn two bnks pyments wthn the set we hve multlterl offset. Although settlement s on gross settlement bss, smultnety mples tht to some extent pyments n the set re self-collterlsng. Ths mens tht to settle pyment n ths set bnk does not need to hve s much lqudty from other sources, n prtculr collterlsed lqudty borrowed from the CB thn t would under RTGS. Thus, the overll lqudty demnds re lower compred to RTGS. Of course, offsettng need not be perfect, wth ncomng pyments n the set beng suffcent to llow ech bnk to settle ts outgong pyments n the set wthout further lqudty. Tht s, offset genertes resdul pyments unless further lqudty s provded. The system opertor wll typclly put celng on the vlue of such resdul pyments. Ths settlement procedure hs obvous prllels wth net settlement. However, pyment offset need not generte the rsk we would ssocte wth DNS. Ths s becuse of the bsence of settlement lgs. The lck of lgs rses from the fct tht settlement s smultneous nd tht no offset pyment s settled untl ll of the resdul pyments hve been settled (ether by RTGS or subsequent offsettng opportuntes). We model blterl rther thn multlterl offset snce, wth ech bnk hvng sngle pyment to mke to ech other bnk, t s cpble of resultng n the theoretcl mnmum of lqudty demnds, zero. (Although we wll return to the ssue of multlterl offset below.) 4. Modfctons to the Model To be ble to nlyse hybrd pyment systems, nd chnges to behvour tht they could brng bout, we must modfy the model n severl key res. Pyments cn now be settled through two strems. There s RTGS strem tht opertes n exctly the sme wy s before. Ths s consstent wth the bsc de tht hybrd system desgn should lwys fully encompss ether RTGS or DNS. The second strem s the centrl queue n whch pyments re plced to wt offsettng

19 pyments. We ssume tht there s no drect cost of submttng pyment nto the centrl queue. When two offsettng pyments re both n the centrl queue t the sme tme they cn be offset nd settled nstntly. There s no lg between the dentfcton of offsettng pyments nd settlement becuse ll pyments hve the sme vlue;.e., there re no resdul pyments to be settled. No pyment cn resde n both strems t the sme tme. So f pyment s plced n the centrl queue nd the offsettng pyment s not there, to cheve quck settlement bnk hs to wthdrw the pyment nd put t n the RTGS strem n the next perod. The second mor chnge s tht we expnd the number of tmes bnk cts. We now hve four stge gme. Ech perod s splt nto two sub-perods. Denote the frst sub-perod n perod () wth () nd the second wth b(b). 9 We ssume tht there s no dely cost ssocted wth pyment settlng n sub-perod tb rther thn sub-perod t. (The subect of whether there s cost to usng the centrl queue s returned to below.) We lso ssume tht there s no rsk of n opertonl event hppenng between sub-perods. The hybrd desgns consdered n ths pper re defned long two dmensons. The frst s the tmng of pyment offset. f offset occurs n ether perod t wll tke plce n both sub-perods n ths perod. We consder three cses: H: offset n the fternoon only H: offset n the mornng only. H3: offset n the mornng nd the fternoon. The other dmenson s the trnsprency of the centrl queue. We consder two dfferent degrees of trnsprency: 9 The results for RTGS tht we found bove lso pply n four stge verson of the gme. n the two stge gme, the only fctor tht s cusng bnks to settle pyments n the mornng rther thn the fternoon s the dely cost. By the sme rgument, n the four stge gme, gven tht pyments wll be settled n ech perod, no bnk chooses to settle pyments n sub-perods nd. Delyng settlement to the second sub-perod genertes no cost for bnk but does open up n opportunty to explot ncomng pyments tht rrve n the frst sub-perod. Of course, f ll bnks behve n ths wy, there wll be no ncomng pyments n the frst sub-perods of the mornng nd the fternoon. Although, when one bnk settles n t whle ll others settle n tb could be n equlbrum. But ths cn be removed f one ssumes tht f bnk s ndfferent between settlng n ech sub-perod (s t would be n ths lterntve equlbrum) t chooses to settle n the second sub-perod.

20 Opque: bnk cn only see ts own pyments n the centrl queue. Vsble: bnk cn see ts own pyments nd pyments ntended for t n the centrl queue. n ny perod tht the offset fclty s not opertng bnk s cton set s ust the set of pyments tht re settled n the RTGS strem. n H, the mornng cton set s tht n equton (). n H, the fternoon cton set s the power set of ll remnng pyments, gven mornng ctons. The set of remnng pyments s the nternl queue mnus the pyments settled by RTGS n the mornng nd those pyments tht re offset n the mornng. When the offset fclty s workng, bnks tke number of ctons. Frst, there re the sets of pyments plced n the centrl queue n ech sub-perod tk, tk q, c, t=,, k=,b. Second, bnks choose the sets of pyments to settle by RTGS n ech sub- tk tk perod,. Assume tht tk = ;.e., no pyment cn be submtted to the q, c centrl queue nd be settled by RTGS n the sme sub-perod. n sub-perod we ssume tht bnk shows preference for submttng more tme-crtcl pyments to the centrl queue: q A, where s the sme cton set s tht n equton ()., c The set of pyments settled by RTGS n sub-perod s member of A A \ q, c. n sub-perod, q, c s tken from the set of pyments tht cn be offset. f bnk submts φ pyments to the centrl queue t s eqully lkely to choose ny of the pyments tht cn be offset. 4. Equlbr under Hybrd Pyment Systems 4.. H: Pyment offset n the fternoon n ths secton we consder nd dscuss mportnt results for ech of the hybrd pyment system desgns. We begn wth H when the centrl queue s opque. The mornng cton s the sme s under RTGS, A. The fternoon ctons re

21 {, q,, q b b, c, s }. f pyment remns unsettled n sub-perod b nd the offsettng pyment hs been receved through the RTGS strem, t s optml for bnk to settle ths pyment by RTGS n b. Ths s shown n Lemm. t follows from Lemm. Lemm : f or nd b,, then. f pyment remns unsettled n sub-perod nd the offsettng pyment hs not yet been receved, there re number of possble outcomes. A bnk mght submt such pyments to the centrl queue f t beleved tht other bnks were gong to do the sme. f t beleved other bnks were not gong to, t would choose to settle such pyments by RTGS n sub-perod b. From the perspectve of sub-perod b, there s no unque outcome. f bnk hs not settled pyment n the mornng, but the offsettng pyment rrved n the mornng, the bnk wll choose to dely settlng the pyment n the RTGS strem untl sub-perod b. The dely between sub-perods entls no cost but llows the bnk to use ny RTGS pyments t receves n sub-perod to mke pyments n b. Ths result s shown n lemm 3. Lemm 3: = for ll. f bnk hs not settled pyment n the mornng nd the offsettng pyment ws not mde n the mornng, t s optml for bnk to submt ths pyment to the centrl queue n sub-perod f ech other bnk submts t lest one pyment. The bnk enoys lqudty svng f the pyment s offset but no cost f the pyment s not offset n nd s settled n b. f ech other bnk submts t lest one pyment, ech of bnk s pyments wll be offset n wth postve probblty. But f ll pyments tht cn be offset re submtted to the centrl queue n, ll of these pyments wll be offset. Thus, n ths equlbrum the centrl queue wll be empty n sub-perod b. b Lemm 4: = for ll. q, c

22 The other possble outcome n perod s tht bnks submt no pyments to the centrl queue. Proceedng to sub-perod b, there s no unque outcome (for resons we dscussed bove). We wll gnore equlbr n whch bnks choose not to use the centrl queue t the erlest opportunty n the nlyss below. A bnk wll settle pyment by RTGS n sub-perod b f nd only f t s not settled n the mornng. f bnk does not settle pyment n the mornng, t wll be settled by RTGS n the fternoon f nd only f the offsettng pyment ws settled n the mornng. f the offsettng pyment ws not settled n the mornng, then both pyments re submtted to the centrl queue n sub-perod nd re offset. Therefore, ( ) ( ) ( ) [ ] ( ) [ ] b = Or ( ) ( ) ( ) [ ] b = () The vlue of the pyments bnk settles n the RTGS strem n the fternoon s ( ) ( ) [ ] (3) Tht s, the more pyments tht re settled n the mornng, the fewer pyments re settled by RTGS n the fternoon. The fternoon collterl condton s ( ) ( ) ( ) [ ] + c The level of collterl tht stsfes the fternoon collterl condton s

23 c = mx 0, ( )( ) ( ) ( ) (4) Or c = 0 A bnk wll know wth certnty tht, for ny mornng cton nd vlue of ncomng pyments, t does not post collterl n the fternoon. Ths s becuse f pyment s offset n the fternoon, becuse t nd ts offsettng pyment were not settled n the mornng, t requres no collterl whle f pyment s settled n the RTGS strem n the fternoon t must be tht the offsettng pyment ws settled n the mornng. Hence, the necessry lqudty s provded by ths ncomng pyment. The mplcton of ths s tht bnks optmston problems re dsentngled from ech others ctons. ts cost functon s ust the frst two elements of equton (9). n Chrt () totl costs under H correspond to totl costs under RTGS mnus expected fternoon collterl cost. Snce the expected fternoon collterl cost s declnng n the vlue of pyments settled n the mornng under RTGS, the optml mornng cton under H s ether equl to tht under RTGS or entls fewer pyments beng settled. Becuse ll bnks hve the sme mornng collterl cost nd dely cost functons, ech bnk chooses to settle the sme vlue of pyments n the mornng, ~. The ntuton behnd ths result s tht the opportunty for pyments to be offset n the fternoon tempts bnks to dely greter number of pyments thn they do under RTGS. The mplcton for welfre s not cler. f the sme number of pyments re settled n the mornng under RTGS nd H, the totl vlue of collterl posted s (wekly) greter under RTGS. Exposure to opertonl rsk s the sme under both desgns. Therefore, H s (wekly) preferred. When fewer pyments re settled n the mornng under H thn under RTGS, whch desgn s preferred depends on the reltve weghts plced on ech crteron. RTGS hs greter speed of settlement but H nvolves less collterl beng posted.

24 As under RTGS, there s n nherent trde-off between lqudty demnds nd speed of settlement under H. The more pyments re settled n the mornng, the more collterl hs to be posted n the mornng. The fewer pyments settled n the mornng, the greter the exposure of the system to opertonl rsk. As before, the mplcton of the trde-off s tht the frst-best s not ttnble when there s pyment offset n the fternoon only nd the centrl queue s opque. Let the centrl queue be vsble. n sub-perod b, bnk fces the sme stuton s t does when the centrl queue s opque. Although bnk cn observe whether offsettng pyments were n the centrl queue n sub-perod, t knows tht postng pyment to the centrl queue n s not bndng commtment on the prt of bnk to keep the pyment n the centrl queue n b f t s not offset mmedtely. Whether bnk keeps pyment n the centrl queue n b depends only on wht t expects the other bnk to do. As before, there s no unque outcome. n sub-perod, bnk wll fnd t optml to submt ny pyment tht could be offset to the centrl queue for the sme resons s outlned bove. Lemm 3 nd 4 lso hold when there centrl queue s vsble. Thus, the sme equlbrum rses s tht when the centrl queue s opque. H s sustnble n equlbrum. A bnk posts collterl only n the mornng. t s such tht the mornng collterl condton bnds;.e., c = ( ) (5) For bnk to repy the CB nd redeem ts collterl t the end of the dy, ts totl ncomng pyments, through the RTGS strem, mnus the pyments t mkes by RTGS n sub-perod b, must be t lest equl to c. The condton s shown n equton (6). b b ( ) + ( ) ( ) ( ) (6)

25 f we substtute equton () nto equton (6) we fnd tht the pyment system s sustnble f ( ) ( ) ( ) ( ) (6b) H s sustnble becuse equton (6b) holds wth equlty for ll bnks. Proposton : The sme equlbrum rses under H when the centrl queue s opque nd when t s vsble. H s sustnble. 4.. H: Pyment offset n the mornng We now turn to H. Lemm confrms tht bnk wll settle ll of ts remnng pyments by RTGS n the fternoon. n sub-perod b, bnks fce smlr stuton s they do n b under H. But n sub-perod, bnk wll choose to submt ll of ts pyments to the centrl queue f ech other bnk submts t lest one pyment, when the centrl queue s opque or vsble. (There re equlbr where bnks choose to not submt ny pyments to the centrl queue n sub-perod. We wll gnore these s we dd under H.) Ths s becuse by dong so bnk mxmses the lkelhood tht ts pyments re offset, whle plcng pyment n the centrl queue nvolves no cost ether n the form of n explct chrge or restrcton on wht bnk cn do wth the pyment n the future. As ech bnk hs sngle pyment to send to ech other bnk, nd tht ll pyments shre common vlue, ll pyments re offset n sub-perod. Ths mples tht no bnk posts collterl n ether perod. The equlbrum corresponds to the frst-best;.e., lqudty demnds nd exposure to opertonl rsk re both mnmsed. Proposton 3: The frst-best s ttned under H when the centrl queue s opque nd when t s vsble. Becuse n equlbrum no bnk posts ny collterl, the sustnblty condton s stsfed.

26 4..3 H3: Pyment offset n the mornng nd fternoon H3 produces the sme equlbrum s under H. n sub-perod, bnk wll choose to submt ts entre nternl queue of pyments to the centrl queue, rrespectve of whether the centrl queue s opque or vsble. (gnorng possble equlbr where no pyments re submtted to the centrl queue n sub-perod.) All pyments re offset. The system s sustnble. Proposton 4: The frst-best s ttned under H3 when the centrl queue s opque nd when t s vsble. 4.3 nformton Asymmetry A key ssumpton n the model s tht the orderng of nternl queues s prvte nformton. Chngng ths ssumpton could hve n mpct on the results nd hence, the optml pyment system desgn Publc nformton n Secton 4 we rgued tht n sub-perod n whch pyments could be offset, bnks ether submt no pyments (the cse we gnored) or ll pyments tht could be offset. No ntermedte cses cn rse n equlbrum becuse bnk knows tht ech of ts pyments tht cn be offset wll be offset wth postve probblty f ech other bnk submts t lest one pyment. A bnk s uncertn bout the dentty of pyments submtted by other bnks n sub-perod becuse ech other bnk s eqully lkely to submt ny of ts pyments tht cn be offset. A bnk s uncertn n the mornng becuse the orderngs of nternl queues s prvte nformton. n ths secton we ssume tht nternl queue orderngs re publc nformton nd see how the results for H nd H3 could be ffected wth the d of n exmple. We show tht ntermedte cses re possble n equlbrum. Let n=3. The nternl queues of the three bnks re respectvely,

27 Q Q Q {, 3} = { 3, } = { 3,3 } 3 = As before, β ({ }) > γ nd β ( ) γ γ. Suppose tht the sets of pyments tht bnks forwrd to the centrl queue n re Q q q q { }, c = { 3, }, c = { 3 } 3. c = n the mornng, ll of the pyments plced n the centrl queue re offset nd no pyments re settled n the RTGS strem n sub-perod b. Under H, the levels of expected fternoon collterl re c, c 0, nd c. Wth H3, ll remnng = pyments re offset n the fternoon nd no collterl hs to be posted by ny bnk. = 3 = The bove does not represent n equlbrum f nternl queues re prvte nformton. When nternl queues re prvte nformton, bnk submts ll of ts pyments to the centrl queue n snce ech could be offset wth postve probblty. When the orderng of nternl queues s publc nformton bnk hs no ncentve to devte from the ctons descrbed bove snce dong so wll (strctly) ncrese ts costs. Bnks nd 3 lso do not hve n ncentve to devte from the bove set of ctons under H nd H3. Consder bnk. (As nd 3 hve the sme cost structure nlysng the ncentves of ust one of them wll suffce.) Under H, f forwrds the pyment 3 nto the centrl queue n but s does not settle t by RTGS n b, the beneft nd cost of devtng re both zero. There s no beneft nd no cost, snce the pyment s not settled n the mornng. f does settle 3 by RTGS n the mornng, the beneft s β ( ) + γ ;.e., the reducton n totl dely costs plus the expected fll Q

28 n fternoon collterl costs. The cost s the ncrese n mornng collterl costs, γ. As t s ssumed tht β ( ) γ γ, does not strctly gn from ths devton. Q Under H3, f devtes by plcng 3 n the centrl queue but does not settle t by RTGS n b, there s no beneft nd no cost. The pyment s not offset or settled by RTGS n the mornng but s offset n the fternoon, ust s before the devton. When 3 s settled by RTGS, there s beneft β Q ) but cost γ. The beneft rses from the fct tht the pyment s settled n the mornng rther thn the fternoon. The cost s due to the pyment beng settled by RTGS rther thn beng offset n the fternoon. The devton s not optml s γ > β ( ). ( Q q, = Q for ll s lso n equlbrum when nternl queues re publcly known. As c wth bnk n the bove cse, none of the bnks hve n ncentve to devte. Therefore, we hve multple equlbr, gnorng those equlbr where bnks fl to submt ny pyments to the centrl queue n. The ntermedte cses we dscussed bove cn rse n equlbrum. The equlbr cn be Preto-rnked wth the one equlbrum correspondng to the frst-best. So, f nternl queues re publc nformton the welfre gns ssocted wth H nd H3 could be overstted. But there could be polcy remedes tht ensure tht ntermedte equlbr cnnot exst. One polcy s to prohbt the wthdrwl of pyments once they hve been plced n vsble centrl queue. Ths enbles bnks to commt to not wthdrwng pyment from the centrl queue even f t not mmedtely offset. A bnk whch observes n ncomng pyment n the centrl queue n wll nturlly respond by forwrdng the offsettng pyment to the centrl queue n b snce t s sure tht the pyments wll be offset. Workng bckwrds, ech bnk hs n ncentve to plce ll of ts pyments n the centrl queue n. 0 0 n repeted gme, bnks could cheve more co-ordnted use of the centrl queue wthout the need for polcy remedy such s prohbted pyment wthdrwl.

29 4.3. Addtonl Uncertnty The result contned n propostons 3 nd 4 tht the sme equlbrum rse for both opque nd vsble centrl queues s perhps unexpected nd leds one to sk whether there somethng bout the form of nformton symmetry n the model tht genertes the result? The nswer s tht the result ppers to be robust s long s one mntns certn other ssumpton mde bove. The key ssumpton s tht submttng pyment to the centrl queue costs bnk nothng. One cn thnk of bnk potentlly fcng number of lyers of uncertnty. The frst, whch we consder bove, s where the overll pttern of pyments nd bnks costs re unverslly known but the tme-crtclty of pyments s the prvte nformton of the bnks sendng them. n such clmte, bnk does not know wth certnty how other bnks ctons mp nto ts own pyoff. There could be ddtonl uncertnty. A bnk could be uncertn bout the exstence of ncomng pyments. The stuton would be complcted further f bnks were to receve pyment requests t severl tmes n the dy;.e., does pyment not exst or s t tht bnk s yet to hve receved the order to send t? On top of ths lyer of uncertnty, bnk could be uncertn bout ts own s well s other bnks pyments. The rgument s tht wth ny type of uncertnty, bnk wll stll forwrd ll of ts pyments to the centrl queue t the erlest opportunty. By dong so t ensures tht ny pyment for whch n offsettng pyment exsts s offset. However, f bnks ncur cost when usng the centrl queue, dfferent knds of uncertnty my led to dfferent equlbr. Suppose tht submttng pyment to the centrl queue s costly for bnk. Assume tht ths chrge s less thn the collterl svng f the pyment s subsequently offset n the sme perod. Ths ensures tht f bnk s suffcently confdent tht pyment t plces n the centrl queue wll be offset n the sme perod t wll use the centrl queue. ntroducng further uncertnty reduces the belef One such cost rses from the fct tht pyments n the centrl queue wll not necessrly be settled nstntly on rrvl n the centrl queue. Hence, levng pyment n the centrl queue wll, n prctce, mpose dely cost. There my lso be opertonl rsk ssocted wth lrge number of pyments resdng n the centrl queue t ny one tme.