142 2021, XXIV, 4
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10.15240/ ul/001/2021-4-009
STOCK PRICE PREDICTION USING MARKOV
CHAINS ANALYSIS WITH VARYING STATE
SPACE ON DATA FROM THE CZECH
REPUBLIC
Milan S oboda1, Pa la Řího á2
1 Uni e si y o Wes Bohemia in Pilsen, Facul y o Economics, Depa men o Economics and Quan i a i e Me hods,
Czech Republic, [email p o ec ed];
2 Uni e si y o Wes Bohemia in Pilsen, Facul y o Economics, Depa men o Economics and Quan i a i e Me hods,
Czech Republic, ORCID: 0000-0002-6632-445X, [email p o ec ed].
Abs ac : The a icle desc ibes empi ical esea ch ha deals wi h sho - e m s ock p ice p edic ion.
The aim o his s udy is o use his p edic ion o c ea e success ul business models. A business
model ha ou pe o ms he s ock ma ke , ep esen ed by he Buy and Hold s a egy, is conside ed
o be success ul. A s ochas ic model based on Ma ko chains analysis wi h a ying s a e space is
used o sho - e m s ock p ice p edic ion. The a ying s a e spa e is de ined based on mul iples
o he mo ing s anda d de ia ion. A o al o 80 s a e space models we e calcula ed o he mo ing
s anda d de ia ion wi h 5-s ep leng hs om 10 o 30 in combina ion wi h he s anda d de ia ion
mul iples om 0.5 o 2.0 wi h he s ep o 0.1. The e iciency o he business models was e i ied o
3 long- e m, liquid s ocks o he Czech s ock ma ke , namely he s ocks o KB, CEZ, and O2 wi hin
a 14-yea pe iod – om he beginning o 2006 o he end o 2019. Business models pe o m bes
when hey use a s a e space de ined on he leng h o a mo ing s anda d de ia ion be ween 15 and
30 in combina ion wi h mul iples o he s anda d de ia ion be ween 1.1 and 1.2. Business models
based on hese pa ame e s ou pe o m he passi e Buy and Hold s a egy. In ac , hey ou pe o m
he Buy and Hold s a egy o bo h he en i e pe iod unde e iew and he yielded i e-yea pe iods
(including ansac ion ees). The only excep ion is he i e-yea pe iods co e ing 2015 o O2 s ocks.
A e he end o he unce ain y pe iod caused by unclea in en ions o he new majo i y s ockholde ,
he s ock p ice ose sha ply. These esul s a e in con lic wi h he e icien ma ke s heo y and
sugges ha in he pe iod unde e iew, he Czech s ock ma ke was no e ec i e in any o m.
Keywo ds: S ock ma ke p edic ion, echnical analysis, Ma ko chains, e icien ma ke heo y.
JEL Classi ica ion: C02, C13, G14, G17.
APA S yle Ci a ion: S oboda, M., & Řího á, P. (2021). S ock P ice P edic ion Using Ma ko
Chains Analysis wi h Va ying S a e Space on Da a om he Czech Republic. E&M Economics
and Managemen , 24(4), 142–155. h ps://doi.o g/10.15240/ ul/001/2021-4-009
In oduc ion
This empi ical s udy deals wi h he sho - e m
p edic ion o s ock p ices on he Czech s ock
ma ke . S ock mo emen s ha e been o in e es
o ade s o a long ime. Using a wide ange
o analy ical me hods, i ies o sa is ac o ily
cla i y pas and p esen changes in s ock
p ices. Based on hese indings, i a emp s o
p edic he u u e de elopmen o s ock p ices.
Ea ly o ecas ing allows ade s o make capi al
gains.
I is necessa y o men ion ha acco ding o
E icien Ma ke Hypo hesis (EMH), s ock p ices
a e unp edic able and ma ke s a e e icien . This
means ha he ma ke esponds immedia ely o
any new in o ma ion. This in o ma ion canno
be p edic ed, i is andomly sen o he ma ke
and he e o e he change in he exchange a e
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is andom and he exchange a es pe o m
a so-called ʻ andom walk’. In e icien ma ke s
he abo e-a e age p o i s canno be achie ed
and acco ding o his heo y, o he app oaches
a e dys unc ional.
The idea o a andom walk was p obably
i s published in he doc o al disse a ion o
he F ench ma hema ician L. Bachelie in 1900.
Acco ding o Coo ne (1964), a e mo e han 60
yea s his Bachelie s udy was i s published in
English and had eco e ed ‘ he igh ul place’ in
inancial econome ics. La e ano he scien is ,
Fama (1965, 1970, 1991), e u ned o his idea
and de eloped i subs an ially. He examined
he co ela ion be ween daily e u ns and p io ’s
pe iod daily e u ns o s ocks composing he
DJI index. The co ela ion coe icien s we e
a ound ze o. He hus concluded ha he e u n
on he s ock had no ela ion o he e u n o
he p io pe iod. Fama’s conclusions we e
con i med by Solnik (1973) who wo ked wi h
he da a om Eu opean s ock ma ke s. The
e icien ma ke hypo hesis (EMH) was also
con i med by o he s udies and esea ches by
in es men economis s.
Robe s (1967) was he i s o iden i y he
pa icula o ms o e iciency as a weak, semi-
s ong and a s ong o m o e iciency. The weak
o m o e iciency means ha he exchange a e
includes all in o ma ion om his o ical da a
and he e o e, me hods o echnical analysis
a e no able o p edic he ma ke . The semi-
s ong o m o e iciency is a si ua ion whe e
he p ice includes bo h his o ical da a and all
public in o ma ion and me hods o undamen al
analysis also ail he e. The s ong o m EMH
demons a es he ac ha e en inside
in o ma ion is inco po a ed in o he s ock p ice.
In a highly e icien ma ke , he e o e, he inside
in o ma ion is wo hless and does no help
in es o s o cap u e abo e-a e age e u ns.
In he beginning, EMH was widely accep ed
by he academic communi y. La e , howe e ,
s udies ques ioning he heo y s a ed o
eme ge. Fo example, Shille (1981) d aws
a en ion o he highe ola ili y o s ock p ices
han can be explained by he ola ili y o
di idends. Haugen (1999) belie es ha ma ke s
o e eac o new unexpec ed in o ma ion
and e e s o a ious s udies ha con i m he
occu ence o anomalies in an e icien ma ke .
The Czech s ock ma ke was also examined.
In mos cases, a weak o m o e iciency is
examined in he Czech s ock ma ke . The
wo ks o Filáček e al. (1998) o Hanousek and
Němeček (2001) inclined o he opinion ha he
Czech s ock ma ke is beha ing ine icien ly.
On he con a y, mo e ecen wo ks by File and
Hanousek (1996) o Di iš and Teplý (2005),
end o demons a e ha he s ock ma ke
beha es e icien ly in a weak o m. The e o e,
he me hods o echnical analysis do no b ing
abo e-a e age e u ns.
The e is no consensus in academia
assessing he ques ion o whe he o no
ma ke s a e e icien . Fo example, esea ch by
Hájek (2007) poin ed ou ha be ween 2000 and
2005, he Czech s ock ma ke was app oaching
a weak o m o e iciency. Simul aneously, he
claimed ha he i e-yea pe iod was oo sho .
And ha s a egies ha exploi ed o sho - e m
dependences can be abno mally p o i able.
Resea ch and discussions ela ed o he
e iciency o s ock ma ke s a e likely o con inue
in he yea s o come. Cu en empi ical esea ch
con ibu es o his discussion. This pape deals
wi h a p oblem o p edic ing he di ec ion o
s ock p ices on he Czech s ock ma ke and
builds on he wo k o S oboda (2016). This
wo k desc ibes a s ochas ic model based on
Ma ko chains analysis. The model is used o
p edic sho - e m u u e s ock p ice mo emen .
The main aim o his empi ical esea ch
is o modi y he s ochas ic model men ioned
abo e and apply his model success ully o he
Czech s ock ma ke . I is possible o expec he
ollowing bene i s o he esea ch:
 Con ibu ion o he discussion o whe he
he Czech s ock ma ke can be conside ed
o ha e a weak o m e iciency.
 Ve i ica ion o he conclusions o he wo k
by S oboda (2016) in he longe e m.
The pape is o ganized in he ollowing
way. The i s pa desc ibes he heo e ical
backg ound and he impo an heo y on Ma ko
chains. The second pa o he wo k cha ac e izes
he da a used o conduc esea ch and desc ibes
he esea ch me hodology. The hi d mos
comp ehensi e pa includes he ob ained
empi ical esul s and hei b ie discussion. The
inal pa con ains he e alua ion o he wo k
pe o med and ou line he di ec ion o u he
esea ch.
1. Theo e ical Backg ound
The esea ch is based on he assump ion ha
he s ock p ice is con inuously c ea ed on he
s ock exchange by he mu ual in e ac ion
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be ween supply and demand. This supply
and demand is gene a ed by di e en ypes
o ade s (long- e m in es o s, specula o s).
These ade s ha e di e en ime ho izons,
hey use di e en me hods o es ima e he
u u e income hey will ob ain by owning s ocks,
and hey ha e di e en isk a e sion, di e en
amoun s o capi al. E e y in es o ollows his
in en ions and has di e en easons o buying
o selling a gi en s ock. Supply o demand o
a gi en s ock may no only be c ea ed due
o he ade ’s subjec i e pe cep ion o he
unde alua ion o o e alua ion o he s ock,
bu also o many o he easons, such as he
need o aise money, modi ica ion o in es men
s a egy, aising new capi al by seconda y
o e ing, o ced pu chases o sales in bo owed
asse s, epu chases o s ocks by he join s ock
company i sel , e c. The simul aneous ac ion o
hese ac o s esul s in he cons an luc ua ion
o he s ock p ice and can be seen as a andom
a iable.
The nex assump ion o echnical analysis is
ha s ock p ices luc ua e in ends. When he
end is iden i ied in a imely manne , ade s
hope o abo e-a e age p o i s. The subjec
o his esea ch is he o ecas ing o sho -
e m p ice mo emen s ( e ia y end), while i
is no he p ice le el ha is impo an , bu he
es ima ion o he magni ude o p ice changes.
The s ock p ice luc ua es in sho - e m ends
and du ing he du a ion o his end he s ock
p ice accumula es a ce ain gain o loss agains
he p ice a he beginning o he end. The
g ea e his change, he g ea e he likelihood o
a change in his end. The key ques ion o us is
how la ge he accumula ed loss o p o i mus be
in o de o he end o change wi h a su icien ly
high p obabili y. Acco ding o he na u e o he
da a (daily opening and closing p ices), i is
p ope o use he heo y o Ma ko chains o
model he p obabili y o a end change.
Ma ko chains (MC) heo y is desc ibed, o
example, in Hillie and Liebe man (2010). MC
a e used o modelling p ocesses which can
be ound in one o ini e (coun able) numbe
o s a es in disc e e ime momen s. MC is
a sequence o disc e e andom a iables x1,
x2, x3, ... wi h he Ma ko p ope y (a he ime
momen n+1 he p ocess will be in s a e i. I is
s ochas ically dependen only on he s a e ha
was ac i e a he p e ious ime momen , i.e.
a he ime n). Fo mally i can be desc ibed as
ollows (1):
P(Xn+1 = xn+1 | X1 = x1, X2 = x2, … , Xn = xn) =
= P(Xn+1 = xn+1 | Xn = xn) (1)
Pa icula ealiza ions xi a e elemen s o
a coun able se S = {si}, i = 1, 2, ..., N which is
called a s a e space. Beha iou o he desc ibed
p ocess is de e mined by:
 Vec o o uncondi ional p obabili ies
p(n)T = [p1(n), p2(n), ..., pN(n)], whe e T means
ansposi ion and pi(n) deno es p obabili y
ha he p ocess is in he momen n in he
s a e i.
 T ansi ion p obabili y ma ix P whose
elemen s pij gi e condi ional p obabili y o
p ocess ansi ion om he s a e i o he
s a e j. Tha could be o mally desc ibed as
pij = P(Xn = sj |Xn-1 = si), whe e i = 1, 2, ... N and
j = 1, 2, ... N.
I we know he p obabili y o pa icula
s a es appea ance a he ime momen when
he p ocess s a s, we can desc ibe he p ocess
beha iou using he ela ions (2):
pT(n) = pT(n – 1)P hus pT(n) = pT(0) P n (2)
I can be de i ed om he ela ions (2)
ha he long- e m beha iou o his s ochas ic
p ocess is de e mined by he ma ix P.
In o de o he model o be success ul, i
is necessa y o de ine he s a e space p ope ly.
The e mus be such s a es in he s a e space
om which he p ocess ans o ms in o s a es
wi h he opposi e end wi h a su icien ly high
p obabili y. In o he wo ds, we need o ind
s a es in which he end is likely o change wi h
a su icien ly high p obabili y. Such s a es could
be used o gene a e buying o selling o de s.
The use o MC o modelling s ock ma ke
beha iou is no a new issue. O e he las
decade, a numbe o pape s ha e been
published using MC heo y o model s ock
ma ke de elopmen . In some o hem he s a e
space is de ined e y simply. Fo example,
Doubleday and Esunge (2011) applied he
Ma ko chains o he DJA index and o a po olio
o selec ed s ocks om his index. They de ined
a s a e space on daily p ice changes. They
had es ed wo ypes o s a e space, wi h wo
s a es and wi h six s a es. Vasan hi e al. (2011)
deal wi h p edic ing he daily de elopmen o
he alue o s ock indices. They p edic ed only
he di ec ion o p ice luc ua ion, i.e. inc ease o
dec ease. In o de o make he p edic ion hey
had used ansi ion p obabili ies which hey had
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calcula ed successi ely om he las 5 yea s,
3 yea s and 1 yea . Pe ko ić e al. (2018) used
a h ee-s a e model o e u ns analysis on he
Belg ade S ock Exchange. Simila me hods
ha e been applied o o he exchanges, such as
Lakshmi and Manoj (2020) used MC analysis
o he Indian s ock ma ke o Ya uz (2019) o
he Tu kish s ock ma ke . The conclusions o
hese wo ks show ha such a simple de ini ion
o s a e space canno be used o ca y ou
success ul business s a egies and a e now
es ed mo e o en on eme ging ma ke s.
The mos p omising me hods o success ul
ading seems o be he applica ion o ad anced
echniques o MC analysis o a di e en ly de ined
s a e space. Nguyen (2018) used he hidden
Ma ko chains o mon hly s ock p ice p edic ion.
He had es ed models wi h wo o six s a es.
These models had been es ed on S&P index.
The esul s showed ha he hidden Ma ko model
ha e ou pe o med he B&H s a egy and ha e
yielded highe pe cen age e u ns. In he s udy by
Huang e al. (2017) a model in eg a ing wo ypes
o MCs was used o s ock p ice p edic ion: egula
and abso bing MC. The abso bing MC p o ides
in o ma ion on accumula ed p ice changes du ing
he leng h o e ia y end. Riedlinge and Nicolau
(2020) p edic ed he de elopmen o he FTSE
100 index one pe iod ahead using mul i a ia e
MCs. The mul i a ia e model enables o include
mo e a iables in he p obabili y model. The
au ho s assumed ha s ock p ices could be
a ec ed by a numbe o a iables and delays by
mo e han one pe iod.
The wo k by S oboda (2016) deal wi h
he ac ha s ock p ices may be e en i e
mo e han one pe iod in such a way ha he
s a e space was de ined on he basis o he
size o daily cumula i e changes in he s ock
p ice. The wo k shows ha in such a de ined
s a e space he e a e s a es in which he e is
a su icien ly high p obabili y o he e ia y
end e e sal. The s a e space in he abo e-
men ioned wo k is de ined on he basis o
mul iples o he s anda d de ia ion om daily
p ice changes. The s anda d de ia ion is
calcula ed o he en i e moni o ed pe iod.
Howe e , he s anda d de ia ion calcula ed o
he whole pe iod does no espec one o he
basic cha ac e is ics o s ock ma ke s ha is
changing ola ili y. The e o e, in his esea ch,
he s a e space de ini ion is based on a mo ing
s anda d de ia ion. I is expec ed ha he s a e
space de e mined in his way will be able o
espond o changing ola ili y and he model will
demons a e be e esul s.
2. Da a
We conduc esea ch on s ocks om he
Czech s ock ma ke , which a e aded on he
P ague S ock Exchange (PSE). The c i e ia o
including s ock ypes in he esea ch we e as
ollows:
 PSE mus be he main ma ke o his s ock
ype.
 The s ocks mus be aded on PSE long
enough.
 Daily olumes mus be high enough, i.e.,
ul illing he assump ion ha he s ock is
aded by all ypes o in es o s and has
su icien liquidi y.
These equi emen s a e me only
by he ollowing ew companies: he
elecommunica ions company O2 CR (O2), he
KB bank (KB) and he ene gy company CEZ
(CEZ). We ha e daily opening and closing p ices
o each s ock ype o a ou een-yea pe iod,
om 2 Janua y 2006 un il 2 Janua y 2020, i.e.
app ox. 3,500 business days. The da a sou ce
is Pa ia Di ec . The companies egula ly paid
di idends in he gi en pe iod. In his esea ch,
he a e - ax di idend is ein es ed.
In he pe iod unde esea ch, on 2016-05-
12, KB spli i s s ocks 5- o -1. All da a (di idends,
opening and closing p ices) ela ing o KB
we e ecalcula ed in o de o main ain he da a
con inui y. On 2015-06-01, O2 was di ided in o
wo companies: O2 and CETIN. The inal p ice
o he O2 s ock he day be o e he di ision o
he company was CZK 177.6. Fo one o iginal
O2 s ock, he s ockholde acqui ed one new O2
s ock and one CETIN s ock. On he i s day a e
he spli , he p ice o he CETIN s ock was CZK
133.5 and he p ice o he O2 s ock was CZK
69.2. A he ime o he di ision o he company
he ma ke was in o med ha a majo i y
s ockholde o CETIN s ocks plans o ge id o
mino i y s ockholde s and he s ocks will la e
be wi hd awn om he ma ke . Main aining da a
con inui y is add essed as ollows. CETIN s ocks
a e sold a he p ice o CZK 133.5 on he i s
day a e he di ision, and his income is u he
coun ed in he same way as he di idend. This
means ha O2 s ocks a e pu chased o hem
acco ding o business s a egy.
The p ocess o he s ock app ecia ion o
he moni o ed s ocks wi h he ein es men o
di idends is shown in Fig. 1.
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Special a en ion in Fig. 1 is d awn o a sha p
inc ease in he app ecia ion o O2 s ocks in he
second hal o 2015. Tha inc ease was caused
by he end o he unce ain y ha had been
e iden as soon as he elease o he in o ma ion
on he majo i y s ockholde ’s ac ions en e ed
he ma ke . In au umn 2013, PPF pu chased
a majo i y s ake om Tele ónica.
3. Resea ch Me hodology
S ock p ices mus i s be ans o med in o
a sui able Ma ko chain. The ans o ma ion
is pe o med in e ms o an app op ia ely
de ined s a e space. A e ob aining he MC,
sui able s a es o gene a ing ading signals
a e selec ed based on he ansi ion p obabili y
ma ix. These gene a ed ading signals help
o c ea e ading s a egies ha se e as he
oo ing o he ading model.
3.1 S a e Space
The de ini ion o he s a e space is suppo ed
by he wo k S oboda (2016). The s ock space
is de ined on he cumula i e changes in he
s ock p ice o e he du a ion o he e ia y
end. The cumula i e change in he s ock p ice
is deno ed by Y , which is in e p e ed as sho
basic indices o daily closing p ices. The base
pe iod is he day o he end change, i.e., he
ansi ion poin om decline o g ow h o ice
e sa. The du a ion o he end is de e mined
by he numbe o consecu i e ising o alling
closing p ices. Fo mally, he calcula ion o he
cumula i e p ice change Y is desc ibed by (3):
(3)
whe e P is he inal daily p ice a ime , P –1
is he inal daily p ice a ime – 1 and P –2 is he
inal daily p ice a ime – 2.
Based on he alues o y (y is a pe cen age
o Y ) a s a e space is de ined. A se wi h
eigh s a es a e used o so he da a. As he
s ock p ice alls, he co esponding condi ions
a e ma ked Di. S a e D1 is he s a e wi h he
smalles p ice d op and s a e D4 is he s a e wi h
he highes p ice d op. Condi ions when he
s ock p ice ises a e ma ked Gi. The G1 s a e
is he s a e wi h he lowes p ice g ow h and,
con e sely, he G4 s a e is he s a e wi h he
highes p ice g ow h. The gene al s a e space
model is de ined by he ollowing p inciple:
D4: y < −3Δ G1: 0 ≤ y < 1Δ
D3: −3Δ ≤ y < −2Δ G2: 1Δ ≤ y < 2Δ
D2: −2Δ ≤ y < −1Δ G3: 2Δ ≤ y < 3Δ
D1: −1Δ ≤ y < 0 G4: 3Δ ≤ y
whe e Δ = kσ ,l ep esen s he wid h o he
in e al, σ ,l is he s anda d de ia ion o daily
changes in he s ock p ice, k is he mul iple o
he s anda d de ia ion. The s anda d de ia ion
is calcula ed as he mo ing s anda d de ia ion
o leng h l, acco ding o he o mula (4):
(4)
Fig. 1: De elopmen o capi al alue
Sou ce: Pa ia, own esea ch
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whe e x – i is he daily change o he s ock p ice
a day – i, l s he leng h o he mo ing s anda d
de ia ion (wid h o he sliding window) and x  ,l is
he mo ing a e age leng hs o l a day . The so-
de ined s a e space model ( a ying model) can
espond o changing ola ili y in s ock ma ke s.
S oboda (2016) in his wo k s a es, ha space
model (un a ying model) was de ined on an
unchanging s anda d de ia ion calcula ed om
all neighbou ing alues. Fo indi idual s ocks,
he ixed s anda d de ia ion is as ollows:
σO2 = 1.548, σCEZ = 1.663, σKB = 1.899. The
de elopmen o a mo ing s anda d de ia ion o
pa ame e l = 20 is shown in Fig. 2.
The s a e assignmen p ocedu e o bo h
models o he s a e space desc ibed abo e is
illus a ed on CEZ s ocks wi h model pa ame e s
k = 1 and l = 20 in Tab. 1.
Fig. 2: Mo ing s anda d de ia ion o l = 20
Sou ce: own
Da e P x [%] y [%] Un a ying model Va ying model
σS a e σ20 S a e
2019-12-12 505.5 0.999 0.999 1.66 G10.71 G2
2019-12-11 500.5 −1.476 −1.476 1.66 D10.66 D3
2019-12-10 508.0 0.594 1.195 1.66 G10.59 G3
2019-12-09 505.0 0.198 0.598 1.66 G10.58 G2
2019-12-06 504.0 0.099 0.398 1.66 G10.57 G1
2019-12-05 503.5 0.299 0.299 1.66 G10.59 G1
2019-12-04 502.0 −0.100 −2.240 1.66 D20.60 D4
2019-12-03 502.5 −1.374 −2.142 1.66 D20.68 D4
2019-12-02 509.5 −0.779 −0.779 1.66 D10.63 D2
2019-11-29 513.5 0.588 0.588 1.66 G10.62 G1
2019-11-28 510.5 −0.293 −0.293 1.66 D10.61 D1
2019-11-27 512.0 0.999 0.999 1.66 G10.71 G2
. . . . . .
Sou ce: own
Tab. 1: P ocedu e o assigning s a es (CEZ, k = 1, l = 20)
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3.2 T ansi ion P obabili y Ma ix
Now ha he MC is a ailable i is p oposed o
ind he p obabili ies o he ansi ion be ween
indi idual s a es. Be o e he calcula ion was
pe o med MC il a ion was ca ied ou . This
il e ing p ocedu e is mean o e eal he
consecu i e iden ical s a es. By il a ion, we a e
able o skip he s a es in which he s ock p ice
s agna es ( espec i ely, hey change li le in he
same end). These s a es a e no conside ed o
be o in e es in e ms o ading. Fo illus a ion
we show a pa o he s ing be o e il e ing: ...
D4, G1, G1, G1, G2, D1, D1, D2, G2, D2, D2, … and
a e il a ion ... D4, G1, G2, D1, D2, G2, D2, ... The
ma ix o ansi ion p obabili ies P is de e mined
o he il e ed MC. The pa ame e s k = 1.0 and
l = 20 o he p esen ed ansi ion p obabili ies
o bo h a ying model and un a ying model
a e demons a ed in Tab. 2. These a e a e age
condi ional ansi ion p obabili ies calcula ed
acco ding o he o mula (5):
(5)
whe e ep esen s condi ional p obabili y
o ansi ion be ween indi idual s a es. The
ΣDi and ΣGi columns indica e he likelihood o
s aying in he end o changing he end. The
ni deno es he a e age numbe o occu ences
o indi idual s a es is indica ed.
F om he alues calcula ed in Tab. 2 we can
obse e:
 Fo bo h models, he p obabili y o end
change in indi idual s a es is high enough,
he un a ying model has a sligh ly highe
p obabili y.
 The alues calcula ed by us o he
un a ying model a e almos iden ical o
he alues in S oboda (2016), he inc ease
in he ime window did no b ing abou
changes in he beha iou o s ock p ices.
 The anomaly can be no iced a he ansi ion
om G4 o G3 in he a ying model, whe e
ze o would be expec ed. The e has been
a a e case whe e a mo ing s anda d
de ia ion has isen su icien ly o be
a ibu ed o a lowe Gi s a e despi e he ise
in he s ock p ice.
+ 1 D4D3D2D1G1G2G3G4ΣDiΣGin
D4
Un a ying 0 0 0 0 0.450 0.334 0.123 0.093 0 1 74.0
Va ying 0 0 0 0 0.538 0.312 0.138 0.012 0 1 112.7
D3
Un a ying 0.333 0 0 0 0.463 0.163 0.033 0.008 0.333 0.667 132.0
Va ying 0.356 0 0 0 0.431 0.180 0.030 0.004 0.356 0.644 185.0
D2
Un a ying 0.063 0.266 0 0 0.524 0.114 0.024 0.009 0.329 0.671 308.0
Va ying 0.093 0.269 0 0 0.430 0.178 0.026 0.004 0.362 0.638 381.7
D1
Un a ying 0.006 0.042 0.260 0 0.598 0.079 0.010 0.005 0.308 0.692 689.0
Va ying 0.013 0.074 0.297 0 0.452 0.137 0.022 0.005 0.384 0.616 602.0
G1
Un a ying 0.005 0.013 0.076 0.594 0 0.265 0.039 0.009 0.687 0.313 668.7
Va ying 0.005 0.029 0.122 0.451 0 0.303 0.077 0.012 0.608 0.392 576.7
G2
Un a ying 0.002 0.014 0.107 0.520 0 0 0.273 0.084 0.643 0.357 312.0
Va ying 0.001 0.023 0.151 0.435 0 0 0.282 0.109 0.610 0.390 392.3
G3
Un a ying 0.004 0.012 0.137 0.496 0 0 0 0.351 0.649 0.351 138.7
Va ying 0.002 0.025 0.168 0.402 0 0 0 0.404 0.596 0.404 200.3
G4
Un a ying 0.014 0.070 0.281 0.635 0 0 0 0 1.000 0.000 95.0
Va ying 0.000 0.046 0.293 0.658 0 0 0.002 0 0.998 0.002 136.7
Sou ce: own
Tab. 2: T ansi ion p obabili ies o he model o k = 1.0, l = 20
s ock
pij
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3.3 T ading S a egies and Models
In he ollowing pa , he e m ading s a egy
iden i ies a se o ules ha de e mine he
pu chase and sale o s ocks. The ading model
includes se e al ading s a egies applied o
selec ed s ocks.
T ading s a egies a e based on he
ollowing p inciple. Pu chasing signals a e
g adually gene a ed by s a es D3, D4 and sales
signals a e gene a ed by s a es G3, G4. By
combining buying and selling s ock s a es, 4
ading s a egies a e ob ained: D3–G3, D3–G4,
D4–G3, D4–G4. S a egies D3–G3 imply ha s a e
D3 gene a es pu chase o de s (i we no longe
hold s ocks) and s a e G3 gene a es a sales
o de (i we hold s ocks). This is in line wi h
he conclusions o he wo k S oboda (2016),
whe e hese 4 ading s a egies gene a ed he
highes e enues.
T ading is always ca ied ou acco ding o
he ollowing ules:
 One ade ( ansac ion) ep esen s he
pu chase and subsequen sale o a s ock.
 I is no possible o make wo pu chases in
a ow.
 I a buying o a selling signal is gene a ed
on a gi en day, he ade is execu ed wi h
he opening p ice om he ollowing day.
 The capi al is always ully in es ed, so i
is heo e ically possible o buy pa s o he
s ocks.
 T ansac ion ees a e no aken in o
conside a ion.
 Di idends and o he income (he eina e
e e ed o as di idends), i we a e en i led
o hem, a e ein es ed a e ax.
 A sho selling (specula ion on p ice
dec ease) is no aken in o accoun .
The alue o he in es ed capi al is calcula ed
acco ding o he ollowing o mula (6):
(6)
whe e C0 = 1.00 s he ini ial alue o capi al
(a uni o capi al is in es ed), Cn s he alue
o capi al a e he n- h ansac ion, Si is he
selling p ice in i- h ansac ion, di ep esen s
he ne di idends (and o he income) in case
ex di idend day occu ed du ing he i- h
ansac ion, Bi is he pu chase p ice in he i- h
ansac ion. I he ansac ion ees we e o be
calcula ed i is su icien o modi y o mula (6)
in o he ollowing o mula (7):
(7)
whe e p s ands o he size o he ee
(in pe cen ).
I would be e y isky o a ade o in es
e e y hing in one selec ed s ock and use one
ading s a egy. To minimize he isk, i is
app op ia e o dis ibu e capi al e enly be ween
he analysed s ocks and be ween indi idual
ading s a egies. The e o e, indi idual ading
s a egies will no be e alua ed as i is mo e
signi ican o e alua e he en i e po olio o
ading s a egies, which is called he ading
model. The ading model includes ou
ading s a egies, de ined on he same s a e
space, which a e applied o h ee s ocks, i.e.,
he in es ed capi al is di ided in o 12 equal
pa s. The e alua ion o he ading model is
ep esen ed by CM and is o mally de e mined
by o mula (8):
whe e
(8)
Acco ding o EMH, ading in s ocks
ollowing h ough wi h he abo e p inciples
should no be success ul and he alue should
no ou pe o m he ma ke . In o he wo ds,
a passi e s ockholding should yield he same
o a highe e u n. Passi e s ockholding can be
e e ed o as a Buy and Hold (B&H) s a egy.
We will conside a ading model success ul i
i ou pe o ms he B&H s a egy. In his pape ,
he B&H s a egy en isages he ein es men
o di idends. I is assumed ha in he beginning
o 2006 s ocks we e pu chased and hen held
un il he end o 2019. Fo he di idends paid,
he s ocks a e pu chased a cu en p ices. By
implemen ing he B&H s a egy, he esul ing
alue o capi al o indi idual s ocks would be
CCEZ = 1.35; CKB = 2.28; CO2 = 3.79 so ha he
a e age alue o capi al is CB&H = 2.47.
4. Resea ch Resul s and Discussion
In he i s place, he summa y esul s o
indi idual models will be p esen ed and hen
one o he models will be analysed in de ail. The
summa y esul s a e lis ed in Tab. 3. Fo each
model, he alue o CM capi al and he a e age
numbe o ealized ades n a e gi en. A o al o
80 a ying s a e space models we e analysed.
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The pa ame e k g adually ook alues om
0.5 o 2.0 wi h he s ep o 0.1. The pa ame e
l o he leng h o he sliding s anda d de ia ion
ook alues om 10 o 30 wi h he s ep o 5. An
un a ying model was also calcula ed o each
alue o he pa ame e s.
Resul s in Tab. 3 show ha on a e age
o all pa ame e s k he e is no signi ican
di e ence be ween he un a ying and a ying
model. The a e age alue o capi al a ies in
a na ow ange be ween 2.19 and 2.26. The
bes esul s a e ob ained when he pa ame e
o he model k is be ween 1.1 and 1.4. While he
un a ying model has a maximum capi al alue
o 2.67 (k = 1.4), he a ying models (excep
l = 10) achie e highe alues: 3.04 (k = 1.1,
l = 15), 2.93 (k = 1.2, l = 15), 2.83 (k = 1.2,
l = 25), 3.37 (k = 1.1, l = 30). Va ying models
combining he pa ame e s l = 15, l = 20, l = 25
and l = 30 wi h he pa ame e s k = 1.1 and
k = 1.2 sligh ly ou pe o m he un a ying model
and ou pe o med he passi e B&H s a egy
(2.47). As expec ed, i is also e iden ha he
numbe o execu ed ades dec eases wi h
inc easing k pa ame e . This is impo an when
calcula ing ading ees.
The Tab. 3 shows he summa y esul s
o he en i e 14-yea pe iod. Resul s may be
skewed in one success ul pe iod, while in o he
pe iods, ading models may no be success ul.
In o de o be conside ed eliable, ading models
should pe o m well o any gi en pe iod. The
i e-yea sliding yields in Tab. 4 and i e-yea
sliding annual yields in Tab. 5 a e calcula ed o
he model wi h he pa ame e s k = 1.2 and l = 25.
This model belongs o he g oup o models ha
demons a ed he bes esul s. I has a e age
esul s in his g oup. The leng h o i e yea s was
chosen because he minimum ecommended
in es men ho izon o in es ing in s ocks is
i e yea s. In addi ion o he o al e u ns o he
ading model, he e u ns o indi idual s ocks a e
also moni o ed and compa ed wi h he e u ns o
he B&H s a egy.
kl = 10 l = 15 l = 20 l = 25 l = 30 Un a ying
CMn CMn CMn CMn CMn CMn
0.50 1.46 200.0 1.50 187.5 1.34 185.5 1.54 180.4 1.70 180.5 1.72 141.8
0.60 1.66 179.4 1.83 168.0 1.85 166.0 1.99 164.7 2.01 162.8 2.24 122.3
0.70 2.05 155.4 2.19 152.1 1.98 145.5 1.96 143.3 2.21 142.5 2.30 105.5
0.80 2.02 134.3 2.32 129.4 2.28 128.0 2.18 123.5 1.84 119.8 2.11 86.3
0.90 2.10 115.1 2.50 111.1 2.27 108.7 2.32 106.9 2.09 104.7 2.38 74.3
1.00 2.31 98.5 2.32 93.4 2.73 92.7 2.18 91.4 2.04 88.3 2.56 63.0
1.10 2.54 83.8 3.04 81.0 2.80 78.3 2.51 75.8 3.37 75.6 2.63 51.9
1.20 2.63 69.7 2.71 68.3 2.93 65.5 2.83 63.9 2.91 63.9 2.56 43.8
1.30 2.67 61.1 2.37 56.7 2.54 54.7 2.42 54.2 2.22 53.1 2.40 36.6
1.40 2.47 52.0 2.23 47.6 2.51 47.4 2.27 46.1 2.29 44.9 2.67 31.5
1.50 2.51 43.3 1.95 37.8 2.18 39.5 2.19 38.8 2.05 38.4 2.36 26.0
1.60 2.31 36.8 2.09 33.8 2.22 34.1 2.04 33.0 2.00 32.7 2.22 22.8
1.70 2.16 31.3 2.31 30.1 2.39 29.5 2.47 28.7 1.88 28.2 1.81 19.8
1.80 2.20 26.4 2.17 25.2 2.29 25.1 2.30 23.8 2.35 25.1 1.83 17.6
1.90 2.15 23.3 1.79 20.2 2.00 20.1 2.10 21.2 2.12 21.1 1.96 16.0
2.00 1.99 19.7 1.70 16.8 1.82 16.6 2.01 18.0 2.64 18.0 1.98 14.8
Mean 2.20 83.1 2.19 78.7 2.26 77.3 2.21 75.8 2.23 75.0 2.23 54.6
S . de . 0.323 0.375 0.392 0.281 0.411 0.295
Sou ce: own
Tab. 3: Final alue o capi al
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