STOCK MARKET SEASONALITY: AY OF THE WEEK EFFECT AN JANUARY EFFECT Lukas Mazal Supervsed by Javer Gardeazabal 008-009 Absrac Ths maser s hess uses a dummy varable approach and an exended dummy varable approach o es for he exsence of calendar effecs n he raes of reurn of common socks. I apples he exended dummy varable approach based on a facor model o reurns of 30 socks raded a he German Sock Exchange and he dummy varable approach o reurns of 8 world ndces. Furhermore, nvesgaes me perssence and evoluon of hese calendar effecs. Fnally, smulaes wo porfolo sraeges based on he Monday effec and he Sepember effec. By esmang a rollng dummy varable regresson hs hess provdes evdence confrmng ha he day of he week effec sared dsappearng n he second half of 990s. The smulaed porfolos are able o ouperform he buy and hold sraegy n all he egh ndces consdered. Ths means exsence of unexploed prof opporunes, whch serously undermnes he effcen marke hypohess.
Sock Marke Seasonaly: ay of he Week Effec and January Effec Acknowledgmens I would lke o hank my personal uor Mr. Javer Gardeazabal for hs suppor and helpful commens ha were nspraonal. He was always accommodang and wllng o dscuss economc and economerc ssues. I also would lke o hank o my frend José Anono Acosa Ramrez for hs fnancal and psychcal suppor and knd help over he enre academc year.
Sock Marke Seasonaly: ay of he Week Effec and January Effec Table of conens Sock Marke Seasonaly: ay of he Week Effec and January Effec... Absrac... Acknowledgmens... Table of conens... 3. Inroducon... 4. Leraure Revew... 5.. Effcen marke hypohess... 5.. Random walk hypohess... 6.3. Calendar effecs... 6.3.. The day of he week effec... 7.3.. The January effec... 7 3. Mehodology... 9 3.. ummy varable approach... 9 3.. Exended dummy varable approach... 0 3.3. Heeroskedascy... 3.4. aa se... 4. Resuls... 3 4.. Exended dummy varable approach: AX Index... 3 4.. Inernaonal evdence of he day of he week effec and he January effec... 6 4.3. Tme perssence of he day of he week effec... 0 4.4. Smulaed porfolo sraeges... 4 5. Concluson... 3 References... 3 3
Sock Marke Seasonaly: ay of he Week Effec and January Effec. Inroducon In recen years, here has been a prolferaon of emprcal sudes ndcang ha he dsrbuon of sock reurns vares by day of he week and monh of he year. The mos noable effecs are a negave reurn for Monday and a posve reurn for January. These calendar anomales are emprcal resuls ha are nconssen wh mananed heores of asse-prcng behavor. They ndcae eher marke neffcency or nadequaces n he underlyng asse-prcng models. Parcularly, hey cas doub on he wdely acceped effcen marke hypohess. These anomales nclude he January effec and he day of he week effec. Ths sudy underakes a furher nvesgaon of he day of he week effec and he January effec. The purpose of hs hess s o nvesgae hese effecs on major world sock markes by usng ndex daa as well as reurns of ndvdual socks. The research queson ha should be answered s: Is here sll seasonaly a world sock exchanges? The research objecves of hs hess are: o renvesgae he day of he week effec by usng an exended dummy varable approach for socks of he German Sock Exchange; o explore nernaonal presence of he day of he week effec and he January effec by usng a smple dummy varable approach; o nvesgae me perssence of he day of he week effec n relaon o he effcen marke hypohess; o examne a marke mng sraegy based on he seasonaly. The frs secon of hs hess presens a bref leraure revew descrbng he effcen marke hypohess and calendar effecs. The second par consders a mehodology used n hs hess. The hrd par of hs hess presens he emprcal resuls and he las secon ncludes concluson ha summarzes resuls and also comprses possble heorecal consequences of he calendar effecs. 4
Sock Marke Seasonaly: ay of he Week Effec and January Effec. Leraure Revew.. Effcen marke hypohess The faher of he effcen marke hypohess Eugene Fama (970) frs defned he erm effcen marke n hs groundbreakng sudy as a marke n whch prces always fully reflec avalable nformaon (p. 384). The effcen marke hypohess predcs ha secury prces follow a random walk and should be mpossble o predc fuure reurns based on publcly avalable nformaon. Ths means ha an effcen marke s one where all unexploed prof opporunes are elmnaed by arbrage. Sharpe e al. (999), summarzes he effcen marke concep sang ha he effcen marke s a marke n whch he secury s prce wll be a good esmae of s nvesmen, where he nvesmen s he presen of he secury s fuure prospecs, as esmaed by well-nformed and sklful analyss who use he nformaon ha s currenly a hand (Sharpe e al., 999, p. 93). The logcal consequence of hs defnon means ha Marke s effcen wh respec o a parcular se of nformaon f s mpossble o make abnormal profs (oher han by chance) by usng hs se of nformaon o formulae buyng and sellng decsons. (Sharpe e al., 999, p. 93). Sharpe e al. (999) dsngushes hree forms of effcency based on a se of nformaon ha s avalable o nvesors. Table provdes hese hree forms of effcency. Form of effcency Weak Semsrong Srong (Sharpe e al., 999, p. 93) Table. Se of Informaon Refleced n Secury Prces Prevous prces of secures All publcly avalable nformaon All nformaon, boh publc and prvae Fama (970) provdes exac formulaon of he good esmae of he nvesmen : Ε ( p j Φ ) [ Ε( rj, Φ )] p j, where E s expeced operaor; p j s a prce of secury j a me ; p j, s s prce a (wh renvesmen of any nermedae cash ncome form he secury); r j, s one perod-perod percenage reurn ( p j, p j, ) p j, ; Φ s a general symbol for whaever se of nformaon s assumed o be "fully refleced" n he prce a. (Fama, 970, p. 384). Ths expresson saes ha he expeced prce of j-h secury n me equals he expeced rae of fuure reurn condonal on he se of nformaon avalable a me mulpled by he prce of j-h secury a me. Furhermore, Fama (970) suggess o x p Ε p Φ j, j, j,. measure undervaluaon or overvaluaon of secury as ( ) Consequenly ( x Φ ) 0 Ε j,, hs means ha any expeced undervaluaon or overvaluaon s equal o zero. In oher words, n average share prces are d correcly. Sraeges based on 5
Sock Marke Seasonaly: ay of he Week Effec and January Effec he nformaon se Φ canno acheve hgher expeced reurns han are equlbrum expeced reurns. There s no known sraegy, whch could connually ouperform sock marke averages, and all excess reurns acheved by nvesors are only by chance... Random walk hypohess The random walk hypohess s closely conneced wh he effcen marke hypohess. Ths hypohess saes ha socks move randomly, because he sock markes are effcen. Thus, he random walk hypohess s a drec consequence of he effcen marke hypohess. The random walk hypohess was nroduced by Kendall (953) and was laer confrmed by Fama (965). The erm random walk was furher popularzed by he 973 book, A Random Walk own Wall Sree (Malkel, 973). Waler Enders (004) defnes random walk as a cumulave sum of a whe nose process. Whereas whe nose s a sequence of random varables ε } such ha E( ε ) E( ε )... 0 ; E ( ε ) E( ε )... σ and E( ε ε s ) E( ε jε j s ) 0 { for all j and s, consequenly he random walk s defned as p ε, where p ln P. However, s generally acceped ha sock marke reurns do no have a zero mean and are heeroskedasc. Therefore, he me pah of sock prces s more appropraely specfed by a random walk plus drf model, where { ε } s heeroskedasc E( ε ) σ. Ths model can be defned as p a ε or afer akng frs dfferences p a ε. Under he random walk hypohess, here s no seasonaly n sock prces, because he sock prces are compleely random. Le us have a model reang any knd of seasonaly by usng dummy varables R α... k k ε. If he random walk hypohess holds, any such model mus have all he parameers referrng o he seasonaly equal o zero. The only non-zero parameer should be he consan erm, whch s he drf..3. Calendar effecs Snce he effcen marke hypohess was nroduced, a grea deal of research was devoed o nvesgang he effcency of capal markes. Snce hen, all knds of calendar anomales n sock marke reurns have been documened exensvely n he fnance leraure. The mos common calendar anomales are he January effec and he day of he week effec. Showng ha marke reurns follow a seasonal paern volaes he assumpon of weak marke effcency n ha by observng he pas developmen of reurns marke parcpans can lead o exraordnary profs. The calendar effecs should be shor lasng, as marke parcpans can learn from pas experence. Hence, f a calendar effec exss, radng based on explong a calendar paern of reurns should yeld exraordnary profs a leas for a shor me. Ye, such radng sraeges affec he marke n ha furher profs should no be possble: he calendar effec should dsappear. 6
Sock Marke Seasonaly: ay of he Week Effec and January Effec.3.. The day of he week effec espe he effcen marke hypohess and s consequences, here s a large body of leraure on he day of he week effec of sock reurns. Ths effec refers o a phenomenon ha he average reurn on Monday s sgnfcanly lower han he average reurn for he oher days. Mos sudes dealng wh hs anomaly use a smple dummy varable approach based on a lnear regresson wh 5 dummy varables referrng o he days of he week. French (980) frs observes negave average daly share reurns for Monday n he perod 953-977 on he New York Sock Exchange. Then, Gbbson and Hess (98) provde furher evdence of negave reurn for Monday. Smlarly, Kao (990) provdes evdence of he day of he week effec from he Tokyo Sock Exchange from he perod 974-987. He concludes ha low Tuesday and hgh Wednesday reurns are observed for he close-o-close reurns. (Kao, 990, p. 04). Wang, L and Erckson (997) confrm he day of he week effec n he perod 96-993 by usng hree ndces from he New York Sock Exchange, Amercan Sock Exchange and NASAQ. They observe he Monday effec n he 96-993 perods. Lakonshok and Smd (988) explore 90 years of he ow Jones Indusral Average Index n he perod 897-986. They observe perssen seasonaly n ow Jones Indusral Average Index reurns confrmng he-day-of-he-week effec wh Monday s average reurns -0.4%. Smlarly, Kem and Sambaugh (983) denfy he Monday effec n S&P Compose Index reurns form 98 o 98. ubos and Louvry (995) provde probably he mos comprehensve sudy confrmng he day-of-he-week effec. They examned eleven ndexes form nne counres n he perod 969-99. They conclude, We fnd negave reurns on Monday, whch are compensaed by abnormal posve reurns on Wednesday. (ubos and Louvry, 995, p. 9). Smlarly, Condoyann e al. (987) used seven ndces from seven counres; ow Jones Indusral Average Index, Ausralan Sock Exchange s All Ordnares Share Prce Index, Torono Compose Index, Pars CAC Indusral, F.T. All-Share (Uned Kngdom), Tokyo New Sock Exchange Index and he Sras Tmes Index (Sngapore) n he perod January 969 o ecember 984. They observed, negave mean weekend reurns do appear o be he norm raher han he excepon n a range of capal markes around he world (p. 74). Mos recenly, Gardeazabal and Regulez (004) reach a smlar concluson. They sugges usng he so called Exended ummy Varable Approach (EVA) derved from he Fama-French hree-facor model (Fama and French, 993). Gardeazabal and Regulez (004) used he daa from he Spansh sock marke. Ther daa se ranges from 998 o 000. They conclude ha by usng he EVA approach hey had found even sronger seasonal effecs han by usng he smple VA..3.. The January effec One of he mos puzzlng emprcal fndngs s ha he sample dsrbuons of monhly sock reurns vary by he monh of he year. The January effec refers o he sgnfcanly hgher average share marke reurns n January when compared wh oher monhs. The leraure on monhly effecs, generally, confrmed hgher reurns n January. Rozeff and Knney (976) frs observed ha he average reurn of an equal-weghed ndex of he New York Sock Exchange n January s sascally sgnfcanly hgher han he average reurn for he oher monhs n he perod 904-974. Haugen and Joron (996) provde evdence confrmng he perssen exsence of he January effec. They conclude ha he January effec sll exss despe he fac ha was well known for reasonably long me and herefore should have dsappeared. Furhermore, hey pon ou ha he January effec s sronger n case of small frms han n case of well-esablshed companes wh hgh capalzaon. They 7
Sock Marke Seasonaly: ay of he Week Effec and January Effec conclude, he January effec s sll gong srong 7 years afer s dscovery (Haugen and Joron, 996, p. 7). Inernaonal evdence of he January effec s provded by Kao and Schallhem (985). 8
Sock Marke Seasonaly: ay of he Week Effec and January Effec 3. Mehodology 3.. ummy varable approach The so called ummy Varable Approach o he sock marke seasonaly s based on esmang a smple regresson model, where each ndvdual dummy varable accouns for he excess reurn for he parcular day. Ths model can be wren n he followng way: R α 33 44 55 v, where R s he reurn of ndex (or secury) n perod, v s a zero-mean dsurbance, s a dummy varable for Monday (.e., f observaon falls on a Monday and 0 oherwse), s a dummy varable for Tuesday, ec. However, s no possble o esmae hs equaon n he above form, because conans all he dummy varables and he consan erm. Therefore, here s a problem of a perfec mulcolneary. In order o cope wh hs problem, vas majory of auhors uses elmnaon of he consan erm or one of he dummy varables from hs regresson equaon. Noneheless, proceedng n hs way solves he perfec mulcolneary, bu leaves he parameers of he model undenfed. Gardeazabal and Regulez (003) sugges mposng a resrcon on he parameers of he model o ackle hs problem. They sae ha hs way we can solve boh he exac mulcolneary problem and he problem of denfcaon of he parameers. Accordng o Gardeazabal and Regulez (003), f α s he mean reurn, mus T T T T hold ha 3 3 4 4 T 5 5 0. Gardeazabal T T T T T and Regulez (003) furher assume ha he proporon of he reurns for he ndvdual days equal /5, whch means E ) E( ) E( ) E( ) E( ) /5. Ths mples ha ( 3 4 5 3 4 5 0. Gardeazabal and Regulez (003, p. 6) sae ha, Ths resrcon says ha he coeffcens of he dummy varables add up o zero and can be nerpreed as a normalzng resrcon. By mposng hs resrcon we can solve boh he exac mulcolneary and denfcaon problems. If we solve for he dummy varable referrng o Thursday 4, we ge 4 ( 3 5 ). By subsung hs resul no ~ ~ ~ ~ he regresson equaon we wll ge R α 33 55 v, where ~. Now, as Gardeazabal and Regulez (003) sae, all he dummy varables are k k 4 expressed as devaons wh respec o he omed dummy 4 and each slope coeffcen k measures he devaon wh respec o he overall mean α. A conssen esmae of 4 can be recovered from he esmaes of he oher coeffcens n he followng. (Gardeazabal and Regulez, 003). The sandard error of he way ( ) 4 3 5 coeffcen can be obaned as ' Σ ˆ where refers o a 4 vecor of ones and Σˆ s a heeroskedascy-conssen esmaor of he covarance marx provded by Whe (980). The sascal nference can be conduced by usng he -es on each parameer. The model s analogous n case of he January effec. In hs case he basc regresson equaon can be defned as R α 0 0 3 3 v 4 4 5 5 6 6 7 7 8 8 9 9 9
Sock Marke Seasonaly: ay of he Week Effec and January Effec 0 where R s he reurn of ndex (or secury) n perod, v s a zero-mean dsurbance, s a dummy varable for January (.e., f observaon falls on a January and 0 oherwse), s a dummy varable for February, ec. We assume ha / ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 0 9 8 7 6 5 4 3 E E E E E E E E E E E E and we resrc he parameers such ha 0 0 9 8 7 6 5 4 3. If we solve for he dummy varable referrng o Augus 8, we ge ( ) 0 9 7 6 5 4 3 8. By subsung hs resul no he regresson equaon we wll ge v R 0 0 9 9 7 7 6 6 5 5 4 4 3 3 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ α where k k 8 ~. A conssen esmae of he 8 can be compued from he esmaes of he oher coeffcens by usng he followng expresson ( ) 0 9 7 6 5 4 3 8. Smlarly as for he day of he week effec, he sandard error of he coeffcen can be obaned as ˆ ' Σ where refers o a vecor of ones and Σˆ s a Whe s heeroskedascy-conssen esmaor of he covarance marx (Whe, 980). 3.. Exended dummy varable approach Accordng o Gardeazabal and Regulez (004), he dummy varable approach o seasonaly leaves oo much varably of sock reurns unexplaned. They sae ha he poron of unexplaned varably ncreases wh sample frequency and nference on daly seasonaly usually leads o weak or null evdence of seasonaly. They propose he so called Exended ummy Varable Approach whch leaves a lower fracon of sock reurns varably unexplaned. In general, her model can be defned n he followng way: J j j j u h R 5 5 4 4 3 3 β α, where h j s he j-h facor s orhogonal on he daly dummes. Ths approach s derved from he Fama-French hree-facor model (Fama and French, 993). The hree facors ha Fama and French (993) used are: he excess reurn on he marke porfolo, R EMR, he reurn on a porfolo of small frms mnus he reurn on a porfolo of large frms, R SMB, and he reurn on a porfolo of hgh book-omarke frms mnus he reurn on a porfolo of low book-o-marke frms, R HML. Takng no accoun only hese hree facors, he above menoned equaon can be wren as: HML HML SMB SMB EMR EMR u h h h R ˆ ˆ ˆ 5 5 4 4 3 3 β β β α, where HML SMB EMR h h h ˆ, ˆ, ˆ are defned as
Sock Marke Seasonaly: ay of he Week Effec and January Effec α hˆ, R EMR 33 44 55 RSMB 33 44 55 EMR α hˆ, R HML 33 44 55 SMB α hˆ. Gardeazabal and Regulez (004) sae ha he model based on he dummy varable approach s msspecfed n he sense ha leaves he rsk facors ou of he regresson equaon (p. 6). However, he OLS esmaor of he parameers s conssen, because he omed facors are orhogonal o he daly seasonals (p. 6). Furhermore, Gardeazabal and Regulez (004) pon ou ha he unexplaned par of sock reurns volaly s lower n case of he exended dummy varable approach, because holds ha J σ v σ u β > 0 j jσ j, where σ ( v E v ) and σ ( ) u E u. HML 3.3. Heeroskedascy I s generally acceped ha varance of sock marke reurns s no consan. Therefore, s necessary o use some heeroskedascy robus mehod. Whe (980) provdes a heeroskedascy-conssen esmaor of he covarance marx. Ths esmaor s defned n he followng way: T T ˆ Σ ( X X ) ( Σe )( ) ( ) ( Σ )( x x X X X X e x x X X ), where e refers o n n n n resduals from an OLS regresson (Greene, 00, p.99). In case of all he regressons esmaed n hs hess, he sascal nference s based on hs Whe heeroskedascyconssen esmaor (Whe, 980). 3.4. aa se The daa se used for he exended dummy varable approach esmaons ncludes daly marke reurns of 30 German companes raded a he German Sock Exchange and daly reurns of he euscher Aken Index (AX), Small-cap euscher Aken Index (SAX) and vdend euscher Aken Index ndex (vax). The daa se ranges from..006 o 3..008. The daa used for he dummy varable approach ncludes daly reurns of he followng 8 ndexes: Fnancal Tmes Index 00, Sandard & Poor's 500 Index, Sandard & Poor's Md Cap Index, Sandard & Poor's Small Cap Index, Russel 3000 Index, NASAQ Cmpose Idex, euscher Aken Index, Md-cap euscher Aken Index, Small-cap euscher Aken Index, Technology euscher Aken Index (TecAX), Swss Marke Index (SMI), IPC Mexco Index, Bovespa Index, Kuala Lumpur Sock Exchange Index (KLSE), Jakara Compose Index (JKSE), Ausrala All Ordnares Index (AOR), Bombay Sock Exchange 30 Index (BSE 30), Hang Seng Index (HSI), Shangha Compose Index (SSE), Ausran Trade Index (ATX), Amserdam Exchange Index (AEX), Belgum Exchange Index (BEL), Sres Tmes Index (STI), Euronex BEL 0 Index (BEL 0), Tel Avv TA-00 Index (TA 00), Tawan Weghed Index (TSEC), Soul Compose Index (KOSPI), NIKKIE 5, CAC 40 Index (CAC). The daa range depends on avalably of he daa for a ceran ndex. The longes daa span s for he Sandard & Poor's 500 Index and ranges from..955 o 8.5.009, he shores daa range s for BOVESPA ndex and s from..997 o 8.5.009. All he marke daa was downloaded from he Yahoo euschland Fnanzen (008) and from he Yahoo Fnance (009). For more deals, please see he reference ls.
Sock Marke Seasonaly: ay of he Week Effec and January Effec The Euro OverNgh Index Average rae was used as a rsk free rae of neres. The Eona s he effecve overngh reference rae for he Euro. I s compued as a weghed average of all overngh unsecured lendng ransacons underaken n he nerbank marke, naed whn he Euro zone by he conrbung banks. (Eurbor FBE, 000). The me seres of he Eona was downloaded from he Eurbor FBE (000). For more deals, please see he reference ls.
Sock Marke Seasonaly: ay of he Week Effec and January Effec 4. Resuls 4.. Exended dummy varable approach: AX Index A model exended by hree facors was esmaed. Ths model s based on he same logc as he model esmaed by Gardeazabal and Regulez (003), he only dfference are facors deployed. Smlarly as n Gardeazabal and Regulez (003), he frs facor s he excess reurn on he marke porfolo. However, he second facor refers o he dfference beween reurns of small-cap SAX and large-cap AX ndex. The hrd facor s he dfference beween reurns of he vax ndex and TecAX ndex. I s generally known ha frms wh hgh book o marke rao have also hgh dvdend yeld. Therefore, we can use he vax ndex, whch s an ndex of companes payng hgh dvdends. In general, he companes of he vax ndex can be regarded as companes, wh low expeced growh of earnngs low prce o earnngs rao, hgh book o marke rao and hgh dvdend yeld. On he oher hand he Tecax ndex represens a porfolo of growh companes. Ths ndex consss of echnology socks, whch means fas growng companes wh very hgh expeced growh of earnngs and consequenly very hgh prce o book rao and low book o marke rao. Fgures o 5 repor he resuls. Each fgure conans he resuls for one explanaory varable. The horzonal lne represens he 95% confdence nerval. For each sock, he combnaon of -sasc n absolue and coeffcen esmae s represened by a blue cross. If he cross falls above he 95% confdence nerval lne, s sgnfcanly dfferen from 0. Fg.. Monday 3
Sock Marke Seasonaly: ay of he Week Effec and January Effec Fg.. Tuesday Fg. 3. Wednesday 4
Sock Marke Seasonaly: ay of he Week Effec and January Effec Fg. 4. Thursday Fg. 5. Frday 5
Sock Marke Seasonaly: ay of he Week Effec and January Effec Under he assumpon of no seasonaly we expec no more han 5% of crosses o be above he lne. Snce we have 30 socks, 5% means crosses. Indeed, here s never more han cross above he lne. Ths suppors evdence of no seasonaly n he reurns of he 30 companes of he AX ndex. 4.. Inernaonal evdence of he day of he week effec and he January effec In order o cope wh he specfc rsk of ndvdual socks, n he sense of he secury characersc lne (Sharpe e al., 999), s much beer o explo he profs of he calendar effecs by radng he marke porfolos represened by he whole ndex. Ths can be easly conduced even by an ndvdual nvesor by radng fuures conracs fxed on varous ndces. Ths way, he nvesor wll ge a suffcen fnancal leverage a a neglgble cos and wll compleely elmnae he specfc rsk of ndvdual socks. Therefore, from he praccal pon of vew s much more excng o nvesgae he seasonal paerns of sock marke ndces raher han seasonal paerns of ndvdual socks. Ths can be done by usng he smple dummy varable approach. The model esmaed n hs hess jons boh he day of he week effec and he January effec by usng dummy varables for monhs and for days. The model has he followng form: R α 5 5 M 6 M 6 T 7 T 7 W 8 W 8 9 H 9 H 0 F F 0 3 u 3 Where k, k M, T, W, H, F refer o daly dummes and k, k,, 3, 4, 5, 6, 7, 8, 9. 0.. refer o monhly seasonal dummes. Tables o 7 provde he resuls of hs regresson model esmaed on he daa of 8 world sock exchange ndces. Index NASAQ -sasc n absolue S&P 500 Table. -sasc n absolue S&P md cap -sasc n absolue S&P small cap 4 4 -sasc n absolue aa range 5..97..985 0.8.99 5.8.995 consan 0.0354.773 0.035.69 0.040.488 0.038.3306 Monday -0.696 6.475-0.0489.7635-0.0959.886-0.69.408 Tuesday -0.049.878 0.09.57-0.034 0.3588 0.030 0.6456 Wednesday 0.080 3.06 0.0354.5739 0.0563.5655 0.04 0.8874 Thursday 0.0689.7474-0.06 0.9333 0.054 0.6609 0.0074 0.493 Frday 0.0688.8735 0.0059 0.63 0.076 0.7947 0.0473.054 January 0.0956.3 0.0069 0.734-0.05 0.8568-0.0696 0.8890 February -0.0435.57-0.0595.6334-0.044 0.7806-0.0758.0668 March -0.0066 0.69 0.080 0.4705 0.03 0.06 0.070 0.34 Aprl 0.0409 0.883 0.056.500 0.0873.369 0.5.560 May 0.0 0.394 0.048 0.7776 0.0564.34 0.0739.390 June 0.0049 0.436-0.0060 0.975-0.0543.074 0.07 0.007 July -0.054.3549-0.064 0.473-0.0563 0.9857-0.359.7909 Augus -0.08 0.65-0.034 0.375-0.094 0.370-0.083 0.456 Sepember -0.0968.73-0.077.80-0.0689.070-0.0509 0.684 Ocober -0.055 0.4694-0.038 0.4039-0.000 0.59-0.035 0.3457 November 0.0398 0.8778 0.0459.0380 0.084 0.3960 0.0400 0.445 ecember 0.0545.635 0.0445.67 0.90.834 0.70.3907 crcal.960.960.9605.9607 6
Sock Marke Seasonaly: ay of he Week Effec and January Effec Index Russell 3000 -sasc n absolue FTSE 00 Table 3. -sasc n absolue AX -sasc n absolue MAX aa range 0.9.987.4.984 6..990 3..000 -sasc n absolue consan 0.059.67 0.08.0096 0.0384.7855 0.07 0.87 Monday -0.043.489-0.0553.7505 0.0499.033-0.0340 0.5597 Tuesday 0.030 0.9740 0.044 0.88 0.087 0.6966-0.070.407 Wednesday 0.039.0889-0.053 0.5670-0.065.577-0.0375 0.753 Thursday -0.044 0.7757-0.0059 0.66-0.048 0.3465 0.065 0.386 Frday 0.0038 0.53 0.050.8590 0.004 0.037 0.70.733 January -0.006 0.0489-0.0099 0.34-0.050 0.4 0.005 0.064 February -0.0453 0.9489 0.0075 0.765-0.0078 0.8-0.009 0.40 March 0.097 0.3776 0.0040 0.0837-0.085 0.59-0.064 0.947 Aprl 0.064.847 0.077.90 0.560.4543 0.390.9799 May 0.063.48-0.005 0.505 0.043 0.48 0.064 0.793 June -0.0346 0.868-0.0545.470-0.039 0.5658-0.0457 0.63 July -0.056 0.338-0.0086 0.936-0.07 0.900-0.0567 0.73 Augus -0.0566.5 0.006 0.0596-0.005 0.9805 0.038 0.554 Sepember -0.0547 0.9784-0.0945.8400-0.304.8848-0.79.559 Ocober -0.0353 0.4387-0.0074 0.0 0.0735 0.7657-0.05 0.005 November 0.040 0.395 0.00 0.300 0.0703 0.975 0.005 0.047 ecember 0.0846.75 0.093.934 0.08.497 0.0564 0.764 crcal.9604.9603.9605.960 Index SAX -sasc n absolue TecAX Table 4. -sasc n absolue SMI -sasc n absolue IPC Mexco -sasc n absolue aa range 3..000 4.3.003 9..990 8..99 consan 0.006 0.079 0.055.885 0.0364.0578 0.0785 3.0933 Monday 0.0340 0.7056 0.347.775-0.086 0.705-0.58.9490 Tuesday -0.058.4854 0.00 0.065 0.09 0.383 0.05 0.906 Wednesday -0.0534.3676-0.008 0.037-0.0076 0.7 0.043 0.855 Thursday -0.09 0.7448-0.03.645-0.0054 0.53 0.0586.498 Frday 0.069.8440-0.038 0.44 0.087 0.848 0.043 0.989 January 0.087.465 0.034 0.03-0.099 0.343-0.069 0.805 February 0.053 0.870-0.000 0.0006-0.0300 0.5957-0.336.646 March 0.0409 0.678-0.0435 0.870 0.033 0.53 0.457.7830 Aprl 0.857 3.079 0.64.99 0.06.53-0.007 0.037 May 0.0405 0.6585 0.0086 0.0644 0.0360 0.7405-0.056 0.3443 June -0.053 0.8957-0.0398 0.34-0.0433 0.8584-0.044 0.3084 July -0.0534 0.8830-0.0677 0.5070-0.030 0.5475-0.0530 0.6780 Augus -0.078 0.359 0.065 0.985-0.0789.0543-0.50.7053 Sepember -0.349 3.0930-0.000 0.768-0.099.398-0.0387 0.4056 Ocober -0.043 0.476-0.0838 0.4063 0.0607 0.760 0.044 0.389 November -0.0334 0.4580-0.076 0.480 0.05 0.9585 0.07.78 ecember 0.0340 0.55 0.047 0.369 0.0600.005 0.570.558 crcal.960.965.9605.9605 7
Sock Marke Seasonaly: ay of he Week Effec and January Effec Index Bovespa -sasc n absolue CAC 40 Table 5. -sasc n absolue Kuala Lumpur -sasc n absolue Jakara Comose aa range 7.4.993.3.990 3..993.7.997 -sasc n absolue consan 0.0535,306 0.00.0836 0.06 0.4899 0.058.496 Monday -0.650.835-0.0483.0599-0.535.793-0.88.5345 Tuesday 0.036 0.74 0.038 0.89-0.004 0.044-0.08 0.759 Wednesday 0.064 0.7866-0.03 0.5907 0.0709.478-0.035 0.4695 Thursday -0.046 0.4884 0.088 0.469-0.066 0.574 0.0875.44 Frday 0.36.669 0.009 0.533 0.5.347 0.504.90 January 0.0.489 0.03 0.938 0.0449 0.480 0.60 0.834 February 0.045 0.337 0.090 0.348 0.57.4494-0.073 0.6546 March 0.0 0.030 0.068 0.9483-0.43.8480 0.055 0.686 Aprl 0.0635 0.4640 0.45.0949 0.0487 0.7077 0.595.5054 May -0.375.0488-0.055 0.739-0.036 0.4759 0.055 0.4553 June -0.056 0.3995-0.0746.346-0.0693. 0.65.096 July -0.0797 0.6450-0.034 0.53-0.065 0.49-0.006 0.0763 Augus -0.979.067-0.0863.97-0.596.9698-0.496 4.950 Sepember 0.060 0.4334-0.908.4544-0.0469 0.39-0.87 0.9437 Ocober -0.047 0.6377 0.0848 0.9668-0.004 0.004-0.59 0.8770 November -0.0094 0.0738 0.038 0.5509 0.005 0.0308 0.085 0.9785 ecember 0.797.9790 0.059 0.9370 0.074.6453 0.39.730 crcal.9607.9605.9606.9608 Index Ausrala All Ordnares -sasc n absolue BSE 30 Table 6. -sasc n absolue HSI -sasc n absolue Shangha Compose -sasc n absolue aa range 3.8.984.7.997..987 4..000 consan 0.033.68 0.0485.4760 0.057.33 0.0369.090 Monday -0.0055 0.080 0.064 0.8447-0.78.94 0.03.35 Tuesday -0.0465.676-0.00 0.670 0.09 0.4388-0.095 0.4594 Wednesday 0.047.056 0.06 0.9870 0.0837.7869 0.08.996 Thursday 0.07 0.76-0.0354 0.57-0.076.7044-0.403.537 Frday 0.00 0.4336-0.077.46 0.093.0964-0.035 0.86 January 0.008 0.0685-0.0567 0.4984-0.0693 0.8003 0.009 0.057 February -0.080 0.7783 0.046 0.463 0.355.774 0.793.40 March 0.07 0.7-0.0774 0.6433-0.0989.438 0.089 0.860 Aprl 0.036.6649-0.0006 0.0047 0.076.0396 0.465.3080 May 0.03 0.4005-0.075 0.59 0.0335 0.4866 0.0055 0.0557 June -0.046.458-0.090 0.539-0.050 0.6393-0.589.579 July 0.049.3 0.033 0.33 0.069.348-0.0376 0.3645 Augus 0.0059 0.646 0.036 0.430-0.050.4675-0.048 0.48 Sepember -0.0395.043-0.07 0.770-0.0467 0.6434-0.065 0.564 Ocober -0.0963.3965-0.566.94-0.039 0.477-0.048.84 November -0.0470.008 0.396.373 0.0064 0.0847 0.034 0.963 ecember 0.06.750 0.99.30 0.0858.46 0.0948.039 crcal.9603.9608.9604.960 8
Sock Marke Seasonaly: ay of he Week Effec and January Effec Index Sras Tmes -sasc n absolue AEX Table 7. -sasc n absolue BEL 0 -sasc n absolue TA 00 aa range 4..988.0.99 9.4.99.7.997 -sasc n absolue consan 0.07.4975 0.077.477 0.095.484 0.0535.4954 Monday -0.08.556 0.0557.086-0.068 0.7048 0.78.308 Tuesday -0.0357.0649 0.044 0.345 0.0003 0.00-0.038 0.707 Wednesday 0.0505.4366-0.0595.3988 0.0034 0.057-0.084.967 Thursday 0.084 0.588-0.0390 0.884-0.0003 0.0078-0.037 0.6699 Frday 0.075.3 0.085 0.655 0.03 0.6853 0.0054 0.0540 January 0.06 0.80-0.089 0.4078 0.03 0.36-0.34.98 February 0.035 0.493-0.006 0.055-0.0358 0.766 0.0849 0.979 March -0.04 0.40-0.039 0.3008-0.03 0.653-0.0333 0.34 Aprl 0.5.936 0.406.9 0.97.64 0.55.783 May -0.0333 0.647-0.0036 0.066-0.08 0.774 0.757.6860 June -0.045 0.4843-0.0 0.366-0.0355 0.7777-0.049.08 July 0.009 0.049-0.0079 0.06-0.0067 0.94-0.0455 0.5074 Augus -0.58.603-0.068 0.3757-0.049 0.787-0.549.786 Sepember -0.08.9-0.34.5875-0..808-0.48.4968 Ocober 0.0035 0.0450 0.0385 0.3897 0.004 0.037-0.538.057 November 0.0650.0843 0.0765.0548 0.0057 0.087 0.635.4998 ecember 0.086.0977 0.093.4308 0.6.484 0.69.304 crcal.9604.9605.9605.9609 Index ATX -sasc n absolue TSEC Table 8. -sasc n absolue Kosp -sasc n absolue Nkke 5 -sasc n absolue aa range..99.7.997.7.997 4..984 consan 0.034.6604 0.0060 0.930 0.0437.0794 0.0089 0.4749 Monday 0.0033 0.079-0.5.7089-0.06 0.8-0.083.0579 Tuesday 0.056 0.3749-0.085.4339-0.0530 0.7334 0.033 0.3574 Wednesday 0.0 0.85 0.077.355 0.057 0.736 0.0359.0037 Thursday -0.0577.466 0.04 0.300 0.035 0.3847 0.057.54 Frday 0.077 0.6766 0.6.003-0.0094 0. -0.033 0.6500 January 0.04 0.6363 0.438.875 0.8.034 0.030 0.4676 February 0.0486 0.7605 0.45.766-0.0868 0.6977 0.0064 0.70 March 0.009 0.386 0.39.09 0.0060 0.0548 0.065.0037 Aprl 0.7.37-0.0355 0.3809 0.0838 0.6474 0.0730.8 May 0.004 0.37-0.0000 0.0004-0.0736 0.5630 0.085 0.549 June -0.0055 0.0970-0.038 0.398 0.0050 0.0355-0.0307 0.689 July -0.034 0.409-0.0776 0.786 0.000 0.0084-0.0357 0.68 Augus -0.076.0-0.048 0.63-0.0695 0.6559-0.077 0.3063 Sepember -0.058.7999-0.300.973-0.64.609-0..7397 Ocober -0.0579 0.5654-0.0768 0.639-0.59 0.784-0.0557 0.637 November -0.03 0.3053 0.0439 0.403 0.3.5470 0.07 0.485 ecember 0.8.9795 0.6.34 0.088 0.579 0.033 0.543 Crcal.9605.9608.9608.9603 Table 9 summarzes he resuls by provdng he number of cases ou of 8 n whch he parameers were sascally sgnfcan. I s apparen from hs able ha he mos sgnfcan day of he week effec was Monday, whch exacly confrms exsng leraure. However, only ndex ou of 8 acheved sascally sgnfcan excess reurns n January. The January 9
Sock Marke Seasonaly: ay of he Week Effec and January Effec effec was overaken by Aprl, ecember, Sepember and Augus. I should be poned ou ha parameers for Aprl and ecember were posve and parameers for Augus and Sepember were negave. Table 9. Number of Sascally Sgnfcan Cases Monday 9 Tuesday 0 Wednesday Thursday Frday 8 January February 0 March 0 Aprl 0 May 0 June 0 July 0 Augus 3 Sepember 7 Ocober 0 November 0 ecember 8 4.3. Tme perssence of he day of he week effec Ths hess also examnes me perssence of he Monday effec of 7 ou of 8 ndces. Accordng o he effcen marke hypohess, once dscovered hs effec should dsappear n a very shor me. The reason why he Monday has been chosen s ha hs effec was he mos sgnfcan. The oher day s effecs were also consdered. However, her evoluon hrough me s very smlar o he Monday effec. Therefore, her me pahs are no repored. The me pah of he Monday effec s compued by esmang a rollng regresson. The regresson equaon s esmaed on he frs 50 radng days, whch approxmaely represens a 5 years perod, and hen hs regresson equaon s reesmaed by movng owards he end of he me seres by 5 radng days. The 7 fgures bellow provde he me pah of he -sascs of he Monday effec for Fnancal Tmes Index 00, Ausrala All Ordnares Index, Hang Seng Index, Sras Tmes Index, Nkke 5, NASAQ Compose Index, and Sandard & Poor's 500 Index. 0
Sock Marke Seasonaly: ay of he Week Effec and January Effec Fg. 6. FTSE 00 Fg. 7. Ausrala All Ordnares
Sock Marke Seasonaly: ay of he Week Effec and January Effec Fg. 8. Hang Seng Index Fg. 9. Sras Tmes Index
Sock Marke Seasonaly: ay of he Week Effec and January Effec Fg. 0. Nkke 5 Fg.. NASAQ Compose Index 3
Sock Marke Seasonaly: ay of he Week Effec and January Effec Fg.. Sandard & Poor s 500 Index I s apparen from he 7 fgures above ha he day of he week effec has very smlar paer for all he ndces examned. I occurred n he 960s, whch can be seen on he las fgure, and was very srong and sable ll he second half of he 990s. Probably he longes me perssence of hs effec s n case of he Sras Tmes Index and Nkke 5, where sayed sgnfcan ll 998. Furhermore, we can observe ha hs effec dsappeared frs from he S&P 500 ndex. 4.4. Smulaed porfolo sraeges The calendar effecs should also be consdered n erms of her exploably. In oher words, has o be assessed f was possble o acheve profs ha would be able o ouperform he buy and hold sraegy. Ths hess ams o consder hs ssue by focusng on he negave Sepember effec and Monday effec only. The same 7 ndces ou of 8 were consdered. Fnancal Tmes Index 00, Ausrala All Ordnares Index, Hang Seng Index, Sras Tmes Index, Nkke 5, NASAQ Compose Index, and Sandard & Poor's 500 Index. Fgures 3 o 9 graphcally compare performance of 3 marke mng sraeges by plong her me pah and a relave performance of a sraegy based on Monday effec and a sraegy based on Sepember effec wh respec o he buy and hold sraegy. The blue lne represens he buy and hold sraegy. The red lne marks a marke mng sraegy based on sellng he ndex on Frday a a closng prce and buyng agan on Monday a a closng prce. The green lne corresponds o a marke mng sraegy based on sellng he ndex n Augus a a closng prce and buyng agan n Sepember a a closng prce. 4
Sock Marke Seasonaly: ay of he Week Effec and January Effec Fg. 3. FTSE 00 Fg. 4. Ausrala all Ordnares 5
Sock Marke Seasonaly: ay of he Week Effec and January Effec Fg. 5. Hang Seng Index Fg. 6. Sras Tmes Index 6
Sock Marke Seasonaly: ay of he Week Effec and January Effec Fg. 7. Nkke 5 Fg. 8. NASAQ Compose Index 7
Sock Marke Seasonaly: ay of he Week Effec and January Effec Fg. 9. Sandard & Poor s 500 Index In case of he FTSE 00, Sras Tmes, Nkke 5, NASAQ and S&P 500, whch means 5 ndces ou of 8, he buy and hold sraegy was he wors performng sraegy. The buy and hold sraegy s no he bes sraegy n any of he consdered ndces. If we had used he sraegy based on shor sellng of he porfolo every Sepember n case of he FTSE 00, we would have moderaed he crash ha followed he ocom bubble. Moreover, he sandard devaon of reurns would have been lower han n case of he buy and hold sraegy, whch mples hgher reurns a a lower rsk. I mgh be approprae o menon ha f we had used he sraegy based on he Monday effec for he Nkke 5, we would have no suffered a loss conneced wh he so called Los decade on he Japan Sock Exchange. The mos pronounced s he day of he week effec n case of he S&P 500 and NASAQ Compose. If we had allowed shor sellng, we would have ouperformed he buy and hold sraegy by almos 400%, whch means wce as bg performance as s acheved by he buy and hold. The Sepember effec would have led o a remarkable ouperformance of 00%. The sraeges based on calendar effecs dd no perform very well on he Ausralan sock marke. In hs case, he addonal profs acheved by hese sraeges would no be probably hgh enough o cover ransacon coss. Table 0 furher summarzes he resuls by provdng he mean daly reurn, s sandard devaon, and a rao of hese wo varables. Ths s n fac he reward-o-varably rao (Sharpe, 966) or he Sharpe rao wh a consan rsk free rae (Sharpe, 994). I s apparen from hs able ha he buy and hold sraegy preformed always wors. The bes sraegy n erms of mean reurn was he sraegy based on he Monday effec. The bes performng sraegy n erms of he sandard devaon of reurns was also he sraegy based on he Monday effec. Ths mples ha a rsk averse nvesor would prefer hs sraegy. I should be poned ou ha hs sraegy was he bes n all he 7 cases ou of 7 n erms of he sandard 8
Sock Marke Seasonaly: ay of he Week Effec and January Effec devaon of reurns. Ths probably means ha Monday reurns are very volale. If we assume a consan rae of subsuon beween rsk and reurn, we can conclude ha he bes performng sraegy was he sraegy based on he raon beween he mean reurn and sandard devaon of reurns, whch was agan he sraegy based on he Monday effec. On base of hs ex pos observaon, can be concluded ha hs sraegy s superor boh n erms of rsk and reurn. I can provde a hgher reurn wh a lower rsk han he buy and hold sraegy. Table 0. Index Sraegy Mean Reurn % Sandard evaon of Reurns Rao of he mean reurn and s sandard devaon Ausrala All Ordnares Buy and Hold 0.03 0.9898 0.035 Long Posons Monday 0.07 0.8789 0.0309 Long Posons Sepember 0.037 0.9537 0.0343 FTSE 00 Buy and Hold 0.083.3 0.05 Long Posons Monday 0.0335 0.9833 0.034 Long Posons Sepember 0.0339.069 0.039 Hang Seng Buy and Hold 0.0506.790 0.083 Long Posons Monday 0.0635.4607 0.0435 Long Posons Sepember 0.050.793 0.090 NASAQ Compose Buy and Hold 0.0375.597 0.098 Long Posons Monday 0.067.04 0.0565 Long Posons Sepember 0.048.099 0.0345 Nkke 5 Buy and Hold 0.000.4709 0.0068 Long Posons Monday 0.040.944 0.086 Long Posons Sepember 0.079.40 0.07 Sras Tmes Buy and Hold 0.069.384 0.003 Long Posons Monday 0.0430.44 0.0386 Long Posons Sepember 0.0340.756 0.066 SP 500 Buy and Hold 0.084 0.984 0.088 Long Posons Monday 0.046 0.8447 0.0504 Long Posons Sepember 0.03 0.9378 0.0343 However, s necessary, o es f he dfferences n reurns and sandard devaons are sascally sgnfcan. A frs, he equaly of varances mus be esed by a F-es, whch s s defned as F. H 0 saesσ σ and s rejeced f F > F( α /, n, n ). Then f s varances are unequal, equaly of means can be esed by he followng -sasc 9
Sock Marke Seasonaly: ay of he Week Effec and January Effec. H 0 sang x ( n ) s ( n ) s where n n ( x x ) s s n n df. s s n n n n n n x s rejeced f > ( α /, df ), Table provdes s of -sascs and crcal s of a relevan dsrbuon. I s apparen from hs able ha varances are always sascally sgnfcanly dfferen from he buy and hold sraegy. However, mean reurns are never sascally sgnfcanly dfferen. Ths mples ha he wo sraeges consdered are no able o acheve sascally sgnfcanly hgher mean reurns. Noneheless, hey can sgnfcanly lower rsk n erms of sandard devaons. Parcularly he sraegy based on avodng Mondays s very powerful n hs sense. I has sascally sgnfcanly lower sandard devaons of reurns han he buy and hold sraegy n all he 7 cases consdered. The sraegy based on avodng Sepember s able o bea he buy and hold sraegy n 6 cases ou of 7, n erms of sandard devaon of reurns. Table. crcal of F-dsrbuon -sasc n absolue crcal of -dsrbuon Index Sraegy F-sasc Ausrala All Ordnares Long Posons Monday.683.044 0.3050.960 Long Posons Sepember.077.044 0.088.960 FTSE 00 Long Posons Monday,3048.04 0.774.960 Long Posons Sepember.67.04 0.884.960 Hang Seng Long Posons Monday,509.045 0.457.960 Long Posons Sepember.076.045 0.00.960 NASAQ Compose Long Posons Monday,870.034,4743.960 Long Posons Sepember.0840.034 0.49.960 Nkke 5 Long Posons Monday,93,046 0.564.960 Long Posons Sepember.0879.046 0.306.960 Sras Tmes Long Posons Monday,409.046 0.678.960 Long Posons Sepember.0845.046 0.85.960 SP 500 Long Posons Monday,3573.085,797.960 Long Posons Sepember,0,085 0.367.960 30
Sock Marke Seasonaly: ay of he Week Effec and January Effec 5. Concluson Ths hess has examned 30 ndvdual socks of he German Sock Exchange for he presence of he day of he week effec and 8 world ndces for presence of he day of he week effec and he monh of he year effec. No evdence of he day of he week effec has been found for socks of he German Sock Exchange where an exended dummy varable approach was appled. The nernaonal evdence n suppor of he calendar effecs s mxed. The day of he week effec was found n case of ndces ou of 8 consdered and he monh of he year effec n case of 7 ou of 8 by usng a smple dummy varable approach. I has been also found ha he day of he week effec became nsgnfcan durng he second half of 990s. Ths was concluded by usng me seres of 7 ndces wh he longes daa range. Furhermore, four marke mng sraeges were smulaed on he daa ses of hese 7 ndces. These sraeges were able o sascally sgnfcanly ouperform he buy and hold sraegy, whch he effcen marke hypohess mplcly recommends, n erms of sandard devaons. They were also able o ouperform he buy and hold sraegy n erms of he mean reurn. However, hs ouperformance was no sascally sgnfcan. Noneheless, sll mples an unexploed prof opporuny ha should have been elmnaed by raonal nvesors once was dscovered. I mgh be approprae o menon ha he day of he week effec was dscovered n 960 and dsappeared around 995. In oher words, ook approxmaely 80 weeks or 900 radng days o he nvesors o explo hs effec and o wpe ou. If we recall he formula from Fama (970) expressng undervaluaon and, appears ha n erms of he day of he week overvaluaon of secury prce ( x Φ ) 0 Ε j, effec he ndces were perssenly, sgnfcanly overd for far oo long me for he marke o be a leas weakly effcen. Ths seems o volae he effcen marke hypohess. 3
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