A Specialized Inventory Problem in Banks: Optimizing Retail Sweeps

Transcription

1 WORKING PAPER SERIES A Secalzed Invenory Prolem n Bank Omzn Real Swee Sure K. Nar and Rcard G. Anderon Workn Paer A //reearc.loued.or/w/2005/ d Arl 2005 FEDERAL RESERVE BANK OF S. LOUIS Reearc Dvon 4 Locu Sree S. Lou MO 6302 e vew ereed are oe o e ndvdual auor and do no necearly relec ocal oon o e Federal Reerve Bank o S. Lou e Federal Reerve Syem or e Board o Governor. Federal Reerve Bank o S. Lou Workn Paer are relmnary maeral crculaed o mulae dcuon and crcal commen. Reerence n ulcaon o Federal Reerve Bank o S. Lou Workn Paer oer an an acknowledmen a e wrer a ad acce o unuled maeral) ould e cleared w e auor or auor. Poo courey o e Gaeway Arc S. Lou MO.

2 A Secalzed Invenory Prolem n Bank Omzn Real Swee Sure K. Nar Rcard G. Anderon Scool o Bune Unvery o Conneccu Sorr C Federal Reerve Bank o S. Lou S. Lou MO 6302 Arl 2005 Deo eld a Federal Reerve Bank are an eenal nu o e une acvy o mo deoory nuon n e Uned Sae. Manan ee deo an moran and comle nvenory rolem or wo reaon. Fr Federal Reerve reulaon requre a deoory nuon old ceran amoun o uc deo a e Federal Reerve Bank o ay auory reerve requremen aan cuomer ranacon accoun demand deo and oer ceckale deo). Second ome nvenory o uc deo eenal or ank o oerae one o er core lne o une urnn aymen ervce o oueold and rm. ncludn wre raner ACH aymen and ceck clearn elemen. Becaue e Federal Reerve doe no ay nere on uc deo ued o ay auory reerve requremen ank eek o mnmze er nvenory o uc deo. In 994 e ankn ndury nroduced a new nvenory manaemen ool or uc deo e real deo wee roram wc avod e auory requremen y reclayn ranacon deo a avn deo. In analy we eamne wo alorm or oeran uc wee roram wn e lm o Federal Reerve reulaon. JEL Code D20 G2 Keyword Real Bankn; Deo Swee; Reulaon D; Requred Reerve; Socac Dynamc Prorammn Proeor and Ackerman Scolar Oeraon and Inormaon Manaemen Dearmen Scool o Bune Unvery o Conneccu. Conac ure.nar@uconn.edu Econom and vce reden Reearc Dvon Federal Reerve Bank o S. Lou. Conac anderon@l.r.or Federal Reerve Bank o S. Lou Reearc Dvon Workn Paer ) Vew ereed eren are olely oe o e auor and no necearly oe o e Federal Reerve Bank o S. Lou e Board o Governor o e Federal Reerve Syem or er a.

3 . Inroducon Deo eld a Federal Reerve Bank are an eenal nu o e une acvy o mo deoory nuon n e Uned Sae. Manan ee deo an moran and comle nvenory rolem or wo reaon. Fr Federal Reerve reulaon requre a deoory nuon old ceran amoun o uc deo a e Federal Reerve Bank o ay auory reerve requremen. Second an nvenory o uc ranacon deo eenal or ank o oerae one o er core lne o une urnn aymen ervce o oueold and rm ncludn wre raner auomaed clearn oue ACH) aymen and ceck clearn elemen. A comeve rm ank eek o mamze er ro wle comlyn w alcale law and reulaon. e Moneary Conrol Ac o 980 auorzed e Federal Reerve o requre ank and deoory nuon o old auory reerve aan ranacon deo e.. ceckn accoun); nce early 992 a 0 ercen reerverequremen rao a aled o deo alance reaer an aromaely $50 mllon. Nonranacon deo e.. avn accoun) and mo oer ank lale are uec o a zero ercen reerve-requremen rao. Federal Reerve reulaon ulae a only wo ank ae may e ued o ay auory reerve requremen deo a e Federal Reerve Bank and vaul ca ca n ank cenral vaul n AM and n ran ec.). Neer o ee ae earn nere or e ank. Penale are leved ee requremen are volaed. Clever ank manaer eek o mnmze er oldn o ee erle reerve. In 994 or uroe a lare commercal ank nvened e real deo wee roram. In uc a roram e ank lnk wo comanon accoun a cuomer ranacon accoun demand deo or mlar ceckale accoun) wc we ereaer reer a e ank ranacon accoun or BA and a newly creaed money marke deo accoun MMDA mlar o a avn accoun). Fund are we reularly rom e BA were end o day alance are uec o a 0 ercen auory marnal reerve requremen o e MMDA wc no uec o any reerve requremen. e MMDA accoun nvle o e cuomer and olely a comonen o e nvenory manaemen ceme; raner eween accoun alo are nvle o e cuomer. Unlke numerou earler reerve-reducon ceme o ank e Federal Reerve Board a no a o wrn oeced o ank reducn er 3

4 Fure Monly wee amoun and cumulave amoun nce 994 $ Bllon Jan-94 Jan-95 Jan-96 Jan-97 Source Monly wee Jan-98 Jan-99 Jan-00 Jan-0 Jan-02 Jan-03 Jan-04 Jan-05 Cumulave wee Board o Governor o e Federal Reerve Syem //reearc.loued.or/are/wdaa.ml requred reerve va uc ceme ee Aend or more deal on Federal Reerve ory w reec o wee roram). $ Bllon Jan-94 Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-0 Jan-02 Jan-03 Jan-04 Jan-05 In or e eence o a real deo wee roram e aly o e ank o reclay or reerve-requremen uroe ranacon deo uec o a 0 ercen reerve requremen a avn deo eccally MMDA) uec o a zero reerve requremen. A real deo wee eenally keen wo e o ook o a o evade e reerve requremen a collecor. Snce de e.. ceck wrn) can only e ervced rom ranacon accoun omal o alway leave ome und n e ranacon accoun and relen e accoun rom e comanon MMDA a requred. o dcourae e ue o e MMDA a ye o avn accoun) a were a ceckn accoun Federal Reerve reulaon Reulaon D) lm o a mamum o 6 e numer o wdrawal a can e made rom an MMDA accoun durn a calendar mon. ee lm do no aly o raner made y eronally vn a ranc or y AM or y one. e conequence o volan e raner lm ar e enre amoun a a een we eween e BA and e MMDA ecome uec o e ame auory reerve requremen a no wee ad een aemed a all. u enerally omal o move all remann alance rom e MMDA o e ranacon accoun on e raner ou o e MMDA accoun. 4

5 Numerou leal nerreaon e.. Ireland 996) rom e Federal Reerve emaze e 6 raner lm. For a rokerae clen w a alance o a mllon dollar a would mean a undred ouand dollar n addonal reerve. ereore e wee mecanm would ave o enure a no more an ve raner were neceary rom MMDA o BA every mon. need or udcou ween requre ocaed comuer oware and alorm. Daa on e areae amoun o deo aeced y real deo wee acvy o mode qualy a e. Commercal ank are no requred o reor daa on wee acvy o e Federal Reerve. Hence n mo cae Federal Reerve a ave mued e amoun en we rom oer deo and reerve daa. Swee acvy rew lowly rom nceon n January 994 rou md-995 and ereaer rew radly a ank concluded a e Federal Reerve would no move o ro uc reerve-requremen reducon ceme. A een n Fure e cumulave amoun o wee nce 994 reaced aou $654 llon y January 2005 or more an 50 ercen o e oal amoun o ranacon alance a ank and r nuon. Wderead ue o wee roram a ncanly reduced e amoun o requred reerve alance a ank manan a e Federal Reerve. A a reul even a areae ranacon deo ncludn e amoun n wee) a ncreaed e oron o oal requre reerve aed y vaul ca a eld almo eady wle e oron aed y Federal Reerve Bank deo o-called requred reerve alance ) a allen; ee Fure 2. In e a coule o year ere a een a l ncreae n reerve alance due o low nere rae. ere ome concern amon econom a rend can ncreae volaly n e ederal und rae a e ank ry o manae er accoun w very low alance Benne and Hlon 997 Wrae 998 Anderon and Race 200 Anderon 2002a). Some econom ave ereed concern a areve manaemen o deo o eld a Federal Reerve Bank va wee roram m ncreae ederal und rae volaly an oucome rearded y e Federal Oen Marke Commee a armul o moneary olcy eecvene. Accordn o Furne 2000) A recenly a a decade ao relavely auory reerve requremen creaed a arly redcale demand or ank reerve makn nere rae aren a relavely eay-o-mlemen olcy coce or e Federal Reerve. Over e la 0 year owever lower reerve requremen and e wderead adoon o 5

6 Fure 2 Comonen o Reerve and Accoun Balance a e Fed $ Mllon Requred reerve alance Aled vaul ca 0 Jan-94 Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-0 Jan-02 Jan-03 Jan-04 Jan-05 Source Board o Governor o e Federal Reerve Syem // // wee accoun ave recaed a dramac all n requred reerve alance. ee lower reerve alance may ncreae uncerany n reerve demand oly comlcan e Fed' aly o aceve e oal o moneary olcy. And Cloue and Dow 2002) ae Several counre ave already moved o moneary yem wou reerve requremen o a er demand or reerve enrely a demand or ece reerve. e Uned Sae alo movn n a drecon due o e adoon o real wee roram y commercal ank. ee roram ave reuled n ncanly lower level o requred reerve leavn ome ank n e oon o no needn o old reerve alance o mee er reerve requremen. Underandn ow demand eave ecomn an moran ue n aled moneary economc. Some even n recen year ave orended an end o real deo wee accoun. Bu n our onon uc even are no lkely o reac ruon. In a lae 990 Federal Reerve urvey a amle o ween ank reonded ey would dconnue real deo wee roram e Federal Reerve ad nere on deo a e Federal Reerve Bank. Lelaon o requre e Federal Reerve o ar ayn nere on erle reerve alance a e Fed a een nroduced u aled o a e Conre e.. e Bank Reerve 6

7 Modernzaon Ac o 2000 and e more recen HR 758 e Bune Ceckn Freedom Ac o e laer aed e Houe o Rereenave u wa no aced on n e Senae. e reaury a no endored e rooal due o oenal udeary urden Aernay 2003). Ceran mall communy ank are uorn e rooal Mau 2003) on e round a wee roram are eenve and comlcaed or mall ank o mlemen wle oer Menze 2003) lke an alernave rooal HR 974) wc would allow 24 raner rom MMDA o BA eac mon nead o e reen 6. Alou real deo wee roram ave een oeran or more an a decade and ank ave earned mllon o dollar y er ue ere are no uled aer modeln er mlemenaon oeraon or omal unn. ee roram ave reduced e amoun o ank erle reerve rom $26.3 llon n Jan 994 o aou $0.7 llon n Feruary 2005 Fure 2). Even aumn a conervave 00 a on read on nere rae e mled earnn are $56 mllon er year. Clearly an neren omzaon rolem w earnn oenal. aer aem o en o rde a n leraure. 2. Syem or Swee Proram ere are a numer o alorm or oeran real deo wee roram. Pera e earle alorm wa e mle wee und rom e ranacon deo no e MMDA a e cloe o une on Frday and revere e wee a e oenn o une on Monday; Monday a ederal ankn olday oone e revere raner o ueday. Becaue e Federal Reerve auory reerve requremen are calculaed rom end-o-day alance on a 7-day week alorm mmedaely reduce e amoun o ranacon deo uec o reerve requremen. Conderale mrovemen owever can e oaned y more ocaed meod a we dcu ne. In ank wee roram are ar o e reaury ca manaemen uncon. Wle a varey o alorm undouedly e anecdoal evdence ue a wo are e mo oular. For dcuon we lael em e reold meod and e cuon meod. Bo ee meod ry o leverae -requency normaon on cuomer eavor daly aern o ne cred and de) o omze e dvon o und over e mon eween e BA and MMDA accoun uec o e conran o no more an raner lm rom MMDA o BA. Noe a ere no lm on raner rom e BA o e MMDA accoun alou eac 7

8 uc raner may ave a mall co or e ank. Recall a ve raner ave already een made or an accoun enerally omal on e raner o move e enre MMDA alance o BA. can e eenve. I an nuonal cuomer a $0 mllon n er accoun and all o e ranerred o BA e reerve wll ave o e ncreaed y $ mllon or wc no nere earned y e ank. All uequen acvy aer e dumn o all MMDA und o BA o de and cred done va e BA. For o meod moran o kee n mnd a e mon-y-mon raner lm moe a comle me-deenden rucure on e uncon a dene e omal oucome. In eneral we wll ow a e reold cuon amoun and omal raner n o drecon deend on e oon o eac ecc une day wn e calendar mon. reold Meod In meod or eac accounolder a reold e or e BA accoun alance. Becaue ere no rercon on e numer o raner rom e BA o MMDA and ecaue we aume zero co o uc raner omaly alway requre a all ncomn und e creded o e BA. I e accoun alance n e BA u ror o e end o e une day eceed a day omal reold ece und are moved rom BA o MMDA eore e cloe o une. I on ncomn de aan e BA mle a neave alance a e end o une und are ranerred rom e MMDA o e BA. In our model we aume a e raner rule a ard conran. A a reul on e raner all und n e MMDA are moved o e BA. In ac a ocac economc decon and ar o e omal oluon. A ank wc cooe no o move all und rom e MMDA o e BA on e raner lacn a e a a even raner wll no e neceary eore e end o e mon wen a new e o raner en. Deermnn e correc oon rce or e eyond e coe o aer and ar o our curren reearc. In alorm e amoun ranerred equal o e reold lu e amoun o neave alance leavn a alance o e reold amoun n BA aer e raner. In oeraon meod eek o manan e BA alance wn an acceale and. e arameer or meod are e reold value. e mnmum level ere zero. e meod can e eneralzed o a uaon were e mnmum level e o a value reaer an zero n wc cae e arameer are e mnmum and mamum reold. 8

9 Cuon Meod In aroac ncomn cuomer und deo) rouou e mon are creded o e MMDA o lon a e raner coun rom MMDA o BA le an. Snce de can only e ervced rom e BA accoun e r de ranacon nae a raner rom MMDA o BA o e de amoun lu a day-ecc cuon amoun. e omal amoun enerally ould e lare enou o ervce a ew more de ranacon. Wen e alance n e BA no ucen o ervce a uequen de ranacon anoer raner rom MMDA o BA naed. raner aan nclude und o ervce e curren de ranacon and a cuon amoun wc n eneral wll der rom e r cuon amoun. aern reeaed a needed u o ve raner. e arameer n meod are e cuon amoun or eac raner. Alou ee wo meod o can e condered mlar o e S) and rq) model n nvenory manaemen e alorm are no omorc a we ave een unale o eal ocac) crcumance n wc e wo meod yeld e ame accoun alance and ro. In o meod e em en nvenored ca a deo a Federal Reerve Bank. e meod der rmarly n wo reec. Fr e alorm der n were e deo are made. In e reold meod o e BA n e cuon meod o e MMDA. In curren work we are elorn omal raee or allocan deo eween e wo accoun. Second ey der n er reamen o e raner rom MMDA o BA. In urer work we are elorn e omal andln o e rk ncurred y no movn all und on e raner. Fnally a noed aove e and r are e o zero ou no neceary and e meodoloy could ene rom no rercn ee o zero. Dcuon w ank reaury manaer ue a mo model ued n racce o mlemen wee roram are very mlc. In e reold meod or eamle oen lle varaon ermed y day; n ome cae manaer allow e reold o vary y raner coun $000 or r raner no BA $2000 or econd raner ec.) u no y day o e mon. Furer manaer may ue e ame reold or all accoun reardle o ze and ranacon aern. In addon e ame reold omeme ued or ranerrn alance ou o BA o MMDA and or movn und ack rom e MMDA. In e ne econ we ow a deren reold or ee wo ye o raner yeld ueror earnn relave o one reold and a reold wc vary y day o mon and accoun acvy yeld urer earnn mrovemen. 9

10 Fure 3 e reul on MMDA and BA alance o un deren reold value or e ame accoun $3000 and $5000 reecvely) MMDA Balance BA Balance MMDA Balance BA Balance Alou no dcued urer n analy dcuon ue a manaer oen ue mlar meod or e cuon aroac w cuon amoun a vary y raner coun ay a cuon o $000 or e r raner $2000 or e econd raner ec.) u no vary y cuomer acvy or day o mon. In e alance o aer we eamne e reold meod. An lluraon w wo deren e o reold or e ame accoun own n Fure 3. e dealed ranacon daa or eac day own n Aend 2 and 3 or e wo cae own n Fure 3. In e cae on e le all raner ecome neceary and ereore e raner dum e enre MMDA alance no BA. All uequen ranacon are n BA. an eenve oucome due o a oor coce o reold. In e cae on e r ecaue o a eer e o reold ued only ve raner are neceary. ereore e BA alance ke low. 0

11 Fure 4 ranon mar or ne ranacon acvy or an accoun Ne Day Wdrawal No Deo Averae Curren day Lare Small acvy Small Lare Amoun Wdrawal Lare 7% 46% 33% 7% 7% -$3500 Small 5% 49% 35% 6% 5% -$500 No acvy 4% 35% 49% 6% 6% $0 Deo Small 4% 44% 4% 6% 5% $500 Lare 9% 50% 34% 3% 4% $ A Socac Dynamc Prorammn Model We develo a ocac dynamc rorammn model or rolem. For racaly and reenaon ere e daa n our model omewa le rc an e daa a a ank reaury manaer would ave avalale or ue n uc a model. Mo ank rean cuomer accoun acvy de and cred) or a lea everal recen mon ncludn daly accoun alance and ne ranacon acvy. e ne ranacon acvy e ne o all deo and wdrawal oed o e accoun durn e day. Pove numer correond o ne deo neave numer o ne wdrawal and zero o no acvy durn e day.) Becaue e ocac model could e very unweldy acual real-numer dollar ranacon value are ued we cooe o model un dcree ranacon nerval. Suoe ere are n ranacon nerval n e model. For eamle we cooe o model w 8 ranacon nerval or ne acvy e nerval may e dened o correond o daly ne ranacon acvy o < o o 0 o o o 5000 >5000. Noe a e nerval reakon do no need o e ymmerc o equal ze or ave a many wdrawal nerval and deo nerval. However moran o ave one nerval or no acvy 0 nce on many day ere may no e any ne acvy. Un e orcal ranacon daa a nn) ze ranon mar creaed. See Fure 4 or an eamle w 5 nerval. e enre o e ranon mar correond o e roaly o avn a ne ranacon n nerval on one day and a ranacon n nerval e ne day were = or all.

12 Fure 5 MMDA and BA alance n varou cae o accoun acvy. e raner no BA reold and e raner ou o BA reold. Balance a ar o day Ne ranacon durn day BA alance 5 mnue Balance a end o day raner MMDA BA eore end o une raner MMDA BA coun Cae m Wdrawal 0 <= < No raner m Cae 2 m Wdrawal < 0 raner coun < 6 raner - ) rom MMDA o BA m-- )] Cae 3 m Wdrawal raner coun = 6 raner enre MMDA alance o BA 0 m Cae 4 m No acvy No raner m Cae 5 m Deo <= No raner m Cae 6 m Deo > raner - ) rom BA o MMDA m )-] Le { n}) e e averae ne ranacon amoun or eac nerval la column o Fure 4) were neave or wdrawal nerval and ove or deo nerval A denoe e MMDA alance a e ennn o e mon lm denoe e MMDA alance and denoe e BA alance on day. Suoe e ne ranacon on day n ranacon nerval and raner rom MMDA o BA ave een made o dae. Le X e e mamum allowale raner. A er curren Federal Reerve reulaon X 6. A e ennn o e mon =0 and we conran numer o e le an or equal o X a e end o e mon. Suoe ere are workn day n e mon. en e ae o e yem n day can e eced y m). Le m ) e e mamal dcouned ne reen value o en n ae m) n erod n e erod rolem wen omal acon are aken n eac day rom rou. e acon eran o coon e correc reold wen raner ecome neceary. We wll ue deren reold or raner no BA and ou o BA. Le r m e e nle-day revenue realzed rom an MMDA alance o m and a BA alance o r m = r m δ ] ) were r e nere rae read a e ank earn on alance eeced marnal ae yeld mnu averae co o und) and δ e marnal) racon o BA alance a e ank need o manan a erle reerve. Under curren reulaon δ aromaely 0. or 0%. 2

13 Le β e e nle-day dcoun acor. en e cae a need o e condered deendn on e ae o e yem and e accoun acvy own n Fure 5. In wa ollow a orand we reer o e ard conran a all und n e MMDA mu e moved o e BA on e raner a a dum o e und ack o e BA. e ollown recurve unconal equaon ecy e model Wdrawal neave) Cae and Cae 2 No dum needed) Suoe e ranacon a wdrawal a > ) en we need o elec a wdrawal reold a olve e ollown recuron. m ) = r m 0 <= < u < 0 β m )] >= 0 β m )] )] ) Cae 3 Dum needed) Suoe a raner rom MMDA neceary and e raner coun X nce all e MMDA alance e ranerred o BA we ave = m β m ) r 0 m ) )] 2) No acvy zero) Cae 4 No acvy) Suoe e ranacon no acvy and e raner coun < X en no raner made. m ) = r β m )] m 3) 3

14 Deo ove) Cae 5 and Cae 6 Deo) Suoe e ranacon a deo we w o deermne a deo reold a olve e ollown recuron m ) = r m β u >= > β m )] 0 m )] )] 4) Boundary condon e oundary condon or model m ) = 0 m 5) We olve model y recurvely olvn e unconal equaon 000) were A e nal alance n e accoun. 0 A We ave ound e aove model a med dcree-connuou DP model dcul o olve. e reold amoun or wdrawal and or deo are connuou real numer wle e value are dcree. Furer e model a everal rde were dervave are no connuou due o o zero-ound rercon uc a BA and MMDA alance canno e neave and o e ard conran o no more an raner er mon rom MMDA o BA. o olve e model we mled e rolem o a o ue dcree raer an med dcree-connuou oluon meod. Our coce o dcree reold value are dcued n e ne econ. 3. e Model w Dcree reold Suoe 2 z and 2 y are dcree rcly ncrean reold were 0< < 2 < z and 0< < 2 < z. Becaue e reold need o e ucen o cover a lea one uure wdrawal and ould e a lea a lare a e aolue value o e malle 4

15 5 averae wdrawal amon e wdrawal nerval. Alo z could e e mamum ole reold a can e ranerred wou reuln n a neave MMDA alance a 0) ma > = m z. Smlarly ) 0 ) ma > = y. We wll ow laer n Fure 7 a ee reold end-on can e eleced o make e comuaon muc more ecen. For now we can mody ) or dcree reold a ollow or wdrawal < < <= < < <= < < <= = )] )] 0 )] 0 )] )] 0 )] 0 )] )] 0 )] 0 ma ) m m m m m m r m z z z z m z β β β β β β M L 6) Smlarly we can mody 4) or dcree deo reold a ollow > > > = )] )] )] )] )] )] )] )] )] ma ) m m m m m m r m y y y y y m y z β β β β β β M L 7)

16 4. Srucural Reul Our model a veron o e equenal anmen rolem Derman Leerman and Ro 972 Ro 983) and very mlar o e ame So wo counn Puerman 994). In a ame wo eam comee o aemle e lare 5-d numer rom a e o 0-9 d randomly and equenally cked y a nner n ve n. e eam may lace eac o e d a are cked y e nner n one o e ve avalale lo or d e un en undred ouand and en ouand oon). e eam a a a larer numer aer e ve n wn. In ame ovouly e r n reul n a 9 one ould u n e r avalale lo and e r n reul n a 0 one ould lace n e la avalale lo. e lacemen o oer d may no e a ovou u urn ou a ere an omal olcy a ece were eac d ould e laced ven a cked n a arcular n. e olcy may no reul n wn every me u wll e e n e lon run e ame reeaedly layed. In rolem ere are wo dmenon e n numer and e oerved numer n a n. e roo o e omal olcy ue e ac a e value o e oluon monoonc n o ee dmenon. o ee a our rolem mlar nk o a ame were e numer o workn day n e mon) n are allowed and n eac n one o 6 raner coun {0 5} may e oerved. e analo o e lacemen o e d e reold value. In our rolem ere are ree dmenon and m). We wll ow ne a e value o e oluon m ) monoonc n eac o ee dmenon. Lemma a) m ) non-ncrean n or all m and ) m ) non-decrean n or all m and c) m ) non-decrean n m or all and Proo We rove ee reul y nducon. For a) noe a e reul rue a =- rom 5). Suoe e reul were rue a. en rom )-4) 6) and 7) ollow a e reul 6

17 rue a. e reul ollow y nducon. For ) noe a a =X e reul rue ecaue o e dumn o e enre MMDA o BA n 2). en uon a e reul were rue a ollow rom 6) and 7) a e reul rue a. e reul en ollow y nducon. c) ollow rom e denon o r m. Suoe e omal reold n ae m) n erod rom 6) or wdrawal. m and m were m = ar ma m ) And rom 7) or deo m = ar ma m ) en un e reul n Ro 983) we can reen e ollown eorem. eorem a) ) m and m and m are non-ncrean n or all m and m are non-decrean n or all m and Proo e roo ollow rom Ro 983 ae 25 Prooon 7.2) and Lemma aove. 7

18 Fure 6 An lluraon o omal reold or wee m=6500 =0 and daa n Fure 4 Lare Wdrawal reold m Small Wdrawal reold m2 raner Coun raner Coun Day o EOM Small Deo reold m4 Lare Deo reold m5 raner Coun raner Coun Day o EOM

19 An lluraon o e omal reold or our cae o oal alance own n Fure 6. Noe a n eac cae e reold value are non-decrean rom e ouwe corner o e ale o e norea corner wc aemen a) and ) o eorem ). Alo e mamum reold n eac cae e mamum ole. For eamle n e o le cae o lare wdrawal reold noe a e MMDA alance $6500 e BA alance $0 and e amoun o e ranacon rom Fure 4 -$3500. u aer ranacon comlee e oal MMDABA alance $3000. e ale akn or all o o e ranerred o BA a e reold own $3000 you ave already made 5 raner and ave 2 day o o o e end o e mon. O coure a e ale ndcae e reold wll e lower you ave ewer raner done y a dae. e reold own n Fure 6 are n ar conra o e mle reold ued y many ank. For eamle a e o reold may e and 2000 or raner coun rou 5. ee reold may e e ame or all day o e mon and or all ranacon ye. Furer ee may e only marnally adued or deren cuomer ye. On e conrary we are uen many more emen ee ne econ) and a e ale ndcae reold a vary y day o mon and ranacon ye. e reul o eorem allow u o lm e earc or reold and make e model muc more ecen. For eamle rom Fure 6 or e cae on e o le MMDA=6500 and BA=0) e earc or e omal reold w 2 day o o raner numer o 3 can e lmed o e rane eween e numer elow and e numer on e r a ). We can ormalze un e ollown reul. Corollary e earc or can e lmed o e rane ) m ) m ) m ) Proo Follow mmedaely rom eorem. 9

20 Fure 7 Alorm or lmn cuon earc or mlar) Se m o 99 or all m; Funcon m) Ben I ) m <>99 en H = ) m ele H = z ; I <>99 en ele = 0 ; m ) L = m ) L m ) = ma L =... H z M H L 0 <= < < 0 0 <= < < 0 H L β β β β m )] m m )] m H L )] )] L H )] )] End; cd = arma d ) ; In order o ue reul we can mlemen e mle alorm own n Fure 7. would allow olvn or larer rolem w a larer numer o ranacon nerval or eamle. A larer numer o nerval wll reul n a ner ranulary o modeln ranacon nerval and averae ranacon amoun. 5. Imlemenaon Iue e mlcaon due o dcree reold and e ecence aned y alcaon o e aove alorm or reduced earc reul n e model akn le an a mnue on a.5gb eronal comuer or a rolem w 7 ranacon nerval or eac value o nal MMDA alance A. However a ew addonal e may e aken o mly mlemenaon. Accoun Semenaon Nowandn e ac a e model a and ecen may no e raccal o run e model or eac accoun older. ereore mlar accoun may e roued no emen and e model run or eac emen nead. Arorae emenaon 20

21 varale may nclude e averae MMDA and BA alance or e accoun and ranacon eavor uc a numer o ranacon er mon averae ze o ne ranacon and e volaly o ne ranacon amoun). Balance Semenaon Even wn an accoun emen no ruden o ave a reold ale a own n Fure 6 or eac value o alance. a would requre numerou ale and wll make mlemenaon dcul. Snce rane o alance wll ave mlar reold amoun make emnen ene o rou alance amoun no emen and ave a nle looku ale o reold or eac alance emen. Anoer aroac may e o ue a rereon model o relae reold value develoed a deren alance emen o e oal alance level n eac emen. rereon equaon could en e ued o reurn a reold value or any ole oal alance on a connuou cale. Scaln o dollar amoun Anoer mlcaon would e o cale and e ennn o e mon MMDA amoun A un ome convenen caln acor. For eamle n a 5 ranacon nerval cae e are and 3750 reecvely and A 500 en 50 may e a convenen caln acor e reae common dvor o e aolue value o ece or e 0) a cover no and 25 reecvely and A o 0. e reold ould en alo e caled un e ame acor. In eamle a reold o 50 wll e convered o. I may e convenen o ue 2 z a are mulle o or eamle 2 =2 3 =3 ec. We ave ound a e numer do no cale eacly aromae caln work well oo. Modeln cuomer eavor An underandn o cuomer eavor crcal o deermnaon o omal reold. For real ank w redomnanly oueold accoun ere are ome ranacon a are more redcale an need o e modeled y way o a ranon mar. For eamle ncome may e deoed weekly and redcale morae aymen are redcale cred card aymen are on cycle dae are redcale a lea or dae no or amoun. Even e amoun can e redced w ome accuracy. Smlarly uly ll are a known dae and arly ale. Only oer undry and le redcale eene need o e modeled un e ranon mar. Our ocac dynamc aroac can ealy accommodae ee renemen o make e model more ecen. 2

22 6. Concluon We ave reened a mle ocac dynamc rorammn model or deermnn e omal reold o reduce erle reerve n real ank. We ave alo reened rucural reul o make e earc or reold more ecen allown or olvn larer rolem. In mulaon un real daa our model wa ound o e eecve n reducn erle reerve and en mlemened. Prooed lelaon aed y e Conre and ned no law m make oolee real deo wee roram y ncrean e numer o raner allowed er mon or allown e Federal Reerve o ay nere on all Federal Reerve Bank deo. Un our model ra orward o ae e mac o ee cane. Increan e numer o allowed monly raner rom 6 o 24 unamuouly relae a ndn conran reuln n a reduced level o erle reerve and ncreaed earnn. e eec o e meaure o ay nere on reerve alance w e Fed deend on ow muc nere acually ad. Mo rooal conran e Fed o ayn ncanly le an e overn ederal und rae. o e een a e rae le an e ank eeced nere-rae read e need or udcou wee would ll e. Bo o ee meaure aed would reduce e earnn realzed y ank rom our model n relave erm nce reold amoun wll e le crcal) u our model would ll e valuale n deermnn eecve wee mecanm. Reerence Aernay W emony o A. Secreary or Fnancal Inuon US Dearmen o reaury on Bune Ceckn Accoun and Reerve Balance eore e Sucommee on Fnancal Inuon and Conumer Cred o e Commee on Fnancal Servce o e US Houe o Rereenave. Marc 5. // Downloaded 3/28/2005 Anderon R. 2002a. Federal Reerve Board Daa on OCD Swee Accoun Proram. Federal Reerve o S. Lou //reearc.loued.or/are/wdaa.ml Downloaded 3/28/

23 Anderon R Real Swee Proram and Money Demand. Federal Reerve Bank o S. Lou Moneary rend Novemer. Avalale a <//reearc/u/m/d/2002.d> Anderon R. Race R.H Real Swee Proram and Bank Reerve. Federal Reerve Bank o S. Lou Revew January/Feruary 83) //reearc.loued.or/ulcaon/revew/0/00ra.d Downloaded 3/28/2005 Benne P. Hlon S Falln Reerve Balance and e Federal Fund Rae. Curren Iue n Economc and Fnance 3 5) -6 Federal Reerve Bank o New York. // Downloaded 3/28/2005. Cloue J. A. J. P. Dow Jr A Comuaonal Model o Bank Omal Reerve Manaemen Polcy. Journal o Economc Dynamc and Conrol Derman C. Leerman G.J. Ro S.M.972. A Sequenal Socac Anmen Prolem. Manaemen Scence Furne Cra H Inerank Paymen and e Daly Federal Fund Rae. Journal o Moneary Economc Ireland O Federal Reerve Board Leal Inerreaon rom 4/29/96. Federal Reerve Board. // Downloaded 3/28/2005. Mau E emony o Amerca Communy Banker on Inere on Bune Ceckn Accoun eore e Sucommee on Fnancal Inuon and Conumer Cred o e Commee on Fnancal Servce o e US Houe o Rereenave. Marc 5. //nancalervce.oue.ov/meda/d/030503em.d Downloaded 3/28/2005 Menze R.M emony o Indeenden Communy Banker o Amerca on Inere on Bune Ceckn Accoun eore e Sucommee on Fnancal Inuon and 23

24 Conumer Cred o e Commee on Fnancal Servce o e US Houe o Rereenave. Marc 5. //nancalervce.oue.ov/meda/d/030503m.d Downloaded 3/28/2005 Puerman M.L Markov Decon Procee Dcree Socac Dynamc Prorammn Jon Wley and Son New York. Ro S Inroducon o Socac Dynamc Prorammn. Academc Pre New York. Wrae J.M I e Fed Ben Swe Ou o Moneary) Conrol? Bune Revew Nov- Dec -0. Federal Reerve Bank o Pladela. // Downloaded 3/28/

25 Aend e Federal Reerve and Real Deo Swee Proram Real deo wee roram were nroduced y a lare U.S. commercal ank n January 994. In uc a roram e ank creae money marke deo accoun a are nvle o cuomer and re-clae cuomer ranacon deo a MMDA a avn) deo o a o avod e 0 ercen marnal auory reerve requremen rao on ranacon accoun; avn deo ncludn MMDA are currenly uec o a zero ercen reerve requremen rao. e 982 Garn-S. German ac wc creaed e money marke deo accoun a a deo nrumen roed e Federal Reerve Board rom clayn uc deo a ranacon accoun or reerve-requremen uroe. roon an eenal elemen o e oeraon o real deo wee roram. Horcally ank ave nroduced many nvenory manaemen ceme o reduce e amoun o deo a Federal Reerve Bank. Bu ror o real deo wee roram e Federal Reerve Board romly ended all uc ceme. In mo cae ruled a e ceme wa rmarly nended o evade auory requremen and ordered a e aeced deo e uec o e ame reerve requremen a oer ranacon deo. One o e more clever elo n wee occurred durn e early 990. In a cae a lare ank ean ween ranacon deo no lare-denomnaon me deo w even-day maure. By lacn oneeven o e we und no eac o even deren lare me deo maurn on deren day o e week ou o rovde ranacon ervce o cuomer wle reducn e amoun o end-o-day alance uec o auory reerve requremen. e Federal Reerve Board ruled a wa olely a ceme o evade reerve requremen a o coure wa) and re-claed e nvolved lare me a ranacon deo uec o e ame reerve requremen a e nal ranacon deo. W earnn move deroyed e ank droed e ween acvy. e -raner rule or MMDA mandaed y e Garn-S. German ac and romulaed o ank a ar o e Federal Reerve Reulaon D. Seccally e law e a mamum o re-auorzed lu rd-ary aymen er mon. I doe no lm e numer o n-eron cuomer wdrawal or raner. I a cuomer make more an 6 uc raner durn a mon ay va ceck) a ank may moe a enaly on e cuomer u e ank need no reor uc a volaon o e Federal Reerve wc n urn unlkely o moe any enaly on e ank. I a ank reularly or requenly erm more an uc wdrawal y akn no acon o requre cuomer comlance w e auory lm e MMDA alance may e claed y e Federal Reerve a ranacon deo raer an MMDA) and ence uec o e ame reerve requremen a oer ranacon deo. e ank alo may e uec o admnrave enale. We model auory reerve requremen a a conan 0 ercen marnal requremen. Acual auory requremen on ranacon deo are ered w a zero ercen rao alyn o aromaely e r $5 mllon a 3 ercen rao alyn o aromaely e ne $50 mllon and nce 992 a 0 ercen rao alyn o e remander. ee reak on are ndeed o row n oal ank deo and cane annually. In 980 e e marnal reerve requremen wa 2 ercen; wa reduced o 0 ercen n 992. For larer ank e lower er maer no a all. e role o e Federal Reerve Board wc e auory reerve requremen wa an moran ue w ome analy and conulan durn er early year o real deo wee roram. e 980 Moneary Conrol Ac e a zero rao or eronal me and avn deo. In 990 e rao or non-eronal me and avn deo and or Eurocurrency lale were reduced rom 3 ercen o zero. Srcly eakn ee deo lale reman 25

26 uec o Federal Reerve auory reerve requremen u a a zero rao and e Federal Reerve Board rean e ower o ncreae ee ercenae n e uure dered. Are e nvle MMDA creaed a ar o a real deo wee roram eronal or non-eronal accoun? Some analy arued a e accoun alou creaed y ank wou er cuomer knowlede are eronal avn deo and ence roeced rom a non-zero rao y e Moneary Conrol Ac. Oer analy arued a e accoun were creaed olely y e ank a a comonen o e ank nvenory manaemen yem or deo a e Federal Reerve Bank ould e rearded a non-eronal deo wc e Federal Reerve Board dered could e made uec o a conderaly er rao. Alou clear a oeraon o a real deo wee roram reduce e amoun o erle reerve eld y ank dcul o emae ow muc o e relaed earnn ave een reaned y ank and ow muc a een realzed y conumer va marke comeon even ou e underlyn nvenory ecnoloy nvle o e cuomer). Recen daa ue a ome oron a een realzed y conumer; ee Anderon 2002). I a een reored a more recenly ome une cuomer ave ecome aware o e weeno-mmda ceme and ave requeed and receved a ncan are o e reuln earnn. I eem ovou a vrually all oueold are unaware o e oeraon o uc wee roram. ere are many addonal and comle aec o ank reerve manaemen a nerac w manan deo a Federal Reerve Bank and real wee roram u ee are eyond e coe o aer. Reerve alance own n Fure 2 comre only aou al o e deo eld y ank a e Federal Reerve. e oer al are mlar deo u encumered y clearn alance conrac w e Federal Reerve; under e erm o uc conrac e ank earn nere on e deo acual amoun mee or eceed conracual mnmum u ncur enale or orall. e unly o deo make reerve manaemen comle ecaue deo eld o ay clearn alance conrac can nead e ued o ay auory reerve requremen. In ac Federal Reerve accounn rule requre a avalale deo a Federal Reerve Bank r e aled o ay auory reerve requremen and only e ece o uc deo may e aled oward ayn a clearn alance conrac. Modeln caale o andln e dynamc o ocac roce are eyond e model n aer. 26

27 Aend 2 Ne ranacon and MMDA and BA alance or an accoun wen a reold o $3000 ued or wdrawal and deo Day Sar o day BA 5 mn eore MMDA EOB) 5 mn eore EOB raner o End o day raner Ne ranacon BA MMDA BA MMDA BA 5 mn eore EOB) Coun

28 Aend 3 Ne ranacon and MMDA and BA alance or an accoun wen a reold o $5000 ued or wdrawal and deo Sar o day 5 mn eore EOB raner o End o day raner BA 5 mn BA 5 mn Day MMDA eore EOB) Ne ranacon BA MMDA BA MMDA eore EOB) Coun