Internalization, Clearing and Settlement, and Stock Market Liquidity 1

Transcription

1 Internalzaton, Clearng and Settlement, and Stock Market Lqudty 1 Hans Degryse, Mark Van Achter 3, and Gunther Wuyts 4 May We would lke to thank partcpants at semnars n Louvan, Mannhem, and Tlburg for ther comments. Hans Degryse holds the TILEC-AFM char on Fnancal Market Regulaton. Gunther Wuyts gratefully acknowledges fnancal assstance from FWO-Flanders under contract G Correspondng author: CentER - Tlburg Unversty, P.O. Box 90153, NL-5000 LE Tlburg, The Netherlands. E-mal: h degryse@uvt nl 3 Unversty of Mannhem, Fnance Area, L5,, Mannhem, Germany. E-mal: [email protected] 4 Unversty of Leuven, Faculty of Busness and Economcs, Department of Accountng, Fnance and Insurance, Naamsestraat 69, 3000 Leuven, Belgum. E-mal: gunther wuyts@econ kuleuven be

2 Abstract We ntroduce a model that lnks stock market lqudty to the costs of clearng and settlement. In partcular, we model how dfferental prcng structures of the clearng and settlement agent stemmng from the nternalzaton of clearng and settlement affects stock market lqudty. We show that when the clearng and settlement agent sets prces such that t breaks even on the order flow per nvestment frm, dfferent stock market equlbra result. Wth substantal costs of non-nternalzed trades, traders from a large nvestment frm announce unattractve prces that are nterestng only for counterpartes of ther own nvestment frm. Traders orgnatng from other brokers fnd these prces not attractve enough as the related costs of clearng and settlement are too large for them. In contrast, the quotes submtted by traders from smaller brokers reman qute lqud as they am to attract counterpartes from all brokers snce ths substantally ncreases ther lkelhood of executon. Further, we analyze the case where the clearng and settlement agent charges the margnal cost for non-nternalzed trades. For suffcently hgh clearng and settlement costs, t may then happen that traders from both brokers target ownbroker counterpartes only. In ths case, the stock market s relatvely llqud wth traders from the large broker quotng more lqud prces than traders from the small broker. Fnally, for suffcently low costs of non-nternalzed trades, welfare s hgher when traders target all possble counterpartes, and not only those of ther own broker. JEL Codes: G10,G15 Keywords: nternalzaton, brokers, clearng and settlement, lqudty

3 1 Introducton The organzaton of a fnancal market s an mportant determnant of ts lqudty. Market mcrostructure, the process by whch nvestors latent demands are ultmately translated nto prces and volumes, has manly focused on prce formaton and prce dscovery, and on the market desgn of fnancal systems. Next to the mplct transacton costs related to tradng, explct transacton costs such as commssons and post-tradng nfrastructure costs are of consderable mportance. Data from Elkns/McSherry, for example, show that explct transacton costs consttute about three quarters of the total transacton costs (see e.g. Domowtz and Stel (00)). Further, accordng to the European Commsson, costs of the post-tradng nfrastructure represent 10 to 0% of total post-tradng transacton costs. Whle t s well-known that post-trade transacton costs are consderable, the market mcrostructure lterature has not yet studed ts mpact on the lqudty of fnancal markets. Ths paper makes a frst step to fll ths vod by analyzng the mpact of dfferences n prcng of clearng and settlement servces on stock market lqudty. These prce dfferences stem from dfferent degrees of nternalzaton of order flow by the post-trade nfrastructure. In partcular, we study how the potental of nternalzng trades affects partcpants wllngness to supply and consume lqudty. Our paper thus studes how the prcng of back offce actvtes nfluences the front offce,.e. the stock market lqudty. Our research s motvated by the recent ncluson of nternalzaton systems at several exchanges and the assocated prcng schedules for tradng servces. Internalzaton occurs when buyer and seller orgnate from the same nvestment frm. Ths may happen when () the nvestment frm trades on ts own account wth hs clent ( clent-to-house transacton ), () two dfferent counterpartes trade through the same nvestment frm ( clent-to-clent transacton ), or () transactons are carred out wthn the same nvestment frm ( house-to-house transacton ). In our settng, nternalzaton reduces the costs of the post-tradng nfrastructure,.e. the costs of clearng and settlement. In the US, the DTCC (Depostary Trust and Clearng Corporaton) whch clears and settles trades of all exchanges observed that an ncreasng number of nvestment frms pre-netted ther trades such that the order flowobserved bythedtccwasnotrepre- sentatve for the entre market. One of the recommendatons the DTCC made was to adapt the clearng and settlement fees n order to reduce the economc ncentve for usng pre-nettng (see e.g. DTCC (003)). In Europe, wth the mplementaton of MFID, the Markets n Fnancal Instruments Drectve, several tradng systems have ntroduced features allowng to nternalze clearng and settlement. Frst, regulated markets have created possbltes for nternalzaton. The London Stock Exchange for example started ts SETS nternalzer n Aprl 007. SETS nternalzer prevents on-book self-executons from passng through to clearng and settlement, thus avodng post-trade nfrastructure

4 costs. As a result, all order book executons where both sdes of the trade orgnate from the same nvestment frm do not pass through to clearng and settlement. The tarff charged s 0.1 bp, whch s 87.5% lower than the headlne rate. 1 Smlarly, Euronext has created an algorthm that nduces buy and sell orders orgnatng from the same nvestment frm to avod the cost of clearng and settlement. Second, systematc nternalzers allow to avod clearng and settlement costs when the trades orgnate from the same nvestment frm. A recent report by Oxera (009) argues that brokers nternalze about 10% of ther trades and they expect ths to ncrease over tme. Our paper addresses how nternalzaton of clearng and settlement may affect stock market lqudty. Our man nsghts can be summarzed as follows. Frst, we fnd that explct transacton costs such as costs of clearng and settlement affect stock market lqudty. In general, hgher costs of clearng and settlement appear to ncrease stock market lqudty. The reasonng s that hgher costs of clearng and settlement nduce more aggressve lmt order prcng to nduce ncomng counterpartes to trade. Ths s n lne wth emprcal evdence of Berkowtz, Logue and Noser (1988) who fnd that larger explct costs decrease mplct transacton costs. Second, nternalzaton reduces the costs of clearng and settlement. Investment frms wth larger market shares are therefore able to create some benefts as they allow to reduce costs of clearng and settlement. However, our results show that when more trades can be nternalzed stock market lqudty decreases. The ntuton behnd ths result s that an ncrease n nternalzaton opportuntes corresponds to a drop n explct transacton costs and therefore reduces the aggressveness of lmt order prces. Thrd, when the clearng and settlement agent sets prces such that t breaks even per broker, dfferent equlbra result. Stock market lqudty s harmed when the costs of clearng and settlement are very hgh (causng the broker-specfc break even prces to dffer substantally). Traders lnked to the large nvestment frm then announce prces that are only attractve to counterpartes of ther own nvestment frm (whch do not bear the hgh clearng and settlement costs). In contrast, the quotes submtted by traders lnked to smaller brokers reman qute lqud as they face another trade-off: submttng aggressve quotes allows them to attract counterpartes from all brokers whch substantally ncreases ther lkelhood of executon. Fourth, n addton to both above mentoned strateges, traders from both brokers may target ther own counterpartes only. Ths happens when the clearng and settlement agent charges the margnal and suffcently hgh cost for non-nternalzed trades. In ths case, the stock market s relatvely llqud wth traders from the large broker quotng more lqud prces than traders from the small broker. Fnally, we perform a welfare analyss comparng the dfferent settngs. We fnd that the equlbra where all traders target counterpartes 1 See page 8 on See page 40 on

5 from all brokers (and not only ther own broker) produce a hgher welfare, compared to equlbra where some (or all) traders am to attract only nternal counterpartes (.e. from ther own broker). To our knowledge no papers exst lnkng the organzaton of the post-tradng nfrastructure to stock market lqudty. Takng a wder perspectve, our paper s related to dfferent sets of lterature. Frst, t relates to the lterature on order submsson strateges n lmt order markets such as Foucault (1995, 1999), Parlour (1998), Handa, Schwartz and Twar (003), Foucault, Kadan and Kandel (005), Goettler, Parlour and Rajan (005), Roşu (009) and Van Achter (009). These papers model how traders choose between market orders and lmt orders n dfferent dynamc settngs. We extend them by ncludng the mpact of heterogenety n post-trade costs on the optmal quote settng behavor of traders belongng to dfferent brokers. Our paper also relates to the lterature on make/take fees as modeled n Foucault, Kadan and Kandel (009). In many markets, provders of lqudty receve a make fee, whereas consumers of lqudty pay a take fee. Foucault, Kadan and Kandel (009) show ths may nduce lqudty cycles to arse. Our paper contrbutes to ths lterature by hghlghtng that outstandng quotes by one broker n the lmt order book may nduce asymmetres for traders afflated to dfferent brokers. When the transacton s nternalzed and mples no costs of clearng andsettlement,thepost-tradecostslowandtsasfthepayabletakefeessmall. In contrast, when a trader of another broker s the counterparty, post-trade costs are hgh andtsasfthepayabletakefeeslarge. Second, our work contrbutes to the lterature on clearng and settlement. The theoretcal papers mostly deal wth the optmal prcng strateges when central securtes depostores (CSDs) nteract, n order to explan the hgh markups for cross-border transfers of securtes or the effects of dfferent degrees of access to the CSDs (see e.g. Rochet (005), Tapkng and Yang (006), Holthausen and Tapkng (007), Tapkng (007), and Koeppl, Monnet and Temzeldes (009)). We model how a cost-based post-trade nfrastructure may affect stock market lqudty n two dfferent ways. Frst, nternalzaton of order flow reduces costs at the CSD and therefore changes the traders aggressveness n the stock market. Second, the way a cost-based prcng structure s mplemented by the CSD may lead to dfferent stock market equlbra. In partcular, a prcng strategy fully reflectng the CSD s margnal cost may lead to an equlbrum where traders opt to only address counterpartes from the same broker. Ths reduces the total number of transactons and decreases market lqudty. Further, the emprcal papers on the posttradng nfrastructure manly nvestgate whether there are economes of scale and scope n the clearng and settlement ndustry (see e.g. Van Cayseele and Wuyts (008)). Our paper shows that transactons may exhbt dfferent degrees of dffculty (.e. cheaper nternalzed clearng and settlement versus more expensve cross-broker clearng and

6 settlement), hngng on the partcular stock market equlbrum that s played. Thrd, some papers connect dfferent phases of the tradng process. Foucault and Parlour (004) model how competton between stock exchanges lnks lstng fees and transacton costs on those exchanges. They fnd that competng exchanges relax competton by choosng dfferent tradng technologes and lstng fees. Berkowtz, Logue and Noser (1988) lnk explct transacton costs to mplct transacton costs and fnd that payng hgher commssons yelds lower executon costs (be t non-commensurate). Our paper also lnks two phases of the tradng cycle,.e. stock market lqudty and post-trade nfrastructure. The remander of ths paper s structured as follows. Secton ntroduces the setup of our model. Sectons 3 to 5 present dfferent prcng schemes mplemented by the clearng and settlement agent, and the correspondng equlbra. Wthn Secton 6, these equlbra are further compared and a welfare analyss s provded. Fnally, Secton 7 concludes. Setup Wedevelopannfnte horzon model to analyze a contnuous lmt order market lstng a sngle securty. Before tradng starts, the clearng and settlement agent decdes upon the prces of clearng and settlement. Traders take these post-trade clearng and settlement prces as gven durng the subsequent tradng day. Each perod n tme =0 1 +, a sngle trader arrves who s wllng to trade one share of the asset. Traders are rsk neutral and expected utlty maxmzers. Further, traders exhbt an exogenously determned tradng orentaton whch makes them ether a buyer or a seller. We assume that the proporton of buyers and sellers n the trader populaton s equal. 3 Buyers have a prvate valuaton for the asset equal to, whereas sellers have a prvate valuaton. We assume both valuatons are non-negatve and 0, whch mples there are always gans from trade between both partes. These dfferences n valuaton are an outcome of taxes, lqudty shocks, or other portfolo consderatons such as dfferences n endowment, or n opnons on the expected value of the asset. Each trader s lnked to one of two possble brokers whch means ther ndvdual orders arealwayssenttothemarketthrough ths partcular broker. More specfcally, a fracton of the total trader populaton s lnked to broker 1, and a complementary fracton 1 s lnked to broker. Throughout ths paper, we manly focus on brokers of dvergent szes. Thus, we assume broker 1 s a large broker servng a relatvely larger fracton of the trader base, whereas broker s a 3 Our model s easly adjusted for the case where the proporton of buyers and sellers s dfferent from 0 5; however t becomes slghtly more complex snce buyers and sellers no longer choose symmetrc strateges. We prefer equal probabltes as ths allows us more easly to dentfy the mpact of dfferent prcng schemes mplemented by the clearng and settlement agent.

7 small one (.e. 1 ). Broker afflatons are ndexed by subscrpt { }. Hence, for a trader arrvng n a random perod, wth probablty t s or a buyer or a seller from the large broker and wth probablty (1 ) t s or a buyer or a seller from the small broker. The post-tradng nfrastructure, whch from now on we denote as CSD (.e. Central Securtes Depostory), handles clearng and settlement mmedately after each transacton, and s consdered to be rsk neutral. The CSD has a cost per leg of the trade for non-nternalzed trades,.e. trades nvolvng dfferent brokers, and a lower cost for nternalzed trades,.e. trades nvolvng the same broker, whch we normalze to zero. In mplementng ts prcng scheme, the CSD always ams to break even on average, but does not necessarly charge ts true costs on each ndvdual transacton. Overall, dependng on the sophstcaton of the set prcng scheme, a CSD can charge dfferent costs based on the sze of the broker and on the type of transacton that s cleared and settled. The frst dstncton mples a dfferent cost for trades from the large vs the small broker. The second dstncton means that the CSD dfferentates between nternalzed and non-nternalzed trades. To properly account for these dstnctons, we consder three dfferent prcng schemes mplemented by the CSD. More specfcally, mcro-foundatons are provded for varous clearng and settlement costs, wth superscrpt { } ndcatng dfferent cases regardng the prcng structure of the CSD for nternalzed ( ) and non-nternalzed ( ) trades, and subscrpt { } referrng to broker sze. The followng table provdes a summary of the three dfferent prcng schemes: Prcng Scheme CSD Unform = = = Broker-Specfc = and = Trade-Specfc = and = The Unform prcng scheme means that the CSD charges the same cost to small and large brokers and to nternalzed and non-nternalzed trades. Ths cost s set optmally such that the CSD breaks even on average. The optmal cost and ts mpact on quotes wll be analyzed n Secton 3. Next, under Broker-Specfc prcng, dscussed n detal n Secton 4, the CSD charges a dfferent cost to the large broker, compared to the small broker. Wthn a broker, however, no dstncton s made between nternalzed or non-nternalzed trades. The fnal scheme, Trade-Specfc prcng, entals that an nternalzed trade wll be charged a dfferent cost, compared to a non-nternalzed trade. The CSD does not dscrmnate between brokers though. In Secton 5, we analyze ths prcng scheme n detal. An arrvng trader bases her order submsson strategy on her observaton of the standng lmt order book (LOB). She has two optons at her dsposal to trade. On the one hand, she could post a quote by submttng a lmt order (LO) whch does not offer

8 certanty of executon. Posted LOs stay n the market only for one perod and are thus take-or-leave offers for the next trader (see Foucault (1999) for a smlar approach). On the other hand, she could submt a market order (MO) whch guarantees mmedate executon but at the cost of a less favorable executon prce. Lqudty-demandng MOs execute aganst standng lqudty-supplyng LOs, so they can only be submtted f a counterparty LO s already present n the LOB. Clearly, the LO s executon probablty s endogenous n the model as t depends on other traders order placement strateges. We wll further dscuss ths ssue below n ths secton. Orders are for one unt of the asset, and once submtted cannot be modfed or cancelled. New n our model and a key contrbuton to the exstng lterature (such as Foucault (1999), Handa, Schwartz and Twar (003), and Van Achter (009)) s that traders also account for the prcng scheme mplemented by the CSD (and the mpled clearng and settlement cost) n choosng ther optmal strategy. More specfcally, t s argued that condtonal upon executon, the utlty of tradng the asset at prce for a buyer at broker for a transacton of type equals ( )=, whle a seller s utlty of tradng at broker wth a transacton of type s ( )= Hence, as non-tradng gans are normalzed to zero, and + reflect the reservaton prce under the approprate prcng structure that buyers are wllng to pay and that sellers are wllng to receve for one share of the asset, respectvely. Traders naturally am to maxmze the expected payoff of ther trade: for a buyer submttng a MO httng a standng quote ; Γ( ) for a buyer submttng a LO at quote ; for a seller submttng a MO httng a standng quote ; Ψ( ) for a seller submttng a LO at quote. accountng for the approprate clearng and settlement cost,andwthγ( ) the executon probablty of a buy LO at quote (the bd prce), and Ψ( ) the executon probablty of a sell LO at quote (the ask prce), as determned by the respectve buyer or seller. In settng the optmal bd or ask quotes when submttng a LO, a trader n general has two possbltes. She could determne quotes that only attract counterpartes from her own broker (we label ths strategy ) or she can opt for a quote that s attractve to all possble counterpartes,.e. traders from her own and from the other broker (we label ths strategy ). Do note that attract n ths context means the targeted ncomng trader s at least wllng to ht the standng LO by submttng a MO. Thus, any trader submttng a LO needs to account for the MO strategy of the subsequently arrvng trader. 4 Gven traders are lnked to ether a large 4 As such, the LO executon probabltes are endogenous, mplyng traders are n a game stuaton. In general, traders optmal order submsson strateges depend on ther LO s probablty of executon, whch n turn s determned by ther order submsson strateges. To properly account for these endogenous lnkages between the MO and the LO placement strateges, they wll be determned smultaneously.

9 or a small broker, four possble combnatons of strateges can be dstngushed: 5 I. traders of both brokers am to address counterpartes of all brokers: { }; II. traders of the large broker only am to address counterpartes of ther own broker, traders of the small broker am to address counterpartes of all brokers: { }; III. traders of the large broker am to address counterpartes of all brokers, traders of the small broker only am to address counterpartes of ther own broker: { }; IV. traders of both brokers only am to address counterpartes of ther own broker: { }. Note that the frst element wthn the mentoned { } always refers to the strategy of traders of the large broker, and the second element to the strategy of traders of the small broker. As wll become clear below, these four possble combnatons of strateges result n four possble sub-equlbra of our game. Indeed, for each combnaton, traders at dfferent brokers may post dfferent bd and ask quotes, and the CSD may charge a dfferent cost. We wll show below, however, that not every potental sub-equlbrum materalzes under every prcng scheme, because some combnaton(s) wll domnate others. All parameters of the model, ncludng,,, and are known to the nvestors. Moreover, they are constant over tme, hence the market s assumed to be n steady state. Ths allows to solve for a statonary equlbrum wthn each prcng scheme as n Foucault (1999) or Van Achter (009). More specfcally, a statonary market equlbrum s defned as a set of mutual order submsson strateges (specfyng an optmal order type, quote and correspondng executon probablty to each possble state of the LOB) such that each trader s strategy s optmal gven the strateges of all other traders. Dvergences n prcng rules mply dfferent types of equlbra arse. Both the magntude of the costs for clearng and settlement as well as the type of equlbrum nfluence stock market lqudty. In Sectons 3, 4 and 5 we provde a thorough analyss of each of the three derved statonary equlbra. 3 CSD Prcng Scheme 1: Unform Prcng Under the unform prcng scheme, whch s denoted by superscrpt, all transactons are handled by the CSD whch charges a unform cost for both brokers to all orders upon executon. Furthermore, at the level of the CSD, we assume that nternalzed 5 Do note that by assumng =1, wthn a broker we have that buyers and sellers have symmetrc strateges. Thus, there s no need to further dfferentate the strateges n ths respect.

10 transactons ental a normalzed zero margnal cost. In contrast, transactons stemmng from traders from dfferent brokers stll mply a cost for the CSD. Hence, wthn ths partcular prcng scheme, the CSD s argued to charge a unform cost to both brokers such that t breaks even on average over all transactons. Thus, t compensates the losses t makes on the dffcult (.e. non-nternalzed) order flow stemmng from dfferent brokers wth gans from the easy order flow stemmng from trades that occur wthn the same broker (.e. nternalzed). In fact, by chargng a unform break even cost per transacton, the CSD does nether dfferentate between dfferent types of transactons, nor between transactons stemmng from dfferent brokers. Denote ths break even cost by, ths prcng scheme then mples that: = = = = Under ths prcng scheme, t s clear that traders from both brokers wll always address all traders. Ths means that the { } combnaton of strateges domnates the three other combnatons. The reason s that as all traders face a unform cost, t s mpossble to set a quote only attractve to traders of one partcular broker. 6 Therefore, when analyzng the equlbrum we only consder the { } combnaton of strateges. 3.1 Equlbrum We now turn to the determnaton of the equlbrum quotes and the optmal.how do traders set ther quotes, takng as gven? Gven that the { } sub-equlbrum wll always preval and that costs and gans are dentcal for traders of both brokers, we must have that bd and ask quotes, set by traders of the large and small broker are dentcal. We denote ths as follows: { } = { } { } { } = { } { } where { } refers to the ask prce ( ) set by a trader from the large broker (subscrpt ) wth unform prcng by the CSD (superscrpt ) and under the { } sub-equlbrum (second superscrpt). The other prces have a smlar notaton. Suppose now a buyer arrves n the market. She wll set the bd prce of her LO such that the next ncomng seller s ndfferent between httng the LO (by submttng a sell MO) or submttng a sell LO herself. Ths mples the expected payoff for the ncomng seller of submttng a MO or a LO must be the same. The followng equaton shows ths 6 Do note that f playng the -strategy would be possble, ths would stll be a sub-optmal strategy as t only reduces executon probabltes wthout nducng any quote advantage.

11 ndfference condton: { } = 1 { } The left hand sde of ths equaton presents the gan from a sell MO, gven the bd prce set by the buyer n the prevous perod. The rght hand sde s the expected gan of a sell LO, whch s the executon probablty of ths order (.e. 1 or the probablty that the next arrvng trader s a buyer who wll ht the standng sell LO snce the seller optmally also sets her ask prce to make the next arrvng buyer ndfferent) multpled by the payoff upon executon of her order corrected for the approprate clearng and settlement cost. Thus, the dea here s that { } s chosen at the lowest level at whch the subsequently arrvng seller s just wllng to submt a MO, whle both accountng for the clearng and settlement cost. In other words, { } equals the seller s cutoff prce and renders ths seller ndfferent between httng the standng LO at { } and submttng her own LO at { }. Submttng a LO at all other quotes s easly proven to be sub-optmal for ths buyer. Smlarly an arrvng seller sets her LO quote n order to make a subsequently arrvng buyer ndfferent between submttng a buy MO at { } or a buy LO at { } : { } = 1 { } Solvng the system of ndfference equatons, and recallng that the CSD sets such that t breaks even on average over all transactons, we obtan the equlbrum for the unform prcng scheme, as shown n Proposton 1: Proposton 1 When the CSD apples a unform prcng scheme,.e. t charges the same cost to both brokers and to nternalzed and non-nternalzed trades, the optmal cost announced by the CSD s: = (1 ) Traders always play the { } sub-equlbrum. The optmal ask and bd quotes of the trader are: Proof. See Appendx. { } = { } { } = + (1 ) 3 { } = { } { } = + + (1 ) 3 By chargng on every transacton (nternalzed and non-nternalzed), the CSD on average ndeed breaks even: t gans on transactons for whch t does not face margnal

12 costs and loses on transactons where actve clearng and settlement takes place. Whle transactons receved from the largest broker more often nduce no costs, as they are more often nternalzed, the CSD stll charges a unform prce to both brokers. We observe that the ask decreases n, whle the bd ncreases n. Thus, larger costs of clearng and settlement appear to nduce more lqud quote-settng behavor and thus mprove stock market lqudty. The reasonng behnd ths remarkable result s that traders submt more aggressve LOs n order to nduce the counterparty to submt a MO (whch ncurs the clearng and settlement cost wth certanty). That s, t s as f the counterparty now has a lower wllngness to trade resultng from the cost of clearng and settlement. Moreover, when both brokers exactly have the same market share (.e. =0 5), the quotes are most lqud. Indeed, f ths condton s fulflled, the cost charged by the CSD per trade s largest leadng to a more aggressve prcng strategy n equlbrum. Further, as could be expected, when one broker attracts the entre market ( =0or =1), clearng and settlement costs do not play a role anymore as all trades are then nternalzed. Ths would mply we are n a model wthout clearng and settlement costs, comparable to Foucault (1999). 4 CSD Prcng Scheme : Broker-Specfc Prcng We now assume that the CSD prce dscrmnates between brokers,.e. sets prces for the large broker and for the small broker (where superscrpt ndcates the analyzed prcng scheme). Ths means that n the notaton of Secton, we have: = = = = As 0 5, trades stemmng from the large broker are more lkely to occur between two traders orgnatng from the same broker as compared to trades stemmng from the small broker. Therefore, t appears reasonable to assume that, such that traders lnked to a certan broker pay a broker-specfc cost on any trade, and ths brokerspecfc cost s lower for the large broker (we wll verfy and confrm ths assumpton later n ths secton). Further, we assume the CSD mplements a prcng scheme such that t breaks even on average for each broker ndvdually (and thus mplctly also overall). A novel mplcaton s then that the quotng behavor of traders lnked to the large broker may dffer substantally from the strateges of traders afflated to the small broker. Consder the followng example to llustrate ths pont. Assume a buyer lnked tothelargebrokerarrvesnthemarket. Ontheonehand,shecouldsubmtaLO. Her quote choce allows her to choose whch counterpartes she wants to address: () by she only attracts counterpartes from the same broker postngalowerbd { }

13 (mplyng a hgher payoff wth a lower executon probablty), whereas () by postng a hgher bd { } she also attracts counterpartes from the other broker (mplyng alowerpayoff wth a hgher executon probablty). Do note { } s the lowest bd quote at whch an ncomng seller from the same (.e. large) broker s wllng to submt a MO, whle accountng for her relatvely low ndvdual clearng and settlement cost and her own LO strategy quotng { }. In turn, { } s the lowest bd quote at whch an ncomng seller from the other (.e. small) broker s wllng to submt a MO, whle accountng for her relatvely hgh ndvdual clearng and settlement cost and her own LO strategy quotng { }. Submttng a LO at any other quote s easly proven to be sub-optmal for ths buyer. 7 As we wll see below, the choce between both quotes hnges on market parameters and on the trader s preferences n the tradeoff between quote level, executon probablty and clearng and settlement cost. Two dstnct sub-equlbra wll result, the resultng quotes of whch reflect the underlyng transacton costs. On the other hand, gven the avalablty of a standng sell LO, she could also submt a MO. As =1, the actons of the sellers lnked to the large broker are completely symmetrc, and could be derved n a smlar way. In turn, the possble strateges for traders lnked to the small broker dffer from those mentoned above as they cannot opt to only address counterpartes merely stemmng from ther own broker by vrtue of the hgher broker-specfc transacton costs they face. Counterpartes from the large broker wll always be wllng to ht ther quotes wth a MO. Ths means that traders from the small broker wll never (be able to) play the strategy. We are thus left wth two possble combnatons of strateges: I. traders of both brokers am to address counterpartes of all brokers (.e. { }) II. traders of the large broker only am to address counterpartes of ther own broker, traders of the small broker am to address counterpartes of all brokers (.e. { }); For both combnatons, we wll now determne the accordng equlbrum quotes set by traders at both brokers and the optmal costs charged by the CSD. 4.1 Equlbrum Whle settng ts optmal cost, the CSD ratonally antcpates the strateges of the traders at the dfferent brokers,.e. whether they play the { } or { } strateges. Therefore, the CSD wll charge a dfferent cost wthn each of the two combnatons. Wedenotethecostchargedtothelargebrokernbothcombnatonsas { } and, respectvely, whle the costs charged to the small broker are { } and { } 7 That s, hgher bd quotes do not ncrease the executon probablty yeldng lower expected payoffs.

14 { }. How does the CSD set these costs? Assume the sub-equlbrum correspondngtothe{ } combnaton of strateges holds. For the large broker, the CSD then determnes the fracton of nternalzed and non-nternalzed trades for that broker under the { } combnaton of strateges. Next, each fracton s multpled wth the cost for the CSD of that type of trade,.e. the proporton of nternalzed trades s multpled by zero, and the proporton of non-nternalzed trades by. The same procedure s appled forthesmallbrokerandforthe{ } combnaton. In ths way, the CSD ensures that t breaks even on average for each broker ndvdually wthn each combnaton. Do note all the underlyng calculatons are provded n detal n the proof of Proposton. Now, we turn to the optmal order submsson strateges of traders at both brokers. We determne ther strateges gven clearng and settlement costs { }, { }, { } and { }. Frst, we dscuss the { } case, n whch traders at both brokers set ther quotes to keep ncomng counterpartes of all brokers at least ndfferent. Thus, a buyer at the large broker keeps the margnal ncomng seller ndfferent,.e. a seller from the small broker snce she faces the hghest clearng and settlement cost when httng ths quote: { } { } = 1 h { } { } Thus, the ncomng seller from the small broker s kept ndfferent between httng the standng quote { } (by submttng a MO sell) accountng for the approprate clearng and settlement cost, and submttng her own sell LO (of whch the executon probablty, the quote and the clearng and settlement cost correctly correspond to the { } strategy ths seller s playng). Smlarly, a seller at the large broker keeps the margnal ncomng buyer ndfferent,.e. a buyer from the small broker: { } { } = 1 h { } { } Now, how wll traders of the small broker set ther LO quotes? Also a buyer from the small broker keeps the margnal ncomng seller ndfferent,.e. the seller of the small broker snce she faces the hghest clearng and settlement cost when httng ths quote: { } { } = 1 h { } { } Smlarly, a seller of the small broker keeps the margnal ncomng buyer ndfferent,.e. a buyer from the small broker: { } { } = 1 h { } { }

15 Next, we focus on the { } combnaton of strateges. Wthn ths combnaton, tradersatthelargebrokersettherquotesonlytokeepthecounterpartesoftherown broker ndfferent. Thus, a buyer at the large broker keeps the ncomng seller from her own broker ndfferent: { } { } = 1 h { } { } Hence, the ncomng seller from the large broker s kept ndfferent between httng the standng quote { } (by submttng a MO sell) accountng for the approprate clearng and settlement cost, and submttng her own sell LO (of whch the executon probablty, the quote and the clearng and settlement cost correctly correspond to the { } strategy ths seller s playng). Smlarly, a seller at the large broker keeps the ncomng buyer from her own broker ndfferent: { } { } = 1 h { } { } At these quotes, only ncomng traders from the large broker are ndfferent, and thus attracted to ht them wth a MO. For the traders orgnatng from the small broker, tradng at these quotes s too costly gven the hgher transacton cost { } they face. Therefore, the executon probabltes are only related to the own broker (.e. ). Now, how wll traders of the small broker set ther LO quotes under the { } combnaton of strateges? We know that wthn ths broker-specfc prcng scheme these traders do not have the possblty to only address traders of ther own broker, as traders from the large broker would automatcally also be nterested n any quote whch makes traders from the small broker ndfferent. Thus, traders at the small broker set ther quote to keep ncomng counterpartes of all brokers at least ndfferent: a buyer keeps the margnal seller ndfferent,.e. a seller from the small broker: { } { } = 1 h { } { } Smlarly a seller keeps the margnal buyer ndfferent,.e. a buyer from the small broker: { } { } = 1 h { } { } Solvng the systems of ndfference equatons for the traders at both brokers and computng the approprate clearng and settlement costs charged by the CSD under both combnatons, we obtan the equlbrum under the broker-specfc prcng scheme as stated n Proposton.

16 Proposton Defne the followng two crtcal values: b { } = (1+ )(1 )( )( ) and b { } = 4(1 )( ). Wth a CSD whch prce dscrmnates between brokers (.e. broker-specfc prcng scheme), traders at both brokers play the followng LO strateges dependng upon the value of the clearng and settlement cost : For low values of,.e. b { }, traders from both brokers target counterpartes of all brokers, thus the { } sub-equlbrum s played. The CSD then announces clearng and settlement costs: { } = (1 ) { } = for the large and the small broker, respectvely. The optmal ask and bd quotes of the trader are: { } = { } = + 3 { } = { } = For ntermedate values of,.e. b { } b { }, there s no prcng strategy such that the CSD breaks even. For hgh values of,.e. b { }, traders from the large broker only target counterpartes of ther own broker, whereas traders from the small broker target counterpartes of all brokers, thus the { } sub-equlbrum s played. The CSD then announces clearng and settlement costs: { } = 1 1+ { } = for the large and the small broker, respectvely. The optmal ask and bd quotes are: { } = + ( )(1 ) + ( + )(1+ ) ( )(1 ) { } = { } = + 3 { } = ( + )(1+ )

17 Proof. See Appendx. Thus, for low clearng and settlement costs, traders at both brokers target counterpartes at all brokers. The margnal trader then needs to be convnced to ht the standng LO. Ths mples the transacton costs for the small broker, whch are relatvely hgh, are reflected n the quotes and the quotes are dentcal for traders from both brokers. The stock market s lqudty s then relatvely hgh as the traders need to quote aggressvely to nduce the traders from the small broker (who face hgh clearng and settlement costs) to partcpate. In contrast, for suffcently large costs of clearng and settlement (nducng larger cost savngs from nternalzaton), traders from the large broker address only ther own counterpartes and prefer not to target traders from the small broker: the gan from ncreased matchng probabltes does not outwegh the concessons n terms of aggressve prcng. The quotes from traders from the large broker then mply a low stock market lqudty as they only address own counterpartes wth relatvely low costs of clearng and settlement. In contrast, the quotes from traders of the small broker are qute aggressve as they need to convnce the margnal traders facng large costs of clearng and settlement to ht ther LOs. 5 CSD Prcng Scheme 3: Trade-Specfc Prcng Under the trade-specfc prcng scheme, we assume the CSD prces accordng to the margnal costs that are assocated wth ndvdual transactons. That s, clearng and settlement costs are set to zero for trades wth both traders stemmng from the same broker, and amount to for trades wth both traders orgnatng from dfferent brokers. As argued before, note that the zero cost attrbuted to nternalzed trades merely represents a normalzaton. More generally, as long as nternalzed trades mply lower margnal costs than non-nternalzed trades, all results mentoned below hold. In terms of the notaton ntroduced n Secton, ths mples: = =0 = = In prncple, all four possble combnatons of strateges that can be played by traders from both brokers are feasble. In the proof of the equlbrum we wll show, however, that the { } combnaton s never optmal. Therefore, we already exclude t n the dscusson below.

18 5.1 Equlbrum The prcng scheme of the CSD (.e. zero cost for nternalzed trades, for nonnternalzed trades), s agan taken as gven by the traders. Note that represents the (exogenous)margnalcostforthecsdandthuswenowdonotneedtocomputet. In contrast to the two prevous prcng schemes, we now only need to determne the optmal quotes for traders of both brokers. Agan, we wll consder each of the possble combnatons of strateges separately. Startng wth the { } combnaton of strateges, traders at the large broker set ther quote to keep the margnal trader ndfferent as they want to address all traders. Thus, they account for the transacton cost. So for buyers and sellers from the large broker, we respectvely have: { } = 1 h { } = 1 { } h { } Thus, wthn the frst ndfference condton for nstance, the ncomng seller from the small broker s kept ndfferent between httng the standng quote { } (by submttng a MO sell) accountng for the approprate clearng and settlement cost, and submttng her own sell LO (of whch the executon probablty, the quote and the clearng and settlement cost correctly correspond to the { } strategy ths seller s playng herself). Smlarly, for traders from the small broker, who keep an ncomng counterparty trader from the large broker ndfferent (thus accountng for the transacton cost ), we have for buyers and sellers: { } = 1 h { } = 1 { } (1 ) h { } (1 ) Next, consder the { } combnaton of strateges. Traders at the large broker set ther quote only to keep counterpartes of ther own broker ndfferent (whch mples the transacton cost does not need to be accounted for). A buyer (seller) at the large broker keeps the ncomng seller (buyer) from her own broker ndfferent, such that: { } = 1 h h { } = 1 { } { } Thus, wthn the ndfference condton stated frst for nstance, an ncomng seller from

19 the large broker s kept ndfferent between httng the standng quote { } (by submttng a MO sell) accountng for the approprate zero clearng and settlement cost, and submttng her own sell LO (of whch the executon probablty, the quote and the zero clearng and settlement cost correctly correspond to the { } strategy ths seller s playng herself). In contrast, traders from the small broker stll am to keep the margnal trader ndfferent. A buyer (seller) from the small broker wll then keep an ncomng seller (buyer) from the large broker ndfferent, leadng to: { } = 1 h h { } = 1 { } { } The reasonng here s smlar to that for the small broker traders under the { } combnaton of strateges, but now the expected LO payoffs of the targeted large broker traders correctly reflect the executon probablty, the quote and the zero clearng and settlement cost correspondng to the { } strategy these traders are playng themselves. Fnally, wthn the { } combnaton of strateges, all traders only keep potental counterpartes of ther own broker ndfferent. Hence, all trades are nternalzed and thus ncur a zero clearng and settlement cost. The ndfference equatons for buyer and seller from the large broker then become: { } = 1 h h { } = 1 { } { } Thus, wthn the frst ndfference condton for nstance, the ncomng seller from the large broker s kept ndfferent between httng the standng quote { } (by submttng a MO sell) accountng for the approprate zero clearng and settlement cost, and submttng her own sell LO (of whch the executon probablty, the quote and the zero clearng and settlement cost correctly correspond to the { } strategy ths seller s playng herself). At these quotes, only traders from the large broker are ndfferent. For traders orgnatng from the small broker tradng at these quotes s too costly gven ther hgher transacton cost. Therefore, the executon probabltes are only related to the own broker (.e. ). Smlarly, the equatons for buyer and seller from the small broker are: { } = (1 ) 1 h h { } = (1 ) 1 { } { }

20 At these quotes, only traders from the small broker are ndfferent. For traders stemmng from the large broker tradng at these quotes s too costly gven ther hgher transacton cost. Therefore, the executon probabltes are only related to the own broker (.e. 1 ). Solvng the above systems of ndfference condtons renders the equlbrum quotes and thus the three dstnct sub-equlbra. Comparng expected profts for each of the sub-equlbra, we are also able to determne when each of the sub-equlbra s vald. All these elements are shown n the equlbrum presented n Proposton 3. Proposton 3 Wth a CSD applyng trade-specfc (margnal cost-based) prcng, traders at both brokers play the followng LO strateges hngng on the value of the clearng and settlement cost : For low values of,.e. ( ), traders from both brokers target counterpartes 3(+ ) of all brokers, thus the { } sub-equlbrum s played. The equlbrum quotes are: { } = + (1 ) 3 { } = + +(1 ) 3 { } = + 3 { } = ( For ntermedate values of,.e. ) ( )(1+ ),tradersfromthe 3(+ ) (1+ )(+ )(3 ) large broker only target counterpartes of ther own broker whereas traders from the small broker target counterpartes of all brokers, thus the { } sub-equlbrum s played. The equlbrum quotes are: { } = + + { } = + + { } = + = { } + { } = + + = { } + + For hgh values of,.e. ( )(1+ ) (1+ )(+ )(3 ), traders from both brokers only target own counterpartes, thus the { } sub-equlbrum s played. The equlbrum

21 quotes are: { } = + + { } = + + { } = +(1 ) 3 { } = (1 ) + 3 Proof. See Appendx. For low clearng and settlement costs, traders at both brokers target counterpartes at all brokers by quotng relatvely lqud prces. Stll, an nterestng dvergence arses, traders from the small broker have to quote more lqud prces (as compared to traders from the large broker) to attan ths goal as they need to convnce traders from the large broker (who face the opportunty to submt a LO featurng lower expected clearng and settlement costs) to accept ther LO. Do note that gven ths quote settng behavor, n case a counterparty from the same broker hts a standng quote, both traders nvolved n the trade receve a bonus as they both do not have to pay. Anncreasenthe large broker s market share evdently nduces traders from the large broker to quote relatvely less lqud prces, whereas traders from the small broker are oblged to quote relatvely more lqud prces to reman attractve to the traders from the large broker. Next, for an ntermedate range of clearng and settlement costs, traders at the large broker alter ther strategy and submt relatvely llqud quotes only targetng traders of ther own broker. In contrast, traders from the small broker stll prefer to target counterpartes at both brokers and thus quote a very lqud quote fully compensatng the clearng and settlement cost a potentally arrvng counterparty from the large broker would face. They do so because the gan from ncreased matchng probabltes stll outweghs the concessons n terms of aggressve prcng. Evdently, ths entals that n case a counterparty from the small broker would ht ths standng quote, both traders nvolvednthetraderecevea bonus astheybothdonothavetopay. Fnally, for suffcently large costs of clearng and settlement (nducng larger cost savngs from nternalzaton), both traders from the large and the small broker only address ownbroker counterpartes by quotng relatvely llqud prces, wth the quotes from the small broker beng more llqud as they face a lower executon probablty. All quoted prces are now ndependent of the cost of clearng and settlement as these strateges am at targetng own-broker counterpartes only.

22 6 Dscusson of the Equlbra 6.1 Stock Market Lqudty and CSD Prcng TO BE COMPLETED 6. Welfare Analyss In ths secton, we characterze ex ante welfare for the dfferent prcng schemes. Our ex ante welfare measure bulds on ratonal trader behavor and s therefore dentcal to the mean realzed ex post welfare. We focus on overall welfare ( ),.e. the sum of all agents expected utltes from tradng (see Glosten (1998), Goettler, Parlour and Rajan (005), Hollfeld, Mller, Sandås and Slve (006), and Degryse, Van Achter and Wuyts (009) for a smlar approach n quantfyng welfare). As the CSD always breaks even, n our model, evdently equals trader welfare. s computed for a random sequence of two trader arrvals under the dfferent prcng schemes and compared to a benchmark maxmum overall welfare measure correspondng to the case featurng a sngle broker ( =1) and zero clearng and settlement costs ( =0), or formally: µ 1 max = [ ] whch represents the probablty a buyer and a seller (or vce versa) sequentally match ( 1 ) multpled by the total tradng gans whch could be made n case of a match. Frst, for the unform prcng scheme, overall welfare equals: = µ 1 [ 4 (1 ) ] Next, when the CSD apples broker-specfc prcng, weneedtomakeadstncton between the dfferent sub-equlbra: For low values of,.e. b { },orthe{ } sub-equlbrum: µ 1 { } = ( ) (1 ) (1 )((1 ) + ) (1 )

23 For ntermedate values of,.e. b { } b { }, we are not able to compute overall welfare as no prcng strategy exsts such that the CSD breaks even for ths nterval. For hgh values of,.e. b { },orthe{ } sub-equlbrum: { } = µ 1 +(1 ) ( ) 1 1+ µ 1 (1 ) 1+ + (1 ) Do note that when 0 (.e. tradng gans fully compensate for the maxmum clearng and settlement cost correspondng to a transacton), we have that { } { }.Thus,the{ } sub-equlbrum results n hgher overall welfare under ths condton. Fnally, under trade-specfc prcngbythecsd, overallwelfare forthedfferent sub-equlbra equals: For low values of,.e. ( ) 3(+ ) or the { } sub-equlbrum: { } = µ 1 [( ) (1 ) ] For ntermedate values of,.e. sub-equlbrum: ( ) ( )(1+ ),orthe{ } 3(+ ) (1+ )(+ )(3 ) { } = µ 1 +(1 ) ( ) ((1 ) ) For hgh values of,.e. ( )(1+ ),orthe{ } sub-equlbrum: (1+ )(+ )(3 ) { } = µ 1 +(1 ) ( ) Agan, do note under the assumpton 0 t s easy to prove that { } { } { }, mplyng the { } sub-equlbrum strctly domnates the two other ones n terms of overall welfare.

24 7 Concludng Remarks Explct transacton costs such as the costs related to clearng and settlement are stll of consderable mportance n today s fnancal markets. Both n the US and Europe, polces have been mplemented n order to reduce the costs of clearng and settlement. In ths paper, we model how nternalzaton of clearng and settlement affects stock market lqudty. Our man nsghts can be summarzed as follows. Frst, we fnd that explct transacton costs such as the costs of clearng and settlement mpact stock market lqudty. In general, hgher costs of clearng and settlement tend to ncrease lqudty. The reasonng s that hgher costs of clearng and settlement nduce more aggressve lmt order prcng to convnce counterpartes to trade. Second, nternalzaton reduces the costs of clearng and settlement. Our results show that when more trades can be nternalzed stock market lqudty decreases. The reasonng behnd ths result s that t represents a drop n explct transacton costs and therefore reduces the aggressveness of lmt order prces. Thrd, when the clearng and settlement agent sets prces such that t breaks even per broker, dfferent equlbra result dependng upon the magntude of the costs of clearng and settlement. Stock market lqudty s harmed when the costs of clearng and settlement are substantal. In that case, the break even prces the clearng and settlement agent charges dffer substantally between nvestment frms wth a large amount of nternalzed trades and other nvestment frms. Traders from the large nvestment frm then announce unattractve prces that are consdered only by counterpartes of ther own nvestment frm. Traders from other nvestment frms fnd these prces not attractve enough as costs of clearng and settlement are too large for them. The quotes from traders orgnatng from smaller brokers reman qute lqud as they face another trade-off: they am to attract counterpartes from all brokers as they beneft more from aggressve quotes snce ths substantally ncreases ther lkelhood of executon. Fnally, we analyze also the case where the clearng and settlement agent charges the margnal cost for non-nternalzed trades and zero costs for nternalzed trades. For suffcently hgh margnal costs of non-nternalzed trades, t may then be optmal for traders from both brokers to target ther own broker counterpartes only. In ths case, the stock market s relatvely llqud wth traders from the large broker quotng more lqud prces than traders from the small broker. Our welfare analyss reveals that overall welfare s lower when some (or all) traders only target counterpartes from ther own broker, compared to the cases where all traders am to attract all potental counterpartes,.e. traders from all brokers.

25 Appendx: Proofs ProofofProposton1. The equlbrum ask and bd quotes follow mmedately from solvng the system of ndfference condtons delneated n the man text. Next,wedervetheprcngstrategy for whch the CSD breaks even when t charges a unform per-transacton cost, whle accountng for the fact that nternalzed order flow does not mply costs. The rato of transactons wth postve margnal costs for the CSD vs-à-vs all possble transactons sent to the CSD equals: 0 5[0 5 (1 )+0 5(1 ) ]+0 5[0 5 (1 )+0 5(1 ) ] 0 5 = (1 ) Note that for each of these transactons, the CSD s actve on both sdes of the market, hence t charges a cost to both legs of the trade. In other words, out of every transacton, on average a fracton (1 ) occurs between traders orgnatng from dfferent brokers, whereas the complementary fracton occurs between traders whch are clent at the same broker (.e., +(1 ) ). A CSD chargng = (1 ) on both legs of every transacton (nternalzed and non-nternalzed) on average breaks even: t gans on transactons for whch t does not face margnal costs and loses on transactons where actve clearng and settlement takes place. Q.e.d. ProofofProposton. Solvng the system of ndfference equatons delneated n the man text, takng clearng and settlement costs as gven, results mmedately n the quotes for the two sub-equlbra. For gven and, we now analyze whch sub-equlbrum holds. Gven the, as we wll show below), traders from the small broker cost structure (.e. have no alternatve strategy than to target all counterpartes. Traders of the large broker smply compare the expected profts they make across the two sub-equlbra,.e. by settng { } or { } for a seller, or { } or { } for a buyer. A buyer prefers to set { } f: 8 1 ³ { } 1 ³ { } 8 An underlyng assumpton n ths dervaton s that f traders are ndfferent between the payoffs of the and the -strategy, the -strategy s preferred.

26 and { } otherwse. Ths translates nto: (3 ) + 4(1 )( ) (1) + As =1, the seller s case s completely symmetrc. Ths condton shows that for a gven and, a unque sub-equlbrum apples. If s larger than the rght hand sde of the stated expresson, traders of the large broker maxmze profts by gong for sub-equlbrum { },.e. to address counterpartes of the own broker only. In other words, f the cost dfferental between the two brokers s too large, t s too costly to also address the traders of the small broker by submttng more lqud quotes. Otherwse they address all counterpartes and { } s played. and In a fnal step of the proof, we now derve the equlbrum prces charged by the CSD. The CSD ratonally antcpates ts set prces determne the strateges of the traders of the dfferent brokers,.e. the sub-equlbrum that apples. We frst consder the equlbrum prces the CSD charges for the { } strateges. For the large broker, the transactons mplyng a cost as a proporton of all possble transactons at ths broker s represented by the followng fracton For the small broker, usng a smlar calculaton, the transactons mplyng a cost as a proporton of all possble transactons s represented by Do note that for both brokers transactons between traders of the same broker count double n these fractons as the broker s handlng both sdes of the transacton and thus reports two trades to the CSD. For 1 we fnd that: 1 1+ that s, the fracton of transactons mplyng a cost wth respect to all possble transactons s evdently lower at the large broker. To compute the prcng strategy at whch 9 0 5[0 5(1 ) ]+0 5[0 5(1 ) ] Ths s 0 5[0 5(1 ) ]+0 5[0 5(1 ) ]+ ( )+ ( ) The numerator gves all transactons n whch both brokers are smultaneously nvolved (these transactons mply a cost ), and the denomnator contans all transactons (so also transactons that are nternalzed wthn one broker and that do not mply a cost when sent to the CSD).

27 the CSD breaks even per broker, we multply these ndvdual fractons by : { } = 1 1+ { } = wth: { } { } as can be expected. Thus, wthn ths sub-equlbrum traders at the large broker bear acostof { } for a transacton, whereas traders at the small broker bear a cost of { } for a transacton. Next, we determne the prcng scheme at whch the CSD breaks even on average for each broker ndvdually wthn the { } combnaton of strateges. For the large broker, the transactons mplyng a cost as a proporton of all possble transactons at ths broker s represented by the followng fracton: 1 For the small broker, the transactons mplyng a cost as a proporton of all possble transactons s represented by: Agan, for both brokers transactons between traders of the same broker count double n these fractons as the broker s handlng both sdes of the transacton and thus reports two trades to the CSD. For 1 we fnd that: 1 that s, the fracton of transactons mplyng a cost wth respect to all possble transactons s evdently lower at the large broker. To compute the prcng strategy at whch the CSD would break even per broker, we multply these ndvdual fractons by : { } = (1 ) { } = wth: { } { } as can be expected. Thus, wthn ths sub-equlbrum traders at the large broker bear for a transacton, whereas traders at the small broker bear a cost of acostof { }

28 { } for a transacton. A comparson of the potental break even prces charged by the CSD shows that for 1 : { } { } { } { } Now, remember from Equaton (1) the exstence condton to have sub-equlbrum { } was { } (3 ) + 4(1 )( ) + { } Now, suppose the CSD prces assumng that the combnaton of strateges { } apples,.e. t charges { } = 1 and { } 1+ =. By substtutng n { } and { }, the exstence condton to have the combnaton of strateges { } then can be reformulated as (1 + )(1 )( )( ) = b { } Smlarly, remember from Equaton (1) the exstence condton to have sub-equlbrum { } was { } (3 ) + 4(1 )( ) + { } Now, suppose the CSD prces assumng that sub-equlbrum { } apples,.e. t charges { } =(1 ) and { } =. By substtutng n { } and, the exstence condton to have sub-equlbrum { } can be rewrtten as { } 4(1 )( ) = b { } For 1, tcanbeshownthatb { } b { }. Q.e.d. ProofofProposton3. Solvng the systems of ndfference equatons delneated n the man text, takng clearng and settlement costs as gven, results mmedately n the quotes for the sub-equlbra. We thus only need to prove exstence. Thus, we now nvestgate under whch condtons the dfferent possble combnatons of strateges correspond to a sub-equlbrum. Frst, the expected lmt order payoffs are computed for the dfferent combnatons of strateges. Next, we wll demonstrate under

29 whch condtons the dfferent sub-equlbra wll hold. Three dstnct possbltes for a sub-equlbrum arse, whch one s played depends on the level of the cost of clearng and settlement. As n the man text, we assume = 1. Ths wll mply we only have to analyze the expected payoffs of one market sde as quotes and expected payoffs ofthe other market sde are completely symmetrc. We frst compute the lmt order payoffs under the four possble combnatons of strateges: { }: The expected payoff of a buyer lnked to the large broker submttng { } under ths combnaton of strateges s: { } = 1 µ + +(1 ) (1 ) 3 Smlarly, the expected payoff of a buyer afflated to the small broker submttng under ths combnaton of strateges s: { } { } = 1 µ { } : The expected payoff of a buyer lnked to the large broker submttng { } under ths combnaton of strateges s: { } = 1 µ + + Smlarly, the expected payoff of a buyer afflated to the small broker submttng under ths combnaton of strateges s: { } { } = 1 µ { } : The expected payoff of a buyer lnked to the large broker submttng { } under ths combnaton of strateges s: { } = 1 µ (1 ) + + (1 ) 3

30 Smlarly, the expected payoff of a buyer afflated to the small broker submttng under ths combnaton of strateges s: { } { } =(1 ) 1 µ (1 ) + 3 { } : The expected payoff of a buyer lnked to the large broker submttng { } under ths combnaton of strateges s: { } = 1 µ + + Smlarly, the expected payoff of a buyer afflated to the small broker submttng under ths combnaton of strateges s: { } { } =(1 ) 1 µ (1 ) + 3 We now derve under whch condtons the dfferent sub-equlbra apply: Sub-equlbrum { } apples when two condtons are jontly satsfed. Frst, traders at the large broker should have no ncentves to devate to the -strategy when traders at the small broker play the -strategy,.e. ths apples when: { } { },or ( ) 3(+ ) Secondly, traders at the small broker should have no ncentves to devate to the -strategy when traders at the large broker play the -strategy: { } { },or ( ) 3(3 ) Gven 0 5, wehavethat ( ) 3(+ ) ths sub-equlbrum holds. s bndng. If ths condton s satsfed,. Sub-equlbrum { } apples when two condtons are jontly satsfed. Frst, traders at the large broker should have no ncentves to devate to the -strategy 10 An underlyng assumpton n ths dervaton s that f traders are ndfferent between the payoffs of the and the -strategy (whch s the case at the cutoff values of ), the -strategy s preferred.

31 when traders at the small broker play the -strategy,.e. ths apples when: { } { },or ( ) 3(+ ) Secondly, traders at the small broker should have no ncentves to devate to the -strategy when traders at the large broker play the -strategy: { } { },or ( )(1+ ) (1 + )(+ )(3 ) Thus, when ( ) ( )(1+ ) 3(+ ) (1+ )(+ )(3 ) thus ths sub-equlbrum holds. the strateges are devaton-proof, and 3. Sub-equlbrum { } apples (usng smlar reasonng) when: and { } { },or ( )( +) ( + )( )(3 ) { } { },or ( ) 3(3 ) For 0 5, both condtons could never be jontly met, hence ths combnaton of strateges wll never realze and forms no sub-equlbrum. 4. Sub-equlbrum { } apples (usng smlar reasonng) when: and { } { } { },or ( )( +) ( + )( )(3 ) { },or ( )(1+ ) (1 + )(+ )(3 ) Further comparson shows that ( )(1+ ) (1+ )(+ )(3 ) thus f t s satsfed ths sub-equlbrum holds. s the most strngent condton, Q.e.d.

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