Understanding the Profitability of Pairs Trading


 Thomas McGee
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1 Undersanding he Profiabiliy of Pairs Trading Sandro C. Andrade UC Berkeley Vadim di Piero Norhwesern Mark S. Seasholes UC Berkeley This Version February 15, 2005 Absrac This paper links uninformed demand shocks wih he profis and risks of pairs rading. Usually employed by sophisicaed invesors, pairs rading is a relaive value sraegy ha simulaneously buys one sock while selling anoher. In a marke wih limied risk bearing capaciy, uninformed demand shocks cause emporary price pressure. A pair of sock prices ha have hisorically moved ogeher diverge when subjeced o differenial shocks. Uninformed buying is shown o be he dominan facor behind he divergence. A sraegy ha sells he higher priced sock and buys he lower priced sock earns excess reurns of 10.18% per annum. The markedomarke reurns of a pairs rading sraegy are highly correlaed wih uninformed demand shocks in he underlying shares. Measuring pairs rading profis represens a succinc way o quanify he coss of liquidiy provision (i.e., he coss of keeping relaive prices in line.) Keywords: Asse pricing, reurn predicabiliy, limis of arbirage JEL number: G12 We hank Will Goezmann wihou his insighs his paper would no exis. Any misakes are ours alone. Conac informaion: Mark S. Seasholes, UC Berkeley Haas School, 545 Suden Services Bldg., Berkeley CA ; Tel: ; Fax: ; c
2 Undersanding he Profiabiliy of Pairs Trading This Version February 15, 2005 Absrac This paper links uninformed demand shocks wih he profis and risks of pairs rading. Usually employed by sophisicaed invesors, pairs rading is a relaive value sraegy ha simulaneously buys one sock while selling anoher. In a marke wih limied risk bearing capaciy, uninformed demand shocks cause emporary price pressure. A pair of sock prices ha have hisorically moved ogeher diverge when subjeced o differenial shocks. Uninformed buying is shown o be he dominan facor behind he divergence. A sraegy ha sells he higher priced sock and buys he lower priced sock earns excess reurns of 10.18% per annum. The markedomarke reurns of a pairs rading sraegy are highly correlaed wih uninformed demand shocks in he underlying shares. Measuring pairs rading profis represens a succinc way o quanify he coss of liquidiy provision (i.e., he coss of keeping relaive prices in line.) Keywords: Asse pricing, reurn predicabiliy, limis of arbirage JEL number: G12 1
3 1 Inroducion This paper sudies he profiabiliy of pair rading. Since he mid1980s such sraegies have consiued a raher secreive and lucraive par of Wall Sree aciviy. Pairs rading is a ype of relaive value sraegy ha buys an overpriced securiy and simulaneously sells a similar, underpriced securiy. Traders ypically rack a pair of securiies whose prices move ogeher. When prices diverge, hey buy he down sock and simulaneously sell he up sock. Traders profi if prices converge bu lose money if prices diverge furher. Pairs rading has generaed hundreds of millions of dollars in profis for companies such as Morgan Sanley and D.E. Shaw. Sudying pairs rading broadens our undersanding of financial markes. Because pairs rading enails risk aking, one can hink of our paper as speaking o he limis of arbirage in acual financial markes. Profis need no be hough of as coming from a narrow Wall S. sraegy. Raher, readers can hink of hese arbirageurs as playing a vial role in he relaive pricing of securiies. Profis are compensaion for performing his service. Equivalenly, readers can hink of profis as compensaion for providing liquidiy during imes of differenial marke sress (e.g., sresses ha affec some socks bu no ohers.) A goal of his paper is o undersand why prices of similar securiies diverge. Since pairs rading sraegies are currenly being employed in sock markes around he world, we choose his framework for sudying price divergence. Measuring pairs rading profis represens a succinc way o quanify he coss of liquidiy provision (i.e., he coss of keeping relaive prices in line.) Surprisingly, relaive value sraegies have received lile aenion in he academic lieraure. The mos noable paper is by Gaev, Goezmann, and Rouwenhors (2003) and offers a comprehensive analysis. 1 The auhors use daily US daa from 1962 o They show a simple pairs rading rule produces excess reurns of 11% per annum. Reurns have high riskadjused alphas, low exposure o known sources of sysemaic risk, cover reasonable ransacion coss, and do no come from shorerm reurn reversals as documened in Lehmann (1990). GGR (2003) inerpre pairs rading profis as poining owards a sysemaic dorman facor relaing o he agency coss of professional arbirage. We conjecure ha uninformed rading shocks can explain he profiabiliy of pairs rading. We would like o carry ou our sudy wih CRSP daa. Unforunaely, one canno idenify aggregae uninformed shocks on he NYSE due o order rouing decisions. For example, one migh hink ha aggregae individual invesor order flow is a good proxy for uninformed demand. However, 1 As of 2Feb2005, he paper has been downloaded 7,533 imes from SSRN. Hereafer we refer o i as GGR (2003). Oher work on pairs rading includes Richards (1999) and Nah (2003). 2
4 brokers filer such orders so ha uninformed orders are likely o go o regional exchanges while oher orders ge sen o he NYSE. Luckily, insiuional feaures of he Taiwan Sock Exchange allow us o idenify a large pool of uninformed buys and sells. Our conribuion o he lieraure is wofold. Firs, we provide an ouofsample es of he pairsrading sraegy described in GGR (2003). Using daily Taiwanese daa from 1994 o 2002, we find excess reurns of 10.18% per annum. The reurns are saisically significan a all convenional levels and annualized Sharpe raios are greaer han one. Furhermore, he reurns canno be explained by exposure o known sources of sysemaic risk. Therefore, our resuls provide addiional suppor regarding he profiabiliy of relaive value sraegies. Again, his profiabiliy can be hough of as compensaion for keeping prices in line. Second, and much more imporanly, we link uninformed rading shocks o he profiabiliy of pairsrading. Such a link is, o our knowledge, new. We show ha uninformed ne buying is significanly correlaed wih a pair s iniial price divergence. Addiionally, we show ha uninformed rading is a significan facor in explaining he sraegy s markedomarke reurns. These resuls sugges ha pairsrading sraegies are profiable because hey idenify siuaions wih emporary price pressure. The sraegy has low risk because a posiion is effecively hedged by an offseing posiion wih similar facor loadings. Execuion is simplified and coss kep o a minimum because he offseing posiion is limied o a single sock. 2 Pairs Trading in Taiwan We collec daily sock prices, reurns, and share informaion a he individual sock level. Daa conain all lised socks on he Taiwan Sock Exchange. Daa are available from a number of vendors including he Taiwan Economic Journal (he TEJ.) The TEJ adjuss daily reurns for capial changes. The full sample conains informaion on 647 differen lised companies in Taiwan. Our sample period begins 5Jan1994 and ends 29Aug Thus, he sample consiss of 2,360 holding/rading days. We follow he same pairs rading sraegy described in GGR (2003). By doing so, our resuls remain free from daa snooping biases, and consiue an ouofsample es of heir resuls. The sraegy calls for observing sock price movemens over a one year formaion period. 2 A he end of he formaion period, pairs of socks are ranked based on comovemen or closeness measures. The wo socks wih he highes degree of comovemen are called he firs pair, he nex wo 2 Throughou his paper, we define one year o be 250 rading days and half a year o be 125 rading days. These numbers remain consan during our enire sudy. 3
5 socks are called he second pair, and so on. In a marke wih 500 lised socks, one mus rank 124,750 differen pair combinaions. We refer o he wo socks in a pair as Sock A and Sock B. Designaions A and B are arbirary since he sraegy can laer buy or sell eiher sock. Afer forming he weny closes pairs, we engage in a six monh rading period. The rading sraegy consiss of hree basic rules: 1) follow he a pair of socks unil prices diverge by a cerain amoun called he rigger value ; 2) a he ime prices diverge, sell he up sock in a pair and buy he down sock in he same pair; and 3) wai unil prices reconverge o closeou a posiion. A he end of he six monh rading period all open posiions are closed ou (possibly a a loss.) A deailed descripion of he mehodology is given in Appendix A. During he rading period a pair of socks can be in one of hree saes: i) no open; ii) shor A and long B; or iii) long A and shor B. We define a risae indicaor funcion o reflec hese hree possibiliies: I AB 0 no open +1 shor A; long B 1 long A; shor B (1) The excess reurn o a given pair posiion (consising of Sock A and Sock B) is: r AB = I AB (r B r A ) (2) Noe ha a pair s excess reurn is zero whenever he posiion is closed since I AB = 0 as indicaed in Equaions (1) and (2). Over one day he porfolio excess reurn o following weny pairs is: r por = pair=1 r AB,pair (3) We repea he sraegy every half year during our sample and end wih a series of nonoverlapping reurns (reurns only come from he six monh rading periods.) Figure 1 shows he iming of he formaion and rading periods. We have sixeen rading periods in our sudy for a oal of 2,000 rading days. Table 1, Panel A gives descripive saisics of he daa and iming of our pairs rading sraegy. 2.1 A pairs rading example We provide an example of he pairs rading sraegy. Figure 2 shows he normalized prices for wo socks during he formaion period. Readers who are skepical ha we picked a paricularly 4
6 nice picure can res assured. Figure 2 shows he firs pair from he firs formaion period. The normalized price series end o move ogeher and boh socks los approximaely 10% during his paricular formaion period. Over he rading period, he socks coninue o move ogeher mos of he ime see Figure 3, Panel A. 3 However, heir normalized price series do diverge by more han he rigger value on four occasions see Figure 3, Panel B. Noice ha he firs wo imes prices diverge, Sock B price is above Sock A price ( I AB = 1 ). A pairs rading sraegy calls for buying Sock A and selling Sock B. The second wo imes prices diverge, Sock A price is above Sock B price ( I AB = +1 ). Therefore, he sraegy akes he opposie posiions. Finally, noice ha he fourh posiion is no closed due o normalized prices reconverging, bu raher by he rading period ending. Being lef wih an open posiion is a risk ha finiehorizon arbirageurs face. Figure 3, Panel C shows he cumulaive profi associaed wih his paricular pair. There are fla (no profi) periods when he pair s posiion is no open (i.e., when I AB = 0.) The sraegy for his paricular pair during his paricular rading period earns an excess reurn of 22.29% over sixmonhs. 2.2 Overview saisics Table 1 gives overview saisics of our pairs rading sraegy. In Panel B we see ha posiions are open 70.34% of he possible sockdays. We follow 20 pairs for 16 halfyear periods, of which only five pairs never open. The average pair opens 2.29 imes during is halfyear rading period. Panel C shows a single pair earns bp per day on average. 4 Since our porfolio is simply an equally weighed average of he weny consiuen pairs, bp is he daily porfolio reurn as well. The Sharpe raio of our sraegy is much higher han he realized Sharpe raio of he Taiwanese sock marke over he same period. 2.3 Profiabiliy and riskadjused reurns We show he pairs rading profis in Taiwan are no coming from exposure o known sources of sysemaic risk. We regress he ime series of pairs rading profis on excess marke reurns, Fama French facors, and a momenum facor. ( r por = α + β 1 r mk r f ) + β 2 (SMB ) + β 3 (HML ) + β 4 (MOM ) + ε p, (4) 3 This example coninues o consider he firs pair of socks from he firs rading period. 4 Each day a posiion is acually open i earns 5.515bp on average ( = 5.515). 5
7 Table 2, Regression 1 shows he pairs rading porfolio earns 3.876bp per day or 10.18% annualized reurns. The risk adjused reurns in Regressions 2 and 3 range beween 2.612bp (6.75% per annum) o 3.886bp (10.20% per annum). While he reurns load significanly on he marke and HML in Regression 3, he magniudes are economically insignifican. The insignifican loading on MOM is comforing because one migh worry ha slow price adjusmen in a marke like Taiwan is driving our resuls. Regression 3 shows his is no he case. Our resuls from Taiwan compare very favorably o resuls in GGR (2003) from he Unied Saes. We find annualized excess reurns of 10.18% and hey find 11.28% annualized excess reurns. They repor an average of pairs raded (in he op 20.) We show ha five pairs never open which equals an average of pairs raded. They repor an average of 1.96 roundrips per pair; we repor 2.29 roundrips per pair. 5 A his poin, i is logical o address he numerous quesions a reader migh have regarding he reurns o he pairs rading sraegy. Given ha our resuls are so similar o hose in GGR (2003) we refer readers o he earlier work. GGR (2003) provides a very horough invesigaion ino possible explanaions of pairs rading profis. The auhors deail he valuearisk of he rading sraegy, he reurns of randomly mached pairs, indusry effecs, a breakdown of long/shor componens, and he effec of shor selling coss. 6 Indusry praciioners already engage in pairs rading in Taiwan. This fac provides addiional suppor of he sraegy s economic significance and viabiliy. A Merrill Lynch repor by Chang e. al. (2001) oulines a pairs rading sraegy ha is very similar o ours. 7 We now urn o explaining he reurns o a pairs rading sraegy wih uninformed rading shocks. 5 Sligh differences when looking a loadings on he marke and FamaFrench facors can be aribued o sudying differen markes. GGR (2003) show a negaive and insignifican marke bea. Our marke bea of o is economically small bu saisically significan. Reurns in GGR (2003) load negaively on a momenum facor while our reurns do no. 6 Following GGR (2003) we employ a boosrap procedure o make sure our profis are no he resul of shorerm reversals sudied in Lehmann (1990). Our boosrap resuls are very close o hose repored in GGR (2003) and available upon reques. 7 Their back esing sraegy opens posiions a a wo sandard deviaion gap and closes hem when prices reconverge o a one sandard deviaion gap. Transacion coss in our sraegy would be even less han ransacion coss in he Merrill Lynch sraegy. 6
8 3 Uninformed demand shocks and pairsrading We hypohesize ha uninformed rading shocks help explain he reurns of relaive value rading sraegies. As in radiional asse pricing models, assume ha sock reurns are deermined by loadings on risk facors plus an idiosyncraic componen. Given a long enough observaion period, wo socks ha have hisorically moved ogeher can be hough of as having similar facor loadings. Assuming facor loadings remain consan in he fuure, he wo sock reurns should coninue o move ogeher. In realiy, fuure sock reurns may no always move ogeher. Divergence in prices may come from idiosyncraic shocks. 8 Suppose we rack a pair of mining companies. Furher suppose ha one firm receives posiive informaion such as finding a new mineral deposi. We expec he sock price of he lucky firm o jump ahead of he sock price of he oher firm. Reurns in he fuure may coninue o move ogeher, bu we expec he price difference o persis. On average, we expec idiosyncraic shocks based on firmspecific informaion o lead o persisen price differences. A change in facor loadings can also cause prices o diverge. Suppose one of he mining companies invess heavily in an inerne sarup company. Such a change may cause prices o diverge oday. In conras o he case of idiosyncraic shock discussed above, we do no expec he price differenial o remain consan in he fuure (even if i is zero oday). Fuure reurns should cease moving ogeher whenever facor loadings change. Now consider a marke wih limied risk bearing capaciy. 9 Uninformed raders place demands ha are uncorrelaed wih asse fundamenals. Opimizing invesors, who are risk averse, accommodae he demands bu require compensaion. Thus, uninformed buying is accompanied by a conemporaneous rise in prices. Following a demand shock, prices meanrever back o preshock (fundamenal) levels. Likewise, uninformed selling is accompanied by a conemporaneous fall in prices and similar meanreversion. In a world wih limied riskbearing capaciy, a pairs rading sraegy effecively maches socks wih similar (hisorical) facor loadings. When prices diverge, he sraegy akes a risky posiion by being he divergence sems from differen uninformed rading shocks 8 Sock prices are normalized a he sar of all observaion periods so ha divergence in prices becomes a meaningful concep. We use he erm price o mean normalized price hroughou his paper. Appendix A gives a full overview of he pairs rading mehodology. 9 Examples include DeLong e. al (1990); Campbell, Grossman, and Wang (1993); Greenwood (2004). 7
9 and no differen informaional shocks. 10 In models such as Greenwood (2004), prices meanrever back owards fundamenals in a linear fashion. The divergence and subsequen meanreversion leads o hree esable hypoheses. The firs is ha differenial demand shocks are correlaed wih iniial price divergence (i.e., he opening of an arbirageur s posiion) The second hypohesis is ha, in he absence of any ype of fuure shock, he fuure reurns o a pair s rading sraegy should be uncorrelaed wih all oher facors in he economy. The zero correlaion comes from he fac ha he socks are good hedges for each oher and prices meanrever smoohly back ogeher. The hird hypohesis applies o a world where fuure shocks may very well affec he underlying socks. If his is he case, he markedomarke reurns of a pair s rading sraegy should be correlaed wih he correcly signed shocks of he underlying socks. Posiive (negaive) shocks o he higher (lower) priced sock cause prices o diverge farher apar han hey already are. An increased divergence makes oday s markedomarke profis negaive. Negaive (posiive) shocks o he higher (lower) priced sock cause prices o meanrever more quickly han normal hus increasing oday s markedomarke profis. 3.1 Uninformed rading daa We use F A o denoe he ne uninformed rading in sock of Company A on day. Readers can hink of F A as represening he flow of money from he uninformed raders ino, or ou of, Company A s sock. Uninformed raders are ne buyers whenever F A is posiive. Uninformed raders are ne sellers whenever F A is negaive. To calculae F A we collec daily holdings daa for each lised firm. The holdings daa cover 608 lised socks beween 5Jan1994 and 29Aug We follow Andrade, Chang, and Seasholes (2004) and define F A as he daily change in he aggregae ne shares held long on margin divided by oal shares ousanding. Normalizing by oal shares ousanding provides a sraighforward mehod o compare F across socks. Suppor for he idenificaion of uninformed shocks is given in Andrade, Chang, and Seasholes (2004). For example, he ne rades come almos exclusively from individual invesors. The rades underperform he marke, are noisy, have relaively low pairwise correlaion 10 Monioring news sories can help an arbirageur idenify siuaions when prices diverge due an informaional even. Thus, using news as well as prices should lead o higher reurns han using prices alone. In his way, he pairs rading profis shown in his paper represen a lower bound of wha Wall S. firms acually earn. 8
10 across socks, and are direcly ied o emporary price shocks. Andrade, Chang, and Seasholes (2004) provide hree levels of analysis. All hree are consisen wih using F as a proxy for uninformed rading. The paper also provides full saisics for F which represens a sizable fracion of daily rading in Taiwan Uninformed rading and iniiaing posiions We consider each of he 732 posiions opened during our sample period and es if uninformed rading shocks are correlaed wih rading posiions being opened. In oher words, we ask if uninformed rading shocks are linked wih iniial divergence in prices? Figure 3, Panel B provides a nice reference poin. We see ha four posiions are opened during his paricular 125 day rading period. On hese four days, we wan o know if ne rading is correlaed wih sock reurns. Table 3 displays he average reurn on he day posiions open (i.e., when divergence reaches he rigger value.) 12. We see on days posiions open, socks diverge by 4.15% on average. Such a resul indicaes he rigger value is no reached in a slow and smooh manner. Raher, large, differenial shocks cause a pair o hi is rigger value. 13 The rising sock ( Up Sock ) in a pair accouns for 71.08% of he divergence on average (i.e., 71.08% = 2.95% 4.15%), while he falling sock ( Down Sock ) accouns for he remaining 28.92%. Table 3 shows he uninformed buying differenial is 11.15bp ( Up Down ). There is clear and srong uninformed buying of he Up Sock. Uninformed invesors in our sample buy 11.36bp of oal marke capializaion on he day a posiion is opened. On average, here is neiher buying nor selling of he sock ha goes down (0.21bp is no significanly differen from zero.) We see an average correlaion of beween reurns and ne uninformed rading in he Up Sock. 14 he Down Socks. Ineresingly, he high correlaion remains when looking a The resuls in Table 3 can be explained wih sraighforward economic inuiion. If here is 11 Readers who are no enirely comforable wih he idenificaion are welcome o associae he pairs rading profis wih shocks o long margin holdings raher han uninformed rading. We are careful o check ha facors such as ineres rae movemens are no driving our resuls. 12 Afer he posiion opens, he sraegy calls for selling he up sock and buying he down sock. 13 The value of 4.15% should no be greaer han he average rigger value of 6.24% from Table 1 since 4.15% represens one day s divergence and 6.24% is a cumulaive divergence needed o open a posiion. 14 Tsaisics are calculaed using he following esimaion of he sandard error: s.e. = ρ T 2 1 ρ 2 number of observaions. where T is he 9
11 large uninformed selling pressure, any ouside invesor can easily sep in o buy he excess supply. However, large uninformed buying pressure creaes a differen siuaion. No all invesors own he sock so no all invesor can sep in o mee he excess demand. Arbirageurs who do no currenly own he sock mus sell shor. To he exen ha shor selling is only available o sophisicaed invesors, he abiliy of he marke o provide liquidiy appears limied Survival analysis and iniiaing posiions A more precise way o gauge which facors affec posiion openings is hrough survival analysis. We measure he amoun of ime (in days) from he sar of each rading period o he ime a pair firs opens. We concenrae on ime unil firs opening because we know every pair of socks sars he rading period wih boh prices normalized o one. There are 320 differen rading periodpair combinaions of which 315 open a leas once before he end of he rading period. There are five pairs ha never open and we keep hese in he sample. Figure 4 shows he disribuion of a pair s ime o firs opening measured in days. A fied gamma disribuion is shown as well. The average ime o firs opening is 20.5 days. The figure clearly shows he five pairs ha never open during heir 125day rading periods. Table 4 presens he resuls of he hazard analysis and repors coefficiens. Esimaion is by maximum likelihood and he baseline hazard funcion is parameerized as a generalized gamma funcion. A negaive coefficien indicaes pairs open more quickly as he covariae (RHS variable) goes up. 16 In Regression 1, a negaive coefficien for F up indicaes pairs open more quickly when he covariae (RHS variable) is high. Shocks o F down have an insignifican effec. Since normalized prices are he cumulaive reurns from he sar of he rading period, we also include cumulaive values of F up and F down. Regression 2 shows ha cumulaive shocks are insignifican and only conemporaneous shocks maer. During periods of high marke volailiy pairs open more quickly as can be seen from he negaive coefficien on 15 Noe he GGR (2003) are careful o check he pairs rading profis exceed all ransacion coss including shor selling coss. 16 The coefficiens may iniially cause confusion. In hese regression, he subjecs saus is alive as long as he pair remains closed. As soon as he rigger value is hi, he saus swiches o dies. A negaive coefficien can be hough of as a decrease in expeced life. Noice ha our specificaion includes imevarying covariaes. 10
12 ( r mk r f ) 2 in Regression 3 and Regression 4. To assess he economic significance of he survival analysis we noe ha a wo sandard deviaion shock o F up increases he probabiliy of opening a posiion by approximaely 50% from he baseline case. 17 A wo sandard deviaion shock o marke volailiy (he only oher significan covariae) increases he probabiliy of a posiion being opened by 25% from he baseline case. We conclude ha ne uninformed buying has a saisically and economically significan impac on a pair becoming open. 3.4 Markedomarke reurns and uninformed rading In an effor o explain he markedomarke reurns, we regress he reurns of our pairs rading porfolio on excess marke reurns, FamaFrench facors, a momenum facor, and measures of uninformed rading: r por = α + X β + ε p, (5) Since he pairs rading sraegy can buy or sell Sock A (while doing he opposie wih Sock B) a any ime, here is no reason o hink a measure of aggregae marke buying or selling can explain reurns. We es his by including a markewide measure of uninformed rading on he righ hand side of Equaion (5): F mk = sk=1 F sk (6) We also calculae a measure of he uninformed rading ha is specific o our porfolio called F por which we include on he righ hand side of Equaion (5). This is a signed measure ha depends on wheher posiions are open and long Sock A, open and shor Sock A, or no open a all: F por = pair=1 ( I AB,pair F B,pair ) F A,pair Table 5, Regression 1 shows he markewide measure of uninformed rading is significan when i s he only righhand side variable. Regression 2 shows he porfoliospecific measure 17 The increase in probabiliy can be seen using σ(f up ) = and he coefficien repored in Table 4. Resuls can also be obained using approximae ime raios. A Probi analysis gives qualiaively similar resuls. (7) 11
13 is far more significan. The coefficien on F por is wih a saisic. In Regression 3, he porfolio measure remains highly significan when sandard measures of risk are added o he regression. ( The markewide ) measure ceases o maer. This resul is no surprising as F mk and r mk r f have a correlaion coefficien. I urns ou he par of F mk ha is orhogonal o excess marke reurns is no significan eiher (resuls no shown.) This high level of significance for F por remains when lags of iself and oher risk facors are included in Regression 4. Ineresingly, he negaive coefficiens on he lagged variables ( ) F por 1 and F por 2 also suppor our hypohesis ha pairs rading profis are direcly linked o uninformed rading. To see his, consider a porfolio ha has only one open posiion. The porfolio is long Sock B and shor Sock A (i.e., I AB,pair = +1). In his case, a posiive shock o F por indicaes ha eiher F B is posiive or F A negaive. Since uninformed rading shocks are posiively correlaed wih reurns, his means he relaive prices converge and he markedomarke profi is posiive. Conversely, a negaive shock o F por indicaes a furher widening of prices and a markedomarke loss. When considering lagged shocks, he opposie siuaion applies. A negaive shock yeserday widened prices. A widening of prices leads o higher average profis in he fuure (which includes oday). The same can be said for negaive shocks wo days ago as well. The higher average profis in days following an adverse shock can be seen in he negaive coefficiens in Table 5, Regression 4. We end wih a final noe on inerpreing he resuls in Table 5. The F por variable is no a sysemaic risk facor herefore he consan canno be inerpreed as an alpha (i.e., a measure of abnormal reurns.) This also implies ha we do no necessarily expec he consan value o go o zero even if F por has large explanaory power (which i does.) The purpose of he regression is simply o show ha he markedomarke riskadjused reurns are highly correlaed wih uninformed rading (F por ). I is, however, imporan o noe ha he adjused R 2 in Table 5, Regression 4 is approximaely hree imes higher han any of he adjused R 2 in Table 2. 4 Conclusion Our paper seeks o undersand he profiabiliy pairs rading (a ype of relaive value sraegy.) We show ha such sraegies work in Taiwan as hey do in he Unied Saes. Average profis and rading frequency are nearly idenical across he wo counries. The hrus of his paper is he conjecure ha pairs rading profis are compensaion for providing liquidiy in 12
14 markes wih limied risk bearing capaciy. Specifically, liquidiy is demanded by uniformed raders and his demand is observed as emporary pressure in sock prices. We find ha iniial price divergence (he opening of a pairs rading posiion) is highly correlaed wih uninformed shocks o he underlying socks. A survival analysis shows ha posiions open more quickly when one of he socks experiences large uniformed buying. The markedomarke reurns of a pairs sraegy are highly correlaed wih he correcly signed shocks o he underlying socks. These resuls holds for boh raw and riskadjused reurns. Taking ino accoun uninformed rading increases adjused R 2 by a facor of hree. Our resuls provide addiional suppor for heoreical models wih limis o arbirage. In such models, markes have limied riskbearing capaciy arising from agency consideraions and specializaion. As a resul, noninformaional shocks have an economically and saisically significan impac on asse prices. The profis o he pair s rading sraegy represen anoher way o quanify he economic impac of he limied riskbearing capaciy. Relaed fuure research is almos limiless. One could correlae he magniude of pairs rading profis wih insiuional deails of markes around he world or in a single marke over ime. Markes in which he underlying securiies are more difficul o price should give rise o greaer pairs rading profis. Markes wih more pervasive uninformed shocks should generae greaer pairs rading profis (or similarly sized profis spread across more arbirageurs.) Our paper sudies he shocks ha cause a pair s underlying socks o diverge. An equally ineresing paper could rack he acual rades of arbirageurs (liquidiy providers) who ake he oher side of he ransacions. 13
15 References [1] Andrade, Sandro C., Charles Chang, and Mark S. Seasholes, 2004, Uninformed Trading and Asse Prices, Working paper, U.C. Berkeley. [2] Campbell, John Y., Sanford J. Grossman, and Jiang Wang, 1993, Trading Volume and Serial Correlaion in Sock Reurns, Quarerly Journal of Economics, 108, 4, Nov., [3] Chang, Ken, Denise Hu, Todd Kennedy, and Russell Cummer, 2001, Saisical Arbirage: Daily Repor on Pair Trades for he AsiaPacific Region, Merrill Lynch, 25 Sepember. [4] DeLong, Bradford, Andrei Shleifer, Lawrence Summers and Rober Waldman, 1990b, Noise Trader Risk in Financial Markes, Journal of Poliical Economy, 98(4), [5] Gaev, Evan, William N. Goezmann, and K. Geer Rouwenhors, 2003, Pairs Trading: Performance of a Relaive Value Arbirage Rule, Working paper, Yale Universiy. [6] Greenwood, Robin, 2004, Shor and LongTerm Demand Curves for Socks: Theory and Evidence on he Dynamics of Arbirage, Forhcoming Journal of Financial Economics. [7] Lehmann, Bruce N., 1990, Fads Maringales, and Marke Efficiency, Quarerly Journal of Economics, 105, 1, Feb, [8] Nah, Purnendu, 2003, High Frequency Pairs Trading wih U.S. Treasury Securiies: Risks and Rewards for Hedge Funds, Working Paper, London Business School. [9] Richards, Anhony J., 1999, Idiosyncraic Risk: An Empirical Analysis wih Implicaions for he Risk of RelaiveValue Trading Sraegies, Working Paper,IMF. [10] Spiro, Leah Nahans and Jeffrey M. Laderman, 1998, How Longerm Rocked Socks, Too, Business Week, November 9, p
16 Appendix A Background on Forming Pairs We follow he same pairs rading sraegy described in Gaev, Goezmann, and Rouwenhors (2003). We begin by defining a one year formaion period (or observaion period) during which we observe normalized sock prices. 18 Consider he sock of company A. Is normalized price begins he observaion period wih a value of one. Sock A s normalized price is hen increased or decreased each day by is daily reurn (compounded.) P A ( ) 1 + r A τ τ=1 (8) A he end of he oneyear formaion period, we calculae he ime series of normalized sock price deviaion for every pair of socks. In a marke wih 500 lised socks, his mehodology enails calculaing 124,750 series of deviaions. We rank pairs of socks from lowes o highes based on he sum of squared deviaions. A he end of each formaion period, we consider he weny closes pairs based on our Closeness AB measure. Here A represens one sock in he pair and B represens he oher. Order is unimporan since we can laer buy or sell eiher Sock A or Sock B: Closeness AB = 250 =1 ( P A P ) 2 B (9) Following he formaion period, we rack each of he weny pairs for he nex half year (125 rading days.) A he beginning of his so called rading period we again renormalize all prices o be one. We wai unil prices have diverged sufficienly before iniiaing a posiion. The rigger value which promps opening a posiion is based on wo sandard deviaions of hisorical price divergence (hisorical in his case means measured over he formaion period ha immediaely precedes he rading period.) T rigger AB = ±2 sdev ( P A P ) B (10) 18 Throughou his paper, we define one year o be 250 rading days and half a year o be 125 rading days. These numbers remain consan during our enire sudy. 15
17 Once a posiion is opened, he sraegy sells he higher priced sock (Up Sock) and buys he lower priced sock (Down Sock). We creae a risae indicaor variable o denoe he pair s posiion during each day of he rading period: I AB 0 no open +1 shor A; long B 1 long A; shor B The pair s normalized prices and reurns are markedomarke each day and he posiion is held open unil prices reconverge. If a posiion is sill open when he rading period ends, i is closed and a gain or loss is recorded. r AB = I AB (r ) B r A Over one rading period, he porfolio reurn o following weny pairs is: r por = pair=1 r AB,pair 16
18 Figure 1 Timing of Formaion and Trading Periods This figure shows he iming of formaion and rading periods. We define one year o be 250 rading days and half a year o be 125 rading days. The sample period sars 05Jan1994 and ends 29Aug2002 (equals 2,360 days.) We have 16 nonoverlapping rading periods, lose 250 days due o he firs formaion period, and have 110 days lef over a he end. Noe: 250+(16 125)+110 = 2,360. Daa are from he Taiwan Economic Journal. formaion period #1 rading period #1 formaion period #2 rading period #2 formaion period #3 rading period #3 formaion period #16 rading period #16 =0 =125 =250 =375 =500 =625 =1,875 =2,000 =2,125 =2,250 ime in days 17
19 Figure 2 Formaion Period Example This figure shows he normalized price series of wo socks during a formaion period. A normalized price series sars a and increases (decreases) by he sock s gross reurn compounded daily. This paricular graph shows he firs mached pair from he firs formaion period in our sample. The sample period sars 05Jan1994 and ends 29Aug Daa are from he Taiwan Economic Journal Sock A Sock B Normalized Price 18
20 Figure 3 Trading Period Example These figures deail he reurns o a relaive value rading sraegy. Panel A shows he normalized price series of a single pair of socks. A normalized price series sars a and increases (decreases) by he sock s gross reurn compounded daily. Panel B shows when, and for how long, posiions remain open. Panel C shows he cumulaive reurn o his pairs rading sraegy. All hree figures are based on he firs mached pair from he firs formaion period in our sample. The sample period sars 05Jan and ends 29Aug Daa are from he Taiwan Economic Journal Panel A: Normalized Price Series 1.20 Normalized Price Sock A Sock B Panel B: Posiion Series Inidicaor of Pair's Posiion Sock A: shor Sock B: long Sock A: long Sock B: shor Panel C: Cumulaive Reurns 30.0% 20.0% Cumulaive Reurn 10.0% 0.0% % 19
21 Figure 4 Time o Firs Opening This figure shows how many days i akes a pair of socks o firs open afer he sar of a rading period. Equivalenly, we can view he graph as he amoun of ime a pair remains closed. A pair opens when prices diverge by wo or more hisorical sandard deviaions. Deails relaing o forming pairs are given in he ex. A fied gamma funcion is also shown along wih he empirical disribuion. The sample period sars 05Jan1994 and ends 29Aug Daa are from he Taiwan Economic Journal. 18.0% 16.0% 14.0% Empirical Disribuion 12.0% Fied Gamma Disribuion 10.0% PDF 8.0% 6.0% 4.0% 2.0% 0.0% >125 Days afer Formaion 20
22 Table 1 Descripive Saisics This able gives descripive saisics of our relaive value (pairs rading) sraegy. The sample period sars 05Jan1994 and ends 29Aug Daa are from he Taiwan Economic Journal. Panel A: Daa Descripion Toal days in sample 2,360 Days in each formaion period 250 Days in each rading period 125 Number of rading periods in sample 16 Toal rading days in sample (16*20=) 2,000 Days los due o iniial formaion period 250 Days los a end of sample (unused daa) 110 Check of oal days ( = ) 2,360 Panel B: Descripion of Pairs Trading Sraegy Number of pairs during one rading period 20 Max number of open pairdays (20 * 16 * 125 =) 40,000 Acual number of open pairdays 28,135 Fracion of ime posiions are open 70.34% Number of pairposiions opened during sudy (openings) 732 Number of pairs ha never open 5 Average rigger value ( 2σ ) Average number of days a posiion is open (28,135 / 732) 38.4 days Average number of posiions opened during one rading period (732 / 16=) Average number of posiions opened for one pair during one rading period (45.75 / 20 =) 2.29 Panel C: Overview of Pairs Trading Profis Average daily reurn of one pair during rading period 3.879bp Average daily reurn of pairs rading porfolio 3.879bp Sdev of daily reurns bp Sharpe raio of reurns (daily) Average reurn of pairs rading porfolio 10.18% Sdev of reurn (annualized) 9.15% Sharpe raio of reurns (annualized)
23 Table 2 Profiabiliy and Pairs Trading This able shows resuls from regressions of reurns o a pairs rading porfolio on risk facors. The sample period sars 05Jan and ends 29Aug Daa are from he Taiwan Economic Journal. Tsaisics (shown in parenheses) are based on sandard errors ha are robus o heeroskedasiciy and auocorrelaion. Adjused Rsquared values from a sandard OLS regression shown a boom. por Dependen Variable: The Reurn o he Pairs Trading Porfolio ( r ) Reg 1 Reg 2 Reg 3 Consan bp bp bp (2.88) (2.92) (2.14) mk f r r (2.46) (4.00) SMB (1.83) HML (4.56) MOM (0.77) Adj Rsq N (days) 2,000 2,000 2,000 22
24 Table 3 Opening Posiions, Sock Reurns, and Ne Uninformed Trading This able shows sock reurns and ne uninformed rading on days pairs posiions are opened. A pair represens wo socks (labeled A and B ) ha have hisorically moved ogeher. Designaing one sock as A and he oher as B is no significan since he sraegy can buy or sell eiher sock. A rading posiion is opened when he sock prices separae by more han wo hisorical sandard deviaions (called he rigger value as defined in he ex.) On he day a pair s posiion is opened, he Up Sock price is above he Down Sock price and he difference is labeled Up  Down. Below, r is he daily reurn differenial or he reurn of one sock in he pair. F is ne uninformed rading differenial or ne uninformed rading in one sock. F is measured as a fracion of shares ousanding. The sample period sars 05Jan1994 and ends 29Aug Daa are from he Taiwan Economic Journal. Average r Average F corr ( r, F ) Up  Down 4.15% 11.15bp (sa) (46.32) (8.41) (3.19) Up Sock 2.95% 11.36bp (sa) (28.01) (9.06) (9.10) Down Sock 1.20% 0.21bp (sa) (12.96) (0.33) (7.15) N
25 Table 4 Survival Analysis of Time o Firs Opening This able repors coefficiens from a survival analysis as hey relae o a pair s ime o firs opening. F is ne uninformed rading in a pair s up or down sock. The sample period sars 05Jan1994 and ends 29Aug Daa are from he Taiwan Economic Journal. Tsaisics (shown in parenheses) are based on sandard errors ha are robus o heeroskedasiciy and allow for clusering by pairs. Reg 1 Reg 2 Reg 3 Reg 4 up F (4.24) (3.67) (4.35) (4.40) down F (0.42) (0.55) (0.27) (0.33) 1 up F τ = 0 τ 1 down F τ = 0 τ (0.15) (1.36) mk f ( r ) 2 r mk ( ) 2 (3.89) (3.46) F in bp (0.94) Num. of Pairs
26 Table 5 Uninformed Trading and Profiabiliy This able shows resuls from regressions of reurns o a pairs rading porfolio on risk facors and a measure of ne uninformed rading ( F ). Here, F mk is an equally weighed measure of ne rading across all firms. And, F por is a correcly signed measure of he ne rading for socks in he pairs rading porfolio. The sample period sars 05Jan1994 and ends 29Aug Daa are from he Taiwan Economic Journal. Tsaisics (shown in parenheses) are based on sandard errors ha are robus o heeroskedasiciy and auocorrelaion. Adjused Rsquared values from a sandard OLS regression shown a boom. por Dependen Variable: The Reurn o he Pairs Trading Porfolio ( r ) Reg 1 Reg 2 Reg 3 Reg 4 Consan bp bp bp bp (2.48) (2.64) (2.56) (2.45) mk F (2.23) (2.80) (0.09) por F (8.66) (9.16) (9.38) por F (1.55) por F (2.00) mk f r r (3.61) (4.34) SMB (2.15) (2.37) HML (4.51) (4.54) MOM (0.37) (0.45) Adj Rsq N (days) 2,000 2,000 2,000 2,000 25
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