Network Risk Parity: graph theory-based portfolio construction

Cici e i, Vi o; Pallo a, Albe o
A icle — Published Ve sion
Ne wo k Risk Pa i y: g aph heo y-based po olio
cons uc ion
Jou nal o Asse Managemen
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Cici e i, Vi o; Pallo a, Albe o (2024) : Ne wo k Risk Pa i y: g aph heo y-based
po olio cons uc ion, Jou nal o Asse Managemen , ISSN 1479-179X, Palg a e Macmillan, London,
Vol. 25, Iss. 2, pp. 136-146,
h ps://doi.o g/10.1057/s41260-023-00347-8
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/316658
S anda d-Nu zungsbedingungen:
Die Dokumen e au EconS o dü en zu eigenen wissenscha lichen
Zwecken und zum P i a geb auch gespeiche und kopie we den.
Sie dü en die Dokumen e nich ü ö en liche ode komme zielle
Zwecke e iel äl igen, ö en lich auss ellen, ö en lich zugänglich
machen, e eiben ode ande wei ig nu zen.
So e n die Ve asse die Dokumen e un e Open-Con en -Lizenzen
(insbesonde e CC-Lizenzen) zu Ve ügung ges ell haben soll en,
gel en abweichend on diesen Nu zungsbedingungen die in de do
genann en Lizenz gewäh en Nu zungs ech e.
Te ms o use:
Documen s in EconS o may be sa ed and copied o you pe sonal
and schola ly pu poses.
You a e no o copy documen s o public o comme cial pu poses, o
exhibi he documen s publicly, o make hem publicly a ailable on he
in e ne , o o dis ibu e o o he wise use he documen s in public.
I he documen s ha e been made a ailable unde an Open Con en
Licence (especially C ea i e Commons Licences), you may exe cise
u he usage igh s as speci ied in he indica ed licence.
h p://c ea i ecommons.o g/licenses/by/4.0/
Vol:.(1234567890)
Jou nal o Asse Managemen (2024) 25:136–146
h ps://doi.o g/10.1057/s41260-023-00347-8
ORIGINAL ARTICLE
Ne wo k Risk Pa i y: g aph heo y‑based po olio cons uc ion
Vi oCici e i1· Albe oPallo a2
Re ised: 31 May 2023 / Accep ed: 11 Decembe 2023 / Published online: 20 Feb ua y 2024
© The Au ho (s) 2024
Abs ac
This s udy p esen s ne wo k isk pa i y, a g aph heo y-based po olio cons uc ion me hodology ha a ises om a hough -
ul c i ique o he clus e ing-based app oach used by hie a chical isk pa i y. Ad an ages o ne wo k isk pa i y include: he
abili y o cap u e one- o-many ela ionships be ween secu i ies, o e coming he one- o-one limi a ion; he capaci y o le e age
he ma hema ics o g aph heo y, which enables us, among o he hings, o demons a e ha he esul ing po olios is less
concen a ed han hose ob ained wi h mean- a iance; and he abili y o simpli y he model speci ica ion by elimina ing he
dependency on he selec ion o a dis ance and linkage unc ion. Pe o mance-wise, due o a be e ep esen a ion o sys em-
a ic isk wi hin he minimum spanning ee, ne wo k isk pa i y ou pe o ms hie a chical isk pa i y and o he compe ing
me hods, especially as he numbe o po olio cons i uen s inc eases.
Keywo ds Po olio cons uc ion· G aph heo y· Hie a chical clus e ing· Eigen alues
JEL classi ica ion G11
In oduc ion
The seminal asse alloca ion model o Ma kowi z (1952) has
been he s onghold o po olio cons uc ion since 1952.
Howe e , ex ensi e esea ch documen s h ee main limi a-
ions, namely p oducing uns able, concen a ed, and unde -
pe o ming po olios. Michaud (1989) p o ides a de ailed
explo a ion o hese issues. The inance li e a u e has p o-
posed nume ous solu ions, including he wo ks o Black and
Li e man (1992) and Ledoi and Wol (2004), among o he s.
The li e a u e on po olio cons uc ion has ecen ly
ocused on wo ields o applica ion: clus e ing and g aph
heo y. On he clus e ing side, he main ep esen a i e is he
hie a chical isk pa i y (HRP) o de P ado (2016), which
applies hie a chical clus e ing o secu i ies based on co -
ela ions. This g oups simila in es men s while dis anc-
ing dissimila ones, wi h op imal alloca ion achie ed ia
in e se- a iance alloca ion. Ra ino (2017) u he p o-
ided a comp ehensi e amewo k o po olio cons uc ion
employing hie a chical clus e ing.
Pe al a and Za eei (2016) ma ked he pionee ing s ep wi h
he inclusion o g aph heo y in he po olio cons uc ion
li e a u e. They demons a ed he close ela ionship be ween
g aph cen ali y measu es and op imal po olio weigh s.
Fu he mo e, hey designed po olios by equally dis ibu -
ing capi al o he mos cen al secu i ies in low- ola ili y
pe iods and ebalancing o he leas cen al secu i ies du -
ing high- ola ili y pe iods. Vỳ os e al. (2019) cons uc ed
op imal po olios based on ou g aphical ep esen a ions
o secu i ies: a comple e g aph, a minimum spanning ee,
a plana maximally il e ed g aph, and a h eshold signi i-
cance g aph.
This pape aims a in oducing ne wo k isk pa i y (NRP),
a no el g aph heo y-based po olio cons uc ion me hod
ha p oduces ully in es ed, long-only po olios. We de i e
ne wo k isk pa i y by d awing a pa allel wi h hie a chical
isk pa i y. Indeed, while hie a chical isk pa i y is based on
hie a chical clus e ing, Ne wo k isk pa i y builds po o-
lios wi h g aph heo y. The connec ion poin be ween bo h
me hodologies is ha hey calcula e po olio weigh s p o-
po ionally o he in e se o he eigen alues o a modi ied
co a iance ma ix. In hie a chical isk pa i y, he op imal
* Albe o Pallo a
[email p o ec ed]
Vi o Cici e i
[email p o ec ed]
1 Be lin, Ge many
2 Middlesex Uni e si y, The Bu oughs, Hendon,
LondonNW44BT, UK
137Ne wo k Risk Pa i y: g aph heo y-based po olio cons uc ion
weigh s a e calcula ed as he in e se o each secu i y’s a i-
ance. This a iance is ob ained om he main diagonal o
a quasi-diagonal co a iance ma ix, which is modi ied o
inco po a e he hie a chical s uc u e de i ed using hie a -
chical clus e ing. We demons a e ha he weigh ing sys em
employed by HRP is equi alen o aking he in e se o he
eigen alues o he same co a iance ma ix, a e an essen-
ial quasi-diagonaliza ion s ep. On he o he hand, ne wo k
isk pa i y ope a es on a di e en p inciple. I u ilizes an
adjacency ma ix based on co a iances, whe e one se o
eigen ec o s is de e mined by he eigen ec o cen ali y, a
measu e o he in luence o a node in a ne wo k. In NRP, he
op imal po olio weigh s a e calcula ed as he in e se o he
eigen ec o cen ali y. A so max no maliza ion is applied o
hese weigh s o ensu e a ully in es ed, long-only po olio.
Th ee ad an ages o ne wo k isk pa i y s em om
ansposing clus e ing-based me hodologies in o a g aph
heo y amewo k. Fi s ly, NRP, g ounded on he p inciple
o eigen ec o cen ali y, encapsula es ela ionships among
secu i ies in a one- o-many ashion, esul ing om he de i-
ni ion o eigen ec o cen ali y ha embeds he impo ance
o neighbo ing nodes oo. Hie a chical clus e ing, on he
o he hand, cap u es only one- o-one ela ionships due o i s
agglome a i e clus e ing app oach, which looks a pai wise
dis ances. Secondly, hie a chical clus e ing depends on he
de e mina ion o a dis ance and a linkage unc ion. NRP,
ins ead, is based on minimum spanning ee (MST), which
solely depends on he unc ion used o con e co ela ions
in o dis ances. Hie a chical isk pa i y in pa icula has been
c i icized o i s adop ion o a single linkage unc ion, which
di e en ia es secu i ies based on he dis ance o he nea es
poin s wi hin clus e s, he eby causing a chaining e ec ha
expands he ee and impac s po olio weigh s (Papenb ock
2011). Thi d, le e aging on g aph heo y, we can p o e ha
he po olio weigh s o NRP and HRP1 ha e a lowe bound
la ge han ze o, hus assigning a posi i e weigh o each
po olio cons i uen and imp o ing po olio di e si ica ion,
a pi all o he classic mean- a iance app oach.
Using a boo s apping app oach, we compa e he Sha pe
a io o NRP wi h ha o HRP, isk pa i y (RP), Ma kow-
i z’s minimum a iance op imiza ion (MVO), and equally
weigh ed (EW) po olios. While NRP consis en ly ou pe -
o ms he o he me hods, compa ed o HRP, i s pe o mance
depends on he numbe o s ocks in he po olio, ou pe -
o ming HRP as he po olio size inc eases.
The es o he pape is o ganized as ollows: Sec ion2
in oduces NRP and compa es i o HRP. Sec ion3 p esen s
he empi ical esul s in e ms o boo s apped Sha pe a io
and weigh s. Sec ion4 concludes and discusses u u e wo k.
Me hodology
Co a iance
The s a ing poin o ne wo k isk pa i y, simila o mos
po olio cons uc ion me hodologies, is o es ima e he
co a iance ma ix o asse e u ns. We use hou ly secu i y
p ices and de ine log- e u ns as:
whe e
i,𝜏
is he e u n o he i- h asse a ime
𝜏
,
pi,𝜏
is he
p ice o he i- h asse a ime
𝜏
, and T is he numbe o hou s
in each mon h. We es ima e he condi ional co a iance
by employing he me hodology o Ba ndo -Nielsen and
Shepha d (2004), who show ha condi ional co a iances can
be es ima ed h ough non-pa ame ic ealized co a iances,
RC
, which con e ge in p obabili y o he quad a ic a ia-
ion o he p ice p ocess unde e y gene al assump ions.
Thus, we calcula e mon hly ealized co a iances,
RC
, as he
agg ega ion o c oss-p oduc s o hou ly e u ns, such ha :
This ensu es posi i e de ini e ealized co a iance ma ices
and makes co a iance ully obse able and moldable wi h
any ime se ies model. The use o hou ly e u ns in con-
s uc ing ou po olios helps o p o ide a highe numbe o
obse a ions, which in u n inc eases he obus ness o ou
co a iance ma ix es ima ion.2 I should be no ed, howe e ,
ha ne wo k isk pa i y is a co a iance-agnos ic po olio
cons uc ion me hod. While accu a e co a iance es ima-
ion is c ucial o any po olio cons uc ion app oach, he
p ima y ocus o his pape is no o p o ide an imp o ed
es ima e o co a iances.
Re iew o Hie a chical Risk Pa i y
Hie a chical Risk Pa i y p oduces po olios om a h ee-
s ep p ocess. Fi s ly, hie a chical clus e ing is pe o med on
co ela ion-based dis ance measu es, s a ing om he co -
ela ion ma ix o asse e u ns. Since clus e ing equi es a
dis ance measu e o encapsula e he ela ionship be ween
i,𝜏=ln
(p
i,𝜏
p
i,𝜏−1)
∀𝜏=1, …,T;∀i=1, ...,
n
(1)
RC
=
N
∑
𝜏=1
𝜏 �
𝜏
.
1 As he p ocess o clus e ing a co ela ion ma ix can be e-w i en
as a ee on a comple e dig aph, he weigh loo applies o HRP as
well.
2 In his s udy, we agg ega e hou ly e u ns in o mon hly co a iances,
and we assume he po olios a e ebalanced mon hly.
138 V.Cici e i, A.Pallo a
secu i ies, we ans o m co ela ions in o Euclidean me -
ics.3 Fo his ans o ma ion, Man egna (1999) de ines he
dis ance
di
,
j
as
di
,
j
=1−𝜌
i
,
j
2
.4 The AGNES hie a chical clus-
e ing algo i hm is hen applied o hese dis ances, sepa a -
ing secu i ies in o clus e s o ganized in a linkage ma ix. In
he second s ep, he linkage ma ix unde goes quasi-diag-
onaliza ion so ha he la ges alues align along he main
diagonal. Finally, in he hi d s ep, op imal weigh s a e cal-
cula ed as he in e se o each secu i y’s a iance, which is
loca ed on he main diagonal a e he quasi-diagonaliza ion
s ep.
G aph heo y backg ound
Fo a comp ehensi e in oduc ion o g aph heo y, e e o
Bollobás (1998, 2001). Conside a di ec ed weigh ed g aph
G=(V,E,W)
o med by a ini e se o e ices V5, a se
o di ec ed edges
E⊂V×V
, whe e each
(
e
x
,e
y)
∈
E
ep-
esen s a link om
x∈V
o
y∈V
, and a se o weigh s
W∶E→ℝ++
de ined on each edge. Two nodes
( x, y)∈V
a e said o be adjacen i he e exis s an edge
(ex,ey)∈E
.
The adjacency ma ix o he g aph
AG
is de ined as he
squa e ma ix whose en ies a e
ai,j=wi
,
j
i
i, j∈E
and
wi,j
a e he weigh o he edge,
ai,j=0
o he wise.
In a inancial se up, a g aph can be used o ep esen a
inancial ma ke , whe ein a secu i y is ep esen ed by a e -
ex, and he ela ionship be ween each pai o secu i ies is
ep esen ed by edges. The simples way o measu ing he
ela ionships among secu i ies is o use linea co ela ions.
As in clus e ing, he adjacency ma ix mus be de ined on
a me ic; hence, we apply he same
di
,
j
=1−𝜌
i
,
j
2
ans o -
ma ion. Mo eo e , he diagonal o he adjacency ma ix is
se o ze o o a oid sel -loops6. As such, a non-ze o en y in
he adjacency ma ix indica es he exis ence o a inancial
ela ionship be ween pai s o secu i ies wi h s eng h
di
,
j
.
A g aph buil on a co ela ion ma ix is a comple e
dig aph, which is a g aph whe e all pai s o e ices a e
connec ed by a pai o unique edges. This is he sou ce o
he link o HRP o g aph heo y. Howe e , co ela ion ma i-
ces lack he no ion o hie a chy. Simon (1991) a gues ha
complex sys ems, such as inancial ma ke s, can be a anged
in a na u al hie a chy comp ising nes ed subs uc u es. The
goal o codependence analysis is choosing which c oss-secu-
i y ela ions eally ma e . F om a ee ep esen a ion s and-
poin , his means choosing which links in he ee a e signi i-
can and emo ing he o he s. de P ado (2016) a gues ha
he lack o hie a chical s uc u e makes po olio weigh s
a y in unin ended ways in an asse alloca ion p oblem.
Fo his eason, comple e dig aphs do no add addi ional
in o ma ion compa ed o co ela ion ma ices, while o he
subg aphs – such as spanning ees7 – can be e se e inan-
cial needs by inco po a ing a hie a chical ep esen a ion and
choosing he links be ween secu i ies ha eally ma e .
We employ he minimum spanning ee (MST) (see
Appendix B o a g aphical ep esen a ion), a subse o
a comple e dig aph ha includes all e ices bu selec s
he minimum possible numbe o edges by sol ing
minS∑e∈s
W(e
)
, whe e
S≤E
is he numbe o links in he
MST and e ep esen s each link in each ealiza ion,
s∈S
.
To ind minimum spanning ees, we employ he algo i hm
by K uskal (1956).8 I is wo h no ing ha he e a e se e al
pa allels be ween K uskal’s algo i hm o minimum span-
ning ees and AGNES algo i hm o hie a chical clus e ing.
Fi s , bo h algo i hms s a wi h a se o ully disconnec ed
nodes and i e a i ely build clus e s. Second, bo h algo i hms
use a g eedy app oach o o m clus e s. In K uskal’s algo-
i hm, he edges o he g aph a e so ed by weigh based
on he dis ance be ween secu i ies, and a each s ep, he
algo i hm adds he nex edge (wi h he lowes weigh ) ha
connec s wo p e iously unconnec ed clus e s. Simila ly, in
AGNES, a each s ep, he algo i hm me ges he wo clus e s
ha a e closes o each o he , based on some dis ance me -
ic. Finally, bo h algo i hms p oduce a hie a chy o clus e s,
which can be isualized in he o m o an MST o a dend o-
g am, espec i ely.
A minimum spanning ee allows quan i ying how secu i-
ies in luence each o he by means o a cen ali y measu e.
A cen ali y measu e
C∶V
→
ℝ+
is a unc ion ha assigns
a non-nega i e alue o each node such ha he highe he
alue, he mo e he node is connec ed o o he s. One such
cen ali y measu e is he eigen ec o cen ali y, acco d-
ing o which a secu i y displays a high cen ali y ei he by
di ec links o o he secu i ies o by being connec ed o o he
secu i ies ha a e hemsel es highly connec ed. As such, he
highe he eigen ec o cen ali y, he mo e cen al a secu i y
3 The calcula ion o a dis ance measu e equi es ha he quan i ies
a e Euclidean me ics, which co ela ions a e no .
4 Acco ding o his dis ance measu e, wo pai s o secu i ies wi h
he same linea co ela ion bu wi h he opposi e sign a e conside ed
equally dis an . While his is a gene al limi a ion, we ega d i as
insigni ican as long as he ocus is on equi y e u ns, which gene ally
exhibi posi i e linea co ela ions.
5 in his manusc ip , we in e changeably use he e ms "nodes" and
" e ices" o e e o he elemen s o he se V.
6 A sel -loop is a link ha connec s a e ex o i sel . We elimina e
sel -loops since sel - ela ions a e insigni ican in po olio cons uc-
ion.
7 A spanning g aph is a subg aph ha con ains all e ices o he
comple e g aph. A ee is an acyclical and connec ed g aph, whe e all
nodes a e connec ed by a single edge. I can be p o ed ha e e y ee
is a g aph and e e y non- i ial g aph con ains a leas one ee.
8 S a ing om a ee in each e ex, he K uskal’s algo i hm
emo es any link wi h a minimum weigh be ween he e ices, com-
bining he ees o which he link has been emo ed.
139Ne wo k Risk Pa i y: g aph heo y-based po olio cons uc ion
is in he ee. Eigen ec o cen ali y is based on he idea ha
a node’s impo ance is de e mined by he impo ance o he
nodes ha i is connec ed o. In o mula, he eigen ec o
cen ali y o a node
𝜁( )
is gi en:
whe e N( ) is he se o neighbo s o node and
𝜆max
is he
la ges eigen alue.
Ne wo k Risk Pa i y
We de i e he ne wo k isk pa i y me hodology om a pa -
allel wi h hie a chical isk pa i y, assuming ha i s co a i-
ance ma ix C is an n-dimensional diagonalized ma ix wi h
ull- ank. Applying he spec al decomposi ion heo em o
C yields:
whe e
uj
is he j- h eigen ec o associa ed wi h he eigen-
alue
𝜆j
o ma ix C. This ela ion leads o he eigen ec o
equa ion
Cui=𝜆iui
, o all
i=1, ..., n
. The eigen alues
𝜆i
and eigen ec o s
ui
a e ob ained by sol ing he cha ac e is ic
equa ion
de (C−𝜆I)=0
. Sol ing o he eigen alues yields
(see Appendix A o he p oo ):
whe e
𝜎i
a e he elemen s o he diagonal ma ix C. In o he
wo ds, he op imal in e se- a iance alloca ion o Hie a chi-
cal Risk Pa i y co esponds o he eigen alues o he quasi-
diagonal co a iance ma ix C, assuming ha he diagonali-
za ion s ep e ec i ely esul s in a diagonal ma ix.
Rea anging he eigen ec o cen ali y de ini ion o he
MST in ma ix o m, we ha e:
whe e
𝜁
is he eigen ec o o he adjacency ma ix A asso-
cia ed wi h i s la ges eigen alue
𝜆max
. In NRP we ake a
somewha simila app oach o HRP by using he eigen ec o
cen ali y and calcula e he po olio weigh s as:
To ob ain ully in es ed po olios such ha
∑w
=
1
, we
apply he so max no maliza ion
𝜎
(w)=
ew
∑
(ew
)
. As Laloux
(1999) showed ha he la ges eigen alue o a co ela ion
ma ix can be seen as a ep esen a i e o sys ema ic isk, and
as he eigen ec o cen ali y is associa ed wi h he la ges
eigen alue o he adjacency ma ix, he eigen ec o
(2)
𝜁
( )=
1
𝜆max
∑
�
∈N( )
𝜁( �
)
(3)
C
=UΛUT=
n
∑
j=1
𝜆juju
T
j
(4)
𝜆i=𝜎i,∀i=1, ..., n
(5)
A𝜁=𝜆max 𝜁
(6)
w
∗
i=
1
𝜁
i
∀i=1, ..., n
.
cen ali y can be unde s ood as a gage o a secu i y’s con i-
bu ion o sys ema ic ma ke isk. In he NRP app oach, hese
secu i ies a e assigned lowe weigh s in he po olio,
he eby aiming o minimize he po olio’s exposu e o sys-
ema ic isk.
Despi e bo h HRP and NRP use he same dis ance me ics
calcula ed as
d(i
,
j)
=1−𝜌
i
,
j
2
, whe e
𝜌i
,
j
is he Pea son co -
ela ion coe icien , hey esul in di e en po olios due o
he di e en ways he no ion o hie a chy is imposed on he
co ela ion ma ix. Hie a chical isk pa i y uses hie a chical
clus e ing wi h Euclidean dis ance and a single linkage c i e-
ion. Ne wo k isk pa i y, ins ead, uses K uskal’s algo i hm
o build a minimum spanning ee and selec he meaning-
ul in e connec ions be ween secu i ies. Th ee bene i s a e
associa ed wi h he la e app oach.
Fi s , in NRP ela ionships among secu i ies a e one- o-
many a he han one- o-one as in HRP. In ac , he eigen ec-
o cen ali y akes in o accoun he impo ance o neighbo -
ing nodes oo. On he o he hand, hie a chical clus e ing
uses a one- o-one app oach o clus e secu i ies.
Second, HRP depends on he speci ica ion o a dis ance
unc ion o ans o m linea co ela ion in o me ics, and
o a dis ance unc ion and a linkage unc ion o clus e ing
pu poses. NRP, on he o he hand, depends only on he i s .
In pa icula , HRP has been c i icized o using he single
linkage unc ion, which sepa a es objec s depending on he
dis ance be ween he wo closes poin s wi hin clus e s. This
causes a chaining e ec ha widens he ee and esul s in
an unequal dis ibu ion o he po olio weigh s (Papenb ock
2011).
Thi d, he usage o g aph heo y allows us o analy ically
p o e he le el o concen a ion o he op imal po olios, as
will be shown.
Po olio weigh s lowe bound
Le e aging he concep o he deg ee d( ) o a e ex in
a g aph G, de ined as he numbe o e ices in G ha a e
adjacen o , i can be es ablished – using he Ge shgo in
ci cle heo em9 - ha he maximum eigen alue
𝜆max
o he
9 The Ge shgo in ci cle heo em p o ides a way o bound he eigen-
alues o a squa e ma ix wi h he sum o he absolu e alues o he
ma ix’s en ies along i s ows o columns. Speci ically, he heo em
s a es ha each eigen alue o a ma ix A lies wi hin a leas one o he
n Ge shgo in disks, which a e de ined as ollows.
Fo
i=1, 2, …,n
, le
Ri
deno e he sum o he absolu e alues o he
o -diagonal en ies in he i h ow o A, and le
aii
deno e he diagonal
en y o A in he i h ow. Then, he i h Ge shgo in disk is he closed
disk in he complex plane wi h cen e
aii
and adius
Ri
. Tha is,
The Ge shgo in ci cle heo em hen s a es ha e e y eigen alue o A
lies in a leas one o he Ge shgo in disks, ha is,
Di=z∈ℂ∶|z−aii|≤Ri.

140 V.Cici e i, A.Pallo a
adjacency ma ix A is bounded abo e by he maximum
deg ee d( ) o he g aph. As
w=𝜁−1
and
𝜁
co esponds o
he la ges eigen alue
𝜆
, hen
w
≥max
(d)−1
, meaning ha
all po olio asse s ha e a weigh ha is g ea e han he
in e se o he maximum deg ee o he MST. In pa icula ,
as
d( )>0
o all non-emp y g aphs, he op imal weigh s o
NRP a e lowe bound by he so max-no malized deg ee o
he MST. As HRP is based on he in e se- a iance alloca-
ion o a quasi-diagonalized co ela ion ma ix which we
show o be p opo ional o
𝜆−1
,10 gi en ha he Ge shgo in
heo em holds o any squa e ma ix, and gi en ha he co -
ela ion ma ix can be ep esen ed as a comple e dig aph,
he po olio weigh s o HRP ha e a lowe bound g ea e
han ze o as well.
Con e sely, he spec al adius o he g aph
𝜌(G)
11 is
bounded by
𝜌
(G)≥
√
𝜆
max
−
1
, whe e
𝜆max
is he la g-
es eigen alue o adjacency ma ix. This sugges s ha he
spec al adius can be seen as he deg ee o concen a ion
ha can occu , as he weigh s a e in e sely p opo ional o
he eigen alues, which g ow la ge in highly ola ile pe i-
ods (hence, esul ing in mo e sp ead ou po olio weigh s).
This lowe bound mechanism con ibu es o c ea ing mo e
di e si ied, less concen a ed po olios compa ed o hose
de i ed om mean- a iance op imiza ion me hods.
O no e is ha he la ges eigen alue o a co a iance
ma ix, a measu e ha a ies o e ime, se es as a p oxy
o sys ema ic isk (Laloux 1999), and i ends o inc ease
du ing high- ola ili y egimes. Viewing his om he pe -
spec i e o he minimum spanning ee (MST), he spec al
adius ends o ha e a highe uppe bound du ing highly
ola ile pe iods as he maximum eigen alue g ows highe .
This means ha as ola ili y inc eases, he spec al adius
also inc eases, he eby leading o a dec ease in he op imal
po olio weigh s and esul ing in mo e di e si ied po o-
lios. This abili y o adap o changing ma ke condi ions is
a key s eng h o NRP and HRP. As MVO p opaga es he
e o s o he ill-es ima ed co a iance ma ix h ough i s
in e sion, i s pe o mance becomes ela i ely wo se du -
ing highly ola ile pe iods, due o addi ional concen a ion.
Whe eas MVO ails o pe o m op imally du ing high- ol-
a ili y pe iods (when mos needed), NRP and HRP employ
a sel -de ensi e mechanism esul ing in be e -di e si ied
po olios, hanks o he dynamic adjus men o he spec al
adius o he g aph.
Empi ical esul s
In his sec ion, we illus a e some empi ical esul s ega d-
ing he applica ion o ne wo k isk pa i y. In his sec ion,
we conside long-only,12 ully in es ed po olios ebalanced
a mon hly equency.13 We use he hou ly p ices o all he
s ocks composing he S &P 500 index om Janua y 2010
o Ma ch 2023, agg ega ed o a mon hly equency in he
p ocess o calcula ing condi ional co a iances. To ensu e
no look-ahead bias in ou s udy, we implemen ed a oll-
ing window app oach. Fo each mon hly ebalancing, we
u ilized he pas 2 yea s’ da a. To ob ain a obus esul ,
we use a boo s apping app oach ha andomly selec s
n=(20, 50, 100, 200)
s ocks – wi hou eplacemen – ou o
he en i e sample and i e a es h ough 10000 simula ions.
We compa e he pe o mances o he compe ing me hods in
e ms o he Sha pe a io.14
Table1 epo s he a e age Sha pe a ios o he boo -
s apped po olios o he di e en numbe s o cons i uen s.
Table 1 A e age mon hly Sha pe a ios o boo s apped po olios o
di e en numbe s o cons i uen s(bes pe o ming in bold ace)
n cons i uen s NRP HRP RP EW MV
n
=
20
0.853 0.892 0.653 0.817 0.430
n=50
0.863 0.882 0.712 0.809 0.576
n
=
100
0.886 0.845 0.725 0.799 0.434
n=200
0.891 0.866 0.723 0.794 0.396
10 O exac ly equal in case o a ully diagonalized ma ix.
11 The spec al adius o a ma ix is he maximum absolu e alue
o i s eigen alues. Mo e o mally, i a ma ix A is a squa e ma ix
wi h eigen alues
𝜆1,𝜆2,…,𝜆n
, he spec al adius
𝜌(A)
is de ined as:
𝜌(A)=max1≤i≤n|𝜆i|.
12 To ob ain a long-sho applica ion, he adjacency ma ix would
need o be adjus ed o accoun o sho posi ions, he no maliza ion
o weigh s would ha e o be e isi ed o ca e o he possibili y o
nega i e weigh s, and he isk con ibu ion calcula ion would equi e
an o e haul o inco po a e he po en ial nega i e con ibu ion o sho
posi ions.
13 We agg ega e hou ly da a in o mon hly ealized co a iance ma i-
ces o ensu e a mo e obus es ima ion o he co a iance ma ix.
Howe e , one mus no ice ha NRP is agnos ic o he choice o he
co a iance es ima o . The di e en equency g anula i y be ween
hou ly and mon hly da a can esul in de ia ions om no mali y.
Howe e , he use o non-no mal e u ns in luences he po olio op i-
miza ion p ocess only o me hodologies ha depend on he assump-
ion o no mally dis ibu ed e u ns, such as Ma kowi z’s minimum
a iance. Hie a chical isk pa i y and ne wo k isk pa i y me hodolo-
gies, which a e he ocus o his s udy, do no ely on his assump-
ion. Addi ionally, he use o boo s apping in ou simula ions helps
o add ess he po en ial issues s emming om non-no mali y, as i
allows us o cap u e he empi ical dis ibu ion o he da a.
14 Sha pe a ios a e calcula ed ollowing he mon hly ebalancing
scheme, hence using mon hly po olio e u ns and mon hly s anda d
de ia ions eco ded a he ime o ebalancing.
whe e
𝜆(A)
deno es he se o eigen alues o A.
𝜆
(A)⊆
n
⋃
i=1
Di
,
Foo no e 9 (con inued)
141Ne wo k Risk Pa i y: g aph heo y-based po olio cons uc ion
Ne wo k isk pa i y and hie a chical isk pa i y ou pe o m
he o he me hodologies in all sample sizes. In pa icula ,
hie a chical isk pa i y achie es he highes Sha pe a io o
smalle po olio sizes, such as
n=(20, 50)
, while ne wo k
isk pa i y is he bes pe o ming as he numbe o po olio
cons i uen s inc eases.15 This beha io is due o he ac ha
he highe he numbe o po olio cons i uen s, he mo e
he minimum spanning ee esembles a inancial ma ke in
e ms o he abili y o place an asse a he cen e o he ee
a he han in he pe iphe y. This is explained by he ac ha
he MST s uc u e used in NRP is an imp o ed ep esen a-
ion o sys ema ic isk and inancial ma ke s. MST p ese es
he mos signi ican ela ionships be ween asse s, e ec i ely
il e ing ou less meaning ul co ela ions. This app oach
enables us o unco e he ma ke ’s inhe en s uc u e by
iden i ying clus e s o asse s ha exhibi simila beha io ,
o e ing a mo e comp ehensi e unde s anding o sys em-
a ic isk. Fu he mo e, inancial ma ke s a e complex ne -
wo ks cha ac e ized by mul idimensional in e -dependencies
be ween asse s. These connec ions ex end beyond pai wise
ela ionships, some hing he MST s uc u e, wi h i s g aph
ep esen a ion, can cap u e mo e accu a ely. The e o e, he
la ge he numbe o po olio cons i uen s, he mo e e ec-
i ely he eal asse connec ions a e ep esen ed, as he MST
can be e link asse s based on simila i ies.
Ne wo k isk pa i y ou pe o ms he o he compe ing
me hods in all po olio sizes, while MVO is he wo s pe -
o me ac oss all sample sizes. In ac , MVO o en leans
owa d po olios ha capi alize on idiosync a ic isk, due o
he in e sion o an ill-es ima ed co a iance ma ix ha p op-
aga es he e o s. This o en esul s in concen a ed po o-
lios wi h la ge weigh s assigned o a ew asse s exhibi ing
lowe ola ili y. On he o he hand, NRP and HRP s i e o
mo e balanced po olios as hey a e speci ically designed o
dis ibu e isk equally ac oss all asse s in he po olio. This
mo e equal isk dis ibu ion inhe en ly aligns he po olio
o he b oade , sys ema ic ma ke mo emen s a he han
he idiosync a ic mo emen s o indi idual asse s. By educ-
ing exposu e o idiosync a ic isk, hese s a egies a e less
suscep ible o he unexpec ed pe o mance o a small sub-
se o asse s and he e o e end o be mo e s able. Fo NRP
and HRP, his is u he suppo ed by he spec al adius
ac ing as a sel -de ensi e mechanism, as du ing pe iods o
high ola ili y, he spec al adius also inc eases, he eby
esul ing in mo e e enly sp ead po olio weigh s and be e
di e si ica ion. MVO, ins ead, du ing pe iods o high ola-
ili y, esul s in e en mo e concen a ed po olios, as he
co a iance ma ix is e en mo e ill-es ima ed.
Figu e1 illus a es he his og ams o he boo s apped
Sha pe a ios o each asse alloca ion s a egy agains he
pe o mance o he equally weigh ed po olio16 o he case
n=100
.
To p o ide some con ex ega ding he mos se e e down-
u ns ac oss he boo s apped samples, o he wo s boo -
s ap simula ion o each s a egy, we compu ed he maxi-
mum d awdown using
n=20
. I was ound ha all po olios
eached hei maximum d awdowns du ing he heigh o he
COVID-19 c isis in Ma ch 2020. In pa icula , he NRP and
HRP wi nessed maximum d awdowns o 42% and 39%. RP
and EW po olios su e ed a maximum d awdown o 48%
and 46%, espec i ely, while MVO had he la ges d awdown
o 61%. Fo compa ison, he S & P 500 expe ienced a maxi-
mum d awdown o 34% du ing he same COVID-19 pe iod.
Finally, Figu e2 plo s he boxplo s o weigh s on he
boo s apping uns wi h
n=20
. We highligh ed in ed he
pa s co esponding o a po olio weigh o 0, as his is he
i s ed lag o iden i y ill-concen a ed po olios. As i is
isible, only he mean- a iance alloca ion incu s in po olio
weigh s equal o 0 as well as se e al imes eaches po o-
lio weigh s well abo e 60%, hin ing a poo di e si ica ion
and majo concen a ion. On he o he hand, he weigh s o
NRP and HRP appea o be be e di e si ied and less con-
cen a ed, as well as exhibi a simila pa e n, in ligh o he
common lowe bound mechanism. Risk pa i y, on he o he
hand, is be e di e si ied han mean- a iance, bu i eaches
peaks up o 60%.
Appendix7 p o ides a commen a y on he a e age boo -
s apped pe o mances o he compe ing me hods agains
he ma ke en i onmen s ha cha ac e ized he pe iods
analyzed.
Conclusions
In his manusc ip , we ha e p esen ed ne wo k isk pa i y
(NRP), a no el po olio cons uc ion me hodology based on
g aph heo y. NRP se es as he coun e pa o hie a chical
isk pa i y (HRP), p o iding a complemen a y app oach o
po olio alloca ion. By u ilizing he concep o eigen ec o
15 We un a hypo hesis es ing on he null hypo hesis ha he Sha pe
a ios o NRP a e equal o hose o HRP. The null hypo hesis is
ejec ed o all numbe s o cons i uen s a 1% signi icance le el,
excep o
n=40
, which is only ejec ed a 5%.
16 Ou in en is no o use he equally weigh ed po olio as a bench-
ma k agains which an ac i e po olio manage is measu ed. Ins ead,
we ega d i as a undamen al pe o mance h eshold ha any asse
alloca ion s a egy should su pass, gi en ha i ep esen s a simple
alloca ion app oach in con as o he in icacies o compe ing me h-
ods.
142 V.Cici e i, A.Pallo a
Fig. 1 His og ams o boo s apped Sha pe a ios agains equally
weigh ed. No e: in ed, we epo he dis ibu ion o he Sha pe a ios
achie ed by he equally weigh ed po olio, while in blue, he dis i-
bu ions o he Sha pe a ios achie ed by each asse alloca ion s a egy
ac oss he boo s apped samples. As i is isible, ne wo k isk pa i y
and hie a chical isk pa i y a e mo e skewed owa d highe Sha pe
a io alues, while he opposi e holds o isk pa i y and Ma kowi z’s
minimum a iance
Fig. 2 Boxplo s o boo s apped weigh s o show he majo di e si-
ica ion bene i s associa ed wi h NRP and HRP. No e: The boxplo s
o he po olio weigh s ac oss he boo s ap un wi h
n=20
(chosen
o p ese e he eadabili y o he plo ). In ed, we highligh he poin s
whe e he boxplo ouches a alue o ze o, a esul ha is no achie -
able in ligh o he lowe bound ule applicable o g aph-based alloca-
ions. The i ial equally weigh ed case is no included
143Ne wo k Risk Pa i y: g aph heo y-based po olio cons uc ion
cen ali y and he minimum spanning ee (MST), NRP
o e s se e al ad an ages o e adi ional me hods.
One key ad an age o NRP is i s abili y o cap u e one-
o-many ela ionships be ween secu i ies. Unlike one- o-one
ela ionships cap u ed by hie a chical clus e ing in HRP,
NRP conside s he impo ance o neighbo ing nodes as well.
This b oade pe spec i e allows o a mo e comp ehensi e
unde s anding o he in e connec edness and dependencies
wi hin a inancial ma ke .
Ano he ad an age is he simplici y o he g aph-based
app oach compa ed o he dis ance and linkage unc ions
used in hie a chical clus e ing. NRP elies solely on he adja-
cency ma ix and he MST, educing he complexi y o he
me hodology while s ill achie ing e ec i e isk alloca ion.
Fu he mo e, he g aph heo y amewo k o NRP allows
o an analy ically p o able lowe bound on he op imal
weigh s o he po olio. This lowe bound ensu es ha he
po olio emains well-di e si ied, mi iga ing he isk o
concen a ion on a ew secu i ies and a oiding he pi alls
associa ed wi h Ma kowi z’s minimum- a iance app oach.
Empi ical esul s based on boo s apped samples o S
& P 500 s ocks demons a e he supe io i y o NRP o e
isk pa i y, Ma kowi z’s minimum- a iance, and equally
weigh ed po olios in e ms o he Sha pe a io. No ably,
he pe o mance o NRP and HRP a ies wi h he numbe
o s ocks in he po olio, wi h HRP showing s eng h in
smalle po olios and NRP excelling as he numbe o con-
s i uen s inc eases.
Fu u e esea ch should es di e en subg aph ep e-
sen a ions, wi h majo ocus posed on how he co a iance
ma ix is es ima ed o o ecas as well as on he inclusion
o di e en asse classes o he han equi y. Fu he mo e,
u u e esea ch could ocus on an explo a ion o co a iance
mis-es ima ions and he po en ial imp o emen s b ough by
dis ance and adjacency ma ices compa ed o sample co a i-
ances. Finally, as he cu en o mula ion o ne wo k isk
pa i y assumes a long-only po olio s uc u e, he ex ension
o long-sho po olios could be an in e es ing a enue o
u u e esea ch.
A p oo o equa ion (4)
In his appendix, we b ie ly illus a e he algeb ical s eps o
p o e ha he op imal weigh s gene a ed by hie a chical isk
pa i y a e p opo ional o he in e se o he eigen alues o
he modi ied co a iance ma ix.
Le ’s assume ha he modi ied co a iance ma ix C is
squa ed, ull- ank, and ully diagonal. Fo ins ance, in he
wo-dimensional case:
Applying he spec al decomposi ion heo em yields:
which esul s in
=Cu
i
=𝜆
i
u
i
,∀i=j
. The eigen alues and
eigen ec o s a e ob ained by sol ing
de (C−𝜆I)u=0
.
Wi hou loss o gene ali y, le ’s assume a wo-dimensional
co a iance ma ix C, om whe e he eigen alues a e gi en
by:
which esul s in:
and since in a 2x2 ma ix he solu ion o he de e minan is
gi en by he p oduc o he diagonal e ms minus he p oduc
o he an i-diagonal ones, one ge s:
inally, sol ing he second deg ee equa ion:
Hence, he wo solu ions a e:
Gene alizing o he
i=1, 2, ..., n
-dimensional ma ix C
yields:
◻
B Plo s o minimum spanning ee
See Figs.3, 4.
C
=
[
𝜎10
0𝜎
2]
C
=UΛUT=
∑
0
<j<k
𝜆jujuT
j
de
(C−𝜆I)=de
([
𝜎10
0𝜎
2]
−
[
𝜆0
0𝜆
])
=
0
de [
𝜎1−𝜆0
0𝜎
2
−𝜆
]
=
0
(
𝜎
1
−𝜆)(𝜎
2
−𝜆)=𝜆
2
−𝜆(𝜎
1
+𝜎
2
)+𝜎
1
𝜎
2
=
0.
𝜆
1,2 =
𝜎1+𝜎2±
√
(𝜎1+𝜎2)2−4𝜎1𝜎2
2
=
𝜎1+𝜎2±�𝜎2
1+𝜎2
2+2𝜎1𝜎2−4𝜎1𝜎2
2
=
𝜎1+𝜎2±
√
(𝜎1−𝜎2)2
2
.
𝜆
1=
𝜎
1
+𝜎
2
+𝜎
1
−𝜎
2
2
=𝜎1∨𝜆2=
𝜎
1
+𝜎
2
−𝜎
1
+𝜎
2
2
=𝜎2
.
𝜆i=𝜎i,∀i∈(1, n).