The pricing discount for limited liquidity: Evidence from SWX Swiss Exchange and the Nasdaq

Transcription

1 The prcng dscount for lmted lqudty: Evdence from SWX Swss Exchange and the Nasdaq September 6, 2003 Claudo Loderer* Insttut für Fnanzmanagement, Unverstät Bern emal: Lukas Roth Insttut für Fnanzmanagement, Unverstät Bern emal: Abstract We nvestgate the prcng dscount for lmted lqudty. Unlke prevous studes that have examned the relaton between hstorcal returns and lqudty, ours looks drectly at current stock prces. Ths approach requres less data and yelds up-todate nformaton about lmted lqudty dscounts. We analyze data from the Swss exchange and the Nasdaq durng , and fnd a statstcally and economcally sgnfcant prce-lqudty relaton n both markets. We test the robustness of that relaton wth a procedure that does not rely on specfc dstrbutonal assumptons. Our fndngs are unaffected. Accordngly, the dscount suffered by the least lqud securtes s about 30%. Keywords: Lqudty; Asset prcng; Bd-ask spread JEL classfcaton: G12, G31 * Correspondng author: Engehaldenstrasse 4, 3012 Bern, Swtzerland. Comments are welcome. We are grateful to the SWX Swss Exchange and ts former CEO, Antonette Hunzker, for provdng the ntradaly bd-ask spread data used n ths study. We also wsh to thank an anonymous referee for the valuable crtcsm and Nancy Macmllan for the great edtoral help. We have benefted from the help and comments of Petra Joerg, Govann Leonardo, Urs Peyer, Karl Pchler, Tna Purtschert, Kurt Schmdheny, Urs Wälchl, and Pus Zgraggen.

2 The prcng dscount for lmted lqudty: Evdence from SWX Swss Exchange and the Nasdaq Abstract We nvestgate the prcng dscount for lmted lqudty. Unlke prevous studes that have examned the relaton between hstorcal returns and lqudty, ours looks drectly at current stock prces. Ths approach requres less data and yelds up-to-date nformaton about lmted lqudty dscounts. We analyze data from the Swss exchange and the Nasdaq durng , and fnd a statstcally and economcally sgnfcant prce-lqudty relaton n both markets. We test the robustness of that relaton wth a procedure that does not rely on specfc dstrbutonal assumptons. Our fndngs are unaffected. Accordngly, the dscount suffered by the least lqud securtes s about 30%. page 2

3 The prcng dscount for lmted lqudty: Evdence from SWX Swss Exchange and the Nasdaq 1 Introducton The purpose of ths paper s to fnd out how to value frms that trade n less than perfectly lqud markets and the nvestment projects these frms undertake. To do so, we emprcally examne the dscount for lmted lqudty that market partcpants mpose when prcng stocks. When dscussng the valuaton of nvestment projects, most fnance textbooks recommend estmatng the projects future mean net cash flows and dscountng them wth an approprate rsk-adjusted dscount rate. No reservatons are generally made. A gven project should have the same value regardless of what frm undertakes t. As t turns out, ths logc mplctly assumes unlmted nvestor clenteles, snce unnhbted tradng among nvestors s needed to guarantee that a gven project commands the prce that theory mples. If the project s valued more hghly, nvestors wll avod nvestng n t and wll try to short t. If t s valued less hghly, nvestors wll flock to t hopng to earn an abnormal return. Whatever the msprcng, nvestors reacton wll tend to force the prce to converge to ts ntrnsc value (see also Shlefer and Vshny, 1997). 1 The problem s that fnancal assets trade n markets wth lmted lqudty. Thus market tradng mght not correct msprcng. Yet f nvestors are not sure that the prces they can deal at are rght, they wll be reluctant to trade. In partcular, they wll hestate to buy, unless offered an llqudty dscount, partcularly for frms where the probablty of msprcng s hgher. Illqud frms are those that are more expensve to trade. The more substantal the mpedments to tradng, the hgher the requred dscount. Practtoners have long recognzed ths phenomenon. In Swtzerland, they recommend addng a lmted-tradablty premum between 1% and 3% to the dscount rate used n frm valuaton. The emprcal lterature reports evdence consstent wth an llqudty dscount. Several studes nvestgate the relaton between hstorcal returns and varous lqudty 1 Black (1986) makes essentally the same argument when dscussng the dfference between prce and value of a stock. page 3

4 proxes, and generally fnd a negatve return-lqudty relaton. 2 We are also nterested n that relaton. However, we nvestgate the prcng relevance of lmted lqudty by lookng drectly at stock prces. The beneft of ths approach s that t uncovers the current relaton between value and lqudty (as opposed to the hstorcal relaton, as wth the return-based approach). Ths could be an mportant beneft snce that relaton appears to change over tme. Our approach also avods the problem of nsuffcent tme-seres data to measure mean returns. Our nvestgaton starts wth an analyss of the frms traded on the SWX Swss Exchange n the year We then replcate the analyss wth two control samples. The frst ncludes once agan the frms traded on the SWX Swss Exchange n and n the year The second ncludes the frms traded on the Nasdaq n Ths second sample enables nternatonal comparsons between a well researched captal market (the U.S. market) and a farly unknown one (the Swss market). We fnd sgnfcant evdence consstent wth the exstence of a dscount for lmted lqudty. The dscount can be szable; for the least lqud stocks n our sample, t equals about 30%. The remander paper s organzed as follows. The next secton dscusses lmted lqudty and ts mpact on frm value n more detal. Secton 3 summarzes the test desgn. Secton 4 presents the data, ther sources, and ther characterstcs. Secton 5 examnes the results of the nvestgaton, and secton 6 draws conclusons and formulates practcal mplcatons. 2 Theoretcal consderatons 2.1 Lmted lqudty and project value To calculate the value of an nvestment project, 3 textbooks generally recommend projectng the mean net cash flows the project wll generate and dscountng them wth an approprate rsk-adjusted dscount rate. 4 As an llustraton, There s also nconsstent evdence. See, n partcular, Chen and Kan (1995). The term nvestment project s used very generally and ncludes the buyng and sellng of frms themselves. To keep thngs smple, we gnore Fama s (1996) reservatons about the textbooks rendton of the theory of valuaton. page 4

5 assume, for smplcty, that nvestments are undertaken by corporatons and are all equty-fnanced. Accordngly, the followng project: Intal nvestment USD 10 mllon Expected perpetual net cash flow USD 8 mllon Rsk-adjusted dscount rate 20% has a value of: 8 NPV = 10 + = USD 30 mllon. 0.2 Yet assumng ths s ndeed how market partcpants compute the value of nvestment opportuntes, what guarantees that the project s really worth USD 30 mllon? The answer s tradng by market partcpants. If the frm s aggregate equty value ncreases by less than USD 30 mllon when ths nvestment s announced, the company s stock wll be undervalued. Investors could then buy a majorty poston n the frm and ether sell the project as a separate entty n the hope of obtanng a hgher prce, or hold ther stake and pocket ther share of the USD 8 mllon cash the project s expected to generate every year. The ncentve to buy goes away when the project rases the market value of the frm s equty by at least USD 30 mllon. Conversely, f the frm s share prce ncreases by more than USD 30 mllon, shareholders wll be nclned to sell, and other nvestors wll ether be dscouraged from buyng or be tempted to short shares n the expectaton of a prce declne. Ths wll cause the share prce to fall. The ncentve to sell or short wll go away when the share prce correctly reflects the project s value. For the tradng mechansm to be effectve, however, the project must be tradable ether drectly (as a separate frm) or ndrectly (as part of a traded frm). In realty, tradablty s generally lmted. Lmted tradablty can occur because: (a) the frm that undertakes the project s not traded and managers do not want to spn off, dvest, or otherwse sell the project; (b) the frm s nvestor clentele s fnte and unwllng, as a result of rsk averson, to engage n large transactons (Merton, 1982, and Black, 1986); (c) the clentele, beng unaware of the project s msprcng or facng lmted wealth, does not trade or s unable to to so (see Shlefer and Vshny, 1997); (d) nvestors follow a buy-and-hold polcy wth a long horzon. The greater the reluctance to trade or the tghter the lmts to tradng, the larger the lkelhood that page 5

6 the company s share prce wll not correspond to ts true, ntrnsc value. But f nvestors cannot be sure that the share prce of a gven frm reflects ts true value, they wll also pay less for ts shares, whch wll therefore sell at a dscount from true value. The same can be sad about the project n our precedng example. The project wll probably be worth close to USD 30 mllon to a hghly lqud frm. To a frm that does not trade very often, however, t wll not be worth that much, smply because no effectve tradng mechansm guarantees a value of USD 30 mllon. Lmted tradablty s synonymous wth lmted lqudty. Assets that are expensve to trade are also expensve to lqudate or to turn nto cash. In keepng wth the lterature, we wll therefore speak about lmted lqudty (or llqudty). The mplcaton of ths phenomenon s that there should be a negatve relaton between asset value and llqudty. An example of what can happen to prces when assets are llqud s the case of Nestlé s regstered and bearer shares n the 1980s. Any nvestor could hold bearer shares. In contrast, untl 1988, only domestc nvestors acceptable to the board could hold regstered shares. The nvestor clenteles of these shares was therefore lmted (Merton, 1982).e., ther tradablty was restrcted. Not surprsngly, the prce of Nestlé regstered shares (properly adjusted to reflect the same clam to future cash flows) traded at a 50% dscount from the bearer shares (see Loderer and Jacobs, 1995, and Stulz and Wasserfallen, 1995). 2.2 Lmted lqudty and the bd-ask spread Ultmately, all the mpedments to tradng ncrease the effectve costs of that actvty. It s debatable, however, whether transacton costs sgnfcantly affect longrun returns (for a survey of the lterature see Easley and O Hara, 2001). In Constantndes (1986), nvestors trade nfrequently, meanng that transacton costs are small compared to holdng-perod returns. In Amhud and Mendelson (1986), the opposte s true. The sgnfcance of transacton costs therefore seems to depend on nvestors horzons. Consstent wth ths argument, Chalmers and Kadlec (1998) report evdence that t s the amortzed rather than the regular spread that appears to page 6

7 get prced the amortzed spread measures the annualzed cost of the spread to nvestors as a group. 5 The logc of the Chalmers and Kadlec analyss, however, s not fully convncng. 6 Addng up transacton costs across trades seems to exaggerate the sgnfcance of transacton costs. To see why, gnore nterest rates and assume that nvestors trade a pot that contans $1,000, whch can be dstrbuted at year-end. Suppose each trade of the pot costs $1. If so, the last nvestor wll be wllng to pay $999 for t. And the next-to-last wll pay $998 so as to be able to cover hs transacton costs. Rollng back the argument, would seem to mply that the current pot prce equals $1,000 n, where n s the antcpated number of trades. Of course, the prce mpled by ths logc s too low. Gven a prce lower than $999, any nvestor can step n, pay the market prce, wat untl the end of the year, and make an arbtrage proft. The market prce wll be $999 and thereby reflect the costs of just one transacton. The same happens n the real world. Investors can step n and ether hold a gven asset over the long run or lqudate t. Thus, the relevance of transactons costs cannot be measured by addng up these costs across trades over an arbtrary horzon. It would be reductve, however, to thnk of transacton costs smply as a tax on gross returns. Transacton costs also have an ndrect effect, whch arses from the fact that, f they dscourage tradng, there wll be lmts to prce equlbraton. As we just argued, n the absence of the equlbratng mechansm of tradng, prces can be arbtrary and nvestors wll not always be able to trust them. Ths wll nduce a prcng dscount too. In our nvestgaton, we use the bd-ask spread as an ndcator of the cost of tradng a gven securty the cost of mmedate executon accordng to Amhud and Mendelson (1986). 7 The bd-ask spread s therefore our man proxy for lmted The amortzed spread equals the product of the effectve spread and the number of shares traded summed over all trades for each day, and expressed as an annualzed fracton of equty value. It s also counterntutve: assets that are almost never traded can n prncple end up beng more lqud than assets that are heavly traded. Several factors determne the bd-ask spread n a securty: order handlng costs, non compettve prcng, nventory rsk, the optons mplctly granted to the rest of the market, and asymmetrc nformaton (see Stoll, 2002; see also Huang and Stoll, 1997, and Glosten and Harrs, 1988). Whch component matters most n affectng prces s mmateral here. All we want to know s whether the cost of mmedacy (the cost of lqudty) affects securty prces. page 7

8 tradablty or lqudty (see also the dscusson n Amhud, 2002). 8 In dong so, we concede that t s debatable whether bd-ask spreads measure lqudty correctly (see Grossman and Mller, 1988, and Easley and O Hara, 2001). 9 Other (or addtonal) measures have been suggested, ncludng: effectve spread; realzed spread; amortzed spread (Chalmers and Kadlec, 1998); share turnover (.e., number of shares traded dvded by number of shares outstandng; or value of tradng dvded by value of shares outstandng; Datar, Nar, and Radclffe, 1998); dollar volume (Brennan, Chorda, and Subrahmanam, 1998); market depth; and prce mpact of tradng (defned by Breen, Hodrck, and Korajczyk, 2002, as the relaton of prce changes and net turnover; and by Amhud, Mendelsohn, and Lauterbach, 1997, as the rato of daly volume dvded by the absolute value of the daly return). 10 We wll therefore nqure nto tradng volume as an alternatve measure of lqudty. 3 Test desgn We nvestgate the relaton between lqudty and stock prces wth a crosssectonal regresson approach. To set a general framework, let us look at stock prces as the present value of a constantly growng dvdend flow and wrte: Stock prce (ex-dvdend): P = ( g) DIV 0 1+ k g, (1) where DIV 0 s the current dvdend per share, g the expected rate of growth of dvdends, and k the requred rate of return on the stock. Under the assumpton of a constant payout rato (π), ths model s often rewrtten by replacng the current dvdend payment by the product of the frm s current earnngs per share (EPS) and Several factors affect the bd-ask spread n a securty, namely the dealers order handlng costs, non-compettve prcng, the nventory rsk of the supplers of mmedacy, the value of the opton granted to nvestors, and the costs of havng to trade wth nvestors wth better nformaton (Stoll, 2002). See also Amhud and Mendelson (1986, 2000), Amhud, Mendelson, and Uno (1999), Bajaj, Dens, Ferrs, and Sarn (2001), Brennan and Subrahmanyam (1996), Easley, Hvdkjaer, and O Hara (2002), Eleswarapu (1997), Koepln, Sarn, and Shapro (2000), and Slber (1991). Studes that have found ndrect evdence of the prcng relevance of lqudty nclude Shlefer (1986), Harrs and Gurel (1986), Bagwell (1992), Loderer and Jacobs (1995), Stulz and Wasserfallen (1995), and Galloway, Loderer, and Sheehan (1998). See also Jones and Lpson (1999) and Engle and Patton (2004). page 8

9 the payout rato. Dong so and rearrangng terms we obtan an expresson for P/E ratos, namely: P/E rato: P EPS ( 1+ g) π =. (2) k g 0 Accordng to ths expresson, P/E ratos are, among other thngs, a postve functon of the payout rato and the expected rate of earnngs growth, and a negatve functon of the requred rate of return on the stock. From equaton (2), a cross-sectonal regresson of P/E ratos aganst ther possble determnants should nclude payout ratos, expected rates of earnngs growth, and requred rates of return. We perform such a regresson. Snce n a cross-sectonal comparson rsk-free rates and market rsk premums are the same across observatons, f we want to control for dfferent requred rates of return, k, we have to control for dfferental rsk. We therefore nclude a rsk varable n the regresson arguments. Moreover, snce the evdence suggests that average stock returns (and therefore stock prces) are affected by frm sze, we add ths varable as well. More mportantly, n our effort to assess the prcng relevance of lqudty, we examne whether the ncluson of lqudty adds to the explanatory power of the regresson. We therefore model P/E ratos wth the followng functon: P/E rato = f(growth, Payout, Rsk, Sze, Lqudty ), (3) where the ndex dentfes a partcular frm and the arguments are defned as follows: Growth Payout Rsk Sze Lqudty = expected rate of earnngs growth; = frm s payout rato; = rsk of the stock n queston; = market value of the frm s equty; = stock s lqudty. Equaton (3) s consstent wth the regresson specfcatons tested n the return-lqudty lterature. From that lterature, the coeffcents assocated wth growth, sze, and lqudty should be postve; based on equaton (2), payout should also have page 9

10 a postve coeffcent, although there s lttle f any evdence to support ths predcton; and rsk should have a negatve coeffcent. Note that these are all relatons that nvolve prce, not EPS. The excepton s the relaton between rsk and P/E ratos. All else beng the same, rsk correlates negatvely wth prce and postvely wth EPS. The other relatons do not share a smlar ambguty. In partcular, the extant emprcal fndngs of a postve relaton between hstorcal returns and lqudty mply that a negatve relaton between P/E ratos and lqudty would have to be a relaton between lqudty and prce, not lqudty and EPS. We wll rely on ths argument when nterpretng the emprcal results. 4 Orgnal-sample characterstcs The orgnal sample covers the frms traded on the SWX Swss Exchange. Snce frms typcally have more than one class of stock outstandng, we focus on the class wth the lowest bd-ask spread. Gven that we are nterested n establshng a relaton between lqudty and stock prces, ths selecton crteron should not carry any substantal prejudce. There are 250 frms lsted on the SWX n June The I/B/E/S database from whch we gather EPS-forecast nformaton, however, provdes data only for a subset of them; we exclude 103 frms wth no such forecasts. Fnancal analysts do not follow these companes closely, possbly because they have a market captalzaton of less than CHF 23 mllon (USD 16 mllon assumng an exchange rate of CHF 1.4 to the dollar). We also exclude 15 frms wth negatve EPS to avod negatve P/E ratos. These latter frms represent a problem because, as argued n the followng secton, the regresson specfcaton we eventually choose s one where we take the natural logarthm of P/E ratos. Another 8 frms lack other data requred n the analyss. That leaves us wth a sample of 124 frms. The cross-sectonal analyss n equaton (3) s performed as of June 30, The varables n the equaton are measured as follows: P/E rato: stock prce observed on June 30, 2000 dvded by EPS reported for We refer to the natural logarthm of ths varable as ln(p/e). Prce data are provded by SWX; EPS data are from the I/B/E/S database; page 10

11 Expected earnngs growth: measured n two ways. The frst s the growth-rate predcton (relatve to 1999) mplct n fnancal analysts average EPS forecasts for the year 2000 as reported on June 30, 2000 n the I/B/E/S database. We label ths varable EPSg2000. The second proxy s the growth-rate predcton mplct n fnancal analysts average EPS forecasts for 2001 (relatve to 2000) as reported on June 30, 2000 n the same database. We label ths varable EPSg2001; Payout rato: dvdends pad n 1999 on the stock n queston dvded by EPS reported for the same year. Dvdend data are from the yearly stock gude Schwezerscher Aktenführer. We label ths varable PAYOUT; Rsk: slope coeffcent of a market model estmated wth monthly returns over the perod to In that model, the market portfolo s approxmated wth the Swss Performance Index (SPI), an ndex that contans all frms traded on the SWX. The computaton assumes nternatonally segmented markets (Stulz, 1995a; Stulz, 1995b). We refer to ths varable as BETA. Alternatvely, we measure rsk wth the standard devaton of return over the same perod of to and label that varable STDEV; Frm sze: natural logarthm of the product of number of shares outstandng tmes average monthly stock prce observed durng the frst sx months of We refer to ths varable as LNSIZE. 11 As mentoned above, f the frm n queston has more than one class of stock outstandng, we take the most lqud one (n terms of bd-ask spread). Informaton on the number of shares outstandng s from Schwezerscher Aktenführer; Lqudty: average relatve bd-ask spread, defned as the rato of the dfference between ask and bd prce, dvded by the bd prce. 12 Intradaly bd-ask spread data are from the SWX. Tradng on the SWX s electronc. Its bd-ask spreads are therefore not necessarly quotes posted by market makers but could reflect lmt orders of ndvdual nvestors. The average we use s computed over the perod to usng two daly observatons (one at 10 a.m., the other at 4 p.m.). From the SWX database, we drop all observatons for whch the bd s smaller than or equal to the ask quote (ths error n the database occurs n 10,294 out of 176,869 cases, wth a frequency of 5.8%). We refer to ths varable as RELSP. Descrptve statstcs for the varables n the regresson are reported n Table 1 below. As can be seen from the standard devatons, there s farly wde varaton n both P/E ratos and bd-ask spreads. Hence, there s somethng to explan, and lqudty could contrbute to that explanaton. All the other explanatory varables n the table also dsplay wde varaton. Moreover, all varables have sample dstrbutons that are skewed rght. The bd-ask spread, for nstance, has an average of 1.68% and a medan of 1.37%. The medan equty value s CHF 1,027 mllon (USD 734 mllon wth an exchange rate of CHF 1.40 to the dollar), the medan P/E The results are vrtually unchanged when we measure LNSIZE by the market captalzaton on June 30, 2000 nstead. Evdence of a lqudty effect n Swss stock prces s reported n Gardol, Gbson-Asner, and Tuchschmd (1997). Lqudty s measured there as the proporton of freely negotable shares n the captal structure. page 11

12 rato s about 18.93, and medan expected earnngs growth equals 14.4% for the year 2000 and 16.2% for Emprcal results Ths secton presents and dscusses the emprcal results. For ease of exposton, t falls nto three parts. In the frst, we examne the orgnal sample. We start wth the analyss of a smplfed verson of equaton (3) and ts statstcal propertes. Then we use that nformaton to test the prce-lqudty relaton postulated n equaton (3) wth an ordnary least squares (OLS) regresson approach. Fnally, we reexamne the relaton wth a procedure that s robust wth respect to dstrbutonal and other standard assumptons concernng the regresson resduals. Part two nvestgates whether the results can be duplcated wth two holdout samples. The fnal part nterprets the results. 5.1 Orgnal sample Prelmnary analyss We frst test the followng specfcaton of equaton (3): P/E = α + α EPSg α EPSg α RELSP + α RELSP + ε,(4) where the varables are defned as above, the subscrpt refers to frm, the α s are regresson coeffcents, and ε s an error term wth the usual ordnary least squares (OLS) propertes. Ths ntal specfcaton postulates a nonlnear prce-lqudty relaton, consstent wth the results reported n the lterature (see especally Amhud and Mendelson, 1986, and Brennan and Subrahmanyam, 1996). The estmaton results are shown n Table 2. All varables n the regresson have hghly sgnfcant coeffcents wth confdence 0.95 or better. The F-test value for the regresson as a whole s also hghly sgnfcant, and the adjusted R 2 ndcates that the estmated relaton explans a szable fracton (86.5%) of the cross-sectonal varaton n P/E ratos. Both measures of expected earnngs growth have the postulated postve coeffcent. Specfcally, the expected growth rate one year ahead (EPSg2000) has a coeffcent of (t-value = ), compared wth the page 12

13 coeffcent of (t-value = 2.930) found for the growth rate expected two years hence (EPSg2001). The obvous nterpretaton s that hgher expected earnngs growth sgnals hgher future resdual cash flows. Moreover, the bd-ask spread (RELSP) has a negatve and ts squared term (RELSP 2 ) a postve coeffcent. The net effect s negatve, at least over a range of bd-ask spread values up to 11%. Ths s partly consstent wth the hypotheszed sgn. A margnal 1 percent ncrease n the bd-ask spread, for nstance, would depress the P/E rato by about 12 [= 1, , = 12]. The table further reports a farly large postve and sgnfcant ntercept estmate, an ndcaton that we may have left out some relevant varables. As ndcated n equaton (3), frm sze, rsk, and payout rato could be among them. In spte of ts explanatory power, one problem wth specfcaton (4) s that t yelds nonnormal resduals. The studentzed range of these resduals s n fact 10.85, a value that rejects normalty wth confdence better than 0.99 and questons the sgnfcance tests reported n Table 2. As t turns out, takng the natural logarthm of the P/E ratos yelds resduals that follow a more normal dstrbuton. The studentzed range of the resduals from that specfcaton (not shown) s 5.56, whch fals to reject normalty at the usual confdence levels. Related to ths, there could be an outler problem. Two observatons appear to serously affect our estmates. 13 The frst refers to Unaxs, a frm that resulted from the restructurng of Oerlkon Bührle. Because of that restructurng and the assocated wrteoffs, Unaxs s 1999 earnngs are close to zero. The P/E rato s consequently very hgh (1,108). The second observaton concerns the frm EL Smplon. Its bd-ask spread s more than 12%, double the next hghest value n the sample. When we reestmate the regresson equaton (4) wthout these observatons, the coeffcent of RELSP 2 becomes nsgnfcantly dfferent from zero. Yet another problem wth regresson (4) s heteroskedastcty. A Cook- Wesberg test has a ch-squared value of , whch rejects homoskedastcty wth confdence better than Ths bases the estmated standard error of the regresson and represents one more reason to queston the sgnfcance tests reported n the table. 13 These two observatons have a so-called hgh leverage. For a dscusson of leverage n the context of outler analyss, see Judge, Hll, Grffths, Lütkepohl, and Lee (1988), p page 13

14 5.1.2 OLS-regresson results The precedng results mply a regresson specfcaton n whch the dependent varable s the natural logarthm of the P/E rato, the regresson arguments nclude frm sze (LNSIZE), payout rato (PAYOUT), and rsk (BETA), and the regresson s estmated wth a Whte correcton procedure. The regresson specfcaton we are nterested n s therefore the followng: ln(p/e) = α 0 + α EPSg α 1 +α BETA + α EPSg α LNSIZE + α 6 3 PAYOUT + RELSP + ε (5) where the dfferent varables are as defned above. Because ths specfcaton s nonlnear, we drop the squared value of the relatve spread (RELSP 2 ) from the regresson arguments. The estmaton results are shown n column (1) of Table 3. We show those obtaned when droppng the two observatons Unaxs and EL Smplon. We wll come back to the ssue of outlers and, more generally, nonnormal resduals n secton The regresson ntercept s stll sgnfcantly dfferent from zero. If we nspect the column of results further, we see that expected earnngs growth once more has the predcted postve and sgnfcant coeffcent, regardless whether we look one or two years ahead (EPSg2000 and EPSg2001, respectvely). Of the new varables, PAYOUT has an nsgnfcant coeffcent, whereas BETA has a negatve and sgnfcant one. Ths s consstent wth the noton that BETA s a relevant rsk measure. We also fnd that the log of the market value of equty (LNSIZE) has a postve and sgnfcant coeffcent. The postve sgn could ndcate that larger frms are less rsky and therefore command hgher stock prces (see also the lterature on sze effects, ncludng Banz, 1981; Fama and French, 1993). The coeffcent of the bd-ask spread (RELSP), however, s nsgnfcantly dfferent from zero at the usual confdence levels. One possble reason s that there s no prce-lqudty relaton. Another s that LNSIZE s a measure of both rsk and lqudty. Consstent wth the latter nterpretaton, the correlaton coeffcent between equty value and bd-ask spread s Large frms tend to have tghter bd-ask spreads, whch means that frm sze could also be a proxy for lqudty. For practcal valuaton purposes, the analyss could stop here, snce the results n column (1) of Table 3 tell us that large frms are more hghly valued than small page 14

15 frms. Whether the reason s rsk or lmted lqudty would be mmateral. We could therefore mpute the dscount mposed on smaller frms by the market, and use that dscount to value the nvestment projects undertaken by smaller frms. As t turns out, we obtan smlar results when we replcate the regresson wth a dfferent measure of sze, namely the book value of the frm s assets. The beneft of that specfcaton s that t yelds a sze coeffcent that can be appled also n the case of nontraded frms. Our purpose, however, s to assess the mpact of lmted lqudty on value. We therefore have to dsentangle the possble rsk and lqudty effects n the coeffcent of LNSIZE n our regresson. We unravel the two effects n queston by regressng frm sze on bd-ask spread. That regresson tells us what part of the crosssectonal varaton n frm sze s due to the cross-sectonal varaton n lqudty and what part s not. Specfcally, we estmate the followng regresson equaton: LNSIZE = µ + µ RELSP + η, (6) 0 1 where s once agan an ndex that dentfes a partcular frm n our sample of 122, and η s an error term wth the usual OLS propertes. Estmated over the perod, ths pooled tme-seres, cross-sectonal regresson has an adjusted R 2 of 63.5% and a bd-ask spread coeffcent of wth a t-statstc of (not shown). By constructon, the resduals from ths regresson are unrelated to lqudty (as measured by the bd-ask spread) and therefore capture sze effects net of lqudty effects. We call these resduals RES-SIZE/RELSP and use them n our regresson n leu of the varable LNSIZE. We hasten to add that one could also regress the bd-ask spread aganst frm sze and use the resdual from that regresson to capture possble lqudty effects unrelated to sze. Our nterest, however, s not so much n dstngushng sze-related from sze-unrelated lqudty effects, but rather the lqudty-related from the lqudty-unrelated sze effects. The new regresson specfcaton s therefore the followng: ln(p/e) = α 0 + α EPSg α 1 +α 4 RES-RELSP + α 2 EPSg α 5 RELSP + ε 3 BETA (7) Note that we dropped the varable PAYOUT as t does not have a sgnfcant coeffcent. Estmaton results for ths specfcaton are dsplayed n column (2) of page 15

16 Table 3. As shown there, the adjusted R 2 s slghtly hgher than that n column (1) of Table 3, and the F-statstc remans hghly sgnfcant. All the varables have coeffcents that are sgnfcantly dfferent from zero wth confdence of at least 0.99, and all have the predcted sgn. Specfcally, the coeffcents for earnngs growth are postve whether we look one or two years ahead; the coeffcent of rsk (BETA) s negatve; and the coeffcent of RES-SIZE/RELSP s postve, wth a value of and a t-statstc of Ths latter result confrms our orgnal asserton that (net) sze could capture aspects of rsk not measured properly by the frm s equty beta; accordngly, larger frms are less rsky and therefore command, all else beng the same, hgher prces. More mportantly, the bd-ask spread n column (2) of Table 3 has a negatve and sgnfcant coeffcent ( wth a t-statstc of 5.016). As predcted, a larger spread reduces equty value Issues of statstcal robustness Before proceedng wth the analyss, we dscuss the normalty of the resduals. Fgure 1 below plots the kernel densty estmates of the regresson resduals and compares them wth the theoretcal normal densty. The relevant regresson coeffcents are those n column (2) of Table 3. Vsual nspecton suggests the presence of skewness. A skewness test s sgnfcant wth confdence better than The problem wth nonnormal resduals s that they are nconsstent wth our t- and F-tests. It s not clear from nspecton, however, whch way the bas goes. Varous approaches have been suggested to obtan regresson coeffcent estmators that are robust wth respect to the dstrbuton of the error terms. 14 We employ a combnaton of the algorthms developed by Huber and Tukey as mplemented n the Intercooled Stata 6.0 statstcal package. Ths method starts wth the OLS regresson estmates, calculates the resduals, and weghs the observatons n accordance wth the sze of ther resduals. Observatons wth large resduals are gven low weghts. The regresson equaton s then reestmated as a weghted least squares regresson. Resduals from ths regresson are computed and used to revse the weghts assgned to the observatons. The weghted least squares regresson s reestmated usng the revsed weghts. Ths procedure s repeated untl the regresson parameters converge. 14 For a dscusson of the ssues nvolved, see Chapter 22 of Judge, Hll, Grffths, Lütkepohl, and Lee (1988) or Chapter 6 n Hamlton (1992). page 16

17 We apply ths procedure to compute the regresson coeffcents. The varable RES-SIZE/RELSP, whch we use to capture net sze effects, s also estmated wth a robust regresson procedure. The results are shown n Table 4. A few thngs are worth notng when comparng these results to those obtaned prevously n column (2) of Table 3. Frst, none of the coeffcents have changed n sgn. Second, the probablty values of the t-tests of sgnfcance for the ndvdual parameters are essentally the same. 5.2 Holdout samples Holdout-sample characterstcs To test our results, we use two holdout samples. The frst covers the SWX durng a 5-year perod precedng the orgnal sample year and durng the year thereafter,.e., and The second refers to the Nasdaq durng Table 5 reports the selecton crtera and the resultng annual szes of these samples. To be ncluded n our samples, companes have to be n the I/B/E/S database of earnngs forecasts, they have to report postve earnngs durng the year n queston, they have to have suffcent monthly return data to compute ther beta coeffcents, and they have to have suffcent bd-ask spread data. Betas are computed wth monthly returns over the past 4.5 years; we requre 50 of the possble 54 monthly observatons to nclude a partcular frm n the sample. Bd-ask spreads are computed as daly averages over the past 2.5 years; we requre 600 of the possble 625 daly observatons to add a partcular frm to the sample. Mssng values can occur by chance or because a frm s not quoted durng the perod n queston. The results do not change sgnfcantly when we apply less restrctve sample selecton crtera. As one can see from the table, the resultng sample szes oscllate between 113 and 128 for SWX companes, and between 450 and 664 for Nasdaq companes. The data used n the analyss of these holdout samples are defned smlarly as n the orgnal sample except for the followng tems. For the SWX sample, the bdask spread s taken from Datastream and corresponds to the closng quote. Moreover, earnngs forecasts for are from Datastream. For the Nasdaq sample, EPS data are from I/B/E/S, everythng else s taken from the Center for Research n Securtes Prces (CRSP) tapes. The market portfolo used to measure rsk s the Swss Performance Index (SPI) and the Nasdaq Composte Index for SWX and page 17

18 Nasdaq frms, respectvely. The results are the same when we use the CRSP Value- Weghted Index for Nasdaq frms. Table 6 presents descrptve statstcs for the two holdout samples. A comparson ndcates that Nasdaq frms have slghtly hgher P/E ratos throughout the sample perod. Accordng to the expected earnngs-growth fgures, ths seems to occur because fnancal analysts project hgher earnngs growth for Nasdaq than for SWX frms, partcularly two years ahead (EPS g,t+2 ). The table also shows that companes n ether sample have beta estmates smaller than one. That does not mean that these frms represent lttle rsk n an absolute sense. Remember that these coeffcents are computed usng the ndex of SWX frms and of Nasdaq frms as market proxes, respectvely. Consequently, our holdout frms have below-average rsk n a relatve sense. In other words, they are less rsky than the average frm n ther respectve ndexes. Gven that the sample selecton crtera weed out frms wth negatve earnngs and frms wth nsuffcent data (and hence rsker frms), ths phenomenon should not be surprsng. The return standard devatons (STDEV) suggest that Nasdaq frms are rsker than SWX frms, partcularly durng the second half of the sample perod. Fnally, the table documents that the representatve Nasdaq frm has hgher relatve bd-ask spreads than the typcal SWX frm, although manly durng the frst half of the sample perod Regresson results We frst replcate the analyss n column (1) of Table 3 (not reported). The results are farly smlar to what we obtan there. In partcular, whereas the coeffcent assocated wth sze (LNSIZE) s postve and sgnfcant n ether sample, that of relatve spread (RELSP) never has the predcted negatve sgn. To dstngush between possble lqudty effects and sze effects unrelated to lqudty, we then estmate the regresson equaton (6) wth a robust estmaton technque. The results are once agan mostly consstent wth those found for the orgnal sample n Table 4. We report them n Table 7, subdvded n two panels: Panel A refers to SWX companes, Panel B to Nasdaq companes. As shown there, the regresson has a sgnfcant F-value wth confdence n excess of 0.99 n all years and samples. Moreover, ts explanatory power s farly large, especally n the SWX sample the adjusted R 2 s between 65% and 97% for SWX companes, and between 43% and 71% for Nasdaq companes. As for the regresson coeffcents, we fnd that page 18

19 hgher expected earnngs growth one and two years ahead (EPSg,t+1 and EPSg,t+2, respectvely) appears to sgnfcantly boost stock prces. More mportantly, and consstent wth the clam that lower lqudty reduces asset value, the relatve bd-ask spread (RELSP) s negatvely related to stock prces n both samples. Ths relaton s statstcally sgnfcant wth confdence better than Furthermore, the resdual of the regresson of frm sze aganst the relatve bd-ask spread (RES-SIZE/RELSP) has a postve and sgnfcant coeffcent n both samples as well. As mentoned above, larger (adjusted) frm sze could reflect lower rsk. Rsk (BETA), however, has the expected negatve coeffcent only n the Swss sample. In contrast, Nasdaq frms exhbt rsk coeffcents that are postve wth confdence of at least Chalmers and Kadlec (1998) report a smlar puzzlng fndng n ther cross-sectonal analyss of stock returns and lqudty. The coeffcents on beta they fnd are negatve and sgnfcant. 15 The Chalmers and Kadlec sample covers U.S. domcled common stocks lsted on ether the Amex or the NYSE durng At the bottom of each panel, we examne the statonarty over tme of the lqudty coeffcents. The number shown s the t-value of a sgnfcance test of the change n that coeffcent from one year to the next. The null hypothess s that the change s zero. As one can see, almost all annual changes are sgnfcant wth confdence The relevance of lqudty n asset prcng therefore appears to vary over tme. Ths suggests that a cross-sectonal analyss of the prcng relevance of lqudty of the type performed here can ndeed be a valuable complement to the return-based approach snce t yelds up-to-date (as opposed to hstorcal) measures of the mpact of lqudty on asset prces. Note, however, that changes n lqudty coeffcents do not necessarly mply changes n lqudty-related dscounts n asset prces. The reason s that lqudty tself can change from one year to the other. A look back at the descrptve statstcs n Table 6 reveals that bd-ask spreads do n fact change over tme. For nstance, on the Nasdaq, they have experenced a steady declne from an average 2.58% n 1995 to an average 1.01% n We repeat the analyss wth a dfferent measure of rsk, namely the stock return varance as opposed to the stock s beta (not shown). The results are essentally the same as those we just examned, ncludng those pertanng to the coeffcent of 15 Eleswarapu and Renganum (1993) report a consstent fndng for a smlar tme perod. page 19

20 rsk. In the Swss sample, hgher return varance reduces stock prces. The opposte s true n the Nasdaq sample. It could be that hgher return varance measures better growth opportuntes. Snce Nasdaq frms would seem to nclude more start-ups than the SWX frms, that could explan our fndng. To check ths nterpretaton, we nclude both measures of rsk n the regresson for Nasdaq frms. We expect a postve coeffcent on return varance and a negatve one on beta. Contrary to that, however, the coeffcent on beta remans postve whereas that assocated wth return varance becomes generally negatve (not shown). As mentoned above, the bd-ask spread s not the only measure of lqudty one can thnk of. Tradng volume, turnover, number of shareholders, and number of market makers, among others, have been proposed as alternatve or addtonal measures n the lterature. 16 To test whether the prce-spread relaton uncovered above s ndeed a prce-lqudty relaton, we repeat the analyss usng tradng volume as a measure of lqudty. Tradng value s the average daly value traded n a partcular stock over the past 2.5 years (the results do not change when we measure the average over the past 6 months). The results of ths nvestgaton are reported n Table 8. In the regresson, we replace the relatve bd-ask spread varable wth the natural logarthm of tradng volume (LNVOLUME). We also estmate the varable that gauges the net effect of frm sze dfferently. As one mght remember, that varable s the resdual of a unvarate regresson of frm sze aganst bd-ask spread. When the lqudty proxy s tradng volume, we estmate that resdual by regressng frm sze (LNSIZE) aganst tradng volume (LNVOLUME). We refer to the resultng resdual as RES-SIZE/LNVOL. Snce lqudty would seem to correlate postvely wth tradng volume, we expect ths lqudty proxy to have a postve coeffcent. The results confrm what we have found so far. The explanatory power of the regressons remans essentally unchanged. Furthermore, all regressors have sgnfcant coeffcents wth the sgn observed n prevous tables (except for the coeffcent of the rsk varable BETA that tends to become less sgnfcant n the SWX sample). Specfcally, the coeffcents of projected earnngs growth (EPSgt+1 and EPSgt+2, respectvely) and frm sze (RES-SIZE/LNVOL) have a postve sgn. Rsk (BETA) has a negatve sgn n the SWX sample and a postve one n the Nasdaq 16 See, for nstance, the dscusson n Amhud und Mendelson (1986) and, especally, Baker (1996). page 20

21 sample. More mportantly, the mpact of hgher lqudty on P/E ratos (and therefore prces) remans postve: hgher tradng volume s assocated wth hgher P/E ratos. Ths s n lne wth the fndng that larger relatve spreads (and therefore lower lqudty) depress P/E ratos (and therefore prces). 5.3 Interpretaton Impled llqudty dscounts The last ssue to address s ultmately the most mportant one: the economc sgnfcance of the results. The analyss so far reveals that lqudty appears to have a sgnfcant mpact on P/E ratos. Gven that we control for varous addtonal factors of nfluence, the coeffcents of the lqudty proxes measure the margnal contrbuton of lqudty to the P/E rato of a gven company. Snce there s lttle theoretcal reason to beleve that market lqudty affects frm earnngs, we have argued that ths contrbuton nvolves prces rather than earnngs. Varous studes that fnd a postve relaton between bd-ask spreads and hstorcal stock returns support ths nterpretaton. To examne the economc sgnfcance of lqudty, we wrte the estmated regresson equaton (7) as: αˆ + αˆ EPSg αˆ EPSg2001+ αˆ BETA + RELSP 0, (8) + αˆ 4 RES- SIZE/RELSP + αˆ 5 RELSP ( P/E) = exp where a hat denotes estmated coeffcents, and we drop for smplcty the ndex. The subscrpt RELSP 0 ndcates that ths expresson holds for the general case of a frm wth lmted lqudty, as measured by a bd-ask spread dfferent from zero. As a comparson, we can wrte the estmated P/E rato for a frm wth perfect lqudty as: αˆ + αˆ EPSg αˆ EPSg2001+ αˆ BETA + RELSP = 0. (9) + αˆ 4 RES- SIZE/RELSP ( P/E) = exp page 21

22 Takng the dfference between equatons (9) and (8) and dvdng by equaton (9) yelds an expresson for the estmated dscount nduced by lmted lqudty, namely: ( P / E) 0 ( P / E) ( P / E) RELSP = RELSP 0 = 1 exp( αˆ 5 RELSP). (10) RELSP= 0 Thus, to compute the prcng dscount generated by a gven degree of llqudty, all we have to do s nsert the approprate bd-ask spread and the estmated coeffcent ˆα 5 n equaton (10) above. For nstance, for the partcular case of frms wth a bd-ask spread equal to the medan spread n the orgnal sample of SWX frms (RELSP = 1.37% n Table 1), we obtan a pont estmate of 1 exp( ) = 12.93%, where the computaton uses the robust regresson coeffcent estmate ˆα 5 reported n Table 4, namely The result says that, compared wth a stuaton of perfect lqudty, the stock prce of a frm wth a bd-ask spread of 1.37% would trade at a 12.93% dscount. To obtan an nterval estmate, we fnd the 95% confdence nterval for the coeffcent estmate and nsert t nto equaton (10). For our orgnal sample, ths nterval s (5.76%, 19.55%). Remember that these dscounts are computed relatve to a stuaton of perfect lqudty. We can replcate ths computaton for the medan frms n our holdout samples. Specfcally, for each sample and each year, we take the medan bd-ask spread and the lqudty coeffcent reported n Table 7 for the year n queston, and compute the llqudty dscount accordng to equaton (10). The results are reported n Fgure 2. As one can see, the medan dscount for SWX frms ncreases durng the sample years from 7.3% to 21.1%. In comparson, the medan llqudty dscount among Nasdaq frms s more or less constant at about 27% durng the same perod. To examne less lqud frms more closely, we compare the ten frms wth the largest bd-ask spread wth the ten wth the smallest spread n the orgnal sample of SWX frms. Table 9 provdes statstcal nformaton for these frms. The medan spread of the least lqud frms s about 26 tmes larger than that of the most lqud frms (4.016% compared to 0.156%), and n one case, AGEFI, the spread exceeds 6%. Somewhat surprsngly, the least lqud frms are also less rsky (medan beta of 0.42 page 22

23 vs. 1.13). 17 Moreover, they are almost 1,000 tmes smaller (USD 54.1 mllon compared wth USD 52.7 bllon, assumng an exchange rate of CHF 1.4 to the dollar). To get a better sense for these equty captalzaton numbers, consder that, on May 16, 2001, the dstrbuton of Nasdaq frms by market value of equty was as follows: Market captalzaton (mllon USD) Number of frms Fracton of total market captalzaton , % % 100 1,000 1, % 1,000 5, % 5,000 20, % 20,000 50, % 50,000 plus % Sum 4, % Wth ths dstrbuton, the ten least lqud frms n our orgnal sample would belong to the smallest 42% [= 1,858/4,424] of the dstrbuton of Nasdaq frms. In comparson, wth a medan captalzaton of USD 52.7 bllon, the most lqud frms would fall nto the top 0.2% [= 9/4,424] of that dstrbuton. Thus the most lqud frms on the SWX are farly large by U.S. standards also, whereas the least lqud frms would be small n ether country. 18 We can use our data to compute the llqudty dscount for these two extreme groups of frms. From equaton (10) and the respectve medan bd-ask spreads of 4.016% and 0.156%, we fnd that, n relaton to perfect lqudty, the dscount of the least lqud frms equals 33.4% [= 1 exp( ) ]; n contrast, the most lqud frms sell at a 1.6% [= 1 exp( ) ] dscount. The llqudty Snce betas are estmated wth monthly returns, nfrequent tradng cannot be the only reason for ths result. As one mght remember, equty value s based only on the most lqud class of stock outstandng for frms wth more than one class. Ths tends to reduce the market captalzaton of Swss frms n the comparson, although not by much. Frst, there are only 15 frms wth multple classes of stock outstandng n the orgnal sample. And second, the market captalzaton of the omtted classes s farly small. page 23

24 dscount of the least lqud companes s therefore szable and should provde frms strong ncentves for correctve acton Is the mpled llqudty dscount reasonable? The llqudty dscount mpled by our estmates appears to be substantal. To assess ts magntude, we can compare t to what the extant lterature suggests. 19 One possble benchmark s papers that nvestgate the relaton between returns and lqudty, n partcular the classcal Amhud and Mendelson (1986) study of NYSE stocks. Accordng to that paper, a 1% ncrease n the relatve spread s assocated wth an ncrease n expected return of 2.53%. Takng nto account that the relatve spreads n the portfolos n that analyss range from 0.49% to 3.21%, ths fgure mples an expected return dfferental of 6.88% between the portfolos wth hgh and low spreads, respectvely. 20 If we consder a stock that pays a perpetual annual dvdend of $1, ths return dfferental yelds an llqudty dscount of roughly 58%. The results n Brennan and Subrahmanyam (1996) ten years later are strkngly smlar: the return dfferental between the most lqud and the least lqud quntles of frms n ther sample of NYSE frms s 662 bass ponts, whch mples a smlar llqudty dscount. Consequently, when taken at face value, the dscount derved n these studes appears to be even larger than what we fnd here. Another place to look for a comparson are studes of restrcted stock sales, such as the prvate placement of a large block of stock wth selected nsttutonal nvestors (see Bajaj, Dens, Ferrs, and Sarn, 2001). A comparson of the prces of publcly traded shares wth the prces at whch these ssues are placed yelds a possble measure of the llqudty dscount (see, for nstance, Slber, 1991). Accordng to Koepln, Sarn, and Shapro (2000), the mean and medan dscounts reported by these studes vary between 22% and 35%, whch would be consstent wth what we fnd. The problem wth these estmates s that they may be confounded by factors other than lmted lqudty for nstance, the possble commtment by the block buyers to be actve montors What follows s taken from Brennan and Tamarowsk (2000), p. 35. Practtoners n Swtzerland recommend addng a lmted lqudty premum of between 1% and 3% when captalzng company cash flows (see the dscusson n Helblng, 1989, and Boemle, 1995). See the dscusson n Koepln, Sarn, and Shapro (2000), p. 95. See also Hertzel and Smth (1993). page 24