NICOLAS P.B. BOLLEN TOM SMITH ROBERT E. WHALEY * MODELING THE BID/ASK SPREAD: On the Effects of Hedgng Costs and Competton ABSTRACT The need to understand and measure market maker bd/ask spreads s crucal n evaluatng the merts of competng market structures and securty desgns. Pror studes of bd/ask spreads suffer from several forms of msspecfcaton, ncludng nadvertent and erroneous use of weghted least squares regresson. Ths study develops a smple, parsmonous model of the determnants of spread, and then tests t emprcally on a sample of NASDAQ stocks. The model performs well and avods the dstortons of pror work. The study demonstrates the mportance of proper model specfcaton n provdng meanngful nference regardng the determnants of spread. August 7, 2001 * Bollen s from the Owen Graduate School of Management, Vanderblt Unversty; Smth s from the Australan Graduate School of Management, Unversty of New South Wales; and Whaley s from the Fuqua School of Busness, Duke Unversty.
MODELING THE BID/ASK SPREAD: On the Effects of Hedgng Costs and Competton The past thrty years have wtnessed a prolferaton of new fnancal products and exchanges worldwde. Wth each new product come decsons regardng market structure. Should the exchange assgn the product to a sngle specalst responsble for makng a market, or should t encourage competton among a number of wllng market makers? Should tradng take place at a specfc locaton wth face-to-face contact among traders and market makers, or should t take place n an anonymous, electronc format? And, wth each new product ntroduced n these markets come decsons regardng optmal desgn. What should be the sze of a standard tradng unt? What should be the securty s mnmum prce varaton (.e., tck sze)? Ultmately, nformed decson-makng regardng these ssues must nvolve a careful balancng of the proft motves of key market partcpants. Investors (.e., demanders of lqudty), on the one hand, want to maxmze ther profts after tradng costs. Market makers (.e., supplers of lqudty), on the other, want to maxmze profts from ther busness operatons. A common focus of both of these sets of market partcpants s a securty s bd/ask spread (.e., the prce of lqudty). The hgher the spread, the greater the market maker proft per unt traded, but the fewer the number of unts demanded. Understandng and measurng the determnants of the bd/ask spread are, therefore, crtcal n evaluatng the merts of dfferent market structures and products. The purpose of ths paper s to develop and test a new model of the market maker s bd/ask spread. As we show, the model s smple, parsmonous, and well grounded from a theoretcal perspectve. Usng a sample of 1,689 NASDAQ stocks durng November 1998, we also show that the model s strongly supported emprcally. The outlne of the paper s as follows. To begn, we descrbe the current state of the theoretcal and emprcal lterature on market maker spreads. Ths dscusson has two parts. In Secton I, we provde a general framework for categorzng the costs assocated wth market makng. Snce much of our contrbuton conssts of methodologcal mprovements on the state of the art n bd/ask spread analyss, a detaled revew of eght pror studes s provded n Secton II. Secton III contans the formal development of our 1
theoretcal model and contrasts ts structure wth the models used n earler work. Secton IV contans an emprcal assessment of the model, and examnes the mportance of proper model specfcaton n provdng meanngful nference regardng the determnants of spread. We hghlght problems of varable selecton, model specfcaton, and estmaton that can dstort nference. Secton V contans a bref summary. I. A FRAMEWORK OF ANALYSIS A number of theoretcal models specfy the cost components of the market maker s bd/ask spread. Stoll (1978a) posts that market maker costs fall nto three categores: order-processng costs, nventory-holdng costs, and adverse nformaton costs. Order-processng costs are those drectly assocated wth provdng the market makng servce and nclude tems such as the exchange seat, floor space rent, computer costs, nformatonal servce costs, labor costs, and the opportunty cost of the market maker s tme. Snce these costs are largely fxed, at least n the short run, ther contrbuton to the sze of the bd/ask spread should fall wth tradng volume the hgher the tradng volume, the lower the bd/ask spread. To some degree, however, ths relaton may be obfuscated by the fact that market makers often make markets n more than one securty. In such cases, fxed order-processng costs can be amortzed over total tradng volume across securtes. In addton, n a hghly compettve market, bd/ask spreads should equal the expected margnal cost of supplyng lqudty, n whch case order-processng costs may be rrelevant n determnng spread. 1 Inventory-holdng costs are the costs that a market maker ncurs whle carryng postons acqured n supplyng nvestors wth mmedacy of exchange (.e., lqudty). Here there are two obvous consderatons: the opportunty cost of the funds that are ted up n carryng the market maker s nventory and the rsk that the nventory value wll change adversely as a result of securty prce movements. Wth respect to the cost of funds, however, t s mportant to recognze that market makers try to reduce or close out postons before the close of tradng each day. If postons are opened and closed n the 1 Anshuman and Kalay (1998) show that, f the startup costs to creatng a competng exchange are sgnfcant, the tck sze (.e., the securty s mnmum prce ncrement, can be set hgh enough that market makers can recoup ther fxed costs as well as earn an economc proft. 2
same day, the margnal cost of fnancng s zero. But, even f nventory s carred overnght, t s not clear whether t represents a cost or a beneft. If durng the day, the preponderance of customer orders are buys, the market maker may well be short nventory, n whch case he wll earn (rather than pay) nterest overnght. Prce-change volatlty, on the other hand, appears to have an unambguous effect on the bd/ask spread. Market makers can hedge the value of ther nventory usng dervatve contracts wrtten on the underlyng securtes or on other securtes whose prce movements are hghly correlated wth the prce movements of the securtes n the nventory. The hedge wll not be costless, however, and wll depend on, among other thngs, the prce-change volatlty of the securtes n the market maker s nventory. The hgher the prce-change rsk, the hgher the bd/ask spread. The thrd category s adverse selecton costs. These costs arse from the fact that market makers, n supplyng mmedacy, may trade wth ndvduals who are better nformed about the expected prce movement of the underlyng securty. For an ndvdual stock, for example, t s easy to magne that certan ndvduals possess nsder nformaton. Advance news of earnngs, mergers, acqustons, restructurngs, spn-offs, and management changes are only a few examples that come to mnd. Whle the ntuton underlyng why adverse selecton may be an mportant determnant of spread s clear, the selecton of an accurate measure of adverse selecton costs s not. Probably the best proxy s stock prce-change volatlty. 2 Adverse selecton s related to nformaton flow, and the greater the nformaton flow, the hgher the prce-change volatlty, and hence the hgher the bd/ask spread. It should also be noted that nformaton flow may affect the bd/ask spread through tradng volume, whch s a proxy for the effects of order-processng costs. But, here the relaton s drect the hgher the nformaton flow (.e., adverse selecton), the hgher the bd/ask spread. 2 Other proxes for the effects of adverse selecton costs have also been used. Branch and Freed (1977), for example, use the number of securtes n whch a dealer makes a market to proxy for adverse selecton the larger the number of securtes managed, the less nformed the dealer s, on average, about a partcular stock. Stoll (1978b), on the other hand, uses a measure of turnover (.e., dollar tradng volume dvded by market captalzaton) the hgher the turnover, the greater the adverse selecton. Glosten and Harrs (1988) use the concentraton of ownershp by nsders the hgher the concentraton, the greater the possblty of adverse selecton. Fnally, Harrs (1994) uses the market value of shares outstandng the larger the frm, the more well known, and hence the lower the possblty of adverse selecton. 3
In addton to these costs, the level of the market maker s bd/spread s lkely to be affected by the level of competton, partcularly n an envronment n whch barrers to entry n the market for markets are beng slowly but surely elmnated. As competton ncreases, the bd/ask spread approaches the expected margnal cost of supplyng lqudty, that s, the sum of nventory-holdng costs and adverse selecton costs. The larger the number of market makers, the greater the competton, and the lower the bd/ask spread. II. A RECONCILIATION OF PAST WORK The model specfcatons used n pror studes to examne the emprcal relaton between the bd/ask spread and ts determnants are nested n the followng general regresson equaton: where spread), SPRD = a + aopc + aihc + aasc + acomp ε, (1) 0 1 2 3 4 + SPRD s the dfference between a securty s bd and ask quotes (.e., bd/ask OPC s order-processng costs, adverse selecton costs, and IHC s nventory-holdng costs, ASC s COMP s the degree of competton. We now revew the regresson specfcatons used n eght dfferent studes and reconcle ther models and results wthn our general framework. For ease of comparson and exposton, the model specfcatons and emprcal results of past studes are summarzed n Table 1. Also for expostonal convenence, we now refer to the underlyng securty as beng a share of common stock. We do ths only because all of the studes dscussed focus on bd/ask spreads n stock markets. It s not ntended to dmnsh the mportance of studes of spreads n other markets. 3 Demsetz (1968) s the frst emprcal study to nvestgate the determnants of the market maker s bd/ask spread. He regresses a stock s spread on the logarthm of the number of trades/the logarthm of the number of shareholders, share prce, and number of 3 Neal (1987), for example, examnes bd/ask spreads of ndvdual stocks traded on the AMEX. George and Longstaff (1993) examne spreads of S&P 100 ndex optons traded on the Chcago Board Optons Exchange (CBOE), and Smth and Whaley (1994) examnes the spread of the S&P 500 futures contract traded on the Chcago Mercantle Exchange (CME). 4
exchanges. The numbers of trades/shareholders, he argues, are drect proxes for the transacton cost rate. The hgher the transacton cost rate, the lower the cost of watng, and hence the lower the bd/ask spread. Under ths lne of argument, these tradng frequency varables fall nto the nventory-holdng cost category. To see ths, recall that prce volatlty s the only unambguous determnant of nventory-holdng costs. Prce volatlty s related not only to the varance of prce changes but also to the amount of tme that the market maker expects to hold an open poston. as follows: Demsetz ratonale for ncludng share prce as a determnant n the regresson s Spread per share wll tend to ncrease n proporton to an ncrease n the prce per share so as to equalze the cost of transactng per dollar exchanged. Otherwse, those who submt lmt orders wll fnd t proftable to narrow spreads on those securtes for whch spread per dollar exchanged s larger. (Demsetz (1968, p.45)) Ths lne of reasonng suggests that relatve spread (.e., bd/ask spread dvded by bd/ask mdpont) should be equal across stocks, holdng other factors constant. Ths mples, of course, the hgher the prce, the hgher the spread. Ths lnkage between spread and prce appears to be based on the noton that share prce s a proxy for the market maker s captal nvestment. The hgher the prce, the greater the nvestment n nventory, the hgher the carryng costs, and the hgher the spread. But, as we have already noted, many market makers are n and out of postons durng the tradng day, n whch case captal nvestment n nventory s not necessary let alone costly. Moreover, even f nventory s carred overnght, the market maker s not always n a net long poston. A net short poston generates cash that could be used to earn nterest ncome. That s not to say that we do not expect to fnd a postve relaton between spread and prce n Demsetz regresson, however. We do. The reason s that prce may be actng as a proxy for nventory prce rsk n Demsetz regresson. To some degree, tradng frequency captures one component of nventory prce rsk, that s, the amount of tme the market maker expects the poston to be open. But, another component, the varance of prce changes, s not ncluded. Snce the standard devaton of prce change s the product of the standard devaton of stock return and share prce, a postve relaton between 5
spread and share prce may appear n Demsetz regresson smply as a result of the fact that share prce s s correlated wth share prce volatlty. Fnally, Demsetz ncludes a competton varable, whch he measures as the number of exchanges on whch the stock was lsted. To be sure, competton should reduce spreads. In Demsetz nvestgaton, however, we should not expect the relaton to be very strong. The reason s that Demsetz sample ncludes only New York Stock Exchange (NYSE) stocks. For these stocks, the lon s share of tradng occurs on the NYSE and very lttle occurs on other exchanges. Consequently, the effects of competton, at least as measured by the number of exchanges makng markets, wll be undermned. A better measure of competton mght be the number of market makers standng at the specalst s post, but such data are dffcult to obtan. Demsetz sample ncludes 192 New York Stock Exchange (NYSE) stocks on two days durng early 1965. More specfcally, hs cross-sectonal regresson s based on observatons created by averagng the respectve varables for each stock across January 5 and February 28, 1965. He fnds that spread vares nversely and sgnfcantly wth the log of the number of trades n one regresson, and the log of the number of shareholders n another, and drectly and sgnfcantly wth prce per share. The coeffcent on the competton varable s negatve, but nsgnfcant. Tnc (1972) argues that Demsetz results are undermned by the fact that several stock characterstcs affectng spread are ether not consdered or measured mprecsely. One mportant mssng factor s nventory prce rsk. Tnc argues that a specalst s unable to hold a dversfed poston and must be compensated for bearng prce rsk. We agree. Unfortunately, he uses the standard devaton of prce, not prce change, as hs measure of rsk. Ths ntroduces an errors-n-the-varables problem that wll tend to negate the mportance of rsk. Tnc also provdes an mproved measure of competton. He uses a Herfndahl Index of concentraton, whch ncludes not only the number of markets but also the overall sze and dstrbuton of tradng actvty among markets. He also re-specfes the tradng frequency varables that affect nventory-holdng costs. In place of number of trades or number of shareholders, he uses the logarthm of number of shares traded, the 6
number of nsttutons holdng the stock, and the percentage of tradng days on whch at least one trade occurs. In Table 1, we nclude tradng volume n the order-processng cost category. Holdng tradng frequency constant, spreads should fall wth number of shares traded. Tnc s sample contans 80 NYSE stocks over 19 tradng days n March 1969. Lke Demsetz, he averages hs observatons across days n the sample. Among other thngs, he fnds that the relaton between spread and the standard devaton of prce s postve, but nsgnfcant. One possble explanaton s the errors-n-the-varables problem alluded to earler. Another s that both standard devaton of prce and prce are ncluded as explanatory varables. It s possble that the share prce varable s better proxy for rsk than the standard devaton of share prce. Tnc also fnds that spreads ncrease sgnfcantly as tradng n a partcular securty concentrates n one market. The use of the Herfndahl Index better captures the effects of competton. The logarthm of number of shares traded, the number of nsttutonal trades holdng the stock, and the percentage of tradng days on whch at least one trade occurs enter the regresson wth sgnfcant negatve coeffcents. Tnc and West (1972) crtcze the Demsetz study n a manner smlar to Tnc (1972), but remedy the problems dfferently. They argue that prce rsk should be ncluded and measure t as the rato of the dfference between hgh and low prces to the average share prce. They argue that the number of exchanges makng markets n NYSE stocks s a poor proxy for measurng competton, so they focus on the NASDAQ market, where barrers to entry are much lower. Based on a sample of 300 NASDAQ stocks on the frst 5 tradng days n November 1971, they fnd that the rsk measure has a postve but nsgnfcant coeffcent. Agan, errors-n-the-varables may be the problem. Rsk measures based only on extreme ponts of a dstrbuton are notorously unrelable. [need a cte here] Moreover, the share prce varable, whch has a postve and sgnfcant coeffcent, may be proxyng for rsk. The relaton between spread and competton s negatve and sgnfcant, re-affrmng the mportance of the level of competton n determnng spreads. In a comparatve study, Tnc and West (1974) examne spreads of stocks lsted on the Toronto Stock Exchange (TSE). Ther purpose s to determne whether spreads set 7
on the TSE are hgher or lower, holdng other factors constant, than spreads on the NYSE and NASDAQ gven that the markets have dfferent structures (.e., rules and regulatons). Unlke the NYSE, for example, the TSE dealers are regulated to be relatvely passve partcpants and are not charged wth the responsblty of mantanng prce contnuty. Polcy ssues asde, ths study s mportant from a methodologcal standpont because t s the frst to ntroduce relatve spread (.e., the bd-ask spread dvded by the bd-ask mdpont) as a dependent varable. Tnc and West begn by estmatng a model of absolute spread, wth prce per share, log of tradng volume, prce volatlty (as measured by the hgh-low prce range dvded by prce), tradng contnuty (as measured by the number of days the stock s traded durng the sample perod dvded by the total number of days n the sample perod), and the number of markets n whch the securty s traded beng used as explanatory varables. Usng a sample of 177 TSE stocks traded durng the perod December 1 through 13, 1971, 4 they fnd that the coeffcents on share prce and prce volatlty are postve and sgnfcant, the coeffcent on tradng volume s negatve and sgnfcant, and the coeffcents on tradng contnuty and number of exchanges are negatve and nsgnfcant. The adjusted R-squared n the regresson s.499. They then proceed by estmatng an alternatve functonal form of the general relatonshp that uses relatve spread as the dependent varable and drops prce per share as an explanatory varable, and fnd that the adjusted R-squared s a whoppng.804! The most powerful explanatory varable turns out to be prce volatlty, whose coeffcent s postve and hghly sgnfcant. The coeffcents on the log of tradng volume and tradng contnuty are negatve and sgnfcant, and the coeffcent on the number of markets s negatve and nsgnfcant. Two comments are n order here. Frst, f the regresson equaton for the absolute spread s correctly specfed based on theoretcal arguments, the regresson equaton for the relatve spread s not. For the relatve spread regresson to be correctly specfed, all of the explanatory varables must be deflated by share prce. Second, gven the hgh level 4 The observatons n the regresson are, for the most part, smple averages of the varables over the nne tradng days n the sample perod. The excepton s prce volatlty, whch s measured by the hgh-low range over the perod dvded by the average share prce over the perod. 8
of the adjusted R-squared (.e.,.804), the relatve spread regresson gves the msleadng mpresson of beng a powerful explanatory model of bd/ask spread. 5 To understand how msleadng the stuaton can be, assume that all stock spreads n the sample are equal to 1/8 th. In such an envronment, the absolute spread regresson model would have no explanatory power. All of the coeffcent values would equal zero, except the ntercept term whose value would be 1/8 th. Now, consder the relatve spread regresson. Snce the absolute spread s assumed to be constant, the varaton n the dependent varable s drven only by varaton n the nverse of share prce. It should not be surprsng to fnd that explanatory varables such as prce volatlty are sgnfcant n a statstcal sense, not because the regresson s tellng us anythng meanngful about spreads, but rather because the prce volatlty varable has share prce n ts denomnator. Indeed, as the varaton n the share prce range across becomes small, the goodness-of-ft of the relatve spread regresson wll become perfect. Benston and Hagerman (1974) examne month-end spreads of 314 NASDAQ stocks over a fve-year perod from January 1963 through December 1967. They specfy ther model n the log-form wth the absolute spread beng a functon of the number of shareholders, prce per share, dosyncratc varance, and the number of dealers. They fnd that all of the spread determnants enter the regresson wth the expected sgn and are statstcally sgnfcant. The key nnovaton n the Benston/Hagerman study s n the measure of prce rsk a stock s dosyncratc rsk. Ther ratonale s that the market maker needs to be rewarded for prce rsk only to the extent that he s not well dversfed or s exposed to traders wth superor nformaton (.e., adverse selecton costs). 6 To measure ths lack of dversfcaton, they use the squared resdual from a regresson of stock returns on market returns over the 60-month perod. Gven the state of the lterature at the tme, the ntroducton of a well-reasoned measure of prce rsk was certanly warranted. Earler measures of rsk lacked theoretcal justfcaton and were prone to measurement error. Nonetheless, the Benston/Hagerman 5 For a lucd dscusson of the ptfalls of usng rato varables n a regresson framework, see Kronmal (1993). 6 Benston and Hagerman were the frst emprcal nvestgators to nclude a specfc dscusson of the mpact of adverse selecton, a concept that had been ntroduced a few years earler by Bagehot (1971). 9
measure s not beyond reproach. Frst, there s no partcular reason for usng the varance as opposed to the standard devaton of the resdual n the market model regresson. In both cases, the coeffcent wll measure rsk averson. Standard devaton, however, offers the advantage of beng specfed n more ntutve unts of measurement stock return rather than stock return squared. Second, the dollar spread of a stock s a functon of prce change volatlty not return volatlty. Consequently, the coeffcent of a return volatlty varable wll not be constant across stocks as ther regresson model assumes, and wll be drectly proportonal to share prce or the square of share prce dependng upon whether the standard error of the resdual or the varance of the resdual s used as the rsk measure. Branch and Freed (1977) conduct a comparatve analyss of spreads on the NYSE versus the Amercan Stock Exchange (AMEX). From a methodologcal standpont, the key nnovaton of ths work les n ther advocacy of relatve spread rather than absolute spread as the varable of nterest. They pont out that pror studes are dvded n the use of absolute or percentage spreads, but then argue that the use of absolute spreads obscures any non-lnearty n the relaton between prce and spread. Ther model specfes that relatve spread s a functon of the tradng volume, the number of exchanges makng a market, prce rsk (as measured by the absolute prce change from the prevous day dvded by share prce), and the number of securtes handled by the market maker. The reasons for ncludng these varables have already been dscussed. Branch and Freed also nclude the nverse of share prce as a determnant of relatve spread. They argue that prce dscreteness, prce clusterng, the fxed costs of executng a transacton, and the desre of specalsts to set ther spreads hgh enough to permt room to maneuver place a mnmum value on the spread, and that the mnmum value s proportonately hgher for low-prced shares than hgh-prced shares. The symptoms/problems assocated wth usng a relatve spread regresson specfcaton have already been noted. The stuaton s partcularly egregous here gven that the nverse of share prce appears on both sdes of the regresson equaton. Not surprsngly, the most sgnfcant varable n ther regressons estmated usng a sample of 1,734 NYSE stocks on January 24, 1974 and a sample of 943 AMEX stocks on May 28, 1974 s the nverse 10
of share prce. Moreover, the second most sgnfcant relaton s ther measure of prce rsk, whch has share prce n the denomnator. The models tested thus far were developed through economc reasonng rather than formal mathematcal modelng. Gven ther ad hoc nature, they are open to crtcsms regardng model specfcaton and varable selecton. In an attempt to analyze the supply of dealer servces (.e., lqudty) more rgorously, Stoll (1978a) develops an explct theoretcal model. The model shows that the relatve bd/ask spread of a securty equals the sum of nventory-holdng costs, adverse selecton costs, and order-processng costs, where each cost component has a precse defnton. Interestngly, the nventoryholdng cost expresson ncludes a term that equals the product of return volatlty and the expected tme the market maker expects the poston to be open, a dstncton that had otherwse gone unnotced n the lterature. Stoll (1978b) conducts emprcal tests of ths theoretcal model of the prcng of dealer servces. As n earler work, assumptons regardng model specfcaton and varable selecton are necessary. As n Benston and Hagerman (1974), for example, Stoll specfes hs regresson model n log-lnear form. And, because many of the varables n Stoll s theoretcal model are not drectly observable, proxy varables are substtuted. Dollar tradng volume, for example, s used to proxy for the length of the market maker s holdng perod (one of the components of nventory-holdng costs), and turnover (as measured by dollar tradng volume dvded by dollar amount outstandng) s used to proxy for adverse selecton costs. Prce per share, Stoll argues, captures the effects of order-processng costs, that s, the fxed cost per trade s spread over more dollars for hgh prced shares. Stoll s sample ncludes 2,474 observatons of NASDAQ stocks durng sx consecutve tradng days (July 9 through 16) n 1973. The explanatory of the regresson model s hgh, wth an adjusted R-squared of.822. As n other studes that use relatve spread as the dependent varable, the sngle, most powerful, explanatory varable s share prce, whch enters the regresson wth a negatve coeffcent. But, also as n these other studes, t s dffcult to dsentangle whether ths means that dfferences n share prce explan dfferences n spread or dfferences n the nverse of share prce. A smlar 11
queston can be asked about the dollar tradng volume, whose coeffcent s negatve and sgnfcant. Is the coeffcent s sgn and sgnfcance drven by an nverse relaton between bd/ask spread and tradng frequency or by the correlaton between prce and the nverse of prce? Harrs (1994) argues that the sgnfcance of the prce varable n the relatve spread regressons s drven by mnmum prce varaton and dscrete bd-ask spreads. To llustrate hs pont, he estmates two regressons. In the frst, he specfes relatve spread to be a functon of share prce, the standard devaton of stock return, the log of total market value, the nverse of the square root of the tradng frequency, the log of the dollar tradng volume, and the log of the rato of the tradng volume on the prmary exchange to the tradng volume across all exchanges. The motvatons for ncludng the dfferent varables are as follows. The standard devaton of stock return s a proxy for nventoryholdng costs, as shown n Stoll (1978a). The log of the stock s market captalzaton serves as an nverse proxy for adverse selecton costs. The larger the frm, the greater the degree of publc nformaton, the smaller the nformatonal asymmetres among nvestors, and the smaller are the adverse selecton costs. The log of dollar tradng volume measures tradng actvty and should be nversely related to relatve spread. The greater the tradng actvty, the lower are the fxed costs (.e., order-processng costs) per share. The log of the rato of tradng volume on the prmary exchange to consoldated tradng volume measures competton. Harrs argues that the nverse of the square root of the number of trades on the prmary exchange should be ncluded as an ndependent varable because t serves as a proxy for the proftablty market makng n the stock. The fewer the number of trades, the less compettve the market. In the absence of competton, market makers can more easly quote a larger spread to cover ther fxed costs and extract a surplus. 7 Whle ths argument has mert, the nverse of tradng frequency also acts as a proxy for the expected amount of tme that the market maker expects hs poston to be open, one of the 7 The motvaton for applyng the square root transformaton s that the rate of market maker partcpaton n trades declnes wth tradng actvty as more publc orders cross. It can also be justfed by nformatontheoretc consderatons: If each transacton conveys a bt of nformaton about value, the standard error of the dealer s nferred estmate of value wll be proportonal to the nverse square root of the transacton frequency. 12
components of nventory-holdng costs. The number of mnutes n a tradng day, 390, tmes the nverse of the number of trades equals the average tme between trades. Harrs estmates the regresson model usng a sample of 529 NYSE and AMEX stocks durng the second quarter of 1989. Observatons for each regresson varable are averages across the days n the quarter. The results of the frst regresson show that share prce, return volatlty, the log of market captalzaton, and the nverse of the square root of tradng frequency have postve and sgnfcant coeffcents. The coeffcents of the log of dollar tradng volume and the log of prmary exchange volume are negatve and sgnfcant. The adjusted R-squared of the regresson ft s.804. In the second regresson, Harrs replaces the prce varable wth the nverse of prce and fnds that the results change dramatcally. Frst, the goodness-of-ft (.e., the adjusted R-squared) ncreases to a level of.987. Second, the coeffcent on the nverse of share prce s postve and hghly sgnfcant. Thrd, the standard devaton of stock return and the nverse of the square root of the tradng frequency have postve and sgnfcant effects on relatve spread, and, fourth, all other varables become nsgnfcant. These results appear to ndcate that many stocks have spreads equal to the mnmum prce varaton and, to the extent there s explanable varaton n spread, t s drven by nventory-holdng costs (e.g., the standard devaton of return and the nverse of the square root of tradng frequency are the only sgnfcant determnants). III. MODEL SPECIFICATION Thus far, we have accomplshed two goals. Frst, we developed a general framework for consderng the factors that drve the level of market maker bd/ask spreads, and, second, we reconcled the results of past studes wthn ths framework. We now turn to developng a formal model of the market maker s bd/ask spread. To begn, we assume that the market maker sets hs bd/ask spread n accordance wth the general framework descrbed by (1). Total order-processng costs are assumed fxed, so order-processng costs per share are drectly proportonal to the nverse of tradng volume (denoted InvTV ). Lkewse, the competton varable s measured as the nverse of the number of dealers makng a market n the securty (denoted InvND ). 13
Unlke past studes, we do not attempt to dstngush between nventory-holdng costs and adverse selecton costs. Both are compensaton for the rsk of an unfavorable prce movement whle the securty s beng held n nventory. In a compettve market, the amount of the compensaton wll equal the margnal cost of hedgng prce rsk. Ths hedgng cost (denoted HC ), n turn, can be modeled as an opton value. 8 If the market maker has no nventory and accommodates a customer order by buyng at the bd, he needs protecton aganst the prce fallng below hs purchase prce before he s able to unwnd hs poston. Conversely, f the market maker has no nventory and accommodates a customer order to buy by sellng at the ask, he needs protecton aganst the prce rsng above hs sales prce. In the frst case, the market maker needs to buy an at-the-money put wrtten on the securty, and, n the second, he needs to buy an at-themoney call. Under Black-Scholes (1973) opton valuaton assumptons, the hedgng cost can be shown to be ( σ ) HC = P 2 N.5 T 1, (2) where P s the securty prce at whch the market maker opens hs poston, N(.) s the cumulatve unt normal densty functon, σ s the standard devaton of return, and T s the expected length of the market maker s holdng perod. 9 In the dervaton of (2), the nterest rate s assumed to be equal to zero snce most open postons are closed by the end of the tradng day. Wth the nterest rate at zero, the cost of hedgng an unantcpated prce drop (.e., the put opton value) s the same as the cost of hedgng an unantcpated prce ncrease (.e., the call opton value). Gatherng terms, our model of the market maker s bd/ask spread s SPRD = α + α InvTV + α HC + α InvND + ε. (3) 0 1 2 3 The coeffcent α 1 should be postve and may be qute large. After all, t represents the market maker s total order-processng costs. If the market s compettve, however, the 8 Here we model the sze of the bd/ask spread as the value of an at-the-money futures opton wth an exercse prce equal to the prce at whch the market maker provdes mmedacy. In contrast, Copeland and Gala (1983) model the bd/ask spread as a straddle n whch the market maker provdes a prospectve trader wth an out-of-the-money call opton to buy at the ask prce and an out-of-the-money put opton to sell at the bd prce. 9 The market maker s holdng perod s, of course, stochastc and depends on factors such as order flow. 14
market maker may not have the ablty to recover fxed costs, n whch case the coeffcent wll be ndstngushably dfferent from zero. The coeffcent α 2 should be postve the hgher the hedgng costs, the greater the bd/ask spread. Indeed, f an accurate proxy for the expected length of market maker s holdng perod can be obtaned, the coeffcent value should be one. Perhaps most mportant, however, s that, unlke past studes, we have dentfed the theoretcal structural relaton between bd/ask spread and many of ts determnants. As (2) shows, the margnal costs of nventory-holdng and adverse selecton are a specfc functon of share prce, return volatlty, and the tme that the market maker expects the poston to be open. Enterng the varables separately on the rght-hand-sde of the regresson equaton, as has been done n past work, obfuscates ther role. Fnally, the coeffcent α 3 should be postve the greater the number of dealers, the lower the nverse of the number of dealers, and the lower the spread. Another vrtue of our theoretcal model of the bd/ask spread s that, unlke the model s used n past studes, t s structurally consstent wth the presence of an exchange-mandated tck sze. The tck sze of a securty s ts mnmum allowable prce ncrement. The mportance of the tck sze n ths context s that t sets the lower bound of the market maker s bd/ask spread. In (3), the ntercept term α 0 represents a stock s mnmum prce ncrement because the values of all three varables on the rght hand-sde of (3) are near or at zero for actvely traded securtes. Fnally, t s worth notng that, although our theoretcal model s specfed n terms of the absolute spread, t can also be estmated usng relatve spreads. But, care must be taken to preserve the underlyng economc relaton, that s, all varables n the regresson must be deflated by prce per share, SPRD 1 InvTV HC InvND = α0 + α1 + α2 + α3 + υ. (4) P P P P P Ths means that there s no ntercept term n the relatve spread regresson (4) and that the nverse of share prce must appear as an explanatory varable. 10 Note that estmatng (4) s tantamount to runnng a weghted least squares (WLS) regresson of (3), where the 10 For a lucd examnaton of the problems encountered s usng ratos as varables n an ordnary least squares regresson, see Kronmal (1993). 15
respectve weghts are the nverse of share prce. The choce between the two estmaton methods must be based on the propertes of the resduals. IV. AN EMPIRICAL EVALUATION The focus now turns to evaluatng the emprcal performance of our theoretcal model (3). The frst part of ths secton provdes a descrpton of the data used n our analyses, and the second part ncludes some summary statstcs descrbng the sample. The thrd part contans the regresson results. A. Data The trade and quote data used n ths study were downloaded from NYSE s TAQ data fles. Although the fles contan nformaton for all U.S. exchanges, our sample contans only NASDAQ stocks due to restrctons on the avalablty of nformaton on the number of dealers. 11 The mnmum prce ncrement for NASDAQ stocks durng November 1998 was 1/16 th. For all tme-stamped trades, we matched the quotes prevalng mmedately pror to the trade. From ths matched fle, we then computed sx summary statstcs for each stock each day: (a) the number of trades, (b) the end-of-day share prce (.e., the last bd/ask mdpont pror to 4:00PM EST), (c) the number of shares traded, (d) the equalweghted quoted spread, (e) the volume-weghted effectve spread, and (f) the average tme between trades. Thus far, we have sad lttle about the types of spread measures that have been used n past studes. All of the studes cted n Secton II use quoted spreads, where the quoted spread s defned as Quoted spreadt = ask prcet bd prcet, (4) where the subscrpt t represents the t-th trade of a partcular stock durng the tradng day. The ntuton for ths measure s that, f a customer buys a stock and then mmedately sells t, he would pay the quoted ask prce and receve the quoted bd, thereby ncurrng a loss (.e., a tradng cost) equal to the bd/ask spread. Ths measure assumes that customers 11 Informaton on the number of dealers makng markets n NASDAQ stocks s avalable on NASDAQ a webste. Although NYSE s TAQ fles breaks down tradng volume by exchange, t does not break down tradng volume by the dfferent market makers standng at the specalst s post. 16
cannot trade wthn the quoted spread. It also assumes only market makers set the prevalng quotes and stand on the other sde of customer trades. In general, past research has used the quoted spread at the end of the tradng day as ther varable of focus. We use an equal-weghted average of the quoted spreads (EWQS) appearng throughout the tradng day. The effectve spread 12 crcumvents the frst of the two weaknesses of the quoted spread. It s based on the noton s that the trade s only costly to the nvestor to the extent that the trade prce devates from the true prce, as proxed by the bd/ask prce mdpont, Mdpont t (bd prcet + ask prce t) = (5) 2 On a round-turn, the cost would be ncurred twce, hence the measure of the effectve spread s Effectve spreadt = 2trade prcet mdpontt. (6) Naturally, f all trades took place at the prevalng bd and ask quotes, the effectve spread would be equal to the quoted spread. On the other hand, f some trades take place wthn the spread, the effectve spread wll be smaller than the quoted spread. The effectve spread measure assumes that, f a trade takes place above the bd/ask mdpont, t s a customer buy order, and, f t takes place below the bd/ask mdpont, t s a customer sell order. The absolute devaton of the trade prce from the bd/ask mdpont, therefore, can be nterpreted as the cost ncurred by the customer and/or the revenue earned by the market maker. Furthermore, the product of one-half the effectve spread tmes the tradng volume can be nterpreted as the market maker revenue from the trade. Whle the effectve spread s a better measure for customer trader costs than the quoted spread, t remans overstated n the sense that t fals to account for the fact that trades may be executed between customers and may not nvolve the partcpaton of the market maker at all. For such a trade, the effectve spread equals zero, that s, the 17
prce concesson conceded by one customer s awarded the other. Absent knowng the dentty of both partes on each sde of the trade, however, no better measure s possble. The volume-weghted effectve spread (VWES) s a volume-weghted average of the effectve spreads of the trades occurrng throughout the day. Wth the sx summary statstcs compled for each stock each day, we computed average values for each stock across all days n the month. To mtgate the effects of outlers, we then constraned the sample to nclude only stocks whose average prce was at least equal to $5 per share and whose shares traded at least fve tmes each day durng all twenty tradng days durng the month. The fnal sample contans 1,689 stocks. Three addtonal measures were then appended to each stock trade record. Frst, the number of dealers makng a market n the stock were gathered from the webste www.nasdaqtrader.com. As a matter of routne, NASDAQ reports ths nformaton each month and makes t freely avalable of ther webste. Second, the rate of return volatlty for each stock was computed usng end-of-day share prces. The daly return standard devaton was annualzed usng the factor, 252. Fnally, the hedgng cost for each stock was computed usng the opton valuaton equaton (2), where P s the stock s average share prce, σ s the annualzed return volatlty, and T s the average tme between trades 13. Usng the average tme between trades as a proxy for the expected length of the market maker s holdng perod understates the hedgng cost. Implctly, t assumes that there s only one dealer makng a market n the stock. If tradng volume was unformly dstrbuted across all dealers, we could multply the average tme between trades by the number of dealers n measurng the expected holdng perod. But, ths value would cause hedgng costs to be overstated, snce only a handful of dealers account for the lon s share of the tradng volume of a stock. Thus, wthout a more refned breakdown of tradng 12 Ths measure of effectve spread has been adopted n a number of studes n the stock market ncludng Chrste, Harrs, and Schultz (1994) and Huang and Stoll (1994). Lghtfoot et al (1986) use t n ther examnaton of bd/ask spreads n the opton market. 13 Snce volatlty s expressed on an annualzed bass, the tme between trades must be measured n years. To accomplsh ths task, we dvded the average number of mnutes between trades by 390 (.e., the number of mnutes n a tradng day) and then by 252 (.e., the number of tradng days n a year). 18
volume across market makers, we opt for the more conservaton measure. 14 B. Summary statstcs Table 2 contans summary statstcs across the stocks n sample. In the top panel, we report spreads measures. The equal-weghted quoted spread (EWQS) for the stocks n the sample has a mean value of.2375 or nearly four tcks. The volume-weghted effectve spread (VWES), on the other hand, s about 2/3 rds the sze at.1624. The dfference between these values ndcates that a large number of trades are beng executed at prces wthn the prevalng bd/ask quotes. Summary statstcs for relatve spreads are also provded n the table. The relatve volume-weghted effectve spread (RVWES), for example, averages 1.14%, and has a range from.06% (MSFT) to 6.06% (BOSHQ). Clearly, tradng less actve stocks can be qute costly. The second panel contans summary statstcs for the varables used as determnants of the spread. The average share prce s $19.72 per share, and ranges from $5 (ESFT) to $177.91 (YHOO). The average number of shares traded per day s about 350,000, wth a mnmum of 441 (BOSHQ) and a maxmum of 19,592,491 (DELL). The average number of dealers for each stock s 45, reflectng the hgh degree of competton among market makers on the NASDAQ tradng system. The number of dealers (ND) ranges from 5 (DXCPO) to 456 (DELL). The average annualzed return volatlty s.6991, reflectng the hgh degree of rsk of the technology-laden NASDAQ market. The average tme between trades (T) s 7.4 mnutes, and ranges from 0.02 (DELL) to 36.82 mnutes (CTBC). Fnally, the hedgng cost (HC) s.0305, ndcatng that a premum of at least three cents s necessary for market makers to cover ther margnal costs of operaton. The cost ranges from.08 cents (SHVA) to 28.30 cents (ABBBY). Table 3 contans estmates of the correlaton among the varables used n the analyss. A number of nterestng results appear. Frst, the correlatons between absolute spreads and relatve spreads are extremely small.247 for the equal-weghted quoted spreads and.244 for the volume-weghted effectve spreads. Recall that past studes of market maker spreads are splt n the use of absolute spread and relatve spread as the varable of focus. The low level of correlaton between the two measures ndcates that 14 One way to attack the problem s to set the coeffcent 2 α equal to one n equaton (3) and use non-lnear regresson to estmate of the average holdng perod across stocks. Assumng a constant holdng perod 19
the two varables are descrbng dfferent phenomenon. Snce the objectve of the studes has been to explan the level of spread, focusng on absolute spread seems the most sensble approach. Second, the correlatons among the regressors n (3) are small, provdng assurance that multcollnearty s not affectng our regresson estmates n any serous way. The correlaton between the nverse of tradng volume and hedgng cost s.086; the correlaton between the nverse of tradng volume and the nverse of the number of dealers s.239; and the correlaton between hedgng cost and the nverse of the number of dealers s.189. Fnally, t s nterestng to note that hedgng cost s hghly correlated wth both absolute spread measures,.750 and.745 for the EWQS and VWES, respectvely, whle t s vrtually uncorrelated wth the relatve spread measures,.083 and.046 for REWQS and RVWES, respectvely. Ths underscores the mportance of proper model specfcaton. Hedgng costs are clearly mportant n the determnaton of the bd/ask spread, but the relaton gets masked when only of the two varables (.e., spread) s scaled by share prce. C. Regresson results Table 4 contans a summary of the regresson results. All of the t-ratos are corrected for heteroscedastcty n the resduals. In all, fve dfferent regressons are estmated. The frst regresson uses the equal-weghted quoted spread as the dependent varable. As the table shows, all of the coeffcents are postve and sgnfcant n a statstcal sense. The sngle most mportant explanatory varable appears to be hedgng cost. Its coeffcent estmate s consderably larger than one, ndcatng that, as expected, the average tme between trades s a downward based estmate of the expected length of the market maker s expected holdng perod. The sgnfcance of the coeffcent α 3 ndcates that the nverse of the number of dealers also plays an mportant role n explanng the absolute level of the bd/ask spread. Clearly ncreased competton drves spreads to lower levels. The effects of competton are also lkely contrbutng to the fact that the nverse of tradng volume has the lowest sgnfcance of the varables n the regresson. Recall that n a hghly compettve market, market makers can recover only ther margnal costs. Fnally, recall that our model (3) s structured n a way that the level across stocks, however, s probably unrealstc. 20
of the ntercept term equals the mnmum tck sze. The estmate of the ntercept term α 0 s.0578, and s not sgnfcantly dfferent from the mnmum tck sze n the NASDAQ market,.0625. The second regresson uses effectve spread rather than quoted spread as the dependent varable. Snce many trades take place wthn the prevalng prce quotes, effectve spread s a more accurate measure of market maker revenue. Lke n the frst regresson, hedgng cost and the nverse of the number of dealers have the strongest explanatory power. Interestngly, the coeffcent of hedgng cost drops to a level of 2.6035. Apparently, the average tme between trades s not as poor a measure of holdng perod as we expected, gven the large number of dealers makng markets. Note also that the ntercept term s sgnfcantly less than the mnmum tck sze. Ths s not surprsng snce the effectve spread can have values as low as zero. 15 The estmate,.0403, represents the level of revenue per share that the market maker can expect to earn for provdng lqudty n an extremely actve stock. The thrd regresson s the same as the second, except that all varables are scaled by share prce and the ntercept term s suppressed. The choce between usng model (3) or model (4) should be based prmarly on whch regresson has the most well behaved resduals. Snce we are correctng the standard errors for heteroscedastcty n both cases, however, the nferences should not be strkngly dfferent. Indeed, the results of the thrd regresson are very smlar to those of the second. They should be. They are the same model. The fourth regresson llustrates what can happen f one nadvertently ncludes an ntercept term n the relatve spread regresson. Whle the coeffcent estmates of the explanatory varables are about the same order of magntude as those n the thrd, the meanng of the coeffcent of the nverse of share prce s lost. Nether that coeffcent nor the ntercept term provdes any meanngful nformaton about the mnmum bd/ask spread. It s also worthwhle to note the adjusted R-squared s consderably larger n the relatve spread regresson than n the absolute spread regresson (.e.,.8119 versus.6540). Ths comparson s meanngless, however, and does not n any way suggest that the 15 The effectve spread equals zero n nstances n whch the quoted spread s an even number of tcks and the trade takes place at the mdpont. 21
relatve spread performs better. Our objectve s to explan the varance of spread, not the varance of the rato of spread to prce. The ffth regresson s llustrates what can happen when the determnants of spread are specfed n an ad hoc fashon. The studes revewed n Secton II fall nto ths category. Whle each study provdes well-reasoned arguments regardng the choce of determnants, the decson regardng the structural relaton between the spread and ts determnants was arbtrary. Some used a lnear model; others a non-lnear model. Some used absolute spread; others used relatve spread. In Secton III, we developed a smple parsmonous theoretcal model of the market maker s bd/ask spread. Its structure accounts for the mnmum prce varaton of the stock and provdes an explct measure of the dollar cost of nventory-holdng and adverse selecton. Suppose we had not developed a theoretcal model but had made well-reasoned arguments for ncludng share prce, return volatlty, and the average tme between trades (.e., the determnants of hedgng cost) together wth the nverse of tradng volume and the nverse of the number of dealers as the determnants of spread. Furthermore, lke n some of the past studes, we assume the relaton s lnear. The results are reported n the ffth panel of Table 4. All of the varables enter the model wth ther expected sgns. Moreover, each varable s sgnfcant n the statstcal sense. Of course, the ntercept term no longer can be nterpreted as beng the mnmum spread. The most mportant result s that the adjusted R-squared for ths regresson s only.5175, when the adjusted R-squared for our smpler model (3) s.6540. Although both regressons contan the same ndependent varables, knowng the proper varable defntons and model structure substantally mproves performance. V. SUMMARY Ths study develops and tests a new model of the market maker s bd/ask spread. The model s smple and parsmonous, showng that the market maker s bd/ask spread s only a functon of the mnmum tck sze, the nverse of tradng volume, expected hedgng cost, and the nverse of the number of dealers. The nverse of tradng volume helps solate the market maker s order processng costs, and the expected hedgng cost accounts for the market maker s exposure to nventory prce rsk and adverse selecton. The expected hedgng cost s modeled as an at-the-money opton and s a functon of 22