Pairs Trading using a Cointegration Approach: An Empirical Investigation using US Stock Market data

Mas e Deg ee P og am in S a is ics and In o ma ion Managemen Pai s T ading using a Coin eg a ion App oach An Empi ical In es iga ion using US S ock Ma ke da a Samuel Tomé Alexand e Lou enço Mas e Thesis p esen ed as pa ial equi emen o ob aining he Mas e Deg ee in S a is ics and In o ma ion Managemen NOVA In o ma ion Managemen School Ins i u o Supe io de Es a ís ica e Ges ão de In o mação Uni e sidade No a de Lisboa MEGI NOVA In o ma ion Managemen School Ins i u o Supe io de Es a ís ica e Ges ão de In o mação Uni e sidade No a de Lisboa Pai s T ading using a Coin eg a ion App oach An Empi ical In es iga ion using US S ock Ma ke da a by Samuel Tomé Alexand e Lou enço Mas e Thesis p esen ed as pa ial equi emen o ob aining he Mas e ’s deg ee in S a is ics and In o ma ion Managemen , wi h a specializa ion in isk managemen . Supe ised by P o . Dou o Jo ge Miguel Ven u a B a o, PhD NOVA IMS & Uni e si é Pa is-Dauphine PSL June, 2024 i STATEMENT OF INTEGRITY I he eby decla e ha ing conduc ed his academic wo k wi h in eg i y. I con i m ha I ha e no used plagia ism o any o m o undue use o in o ma ion o alsi ica ion o esul s along he p ocess leading o i s elabo a ion. I u he decla e ha I ha e ully acknowledged he Rules o Conduc and Code o Hono om he NOVA In o ma ion Managemen School. [Lisboa, 2024] ii ACKNOWLEDGEMENTS Fi s o all, I wan o hank all my amily, who had suppo ed me all along his jou ney and opened my eyes o eali y, when i was necessa y. P ima ily I wan o hank my b o he , who has been a ole model o me since I was bo n, suppo ing me in e e y s ep ha I ook. Then, a hank o my sis e Ru e, who is a ema kable s uden and a e y kind and esilien pe son, made me a be e pe son in ha aspec oo. Thank my sis e Luísa who e e y day shows me wha is he will o li e, despi e e e y ad e si y. Las ly, I hank my pa en s who ga e me he bes educa ion hey could and made me who I am oday. Thank you o my a he who is my idol, being he mos ha dwo king pe son I know, and o my mo he , he kindes pe son I know, who is always he e o me. Secondly, I wan o hank all my iends and my gi l iend, who suppo ed me in his hesis, gi ing me he s eng h o no gi e up, and comp ehension in he momen s ha I could no be p esen in he las mon hs. Finally, hanks o my supe iso , P o esso Dou o Jo ge B a o, who helped me along he way and was pa ien and lexible wi h me. iii ABSTRACT This disse a ion will s udy he e u n o a pai s ading s a egy in he US s ock ma ke . Pai s ading is a ma ke -neu al ading s a egy ha exploi s he p ice mo emen s be ween wo his o ically co ela ed secu i ies. The pu pose o his hesis is o e alua e he pe o mance o he coin eg a ion app oach in his ma ke -neu al s a egy and o assess i i has an edge o e o he s a egies. This esea ch employs a quan i a i e app oach, using his o ical p ice da a o 50 secu i ies om a ious sec o s o e 8 yea s. The s udy applies a coin eg a ion app oach o iden i y pai s and execu e ades, wi h he app oach done wi h he Engle-G ange me hod. Pe o mance me ics such as e u ns, Sha pe a io, and maximum d awdown a e analyzed. The indings in his s udy indica e ha pai s ading can gene a e consis en e u ns wi h ela i ely low ola ili y in s able ma ke condi ions. Howe e , he s a egy's pe o mance de e io a es du ing pe iods o high ma ke ola ili y. This s udy obse ed ha he pai s ading s a egy had be e e u ns wi h coin eg a ed pai s and con ibu ed o he exis ing li e a u e on he coin eg a ion app oach by o e ing a comple e e alua ion o pai s ading pe o mance and e u ns wi h US s ock ma ke da a. KEYWORDS Pai s T ading ; Coin eg a ion ; Leas Squa es ; Ma ke -neu al s a egy ; S a is ical A bi age i TABLE OF CONTENTS 1. In oduc ion .................................................................................................................. 1 2. Li e a u e e iew........................................................................................................... 3 3. Me hodology ................................................................................................................. 6 3.1. Coin eg a ion App oach ........................................................................................ 6 3.2. Dis ance App oach ................................................................................................ 8 3.3. O he App oaches .................................................................................................. 9 3.4. Selec ion o App oach ......................................................................................... 10 3.5. Leas Squa es Reg ession .................................................................................... 10 3.6. Risks .................................................................................................................... 12 3.7. Da a...................................................................................................................... 12 4. Empi ical S udy .......................................................................................................... 13 4.1. Resea ch Ques ions ............................................................................................. 13 4.2. Assump ions ........................................................................................................ 13 4.3. Coin eg a ion es s ............................................................................................... 13 4.4. Leas Squa es eg ession o pai s ading ........................................................... 16 4.5. Sp ead and Z-sco e .............................................................................................. 18 4.5.1. T ading he sp ead ........................................................................................ 19 4.5.2. Pe o mance o he s a egy .......................................................................... 21 4.6. Risk-Adjus ed Pe o mance Me ics ................................................................... 25 4.6.1. Capi al Asse P icing Model (CAPM) .......................................................... 25 4.6.1.1. Be a .................................................................................................... 26 4.6.2. Sha pe Ra io ................................................................................................. 27 5. Resul s and discussion ................................................................................................ 28 5.1. Coin eg a ion esul s ............................................................................................ 28 5.2. Risk-Adjus ed Pe o mance Me ics Resul s....................................................... 30 5.3. Pai s ading s a egy esul s ................................................................................ 31 6. Conclusions and u u e wo ks .................................................................................... 34 Bibliog aphical Re e ences ............................................................................................. 37 Appendix A ..................................................................................................................... 40 i LIST OF FIGURES Figu e 1: Coin eg a ion Resul s................................................................................................ 15 Figu e 2: S ock P ices .............................................................................................................. 17 Figu e 3: Leas Squa es Reg ession ......................................................................................... 18 Figu e 4: Sp ead om no malized po olio ............................................................................ 19 Figu e 5: Z-sco e ...................................................................................................................... 20 Figu e 6: Z-sco e and ading signal ........................................................................................ 21 Figu e 7: Re u n o aded sp ead ............................................................................................. 22 Figu e 8: P&L o aded sp ead wi h ein es men .................................................................. 23 Figu e 9: D awdown ................................................................................................................. 24 Figu e 10: T ading signal and cumula i e P&L ....................................................................... 24 Figu e 11: Equi alen po olio app oach code ........................................................................ 25 Figu e 12: Lib a y´s used ......................................................................................................... 43 Figu e 13: Coin eg a ion Code ................................................................................................. 43 Figu e 14: P ices Loading ........................................................................................................ 43 Figu e 15: T aining se and LS eg ession ............................................................................... 44 Figu e 16: Sp ead Equa ion ...................................................................................................... 44 Figu e 17: No malized Po olio and No malized Sp ead ....................................................... 44 Figu e 18: Z-sco e code ............................................................................................................ 45 Figu e 19: Sp ead S anda d De ia ion ..................................................................................... 45 Figu e 20: Z-sco e and ading signal code .............................................................................. 45 Figu e 21: Re u n o aded sp ead code .................................................................................. 46 Figu e 22: P&L o aded sp ead wi h ein es men code ....................................................... 46 Figu e 23: D awdown code ...................................................................................................... 46 Figu e 24: T ading signal wi h Z-sco e code ........................................................................... 46 Figu e 25: Equi alen po olio app oach ................................................................................ 46 5 o 12 mon hs o iden i y equi y pai s using he dis ance me hod, ollowed by a ading pe iod whe e posi ions a e opened when p ices di e ge by mo e han wo s anda d de ia ions and closed when hey con e ge. S udies like Pe lin (2007) and Papadakis and Wysicki (2008) ha e u he explo ed pai s ading ac oss di e en ma ke s and condi ions, showing a ied p o i abili y and ac o s in luencing e u ns. Engelbe g, Gao, and Jaganna han (2009) added in o ma ional e en s o he s a egy, inding ha di e ences in how quickly new in o ma ion is inco po a ed can a ec p o i abili y. Recen esea ch, such as F anco (2014) and Ribei o (2015), con i ms a signi ican dec ease in pai s ading pe o mance in he las decades, pa icula ly du ing high ansac ion cos pe iods. Howe e , hey also sugges ha he s a egy emains p o i able unde ce ain condi ions, such as du ing ma ke c ises o when limi ed o same-indus y pai s. O e all, pai s ading emains a obus and in iguing s a egy, hough i s p o i abili y has been challenged by ma ke changes and inc eased compe i ion om hedge unds. Li e a u e con inues o e ol e wi h new me hodologies and inno a ions, keeping he academic and ading communi ies engaged. 6 3. METHODOLOGY The me hodology is based on coin eg a ion app oach and leas squa es eg ession. The ime ho izon is 8 yea s, om 2016 o 2024. The e will be analyzed 25 pai s, so 50 s ocks om he US s ock ma ke . The e a e some pai s whe e he ime ho izon is no he abo e since some companies made hei IPO a e ha da e. The objec i e he e is o demons a e ha he s a egy in pai s ha a e coin eg a ed has a be e pe o mance han he ones ha a e no . 3.1. COINTEGRATION APPROACH Fo ecas ing s ock ma ke p ices and e u ns is pa icula ly di icul , gi en he nonlinea i y, ola ili y, and complexi y o he ime se ies and he gene ally accep ed, semi-s ong o m o ma ke e iciency (Fama, 1970; B a o, 2024). P e ious esea ch iden i ies se e al uni a ia e o mul i a ia e echniques conside ing he lagged alues o he ime se ies, and undamen al and echnical analysis ac o s, conside ing bo h s a is ical lea ning and a i icial in elligence me hods, including model combina ion app oaches (B a o e al. 2020, 2021, 2023; Asho eh e al. 2021, 2022). Coin eg a ion is an econome ic model which eplies o he long- un equilib ium be ween economic ime se ies. I we ha e wo o mo e ime se ies ha a e nons a iona y, bu a linea combina ion o hem is s a iona y, hen hey a e p obably coin eg a ed (Wei, 2006). The concep o s a iona i y is co ela ed o he p ope ies o hese s ochas ic p ocesses. I he da a a e assumed o be s a iona y i he means, a iances, and co a iances o he se ies a e independen o ime, a he han he en i e dis ibu ion we assume weak s a iona i y. Nons a iona i y in a ime se ies occu s when he e is no cons an mean o no cons an a iance, howe e , he mos impo an one is he uni oo . Rega ding he uni oo , a sequence ha has one o mo e speci ic oo s ha a e equal o one is named a uni oo p ocess. The simple model ha should ha e a uni oo is he AR(1) model. In he empi ical esul s, o he p og amming phase in Rs udio, I´ e used he package, egcm, ha compu es he Engle-G ange coin eg a ion es , and many o he uni - oo es s, like Augmen ed Dickey-Fulle (ADF) Tes , Phillips-Pe on (PP) Tes , Johansen's T ace Tes (JOT) and o he s. Rega ding he Engle-G ange me hod (Engle-G ange , 1987), i is based on he ac ha eg essing non-s a iona y se ies on o he se ies, can lead o misleading esul s. Howe e , i all a iables a e ound o ha e a uni oo (i.e., hey a e in eg a ed o o de one, I(1), he eg ession 7 can s ill be signi ican i he a iables a e coin eg a ed. Then o check o coin eg a ion, we es ima e he leas squa es eg ession equa ion ( ha I will alk abou la e ) and analyze he esiduals o a uni oo . I he esiduals a e s a iona y, I(0) indica es ha he a iables a e coin eg a ed and sha e a long- e m equilib ium ela ionship. The Engle-G ange me hod uses his app oach, so i es s o no coin eg a ion by examining he s a iona i y o he eg ession esiduals. This me hod uses he o dina y leas squa es (OLS) o es ima e he ela ionship be ween he a iables and apply uni oo es s o he esiduals o check o s a iona i y. I he null hypo hesis o a uni oo is ejec ed, i suppo s coin eg a ion. Rega ding he uni oo es s, he Augmen ed Dickey-Fulle (ADF), one o he mos popula ones, de eloped by (Dickey and Fulle , 1979), es s o he exis ence o a uni oo in a ime se ies sample, which helps de e mine i he se ies is non-s a iona y. The me hodology elies on he ac ha he ADF es augmen s he basic Dickey-Fulle es by including lagged di e ences o he dependen a iable o accoun o highe -o de co ela ion and, secondly, he null hypo hesis. Due o he in alidi y o he DF s a is ic in he p esence o se ial co ela ion, he es is ypically applied in i s augmen ed o m. (𝐻0) is ha he se ies has a uni oo (i.e., i is non- s a iona y) and las ly al e na i e hypo hesis (𝐻1) is ha he se ies is s a iona y. In e p e a ion o his es depends on whe he he p- alue is less han he signi icance le el (e.g., 0.05), i i ´s less han he signi icance le el, we ejec 𝐻0 and conclude ha he se ies is s a iona y. I he p- alue exceeds he signi icance le el, we ail o ejec 𝐻0 and conclude ha he se ies is non-s a iona y. Ano he es is he Phillips-Pe on (PP) Tes . I is simila o he ADF es , he PP es checks o a uni oo in a ime se ies sample. Rega ding he me hodology, he PP es adjus s o se ial co ela ion and he e oskedas ici y in he e o s by modi ying he es s a is ics. I uses non- pa ame ic s a is ical me hods o accoun o he au oco ela ion and he null hypo hesis (𝐻0) is ha he se ies has a uni oo (i.e., i is non-s a iona y). In e p e a ion is he same. Cheung, Y. W., & Lai, K. S. (1997). Conce ning he Pan ula, Gonzales-Fa ias and Fulle (PGFF) Tes , is ano he app oach o es ing o uni oo s and de e mining s a iona i y in a ime se ies. The PGFF es in ol es es ing he p esence o a uni oo by conside ing bo h le el and end s a iona y al e na i es. This es can accoun o s uc u al b eaks in he ime se ies da a and he null hypo hesis (𝐻0) is ha he se ies has a uni oo . 8 The Ellio , Ro henbe g, and S ock DF-GLS (ERSD) Tes is a mo e e icien e sion o he ADF es o es ing o a uni oo in a ime se ies. I applies a Gene alized Leas Squa es (GLS) de ending p ocedu e o he da a be o e pe o ming he Dickey-Fulle es . The null hypo hesis (𝐻0) is ha he se ies has a uni oo . Johansen's T ace Tes (JOT) is used o de e mine he numbe o coin eg a ion ec o s in a mul i a ia e ime se ies. The es is based on a Vec o Au o eg essi e model and assesses he ank o he coin eg a ion ma ix. I es s he null hypo hesis ha he numbe o coin eg a ion ec o s is less han o equal o a gi en numbe agains he al e na i e hypo hesis o mo e coin eg a ion ec o s. In his es , i he es s a is ic is g ea e han he c i ical alue, we ejec 𝐻0 and conclude ha he e a e mo e coin eg a ion ec o s. I he p- alue is less han he signi icance le el, i indica es he p esence o coin eg a ion. Las ly, he Schmid and Phillips Rho (SPR) Tes is ano he me hod o es o uni oo s in a ime se ies. I is an ex ension o he Phillips-Pe on es and adjus s o po en ial se ial co ela ion in he e o e ms and he null hypo hesis (𝐻0) is ha he se ies has a uni oo . These es s a e designed o check o uni oo s (s a iona i y s. non-s a iona i y) and coin eg a ion in ime se ies da a. They a e essen ial ools in econome ics o unde s anding he p ope ies o ime se ies and o building eliable models. Each es has i s s eng hs and weaknesses, and o en mul iple es s a e used oge he o make a mo e obus conclusion abou he da a. 3.2. DISTANCE APPROACH The dis ance app oach is a popula and ela i ely s aigh o wa d me hod used in pai s ading popula ized in he esea ch o Ga e e al. (2006). I ocuses on he p ice di e ence o sp ead be ween wo co ela ed asse s, usually s ocks, o iden i y ading oppo uni ies based on he assump ion ha he sp ead will e e o a his o ical mean. This me hod le e ages he concep o mean e e sion, which is cen al o many pai s ading s a egies. In he dis ance app oach, ade s iden i y pai s o s ocks ha ha e shown a s able long- e m ela ionship. The basic p emise is ha while he p ices o he wo s ocks migh di e ge empo a ily due o a ious ma ke ac o s, hey will e en ually con e ge again because o hei unde lying co ela ion. 9 The app oach is spli in o wo phases. The e is a o ma ion pe iod in which likewise mo ing pai s a e selec ed and hen a ading pe iod in which ading signals a e c ea ed and used o ake a posi ion in he s ocks and consequen ly o close hem. (Ga e e al., 2006) use daily da a om 1962 o 2002 on all liquid US s ocks. Fo e e y s ock, a cumula i e o al e u n index is o med and no malized o he 12 mon hs o ma ion pe iod in he o ma ion pe iod In ha esea ch, i is s a ed ha he Top 20 pai s ha ha e he smalles his o ic dis ance o he 12-mon h o ma ion pe iod a e now selec ed o he ollowing 6-mon h ading pe iod. In addi ion o ha , hey also applied sec o il e c i e ia o il e ou pai s ha belong o he same indus y, c i e ia ha I will apply in my selec ion o pai s. Du ing he ading pe iod, he p ice se ies o he secu i ies a e no malized again on he i s day. Posi ions a e opened when he his o ic sp ead o he pai s di e ges by mo e han wo s anda d de ia ions om hei mean and a e closed once hey e e o he his o ic mean. This me hod adds a quali a i e elemen o he quan i a i e app oach. 3.3. OTHER APPROACHES Like he coin eg a ion and dis ance app oaches, he ime se ies app oach in pai s ading elies on s a is ical me hods o iden i y and exploi ela ionships be ween wo secu i ies. In ime se ies analysis, ade s ypically use s a is ical models o o ecas u u e p ice mo emen s based on his o ical da a. The ime se ies app oach can be in eg a ed in o pai s ading by ocusing on modeling and o ecas ing he sp ead be ween wo co ela ed asse s, Pa e son, K. D. (2000). The s ochas ic con ol app oach e e s o a me hodology used in inancial modeling and decision-making unde unce ain y, Mudchana ongsuk, e al (2008). I in ol es u ilizing ma hema ical echniques om s ochas ic analysis and con ol heo y o op imize decision- making p ocesses in si ua ions whe e ou comes a e in luenced by andom a iables. This app oach is pa icula ly ele an in inance, whe e ma ke dynamics a e o en unp edic able and subjec o s ochas ic p ocesses. By applying s ochas ic con ol echniques, p ac i ione s aim o de ise s a egies ha maximize desi ed objec i es while accoun ing o he inhe en andomness and unce ain y p esen in inancial ma ke s. 10 Las ly, he copula app oach is a s a is ical me hod used o model he dependence s uc u e be ween mul iple a iables, pa icula ly in inance and isk managemen . The objec i e o he copula app oach is o apply he op imal copula be ween wo s ock e u ns and de ec ela i e posi ions be ween pai s, Liew and Wu (2013). I in ol es sepa a ing he ma ginal dis ibu ions o indi idual a iables om hei join dis ibu ion, allowing o a lexible and comp ehensi e analysis o hei in e ela ionships. Copulas a e unc ions ha link he ma ginal dis ibu ions o he join dis ibu ion, cap u ing he dependence pa e ns ega dless o he speci ic dis ibu ions o he indi idual a iables. This app oach is aluable o assessing and managing a ious ypes o isk, such as ma ke isk, c edi isk, and ope a ional isk, by accu a ely modeling he dependency be ween di e en ac o s and asse s. Addi ionally, copulas a e used in po olio op imiza ion, p icing o complex inancial p oduc s, and measu ing sys emic isk in inancial sys ems. 3.4. SELECTION OF APPROACH K auss (2017) conduc ed a s udy ha compa ed he pe o mance o a ious pai s ading s a egies. Fo he dis ance app oach, he annualized e u n o equi ies anges be ween 7% and 11%. Al hough Vidyamu hy's (2004) pape , which is he mos equen ly ci ed o he coin eg a ion app oach, does no p o ide empi ical esul s, Caldei a and Mou a (2013) epo an annualized e u n o 16,38% o he coin eg a ion me hod in equi ies despi e he esea ch only examine he pe iod om 2005 o 2010 o B azilian s ocks. Despi e his, se e al indica o s sugges ha he coin eg a ion app oach may be supe io o he dis ance app oach. The dis ance app oach in ol es a simple me hodology, me ely measu ing he dis ance be ween p ice indices. In con as , he coin eg a ion app oach, which in ol es eg ession analysis and s a iona i y es s o selec adeable pai s, is conside ed mo e obus om an econome ic pe spec i e, consequen ly, I will use in his s udy he coin eg a ion app oach. 3.5. LEAST SQUARES REGRESSION As s a ed be o e, pai s ading is a ep esen a i e ma ke -neu al ading s a egy ha simul aneously, longs an unde alued s ock and sho s an o e alued s ock. This s a egy is a o m o s a is ical a bi age ading ha assumes he mo emen s o he p ices o he wo asse s 11 will be like p e ious ends. I ollows he hypo hesis ha p ices will e u n o he long- e m equilib ium. This s a egy s a ed om he idea ha a bi age oppo uni ies exis when he p ice gap be ween wo asse s expands o o pas a ce ain le el. I is also based on he belie ha his o ical p ice mo emen s will no change signi ican ly in he u u e. Ga e , e al. (2006). Leas Squa es Reg ession is a me hod used o es ima e he ela ionship be ween wo a iables by i ing a line ha minimizes he sum o he squa ed di e ences be ween he obse ed alues and he alues p edic ed by he line. Bjö ck, Å. (1990). In he con ex o pai s ading, i is used o model he linea ela ionship be ween he p ices o wo asse s. The basic o m o he eg ession model is: 𝑦𝑖 =𝛽0+ 𝛽1𝑥𝑖+𝜖𝑖 (1) Equa ion (1) de ines he simple linea eg ession model. I is also called he wo- a iable linea eg ession model o bi a ia e linea eg ession model because i ela es he explana o y (𝑥) and explained (𝑦) a iables. The a iable 𝜖, called he esidual e m o dis u bance in he ela ionship, ep esen s ac o s o he han 𝑥 ha a ec 𝑦 (Woold idge, 1996). The Engle-G ange wo-s ep me hod di ec ly uses leas squa es eg ession as i eg esses 𝑦𝑡 on 𝑥𝑡 using o dina y leas squa es (OLS) o ob ain he esiduals 𝜖𝑡. Then i es s he esiduals 𝜖𝑡 o s a iona i y using a uni oo es , like he Augmen ed Dickey-Fulle es . I he esiduals a e ound o be s a iona y, he se ies 𝑥𝑡 and 𝑦𝑡 a e conside ed coin eg a ed. Rega ding i s di ec implica ion in he pai s ading s a egy, i ´s e y impo an in he s ep o calcula ing he sp ead. So, he mechanism is he ollowing, i s , a pai o s ocks wi h simila ends is iden i ied. Second, eg ession analysis such as o dina y leas squa es (OLS), o al leas squa es (TLS), and e o co ec ion models (ECM) is used o calcula e he sp ead o hese s ocks. Finally, i he sp ead hi s p ese bounda ies, in es o s will open a po olio ha akes a long posi ion on he unde alued s ock and sho s he o e alued s ock. Subsequen ly, i he sp ead e e ses o he mean, in es o s will close he po olios ha a e opposi e posi ion o he open po olio. In his case, he in es o ob ains an a bi age p o i by execu ing his s a egy. Howe e , he e is a isk when he sp ead does no e e se o he mean. In such a si ua ion, in es o s a e a high isk because hey canno close he po olio. 12 3.6. RISKS Pai s ading can be ela i ely low isk i pai s a e well-selec ed and he ela ionships uly a e mean e e ing, so ha ´s because my hesis is based on he coin eg a ion app oach. I can also be neu al o o e all ma ke mo emen s, ocusing only on he ela ionship be ween he pai . By se ing a s op-loss bounda y, in es o s can hedge he isk. Many esea che s ha e applied a ious s a is ical me hods o imp o e he e iciency and pe o mance o pai s ading. They ocused on using he sp ead as a ading signal. The challenges ha I will ha e he e in ol e model isk (inco ec model o pa ame e s), highe d awdowns ha may lead o ex ended non- ealized losses, and he b eakdown o his o ical ela ionships due o s uc u al ma ke changes, i a black swan e en occu s in one o hem, like aud o some simila e en like he En on case ha leads o bank up cy. This isk is why he e u ns on indi idual s ocks canno be aken as s a iona y. Howe e , as we a e sho in he o e alued s ock, his isk is diminished. 3.7. DATA The da a consis s o i y s ocks om he S&P500, o ming wen y- i e pai s ha will be analyzed. The da a is om 01-01-2020 o 30-04-2024 and i was ob ained h ough Yahoo Finance. I was adjus ed o spli s and o he co po a e ac ions. The s ocks ha I chose we e based on he c i e ia ha hey mus be om he same sec o o acili a e coin eg a ion. The e a e i y s ocks bu ha shouldn’ be a p oblem as Alexande e al. (2002) p o e, e icien long- sho hedge s a egies can be accomplished wi h ela i ely ew s ocks, he e o e his is no a p oblem. Fu he mo e, hedge unds usually only use pai s ha belong o he same ac i i y sec o , and I will do ha oo. I ha e chosen ma ke da a om he US s ock ma ke because i is he s ock ma ke wi h highe liquidi y, mo e da a, and mo e eliable nowadays. 13 4. EMPIRICAL STUDY 4.1. RESEARCH QUESTIONS In his hesis on pai s ading using he coin eg a ion app oach, i is essen ial o o mula e esea ch ques ions ha guide he in es iga ion in o he me hodology, e ec i eness, and applica ion o his ading s a egy. The e o e, o s a he empi ical s udy, his pape will ocus mainly on he ollowing esea ch ques ions: How does he pai s ading s a egy wi h he coin eg a ion app oach pe o m, using US s ock ma ke da a, in he cu en inancial ma ke s en i onmen ? Wha is he isk-adjus ed e u ns, Sha pe a ios, and o he pe o mance me ics o pai s ading using coin eg a ion? Who p o ides a be e e u n, coin eg a ed o non-coin eg a ed pai s? 4.2. ASSUMPTIONS Fi s o all, he assump ions made in his wo k we e made o make his a be e esea ch and o bes se e his wo k. The ansac ion cos s we e assumed a 0, due o he di icul y o quan i ying hem. Then, he be a was assumed o be 0, as his is a ma ke -neu al s a egy, o a s a is ical a bi age, in my poin o iew, he be a a 0 makes sense due o he absence o isk. Wi h he be a a 0, he CAPM will be he alue o he isk- ee a e, which I assumed o be he 10-yea easu y bond on 30 Ap il 2024, which s ands a 4,69%. Rega ding he capi al in es ed, he e u ns a e shown wi h only 1 uni , so I will no assume in his wo k ha money was in es ed, I will wo k wi h uni s and pe cen ages. 4.3. COINTEGRATION TESTS As s a ed be o e, he e is a e y con enien package, egcm, ha compu es he Engle-G ange coin eg a ion es and many o he uni - oo es s. In pa icula , gi en wo se ies 𝑥(𝑡) and 𝑦(𝑡), i sea ches o pa ame e s 𝛼, 𝛽, and 𝜌 such ha 𝑦𝑡 =𝛼 + 𝛽𝑥𝑡+𝑟𝑡 (2) 𝑟𝑡 =ρ𝑟𝑡−1 +𝜖𝑡 (3) 14 whe e 𝑟𝑡 is he eg ession esidual and 𝜖𝑡 he inno a ion. I |ρ|<1, hen 𝑥𝑡 and 𝑦𝑡 a e coin eg a ed (i.e., 𝑟𝑡 doesn’ con ain a uni oo ). O cou se, he di icul y is in assessing exac ly how much smalle han 1 he alue o |𝜌| has o be. This is he ask o he uni - oo es . The mos common example o a uni - oo es is he Augmen ed Dickey-Fulle (ADF) es as I s a ed be o e in he me hodology, which conside s a null hypo hesis ha a uni oo is p esen and an al e na i e hypo hesis ha he se ies is s a iona y (so a small p𝑝- alue means an indica ion o s ong s a iona i y). Complemen a y es s o he Augmen ed Dickey-Fulle (ADF) app oach include o ins ance Kwia kowski e al. (1992). Ano he in e es ing quan i y o measu e mean- e e sion is he hal -li e, de ined as he ime i akes o he sp ead o mean- e e hal o i s dis ance a e ha ing di e ged om he mean o he sp ead. The egcm lib a y in R is designed o he es ima ion and es ing o coin eg a ing ela ionships be ween pai s o inancial ime se ies. egcm s ands o "Engle-G ange Coin eg a ion Models," as s a ed be o e, which e e s o he s a is ical me hodology used o iden i y coin eg a ed pai s. This package simpli ies he p ocess o inding and es ing coin eg a ed pai s, which is pa icula ly use ul in pai s ading s a egies. Le ´s check i Visa and Mas e ca d a e coin eg a ed. In Figu e 1 below, he p ice se ies shows a eg ession model o Visa in o Mas e ca d because he s ock p ice o Mas e ca d is highe . Residual se ies is he di e ences in p ices, so he sp ead and he inno a ions a e he changes in esiduals. 21 posi ion is closed i he Z-sco e is non-nega i e, and i a sho posi ion is held (signal[ -1] = - 1), he posi ion is closed i he Z-sco e is non-posi i e. Figu e 6: Z-sco e and ading signal 4.5.2. PERFORMANCE OF THE STRATEGY Finally, we can compu e he e u n p o i and loss o he s a egy. A simple way o doing his is di ec ly om he sp ead and he ading signal. Acco ding o Figu e 21, he sp ead e u n is compu ed as he di e ence be ween consecu i e alues o he sp ead. The aded e u n is calcula ed by mul iplying he sp ead e u n by he lagged signal. The lag(signal) shi s he signal by a one- ime s ep o align he posi ion wi h he subsequen e u n. This ensu es ha he posi ion aken a ime -1 a ec s he e u n a ime . In Figu e 7 below, i ´s possible o analyze his e u n p o i and loss o he s a egy, hus he e u n o he sp ead be ween wo s ocks. Some spikes indica e he ola ili y in he ma ke s a ha momen , as a he beginning o 2019 and Ma ch 2020, wi h he COVID-19 c isis. 22 Figu e 7: Re u n o aded sp ead Conce ning he weal h o cumula i e p o i and loss, he e a e wo di e en ways o compu e i . Wi h ein es men o no compounded. The same ini ial budge o , say, only 1€ is in es ed e e y ime a new ade. Wi h ein es men , all he exis ing weal h a each ime is ully ein es ed. I will only show he compounded way because as we will see la e , i is he mos p o i able, howe e , he no compounded is mo e eliable because e u ns on indexes like S&P500 a e no compounded. In Figu e 8 he e is he plo o he cumula i e p o i and loss o he aded sp ead, showing esilience in he esul s, e en a e he end o he aining pe iod, showing obus esul s on his s a egy and mo e speci ically in his coin eg a ed pai . 23 Figu e 8: P&L o aded sp ead wi h ein es men We can obse e ha weal h inc eases s eadily du ing he aining pe iod and in his case, i con inues o ise a e ha . Howe e , in some cases, i la ens ou du ing he es o ou -o - sample pe iod, which is he one ha coun s. The same plo s can be ob ained wi h he package Pe o manceAnaly ics, and he a gumen geome ic de e mines whe he i is compounded o no . Howe e , I ha e used his lib a y o plo he d awdowns o he aded sp ead e u ns. The below Figu e 9 shows he d awdown o he s a egy. This plo shows pa o he isk associa ed wi h his s a egy. La ge d awdowns indica e highe isk as hey can lead o ex ended non- ealized losses in some pe iods, he e o e, a s a egy wi h smalle and sho e d awdowns is conside ed less isky. I ´s no able he co ela ion be ween he d awdown and he imes ha he e a e no any mean e e ing sp ead. Fo example, in 2021 we see a big d awdown and he z-sco e eaches a s anda d de ia ion o 3. 24 Figu e 9: D awdown Now I will inpu he ading signal wi h he z-sco e. The e is a wo-panel plo o isualize he Z-sco e wi h ading signals and he cumula i e p o i and loss o he aded sp ead. In Figu e 10 we can see he Z-sco e and ading signal wi h he cumula i e p o i and loss o aded sp ead. This p o ides a clea iew o he s a egy's p o i abili y and g ow h o e he selec ed pe iod. I helps o assess he long- e m iabili y o he ading s a egy. We can see he ela ion be ween ading signals and he p o i and loss o he s a egy. In he pe iod om July 2021 o Janua y 2022, only one ading signal was gene a ed, esul ing in a ime o declining e u ns. Figu e 10: T ading signal and cumula i e P&L A mo e e icien way o igu e ou he e u n and P&L o he s a egy is ia he equi alen po olio. The p e ious way o calcula ing he P&L di ec ly om he sp ead can lead o w ong esul s i γ o μ change o e ime. I ha happens, hen he sp ead may look e y nice bu i ’s no ealis ic because once I am in a ade, shouldn’ change 𝛾 o 𝜇. Le ’s use now he equi alen 25 po olio app oach. Fi s , I combine he no malized po olio weigh s wi h he lagged ading signal o c ea e he po olio weigh s. A e ha , un he log e u ns om he log-p ices and calcula e he po olio e u ns by applying he po olio weigh s o he log- e u ns. The plo shows he Z-sco e again and he cumula i e p o i and loss o he aded sp ead. Figu e 11 shows ha wi h his equi alen po olio app oach, he pe o mance o his pai s ading s a egy is be e , accumula ing p o i s om 1 o o e 1.8. Wi hou his app oach, he e u ns we e sligh ly abo e 1.6, as shown in Figu e 10. Figu e 11: Equi alen po olio app oach code 4.6. RISK-ADJUSTED PERFORMANCE METRICS 4.6.1. CAPITAL ASSET PRICING MODEL (CAPM) Rega ding isk-adjus ed pe o mance me ics, I ha e used some well-known me ics, sha pe a io, and CAPM. The Capi al Asse P icing Model (CAPM) de eloped by William Sha pe, John Lin ne , and Jan Mossin in he 1960s, ep esen s a pi o al de elopmen in inancial economics, Sha pe (1964) and Lin ne (1965). I is a inance model ha es ablishes a linea ela ionship be ween he equi ed e u n on in es men and isk. CAPM ex ends he p inciples o po olio heo y by Ha y Ma kowi z, p o iding a amewo k o quan i y he ela ionship be ween sys ema ic isk and expec ed e u n o asse s, Ma kowi z (1959), who a gues ha in es o s a e isk a e se and will choose a po olio by ading o be ween isk and e u n o one in es men pe iod. The e o e, in es o s will choose e icien po olios ha minimize he 26 a iance o po olio e u n, gi en a speci ic le el o expec ed e u n, o maximize expec ed e u n, gi en a speci ic le el o a iance. In heo y, he capi al asse p icing model is employed o se he in es o equi ed a e o e u n on a isky secu i y gi en he non-di e si iable i m-speci ic isk, as he sys ema ic isk will be elimina ed in a well-di e si ied po olio. The CAPM equa ion acco ding o Sha pe and Lin ne 's assump ions o isk- ee bo owing and lending is exp essed as ollows: 𝐸(𝑅𝑖) = 𝑅𝑓+𝛽𝑖[𝐸 (𝑅𝑚)−𝑅𝑓] (8) Whe e E(Ri) is he expec ed e u n on asse I, 𝑅𝑓 is he isk- ee a e, βi is he be a o asse I and E(Rm) is he expec ed e u n o he ma ke po olio Whe e ma ke Be a: 𝛽𝑖 = 𝐶𝑂𝑉 (𝑅𝑖;𝑅𝑚) 𝜎² (𝑅𝑚) (9) Whe e 𝐶𝑂𝑉(𝑅𝑖,𝑅𝑚) is he co a iance be ween he e u n o he s ock (Ri) and he e u n o he ma ke (Rm) and 𝜎² (𝑅𝑚) is he a iance o he ma ke e u n. 4.6.1.1. BETA The be a o a pai s ading s a egy is ypically e y low and o en close o ze o. This is because pai s ading is a ma ke -neu al s a egy, meaning i is designed o be independen o he o e all ma ke mo emen s. The s a egy in ol es aking simul aneous long and sho posi ions in wo co ela ed s ocks, wi h he expec a ion ha hei p ice mo emen s will con e ge, i espec i e o ma ke di ec ion. The key poin o be a in pai s ading is ma ke neu ali y since pai s ading aims o hedge ou ma ke isk by aking opposi e posi ions in wo co ela ed s ocks. This neu alizes he e ec o b oade ma ke mo emen s on he s a egy's pe o mance. Then, he e u ns o he pai s ading s a egy a e d i en by he ela i e p ice changes be ween he wo s ocks a he han he o e all ma ke mo emen s, he be a o he s a egy wi h espec o he ma ke index is usually e y low, close o ze o and he sho posi ion in one s ock o se s he ma ke isk o he long posi ion in he o he s ock, which con ibu es o he low be a. The be a o a pai s ading s a egy is usually close o ze o, highligh ing i s ma ke -neu al cha ac e is ic. This indica es ha he s a egy's pe o mance is independen o o e all ma ke mo emen s, elying ins ead on he ela i e p ice mo emen s o he pai ed s ocks. 27 To conclude, I will conside be a as 0 since his is a ma ke -neu al s a egy, being he alue o CAPM he cu en isk- ee a e. Fo ha I will assume he 10-yea easu y a e as o 30 Ap il 2024 as a isk- ee a e, ha is 4,69% and i will be used as a benchma k o compa e he esul o he pai s ading s a egy. 4.6.2. SHARPE RATIO Roy (1952) was he i s o sugges a isk- ewa d a io o e alua e a s a egy’s pe o mance. Sha pe (1966) applied Roy’s ideas o Ma kowi z’s mean- a iance amewo k, in wha has become one o he bes known pe o mance e alua ion me ics. Sha pe a io has become he ‘gold s anda d’ o pe o mance e alua ion. Sha pe a ios a e g ea ly a ec ed by some o he s a is ical ai s inhe en o hedge und s a egies in gene al. The Sha pe a io compa es how well an equi y in es men pe o ms o he a e o e u n on a isk- ee in es men , such as U.S. go e nmen easu y bonds o bills. When compa ing unds o po olios, in es o s should conside bo h absolu e e u ns and isks. While one po olio o und could ha e highe e u ns, i is only a good in es men i hose highe e u ns do no come wi h addi ional isk. The o mula o he sha pe a io is he ollowing: Sha pe Ra io = 𝑅𝑝−𝑅𝑓 𝜎𝑝 (10) Whe e 𝑅𝑝 is he po olio e u n, 𝑅𝑓 is he isk- ee a e and 𝜎𝑝 is he s anda d de ia ion o he po olio. A good Sha pe Ra io la gely depends on he in es o 's isk ole ance and in es men objec i es. Gene ally, any alue g ea e han 1 is conside ed accep able, while a a io highe han 2. is conside ed e y good, and a a io o 3 o highe is conside ed excellen . Howe e , i 's impo an o no e ha a highe Sha pe Ra io may no always be be e , as i could be a esul o aking on excessi e isk. Ul ima ely, in es o s should e alua e hei po olios based on hei unique in es men goals and isk ole ance, using he Sha pe Ra io as one o many ools o making in o med in es men decisions. Fo he Sha pe Ra io calcula ion, I ha e used he annual e u n wi h ein es men as he po olio e u n, he isk- ee a e is he 10-yea easu y a e and I ha e es ima ed he s anda d de ia ion in Rs udio o e e y pai o s ocks. 28 5. RESULTS AND DISCUSSION 5.1. COINTEGRATION RESULTS Fo ins ance, le ´s check he case o Visa and Mas e ca d. Based on he p o ided eg ession and uni oo es s, I need o assess whe he he e is a coin eg a ion ela ionship be ween MA and V based on he in o ma ion abou he esiduals. Reg ession Model and Residuals The gi en eg ession model: MA[i]=1.8454×V[i]−47.0284+R[i] whe e R[i] =0.9660R[i−1]+ϵ[i],ϵ[i]∼N(0,2.23422) sugges s ha MA is eg essed on V and he esiduals R[i] a e supposed o ollow an AR(1) p ocess. Howe e , he e's a wa ning ha he esiduals do no exhibi a pe ec AR(1) s uc u e. Now, le 's e iew he uni oo es s applied o he esiduals ha can be seen in Table 1 below: V-MA S a is ics P-Value -5.618 0.00010 -63.894 0.00010 0.963 0.00010 -4.203 0.00219 -38.439 0.00010 -50.648 0.00372 Table 1: Uni oo es s Rega ding he Augmen ed Dickey-Fulle (ADF), ha is he es ha I will ely mo e on since i es s he abili y o accoun o highe -o de au oco ela ion, has lexibili y in modeling di e en ypes o ime se ies da a, and i s well-es ablished p esence in he li e a u e and i s ease o use, make a good choice o ely on es ing o uni oo s. I has a es s a is ic o -5.618 and a p- alue o 0.00010, which indica es s ong e idence agains he null hypo hesis o a uni oo , sugges ing ha he esiduals a e s a iona y. The P- alue is smalle han he mos common signi icance le el o 0.05, and i ´s e en mo e mino han he 0.01 le el. Rega ding he s a is ics, he alue o -5.618 is smalle han he c i ical alue o he ADF es o he signi icance le el o 0.01 which is -3.43. The c i ical alues o he ADF es a e in he ollowing Table 2, acco ding o Cook (2001). In his esea ch, Cook used se e al sample sizes. I will use he 80- sample size, e en i mine is only 50. Fo all sample sizes (including I = o) and a all le els o 29 signi icance, he MK c i ical alues a e g ea e in absolu e alue han hose o Cheung and Lai (1995). Consequen ly, he MacKinnon alues u he exagge a e he low powe p oblem associa ed wi h ADF es s. Howe e , MacKinnon used o calcula e c i ical alues o any sample size, and his has become s anda d p ac ice in empi ical wo k, so I will use i oo. On he le , we ha e he signi icance le el, and on he igh , he ma ched c i ical alue o his es . Signi icance Le el C i ical Value 1% -4,076 5% -3,466 10% -3,159 Table 2- C i ical alues A mo e nega i e es s a is ic alling in he ejec ion egion means we ha e s a is ically signi ican e idence o conclude ha he se ies is s a iona y and does no ha e a uni oo . By ocusing on mo e nega i e alues, hese es s help iden i y when a ime se ies is likely o be s a iona y, which is c ucial o making meaning ul in e ences and a oiding alse eg essions in ime se ies analysis. Fo he coin eg a ion es s, a p- alue less han 0.05 indica es s ong e idence agains he null hypo hesis, allowing me o ejec i and I will ely on ha alue. Howe e , a p- alue less han 0.01 indica es e en s onge e idence agains he null hypo hesis. I i ´s g ea e han 0.05 and less han 0.10, he e idence is weak, and I will no ha e con idence in ha da a. Rega ding he coin eg a ion es s, i ´s no usual o a pai o wo s ocks o pass on e e y uni oo es , so I will accep ha a pai is coin eg a ed i 3 o he 5 es s a e below he 0.05 p- alue (95% con idence). Rega ding he o he es s, he Phillips-Pe on (PP) es also con i ms he s a iona i y o he esiduals wi h a s a is ic o -63.894 and a p- alue o 0.00010. Then Pan ula, Gonzales-Fa ias, and Fulle 's (PGFF) es unusually shows a s a is ic o 0.963 bu a e y low p- alue o 0.00010, ypically sugges ing ejec ion o he null hypo hesis o a uni oo , hough he in e p e a ion migh need con ex . Ellio , Ro henbe g, and S ock DF-GLS (ERSD) es wi h a s a is ic o - 4.203 and a p- alue o 0.00219 also sugges s ha he esiduals a e s a iona y. Johansen's T ace Tes (JOT) and Schmid and Phillips Rho (SPR): Bo h o hese coin eg a ion-speci ic es s indica e no uni oo , suppo ing he p esence o coin eg a ion. Rega ding he case o he PGFF es ha has a posi i e s a is ic, I will dis ega d ha and conside an ou lie , ega ding he o he decen alues o ejec he null hypo hesis. 30 The e o e, despi e he wa ning ha he esiduals a e no pe ec ly au o eg essi e (AR(1)), almos all uni oo es s on he esiduals s ongly sugges ha hey a e s a iona y. The e o e, i can be concluded ha Mas e ca d and Visa a e coin eg a ed. Rega ding he coin eg a ion es s, we can conside ha 8 pai s ha e some so o coin eg a ion, so a p- alue smalle han he signi icance le el o a s a is ic smalle han he c i ical alue. We ha e o he pai s ha a e no coin eg a ed bu ha e shown a good pe o mance. Bu in he nex s ep, we will see he esul s. 5.2. RISK-ADJUSTED PERFORMANCE METRICS RESULTS We ha e o check he isk o he po olio. The isk pe o mance me ics o he in es men po olio we e calcula ed o he pe iod om Janua y 1, 2016, o Ap il 30, 2024. These me ics p o ide insigh s in o he isk-adjus ed e u ns and o e all ola ili y o he po olio. The key me ics conside ed include he Sha pe Ra io, s anda d de ia ion, and CAPM. As I s a ed be o e, o he Sha pe Ra io calcula ion, I ha e used he annual e u n wi h ein es men as he po olio e u n, he isk- ee a e is he 10-yea easu y a e, which s ands a 4,69% and hen I ha e es ima ed he s anda d de ia ion in Rs udio o e e y pai o s ocks. The s anda d de ia ion was calcula ed om he sp ead o his s a egy in each pai . The sp ead is how much he p ice o wo s ocks will de ia e om each o he , so i will de he s anda d de ia ion o he s a egy. As men ioned be o e, I ha e assumed he be a a 0 o his pai s ading s a egy, so he alue o CAPM will be he isk- ee a , which s ands a 4,69% on 30 Ap il 2024. Fo he coin eg a ed pai s, in Table 3, we can see ha he pai wi h he bes Sha pe Ra io, so, he pai wi h he bes pe o mance and less isk, was Visa and Mas e ca d (V-MA). The pai wi h he lowes Sha pe Ra io was he NVO-LLY, wi h a alue o 0,05. This shows he co ela ion be ween s anda d de ia ion and Sha pe a io, as his pai had he highes alue in s anda d de ia ion and he lowes in Sha pe a io, being a pai wi h highe isk han he emaining ones. 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In oduc o y Econome ics: A Mode n App oach 3 d ed. 40 APPENDIX A PYPL-SQ V-MA ADBE-ADSK Uni Roo Tes s o Residuals S a is ics P-Value S a is ics P-Value S a is ics P-Value Augmen ed Dickey-Fulle (ADF) -2,899 0,13613 -5.618 0.00010 -3,664 0,02055 Phillips-Pe on (PP) -22,654 0,03519 -63.894 0.00010 -25,024 0,02152 Pan ula, Gonzales-Fa ias and Fulle (PGFF) 0,988 0,04338 0.963 0.00010 0,989 0,05725 Ellio , Ro henbe g and S ock DF-GLS (ERSD) -2,995 0,01597 -4.203 0.00219 -3,033 0,0143 Johansen's T ace Tes (JOT) -19,739 0,06185 -38.439 0.00010 -15,624 0,19575 Schmid and Phillips Rho (SPR) -33,325 0,01401 -50.648 0.00372 -28,89 0,02589 JPM-GS MRK-ABBV BK-FAF KO-PEP F-GM S a is ics P-Value S a is ics P-Value S a is ics P-Value S a is ics P-Value S a is ics P-Value -2,034 0,50211 -2,096 0,47323 -2,273 0,39057 -3,686 0,01945 -2,048 0,49574 -8,245 0,48574 -8,589 0,46715 -13,704 0,19541 -29,821 0,00926 -8,338 0,48073 0,996 0,40222 0,996 0,38186 0,994 0,21177 0,985 0,02218 0,996 0,48476 -1,434 0,34067 -1,912 0,16419 -1,73 0,21792 -3,498 0,00733 -1,141 0,46227 -8,208 0,80636 -12,234 0,44343 -11,835 0,47306 -18,578 0,08778 -15,236 0,22084 -6,062 0,71896 -10,637 0,42453 -10,385 0,43614 -29,358 0,02426 -6,354 0,69717 GOOG-META JNJ-PFE NVDA-AMD AVGO-QCOM INTC-MU S a is ics P-Value S a is ics P-Value S a is ics P-Value S a is ics P-Value S a is ics P-Value -2,151 0,44741 -1,695 0,65842 -0,672 0,94361 -1,584 0,70978 0,686 0,99659 -8,454 0,47444 -9,256 0,4311 -2,316 0,91346 -5,552 0,69044 1,509 0,99732 0,996 0,39396 0,995 0,36028 0,999 0,84704 0,997 0,57235 1,001 0,99385 -1,662 0,24618 -1,537 0,29792 -0,941 0,54731 -1,34 0,3797 1,23 0,99021 -15,726 0,19171 -8,802 0,75189 -33,024 0,00275 -16,98 0,14181 -7,926 0,82765 -9,48 0,47778 -8,209 0,55888 -2,528 0,95851 -5,097 0,79092 -6,644 0,67555 PANW-CRWD BMBL-MTCH SPGI-MCO AAPL-MSFT NVO-LLY S a is ics P-Value S a is ics P-Value S a is ics P-Value S a is ics P-Value S a is ics P-Value -1,558 0,71967 -2,395 0,32847 -3,9 0,00977 -1,241 0,83804 -3,406 0,04135 -4,629 0,76381 -11,064 0,33452 -45,294 0,00123 -7,561 0,53288 -37,15 0,00546 0,996 0,62086 0,986 0,27946 0,969 0,00134 0,995 0,32028 0,981 0,00673 -0,798 0,61468 -1,596 0,27805 -2,306 0,0781 -1,049 0,5007 -2,684 0,03395 -5,442 0,96755 -20,057 0,05603 -30,662 0,00495 -11,318 0,51443 -44,254 0,0001 -2,99 0,92651 -6,166 0,71848 1,569 0,99987 -4,616 0,82614 -11,688 0,3762 41 WFC-C BKNG-ABNB WMT-COST XOM-CVX NFLX-DIS S a is ics P-Value S a is ics P-Value S a is ics P-Value S a is ics P-Value S a is ics P-Value -2,712 0,19101 -2,117 0,45953 -2,145 0,45032 -2,438 0,31329 -1,924 0,55317 -9,384 0,42424 -10,031 0,3902 -10,195 0,38041 -8,532 0,47025 -7,522 0,53596 0,996 0,46758 0,986 0,24303 0,995 0,29086 0,997 0,60348 0,995 0,37004 -1,195 0,43992 -2,015 0,14009 -1,333 0,38261 -0,81 0,60412 -1,642 0,25461 -8,559 0,77487 -7,64 0,85199 -14,887 0,24671 -9,388 0,69665 -10,371 0,60385 -12,801 0,32498 -7,331 0,63864 -6,474 0,6882 -12,989 0,31634 -7,06 0,64452 Pai s ading s a egy in es men e u n Wi hou ein es men Wi h ein es men Annual e u n wi hou / ein es men % Annual e u n wi h/ ein es men % PANW-CRWD 1,1260 1,0096 1,51% 0,11% BMBL-MTCH 1,7645 1,9060 9,17% 10,87% AAPL-MSFT 1,5186 1,6185 6,22% 7,42% JPM-GS 1,3031 1,3187 3,64% 3,82% MRK-ABBV 1,2509 1,2147 3,01% 2,58% BK-FAF 1,7390 1,8721 8,87% 10,46% F-GM 1,2961 1,2389 3,55% 2,87% GOOG-META 1,6795 1,7873 8,15% 9,45% JNJ-PFE 1,4594 1,5277 5,51% 6,33% NVDA-AMD 1,1729 0,9781 2,07% -0,26% AVGO-QCOM 1,3188 1,2692 3,83% 3,23% INTC-MU 1,4681 1,3729 5,62% 4,47% WFC-C 1,3251 1,2977 3,90% 3,57% BKNG-ABNB 1,7176 1,9196 8,61% 11,04% WMT-COST 1,4191 1,4653 5,03% 5,58% XOM-CVX 1,1179 1,0967 1,41% 1,16% NFLX-DIS 1,7015 1,7702 8,42% 9,24% MCD-SBUX 1,2849 1,2797 3,42% 3,36% PM-MO 1,4333 1,4806 5,20% 5,77% PYPL-SQ 1,7868 1,9335 9,44% 11,20% V-MA 1,6485 1,8925 7,78% 10,71% ADBE-ADSK 1,9520 2,3942 11,42% 16,73% SPGI-MCO 1,7584 2,1041 9,10% 13,25% NVO-LLY 1,4232 1,4064 5,08% 4,88% KO-PEP 1,5030 1,6280 6,04% 7,54% Mean 1,6787 1,8931 5,84% 6,62% Table 8: Pai s ading s a egy in es men e u n MCD-SBUX PM-MO S a is ics P-Value S a is ics P-Value -1,8 0,60997 -2,918 0,1305 -8,632 0,46482 -16,456 0,12084 0,995 0,31055 0,992 0,14275 -1,702 0,22963 -2,851 0,02235 -9,989 0,63993 -12,069 0,45568 -4,053 0,8671 -19,296 0,11709 Table 7: Uni Roo es o esiduals 42 Coin eg a ed Pai s Wi hou ein es men Wi h ein es men Annual e u n wi hou / ein es men % Annual e u n wi h/ ein es men % PYPL-SQ 1,7868 1,9335 9,84% 11,20% V-MA 1,6485 1,8925 8,11% 10,71% ADBE-ADSK 1,9520 2,3942 11,90% 16,73% SPGI-MCO 1,7584 2,1041 9,48% 13,25% NVO-LLY 1,4232 1,4064 5,29% 4,88% KO-PEP 1,5030 1,6280 6,29% 7,54% Mean 1,6787 1,8931 8,48% 10,72% Non Coin eg a ed Pai s Annual e u n wi h/ ein es men % S anda d De ia ion Sha pe Ra io PANW-CRWD 0,12% 8,72% -0,52 BMBL-MTCH 11,33% 8,66% 0,77 AAPL-MSFT 7,73% 5,87% 0,52 JPM-GS 3,98% 7,89% -0,09 MRK-ABBV 2,68% 10,54% -0,19 BK-FAF 10,90% 8,74% 0,71 F-GM 2,99% 9,96% -0,17 GOOG-META 9,84% 4,61% 1,12 JNJ-PFE 6,60% 4,38% 0,44 NVDA-AMD -0,27% 18,32% -0,27 INTC-MU 4,66% 7,85% 0,00 BKNG-ABNB 11,49% 4,99% 1,36 WMT-COST 5,82% 3,86% 0,29 XOM-CVX 1,21% 10,12% -0,34 NFLX-DIS 9,63% 12,63% 0,39 MCD-SBUX 3,50% 6,80% -0,18 PM-MO 6,01% 6,43% 0,20 Mean 5,17% 7,39% 0,21 Table 10: Sha pe Ra ion non-coin eg a ed pai s Table 9: Pai s ading e u n o coin eg a ed pai s 43 Figu e 14: P ices Loading Figu e 12: Lib a y´s used Figu e 13: Coin eg a ion Code 44 Figu e 15: T aining se and LS eg ession Figu e 16: Sp ead Equa ion Figu e 17: No malized Po olio and No malized Sp ead 45 Figu e 18: Z-sco e code Figu e 19: Sp ead S anda d De ia ion Figu e 20: Z-sco e and ading signal code 46 Figu e 22: P&L o aded sp ead wi h ein es men code Figu e 23: D awdown code Figu e 24: T ading signal wi h Z-sco e code Figu e 25: Equi alen po olio app oach Figu e 21: Re u n o aded sp ead code