Gambe a, Vaughn; Kwon, Roy
A icle
Risk e u n ade-o in elaxed isk pa i y po olio
op imiza ion
Jou nal o Risk and Financial Managemen
P o ided in Coope a ion wi h:
MDPI – Mul idisciplina y Digi al Publishing Ins i u e, Basel
Sugges ed Ci a ion: Gambe a, Vaughn; Kwon, Roy (2020) : Risk e u n ade-o in elaxed isk pa i y
po olio op imiza ion, Jou nal o Risk and Financial Managemen , ISSN 1911-8074, MDPI, Basel, Vol.
13, Iss. 10, pp. 1-28,
h ps://doi.o g/10.3390/j m13100237
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Jou nal o
Risk and Financial
Managemen
A icle
Risk Re u n T ade-O in Relaxed Risk Pa i y
Po olio Op imiza ion
Vaughn Gambe a * and Roy Kwon *
Depa men o Mechanical and Indus ial Enginee ing, Uni e si y o To on o,
5 King’s College Rd, To on o, ON M5S 3G8, Canada
*Co espondence: [email p o ec ed] (V.G.); [email p o ec ed] (R.K.)
Recei ed: 28 Augus 2020; Accep ed: 1 Oc obe 2020; Published: 4 Oc obe 2020


Abs ac :
This pape o mula es a elaxed isk pa i y op imiza ion model o con ol he balance
o isk pa i y iola ion agains he o al po olio pe o mance. Risk pa i y has been c i icized as
being o e ly conse a i e and i is imp o ed by e-in oducing he asse expec ed e u ns in o
he model and pe mi ing he po olio o iola e he isk pa i y condi ion. This pape p oposes
he inco po a ion o an explici a ge e u n goal wi h an in ui i e a ge e u n app oach in o
a second-o de -cone model o a isk pa i y op imiza ion. When he a ge e u n is g ea e han isk
pa i y e u n, a iola ion o isk pa i y alloca ions occu s ha is con olled using a compu a ional
cons uc o ob ain nea - isk pa i y po olios o e ain as much isk pa i y-like ai s as possible.
This model is used o demons a e empi ically ha highe e u ns can be achie ed han isk pa i y
wi hou he isk con ibu ions de ia ing d ama ically om he isk pa i y alloca ions. Fu he mo e,
his s udy e eals ha he elaxed isk pa i y model exhibi s ad an ageous ai s o obus ness o
expec ed e u ns, which should no de e he use o expec ed e u ns in isk pa i y model.
Keywo ds:
po olio op imiza ion; isk pa i y; equal- isk budge ; elaxa ion; asse alloca ion; ma ginal
isk con ibu ion; obus op imiza ion; Ma kowi z; isk-based; obus
1. In oduc ion
While und managemen and po olio op imiza ion ha e been ex ensi ely s udied in li e a u e,
one aspec known as isk pa i y has gained conside able ac ion. Risk pa i y was de eloped o
emo e he unce ain y o es ima ed e u ns and o p o ec agains losses associa ed wi h po olio
concen a ion. A p oac i e isk pa i y app oach has been shown o educe d aw-downs o e mo e
isky s a egies, which gi es he s a egy i s impo ance, as ou ed by Qian (2005);
Mailla d e al. (2010)
and Lee (2011). Howe e , he inc ease in d aw down p o ec ion limi s upside g ow h in s ong bull
ma ke s, so i is commonly used o p o ec om la ge losses a he han o seek g ea e e u ns.
The conse a i eness may be oo s eep o some p ac i ione s, so i would be use ul o a p ac i ione
o inc ease e u ns while s ill e aining o some deg ee he bene i s o isk pa i y alloca ions. In his
ega d, one can s ill gua d agains unexpec ed ma ke down u ns by di e si ying he isk ac oss all
he asse s.
1.1. Po olio Op imiza ion and Risk Pa i y Backg ound
Since he incep ion o mode n po olio heo y (MPT) and mean- a iance op imiza ion (MVO),
in oduced by Ma kowi z (1952) academic li e a u e has expanded wi h insigh s in o imp o emen s
o his undamen al basis o po olio alloca ion. Choosing he bes po olio o asse s and hei
indi idual weigh s, x, ou o all possible po olios being conside ed is he essence o po olio
op imiza ion. Each asse has an expec ed e u n, µ, and a s anda d de ia ion o e u ns, σ, compu ed
om his o ical da a. The ela ionship be ween he asse s is go e ned by he a iance–co a iance (VCV)
J. Risk Financial Manag. 2020,13, 237; doi:10.3390/j m13100237 www.mdpi.com/jou nal/j m
J. Risk Financial Manag. 2020,13, 237 2 o 28
ma ix o asse e u ns
Σ
, which indica es he co ela ions be ween asse s ha a sol e uses o minimize
he po olio a iance,
σp
. MVO po olios seek he lowes amoun o isk o a gi en le el o e u n
and is ep esen ed in Model
(1)
. This model uses bo h pe o mance aspec s and isk aspec s o achie e
he op imal asse alloca ions by conside ing he ade-o be ween isk and e u n ha is go e ned by
he asse co ela ions.
min
x
1
2xTΣx
s. . µxT≥
1xT=1
x≥0
(1)
When achie ing he minimum a iance (MV) o a a ge e u n he op imiza ion does no
conside he indi idual isk o asse s, only he o al isk o he po olio. Due o his ea u e o he
MVO, po olios can be alloca ed in o a ew asse s as he amewo k concen a es in o asse s wi h
low ola ili y o ice e sa o maximum e u n (Ma kowi z 1952). This is he leading c i icism o
mean- a iance po olios. The unce ain y o he es ima ed pa ame e s can lead o undi e si ied
po olios in e ms o asse weigh s and isk. MVO po olios a e ypically high in es ima ion e o and
a e sensi i e o inpu s due o he es ima ed expec ed e u ns as shown by Bes and G aue (1991) and
Chop a and Ziemba (1993). This leads o uns able po olios depending on he quali y o he es ima ed
pa ame e s and i is shown ha small changes in he inpu es ima ed e u ns can lead o signi ican
changes o asse alloca ions (Me on (1980); Black and Li e man (1992)).
1.1.1. Shi o Risk Based Ideology
The con ounding e ec s o he unce ain y in MVO has led o he s udy o echniques ha y
o elimina e he need o es ima ed pa ame e s, mainly expec ed e u ns and co a iances. As shown
by Chop a and Ziemba (1993), he es ima ed VCV ma ix causes less ins abili y hen he es ima ed
expec ed e u ns and i is sugges ed by Chop a and F ahm and Wieche s (2011) ha simply emo ing
he need o es ima ed expec ed e u ns om he op imiza ion is possible and leads o p ima ily
isk-based op imiza ions ha a e mo e s able. In MVO, his equa es o he minimum- a iance po olio
which i sel concen a es in o he asse s wi h he lowes ola ili y. Howe e , his s ill p oduces
undi e si ied po olios wi h e en lowe e u ns.
The a ailable me hods o asse alloca ion ha e e ol ed p ima ily om ex ensions and changes o
he o iginal mean- a iance amewo k and e o s ha e been made o co ec o he MVO’s endency
o p oduce o e -concen a ed po olios. This has led o he de elopmen o equal- isk con ibu ion
(ERC) po olios as measu ed by he s anda d de ia ion (Mailla d e al. (2010); Roncalli (2014)) wi hou
he need o inco po a e expec ed e u ns and esul s in di e si ica ion.
The isk con ibu ion o an asse is de ined by he p oduc o i s weigh in he po olio and i s
ma ginal isk con ibu ion (MRC). As discussed in Bai e al. (2016) and con i med by Roncalli and
Weisang (2016), Eule ’s Decomposi ion can be used o decompose he o al po olio isk measu e,
s anda d de ia ion
σ2
p
, in o indi idual asse isk con ibu ions
σi
, whe e
n
is he numbe o asse s in
he po olio. The asse s ma ginal isk con ibu ions will de ine how indi idual asse s a e app oached
in he op imiza ion beyond jus using he o al po olio isk which is asse agnos ic in an MVO.
By en o cing a isk bound on each asse as explo ed by Haugh e al. (2015) and Cesa one
and Ta della (2017) o in his case, equal isk budge s, e e y a ailable asse is p esen in some
magni ude, ensu ing di e si y. The di e ences be ween isk con ibu ions a e minimized wi h a
s a egy commonly known as isk pa i y op imiza ion, which leads o his desi ed di e si ica ion
ai . Risk pa i y op imiza ion p o ec s agains o e -concen a ion in o indi idual asse s; i is he
o ced cons uc ion o di e si ied po olios in which esou ces a e alloca ed based on he measu e
o isk whe e all asse s con ibu e equal isk
σi(2)
, a he hen equal weigh . Risk-based po olios
J. Risk Financial Manag. 2020,13, 237 3 o 28
do no equi e an explici es ima ion o asse expec ed e u ns and he sum o he asse absolu e isk
con ibu ions (ARC) equa es o he o al po olio isk (3).
σi(x) = xi(Σx)i
√xTΣx=√xTΣx
nwhe e ∀i i =1...n,n=# o asse s (2)
σp=√xTΣx=
n
∑
1
σiwhe e ∀i i =1...n(3)
Va ious me hods exis o p oducing isk pa i y po olios as explo ed by Mailla d e al. (2010);
Feng and Paloma (2015) and Bai e al. (2016), including he leas squa es app oach and he log ba ie
me hods. Risk pa i y o e comes he d awbacks o Ma kowi z po olios by a emp ing o educe he
eliance on noisy es ima ed pa ame e s which may mislead he op imiza ion o ely on un eliable
es ima ed asse e u ns. Risk pa i y op imiza ion elimina es he expec ed e u ns om he model
en i ely. T adi ionally, he isk pa i y o mula ion is a leas squa es ou h-o de objec i e p oblem as
composed in model
(4)
. The isk con ibu ion o each asse is ep esen ed by
xi(Qx)i
and is compa ed
o ano he asse s isk con ibu ion deno ed by
xj(Qx)j
. The model will compa e isk con ibu ion o
each asse wi h e e y o he asse and each an objec i e alue o 0 indica ing all isk con ibu ions a e
he same.
min
x
n
∑
i=1
n
∑
j=1
(xi(Σx)i−xj(Σx)j)2
s. . 1Tx=1
xi≥0
(4)
This polynomial objec i e is non-linea and non-con ex and conce ns o e i s nume ical
complexi y a e aised when conside ing addi ional cons ain s in he model. To alle ia e hese
conce ns, Lobo e al. (1998) discuss he applica ions o a second-o de cone p og amming (SOCP)
model and a clea implemen a ion o isk pa i y is shown by Mausse and Romanko (2014) who
use an equi alen SOCP o mula ion as i appea s in Model (0). Model (4) is e o mula ed by
ans o ma ion o he cons ain s in o cone cons ain s and a ans o ma ion o he objec i e in o
linea o m. This SOCP o mula ion is an epig aph o he leas squa es me hod, bu i is eadily and
e icien ly sol able by quad a ic sol e s due o i s con ex na u e (Ben-Tal and Nemi o ski (1998);
Alizadeh and Gold a b (2003)
) and is mo e iendly and lexible o addi ional cons ain s o deal wi h
he expec ed e u n in o ma ion. The SOCP model is sui able since he model is using he s anda d
de ia ion o e u ns as he isk measu e which maps o he second o de cons ain s
(7)
and
(8)
.
Al e na i e isk measu es a e no compa ible wi h his model.
Model (0) min
xψ−γ(5)
s. . ζi= (Σx)ii = 1...n (6)
xTΣx≤nψ2(7)
xiζi≥γ2i = 1...n (8)
1Tx=1 (9)
xi,ζi≥0 i = 1...n (10)
ψ,γ≥0 (11)
J. Risk Financial Manag. 2020,13, 237 4 o 28
As he linea objec i e o he SOCP o mula ion app oaches ze o, he isk con ibu ion be ween
each asse becomes equal. Each asse ’s absolu e isk con ibu ion (ARC), deno ed
γi
, is compa ed o he
a e age isk o he po olio,
ψ
. The a e age isk o he po olio
ψ
in
(7)
is an uppe bound on he isk
con ibu ion o any indi idual asse and dic a es ha any one asse ’s isk con ibu ion should be equal
o o less han he a e age isk. The a e age isk o he po olio equa es o an equal isk magni ude,
so he objec i e is e ec i ely en o cing he isk pa i y alloca ions. To acili a e his, he lowe bound o
an indi idual asse ’s isk con ibu ion is
γ
, en o cing ha
ψ≥γ
. The o al isk con ibu ion o each
asse is compu ed using he asse ’s MRC and asse weigh h ough cons ain s
(6)
and
(8)
, espec i ely.
When
ψ
=
γ
, i is ob ious ha i is a isk pa i y po olio and he model becomes a minimiza ion o
he di e ence be ween hese wo isk alues. The SOCP e ec i ely minimizes he di e ence be ween
each asse ’s isk con ibu ion and he a e age isk o he po olio ins ead o compa ing each asse
o ano he , educing he compu a ional complexi y along he way. Las ly, cons ain
(9)
de ines he
common po olio budge equi emen s.
Due o he abili y o en o ce di e si y, isk pa i y achie es nei he minimum isk no
maximum e u ns and i is shown by Mailla d e al. (2010) ha he pe o mance lies be ween he
minimum- a iance and he equal weigh po olio. An in-sample analysis was conduc ed o e a
20 yea pe iod on a 50-asse se and plo ed in Figu e 1. This plo demons a es how isk pa i y akes
ad an age o iskie asse s much like an equal weigh po olio and p oduces e u ns abo e minimum
a iance. This end is e e sed in ma ke c ashes seen immedia ely ollowing he 2008 ma ke c ash
whe e equal weigh and isk pa i y po olio losses con e ge below he minimum a iance po olio
due o added weigh in he highe isk asse s.
Figu e 1. Risk pa i y compa a i e pe o mance—Bull Ma ke —2009–2018.
Risk pa i y is bo h a meaning ul and e ec i e app oach o po olio cons uc ion. I has emo ed
expec ed e u ns o alle ia e he conce ns o ins abili y and balance e u n wi h di e si ica ion o isk,
he eby a oiding concen a ion in o isky asse s.
1.1.2. Mo i a ion and he U iliza ion o Re u n Es ima es
Few s udies exis on he in oduc ion o pe o mance aspec s in o isk pa i y. Roncalli (2014)
shows how o build isk pa i y po olios ha depend on he expec ed e u ns by using a isk measu e
ha inco po a es expec ed e u ns a he han jus he s anda d de ia ion o ealized e u ns. Roncalli
demons a es his isk-budge ing echnique whe e each asse has a pe o mance and ola ili y aspec
and he isk budge limi s de ine he weigh alloca ions based on ha ade-o . Much like he isk
pa i y op imiza ion, when a isk budge is g ea e han ze o, i mus hold he asse in some weigh ,
en o cing di e si y. His model se es o bene i om he addi ional in o ma ion ha he expec ed
e u ns p o ide o ind mo e accu a e isk pa i y po olios. He inds ha some me i exis s in he
balance o isk-based alloca ion and pe o mance-based alloca ion, which his pape seeks o exploi .
Lee (2011) had a simila sen imen p io , ejec ing ha isk pa i y imp o es e iciency and a guing ha
i is he s a ing poin in he absence o s onge in es men iews which Roncalli supplemen s wi h
expec ed e u ns; he model p esen ed in his pape is d i en om his inclusion.

J. Risk Financial Manag. 2020,13, 237 5 o 28
Feng and Paloma (2015) explo e di e en isk pa i y o mula ions and apply di e en
pe o mance objec i es and isk measu es, such as ola ili y, VaR and CVaR, o a successi e con ex
op imiza ion o sol e o long and long/sho po olios. They show he non-con exi y o he a ious
isk pa i y o mula ions ha lead o he necessi y o a nume ical algo i hm o sol e o ind he mos
op imal solu ions. This gi es ise o he use o a second o de epig aph o he leas -squa es isk pa i y
by Mausse and Romanko (2014) as he basis o ou model. This second-o de cone p og am (SOCP)
model pe mi s he addi ion o u he open- aced cons ain s, such as he a ge e u n used o o ce a
iola ion, and is sol able ia mode n quad a ic sol e s in he long-only domain, alle ia ing some o
he nume ical di icul ies o p ac ical implemen a ion.
A dia e al. (2018) de elops a me ic o op imize he balance be ween pe o mance and isk aspec s.
The me ic measu es concen a ion o weigh and how much i di e ges om he isk con ibu ions,
measu ing he misma ch be ween pe o mance and isk con ibu ions. They use he amewo k o
op imize o he s a egies o ma ch he unde lying pe o mance o isk pa i y in e ms o ola ili y
and e u n and no o speci ically enhance e u ns as is he aim o his s udy. Along wi h A dia,
his pape in uses he pe o mance and isk aspec s o achie e his. Ra he han inding an op imal
mean- a iance ade-o he no ion o a ge ing nea isk pa i y, po olios a e in oduced o cap u e
he use ul ai s o isk pa i y o he highes deg ee possible o passing con ol o he p ac i ione .
To alle ia e he mean- a iance aspec , Pe che e al. (2015) ecognizes ha d i ing he op imiza ion on
he VCV ma ix con e ges an MVO owa ds minimum- a iance and con i ms ha using he diagonal
o he VCV ma ix con e ges owa ds equal isk alloca ions ins ead, due o a less agg essi e d i ing
ma ix. Me hods a e implemen ed in his s udy ha u ilize hese indings o inco po a e he es ima ed
expec ed e u ns o il he s anda d isk pa i y po olio away om i s cu en conse a i e alloca ion
and seek g ea e e u ns.
1.1.3. Pa ame e Unce ain y and Robus Op imiza ion
Rejec ing he s a is ical unce ain y in models using es ima ion is impossible; howe e , signi ican
wo ks ha e been p oduced o depa om he simple agnos ic da a-d i en desc ip i e s a is ics
which need o be men ioned he e. Bo h he sample mean e u n and sample co a iance ma ix a e
es ima ed using adi ional i s and second o de me hods o e he his o ical e u ns o each asse .
The in oduc ion o es ima ed expec ed e u ns in o he isk pa i y model aises conce ns o e i s
es ima ion e o as well. Es ima ed pa ame e s om aw da a ha e a deg ee o unce ain y, which can
ende he op imal solu ions a poo i o he model. The e ha e been mul iple app oaches in oduced
o deal wi h es ima ion e o (Black and Li e man (1992); Roncalli (2014); Jo ion (1986); DeMiguel
and Nogales (2006); bauwens2006mul i a ia e), he mos popula app oaches being au o eg essi e
condi ional he e oscedas ic models, sh inkage me hods and obus op imiza ion, as discussed by
Gold a b and Iyenga (2003) and Tü üncü and Koenig (2004).
Condi ional he e oscedas ic models elax he assump ion o cons ancy o he co a iance ma ix
and ins ead ollow a lexible dynamic s uc u e (Engle 2002). Using many asse s equi es a mul i- a ia e
gene alized au o eg essi e condi ional he e oscedas ici y (M-GARCH) model ha o ecas s changes
o he ola ili y o inancial ime se ies in he sho e m (Bauwens e al. 2006). The p oposed
model will u ilize a longe e m olling p ocedu e ha es ima es he co a iance o e a 3-yea
pe iod. Holding a po olio o e he 6–12-mon h pe iod be o e e-balancing may no add signi ican
alue o he model and he p oposed model a oids he added complexi y. Al e na i ely, sh inkage
me hods
(Ledoi and Wol (2003)
;Kwan (2011)) imp o es he condi ioning o he co a iance ma ix.
Sample co a iance ma ices a e subjec o es ima ion e o o he kind mos likely o pe u b a
mean- a iance op imize . Ins ead i uses a weigh ed a e age o he sample and a ge co a iance
ma ices o gene a e a nea - ue co a iance ma ix by pulling he mos ex eme coe icien s owa ds a
mo e cen al alue and sys ema ically educing es ima ion e o (Ledoi and Wol 2004). Two aspec s
a ise he e— his is usually equi ed when he e a e insu icien obse a ions o he unde lying a iables
o he numbe o asse s Daniels and Kass (2001). This is no he case in he p oposed model analysis
J. Risk Financial Manag. 2020,13, 237 6 o 28
whe e he e a e 15x he numbe o obse a ions o asse s. Secondly, he isk pa i y-like beha io o he
p oposed model na u ally limi s he ex emes in a iance alloca ion which coun e ac s he unde lying
pu poses o using sh inkage me hods. The s onges a gumen agains using ei he me hod is he
na u ally isk limi ing beha io o he isk pa i y like model ha a oids concen a ion in o high e u n
asse s and sub-sequen ially he unce ain y ha comes wi h hem.
Robus op imiza ion seeks o op imize a po olio o he wo s -case ealiza ion o he es ima ed
pa ame e s. S ong discussions on he ac ion o obus po olios a e au ho ed by Ce ia and S ubbs
(2006), Fabozzi e al. (2007), Kim e al. (2014b) and Kim e al. (2014a). San os (2010) con i ms ha
he me hods by Ce ia e al. and Tü üncü and Koenig (2004) p oduce be e po olios in e ms o he
Sha pe a io and u no e when compa ed o mean- a iance po olios. This indica es ha obus
op imiza ion is an e ec i e way o alle ia e p oblems wi h es ima ion e o s in e u ns.
Much like minimum a iance and equal weigh po olios, isk pa i y po olios a e assumed
o be obus due o he di e si ica ion o isk. Poddig and Unge (2012) show ha he ou come o
isk pa i y po olios a e a less in luenced by es ima ion e o s han MVO po olios. This pape
demons a es ha obus op imiza ion adds a laye o complexi y o he op imiza ion, which sees li le
bene i (Sec ion 5.6). By using isk pa i y as he unde lying op imiza ion o he enhanced model, i is
a gued ha i e ains some inhe en obus ness o es ima ed expec ed e u ns and, o some deg ee,
nulli ies he e ec s o unce ain y. Fu he de ails can be ound in Be simas e al. (2011), including a
comp ehensi e lis o li e a u e on obus op imiza ion.
1.2. Pu pose and Con ibu ions
P o ec ing om he downside wi h di e si ied po olios limi s he upside, esul ing in poo
pe o mance in s able g ow h ma ke s. The pu pose is o de elop a ool which elaxes he isk pa i y
op imiza ion by allowing isk pa i y o be iola ed while biasing he sol e owa ds isk pa i y. This will
imp o e upon he conse a i e na u e o he model, bu no d ama ically so, asse ing ha inding
nea isk pa i y po olios will p o ide imp o ed bull ma ke pe o mance wi hou a d ama ic inc ease
in isk.
Risk pa i y was c ea ed o o e come wo aspec s, i s o imp o e he di e si ica ion o e he
common MVO and so educe he nega i e e ec s om es ima ed pa ame e unce ain y. The second
is o o e come he e ec s o cap weigh ed po olios, such as wi h he SP500, which is al eady
well di e si ied. The SP500 is he backbone o in es ing, whe e illions o dolla s a e aded, so by
di e si ying and a oiding cap weigh ed concen a ions, his will sa e i om a 2001 do -com s yle
bubble h ough isk budge ing. The e a e wo main a gumen s ha mus be add essed o achie e
he esea ch goals. The e u ns mus be added back in o he isk pa i y op imiza ion o inc ease
he agg essi eness o he model and yield highe e u ns, which can concen a e he isk alloca ions
mo i a ing he second aspec , d i ing he isk con ibu ions back owa ds equal isk o a oid his
concen a ion. The e is no SOCP based model wi h a e u n goal ha hos s ex a cons uc s o keep
he isk alloca ions nea isk pa i y. The e a e wo ks ha inco po a e he po en ial use o e u ns
o imp o e isk measu es bu no hing o a ec he esul ing isk alloca ions speci ically. This pape
p oposes he inco po a ion o a e u n goal in o a second-o de -cone p og am (SOCP) model o a isk
pa i y op imiza ion and include a compu a ional cons uc o ob ain nea isk pa i y po olios o e ain
as much isk pa i y like ai s as possible.
This pape is o ganized as ollows. Fi s , Sec ion 1in oduces he backg ound on po olio
op imiza ion and a ious s a egies ha se he ounda ion o he p oposed model. Sec ion 2
in oduces he me hods and e alua ion me ics. Sec ion 3desc ibes he elaxed isk pa i y model
and in oduces he a ionale o he elaxa ion echnique employed. Sec ion 4de ails he isk
con ibu ion cha ac e is ics o he model and pa ame e op imiza ion in-sample. Sec ion 5p esen s he
compu a ional expe imen s including he ou -o -sample pe o mance, ela i e model pe o mance
and isk o e u n p ope ies o he model. Las ly, Sec ion 6discusses he model’s esul s and makes
conclusions owa ds he esea ch ques ion.
J. Risk Financial Manag. 2020,13, 237 7 o 28
2. Me hods and Da a
Risk-pa i y is a s a egic alloca ion o long- e m p o ec ion om nega i e ma ke e en s. Wi h his
in mind, he model is conside ed unde a long in es men ho izon; including he pe iods pos he
2001 do -com c ash and he 2008 inancial mel down. The mos ecen bull ma ke om 2009 o 2018
is examined since i ep esen s he cu en inancial en i onmen and egula o y p ac ices ha ha e
changed since he 2008 inancial c ash. The analysis uses a di e si ied se o 50 US S ocks (Table 1) as
cons uc ed in Cos a and Kwon (2019) selec ed om he S&P 500 index encompassing he en GICS
indus y sec o s. The 10-yea US T-Bill a e is in e p e ed as he isk- ee a e and is used o compu e
he ex-pos Sha pe a io.
Table 1. Summa y o asse s and es ima ed annualized p ope ies be ween 1992–2018.
Consume Disc. Consume S aple Heal hca e In o ma ion Tech. U ili ies
µ σ µ σ µ σ µ σ µ σ
F 12.1 15.5 KR 14.8 10.9 MRK 11.0 9.8 IBM 12.1 10.1 ED 9.9 6.6
GT 10.2 16.3 MO 17.2 9.5 PFE 13.7 9.5 MSI 14.7 14.9 ETR 11.2 7.8
FL 14.1 15.6 CL 14.0 8.4 LLY 13.6 9.8 HPQ 15.3 13.3 DTE 11.7 7.2
DIS 14.5 10.1 KO 10.7 8.1 BMY 10.5 9.6 TXN 26.0 14.6 CNP 11.1 10.9
MCD 16.2 8.2 PEP 12.0 7.6 JNJ 13.3 7.5 XRX 8.9 15.1 AEP 10.9 8.1
PG 13.2 8.0 QCOM 32.4 18.1 LNT 11.1 7.3
Ene gy Financial Indus ial Ma e ial Telecommunica ion
µ σ µ σ µ σ µ σ µ σ
CVX 13.7 8.4 AON 15.5 10.4 BA 17.5 11.1 PPG 14.6 9.6 T 10.2 9.1
MRO 11.8 12.9 BK 17.7 11.6 LMT 15.6 9.2 AA 10.9 14.4 VZ 11.1 8.5
OXY 15.6 10.6 AXP 18.9 11.9 MMM 13.6 8.2 DD 31.0 29.4 S 10.2 17.3
XOM 11.2 7.6 C 16.1 18.3 CAT 21.1 12.0 IP 9.0 12.3
HAL 16.2 15.3 WFC 18.1 12.5 DE 19.8 11.8
The Sha pe a io as in oduced by Sha pe (1994), is a inancial a io ha measu es he excess
e u n pe uni o de ia ion commonly e e ed o as he isk adjus ed e u n. This a io indica es he
quali y o he e u n based on how much isk was aken on and gene ally a highe Sha pe a io is
desi ed. The ex-pos Sha pe a io is compu ed in his s udy om he annualized mean and s anda d
de ia ion o excess e u ns based on unde lying weekly e u n and isk- ee a e da a.
The po olio u no e de ailed by San os (2010) indica es how much he po olio componen s
ha e changed o e some pe iod o ha e he op imal po olio. This alue is an indi ec measu emen o
he magni ude o he ansac ion cos and is expec ed o be g ea e hen ze o since his model employs
an ac i e s a egy. Las ly, o e alua e how closely he po olio aligns wi h he isk pa i y alloca ions he
Euclidean dis ance be ween each asse s ARC o he expec ed isk pa i y isk con ibu ions is compu ed.
A mean squa ed e o (MSE) measu e p o ides a single compa able dis ance alue hence o h e e ed
o as dis ance and labeled d. A smalle dis ance o isk pa i y is desi ed when we iola e isk pa i y o
achie e he highe a ge goals.
3. Relaxed Risk Pa i y Model De elopmen
This p oposed model is an ex ension o he Mausse and Romanko (2014) isk pa i y SOCP wi h
aspec s om Roncalli (2014) and Pe che e al. (2015). The use o he expec ed e u n pa ame e s is
expec ed o yield alue o he cons uc ion o nea isk pa i y po olios. Unlike in Roncalli’s me hod,
he expec ed e u ns a e implemen ed in a manne such ha he op imal po olio’s e u n pe o mance
can be inc eased. This a ge s enhanced e u ns and e ains isk di e si ica ion, which is he a ge ed
p ope y o isk pa i y.
J. Risk Financial Manag. 2020,13, 237 8 o 28
3.1. Enhanced Risk Pa i y SOCP Fo mula ion
An addi ional a ge e u n cons ain
(16)
is added o he SOCP model, Model (A), p oposed by
Mausse and Romanko (2014). The a ge e u n is needed o a ge e u ns g ea e han isk pa i y
so he op imiza ion can iola e he isk pa i y ai by es ic ing he po olio om eaching equal
isk alues. Te ms
(12)
–
(15)
and
(17)
a e iden ical o he SOCP Model (0) in Sec ion 1.1.1, whe e hey
a e discussed.
Model (A) min
xψ−γ(12)
s. . ζi= (Σx)ii = 1...n ma ginal isk (13)
xTΣx≤nψ2 isk bound (14)
xiζi≥γ2i = 1...n o al isk (15)
µTx≥R a ge e u n (16)
1Tx=1 po olio budge (17)
xi,ζi≥0 i = 1...n
ψ,γ≥0
Figu e 2a p esen s he isk alloca ions o he easible isk pa i y solu ion o he isk pa i y SOCP.
The isk pa i y isk con ibu ion a ge can be seen as he do ed ho izon al dashed line and he asse s
labeled X1 h ough X50. Choosing a e u n alue g ea e hen he expec ed isk pa i y e u n esul s
in Figu e 2b and he e is a de ia ion om he isk pa i y a ge line o all asse s while concen a ion
occu s in o a ew. The model is conside ing he expec ed e u n as one o he d i e s o he isk- e u n
ade-o , so pa allels he MVO esul . This concen a ion o isk is due o he mean- a iance aspec
o he model, which does no conside he dis ibu ion o addi ional isk needed o achie e he
a ge e u n.
Figu e 2. Risk con ibu ion concen a ion in Model (A)—50 asse s, 1997 o 2018.
Concen a ed isk is coun e -p oduc i e o achie ing nea - isk pa i y po olios and hei
p ope ies. Highe isk concen a ions may expe ience mo e ola ile beha io and see g ea e losses
when he s abili y o a ma ke is h ea ened. This cha ac e is ic is mi iga ed h ough he use o a isk
pa i y op imiza ion; howe e , he addi ional isk needs conside ed.
3.2. Relaxa ion o he Risk Pa i y A ibu e
The SOCP is a o m o elaxa ion i sel as he leas squa es o mula ion objec i e is e o mula ed
in o he cons ain s. A easible isk pa i y solu ion o a long only model may no exis when a change
o he s uc u al cons ain s o he SOCP isk pa i y model is imposed (Mausse and Romanko 2014).
As shown by Bai e al. (2016) and Lemma 5.1, he long-only isk pa i y op imiza ion has a unique
solu ion. So, any s uc u al changes ha es ic one om eaching his solu ion will esul in a iola ion
o he isk pa i y p ope y. This sugges s he enhanced e u n model in Model (A) canno ind an
exac isk pa i y solu ion o all a ge e u n goals R and ha a elaxa ion is necessa y o ind he
J. Risk Financial Manag. 2020,13, 237 15 o 28
asse s. Abo e his poin , he model’s pe o mance aspec s ake p ecedence and he penal y
λ
loses
i s e ec i eness o a ge nea - isk pa i y po olios. The dis ance om isk pa i y is a ec ed by bo h
he a ge e u n and he penal y s eng h and can be obse ed in Figu e 8spanning a ious a ge
e u ns. This plo indica es he a e age dis ance o isk pa i y in-sample o he inc easing a ge e u n
mul iplie m.
Figu e 8. Dis ance o Risk Pa i y Mean-Squa ed-E o (MSE)—2009–2018—Bull Ma ke .
Fo his model he e is a dec ease in dis ance om isk pa i y, as seen in Figu e 8, when
λ>
0.
The po olio becomes nea e o isk pa i y wi h
λ≤
1 up un il a 1.6x a ge e u n goal. Wi hin his
ange o e u n and penal y
λ
, he p ac i ione can expec an imp o emen in he isk pa i y p ope ies
o an enhanced po olio o e an un- egula ed model. The plo e eals a
λ=
100 ha he unde lying
isk pa i y model is iola ed a a a ge e u n mul iplie m = 1.0x, whe e he dis ance o isk pa i y
dMSE
is much g ea e han ze o. This is con i med by he da a in Table 3and indica es ha he model
canno ind isk pa i y po olios a he same a ge e u n as isk pa i y wi h high lambda alues.
Using
λ=
100 would ne e be ad ised o used in p ac ice and is he e o demons a e a na u al limi
o he pa ame e be o e he unde lying isk pa i y is a ec ed— his appea s o be
λ=
1 om Table 3.
This ag ees wi h he plo in e ealing alues o λless hen one o p oduce supe io esul s.
Penal y inc eases he esul in a educ ion o he Sha pe a ios in Table 3. This is due o he
educ ion in alloca ion in high- isk high e u n asse s and a mino inc ease in a iance due o some
concen a ions equi ed o mee he a ge e u n. The dis ance measu e
dMSE
is abula ed in Tables
3and 4as he dis ance o isk pa i y o a 20 yea pe iod o 1997 o 2018 and a bull ma ke pe iod
2009–2018, espec i ely.
Table 3. In-Sample MSE Dis ance o RP wi h inc easing λ—1997–2018.
Ta ge λ= 0 λ= 0.5 λ= 1 λ= 100
mSha pe dMSE Sha pe dMSE Sha pe dMSE Sha pe dMSE
1.0x 0.622 0 0.622 0 0.622 0.003 0.633 0.036
1.1x 0.691 0.08 0.681 0.071 0.679 0.072 0.683 0.080
1.2x 0.74 0.145 0.721 0.134 0.718 0.136 0.717 0.152
1.4x 0.762 0.271 0.753 0.265 0.752 0.269 0.748 0.302
1.6x 0.725 0.404 0.738 0.408 0.736 0.413 0.734 0.429
1.8x 0.649 0.565 0.668 0.56 0.667 0.561 0.667 0.567
2.0x 0.569 0.756 0.584 0.75 0.584 0.751 0.585 0.753
No e: Rolling Ho izon = 6 mon hs, λ= 0.2, Look-back = 3 yea s.
Bo h ables indica e ha a penal y
λ<
1 p oduces smalle dis ances o isk pa i y and imp o ed
Sha pe a ios indica e ha enhancing isk pa i y is possible using po olios nea isk pa i y. The Sha pe
a ios do no always imp o e o e an un- egula ed model (
λ
= 0) a highe a ge e u ns, bu his is
no he p ima y goal, a he he educ ion in dis ance o isk pa i y is. Speci ically, in a bull ma ke
pe iod in Table 4, he Sha pe a io can be seen o imp o e up un il he a ge m= 1.2x when λ=0.5.

J. Risk Financial Manag. 2020,13, 237 16 o 28
Table 4. In-Sample MSE Dis ance o RP wi h inc easing λ—2009–2018.
Ta ge λ= 0 λ= 0.5 λ= 1 λ= 100
mSha pe dMSE Sha pe dMSE Sha pe dMSE Sha pe dMSE
1.0x 1.331 0 1.331 0 1.335 0.005 1.389 0.037
1.1x 1.377 0.075 1.395 0.052 1.397 0.051 1.441 0.054
1.2x 1.399 0.130 1.425 0.097 1.428 0.096 1.471 0.095
1.4x 1.396 0.230 1.424 0.194 1.428 0.194 1.460 0.203
1.6x 1.363 0.328 1.338 0.313 1.340 0.317 1.345 0.333
1.8x 1.306 0.436 1.246 0.448 1.242 0.452 1.237 0.460
2.0x 1.126 0.610 1.133 0.614 1.132 0.615 1.132 0.616
No e: Rolling Ho izon = 6 mon hs, λ= 0.2, Look-back = 3 yea s.
This s uc u al s udy o he model demons a es ha an inc easing penal y s eng h
λ
can
posi i ely a ec he e icien on ie by educing he o al po olio isk. Fo high a ge e u ns,
his penal y app oaches he mean- a iance po olio wi h a highe dis ance o isk pa i y whe e he
desi able isk pa i y p ope ies deg ade. When he model is calib a ed and used wi hin a easonable
ange o enhanced e u n and
λ
penal y s eng h shows a s onge esul han he isk pa i y po olio.
Pa ame e Op imiza ion
A ange explo a ion s a egy is employed o model pa ame e es ima ion. To acili a e he
ou -o -sample analysis, he hype -pa ame e
λ
is ixed o a penal y s eng h ha achie es a balance
be ween dis ance o isk pa i y and pe o mance. A g id-sea ch is ca ied ou ac oss inc easing penal y
s eng h and a ge e u n mul iplie s. As de e mined in Sec ion 4, he penal y
λ
a ec s how he
elaxed isk pa i y model on ie app oaches he e icien on ie . Reasonable gains can be made wi h
app op ia e uning be ween ze o and one. Values o
λ>
1 a e oo s ong and concen a e isk h ough
o e -penaliza ion. A s ong penal y o e -penalizes each indi idual isk asse and esul s in po olios
ha a e no nea e o isk pa i y. A
λ<
1 alue indica es ha a educ ion in s eng h o he egula ing
e m achie es a be e esul . Table 5 esul s indica e a low penal y s eng h o
λ=
0.2 in Model (B)
and p oduces he smalles dis ance o isk pa i y ac oss he a ious e u n le els and will be used in
he emainde o his s udy.
Table 5. Single pe iod wi h inc easing e u n mul iplie and penal y alue λ.
1.0x 1.1x 1.2x 1.4x 1.6x 1.8x
λSha pe dMSE Sha pe dMSE Sha pe dMSE Sha pe dMSE Sha pe dMSE Sha pe dMSE
0 1.331 0 1.377 0.075 1.399 0.13 1.396 0.23 1.363 0.328 1.306 0.436
0.1 1.331 0 1.394 0.061 1.398 0.069 1.421 0.195 1.348 0.309 1.261 0.451
0.2 1.331 0 1.398 0.058 1.398 0.058 1.425 0.194 1.347 0.306 1.273 0.43
0.3 1.331 0 1.396 0.053 1.426 0.097 1.422 0.194 1.339 0.309 1.257 0.439
0.4 1.331 0 1.395 0.052 1.425 0.097 1.423 0.194 1.338 0.311 1.251 0.443
0.5 1.331 0 1.395 0.052 1.425 0.097 1.424 0.194 1.338 0.312 1.248 0.446
0.6 1.331 0 1.395 0.052 1.426 0.097 1.425 0.194 1.338 0.313 1.246 0.448
0.7 1.331 0.001 1.395 0.051 1.427 0.097 1.426 0.194 1.339 0.314 1.245 0.449
0.8 1.332 0.003 1.396 0.051 1.427 0.097 1.426 0.194 1.339 0.315 1.244 0.45
0.9 1.332 0.004 1.396 0.051 1.427 0.097 1.427 0.194 1.339 0.316 1.243 0.451
1 1.335 0.005 1.397 0.051 1.428 0.096 1.428 0.194 1.34 0.317 1.242 0.452
The op imal olling ho izon size and look-back pe iod o he ou -o -sample analysis is de e mined
h ough a g id-sea ch o possible alues. The esul s indica e ha he sho e aining pe iod p oduces
supe io Sha pe a ios o longe aining pe iods and ha he bi-annual e-op imiza ion achie es he
same le el o pe o mance as he annual ho izon a less u no e . This p ocedu e indica es ha a
6-mon h olling ime ho izon wi h a 3-yea look-back pe iod p oduces he bes esul s. I is gene ally
accep ed ha a sho e aining pe iod will be e e lec he cu en ma ke condi ions, which is
desi able o con ol he isk alloca ions. Fo b e i y, he de ails a e excluded.
J. Risk Financial Manag. 2020,13, 237 17 o 28
5. Ou -o -Sample Compu a ional Analysis
5.1. Rolling Ho izon Ou -o -Sample Analysis P ocedu e
A ixed e m e-balancing olling ho izon amewo k ou lined by (DeMiguel e al. 2009) is used
wi h a ying a ge e u n goals m o he ou -o -sample analysis in Sec ion 5. This leads o he
ollowing gene alized wo s age model implemen a ion, as ou lined in Algo i hm 1. The olling
window ho izon leng h is selec ed o ake ad an age o he ime se ies cha ac e is ics, which will be e
e lec he mos ecen ma ke en i onmen s o some look back pe iods. The adap i e a ge e u n
om Sec ion 3.5 upda es each e-op imiza ion pe iod using he 3-yea aining pe iod wi h 6-mon h
olling ho izon.
Algo i hm 1: Mul i Pe iod Ou -o -Sample Op imiza ion.
Resul : xop imal
Ini ialize:
(i) Selec ime ho izon and asse se ;
(ii) Se look-back pe iod s a da e dsiand end da e dei;
(iii) Compu e he numbe o pe iods P;
(i ) Se Penal y S eng h λ;
( ) Se a ge e u n mul iplie m;
( i) i = 0;
while i≤Pdo
compu e µj
ibe ween dsiand dei= expec ed asse e u ns
compu e Σj= co a iance– a iance ma ix o e u ns wi h diagonal pe u ba ion
o en o ce a posi i e semi-de ini e (PSD) ma ix.
1. Pe iod i Risk Pa i y Op imiza ion
xj
p = isk pa i y weigh ings
E[x p]j=µx p− isk pa i y expec ed e u n
2. Se a ge e u n goal wi h mul iplie
m∈IR ≥1.0
Rj
a ge =mE[x p]j
3. Pe iod i Relaxed Risk Pa i y Op imiza ion
xj
op imal = pe iod i elaxed isk pa i y weigh ings
I e a e pe iod s a da e dsiand end da e dei
i=i+1
end
As he olling ho izon mo es each de ined pe iod o wa d, he pa ame e es ima ion window
ollows so only he newes pe iods o da a and pa ame e es ima es a e used. The po olios a e held
cons an o e he nex de ined ou -o -sample pe iod and he pe o mance o each po olio is e alua ed.
This is con inued un il he en i e a ailable ime pe iod is consumed by he olling ime ho izon. Each
oll o his op imiza ion is pe o med using Gu obi (Ve sion 8.01) wi h py hon on an i7-8550u 1.8 GHz
4-Co e 16 GB Windows 10 sys em and sol es in a ma e o minu es wi h n = 50 asse s.
5.2. Ou -o -Sample Pe o mance
The wo-s age olling algo i hm in Sec ion 5.1 is used o e alua e he ou -o -sample pe o mance
o Model (C). The model is applicable o any ime ame applied o i bea ing in mind ha i has
su icien ailing da a o he look back pe iod. The ou -o -sample analysis uses a 3-yea look-back
pe iod wi h weekly da a o es ima e he expec ed e u ns, VCV ma ix and he expec ed isk pa i y
e u n p io o he s a o he i s olling pe iod. The analysis is pe o med o e 20 yea s using a
J. Risk Financial Manag. 2020,13, 237 18 o 28
6-mon h olling ho izon ha begins Janua y 1997 and ends mid-2017, consis ing o a o al o 23 yea s
o weekly e u n da a ac oss 40 in es men pe iods. A ixed 6-mon h olling pe iod is used ha aligns
wi h po en ial eal-wo ld s a egies1. The compa ison o he p oposed model o exis ing s a egies as
used by Loh e e al. (2014) is dis ega ded in his wo k as Loh e demons a es he meaning ulness o
a maximum di e si ied po olio. I is well unde s ood ha he a ionale and posi ioning o he isk
pa i y po olio wi hin he ield is mean o achie e a conse a i e e u n and is unde s ood o ha e
lowe Sha pe a ios. The ela i e ou -o -sample po olio alue o he nominal isk pa i y po olio
alue is depic ed in Figu e 9and he pe o mance is abula ed in Table 6.
The esul s indica e ha he annualized o al e u n o e he in es men ho izon ac oss a ious
ma ke condi ions does indeed imp o e, including he Sha pe pe o mance. The e u ns imp o emen s
a e no d ama ic, which e lec s he conse a i e na u e o using he isk pa i y model in he i s place.
Mode a ely imp o ing beyond isk pa i y in a p edicable manne is wi nessed wi h a consis en li in
cumula i e e u n o he inc ease in a ge e u n mul iplie m.
Table 6. Annualized Ou -o -Sample Pe o mance—P oposed Model.
Ta ge mRe u n % ∆R Vola ili y % ∆Vol Sha pe Dis ance % Tu no e
1.0x 11.37 – 15.11 – 0.788 0.000 8.3
1.2x 12.07 7 15.28 1 0.822 0.250 20.5
1.4x 12.55 12 15.66 3 0.833 0.479 33.9
1.6x 13.06 16 16.19 6 0.839 0.727 47.3
Figu e 9. Relaxed Risk Pa i y Rela i e Ou -o -Sample Cumula i e Pe o mance o Nominal.
A close look a he ela i e cumula i e e u ns pe iod by pe iod in Figu e 9ac oss he di e en
ma ke condi ions shows an accele a ion o g ow h du ing pe iods o s able ma ke condi ions, such as
in 2004–2009, leading o g ea e weal h accumula ion. Al e na i ely, an accele a ion o losses is
wi nessed when he a ge e u n is high due o a mo e concen a ed po olio. This is a ibu ed o he
adap i e a ge e u n, which will inc ease he a ge e u n in each op imiza ion pe iod as he bull
ma ke con inues. The change in annualized e u n and annualized ola ili y in Table 6 e eals ha
he e u n is inc easing a a g ea e magni ude hen he ola ili y shown by examining %
∆
R and %
∆
Vol. This inc ease in a ge e u n will inc ease he dis ance o isk pa i y in bull ma ke s o cap u e he
well pe o ming asse s; howe e , agg essi e a ge e u n mul iplie s will inc ease he dis ance o isk
pa i y beyond a easonable amoun , leading o a d ama ic loss in isk pa i y ai s. Ca e mus be aken
o balance o choose he alues o he pa ame e s used.
These highe e u n mul iplie s subsequen ly educe he isk pa i y p ope ies in bull ma ke s,
which is desi able. This is appa en in Figu e 9, im which, du ing 2000 and mid-2012, he loss in
1
The e-op imiza ion pe iod may align wi h ce ain ma ke e en s which could change he esul ing pe o mance o be e
o wo se. A dynamically adjus ed e-op imiza ion pe iod is no conside ed bu is a poin o u u e in es iga ion.
J. Risk Financial Manag. 2020,13, 237 19 o 28
po olio alue du ing a co ec ion is ampli ied. The subsequen 3-yea pe iods a e ma ke shocks in
Figu e 9, speci ically 2002 o 2005 and 2009 o 2012, indica e ha he ela i e alue s ays consis en
when ola ile pe iods a e included in he pa ame e es ima ion pe iod. The esul ing ela i e g ow h
in po olio weal h is o gone as he model e e s close o he isk pa i y weigh ing. Risk pa i y ails
o capi alize on changing es ima ed e u ns and he esul s he e ein o ce he no ion ha including he
es ima ed e u ns in his con olled manne can mode a ely capi alize on s ong ma ke condi ions
and s ill seek isk pa i y ai s.
5.3. Does he P oposed Model Achie e Be e Ou -o -Sample Pe o mance?
The e u n cha ac e is ics in Table 7 e eal imp o emen s o he pe o mance. The MVO p oduces
he bes Sha pe a ios as expec ed because i is no es ic ed om inding he lowes possible a iance
o he po olio. The addi ional egula ing e m
ρ
used o ind nea isk pa i y po olios educes his
isk, imp o ing he Sha pe a io o e he enhanced isk pa i y model. F om Table 8, he enhanced isk
pa i y clea ly has a educ ion in dis ance and dis ance ola ili y agains he MVO model, which is he
expec ed ou come. Fu he mo e, a s ong educ ion in dis ance o isk pa i y is wi nessed again wi h
he inclusion o he isk egula ing e m
ρ
. As he isk alloca ions app oach nea isk pa i y po olios,
a educ ion in he dis ance ola ili y,
σMSE
, indica es ha la ge de ia ions ha e been educed ac oss
he in es men ho izon, esul ing in u he di e si ica ion o isk.
Table 7. Annualized ou -o -sample pe o mance.
Ta ge Mean-Va iance Enhanced Risk Pa i y Relaxed Risk Pa i y
mRe u n Risk Sha pe Re u n Risk Sha pe Re u n Risk Sha pe
1.0x 9.85 13.33 0.773 11.23 15.44 0.766 11.37 15.11 0.788
1.2x 10.33 13.61 0.793 11.59 15.66 0.778 12.07 15.28 0.822
1.4x 10.96 14.13 0.810 12.14 16.15 0.791 12.55 15.66 0.833
1.6x 11.78 14.69 0.835 12.73 16.87 0.795 13.06 16.19 0.839
Table 8. Ou -o -sample dis ance o isk pa i y.
Ta ge Mean-Va iance Enhanced Risk Pa i y Relaxed Risk Pa i y
m dMSE σMSE dMSE σMSE dMSE σMSE
1.0x 1.781 0.600 0.000 0.000 0.000 0.000
1.2x 1.804 0.593 0.411 0.173 0.331 0.137
1.4x 1.857 0.582 0.710 0.295 0.598 0.258
1.6x 1.949 0.567 0.996 0.428 0.880 0.410
The elaxed isk pa i y model has nea equi alen Sha pe pe o mance o an MVO bu a an
imp o ed annualized e u n wi h smalle dis ances o isk pa i y ac oss all a ge e u ns s udied.
Highe e u ns a e shown o he enhanced a ge e u n model bu a an inc ease in isk. Including
he isk egula ing e m wi h he enhanced isk pa i y model imp o es he isk di e si ica ion and he
esul s indica e a mode a e imp o emen o he annualized e u n. This aligns wi h he hypo hesis o
his s udy, ha be e pe o ming ou -o -sample po olios in e ms o Sha pe a io and dis ance o isk
pa i y a e possible wi h nea isk pa i y po olios.
5.4. Re e sion o Risk Pa i y in Vola ile Ma ke Condi ions
The dis ance om isk pa i y indica es o wha deg ee he po olio may e ain isk pa i y
cha ac e is ics. P og ession along he in es men ho izon e eals ha he model e e s o isk
pa i y alloca ions, Figu e 10, ollowing signi ican ma ke shocks and con i ms he indings om
Figu e 9.
J. Risk Financial Manag. 2020,13, 237 20 o 28
Figu e 10. Dis ance o Risk Pa i y Con ibu ions pe pe iod.
Table 9. Re e sion o isk pa i y o low a ge e u ns.
Ta ge mRisk Sha pe Dis ance
1.0x 15.11 0.788 0.000
0.2x 15.11 0.788 0.000
0.8x 15.11 0.788 0.000
1.2x 15.28 0.822 0.137
Du ing pe iods o ma ke dis ess, he adap i e a ge e u n app oach p oduces a ge e u ns
ha a e below wha he isk pa i y e u n can achie e due o he dep ecia ion o expec ed e u n
es ima es in he look-back pe iod. The ins abili y in e u ns esul s in he model dis ega ding he
mean- a iance ade-o and o ming he op imiza ion wi h only he isk cha ac e is ics; ul ima ely
e e ing he model back o he isk pa i y weigh ing. The mean- a iance con lic becomes negligible
and any addi ional in o ma ion om he expec ed e u ns is no conside ed. This can be seen du ing
2002 o 2004 in which he po olio’s ela i e pe o mance ollows in line wi h he isk pa i y po olio
a e only he 2 yea aining pe iod beyond he c ash. This ac s as a gua d agains concen a ed
po olios and inhe en ly and indi ec ly adds a obus laye o he model since i does no d ama ically
lose alue below he isk pa i y po olio in a ma ke co ec ion. The single-pe iod isk, e u n and
dis ance o isk pa i y is compu ed o demons a e his ai . I is ob ious om he esul s in Table 9 ha
a ge e u n goals wi h a mul iplie be ween 0.2x and 0.8x (indica ing a a ge goal less hen he isk
pa i y e u n) esul s in he isk pa i y alloca ion, which is equi alen o he 1.0x a ge e u n mul iplie .
Al hough use ul, he e is a lag in he models eac ion o he c ash due o he look-back pe iod
leng h. I he ma ke s ays ola ile o long pe iods i will be a isk pa i y bu will no cap u e sho
pe iods o high ola ili y due o he a e aging e ec o he mean asse e u ns o e he his o ical
aining pe iod. This ansla es o a slowe eac ion o he s a o bull ma ke s o going some possible
ea ly gains. This model is pu ely aw-da a-d i en and does no inco po a e p edic i e indica o s o
co ec o his lag ha change he models pa ame e s, based on p edic ed ma ke condi ions, such as
in Cos a and Kwon (2019). Cap u ing ma ke changes lags he cu en ma ke e en s by he size o
he look-back pe iod and olling ho izon. These wo pa ame e s can be dec eased in size o cap u e
cu en ma ke e en s soone , since no p edic i e indica o s a e being used, bu some lag mus s ill
be accep ed.
5.5. Cos Ve sus Bene i o Relaxa ion o Enhanced Po olios
The inc ease in ola ili y is used as an app oxima e measu e o how much isk ha a p ac i ione
akes on o enhance he e u ns om he nominal isk pa i y esul . Conduc ing a es using a 6-mon h
olling ho izon wi h a ange o a ge e u n mul iplie s om 1.0x h ough 1.8x e eals Figu e 11,
depic ing he a ge e u n o po olio ola ili y inc ease. The cu e indica es ha he inc ease in
ola ili y has a non-linea ela ionship wi h e u n. The esul s om bo h he enhanced isk pa i y
model, Model (B), and he elaxed isk pa i y model, Model (C), a e included. The esul s show ha

J. Risk Financial Manag. 2020,13, 237 21 o 28
he use o he addi ional isk egula ing e m is jus i ied o he p oposed model and ha a modes
inc ease in e u n can ac ually lead o po olios wi h equi alen isk as he isk pa i y po olio i sel .
As he isk pa i y p ope ies a e los wi h highe e u ns, he cos inc eases d ama ically. This ein o ces
he idea ha nea - isk pa i y po olios can yield be e e u ns wi hou a d ama ic cos o he isk
pa i y ai s.
Figu e 11. Ou -o -sample ola ili y cos o enhanced e u ns—1997–2018.
A look a he a e age dis ance o isk pa i y o each e-op imiza ion pe iod in Figu e 12 indica es
he loss o isk pa i y p ope ies o e he ho izon. I is ob ious ha a ge ing nea isk pa i y po olios
lowe s he dis ance o isk pa i y, as compa ed o no penal y, and ha he loss in isk pa i y ai s o e
a ho izon o each a ge mul iplie is qui e consis en . The linea ela ionship indica es ha he isk
pa i y p ope ies a e los a a p edic able a e wi h enhanced e u ns. The dis ance om isk pa i y
has an almos equi alen ola ili y o he unde lying isk pa i y po olio a 1.15x e u n, as e ealed
in Figu e 11.
Figu e 12. Risk pa i y p ope y cos o enhanced e u ns (MSE)—1997–2018.
The Sha pe a io p og ession in Figu e 13 o he elaxed isk pa i y is always g ea e hen he
un- egula ed Model (B), con i ming be e Sha pe cha ac e is ics wi h nea isk pa i y po olios o
inc easing a ge e u n mul iplie s opposing he in-sample esul in which he e is a dec ease a
highe a ge e u ns. In all analysis cases, he elaxed isk pa i y model ou -o -sample consis en ly
ou -pe o ms he nominal isk pa i y model and he mo e concen a ed Model (B).
J. Risk Financial Manag. 2020,13, 237 22 o 28
Figu e 13. Relaxed isk pa i y in es men ho izon Sha pe Ra io-1997–2018.
5.6. Inhe en Robus T ai s o Handling Unce ain y
I is well unde s ood in he li e a u e ha he dependence on es ima ed pa ame e s—namely
he expec ed e u n and VCV ma ix o asse e u ns—leads o es ima ion e o because hey a e
de e mined om sample es ima es (Fama and F ench 1993). Chop a and Ziemba (1993) ex end his
idea by showing ha he unce ain y in he es ima ed expec ed e u ns a e much mo e signi ican han
he es ima ed co a iances (Bes and G aue (1991); Michaud (1989)), indica ing ha imp o ing hese
es ima es would d ama ically imp o e he po olio’s s abili y.
I is shown he e ha he elaxed isk pa i y model inhe en ly comba s pe cei ed ins abili ies ha
migh a ise wi hou eso ing o addi ional obus op imiza ion me hods. Risk pa i y alone is no a
obus model bu could exhibi he ai s o a obus model. A obus model embeds he unce ain y
in o he model design and accep s he unce ain y in o he op imiza ion as a de e minis ic a iabili y o
he es ima ed e u n. I is gene ally used o keep he sol e away om noisy es ima ed pa ame e s and
p e en oo much weigh being applied o asse s wi h e y high expec ed e u ns and high s anda d
e o s. I p o ides solu ions immune o unce ain y by using he wo s -case scena io o he es ima ed
e u n pa ame e o each asse acili a ed by allowing he sol e o choose some alue o expec ed
e u n a ound he es ima ed e u n o use as he ue e u n. The implemen a ion is done h ough
applying an unce ain y se a ound he expec ed e u n es ima es and scaled by he s anda d e o o
he asse e u ns.
Robus ness is added o Model (C) h ough he addi ion o an ellipsoidal unce ain y se
(45)
a ound each es ima ed e u n, as desc ibed by Tü üncü and Koenig (2004). Model (D) applies a
egula o y e m and sizing pa ame e o he es ima ed e u ns
(44)
o he po olio. Two egula ing
e ms wi h penal ies a e seen in Model (D), one ac ing on he isk as
ρ
and he o he on he e u n as
κ
.
Model (D) min
xψ−γ(39)
s. . ζi= (Σx)i−ma ginal isk (40)
xTΣx≤n(ψ2−ρ2)− isk bound (41)
xiζi≥γ2− o al isk (42)
λxTΘx≤ρ2−penalized egula ing e m (43)
µTx−κ≥R− a ge e u n (44)
e2xΩx≤κ2−unce ain y se (45)
1Tx=1−budge (46)
xi,ζi≥0
ψ,γ,κ≥0
J. Risk Financial Manag. 2020,13, 237 23 o 28
I is ob ious ha he egula ing e ms o bo h he isk,
ρ
, and he e u n,
κ
, pa allel each o he
in s uc u e. Howe e , hey se e di e en pu poses. No ably, he e u n egula ing e m is d i en
by he s anda d e o
Ω(47)
and a de ined con idence e m
e(48)
, which de e mines he size o he
unce ain y se . The
e
a iable is he in e se cumula i e chi-squa e dis ibu ion a a gi en con idence
le el, deno ed by
α
, ypically 95% o 99%. The dis ance de e mined using he s anda d e o o he
es ima ion is p opo ionally enla ged using he
e
sizing pa ame e . A igh e con idence is indica ed
by a highe con idence le el, which p oduces a la ge ellipsoid ha indica es a g ea e con idence ha
he ue alue is wi hin he ellipsoid.
Ω=diag(Σ)/N whe e N =#samples (47)
e=χ2(α)→e=χ2(0.95, n)(48)
When compa ing he esul s om he elaxed isk pa i y in Model (C) o a obus e sion o
Model (D) a a 95% con idence le el, he imp o emen is negligible wi h limi ed easibili y a highe
a ge e u ns. This esul indica es ha he model is al eady seeing some obus ai s. Model (C) has a
cons ained easible space and he e o e is seeing lowe pe o mance hen an MVO in-sample. This is
a pu pose ul design ai o es ic eaching he MVO alloca ions, so as o achie e he di e si y o
nea - isk pa i y po olios. The p oposed model exhibi s quasi- obus p ope ies due o he simila i y
in s uc u e o he ellipsoidal unce ain y se cons ain s. The on ie in Figu e 14 indica es he easible
ange o Model (D). This in-sample esul is pa alleled in he ou -o -sample pe o mance in Figu e 15,
whe e changes o he pe o mance a e negligible. The obus i ica ion wo ks agains he isk pa i y
a ge by concen a ing isk in o asse s, which explains he small imp o emen o he e u n in Table 10.
Table 10. Annualized ou -o -sample pe o mance.
S a egy Re u n Vola ili y Sha pe
Risk Pa i y 11.23 15.44 0.765
Relaxed Risk Pa i y 12.04 15.62 0.805
Robus Relaxed Risk Pa i y 12.15 15.63 0.810
Mean-Va iance 11.37 14.59 0.809
Figu e 14. In-sample e icien on ie s—1997–2018.
J. Risk Financial Manag. 2020,13, 237 24 o 28
Figu e 15. Ou -o -sample elaxed isk pa i y o obus —1997–2018.
The le el o con idence ha is easible in he ou -o -sample esul is only 5%. This is, in pa , due
o a smalle op imiza ion sample size, which inc eases he
e
alue and wi h i he unce ain y se size.
An agg essi e con idence penalizes he e u ns beyond capabili ies o sa is ying he a ge e u n and,
he e o e, ende s he model in easible. Ta ge ing nea - isk pa i y po olios imp o es he pe o mance
o e isk pa i y alone and exhibi s some obus ai s. Including he ellipsoidal unce ain y se in o he
a ge e u n sees negligible imp o emen o he models ou -o -sample pe o mance. The p esence
o his elaxed isk pa i y has some obus ad an ages, bu no equi alen o adi ional obus
op imiza ion. The obus model s ill has be e pe o mance on a pe cen age poin basis, bu he
elaxed isk pa i y model helps alle ia e some o he conce ns om expec ed e u ns by abso bing
some o he unce ain y, jus no all o i . By limi ing he isk in he model, he isk pa i y ai
o ganically limi s he concen a ion in o asse s wi h highe unce ain y. The weigh and isk alloca ions
in Table 11 ac oss he p oposed elaxed isk pa i y and a obus isk pa i y a e highly co ela ed and
ha e almos iden ical s anda d de ia ions and means, adding mo e suppo o his inding.
Table 11. Summa y S a is ics o Relaxed Risk Pa i y e sus Robus Risk Pa i y
Weigh Alloca ions Risk Alloca ions
Relaxed Risk Pa i y Robus Risk Pa i y Relaxed Risk Pa i y Robus Risk Pa i y
S d De 0.0114 0.0125 0.002533 0.002502
Mean 0.0201 0.0205 0.002831 0.002780
Co ela ion - 0.9896 - 0.991100
I is demons a ed ha nea isk pa i y po olios p oduced h ough he elaxed isk pa i y model
in Model (C) o e some ad an ages nea ing he obus op imiza ion o Model (B). Risk pa i y-like
po olios go a long way o being obus . Coupling expec ed e u ns wi h isk pa i y op imiza ion
should no be de e ed due o expec ed unce ain y in he es ima ed pa ame e s. These esul s p o ide
con idence ha he elaxed isk pa i y model e ains he obus ness o he isk pa i y op imiza ion
wi hou u he in oduc ion o adi ional obus op imiza ion echniques.
6. Discussion and Conclusions
The main con ibu ion o his hesis is a elaxed isk pa i y model ha is s a egically designed o
minimize he dis ance om isk pa i y a he hen o minimize isk o maximize e u n. The model
a ge s nea isk pa i y po olios whe e an imp o emen in he e u ns can be s uc u ally de ined.
The model can inc emen ally de ia e away om isk pa i y alloca ions h ough p o iding a a ge
mul iplie based on a p ac i ione s isk ole ance ha ac s on he p e ious pe iod’s isk pa i y e u n in
an e o o seek a g ea e e u n nex pe iod. I has been shown ha he a ge e u n o he loss in isk
pa i y a ibu es is a nea linea ela ionship, seen in Figu e 12, posing a scena io whe e he p ac i ione
can use he model o gene a e po olios ha s ay nea a desi ed isk ole ance. This pape e-in oduces
a pe o mance goal in o he op imiza ion and conside s a isk pa i y elaxa ion in he long-only domain.