Convertible Bonds and Stock Liquidity. Author. Published. Journal Title DOI. Copyright Statement. Downloaded from. Griffith Research Online

Transcription

1 Converble Bonds and Sock Lqudy Auhor Wes, Jason Publshed 2012 Journal Tle Asa-Pacfc Fnancal Markes DOI hps://do.org/ /s Copyrgh Saemen 2011 Sprnger Japan. Ths s an elecronc verson of an arcle publshed n Asa Pacfc Fnancal Markes, March 2012, Volume 19, Issue 1, pp Asa Pacfc Fnancal Markes s avalable onlne a: hp://lnk.sprnger.com// wh he open URL of your arcle. Downloaded from hp://hdl.handle.ne/10072/46102 Grffh Research Onlne hps://research-reposory.grffh.edu.au

2 Converble Bonds and Sock Lqudy Jason Wes Grffh Unversy 17 January 2011 Ths paper shows ha he probably of exercse of converble bonds ssued agans a frm s sock drecly affecs he lqudy of he sock self. Usng he rao of absolue sock reurn o s dollar volume as a proxy for sock lqudy I demonsrae ha here s a drec and posve relaonshp beween converson probably and sock lqudy whle conrollng for frm sze, book o marke equy value and frm bea. I descrbe he effec of unlsed converble deb on he lqudy of lsed frms n he US, Korea and Sngapore. The effecs of converson probably on sock lqudy are less pronounced for smaller frms, whch helps explan me seres varaons n he lqudy premums for smaller frms over me. The relaonshp beween converbles and sock lqudy s manly arbued o he expeced ncrease n he number of shares avalable for rade upon converson and he expeced change n he capal srucure of he frm. JEL Classfcaon: G12, G15, C25 Key words and phrases: Lqudy and asse prcng, converble bonds, lqudy premum Deparmen of Accounng, Fnance and Economcs, Grffh Busness School, Nahan, QLD, Ausrala, 4111, (w), emal: j.wes@grffh.edu.au

3 1. Inroducon Ths paper examnes how he lqudy of a sock changes n response o he ancpaed exercse of converble bonds ssued by a frm. In parcular, hs sudy proposes ha he probably of converson of a converble bond ssued agans a frm s sock s drecly and posvely relaed o he lqudy of he underlyng sock. The resuls are conssen wh hs hypohess. Resuls from hs sudy also confrm ha he ancpaed ncrease n lqudy s prmarly due o he ncreased volume of shares avalable for rade. The llqudy measure employed for hs analyss s he daly rao of absolue sock reurn o s dollar volume, averaged over some perod, as used n Amhud (2002). The llqudy measure can be nerpreed as he daly prce response assocaed wh one dollar of radng volume. Whle order-based measures of llqudy usng bd-offer spreads and he probably of nformaon based radng can provde a more dealed analyss of marke mcrosrucure effecs, he llqudy measure used here serves as a more general esmae of he prce mpac of converson probables over a longer perod. The resuls show ha boh across socks and over me, he exercse probably of converble bonds s a decreasng funcon of expeced llqudy. An ncrease n he exercse probably of converble bonds has a posve and hghly sgnfcan effec on he lqudy of he sock. The ess used here are he emporal effecs of he lkelhood of converson on he marke lqudy of he underlyng sock no whch he bonds can be convered. The exercse probably s hus an ncreasng funcon of expeced sock lqudy. Afer esablshng a lnk beween he lqudy and exercse probably of converbles, he underlyng cause of he ncrease n lqudy condonal on he converson of bonds no equy s addressed. In parcular, I propose ha ncreased lqudy can be caused by favourable changes n he deb-capal rao of frms and also by marke expecaons of hgher volumes of sock avalable for rade upon converson. The resuls show ha hgher expeced volume of sock avalable for rade s he prmary reason for he ncreased lqudy. Secon 2 nroduces he lqudy measure used n hs sudy along wh a defnon for he probably of converson ha wll be employed n he regresson. Ths secon also presens he research mehodology for deecng a relaonshp beween lqudy and he probably of converson, sock reurns, frm sze, book-o-marke rao and bea. Secon 3 dscusses he mehodology. Secon 4 presens he resuls and proposes reasons for he posve relaonshp beween lqudy and he probably of converson. Secon 5 dscusses he resuls. Secon 6 offers some concludng remarks. 2. Cross-seconal Lnk beween Exercse Probably and Lqudy Recen research evdence srongly suggess ha llqud asses offer greaer reurns han oherwse smlar lqud asses. Whle llqudy s clearly a cenral componen of asse prcng, capurng he essence of wha acually defnes llqudy s a more dffcul prospec. The effec of converble nsrumens on he lqudy of he sock agans whch hey are ssued wll ad n formng a more comprehensve pcure of he 2

4 rue characer of asse llqudy. My paper nvesgaes he lnk beween he lqudy of a sock and he exercse probably of a converble ssued agans ha sock. The resuls solae he prmary elemens ha conrbue o he lqudy of ordnary shares of frms ha ssue converbles agans her sock. The value for he cumulave probably dsrbuon funcon N ( d 2 ) expresson as defned n Ingersol (1977) s used as a proxy for he probably of exercse. The general specfcaon esmaes ncludes conrols for he sze of he frm Sze, he book-equy rao represened as BE and he frm s bea Bea, and s ILLIQ 0 1N d2 ) 2Sze 3BE 4 ( Bea, (1) For he -h sock, where and d 2 2 ln( S0 X ) ( r q ( ) 2) T, T (2) S 0 s he curren value of he sock, X s he srke prce for converson, r s he connuously compounded rsk free rae, s he sock prce volaly and N (.) s he cumulave probably dsrbuon funcon for a varable ha s normally dsrbued wh a mean of zero and a sandard devaon of one. The llqudy varable s esmaed usng ILLIQ 1 n n R V, d d 1, d, (3) where R, d and V, d are, respecvely, he reurn and dollar volume (n mllons) on day d n week, and n s defned as he number of vald observaon days n week for sock. The effec of converson probably, book o marke equy and frm bea on llqudy s examned for a porfolo of frms over me. A cross-secon of each frm a each pon n me s also conduced by addng a varable o represen he sze of he frm o he regresson. Frm sze as an ndependen varable s explcly excluded from he porfolo regresson snce sub-porfolos grouped by frm sze wll be examned separaely. The es procedure for he second se of regressons follows he Fama and MacBeh (1973) mehod. A cross-seconal model s esmaed for each week 1,2,... where weekly values of llqudy are a funcon of sock characerscs: ILLIQ J 0, j, j 1 X U. (4) j, ILLIQ s he llqudy defned n equaon (3) on sock n week and X, s characersc j of sock esmaed from daa n week. The coeffcens j, measure he j 3

5 effecs of sock characerscs on llqudy and U are he resduals. The weekly regressons of model (4) over he perod produce 104 esmaes of each coeffcen j,, j 1,..,4. These weekly esmaes are averaged and ess of sascal sgnfcance are performed. Below we dscuss each of he sock characerscs used n he regressons. 2.1 Converble deb nsrumens Converble bonds are corporae deb nsrumens ssued by a company ha gves he holder he rgh o forgo fuure prncple and/or coupon paymens and receve or conver such paymens no a specfed number of shares n he company s sock, a ceran mes n he fuure. In prncple, a converble bond s a hybrd secury conssng of a sragh bond and a call opon on he underlyng equy. Converble bonds gve her holders he ably o share n prce apprecaon of he company's sock. When frs ssued, converble bonds usually offer lower coupon raes han non-converble ones. Converbles are herefore a cos-effecve mehod of fnancng o he ssuer. However, he acual reurn on he converble bond has he poenal o exceed s saed yeld o maury due o he converbly feaure. The sragh bond value acs as a "bond floor" because f he sock prce remans depressed durng he enre holdng perod, he bond's value s reaned. The presence of downsde proecon makes converbles aracve o many prvae and nsuonal nvesors searchng for yeld enhancemen secures wh a degree of defence agans adverse volaly movemens. Converble bonds end o be ssued by smaller and more speculave companes because s cosly o assess her busness rsks, and here are concerns ha he company s managemen may no ac n he bondholders' neres, see Sen (1992). Converble bonds ofen represen unsecured and generally subordnaed deb. Usually, he ssuer s n a new lne of busness ha makes dffcul for nvesors o assgn a far dscoun rae by assessng he probably of busness falure or defaul. The converbly feaure algns he neress of he holders of converble bonds wh hose of he company's managemen, allowng he nvesors o prof when he company's share prce rses and o mnmze losses when s share prce falls. Converble deb s herefore srucured o allow managers o oban fnancng mmedaely hrough a delayed equy offer. Corporaons use converble deb as a subsue for common equy because also provdes ndrec equy fnancng ha mgaes he adverse selecon coss assocaed wh drec equy offerngs. Sen (1992) refers o hs as he backdoorequy hypohess. In addon, Sen shows ha he lkelhood of converble deb ssues ncreases when he coss of eher sragh deb or common sock ssues are hgh. The backdoor equy hypohess predcs ha converble bonds are a subsue for common equy and ha hs subsuon s mos lkely o occur n frms facng sgnfcan nformaon asymmeres and hgh fnancal dsress coss. The converson prce of a converble s calculaed as he produc of he share prce on he dae of ssue of he converble and he desgnaed converson premum of he bond a ssue. If he sock prce s far below he converson prce, he far value of he converble bond behaves lke he value of a sragh bond. If he sock prce s far above 4

6 he converson prce, he far value of he converble bond behaves lke he value of he sock. I examne he converble bonds of 107 publcly lsed US, Korean and Sngaporean frms who have ssued non-callable converble bonds. The converson premums of he bonds examned n hs sudy le n he range of % wh a medan converson premum of 36%. For a fve-year bond he converson prces for mos of he bonds are no unrealscally hgh. In general, praconers value converble bonds hrough smulaneously prcng and combnng he equy and fxed-ncome componens of he nsrumen. Valung converbles wh callable and converson feaures pror o maury can however, be dffcul. Unlke mos converble bonds ssued, he converble bonds used n hs sudy are fully converble a he holder s dscreon pror o maury, mosly afer an nal non-converson perod. In all cases he nal non-converson perod consues less han 25% of he enre lfe of he bond. Furhermore hese bonds are no callable by he ssuer. Ths allows us o nealy de-couple he bond and equy opon componens of he converble nsrumen o measure exercse probables. The value of he opon componen of he bond s esmaed usng conngen clam valuaon echnques from Ingersol (1977). The opon-lke characerscs of converbles provde a useful measure o use as a proxy for he probably of converson. More mporanly, hs measure provdes a reasonable esmae for he emporal probably of converson of each converble bond because allows he parameers of he opon o change hrough me. The bonds used n hs sudy are also held enrely by fnancal nsuons. Each bondholder sold he convered equy whn hree monhs of converson, and n mos cases, he sock was sold almos mmedaely. 2.2 A measure of llqudy - ILLIQ In capal markes, nvesors requre secures o have a ceran level of lqudy o faclae he radng of secures whou he need for prce dscouns. Invesors herefore may demand a rsk premum for secures ha do no mee her lqudy expecaons and may be wllng o accep lower reurns for more lqud secures. Sudes by Daar, Nak and Radclffe (1998) and Amhud (2002) esablshed hs lnk. A perfecly lqud marke exss f any amoun of a parcular asse can be nsananeously convered no a more lqud form and can hen be convered back no an asse a no cos. There s no wealh depleon from ransacon coss n a perfecly lqud marke. In conras, a perfecly llqud asse canno be raded a any prce. A lqud marke however, s deemed o exs f such a converson resuls n so-called mnmal ransacon coss. Transacon coss conss of boh explc and mplc coss. Explc coss generally nclude he exernally mposed coss of execung a ransacon such as brokerage commssons and axes. The mplc coss assocaed wh a ransacon, whch nclude bd-offer spreads and marke mpac coss, generally arse because of regulaory lms, nformaon dssemnaon, parcpaon and echnologcal consrans. For nsance, mnmum ck szes ha are arfcally large ncrease ransacon coss and ac as a dsncenve o nvesors. We wll consder only he mplc coss of radng n socks n hs paper. 5

7 Lqudy measures appled n recen research may be dvded no wo broad caegores: rade-based measures and order-based measures. Trade-based measures commonly used nclude radng volume, radng frequency and urnover rao. Tradebased measures are smple o calculae usng readly avalable marke daa and have wdespread accepance parcularly among marke professonals. Order-based measures have emerged from he prolferaon of auomaed radng sysems ha allows he use of more order-based lqudy measures. Order-based measures nclude absolue and relave bd-offer spreads and bd and offer volumes a each prce sep used o measure he socalled deph of he marke. Whle order-based measures are suable for nraday lqudy sudes, such measures provde lmed advanages for observng lqudy over longer frequences. I s doubful ha here s a sngle measure ha capures all aspecs of sock prce lqudy. The rade-based measure of he response of prce o order flow frs suggesed n Kyle (1985) and employed by Amhud (2002) s used n hs sudy as a measure of llqudy. The use of hs llqudy measure s applcable n hs sudy for several reasons. An llqud sock, ha s, one ha has a hgh value of ILLIQ, s one where he sock s prce moves a lo n response o lle volume. Conssen wh hs vew, Amhud (2002) shows emprcally ha ILLIQ s posvely relaed o measures of prce mpac and fxed radng coss over he me perod n whch mcrosrucure daa was avalable. Ths measure follows he concep of llqudy proposed n Kyle (1985) whch deecs he response of prce o order flow. Anoher nerpreaon of ILLIQ s relaed o he consensus among raders regardng new nformaon, followng Harrs and Ravv (1993). When a general consensus among nvesors s obaned concernng he mplcaon of news he sock prce changes whou necessarly ncreasng radng volume. If dsagreemen among raders occurs an ncrease n radng volume s generally observed. Thus ILLIQ can also be nerpreed as a measure of consensus belef among nvesors abou new nformaon. Whle a number of facors conrbue o he lqudy of a frm s sock, I propose ha he presence of converble nsrumens wh a hgh lkelhood of converson wll have a posve nfluence on he sock s lqudy. 2.3 A measure for he probably of converson - N d ) Snce he nenon s o deec a relaonshp beween he probably of exercse of a converble and he lqudy of he frm s sock, a proxy for he measuremen of he lkelhood of converson s requred. The mos approprae measure of he probably of exercse of a converble nsrumen s o oban a value for he cumulave probably dsrbuon funcon N ( d 2 ) expresson as defned n Ingersol (1977). The value of a call opon under he Black and Scholes framework usng rsk neural valuaon can be expressed as c e rt ( r q) T [ S0N( d1) e XN( d 2 )], (5) 6

8 where S 0 s he curren value of he sock, X s he srke prce for converson, r s he connuously compounded rsk free rae, s he sock prce volaly and N (.) s he cumulave probably dsrbuon funcon for a varable ha s normally dsrbued wh a mean of zero and a sandard devaon of one. In addon, d 1 d 2 T, (6) where d 2 s defned n equaon (2) assumng a connuous dvdend yeld q. Ths approach reles on he general assumpons underlyng he Black and Scholes (1973) and Meron (1973) models. The Black and Scholes model s nuvely undersood when broken down no qt he wo man pars. The frs elemen of he equaon S 0 e N( d 1 ) derves he expeced benef from acqurng he sock ourgh. Ths s equvalen o mulplyng he sock prce ( r q) T adjused for he connuous dvdend yeld S e by he change n he call premum, wh respec o a change n he underlyng sock prce N ( d 1 ). The N ( d 1 ), commonly known as he dela, measures he sensvy of he converble bond o s underlyng sock value. The second componen of he model Xe rt N d ) s he presen value of payng he exercse prce on he expraon day. In oher words Xe rt N( d 2 ) s he presen value of he srke prce mes he probably ha he srke prce wll be pad. Whle hs vew s relevan for share opons, he underlyng prncples of he formula can also be appled o compue he probably of converson of a converble bond ha has a converson prce X. The value N ( d 2 ) serves as a useful proxy for he probably of exercse snce dynamcally ncorporaes he rsk free rae, volaly of he underlyng sock, dvdend yeld, me o maury of he converble nsrumen and he converson prce. Inuvely, under he Black and Scholes model he expresson N ( d 2 ) s he probably ha he opon wll be exercsed n a rsk neural world. The far value of a converble nsrumen however, s no srcly obaned n a rsk neural world, and s ypcally prced usng a numercal approxmaon such as a rnomal ree usng a combnaon of boh rsk free and cred adjused raes. The far value of a converble nsrumen s obaned n pracce usng a lnear combnaon of he rsk free rae and a cred spread as he approprae dscoun facor. Therefore, whle he expresson for N ( d 2 ) canno srcly be used o assess he rue probably of exercse under eher he rsk neural or real world measure, serves as a useful proxy for he overall lkelhood of exercse snce ncorporaes changes n dvdends, volaly and me. The man advanage n usng hs approach s ha he npus can be synchronsed o each pon n he daa. The Black and Scholes formula ress on he assumpons of consan volaly, dvdend yeld and rsk free rae hrough me. Whle hs provdes only approxmae far values for sock opons we wll gnore he mnor lmaons of he Black and Scholes model n hs sudy. In any case, he value obaned for N ( d 2 ) s only an approxmae represenaon of he probably of converson and does no fully reflec he se of decson varables ypcally appled by a bondholder. 0 7

9 I s mporan o nclude all facors ha may affec he probably of converson usng he mos recen observaons n he marke. For nsance, nvesors may perceve ha hgher dvdend yeld can provde greaer lqudy, gnorng he mpac of axaon. Ths s smlar o he fndngs of Reddng (1997) ha suggess large nvesors prefer companes wh hgh lqudy whle also preferrng o receve dvdends. Hence me dependen dvdend yelds are used n hs analyss. In addon, changes o he volaly of he sock have a large mpac on he value of s assocaed sock opons. Therefore usng he curren volaly s lkely o nfluence he presen lkelhood of converson more han pas volaly. Whle mpled volaly observed n he marke would provde he mos accurae value, n mos cases such observaons were unavalable and hsorcal esmaes were used. To oban he mos accurae values, we updae each facor of he proxy for he converson probably N ( d 2 ) on a weekly bass. 2.4 Cross seconal conrol varables - Sze, BE and Bea The sudy conrols for he frm sze, he book o marke equy and he frm s bea. The Sze varable s he marke value of sock a me. The marke value of he sock s relaed o lqudy snce a larger sock ssue has smaller prce mpac for a gven order flow and a smaller bd-offer spread. Sock expeced reurns are negavely relaed o sze shown n Fama and French (1992) and can also be vewed as beng anoher proxy for lqudy, see Amhud and Mendelson (1986). A relaonshp beween expeced reurn and book o marke equy BE has been shown by Fama and French (1992) due o he funconal relaon beween expeced reurn and he marke value of equy. I nclude he BE varable o ensure he regresson capures he effecs of changes o he book value of asses and s effec on expeced reurns. Book o marke equy value s he rao of oal asses o frm marke value. Book value s calculaed as he marke value of equy plus oal asses mnus he book value of equy. The book o marke values ranged beween 0.31 and The Bea varable s ncluded as a measure of rsk. The effec of bea s expeced o be posve. The values for bea are ncluded o ensure ha he effecs of sysemac rsk are accouned for. The beas are calculaed agans he mos represenave marke capalsaon ndex for each counry. The marke porfolos used were he US S&P500 growh ndex for US socks, he Korean S&P200 growh ndex for Korean Socks and he Sras Tmes Index for Sngapore frms. The deals of each ndex are avalable n appendx A. Beas for each of he 107 socks analysed vares beween 0.89 and Daa and Research Mehod 3.1 Marke and radng ssues The daa used n hs sudy s aken from hree commercal bank daabases conssng of a oal of 107 publcly lsed US, Korean and Sngaporean frms who have ssued non-callable converble bonds. All bonds ssued are converble a he dscreon 8

10 of he holder any me pror o maury, are non-callable by he ssuer and conan provsons for accrued neres beween coupon paymens upon early converson. The opon componen of each converble bond herefore conans an Amercan-syle exercse feaure. Each bond s subjec o rgd coupon paymens rangng from 3 percen o 8 percen. Each bond had a maury of hree years or more from nal ssue and was ssued beween July 1997 and Ocober All converbles provde he opon o be convered no he sock a a specfed converson rao of he ssung company and no anoher company ndrecly. None of he converbles used are subjec o a rache feaure or smlar characersc ha requres he payable yeld of he converble o be ncreased should dvdends on he common sock ncrease. Only frms whose cred qualy as quoed by Sandard and Poors and Moody s remaned he same hroughou he lfe of he converble were used n hs sudy. Ths removes any bas n he resuls caused by ncreased radng acvy followng cred upgrades or downgrades. Over 55% of he bonds were convered no equy eher before or a maury whle he remanng bonds expred ou of he money a maury. No sock spls or rghs ssues were recorded n he daa. 3.2 Daa npu defnons The ILLIQ daa was compued a a weekly frequency for each of he 107 socks over he lfe of he converble bonds n he sample. The ILLIQ daa was ransformed no weekly daa by compung he arhmec average of daly daa across each week. The range of he ILLIQ varable n he sample daa was very large. I s herefore possble ha any esmaed relaonshp beween lqudy and he probably of exercse may be drven by a few exreme realsaons of ILLIQ. To correc for hs possbly, he hghes and lowes 2.5 percen of observaons of ILLIQ were dscarded from he daa. The runcaed daa se conans weekly ILLIQ values ha range from 0 o 1.09 for all socks. The probably of converson as measured by N ( d 2 ) requres several npus ha change hrough me. The volaly, rsk free rae and dvdend yeld mus be updaed a each daa pon o oban a weekly updaed measure for N ( d 2 ). The rsk free rae s maury-mached o observed governmen bond yelds a each pon n he daa se. For nsance, he rsk free rae used o value a converble bond ha has 5 years o maury on 13 July 2001 uses he 5-year governmen bond yeld observed on ha dae. As he maury of each bond s reduced hrough me he rsk free rae s esmaed by nerpolang beween quoed governmen bond yelds usng a cubc splne. The dvdend yeld s also updaed a each daa pon o oban an accurae value for N ( d 2 ). We assume a consan dvdend yeld measured a each pon n he daa se for each sock. Ths s a reasonable assumpon gven ha he dvdend yelds for 80% of he socks n our sample are zero and he remanng 20% are relavely sable a % over he esmaon perod. Fnally, a each pon n he daa se, we assume consan volaly of he sock reurn ou o he maury of he converble bond measured usng an exponenally weghed movng average (EWMA) model o updae he volaly for each weekly daa 9

11 pon. The EWMA mehod s more accurae han consan hsorcal volaly measures because capures changes n volaly ranslang drecly no changes n N ( d 2 ). The EWMA mehod s less relable han usng observed a he money mpled volales however he lack of exchange raded opons for he socks used n hs analyss does no perm he use of such daa. The EWMA model has weghs assgned o he connuously compounded reurn u ha decrease exponenally furher back n me, see Engle (1982). The weghng scheme leads o a smple formula for updang volaly esmaes,, (7) 2 n 2 n 1 2 ( 1 ) un 1 where he esmae for he volaly n on week n s calculaed from he prevous week s esmae of volaly n 1 on day n 1, and he mos recen observaon on changes n he reurn of he sock u n 1. To oban he sarng value of weekly volaly n he EWMA updang scheme, we use m n ( 1 ) un 1, (8) 1 where he number of weekly observaons m used for he nal esmaon s se o be relavely large m=100. The value for n he EWMA model conrols how responsve he esmae of he weekly volaly s o he mos recen observaons of he sock reurn. We apply a value of o esmae he weekly volaly. Fgure 1 shows he EWMA volaly for one of he more volale socks used n hs sudy. The jumps n volaly have an mporan effec on he value of N ( d 2 ) and are updaed o oban he mos accurae value o represen changes n he probably of exercse. Fgure 1 The rsk free rae, dvdend yeld and volaly are updaed weekly usng he mos recenly observed daa. The weekly value of N ( d 2 ) s compued for each sock over he lfe of he daa seres. The daa pons correspondng o he hghes and lowes 2.5 percen for N d ) were dscarded from he sample o preven bas from oulers. The fnal daa seres for N d ) s herefore runcaed n a smlar way o he ILLIQ varable. Ths procedure provdes an accurae pcure of he emporal probably of converson N ( d 2 ) ha akes no accoun changes n he rsk free rae, dvdend yeld and mos mporanly, volaly. From hese assumpons, he probably of converson as measured by N ( d 2 ) ranges from o for all socks used n he sudy. 10

12 4. Resuls 4.1 Lqudy and he probably of converson The emprcal ess use a wo-pass regresson mehod on he esmaes n equaon (9). In he frs pass all frms are grouped whn n a sngle porfolo. The ILLIQ varable was regressed on values of N ( d 2 ), book o marke equy BE and frm bea Bea over he esmaon perod ILLIQ 0 1N d2 ) 2BE 3 ( Bea. (9) Table 1 Table 1 provdes he regresson resuls. I s evden ha a srong posve relaonshp exss beween lqudy and he probably of converson, as measured by N ( d 2 ). The resuls are sascally sgnfcan a he 1 percen level. The book o marke equy and frm bea has no sgnfcan mpac a he 1 percen level. In order o solae he mpac of he probably of converson on lqudy, he dfferences n lqudy beween socks whose bonds have a hgh converson probably compared wh hose wh lle chance s examned. I conduc a second pass regresson whch groups he same se of frms no four porfolos based on he lkelhood of exercse of he converble bonds n he fnal 12 monhs of each bond s lfe. The general regresson remans he same however he frms are sored no four porfolos based on he value of N ( d 2 ) observed over he fnal 12 monhs of he lfe of each bond. The frs porfolo consss of frms where he probably of converson s greaer han 75%, ha s N ( d 2 ) > The second porfolo consss of frms where he probably s beween 50-75%, ha s 0.5 < N ( d 2 ) < The hrd porfolo groups frms whose probably s beween 25-50%, ha s 0.25 < N ( d 2 ) < 0.5. The fourh porfolo groups frms who have ssued bonds wh lle chance of converson, ha s N ( d 2 ) < Frms ha swched from one porfolo o he oher are excluded from he second pass regresson. By sorng on converson probables, we are able o maxmse he cross seconal varaon of ILLIQ o be explaned by N d ). In addon, Amhud (2002) examnes he relaonshp beween reurns and lqudy n a me-varyng framework and he resuls show a sgnfcanly posve relaonshp beween expeced marke llqudy and ex ane sock excess reurn, whch suggess ha expeced sock excess reurns are, n par, a premum for sock llqudy. The effecs of llqudy on sock excess reurns dffer across socks by her sze over me. Ths suggess ha share llqudy affecs smaller socks more srongly han larger socks. I herefore conduc a second pass regresson on he same se of frms grouped no four porfolos by sze, quarles 1 o 4, o deec he mpac of converson probably on sock lqudy for small and large frms. The resuls are presened n Table 2. 11

13 Table2 The se of resuls n Table 2 suggess ha he probably of converson s sascally sgnfcanly relaed o he lqudy of socks agans whch converbles are ssued. The effec s more pronounced for socks wh a hgh probably of converson and also for smaller socks. The nsgnfcan resuls for frms consung he N ( d 2 ) < 0.25 porfolo suggess ha frms wh a low probably of converson are unrelaed o sock llqudy. I s lkely ha he behavour of socks wh hese converble bonds has lle mpac on he lqudy of he sock and snce here s only a small chance hey wll be convered, oher effecs such as frm sze and performance wll have a greaer mpac on underlyng lqudy. The book o marke equy value BE s sgnfcan for he porfolo of all socks, all porfolos of varyng converson probables and he hree smaller quarle sock porfolos. Ths suggess ha he expeced reurn as refleced n he BE varable has a more sgnfcan effec on lqudy for smaller socks and socks wh a hgher han average probably of converson. However he BE resuls are very sgnfcan for socks wh lle chance of converson, N ( d 2 ) < 0.25, suggesng ha nvesors are reang all converbles as sragh deb and any llqud effecs are a resul of frm sze and performance. Perhaps surprsngly bea s nsgnfcan for all porfolos. 4.2 Cross seconal model I consruc a cross seconal model by regressng sock characerscs agans ILLIQ as n equaon (2). The model s esmaed for he wo years mmedaely pror o maury for each bond generang 104 ses of coeffcens. I choose he fnal wo years of each bond snce greaer changes n N d ) agans ILLIQ due o he me decay of he opon can be observed n hs perod. The mean and sandard error of he 104 esmaed coeffcens are calculaed for each sock characersc followed by a -es of he null hypohess of zero mean. The resuls, presened n Table 3 srongly suppor he hypohess ha llqudy s srongly relaed o he probably of converson afer conrollng for frm sze, book o marke equy and frm bea. Table 3 As shown n Table 3 s apparen ha a srong posve relaonshp exss beween he probably of converson and sock lqudy. The hghly sgnfcan values for N ( d 2 ) across each regresson confrms ha sock llqudy and converson probables are nversely relaed, or conversely lqudy and converson probably are drecly relaed. The BE and Bea parameers are sascally nsgnfcan across he specrum of socks used n hs sudy. Frm sze s sascally sgnfcan for each of he regressons. Ths s because frm sze can be vewed as a coarse measure of llqudy as shown n Amhud and Mendelson (1986) and herefore he nverse relaonshp beween Sze and ILLIQ s evden. In addon, small llqud socks should experence sronger 12

14 effecs of marke llqudy hrough a greaer posve effec of expeced llqudy on ex ane reurn and a more negave effec of unexpeced llqudy on conemporaneous reurn. For large lqud socks boh effecs should be weaker snce hese socks become relavely more aracve n mes of gher marke lqudy. From hese resuls I conclude ha a srong posve relaonshp exss beween lqudy and he probably of converson, as measured by N ( d 2 ) along wh a drec and posve relaonshp beween lqudy and frm sze. 4.3 Reasons for Lqudy Changes Changes n capal srucure and share volume The exercse of he converbles allows us o solae he reasons why lqudy may change as he probably of converson changes. For a bond ha s approachng expry wh a hgh probably of converson, he lqudy of he sock s expeced o ncrease. Fgure 2 shows he raded volume of sock agans he share prce and he maury dae for he enre porfolo of socks. A weak relaonshp can be observed beween volume raded and share prce changes, alhough hs relaon s no conssen over he perod. When he volume and sock reurns are combned o measure lqudy hrough he ILLIQ parameer, he correlaon beween sock lqudy and N d ) s much clearer. Fgure 2 Fgure 3 shows he lqudy agans he converson probably for he enre porfolo of socks. Increases n he share prce clearly ranslae no ncreases n he probably of converson, however oher facors such as volaly and he me decay of he opon componen of he converble have an effec. Fgure 3 I propose ha hs change s due o he ancpaed rade of a hgher volume of shares, should he holder wsh o mmedaely sell he newly convered sock, and s also due o he expeced change n he capal srucure of he frm. Gven ha he frm wll have a sgnfcanly lower deb/equy rao upon converson, hs may heghen demand for he sock resulng n even greaer lqudy. Prce reacons o converble deb secury offer announcemens are negave and sascally sgnfcan, see Lews, Rogalsk and Seward (1999). A major feaure of converble deb ssues s he mpac of share dluon upon a lkely converson of he deb o equy. Whle nvesors wll naurally ncorporae he lkely mpac of addonal shares beng ssued followng converson no he share prce, here may be endogenous changes n he lqudy of he sock. Ths could be he resul of a sudden ncrease n he number of radable shares avalable, sgnfcan changes n capal srucure and dvdend paymen expecaons. I s clear from he research of Lews, Rogalsk and Seward (1999) ha some form of share dluon upon converson s expeced o have some mpac on reurns. However he mpac of he posve effecs from a change n capal srucure and ncreased volume of shares s expeced o overrde he negave effecs of an expeced large block rade of shares. Ths expecaon manfess self n he exercse probably of converbles nearng expry. For all converble nsrumens used n hs sudy, he 13

15 proporon of shares o he pre-converson floa s less han 12 percen. Therefore only small effecs from share prce dluon are expeced n hese resuls. For larger proporons beng convered upon expry dsorons n lqudy wll probably be observed. For a bond ha s approachng expry wh a low probably of converson, he expeced change n lqudy s perhaps less clear. The expry of he converble bond can be refnanced wh anoher converble or sragh bond, resulng n mnmal change o he capal srucure of he frm. No ancpaed change n he volume of shares avalable n he marke followng converson should herefore resul n mnmal mpac on lqudy for a converble ha has a low exercse probably. The followng analyss searches for he reasons why lqudy ncreases as he converson probably ncreases. The basc specfcaon esmaes are ILLIQ 0 1ECS 2VOL, (10) where ECS E DT DT E T (11) represens he ancpaed new capal srucure of he frm usng he deb D and equy E a converson dae T, and VOL E conv T oal T (12) represens he rao of he number of shares convered o he exsng oal ordnary shares avalable for rade for he -h frm. The resuls from hs regresson are presened n Table 4. Table 4 Table 4 shows ha boh he expeced change n he proporon of shares avalable for rade and he expeced change n he deb equy rao wll resul n greaer lqudy. However he effec of he expeced ncrease n radeable volume s greaer for mos porfolos of socks. Ths suggess ha sock lqudy ncreases n ancpaon of he expeced marke capalsaon ncrease upon converson for frms wh a hgher probably of converson. As fund managers rebalance her porfolos n ancpaon of he change n he oal equy of frms wh a hgh chance of converson, so he lqudy appears o ncrease. The expeced changes are also greaer for smaller frms as evden from he resuls. The resuls for he porfolo of socks wh lle chance of converson, N d ) < 0.25, are nsgnfcan. 14

16 5. Dscusson The resuls ndcae ha as he probably of exercse of a converble bond ssued agans a parcular sock ncreases, he lqudy of he sock self appears o also ncrease. By ncorporang me o expry, sock prce volaly usng an EWMA model and dvdend paymens n he N ( d 2 ) componen derved from he Black and Scholes model, he neracon of hese facors help predc wh greaer accuracy he probably of converson hrough me. There s a clear relaonshp beween average lqudy and he average lkelhood of converson of a converble for he frms used n hs sudy. There are some ssues o consder however when each facor s examned n solaon. From Consanndes and Scholes (1980), socks wh hgher volaly should resul n lower reurns due o he ax radng opon. Taxes have no been consdered n hs sudy however s expeced ha her mpac on he converbly opon s small, parcularly for he markes used n hs sudy. Two bond yeld premums are known o have a posve emporal effec on sock reurns hrough me, see Fama and French (1989). The defaul yeld premum, whch s he excess yeld on corporae bonds and he erm yeld premum, whch s he dfference beween long-erm and shor-erm bond yelds are he man causes. Amhud (2002) found ha he effec of he defaul premum vares sgnfcanly wh frm sze. Snce he defaul premum sgnfes defaul rsk and fuure adverse economc condons, should have a greaer effec on he expeced reurn of smaller frms. Unforunaely we are unable o es he hypohess n hs sudy due o daa consrans, however we conrol for hs ndrecly by selecng frms whose cred rang remans consan hroughou. Ths may nduce some survvor bas n he resuls however marke observed cred spreads for all frms used n hs sudy are unavalable. A fnancally leveraged frm ends o ssue converble deb only f managemen s relavely opmsc abou s fuure share prce performance, see Lews, Rogalsk and Seward (1999). Converble deb ssue announcemens produce a sgnfcan negave sock prce reacon ha s conssen wh he backdoor-equy hypohess. 6. Concludng Remarks In hs paper I provde evdence ha here s a posve relaonshp beween he probably of converson of a converble bond ssued on a company s sock, and he lqudy of he sock self. I use he daly rao of absolue sock reurn o s dollar volume, averaged over a weekly frequency as a proxy for he lqudy of he sock and N ( d 2 ) as a proxy for he probably of converson. In general I fnd ha as he probably of converson ncreases, he lqudy of he sock also ends o ncrease. Ths effec s more pronounced for larger frms and frms wh a hgher han average chance of converson n he 12 monhs pror o deb maury. Ths resul s sgnfcan across all 107 socks measured n hs sudy. I also conduc a cross secon regresson o measure he effec of converson probably on lqudy for each frm, whle conrollng for frm sze, book o marke equy and frm bea. The resuls from hs model confrm he drec relaonshp beween lqudy and converson probably. Boh he expeced change n he proporon of shares avalable for rade and he change n he deb equy rao are 15

17 denfed as he reasons why lqudy s expeced o ncrease. The expeced ncrease n he volume of radeable shares s he prmary cause of he ncrease n lqudy. 16

18 References Amhud, Y., 2002, Illqudy and sock reurns: Cross secon and me seres effecs, Journal of Fnancal Markes, 5, Amhud, Y. and Mendelson, H., 1986, Asse prcng and he bd ask spread, Journal of Fnancal Economcs, 17, Black, F., and Scholes, M., 1973, The prcng of opons and corporae lables, Journal of Polcal Economy, 81, Breen, W.J., Hodrck, L.S. and Korajczyk, R.A., 2002, Predcng equy lqudy, Managemen Scence, 48 (4), Chorda, T., Subrahmanyan, A., and Anshuman, V.R., 2001, Tradng acvy and expeced sock reurns, Journal of Fnancal Economcs, 48, Consanndes, G.M. and Scholes, M.S., 1980, Opmal lqudaon of asses n he presence of personal axes: mplcaons for asse prcng, Journal of Fnance, 35, Cox, J., Ross, S. and Rubensen, M., 1979, Opon prcng: A smplfed approach, Journal of Fnancal Economcs, 7, Daar, V.T., Nak, N.Y. and Radclffe, R., 1998, Lqudy and sock reurns: An alernave es, Journal of Fnancal Markes, 1, Duffe, D. and Sngleon. K, 1999, Modellng erm srucures of defaulable bonds, Revew of Fnancal Sudes, 12, Engle, R., 1982, Auoregressve condonal heeroscedascy wh esmaes of he varance of UK nflaon, Economerca, 50, Fama, E.F. and MacBeh, J.D., 1973, Rsk, reurn and equlbrum: emprcal ess, Journal of Polcal Economy, 81, Fama, E.F., and French, K.R., 1989, Busness condons and expeced reurns on socks and bonds, Journal of Fnancal Economcs, 25, Fama, E.F., and French, K.R., 1992, The cross secon of expeced sock reurns, Journal of Fnance, 47, Haugen, R.A., and Baker, N.L., 1996, Commonaly n he deermnans of expeced sock reurns, Journal of Fnancal Economcs, 41,

19 Hull, J.C., 2000, Opons, Fuures and Oher Dervaves, Prence-Hall, New Jersey. Ingersoll, J. E., 1977, A conngen clams valuaon of converble secures, Journal of Fnancal Economcs, 4(3), Lews, C., Rogalsk, R., and Seward, J., 1999, Is converble deb a subsue for sragh deb or common equy?, Fnancal Managemen 28:3, Meron, R.C., 1973, Theory of Raonal Opon Prcng, Bell Journal of Economcs and Managemen Scence, 4, Perera, J.P. and Zhang, H.H., 2003, The lqudy premum n a dynamc model wh prce mpac, Workng Paper, Unversy of Norh Carolna. Reddng, L.S., 1997, Frm sze and dvdend payous, Journal of Fnancal Inermedaon, 6, Rendleman, R., and Barer, B., 1979, Two sae opon prcng, Journal of Fnance, 34, Sen, T., 1992, Converble bonds as backdoor equy fnancng, Journal of Fnancal Economcs, Augus,

20 Annualsed Volaly 150% EWMA Volaly 100% 50% 0% Jun-03 Sep-03 Dec-03 Apr-04 Jul-04 Oc-04 Jan-05 May-05 Fgure 1. Weekly exponenally weghed movng average (EWMA) volaly for a frm chosen from he sample esmaed over a wo-year perod. The EWMA updang scheme s used as a proxy for mpled sock volaly o evaluae he probably of converson measured usng N d ) over he esmae perod. Tme 19

21 1.4 Volume Traded and Share Prce Prce Volume Nov-01 May-02 Dec-02 Jun-03 Jan-04 Aug-04 Feb-05 Sep-05 Dae Fgure 2: Traded volume of sock agans he share prce for a porfolo of frms ha have ssued converbles agans her own sock. Boh ses of daa have been normalsed for ease of comparson Lqudy Prob of Conv Lqudy and Converson Probably Nov-01 May-02 Dec-02 Jun-03 Jan-04 Aug-04 Feb-05 Sep-05 Dae Fgure 3: Lqudy agans he probably of converson for a porfolo of frms ha have ssued converbles agans her own sock. Lqudy s measured as he nverse of ILLIQ and he converson probably s measured usng N d ). Boh ses of daa have been normalsed for ease of comparson. 20

22 Consan N d ) BE Bea Adjused R² (9.846) (4.608) (2.379) (3.841) (1.858) (4.273) (5.891) (-1.171) (1.485) (3.891) (1.931) (0.548) Table 1: Regresson of llqudy on converson probably and oher frm characerscs. Regresson of sock llqudy ILLIQ on he probably of converson of converble bonds ssued agans he sock N d ), he book o marke equy value BE and frm bea Bea. The able presens he means of he coeffcens and he -sascs are n parenheses. The porfolo of all socks ncludes he sock characerscs of all 107 frms measured over he perod n whch he converble bond s ssued pror o maury or exercse. The llqudy measure used ILLIQ s rao of absolue daly sock reurn o s dollar volume averaged for each week. The probably of converson s measured usng N ( d 2 ) and s updaed usng marke observed values for he rsk free rae, dvdend yeld and an exponenally weghed movng average (EWMA) updang scheme for sock volaly. Book-o-marke equy value s he rao of oal asses o frm marke value. Book value s calculaed as he marke value of equy plus oal asses mnus he book value of equy. Bea s he frm bea measured as he covarance of he sock and he marke porfolo dvded by he varance of he marke porfolo. The marke porfolo s he bes represenave broad based marke capalsaon ndex for Sngapore, Korea and US frms respecvely. 21

23 Varable All N ( d 2 ) N ( d2) N ( d2) N ( d 2 ) Quarle 1 Quarle 2 Quarle 3 Quarle 4 socks (larges) (smalles) Consan (1.485) (2.762) (1.116) (1.287) (0.547) (2.767) (1.987) (1.340) (1.672) N ( d 2 ) (3.891) (5.772) (4.778) (3.343) (1.882) (5.667) (4.892) (3.991) (3.361) BE (1.931) (2.466) (2.050) (1.326) (3.210) (1.898) (0.991) (0.845) (0.549) Bea (0.548) (1.098) (0.510) (0.659) (0.228) (0.989) (0.078) (0.552) (0.434) Adjused R² Table 2: Cross seconal regresson of llqudy on converson probably and oher frm characerscs. Cross seconal regressons of llqudy on he probably of converson of converble bonds ssued agans he sock, and he sock characerscs of book o marke equy BE and frm bea Bea. The porfolo of all socks ncludes he sock characerscs of all 107 frms measured over he perod n whch he converble bond s ssued pror o maury or exercse. The llqudy measure used ILLIQ s rao of absolue daly sock reurn o s dollar volume averaged for each week. The probably of converson s measured usng N ( d 2 ) and s updaed usng marke observed values for he rsk free rae, dvdend yeld and an exponenally weghed movng average (EWMA) updang scheme for sock volaly. Book-o-marke equy value s he rao of oal asses o frm marke value. Book value s calculaed as he marke value of equy plus oal asses mnus he book value of equy. Bea s he frm bea measured as he covarance of he sock and he marke porfolo dvded by he varance of he marke porfolo. The marke porfolo s he bes represenave broad based marke capalsaon ndex for Sngapore, Korea and US frms respecvely.

24 Consan N d ) Sze BE Bea Adjused R² (7.856) (3.608) (4.702) (3.984) (2.891) (2.379) (3.841) (1.858) (4.273) (5.891) (-1.171) (2.087) (3.099) (2.443) (0.576) (1.485) (3.891) (1.931) (0.548) (1.219) (2.882) (2.088) (0.242) (1.656) Table 3: Cross seconal regresson of sock llqudy on converson probably and oher frm characerscs. Cross seconal regresson of sock llqudy ILLIQ on he probably of converson of converble bonds ssued agans he sock N d ) and he oher sock characerscs of frm sze Sze, book o marke equy BE and frm bea Bea. The able presens he means of he coeffcens and he -sascs. The cross secon of socks ncludes he sock characerscs of all 107 frms measured over he perod n whch he converble bond s ssued pror o maury or exercse. The llqudy measure used ILLIQ s rao of absolue daly sock reurn o s dollar volume averaged for each week. The probably of converson s measured usng N ( d 2 ) and s updaed usng marke observed values for he rsk free rae, dvdend yeld and an exponenally weghed movng average (EWMA) updang scheme for sock volaly. Book-o-marke equy value s he rao of oal asses o frm marke value. Book value s calculaed as he marke value of equy plus oal asses mnus he book value of equy. Bea s he frm bea measured as he covarance of he sock and he marke porfolo dvded by he varance of he marke porfolo. The marke porfolo s he bes represenave broad-based marke capalsaon ndex for Sngapore, Korea and US frms respecvely. 23

25 Varable All N ( d 2 ) N ( d2) N ( d2) N ( d 2 ) Quarle 1 Quarle 2 Quarle 3 Quarle 4 socks (larges) (smalles) Consan (1.485) (1.998) (1.045) (1.452) (0.342) (1.112) (0.674) (1.248) (1.982) ECS (2.931) (2.445) (2.288) (2.378) (0.671) (2.234) (2.096) (2.797) (3.001) VOL (3.891) (4.554) (2.432) (4.385) (1.223) (2.887) (2.781) (3.589) (3.779) Adjused R² Table 4: Cross seconal of regresson of llqudy on he expeced change n capal srucure and radeable share volume. Cross seconal regressons of llqudy ILLIQ on he probably of converson of converble bonds ssued agans he sock N d ), he expeced change n capal srucure ECS and he expeced proporon of shares avalable for rade upon conversonvol. The porfolo of all socks ncludes he sock characerscs of all 107 frms measured over he perod n whch he converble bond s ssued pror o maury or exercse. The llqudy measure used ILLIQ s rao of absolue daly sock reurn o s dollar volume averaged for each week. The probably of converson s measured usng N d ) and s updaed usng marke observed values for he rsk free rae, dvdend yeld and an exponenally weghed movng average (EWMA) updang scheme for sock volaly. The frms are used are from Sngapore, Korea and he US.

26 Appendx A Marke Characerscs of Sngapore and Korea A.1 Sngapore A.1.1 Feaures of he Sngapore Sock Exchange The Sngapore Sock Exchange (SGX) lss he socks of over 500 companes on eher he SGX Manboard or he SGX SESDAQ. The SGX operaes a fully auomaed order drven radng sysem known as he Cenral Lm Order Book (CLOB). Lm and marke orders are execued n a connuous aucon beween 09:00 and 12:30 and hen beween 14:00 and 17:00 Monday o Frday, on he bass of prce hen me prory. There s a pre-open roune from 08:30 o 09:00 and a pre-close roune from 17:00 o 17:06. Shares are manly raded n board los of 1000 uns bu he radng of odd uns s allowed. The Un Share Marke enables he radng of odd los n any quany less han one board lo of he underlyng share n he Ready Marke. For example, for a buy order of 1060 XYZ shares, 1000 shares wll be execued on he Ready Marke and he remanng 60 shares wll be execued on he Un Share Marke. Amalgamaon of rades beween he Ready and Un Share markes s possble, whereby rades execued n he wo markes for he same underlyng share can be consoldaed n a sngle conrac. Board lo radng on he exsng Ready Marke s mananed for hose who wsh o connue radng n he desgnaed board lo szes. The mnmum bd sze for lsed companes radng a a share prce less han SGD$1 s SGD$0.005, and beween SGD$1 and SGD$3, he mnmum bd sze s SGD$0.01. All socks examned n hs paper currenly rade a a prce of less han SGD$3.00. The small ck szes used by he Sngapore Exchange are expeced o remove mos of he nose n our esmaon. In addon o brokerage fees, a clearng fee of 0.05 percen on he value of he conrac, up o a maxmum of SGD$200 s payable. No foregn exchange conrols or capal gans ax s payable on he secures raded by non-resdens. There are no lmaons on he reparaon of ncome, capal and capal gans. A.1.2 Sras Tmes Index The ndex used o represen he marke porfolo s he Sras Tmes Index (STI). The STI s a modfed marke capalsaon-weghed ndex comprsed of he mos heavly weghed and acve socks raded on he Sock Exchange of Sngapore. The Sras Tmes Newspaper of Sngapore comples he STI. The ndex was developed wh a base value of as of Augus 28, A.2 Korea A.2.1 Feaures of he Korean Sock Exchange The Korea Sock Exchange (KRX) Orders submed by nvesors are mached accordng o he prce and me prory, and are execued by means of eher perodc call aucon or connuous acon. Perodc call

27 aucon s ulzed o deermne an mparal far prce afer a perod of rade suspenson or n cases where dealed nformaon on he secures markes s lackng or unavalable. The connuous aucon s a mehod used o deermne prce durng regular radng hours afer he openng prce has been deermned. Ths mehod maches a bd and offer blaerally and when a new bd or offer eners no he sysem, s mached wh any of he exsng offers or bds compled n he order book accordng o prce prory and me prory. If he lowes bd and hghes offer can be execued, a prory s gven o he order receved earler. The radng hours are 09:00-15:00 for he regular sesson, and 07:30-08:30 and 15:10-18:00 for he pre-hours sesson and he afer-hours sessons respecvely. Tradng uns are 10 shares (n case of socks), and 1 share for socks whose closng prces on prevous day are hgher han KRW 100,000. In cases where dffcules n he normal rade execuon of an ssue s ancpaed because of drasc ncrease n he number of negoable shares, he radng un s adjused upward. Odd los wh quanes of less han 10 shares can be raded n he pre-hours and afer-hours sessons. The quoaon prce un mehod s broken no sx dfferen quoaon prce uns are used dependng on he prce of shares. Mos marke daa are avalable on real-me bass hrough he KRX compuer sysem. A.2.2 Korean KRX100 The ndex used o represen he marke porfolo s he KRX 100. Ths ndex chooses s consuens from boh he man board sock marke and he Kosdaq marke. The consuens of he KRX 100 are blue chp socks represenng he KRX marke chosen based on marke capalzaon, radng value, profably, sably and soundness. To ensure ha KRX100 accuraely represens he marke free-floa weghed and capalzaon lm mehods are adoped. 26