An improved model accuracy for forecasting risk measures: Application of ensemble methods

Maka jane, Ka leho; Mmelesi, Kesaobaka
A icle
An imp o ed model accu acy o o ecas ing isk
measu es: Applica ion o ensemble me hods
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An imp o ed model accu acy o o ecasting isk
measu es: applica ion o ensemble me hods
Ka leho Maka jane & Kesaobaka Mmelesi
To ci e his a icle: Ka leho Maka jane & Kesaobaka Mmelesi (2024) An imp o ed model
accu acy o o ecas ing isk measu es: applica ion o ensemble me hods, Jou nal o Applied
Economics, 27:1, 2395775, DOI: 10.1080/15140326.2024.2395775
To link o his a icle: h ps://doi.o g/10.1080/15140326.2024.2395775
© 2024 The Au ho (s). Published by In o ma
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Published online: 08 Sep 2024.
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An imp o ed model accu acy o o ecas ing isk measu es:
applica ion o ensemble me hods
Ka leho Maka jane
a
and Kesaobaka Mmelesi
b
a
Depa men o S a is ics, Uni e si y o Bo swana, Gabo one, Bo swana;
b
School o Economics and
Econome ics, Uni e si y o Johannesbu g, Johannesbu g, Sou h A ica
ABSTRACT
S a is ical-based p edic ions wi h ex eme alue heo y imp o e he
pe o mance o he isk model no by choosing he model s uc u e
ha is expec ed o p edic he bes bu by de eloping a model
whose esul s a e a combina ion o models wi h di e en shapes.
Using di e en ensemble algo i hms o conglome a e he TBATS
and he GEV dis ibu ion, we ound ha he s acking ensemble
algo i hm ou pe o ms o he ensembles hence he o ecas ing
accu acy o isk measu es is imp o ed wi h he s acking ensemble
algo i hm. The isk es ima es sugges ha he e u ns on losses
a e aging 0.014 and 0.018 in es ed a 90 and 99 pe cen espec-
i ely, a e iskie han he e u ns on gains. Back es ing esul s
u he e ealed ha all he isk measu es a e eliable, and he
combined model is a good one o compu ing inancial isk pa i-
cula ly in Sou h A ica. A high con idence le els, all he isk mea-
su es seem o pe o m be e han a lowe con idence le els, as
e idenced by highe p obabili y alues om back es ing using he
Kupiec and Ch is o e sen es a 95 pe cen han a 99 pe cen
le els o signi icance.
ARTICLE HISTORY
Recei ed 18 Janua y 2024
Accep ed 14 Augus 2024
KEYWORDS
combined o ecas ing;
inancial ma ke s; ex eme
alue heo y; machine
lea ning
JEL CLASSIFICATION
C13; C22; C52; C58
1. In oduc ion
F om he iewpoin o inancial isk manage s, a isk measu e can be iewed as a map
om he space o p obabili y dis ibu ions o ac ual numbe s. Risk managemen p o es-
sionals can adjus hei capi al ese es agains he downside isk by using isk measu es
o gi e banks and o he inancial ins i u ions speci ic alues o po en ial losses. Two
p e alen measu es o inancial isk ha ule mode n inancial egula ion a e alue-a -
isk (VaR) and expec ed sho all (ES). In he wo s -case scena io, a a speci ic con idence
le el, VaR gi es banks and in es men ins i u ions a loss le el (Laza & Xue, 2020).
Value-a - isk is no a cogen isk measu e in he adi ional sense and has inhe en laws
because i igno es he shape and s uc u e o he ail. As a esul , ollowing he inancial
c isis o 2007–2008, he BCBS (2014) he ein e e enced (Basel Commi ee on Banking
Supe ision) sugges ed swi ching om VaR, which had a con idence le el o 99 pe cen ,
CONTACT Ka leho Maka jane [email p o ec ed] Depa men o S a is ics, Uni e si y o Bo swana,
Gabo one, Bo swana
JOURNAL OF APPLIED ECONOMICS
2024, VOL. 27, NO. 1, 2395775
h ps://doi.o g/10.1080/15140326.2024.2395775
© 2024 The Au ho (s). Published by In o ma UK Limi ed, ading as Taylo & F ancis G oup.
This is an Open Access a icle dis ibu ed unde he e ms o he C ea i e Commons A ibu ion License (h p://c ea i ecommons.o g/
licenses/by/4.0/), which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly
ci ed. The e ms on which his a icle has been published allow he pos ing o he Accep ed Manusc ip in a eposi o y by he au ho (s) o
wi h hei consen .
o ES, which has a con idence le el o 97.5 pe cen . Gi en ha i s ealisa ion mus be
lowe han ha o VaR, ES is he expec ed e u n on in es men .
On he o he hand, de i ing a model ha in es o s and/o s ock ma ke s will use o
ack, p edic , and o ecas hei daily gains o losses in eal ime is he e o e c ucial. As
a esul , o ecas ing isk is a huge and ac i e esea ch a ea ha has piqued he in e es o
many di e en academic disciplines, including inance, enginee ing, and s a is ics among
o he s. Due o his, a signi ican amoun o li e a u e has ocused on echniques ha can
gene a e eliable o ecas s in a ange o eal-wo ld scena ios (Haji ahimi & Khashei,
2019). The li e a u e ypically poin s o wo main app oaches: (1) de eloping and
p oposing new o ecas ing models, and inc easing he accu acy o ob ained esul s. (2)
Combining di e en o ecas ing model ypes. This is because no single comp ehensi e
model can simul aneously cap u e all o he pa e ns p esen in he da a; he e o e,
hyb idisa ion is usually used.
The e o e, he pu pose o his s udy is o combine he TBATS wi h he gene alised
ex eme alue (GEV) dis ibu ion and compa e he e ec i eness o di e en ensemble
me hods in combining he wo models. In his way, we aim o imp o e and, enhance he
o ecas ing accu acy o isk measu es, namely, Value-a - isk, expec ed sho all, condi-
ional ail expec a ion (CTE), and glue alue-a - isk (Glue-VaR). Gene ally speaking,
se e al models a e es ima ed, and he mos accu a e model is chosen. Because o a ew
possible impac ing ac o s such as sampling a iance, model unce ain y, and s uc u al
changes, he inal model chosen may no be he bes o use in he u u e. Wi h minimal
addi ional wo k, he model selec ion p oblem is made easie by in eg a ing a ious
models. Secondly, ime se ies da a in he eal wo ld a e a ely pe ec ly linea o non-
linea ; because bo h linea and non-linea pa e ns a e equen ly p esen . In his
scena io, nei he he TBATS no he GEV dis ibu ion can be used e ec i ely o model
and o ecas ime se ies da a speci ically he s ock ma ke ime se ies, since he TBATS is
no designed o handle he ail beha iou o he dis ibu ion and ex eme alues. While
he GEV dis ibu ion on he o he hand, canno handle bo h linea and non-linea
pa e ns equally well on i s own. Fo ha eason, we ind i in e es ing o us o accu a ely
ep esen complex ime se ies s uc u es by me ging he TBATS wi h he GEV dis ibu-
ion and see how he new combined model pe o ms on a i e day business inancial ime
se ies exchange/Johannesbu g s ock exchange (FTSE/JSE) all-sha e index.
Thi dly, he o ecas ing li e a u e i ually unanimously ag ees ha he e is no single
ideal s a egy o e e y si ua ion. This is mos ly because eal-wo ld issues a e equen ly
complica ed, making i possible o dis inc pa e ns o be cap u ed di e en ly by
di e en models. U ilising he TBATS helps us o handle mul iple seasonali ies and
i egula ends, making i sui able o us o cap u e he unde lying dynamics o he
da a, while he gene alised ex eme alue dis ibu ion on he o he hand helps us o
accu a ely cap u e ex eme alues in he ails o he dis ibu ion. The in eg a ion o his
wo models helps us o ob ain mo e accu a e es ima es o ail isk, pa icula ly in
scena ios whe e ex eme e en s a e in equen bu signi ican ; le e aging he s eng hs
o bo h models o imp o e o ecas ing pe o mance, pa icula ly in si ua ions whe e
cap u ing ex eme e en s accu a ely is c ucial o isk managemen o decision-making.
Wi h his p oposed h ee-s age hyb id model, in he is s age we amalgama e he
TBATS o ob ain independen ly and iden ically dis ibu ed (i.i.d) esiduals while a he
same ime modelling non-cons an seasonal pa e ns o e ime. Ins ead, he model helps
2K. MAKATJANE AND K. MMELESI
us o adap o changes in seasonali y, making i sui able o ime se ies da a wi h i egula
o e ol ing seasonal pa e ns. On he second s age, we ex ac he block maximas o he
posi i e e u ns (i.e., he gains) and block minimas o nega i e e u ns (i.e., he losses)
om he i.i.d esiduals o he TBATS model and we i he GEV dis ibu ion o model
ola ili y, ex eme ail losses, and gains o he Sou h A ican s ock ma ke . In de ail, we
conside a gene alised ex eme alue dis ibu ion wi h a speci ica ion ha he ex eme
alue sequence comes om he au o eg essi e and he mo ing a e age (ARMA) e o s
ha cap u e empo al dependencies and au oco ela ion in he ime se ies. The depen-
dence is cap u ed by an app op ia e empo al end in he local and scale pa ame e s o
he GEV dis ibu ion. The TBATS-GEVD modelling app oach is belie ed o pe o m
be e in o ecas ing he isk measu es and i is sui ed o explain ex emes be e han he
classical me hods; whe e he classical me hods such as Au o eg essi e in eg a ed mo ing
a e age (ARIMA) among o he s canno cap u e he ail beha iou adequa ely because
hey nei he assume a no mally dis ibu ed no e en a e ailed dis ibu ed (e.g., )
inno a ions as sugges ed by Calab ese and Giudici (2015). This s udy is he i s empi ical
analysis ha employs TBATS in conjunc ion wi h he GEV dis ibu ion o quan i y he
likelihood o u u e ex eme daily losses and gains.
Ex eme Value Theo y (EVT) is one o he mos use ul echniques o p edic ex eme
cases, which ha e a big e ec bu less occu ence p obabili y. In his s udy, he Block
Maxima and Minima (BM) app oach, which is one o he main ex eme alue echniques,
has been used. In his app oach, he e is no p ope block size selec ion me hodology
(Öza i e al., 2019). I has been obse ed ha he block size selec ion o he es ima ion
made in p e ious s udies has been used andomly wi hou elying on any assump ion. I
is necessa y o new applica ions o ind a ce ain block size ha can be used o all o he
da a se s. Hence, he second main con ibu ion o his s udy is he no el y o he
applica ion on how he op imal numbe o blocks o be i ed o he GEV dis ibu ion
a e selec ed. We p opose a me hodology o selec he bes block size o bo h maximal
and minimal e u ns wi h a case s udy on he Sou h A ican ma ke all-sha e index which
makes up he op 99 pe cen o he o al p e- ee- loa ma ke capi alisa ion o all lis ed
companies on he Johannesbu g S ock Exchange.
The es o his s udy is o ganised as ollows sec ion 1.1 ep esen s li e a u e e iew.
Sec ion 2 is he me hodology ollowed. While sec ion 3 p esen s esul and discussion
om da a analysis and inally sec ion 4 p esen s he conclusion o he s udy.
1.1. Li e a u e e iew
Con en ional measu es o isk in ea nings based on his o ical s anda d de ia ion equi e
long ime-se ies da a and a e inadequa e when he dis ibu ion o ea nings de ia es om
no mali y. A me hodology ha is based on cu en undamen als, and quan ile eg es-
sion o o ecas isk e lec ed in he shape o he dis ibu ion o u u e ea nings, was
de eloped by Kons an inidi and Pope (2016). E en hough VaR is widely used, many
s udies ha e ho oughly examined i s d awbacks. When a loss dis ibu ion has a ails,
VaR is no only incohe en bu also ails o accu a ely es ima e he isk o he loss
(Rocka ella & U yase , 2002), and he e o e, his signi ican ly unde mines he eliabili y
o his isk measu e (Chen, 2018). Despi e his, alue-a - isk is s ill a popula isk
indica o because i is so easy o compu e and comp ehend. In addi ion o demons a ing
JOURNAL OF APPLIED ECONOMICS 3

he incohe ence o VaR, A zne e al. (1999) also in oduced he expec ed sho all and
dubbed i he ideal isk indica o . Using cohe en isk measu e heo y, P lug (2000)
u he demons a ed ha ES is a cohe en isk measu e. And, in con as o VaR, which
is mo e equen ly used, expec ed sho all is a isk measu e ha is sensi i e o he shape
o he ail o he dis ibu ion o e u ns on a po olio.
To co ec ly es ima e isk measu es, a dis ibu ion ha cap u es all ex eme and
s ylised ac s is equi ed. Ex eme o “ ail” isk, as desc ibed in BCBS (2019), equi es
isk p ac i ione s o ully unde s and i . I has been de eloped o use he ex eme alue
heo y whe e Fishe and Tippe (1928) and Pickands (1975) a e c edi ed o de eloping
he ield o EVT. By ob aining h ee asymp o ic limi s, Fishe and Tippe (1928) we e
able o desc ibe he dis ibu ion o ex eme alues, assuming ha he a iables a e
independen . An a gumen o modelling ex eme alues using a gene alised ex eme
alue dis ibu ion can be a ibu ed o Ma i z and Mun o (1967). Ex eme alue heo y is
a heo y ha measu es and models ex eme e en s (la ge luc ua ions) ( ails o s a is ical
dis ibu ions), i.e., i is sui able o inancial asse s wi h ex eme e u ns ( e y la ge
luc ua ions in e u ns); hence, i assumes independen and iden ically dis ibu ed
obse a ions. This assump ion o i.i.d does no necessa ily hold o inancial ime se ies
da a (Chen, 2023).
To co ec his, McNeil e al. (2015) p oposed a wo-s age me hodology in he o m o
a gene alised au o eg essi e condi ional he e oscedas ici y (GARCH)-EVT model using
i e index e u ns in hei illus a ions. The i s s ep was o cap u e he he e oscedas ici y
(non-cons an a ia ion o luc ua ions) ea u es by i ing he GARCH model.
The second s ep was o apply he Gene alised Pa e o dis ibu ion (GPD) and he GEV
dis ibu ion o he esiduals ex ac ed om a selec ed GARCH model. The ad an ages o
he GARCH-EVT hyb id model a e in i s capaci y o model he ex eme ail (la ge
luc ua ions) beha iou using he EVT me hods while also cap u ing condi ional he e o-
scedas ici y (changing a ia ion) in he da a h ough he GARCH amewo k.
By implemen ing a dynamic me hod o o ecas ing a 1-day-ahead VaR, which
combines he GARCH models and EVT o examine he ex eme beha iou o majo
economic s ock indices be o e and du ing he ou b eak o he COVID-19 pandemic,
Oma e al. (2020) and Paul and Sha ma (2018) we e able o accu a ely calcula e VaR by
using EVT me hods. Explici ly, hese au ho s adop ed he GPD ia he Peaks-o e -
h eshold (POT) me hod. Comp ehensi e in-sample ola ili y modelling was implemen-
ed wi h skewed s uden dis ibu ion assump ions, and he in o ma ion c i e ions we e
used o es ablish hei goodness o i . Fu he mo e, he VaR quan iles we e es ima ed by
using he condi ional EVT (C-EVT) amewo k o ob ain ou -o -sample VaR o ecas ing
esul s. The combined GARCH and EVT model pe o med ela i ely well in es ima ing
he isk o all s ock indices. The back es ing esul s demons a e ha he exponen ial-
GARCH skewed-s uden ’s- and C-EVT models a e he mos app op ia e echniques o
be e measu ing and o ecas ing VaR in compa ison wi h he con en ional me hods.
O he s udies like hose o Echaus and Jus (2020) examined he abili y o alue-a -
isk es ima es when each es ima e is made wi h an op imal choice o he ails o he
dis ibu ion by he combina ion o GARCH-EVT. He e, 5 me hods we e applied o
desc ibe he ail, namely he dis ance-me ic me hod wi h he mean absolu e penal y
unc ion, he minimisa ion o he asymp o ic mean squa ed e o (AMSE) es ima e, he
pa h-s abili y algo i hm, he ixed-quan ile p ocedu e, and he au oma ed eyeball
4K. MAKATJANE AND K. MMELESI
me hod. The GARCH-EVT app oach in combina ion wi h a no el algo i hm o au o-
ma ically de e mine he op imal h eshold o model he ail dis ibu ion was p oposed by
B uhn and E ns (2022). Fu he mo e, indi idual ma ke isks we e agg ega ed wi h
a -s uden Copula o in es iga e possible di e si ica ion e ec s on a po olio le el
(Ho mann & Bö ne , 2020). The empi ical analysis indica es ha all examined c yp o-
cu encies show high ola ili y in hei p ice mo emen s, whe eby Bi coin ac s as he
mos s able c yp ocu ency. All e u ns dis ibu ions a e hea y- ailed and subjec o
ex eme ail isks.
To e ec i ely cap u e clus e ing ola ili y in he p esence o s uc u al b eaks (s uc-
u al changes) and ail beha io , Maka jane e al. (2021) also de eloped a wo-s age
analysis. The i s s age was o es ima e a Ma ko -swi ching exponen ial GARCH model
o ob ain egime swi ching esiduals ha a e i.i.d. In he second s age, hese au ho s
applied EVT o he uppe egime esiduals and es ima ed bo h he GPD and he GEV
dsi ibu ion. This was o comple e he s udy by McNeil e al. (2015), Sahamkhadam e al.
(2018) and Echaus and Jus (2020) among o he s. To manage he isks o a po olio
made up o commodi ies, cu ency indices, and equi y secu i ies, Koliai (2016) p esen ed
he GARCH-EVT wi h an R- ine copula model. The e ec i eness o he gene alised
Lambda dis ibu ion (GLD), he gene alised Pa e o dis ibu ion, and he gene alised
ex eme alue dis ibu ion we e ad oca ed by Huang e al. (2017) and hese au ho s
simula ed daily VaR and ES o log- e u ns o pla inum, gold, and sil e p ices; gi ing
GPD and GLD be e pe o mance o e GEV dis ibu ion.
The main app oach o o ecas ing is he combina ion o se e al models (Haji ahimi &
Khashei, 2019), which pe o ms be e han using jus one model. Compa ing he hyb id
me hod o he single model, accu a e pe o mance can be p oduced (Büyükşahin &
E ekin, 2019). Howe e , some s udies like o Ibn Musah e al. (2018), o ins ance,
aim o in es iga e he isks connec ed wi h he main Ghanaian s ock exchange while
combining EVT wi h a i icial neu al ne wo ks (ANNs). Recen esea ch conduc ed by
Ilyas e al. (2022) has p oposed a new hyb id me hod, consis ing o a ully modi ied
Hod ick P esco il e (FMHP) o imp o e p edic ion accu acy. This me hod consis s o
h ee main componen s: machine-lea ning-based p edic ion, no el ea u es, and a noise-
il e ing echnique. Mo eo e , he combina ion o gene a i e ad e sa ial ne wo ks
(GANs) wi h ex eme alues p o ed o be mo e e ec i e in modelling ex eme alues
(Boulaguiem e al., 2022).
The use o an ensemble lea ning-based olling window app oach o in es iga e he
ad an age o combining mul iple GARCH- ype models wi h LSTM (he ein e e enced
Long-sho Te m-memo y) o o ecas alue a isk is ad oca ed by Kakade e al. (2022).
These au ho s use co e age es s loss unc ions o he inancial c isis o 2008–2009 and
he COVID-19 ecession o 2020–21 o assess he model’s pe o mance on c ude oil
e u ns. When compa ing he VaR o ecas s o he hyb id models o he con en ional
and GARCH echniques, a no able imp o emen in quali y and accu acy is no ed. I is
disco e ed ha he mos e ec i e es ima o o Value-a -Risk is he Fil e ed His o ical
Simula ion echnique.
Mo eo e , Ba e a e al. (2022) p oposed a non-asymp o ic con e gence analysis o
a wo-s ep app oach o lea n alue-a - isk and expec ed sho all in a nonpa ame ic
se ing using Rademache and Vapnik-Che onenkis bounds. The app oach o hese
au ho s o VaR is ex ended o he p oblem o lea ning a once mul iple VaRs
JOURNAL OF APPLIED ECONOMICS 5
co esponding o di e en quan ile le els. This led o he de elopmen o e ec i e
lea ning sys ems based on leas -squa es eg essions and neu al ne wo k quan iles.
Wi hou access o he la e , a pos e io i Mon e Ca lo (non-nes ed) me hod was
employed o es ima e dis ances o he g ound- u h VaR and ES. Nume ical expe imen s
in a Gaussian oy model and a inancial case s udy, whe e he goal is o de e mine
a dynamic ini ial ma gin, we e used o demons a e his. These nume ical es s indica e
ha , despi e ini ially seeming coun e -in ui i e, lea ning se e al quan iles (mul i-α (I),
mul i-α (II), o mul i-α (III)) can aid in e ec i ely a ge ing ex eme quan iles han
a ypical single quan ile lea ning s a egy.
Recen esea ch has es ablished a lexible likelihood-based amewo k o he join
modelling o VaR and ES, based on he ela ionship be ween he quan ile sco e unc ion
and he asymme ic Laplace densi y. Cap u ing he unde lying combined dynamics o
hese wo quan i ies is o g ea ele ance in inancial applica ions. To ackle his issue, Li
e al. (2020) c ea ed a hyb id model ha e ec i ely cap u es he unde lying dynamics o
VaR and ES. The model is based on he asymme ic Laplace quasi-likelihood and uses he
Long Sho -Te m Memo y ime se ies modelling echnique om machine lea ning. This
model is known as LSTM-AL. In he LSTM-AL model, hese au ho s use he adap i e
Ma ko chain Mon e Ca lo (MCMC) app oach o Bayesian in e ence. Thei empi ical
esul s show ha he p oposed LSTM-AL model has imp o ed he VaR and ES o ecas -
ing accu acy o e a ange o well-es ablished compe ing models.
An ea ly wa ning sys em o ex eme daily losses o inancial ma ke s is o c ucial
modelling sys em. A h ee-s age me hodology was es ablished by Maka jane and Mo oke
(2021) whe e in he i s s age a seasonal Au o eg essi e in eg a ed mo ing a e age
(SARIMA) was es ima ed o il e he se ies o ob ain i.i.d esiduals. By using he Ma ko -
chain Mon e-Ca lo app oach, he Ma ko -swi ching exponen ial GARCH model
coupled wi h gene alised ex eme alue dis ibu ion was i ed o hese i.i.d esiduals
and inally, he logis ic model ee was ime-hono ed o es ablish ea ly wa ning signs.
Al e na i ely, Maka jane and Tsoku (2022) o e come he dimensionali y p oblem in
o ecas ing VaR and ES unce ain y in e als in inancial ime se ies da a by boo s ap-
ping and back es ing densi y o ecas s using Bayesian me hods ha a e based on
a weigh ed h eshold and quan ile o a con inuously anked p obabili y sco e. These
au ho s ound ha ex ension o his non-s a iona y dis ibu ion in li e a u e is qui e
complica ed since i equi es speci ica ions no only on how he usual Bayesian pa a-
me e s change o e ime bu also on hose wi h bulk dis ibu ion componen s. This
implies ha he combina ion o a s ochas ic econome ic model wi h ex eme alue
heo y p ocedu es p o ides a obus basis necessa y o he s a is ical back es ing and
boo s apping densi y p edic ions o VaR and ES.
2. Me hodology
A i e-day inancial ime se ies exchange Johannesbu g s ock exchange/All-sha e index
(FTSE/JSE-ALSI) o he pe iod o 4 Janua y 2010 o 5 Ap il 2024 is used in his s udy.
This consis s o 3646 obse a ions. The use o a i e-day (s) equency is based on he ac
ha , on weekends and holidays, s ock ma ke s a e closed. The e o e, he Sou h A ican
S ock Exchange is no an excep ion. No ading is happening on hese days, hence he
eco ded ma ke p ices a e om Monday(s) – F iday(s); excep whe e he holiday a ises
6K. MAKATJANE AND K. MMELESI
du ing he week. In his case, only ou da a poin s a e a ailable ins ead o i e. To a oid
any exchange a e luc ua ions, he index is kep in i s o iginal cu ency; i.e., he FTSE/
JSE-ALSI used in his s udy is kep in i s ZAR; he Sou h A ican cu ency.
Le X be a s ock p ice index on he day and X 1be a s ock p ice index on he day
1. Le ing o be s ock e u ns a ime ; Algie i and Leccadi o (2020) showed ha
can be modeled by
¼μ þε (1)
whe e μ is a ime- a ying mean o a ime se ies and ε is he e o e m ha should be
modeled by
ε ¼η σ :(2)
In model (2), σ is a ime- a ying dynamics o a ime se ies; while η is an i.i.d p ocess.
2.1. P oposed TBATS—gene alised ex eme alue dis ibu ion
The capaci y o con en ional seasonal and exponen ially smoo h models o handle dual-
calenda , mul i-seasonal, and nonin ege seasonal ime se ies is es ic ed (So okina e al.,
2023). Se e al schola s ha e examined he challenge o handling in ica e seasonal ime
se ies in his con ex , and o add ess his issue, he exponen ially smoo h model was
a modi ied p oblem (Zhao & Zhang, 2022); hence he BATS model was in oduced o
add ess his complex ime se ies pa e n. The basic model o m is exp essed as
BATS p;q;m1;���;mT
ð Þ whe e B is he Box-Cox ans o ma ion o add essing he e o-
genei y, A is he ARMA e o o add essing sho - e m dynamics, damping (i any)
ends, and seasonal componen s, T and S a e he end and seasonal componen s o he
ime se ies espec i ely. In addi ion, p and q o he BATS model a e he Au o eg essi e
p and mo ing a e age q pa ame e s espec i ely. Fu he mo e, m1;���;mT a e he
seasonal pe iods o he ARIMA model. The ma hema ical o mula o he BATS model
acco ding o Munim (2022), is p esen ed as ollows
ωð Þ
¼
ω
1
ω;ω�0
;ω¼0
�:(3)
The majo objec i e o he ime se ies p edic ion echnique known as he TBATS is o
o ecas complex seasonal ends by using exponen ial smoo hing. T igonome ic seaso-
nal unc ions we e used in place o he seasonal componen s o de elop he TBATS
model, which is a modi ied ime se ies me hod based on he BATS model (Thayyib e al.,
2023). Thus, he s uc u e o his model is he ini ial T ha signi ies he “ igonome ic”
unc ion. This is exp essed as TBATS ω;p;q;φ;m1
g:���;mT;kT
gð Þ. Hence, Talkhi
e al. (2024) showed ha he ma hema ical o mula o he TBATS model is as ollows
2.2. Seasonal pe iods
ωð Þ
¼l 1þϕb 1þXM
i¼1Sið Þ
miþd (4)
JOURNAL OF APPLIED ECONOMICS 7
hei capi al equi emen s and his is done by back es ing o co e ma ke isk
due o hei ading ac i i ies. The capi al equi emen size acco ding o is
calcula ed as
Cap ¼ ac �max VaR 0:01ð Þ;1
60 X59
i¼1VaR 10:01ð Þ
� � (23)
whe e, Fac is he mul iplica ion ac o epo ed in Table 1. In o he wo ds, he
equi ed capi al is equal o he mul iplica ion ac o ha is mul iplied by he
highes alue be ween oday’s 99 pe cen VaR and he mean o he las 60 days
99 pe cen VaR. The isk manage should au oma ically upg ade he mul iplica-
ion ac o om 3.0 o 4.0.
3. Resul s and Discussion
In his sec ion o he s udy, we p esen he analysis and discussion o he esul s. These
esul s a e p esen ed in ables and igu es. Following Musho i and Chikob u (2024),
Chikob u and Ndlo u (2023a), and Chinhamu e al. (2015), we analyse he gains and he
losses sepa a ely because we wan o model di e en isk p o iles which leads us o
unde s and he signi icance o a downside isk ha is indica ed by he nega i e e u ns
(losses) and also unde s anding he ail beha iou o hese losses which is c ucial o isk
managemen , pa icula ly o se ing capi al ese es, insu ance, and egula o y compli-
ance. Mo eo e , posi i e e u ns (gains) highligh oppo uni ies o subs an ial p o i s.
The e o e, unde s anding he ail beha iou o gains can u he help in de ising in es -
men s a egies and op imisa ion o po olios. I is wo h no ing ha inancial e u ns
a e asymme ic hence hei dis ibu ions a e o en asymme ic, meaning ha he ails on
he le (losses) and he igh (gains) can ha e di e en shapes and cha ac e is ics.
Modelling hem sepa a ely allows o a mo e accu a e ep esen a ion o hei dis inc
beha iou s. In addi ion, his app oach helps o iden i y be e and unde s and speci ic
ac o s ha d i e posi i e and nega i e e u ns; whe e, nega i e e u ns migh be d i en
by ac o s like ma ke c ashes, economic down u ns, o unexpec ed nega i e news.
Howe e , posi i e e u ns migh be d i en by ma ke booms, ex ao dina y company
pe o mance, o posi i e economic indica o s. Finally, managing he isk o ex eme
losses is c ucial o inancial ins i u ions, especially hose in ol ed in isk-sensi i e
ac i i ies like banking, insu ance, and in es men managemen . While on he o he
side, we wan o iden i y and le e age he oppo uni ies o ex eme gains which a e
bene icial o in es men s a egies, hedge unds, and pe o mance op imisa ion.
Figu e 1 depic s a plo o e u ns o FTSE/JSE-ALSI. Obse ed in his igu e a e he
ela i e dynamics o he FTSE/JSE-ALSI index o he speci ied pe iod. I is obse ed ha
Table 1. The basel II zones and he excep ions based on a 250 in aday ading sample.
Con idence le el
Zone 90% 95% 99% Mul iplica ion ac o
G een 0–32 0–17 0–4 3
Yellow 33–43 18–26 5–9 3.40, 3.50, 3.65, 3.75 o 3.85
Red �44 �27 �10 4
14 K. MAKATJANE AND K. MMELESI

he mean o e u ns is no cons an while he a iance oscilla es a ound high ola ili y
(ins abili y) and low ola ili y ( anquilli y) wi h mo e ex emes owa ds he end o he
sample pe iod. This implies ha FTSE/JSE-ALSI ha e ola ili y clus e ing.
Rega ding he ma ginal dis ibu ion, he quan ile-quan ile (Q-Q) and no mal
his og am plo in Figu e 2 e eal a s ong depa u e om linea i y in he lowe and
uppe ails o he dis ibu ion. This e idence is also seen in Table 2 whe e he
epo ed ku osis is g ea e han h ee and he skewness is less han ze o indica ing
ha e u ns on FTSE/JSE-ALSI a e asymme ical wi h nega i ely skewed inno a ions.
Figu e 2. No mal his og am and no mal quan ile-quan ile plo s.
Table 2. Desc ip i e s a is ics o e u ns se ies.
Mean
s d
de ia ion Skewness Ku osis J-B es S-W es K-S es
FTSE/JSE-Re u ns 0.000311 0.0096 −1.39970428 17.06483815 32932.373
(0.001)
0.898
(0.001)
0.484
(0.001)
No e: NB: alues in () a e p obabili y alues o he S-W and A-D espec i ely.
Figu e 1. Fi e-day e u ns o FTSE/JSE-ALSI plo .
JOURNAL OF APPLIED ECONOMICS 15
The no mal his og am on he le panel o Figu e 2 u he con i ms his nega i e
skewness which also possesses high ku osis ha is abo e h ee. The e o e,
a conclusion is ha FTSE/JSE-ALSI e u ns ha e a ail and lep oku ic beha iou .
Khan e al. (2021) in hei s udy o ex eme alue heo y and COVID-19 ha e ound
a ail beha iou in NIFTY-50 hey ha e used.
As e idenced in Table 2, he deduc ion he e is ha he o e all e u ns on FTSE/JSE-
ALSI a e inc easing du ing he pe iod in ques ion. The mean o he a e age e u ns is
e y small compa ed o he s anda d de ia ion. Mokoena (2016) and Bey ell (2016) in
hei s udies epo ed he same esul s. All in all, one can in e ha he mean o e u ns is
somehow smalle han he s anda d de ia ion o he e u ns. The FTSE/JSE-ALSI e u ns
conside ed e ealed ku osis ha is abo e 3, indica ing ha he e u ns se ies a e all
lep oku ic. Ano he ea u e o he e u ns se ies is he p esence o skewness; hey a e
ound o be nega i ely skewed, implying ha he dis ibu ion o FTSE/JSE-ALSI e u ns
is signi ican ly a e han he Gaussian dis ibu ion. This nega i e skewness indica es
ha he lowe ail o he dis ibu ion is hicke han he uppe ail and declines in e u ns
a e mo e common han hei inc eases. The Ja que-Be a (J-B), Shapi o Wilk (S-W), and
Kolmogo o – Smi no (K-S) es s con i med non-no mali y a a 5% le el o signi icance.
The h ee es s a is ics led o he ejec ion o he null hypo hesis o no mali y, con i ming
he esul s epo ed in Figu e 2. Acco ding o Vee and Gonpo (2014) and Ko kpoe and
Junio (2018), s ock ma ke s a e desc ibed by boom-bus cycles and such cycles eed in o
he ola ili y o he ma ke s.
3.1. T end analysis esul s
T ends a e e iden in inancial ime se ies da a du ing he week, mon h, qua e , e c. On
weekends and holidays, all exchanges a e closed; he e o e, no ading akes place. The
deg ee o ac i i y on weekdays is signi ican ly high. Ac oss FTSE/JSE-ALSI, he day-o -
he-week impac is no iced in Figu e 3. Gene ally, ma ke ac i i y is lowes on Monday
and highes on he las wo wo king days o he week. To be p ecise, ading ac i i y s a s
g owing p og essi ely om Monday o F iday. On weekends and holidays, he e a e no
Figu e 3. Day o he Week pa e n.
16 K. MAKATJANE AND K. MMELESI
ading ac i i ies, leading o low opening s ock p ices on Monday. The weekend e ec
(also called he Monday e ec , he day-o - he-week impac , o he Monday seasonal)
desc ibes how s ocks end o pe o m be e on F idays han on Mondays (S ohsal e al.,
2019). In gene al, o e all he yea s, he e has been an inc easing end in FTSE/JSE-ALSI
p ices and his obse a ion is seen in Figu e A1 in Appendix 1 o he whole sample
pe iod.
The Mann–Kendall es s a is ic and Sen’s slope es ima o a e used o analyse he long-
e m ends o he FTSE/JSE-ALSI. The ou come o he Mann–Kendall es esul s as
epo ed in Table 3 and e ealed ha FTSE/JSE-ALSI has a signi ican mono onic
inc easing long- e m end because he alue o τ>0. Sen’s slope alue also shows
signi ican ly inc easing magni udes o ends, which co espond wi h he Mann–
Kendall es esul s. The cause o his inc ease in he a e age e u ns o he o he ou
days o he week is posi i e, while he a e age e u ns o Monday a e no iceably nega i e,
he e o e, a signi ican posi i e end is depic ed in Table 3. Acco ding o Killian (2023)
and A sin and Oc an (2015), his inc ease is in luenced by he demand ha ou s ips he
supply. Tha is, mo e people wan o buy he sha e p ice ins ead o selling i ; and,
consequen ly, he p ice ises. O he wise, he p ice alls because supply is g ea e han
demand. I mo e buye s mo e in o he ma ke , he demand g ows, and sha e p ices go up
especially i he e is limi ed supply. I supply and demand a e jus abou equal, he sha e
p ice is likely o mo e a ound in a na ow ange o a while, un il one o he ac o s
ou weighs he o he (Khumalo, 2013). O he ac o s include mac oeconomic ac o s such
as in e es a e changes, inancial ou look, and in la ion luc ua ions. I he in e es a e
and in la ion a e go down, and he economic ou look is in a good s a e, demand usually
inc eases, and he sha e p ice is likely o ise (Killian, 2023).
3.2. Tba s-s a iona y gene alised ex eme alue dis ibu ion
To begin he main analysis, a TBATS is ained wi h a a io o 75% aining and 25%
alida ion se s. To accoun o s a iona i y in ou e u ns se ies, we employ he augmen ed
Dickey-Fulle (ADF) and Philips Pe on (PP). The same p ocedu e is done by Chinhamu
e al. (2015). The PP es is ca ied ou using he Ba le Ke nel spec al es ima ing me hod,
while he ADF es is se o lag 0 using he Schwa z In o ma ion C i e ion (SIC). Table 4
p esen s he esul s o he ADF and PP es s, which show ha he uni oo null hypo hesis
is no suppo ed o any es . As a esul , i is possible o ega d he e u n se ies on FTSE/
JSE-ALSI as s a iona y.
Table 4. Resul s o ADF and PP uni oo es s o
FTSE/JSE-ALSI e u n se ies.
Uni Roo es Tes s a is ic p- alue
ADF Tes −13.696 0.001
Philips-Pe on es −45.459 0.001
Table 3. Mann-Kendall es s a is ic and Sen’s slope es ima o .
M-K Tes S a is ic Kendall’s Tau p- alue Sen’s Slope
FTSE/JSE-ALSI 58.055 0.753 0.001 12.635
JOURNAL OF APPLIED ECONOMICS 17
Using he e u ns on FTSE/JSE-ALSI which a e compu ed using model (1), a TBATS model
is i ed o hese e u ns. Asso men o he bes TBATS model au oma ically is accomplished
by using he TBATS unc ion in Py hon o hese e u ns. Employing he maximum likelihood
es ima ion, he ollowing pa ame e es ima es a e ob ained: ^
λ¼0:337515;^
α¼0:021854
^
β¼0:960208 he damping pa ame e es ima e is ^
ϕ¼  0:91980288.
3.3. Selec ion o block minima/maxima
A e i ing ou TBATS model, we ex ac he esiduals which a e i.i.d. As a i s s ep,
25% o he esidual se ies is ese ed o u u e es ing. The ime in e al o he da a se o
be sepa a ed is om 21 Augus 2020 o 5 Ap il 2024. In he second s ep, he da a se
consis ing o 3646 obse a ions is di ided in o blocks o di e en sizes. Unlike Chikob u
and Ndlo u (2023a) who ex ac ed mon hly pe iod minima/maxima om he daily
e u ns o he BTC/USD and ZAR/USD e u ns da a, we de e mine he minimum
block size as 52 (i.e., weekly) and he maximum block size as 59 because his is he ade-
o p oblem be ween he bias and a iance, and Figu e 4 p esen s hese esul s.
Table 5 shows he i s wo blocks esul ing om pa i ioning he esiduals in o 52
blocks by da e o p o ide a de ailed o e iew o he block pa i ioning. A e ecei ing all
he blocks, he maximum and minimum alues o he blocks a e calcula ed as shown in
Table 5, and di e en da a se s o maximum and minimum alues a e ob ained. Figu e 4
u he shows a g aph o all minimum and maximum alues ob ained om 52 blocks. As
shown in Figu e 4, he wo a iables de i ed om he minimum and maximum wa es
beha e in a simila s uc u e o bo h posi i e e u ns and nega i e e u ns.
When a block is es ablished, wo andom a iables a e calcula ed om he
minimum and maximum alues ob ained om ha block, and he cha ac e is ics
o hei dis ibu ions a e eco ded by de e mining how hese a iables a e dis-
pe sed o bes dis ibu ed. The block leng h anges om 20 o 59 o nega i e
e u ns and 18 o 59 o posi i e e u ns (owing o da a spa si y), as de e mined
by which block size is be e explained o p ojec ed. The blocks allo ed o he
Figu e 4. Block maxima (le panel) and block minima ( igh panel).
18 K. MAKATJANE AND K. MMELESI
esiduals om he TBATS a e used o cons uc he da a blocks wi h he numbe s
20, 21, and 22 o nega i e e u ns and 18,19 and 20 o posi i e e u ns. Aside
om he maximum and maximum alues o he gene a ed block sizes, he column
o als o he block sizes o each o he 212 nega i e e u ns da a and he 210
posi i e e u ns da a a e calcula ed s a ing om 20 o 59 and 18 o 59, espec-
i ely. As a esul , i is possible o ma ch he bes and wo s ou comes among he
chosen blocks. Bu o he wise, he ou comes show which blocks in his applied
analysis a e he wo s and he bes . We he e o e ound ha he block size
achie ing he bes p edic ion o he es ima ion me hod ob ained om he max-
imum (minimum) alues ha is, posi i e e u ns is 15 wi h ank 3 and o he
nega i e e u ns is 11 wi h ank 4. These esul s a e epo ed in Table 6.
Since he bes block sizes a e ob ained o bo h nega i e and posi i e e u ns, we now
i sepa a ely hese gains (maxima) and losses (minima) o he GEV dis ibu ion and
Table 7 shows he pa ame e es ima es oge he wi h hei co esponding s anda d
e o s (SE) o bo h gains and losses. I can be seen in Table 7 ha o he nega i e
e u ns (i.e., losses), he posi i e shape pa ame e is es ablished, and he nega i e shape
pa ame e is u he es ablished o posi i e e u ns (i.e., gains). This implies ha
nega i e e u ns o losses o he FTSE/JSE-ALSI a e bes modelled by a ype II F éche -
GEV dis ibu ion, while he gains a e being modelled by a ype III Weibull-GEV
dis ibu ion. In a s udy compa ing he iskiness o Bi Coin/US Dolla and Sou h
A ican and/US Dolla e u ns, Chikob u and Ndlo u (2023a) i ed he GVE dis-
ibu ion o bo h he losses (nega i e e u ns) and gains (posi i e e u ns). The esul s
o hese au ho s po ayed a posi i e shape pa ame e o bo h e u ns, indica ing
a ype II F éche -GEV which is a con as o his s udy which ound a mix u e o
amily dis ibu ions o bo h gains and losses o FTSE/JSE-ALSI.
Table 5. FTSE/JSE-ALSI sample applica ion.
Minima Maxima
Size Da e Block Size Da e Block
1 08/02/2010 2246 1 04/03/2010 2546
2 08/62/2010 2200 2 01/02/2011 2835
3 08/08/2011 2481 3 16/05/2013 4967
4 24/06/2023 4133 4 22/10/2013 5500
5 30/01/2014 4666 5 10/04/2015 8101
6 16/01/2015 6343 6 15/08/2016 6979
7 11/12/2015 4703 7 20/03/2017 7242
8 06/07/2016 5674 8 20/03/2018 9620
9 06/12/2016 5944 9 28/08/2018 8119
10 27/06/2018 6717 10 15/01/2019 8469
11 26/10/2018 5952 11 20/06/2019 8109
12 14/08/2019 6886 12 20/11/2019 8408
13 23/03/2020 4544 13 29/07/2020 6505
14 30/10/2020 4727 14 12/03/2021 6405
15 23/04/2021 5496 15 15/06/2021 6500
16 26/11/2021 5270 16 30/02/2022 7232
17 06/07/2022 5086 17 01/03/2023 5989
18 15/12/2022 4740 18 01/03/2024 7433
19 11/05/2023 5141
20 03/042024 6439
Minimum 2200 Minimum 2546
Maximum 6886 Maximum 9620
JOURNAL OF APPLIED ECONOMICS 19

3.3.1. Goodness o i es
A e model es ima ion, he goodness o i (GoF) es is assessed. The Ande son-Da ling
and Shapi o–Wilk es o GoF es s a e used and he esul s a e p esen ed in Table 8.
None heless, S ephens (1977) ecommended hese es s o he GoF o ex eme alue
dis ibu ions, hence hei use in his s udy. Chikob u and Chi u i a (2015) and Maposa
e al. (2016) also used hese es s o es ing GoF o he GEV and GPD hey es ima ed in
hei s udies. The null hypo hesis is ha he e u ns ollowing a GEV dis ibu ion a e no
ejec ed (p- alues a e no signi ican ). Hence, he conclusion is ha he FTSE/JSE-ALSI
e u ns a e modelled e y well wi h he speci ied dis ibu ion. The es ima ed asymp o ic
laws can eplace he dis ibu ion o ex emes wi h he empi ical dis ibu ions o he ails
by hese esul s. In ex eme ma ke condi ions, i is hus possible o es ima e he po en ial
losses o s ock and commodi y indexes.
Table 6. Bes and wo s block size anking maximum FTSE/JSE-ALSI.
Minimas Maximas
Ranking Di e nce Block size Ranking Di e nce Block size
1 922 18 1 920 18
2 873 20 2 873 20
3 988 11** 3 977 22
4 889 15 4 889 15**
5 900 25 5 910 25
6 915 19 6 915 19
7 880 26 7 889 26
8 899 17 8 899 17
9 950 9 9 942 9
10 935 16 10 935 16
11 942 18 11 922 18
12 879 30 12 809 30
13 895 22 13 805 30
14 925 10 14 905 10
15 905 8 15 955 8
16 932 5 16 932 5
17 911 9 17 931 9
18 919 13 18 929 13
19 882 14
20 879 12
Table 7. Maximum likelihood es ima es o he GEV dis ibu ion.
Nega i e Re u ns (losses) Posi i e Re u ns (Gains)
Ex emes 11 15
^
�1.156 −0.266
se ^
�
�� 0.231 0.049
^
σ−0.033 0.024
se ^
σð Þ 0.028 0.084
^
μ0.0208 0.0084
se ^
μð Þ 0.031 0.015
Table 8. Goodness o i es o GEV dis ibu ion.
Tes Posi i e Re u ns Nega i e Re u ns
Ande son Da ling S a is ic 0.415 0.551
P-Value 0.335 0.156
Shapi o Wilk S a is ic 0.062 0.074
P-Value 0.362 0.247
20 K. MAKATJANE AND K. MMELESI
3.3.2. Compa a i e analysis and combined o ecas model using ensemble me hods
The pu pose o his sec ion is o de e mine he model ha bes mimics he da a and also
p oduces ewe o ecas s. This will help in assis ing he maximum dispa ching o he
Sou h A ican s ock ma ke . The h ee e o me ics namely mean absolu e e o (MAE),
mean squa e e o (MSE), and mean o ecas e o (MFE) a e used o measu e he
pe o mance o each model and he esul s a e summa ised in Table 9. Some en a i e
conclusions a e d awn om his able, which indica es ha no model is supe io o he
o he . Fo posi i e e u ns, MSE selec s he bes as he TBATS while o nega i e he
selec ed model is GEV using he same MSE me ic. Howe e , MAE selec s GEV as he
bes model o posi i e e u ns while MFE selec s bo h models.
Because we ound ha no model is supe io o he o he , we he e o e combine he wo
models and assess he pe o mance o he combina ion. We de e mine he ideal weigh o
agg ega e base lea ne p edic ions in such a way ha he esul ing ensemble minimises
he o al expec ed p edic ion e o ; hence, he aim is o ind he mos e ec i e way o
combine p edic ions. We, he e o e, use he gene alised ensemble me hod (GEM),
s acking ensemble, blending, bagging, and boos ing. These ensemble me hods a e chosen
because hey o e a ious ad an ages such as imp o ed pe o mance, educed o e -
i ing, model obus ness, and lexibili y in model selec ion and combina ion, making
hem popula choices in machine lea ning o achie ing be e esul s. Each ensemble
me hod has i s s eng hs and i is chosen based on he speci ic cha ac e is ics o he
p oblem and he da a a hand.
Table 10 shows he a e age esul s o TBATS-GEV based on he MLE sea ch me hod
along wi h he mean squa ed e o , mean absolu e e o , and mean o ecas e o made by
each ensemble me hod. The supe io i y o he designed ensemble echnique is seen by
Table 10. A e age esul s and c ea ed ensembles on nega i e and posi i e e u ns.
Ensemble Me hod Posi i e Re u ns Nega i e-Re u ns
TBATS-GEV MSE GEM 6.01 3.52
MSE S acking 4.01 3.27
MSE Blending 5.31 3.45
MSE Bagging 6.15 4.87
MSE Boos ing 7.28 6.88
TBATS-GEV MAE GEM 5.87 3.88
MAE S acking 3.01 3.1
MAE Blending 7.31 5.45
MAE Bagging 8.15 7.87
MAE Boos ing 7.13 6.19
TBATS-GEV MFE GEM 7.89 4.82
MFE S acking 4.00 2.22
MFE Blending 5.31 5.01
MFE Bagging 7.25 4.87
MFE Boos ing 7.28 6.88
Table 9. Pe o mance Model selec ion c i e ia.
MSE MAE MFE
Posi i e Re u ns
TBATS 8.779 7.238 6.328
GEV 8.953 6.998 6.328
Nega i e Re u ns
TBATS 8.879 7.028 6.879
GEV 8.053 7.028 6.128
JOURNAL OF APPLIED ECONOMICS 21
compa ing hei o ecas ing e o s. This demons a es imp o emen s in he s acking ensem-
ble algo i hm as compa ed o o he ensemble algo i hms. I is e iden om Table 10 ha he
s acking ensemble ou pe o med o he ensemble me hods, by p oducing ewe o ecas ing
e o s; because, MSE, MAE, and MFE all ad oca e o he TBATS-GEV model ha is
combined using he s acking ensemble algo i hm. We, he e o e, p oceed o es ima e he
ma ke isk using TBATS-GEV combined wi h he s acking ensemble algo i hm.
3.4. Risk es ima es
Since he s acking ensemble algo i hm has ou pe o med o he ensemble algo i hms, he
ou isk measu es discussed in sec ion 2 a e compu ed using he TBAT-GEV dis ibu ion o
e alua e he isk o losses and gains in he inancial sec o in Sou h A ica o e u ns on
i e day FTSE/JSE-ALSI and he esul s a e p esen ed in Table 11. The compu ed alues
sugges ha he losses (i.e., nega i e e u ns) a e iskie han he gains (nega i e e u ns)
since hey ha e highe alues o isk measu es. Fo he gains, a 99%, he VaR alue is
0.000850. This implies ha he e is a lowe likelihood o signi ican losses; indica ing ha he
isk o la ge nega i e e u ns is ela i ely low wi h a alue o 0.85% o VaR a a 99% le el o
signi icance. The 99% le el o signi icance sugges s ha lowe isk is ealised o he ou
es ima ed isk measu es han a 90% and 95%. This is because, a a highe le el o
signi icance, less oom o e o is ole a ed. These esul s a e in con adic ion o he one
ound by Chikob u and Ndlo u (2023b); because he s udy by hese au ho s had ound he
lowe VaR es ima e a 95% le el.
I is wo h no ing ha a 99% le el, Glue-VaR is nega i e gi ing he alue o −
0.000012235. This sugges s ha i one is ocusing on he lowe ail o he dis ibu ion o
FTSE/JSE-ALSI e u ns, he e is a high likelihood o in es o s incu ing losses a his ail o
he dis ibu ion. This nega i e alue e lec s a ela i ely high le el o isk. I indica es ha he
in es men has he po en ial o signi ican losses, especially in ex eme ma ke condi ions
o du ing e en s ha nega i ely impac he in es men . Yu e al. (2019) also e ealed ha
a one-day change in he inancial ma ke ’s alue would no dec ease by mo e han
0.00123%. To ake in o accoun ma ke liquidi y cons ain s and Basel egula ions, 5-day
isk ho izons in addi ion o he mo e ypical 1-day ho izon a e conside ed. Compa ing
losses o gains, he esul s indica e ha he p ospec s o po en ial ex eme losses a e g ea e
han he p ospec s o po en ial ex eme gains.
3.4.1. E alua ing he accu acy o isk measu es
A back es ing p ocedu e is used o e alua e he isk measu es’ accu acy, and he es ima es
a e shown in Tables 12 and 13. While back es ing he es ima ed isk measu es,
a p edic ed loss is compa ed o he ac ual loss, obse ed he ollowing day. An exceedance
Table 11. Compu a ion isk measu es.
Posi i e Re u ns Nega i e Re u ns
P VaR ES CTE Glue-VaR VaR ES CTE Glue-VaR
0.9 0.001544 0.023454 0.023854 0.000630 0.001839 0.027366 0.027366 0.001110169
0.95 0.001230 0.023924 0.023924 0.000363 0.001471 0.027331 0.027331 0.000432336
0.99 0.000850 0.023967 0.023967 0.000134 0.000734 0.027366 0.027366 −0.000012350
22 K. MAKATJANE AND K. MMELESI
occu s when he ac ual loss exceeds he calcula ed isk measu e. An expec ed numbe o
exceedances o e he VaR cu e is p edic ed s a is ically o VaR. VaR0:99, Rydell (2013)
s essed ha he loss exceeds ha alue 99 imes ou o 100, and one exceedance o e e y
100 obse a ions is s a is ically expec ed. These exceedances a e used as he e e ence. Fo
his s udy, he combined TBATS-GEV is conside ed eliable i he numbe o excee-
dances when back es ing he model is wi hin a 10% con idence in e al om he
s a is ically expec ed losses.
The null hypo hesis is ha he isk model is co ec ly speci ied and accu a ely p edic s
inancial isk o FTSE/JSE-ALSI. Ne e heless, bo h he Kupiec and he Ch is o e sen
es s sugges ha he null hypo hesis is no ejec ed and he conclusion is ha , he ou
isk measu es p oduced eliable, e icien , and unbiased isk es ima es o bo h 95% and
99% con idence le els. Fo long- ime pe iods such as mo e han en yea s as is he case in
his s udy, he combined model p oduces sui able isk es ima es. Acco ding o Bee and
T apin (2018), hese long pe iods a e indica ed by high p obabili y alues o losses and
gains a all selec ed con idence le els o all he es s used.
The esul s o he egula o y back es on he isk measu es a e p esen ed in Table 13.
The numbe s in he able display how many imes each isk measu e in each zone has
been back es ed o e he cou se o a yea (250 ading days). The es is anked as ollows:
(1) he g een zone i he e a e 0 o 4 excep ions epo ed wi hin a 250- ade day window;
(2) he yellow zone i he e a e 5 o 9 excep ions; and (3) he ed zone i he e a e mo e
han 9 excep ions. All he isk measu es all in o he yellow zone, which imposes a ine
anging om 0 o 85 imes he isk measu e o calcula e he ma ke isk cha ge.
In e es ingly, he hyb ids wi h 250 days o da a pe o med uni o mly h oughou all
he isk measu es. This gi es an annual a e age penal y ha anges om 20% o 59%.
Gene ally, all he isk measu es showed ha he combined TBATS-GEV is a good
Table 13. Basel III zones back es esul s.
Zone 250 Days VaR 250 Days ES 250 Days CTE 250 Days Glue VaR
Gains G een Zone 3 3 5 6
Yellow Zone 2 2 1 3
Red Zone 2 1 0 1
a e age annual penal y 0.487 0.543 0.655 0.654
Losses G een Zone 5 3 4 4
Yellow Zone 3 1 1 3
Red Zone 1 1 1 0
a e age annual penal y 0.472 0.376 0.55 0.35
Table 12. Back es ing isk measu es.
p- alues o Kupiec es p- alues o Ch is o e sen es
Risk Measu e Le el 0.95 0.99 Risk Measu e Le el 0.95 0.99
GEV VaR Gains 0.517 0.378 VaR Gains 0.207 0.885
GEV ES Gains 0.919 0.877 ES Gains 0.865 0.987
GEV CTE Gains 0.517 0.522 CTE Gains 0.476 0.939
GEV Glue-VaR Gains 0.796 0.95 Glue-VaR Gains 0.871 0.994
GEV VaR Losses 0.529 0.902 VaR Losses 0.801 0.739
GEV ES Losses 0.668 0.872 ES Losses 0.701 0.885
GEV CTE Losses 0.65 0.872 CTE Losses 0.698 0.839
GEV Glue-VaR Losses 0.694 0.682 Glue-VaR Losses 0.98 0.794
JOURNAL OF APPLIED ECONOMICS 23
Appendix
Figu e A1. Annual end o each yea .
30 K. MAKATJANE AND K. MMELESI