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.
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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.
Table 3: Sha pe Ra io
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
PYPL-SQ
11,20%
8,16%
0,80
V-MA
10,71%
1,42%
4,24
ADBE-ADSK
16,73%
5,12%
2,35
SPGI-MCO
13,25%
2,05%
4,17
NVO-LLY
4,88%
8,27%
0,02
KO-PEP
7,54%
2,65%
1,07
Mean
10,72%
4,61%
2,11
37
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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