G een Exchange-T aded Fund Pe o mance
E alua ion Using he EU-EV Risk Model
I ene B i o1
[0000−0002−7075−3265], Jos´e Manuel Aze edo2[0000−0001−6951−4278],
and Ana Isabel Aze edo2[0000−0003−0882−3426]
1Cen e o Ma hema ics, Depa men o Ma hema ics, Uni e si y o Minho,
4800-045 Guima ˜aes, Po ugal,
[email p o ec ed]
2CEOS.PP, ISCAP, Poly echnic o Po o, 4465-004 S. Mamede de In es a, Po ugal,
[email p o ec ed], [email p o ec ed]
Abs ac . This wo k e alua es he pe o mance o g een exchange- aded
unds (ETFs) using he expec ed u ili y, en opy and a iance (EU-EV)
isk model. Da a om 14 g een ETFs analysed in ea lie li e a u e in he
in-sample pe iod om Janua y 2008 o Decembe 2010 a e used.
The g een ETFs a e anked acco ding o hei isk, conside ing he e-
u ns’ expec ed u ili y, en opy and a iance, and he bes - anked ETFs
a e selec ed o cons uc equally weigh ed po olios. Then, he pe o -
mance o he g een ETFs po olios is e alua ed and compa ed wi h hose
o he S&P500 Index. Cumula i e e u ns in in-sample and ou -o -sample
pe iods and pe o mance me ics, such as Maximum d awdown, Sha pe
a io, So ino a io, Be a and Alpha, a e analysed. The esul s show ha ,
in gene al, he equally weigh ed po olios o med wi h hal he numbe
o bes - anked ETFs ou pe o m he benchma k index in he in-sample
pe iod and o speci ic ime anges in he ou -o -sample pe iods.
Keywo ds: ETF ·Po olio pe o mance e alua ion ·EU–EV isk model
1 In oduc ion
Exchange- aded unds (ETFs) a e popula inancial ins umen s, made up o
di e en secu i ies (e.g. s ocks, bonds), ha can be aded on an exchange and
ha o e a g ea di e si ica ion, ax e iciency and low expenses (see e.g. [1],
[2], [3]). Among hese, g een ETFs a e unds ha in es in companies ha
suppo en i onmen ally esponsible echnologies o a e in ol ed wi h esea ch,
de elopmen , p oduc ion and p o ision o al e na i e ene gy and possess posi-
i e en i onmen al, social and co po a e go e nance (ESG) cha ac e is ics. Due
o he inc easing in e es in sus ainable economic de elopmen and due o he
inc eased inancializa ion o g een ene gy coupled wi h he in e es o in es o s
in his asse class, see e.g. he ecen s udy in [4] and e e ences he ein, i is ele-
an o analyse in mo e de ail he isk and pe o mance o g een ETFs, since one
can ind only ew con ibu ions add essing his issue. Sabbaghi [5] in es iga ed
he ime-se ies beha iou o g een exchange aded und e u ns and hei as-
socia ed condi ional ola ili y dynamics using he GARCH me hodology. In [6],
2 I ene B i o e al.
Sabbaghi cons uc ed an equally-weigh ed po olio o g een ETFs and analysed
i s e u n pe o mance (using e.g. mean, s anda d de ia ion, Jensen’s alpha) and
compa ed i o ha o he S&P500 index o e di e en sub-pe iods in ime om
2005 o 2010, be o e and a e he inancial ma ke collapse o 2008. He ound
ha he po olio ou pe o med he S&P500 index p io o he inancial collapse,
howe e he po olio was highly ola ile and unde pe o med he S&P500 index
in he pe iod a e he collapse. Tsolas and Cha les [7] in es iga ed he pe o -
mance o g een ETFs using da a en elopmen analysis. Riz i e al. [4] s udied
he ela ionship be ween g een and g ey ene gy ETFs and concluded ha g een
ene gy is mo e p ominen in de e mining he e u ns in he US equi y ma ke .
The objec i e o he p esen wo k is o analyse he isk and pe o mance o
g een ETFs using he ecen ly p oposed EU–EV isk model and e alua e hei
pe o mance using di e en me ics. The EU–EV isk model, de eloped in [8],[9],
can be used o classi ying s ock isks and o he p eselec ion o he bes anked
s ocks in o de o cons uc op imum po olios wi h a educed numbe o s ocks
(see [10],[11]). The aim is now o apply he EU–EV isk model o g een ETFs
in o de o assess hei isk (using expec ed u ili y, en opy and a iance) and in
o de o in es iga e he capabili y o he isk model o selec he e icien unds
o in es men pu poses. Da a o a sample o 14 g een ETFs om 2008 o 2010
a e used, ha we e analysed in ea lie li e a u e ([6],[7]). The ETFs a e anked
acco ding o hei isk and di e en equally-weigh ed po olios a e o med. The
po olios’ pe o mances a e analysed and compa ed wi h he S&P500 index
conside ing di e en ime pe iods.
This pape is s uc u ed as ollows. Sec ion 2 con ains he de ini ions o he
EU–EV isk model and o he pe o mance measu es and explains how he ETF
po olios a e cons uc ed and hei pe o mances a e e alua ed. In sec ion 3,
he p oposed me hod is applied o he sample o 14 g een ETFs. The isks o
he ETFs a e analysed, di e en po olios a e o med wi h he bes anked
ETFs and hei pe o mances a e compa ed wi h hose o he benchma k in he
in-sample pe iod and in di e en ou -o -sample ime in e als. The pape ends
wi h he Conclusions in sec ion 4.
2 ETF Po olio Cons uc ion and Pe o mance
E alua ion
In o de o cons uc he ETF po olios, he EU–EV isk model will be used
o selec he bes anked ETFs om an ini ial gi en se o ETFs. The selec ed
ETFs will hen be used o build equally weigh ed po olios, whose pe o mance
is hen analysed using di e en pe o mance me ics.
2.1 EU-EV Risk Model
Conside a se o ETFs S={S1, . . . , SI}and he ac ion space A={a1, . . . , aI},
whe e
ai= (xi1, pi1;xi2, pi2;. . . ;xiN , piN )∈A
G een ETF pe o mance e alua ion 3
is he ac ion o selec ing he ETF Si,i= 1, . . . , I, yielding he equency dis-
ibu ion o e u ns, whe e xin a e he ou comes, occu ing wi h p obabili ies
pin,n= 1, . . . , N, ep esen ed by he andom a iable Xi. The EU–EV isk o
he ac ion aidepends on he co esponding e u ns’ dis ibu ion wi h andom
a iable Xias ollows.
De ini ion 1 (EU–EV isk). The EU–EV isk o he ac ion aiwi h associ-
a ed andom a iable Xiis de ined by
Rλ(Xi) = λ
2
H(Xi) + Va [Xi]
max
ai∈A{Va [Xi]}
−(1 −λ)E[u(Xi)]
max
ai∈A{|E[u(Xi)]|},
whe e 0≤λ≤1,u(x) = ln(1 + x), x ≥0
−ln(1 −x), x < 0is he u ili y unc ion and
H(Xi) = −PN
n=1 pin ln pin is he en opy.
The cons an λis a ade-o pa ame e ha combines he expec ed u ili y and
he unce ain y e lec ed by en opy [12] and a iance [13] and i can be used o
exp ess he di e en isk a i udes o decision-make s (see [9] o mo e de ails).
I λ < 0.5, hen mo e weigh is gi en o he expec ed u ili y e m, which co -
esponds o a isk-a e se a i ude [14]. I λ > 0.5, mo e weigh is gi en o he
unce ain y componen and his e lec s a isk-seeking beha iou . The ade-o
pa ame e is used in he model o : building po olios ha s ike a balance
be ween isk and e u n (po olio isk models), unde s anding and na iga ing
unce ain y in inancial ma ke s (en opy and unce ain y), e alua ing he luc-
ua ion and po en ial dange s associa ed wi h asse s and de i a i es ( a iance).
Gaining a solid g asp o hese concep s enables in es o s and inancial p o es-
sionals o make well-in o med decisions, e ec i ely manage isk, and maximise
he po en ial o hei po olios.
The ETFs a e classi ied wi h he EU-EV isk model by anking hem ac-
co ding o he ollowing ule. Gi en wo ETFs Si1and Si2,i1, i2∈ {1, . . . , I},
i
Rλ(Xi1)< Rλ(Xi2),
hen Si1is p e e ed o e Si2(which can be w i en as Si1≻Si2), since Si1has
lowe EU–EV isk han Si2.
Fo λ∈[0,1], we will selec om he se S he I/2 bes anked ETFs and o m
subse s (wi h he ini ial numbe o unds educed o he hal ) o he di e en
alues o λ. Wi h hese subse s we will cons uc equally weigh ed po olios,
whose pe o mance will hen be compa ed wi h hose o he equally weigh ed
po olio consis ing o all I unds and wi h a benchma k po olio. Acco ding
o esul s p esen ed in ea lie li e a u e, o example in [15], [16], using equal-
weigh ed s a egies lead o po olios ou pe o ming alue-weigh ed s a egies,
he e o e we op ed o use equal weigh s in he po olio cons uc ion.
4 I ene B i o e al.
2.2 Pe o mance Measu es
The ollowing pe o mance measu es (see e.g. [17],[18]) will be used o analyse
he pe o mance o he po olios in di e en ime pe iods.
De ini ion 2 (Maximum D awdown). The Maximum D awdown (MDD) is
he la ges pe cen age d op in o al e u ns om he s a o he end o a pe iod,
compu ed o e all in e als o ime ha can be o med wi hin a speci ied in e al
o ime, and i is de ined as ollows. Le x , = 1, . . . , T, ep esen he daily
cumula i e e u ns o he po olio. The Maximum D awdown is gi en by
MDD =PV −LV
PV ,
whe e LV =x∗
, he lowes poin alue ( ough alue), and PV =x∗
,max, he
peak alue, a e he alues ha maximize he d awdown
DD =x ,max −x
x ,max
,
whe e x ,max = max{xs:s= 1, . . . , } o = 1, . . . , T . A lowe MDD alue
indica es a lesse deg ee o isk.
In he ollowing de ini ions, P ep esen s he expec ed e u n o he po olio,
σPis he s anda d de ia ion o he po olio e u ns and is he isk- ee a e.
We will conside = 0, meaning ha he a e o e u n o a ze o isk benchma k
in es men is aken o be equal o ze o.
De ini ion 3 (Sha pe a io, So ino a io, Be a). The Sha pe a io mea-
su es he excess e u n ( he e u n o he po olio less he isk- ee a e o in e -
es ) pe uni o o al isk o he po olio ( he s anda d de ia ion o he po olio’s
e u ns) and is de ined by
Sha pe = P−
σP
.
The Sha pe a io wi h = 0 quan i ies he ela ion be ween he expec ed e u ns
and he s anda d de ia ion o he e u ns o he po olio. Po olios wi h highe
Sha pe a ios pe o m be e acco ding o his measu e.
The So ino a io is a modi ica ion o he Sha pe a io, whe e only he down-
side de ia ion is aken in o accoun , and is exp essed by
So ino = P−
σ−
P
,
whe e σ−
P ep esen s he s anda d de ia ion o he nega i e po olio e u ns. A
highe So ino a io indica es a be e pe o mance.
The isk me ic Be a de e mines he isk o ola ili y o a po olio by com-
pa ing i o he ma ke and is de ined by
Be a =Co ( P, S)
σ2
S
,
G een ETF pe o mance e alua ion 5
whe e Co ( P, S)is he co a iance be ween he expec ed e u n o he po olio
and he expec ed ma ke e u n So he benchma k S, and σ2
Sco esponds o
he a iance o he ma ke e u ns. Po olios ha ing Be a>1can be in e p e ed
o be mo e ola ile o iskie han he benchma k. I Be a<1, he po olio is
less ola ile han he benchma k. I Be a= 1, i has he same ola ili y as he
benchma k.
De ini ion 4 (Alpha). Jensen’s Alpha is a pe o mance me ic ha measu es
he po olio e u n ela i e o he ma ke e u n and is de ined by
Alpha = P−[ +Be a( S− )],
whe e Be a is gi en in De ini ion 3. A alue o Alpha>0indica es ha he
po olio has pe o med be e han he ma ke index. I Alpha<0, he po olio
has unde pe o med he ma ke index. I Alpha= 0, he po olio’s pe o mance
is in line wi h ha o he ma ke .
3 Applica ion o G een ETFs
The aim is o apply he EU-EV isk model o he selec ion o unds om a sample
o 14 g een ETFs in o de o in es iga e i he EU-EV model adequa ely selec s
he ele an ETFs o an e icien po olio cons uc ion wi h a educed numbe
o ETFs. Da a o a sample o 14 g een ETFs (wi h icke symbols PBW, PHO,
PUW, PKN, PIO, PZD, EVX, NLR, FIW, QCLN, CGW, DSI, KLD, PBD),
desc ibed in [6] and [7], in he in-sample pe iod om Janua y 2008 o Decembe
2010 a e used.
3.1 Po olio Cons uc ion
F om each g een ETF Si,i= 1,...,14, he daily closing p ices {Pi0, . . . , PiT },
T+ 1 = 756, om Janua y 2008 o Decembe 2010 a e collec ed. The daily
e u ns a e calcula ed by
i = ln Pi
Pi( −1) ;i= 1,...,14; = 1,...,755.
The equency dis ibu ion o e u ns is de e mined is ollows. The in e al
[ min, max] = [−1.0114,1.0205] whe e min = min
1≤i≤14{ i1, . . . , i755}and max =
max
1≤i≤14{ i1, . . . , i755}, is di ided in o N= 19 subin e als Jn,n= 1, . . . , N, o
leng h ∆= max− min
N= 0.10694. Then, he ela i e equency o he e u n o
Siin he subin e al Jnis calcula ed by
pin =|{ i ∈Jn: = 1, . . . , T}|
T
6 I ene B i o e al.
and he expec ed e u n o Si om he subin e al Jnis es ima ed by
xin =1
|{ i ∈Jn: = 1, . . . , T}| X
i ∈Jn
=1,...,T
i ,
whe e |·| ep esen s he ca dinali y o a se .
The EU–EV isks o each und Si,i= 1,...,14, will hen be de e mined
using he equency dis ibu ions.
Table 1 con ains he EU–EV isks o he 14 g een ETFs o ce ain alues
o λ(λ= 0,0.5,0.75,0.85,1). Fo each alue o λ, he 7 lowe isk alues a e
highligh ed in bold.
Table 1. EU–EV isks o λ= 0,0.5,0.75,0.85,1.
Fund Ticke R(ai)
λ= 0 λ= 0.5λ= 0.75 λ= 0.85 λ= 1
S1PBW 1.0000 0.5934 0.3901 0.3088 0.1868
S2PHO 0.0894 0.0989 0.1037 0.1056 0.1085
S3PUW -0.0072 0.0636 0.0991 0.1132 0.1345
S4PKN 0.1541 0.4380 0.5800 0.6367 0.7219
S5PIO 0.1917 0.1404 0.1147 0.1044 0.0890
S6PZD 0.2849 0.2022 0.1609 0.1443 0.1195
S7EVX -0.0386 0.0147 0.0414 0.0521 0.0681
S8NLR 0.3383 0.2189 0.1593 0.1354 0.0996
S9FIW -0.0410 0.0231 0.0551 0.0680 0.0872
S10 QCLN 0.6343 0.4011 0.2846 0.2380 0.1680
S11 CGW 0.2021 0.1407 0.1100 0.0977 0.0793
S12 DSI 0.0787 0.0757 0.0742 0.0736 0.0727
S13 KLD 0.0806 0.0657 0.0582 0.0552 0.0570
S14 PBD 0.8447 0.5029 0.3320 0.2636 0.1611
Conside ing he ange o λ∈[0,1], one can o m he ollowing 5 di e en
subse s
G7,λ=0 ={PHO,PUW,PKN,EVX,FIW,DSI,KLD}, λ ∈[0,0.0561)
G7,λ=0.5={PHO,PUW,PIO,EVX,FIW,DSI,KLD}, λ ∈[0.0561,0.5175)
G7,λ=0.75 ={PHO,PUW,EVX,FIW,CGW,DSI,KLD}, λ ∈[0.5175,0.8139)
G7,λ=0.85 ={PHO,PIO,EVX,FIW,CGW,DSI,KLD}, λ ∈[0.8139,0.9655)
G7,λ=1 ={PIO,EVX,NLR,FIW,CGW,DSI,KLD}, λ ∈[0.9655,1],
whe e he bes 7 unds wi h lowe isk we e selec ed, wi h λbelonging o he in-
e als [0,0.0561), [0.0561,0.5175), [0.5175,0.8139), [0.8139,0.9655), [0.9655,1].
Each pa icula alue o λin Table 1 belongs o one o hese in e als, so ha he
no a ions G7,λ=0,G7,λ=0.5,G7,λ=0.75,G7,λ=0.85 and G7,λ=1 a e used o ep esen
he di e en se s.
G een ETF pe o mance e alua ion 7
The po olios a e cons uc ed as equal weigh ed combina ions o he se en
bes anked unds om he i e se s. The po olios will be deno ed by G7,λ=0,
G7,λ=0.5,G7,λ=0.75,G7,λ=0.85 and G7,λ=1.
In he ollowing analysis, we will use he S&P500 index as benchma k po -
olio. We will also conside he equal-weigh ed po olio G14 o med wi h all 14
g een ETFs
G14 ={PBW, PHO, PUW, PKN, PIO, PZD, EVX, NLR, FIW, QCLN,
CGW, DSI, KLD, PBD}
and he equal-weigh ed po olio G4 o med wi h he g een ETFs o he se
G4={EVX,NLR,FIW,CGW},
These ETFs we e iden i ied in an ea lie s udy (see [7]) as op e icien ETFs
using da a en elopmen analysis.
3.2 Pe o mance E alua ion
The pe o mance o he i e po olios cons uc ed wi h he bes anked ETFs
will be analysed and compa ed wi h he pe o mance o he G14 and G4po olios
and wi h he S&P500 benchma k po olio, i s in he in-sample pe iod om
Janua y 2008 o Decembe 2010 and hen in di e en ou -o -sample pe iods, om
Janua y 2011 o Decembe 2020, wi h one-yea , wo-yea , i e-yea and en-yea
ime ho izons. Fo ha pup pose, he cumula i e e u ns a e calcula ed in he
men ioned ime pe iods, along wi h he me ics p esen ed in Sec ion 2.2, using
he S&P500 index as benchma k S.
Conside ing he in-sample pe iod om Janua y 2008 o Decembe 2010, Fig-
u e 1 con ains he cumula i e e u ns in ha ime in e al and Table 2, he
esul s o he di e en me ics (whe e in he ollowing ables he alues co e-
sponding o he bes pe o mances a e highligh ed in bold). One can obse e
ha G7,λ=0 is he bes pe o ming po olio acco ding o all pe o mance indica-
o s, yielding he highes cumula i e e u ns and his wi h he lowes ola ili y
acco ding o Be a. The G14 po olio is he second bes pe o ming po olio
aking in o accoun he cumula i e e u ns, he Alpha, Sha pe and So ino me -
ics, howe e , achie ed wi h he highes ola ili y in e ms o Be a and i has
he highes maximum d awdown. The e olu ion o he cumula i e e u ns o he
emaining po olios is e y simila o he e olu ion o he benchma k’s cumu-
la i e e u ns. Since he ETFs o G7,λ=0 we e selec ed wi h he EU-EV model
p i ileging highe expec ed u ili y and almos igno ing a iance and en opy in
he gi en pe iod, one would expec a be e pe o mance his po olio in e ms
o highe e u ns wi h espec o he o he po olios in he same pe iod and his
is consis en wi h he ob ained esul s.
In he one-yea and wo-yea ou -o -sample pe iods om 2011 o 2012, he
esul s seem o indica e ha G7,λ=0 ollowed by S&P500 a e he bes pe o ming
po olios. This is e iden om he pe o mance measu es in Table 3 and Table 4,
8 I ene B i o e al.
Fig. 1. Cumula i e e u ns om Janua y 2008 o Decembe 2010.
Table 2. Pe o mance measu es om Janua y 2008 o Decembe 2010.
MDD Sha pe So ino Be a Alpha
S&P500 0.5325 -0.0019 -0.0024 1.0000 0.0000
G14 0.5704 0.5363 0.7863 1.0164 0.2402
G40.5647 0.0510 0.0632 0.9447 0.0168
G7,λ=0 0.4890 0.9670 1.6931 0.8925 0.6630
G7,λ=0.50.5522 0.1161 0.1480 0.9771 0.0381
G7,λ=0.75 0.5514 0.1111 0.1396 0.9798 0.0365
G7,λ=0.85 0.5480 0.0807 0.1035 0.9409 0.0255
G7,λ=1 0.5504 0.0425 0.0538 0.9779 0.0133
whe e only he maximum d awdown is now he lowes one o S&P500. The
po olio G7,λ=0 achie es highe cumula i e e u ns o a la ge ime in e al in
2011 (see Figu e 2). Howe e , he e a e also ime pe iods whe e S&P500 exhibi s
highe e u ns (in he i s qua e and in he las qua e o ha yea ). In 2012
(see Figu e 3), he S&P500 benchma k po olio ou pe o ms he o he po olios
in e ms o cumula i e e u ns o a wide ime ange in 2012 (only in he las
qua e o 2012 he po olio G7,λ=0 su passes again he benchma k). Bo h in he
one-yea and wo-yea pe iods, he po olios G14,G4and G7,λ=1 unde pe o m
he emaining po olios. All po olios a e mo e ola ile han S&P500.
In he i e-yea ou -o -sample pe iod om 2011 o 2015, now, in con as o
he o he ou -o -sample pe iods, i is S&P500 ha exhibi s he bes Sha pe and
So ino a ios and he lowes maximum d awdown (see Table 5), only he Alpha
alue is highe o G7,λ=0, which also yields he highe cumula i e e u ns (see
Figu e 4).
G een ETF pe o mance e alua ion 9
Fig. 2. Cumula i e e u ns o 2011.
Table 3. Pe o mance measu es o 2011.
MDD Sha pe So ino Be a Alpha
S&P500 0.1939 0.0663 0.0850 1.0000 0.0000
G14 0.3032 -0.5402 -0.7855 1.1621 -0.1742
G40.2492 -0.6656 -0.0159 1.0523 -0.1741
G7,λ=0 0.2594 0.0820 0.1309 1.0721 0.0165
G7,λ=0.50.2606 -0.3128 -0.4320 1.0934 -0.0947
G7,λ=0.75 0.2476 -0.2355 -0.3283 1.0820 -0.0752
G7,λ=0.85 0.2326 -0.2671 -0.3708 1.0514 -0.0803
G7,λ=1 0.2349 -0.5135 -0.6958 1.0249 -0.1337
Table 4. Pe o mance measu es om Janua y 2011 o Decembe 2012.
MDD Sha pe So ino Be a Alpha
S&P500 0.1939 0.3610 0.4598 1 0
G14 0.3090 0.0110 0.0159 1.1402 -0.0714
G40.2492 -0.1570 -0.2143 1.0661 -0.1003
G7,λ=0 0.2594 0.5742 0.9511 1.0131 0.1265
G7,λ=0.50.2606 0.1333 0.1797 1.0925 -0.0443
G7,λ=0.75 0.2476 0.1929 0.2620 1.0814 -0.0318
G7,λ=0.85 0.2326 0.1910 0.2595 1.0489 -0.0325
G7,λ=1 0.2349 -0.0527 -0.0707 1.0328 -0.0772
Conside ing he e olu ion o he cumula i e e u ns in he en-yea ou -o -
sample pe iod om 2011 o 2020 (see Figu e 5), one can obse e ha om 2016
onwa ds he cumula i e e u ns o G7,λ=0 inc ease no ably. The po olio G14
