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Coh
ere
nt a
lloca
tion o
f risk cap
ital ∗
Mich
el D
enault
†
´ Eco
le d
es H
.E.C
. (Montr´e
al)
January
20
01
Orig
inal v
ersio
n:S
ep
tem
ber 1
999
Ab
stract
The a
lloca
tion p
roble
m ste
ms fro
m th
e d
iversifi
catio
n e
ffect o
bse
rved
in risk m
easu
rem
ents o
f financia
l portfo
lios: th
e su
m o
f the
“risks”
of m
any
portfo
lios
is la
rger th
an
the
“risk” o
f the
sum
of th
e p
ortfo
lios. T
he
allo
catio
n p
roble
m is
to a
pportio
n th
is
div
ersifi
catio
n a
dvanta
ge
to th
e p
ortfo
lios
in a
fair m
anner, y
ield
ing, fo
r each
portfo
lio, a
risk a
ppra
isal th
at a
ccounts
for
div
ersifi
catio
n.
Our a
ppro
ach
is a
xio
matic, in
the
sense
that w
e fi
rst arg
ue
for th
e n
ece
ssary
pro
pertie
s o
f an
allo
catio
n p
rincip
le, a
nd
then
consid
er p
rincip
les th
at fu
lfill th
e p
ropertie
s. Importa
nt re
sults
from
the
are
a o
f gam
e th
eory
find
a d
irect a
pplica
tion. O
ur m
ain
re
sult is th
at th
e A
um
ann-S
haple
y v
alu
e is b
oth
a co
here
nt a
nd p
ractica
l appro
ach
to fi
nancia
l risk allo
catio
n.
Keyw
ord
s: a
lloca
tion
of ca
pita
l, coh
ere
nt risk
measu
re, risk-a
dju
sted
perfo
rman
ce m
easu
re; g
am
e th
eory
, fuzzy
gam
es,
Sh
ap
ley v
alu
e, A
um
an
n-S
hap
ley p
rices.
∗The a
uth
or e
xpre
sses sp
ecia
l than
ks to F. D
elb
aen, w
ho p
rovid
ed
both
the in
itial in
spira
tion
for th
is work
and
gen
ero
us su
bse
qu
en
t ideas a
nd
ad
vice
, an
d to
Ph. A
rtzner fo
r dra
win
g h
is atte
ntio
n to
Au
bin
’s litera
ture
on fu
zzy g
am
es. D
iscussio
ns w
ith P. E
mbre
chts, H
.-J. L¨
uth
i, D
. Stra
um
ann
, an
d S
. Bern
eg
ger h
ave
been
most fru
itful. Fin
ally
, he
gra
tefu
lly a
cknow
led
ges th
e fi
nan
cial su
pport o
f both
RiskLa
b (S
witze
rland
) and
the
S.S
.H.R
.C. (C
anad
a)
´´
†Assista
nt Pro
fesso
r, Eco
le d
es H
aute
s Etu
des C
om
mercia
les, 3
00
0 ch
. de la
Cˆ
ote
-Sain
te-C
ath
erin
e, M
ontr´
eal, C
an
ada, H
3T 2
A7
; mich
el.d
enau
; (51
4) 3
40
-71
61
1
Intro
ductio
n
Th
e th
em
e o
f this
pap
er is
the
sharin
g o
f costs
betw
een
the
con
stituen
ts o
f a fi
rm. W
e ca
ll this
sharin
g “a
lloca
tion”, a
s it is
assu
med
that a
hig
her a
uth
ority
exists w
ithin
the fi
rm, w
hich
has a
n in
tere
st in u
nila
tera
lly d
ivid
ing
the fi
rm’s co
sts betw
een
the
con
stituen
ts. We w
ill refe
r to th
e co
nstitu
en
ts as p
ortfo
lios, b
ut b
usin
ess u
nits co
uld
just a
s well b
e u
nd
ersto
od
.
As a
n in
sura
nce
ag
ain
st the
un
certa
inty
of th
e n
et w
orth
s (or e
qu
ivale
ntly
, the
pro
fits) o
f the
portfo
lios, th
e fi
rm co
uld
well,
and
wou
ld o
ften
be re
gu
late
d to
, hold
an
am
ou
nt o
f riskless in
vestm
en
ts. We w
ill call th
is bu
ffer, th
e risk ca
pita
l of th
e fi
rm. Fro
m
a fi
nan
cial p
ersp
ectiv
e, h
old
ing
an
am
ou
nt o
f mon
ey d
orm
an
t, i.e. in
extre
mely
low
risk, low
retu
rn m
on
ey in
strum
ents, is se
en
as a
bu
rden
. It is th
ere
fore
natu
ral to
look
for a
fair a
lloca
tion
of th
at b
urd
en
betw
een
the
con
stituen
ts, esp
ecia
lly w
hen
the
allo
catio
n p
rovid
es
a b
asis
for p
erfo
rman
ce co
mp
ariso
ns
of th
e co
nstitu
en
ts b
etw
een
them
selv
es
(for e
xam
ple
in a
rora
c
ap
pro
ach
).
Th
e p
rob
lem
of a
lloca
tion
is inte
restin
g a
nd
non
-trivia
l, beca
use
the
sum
of th
e risk
cap
itals o
f each
con
stituen
t, is usu
ally
larg
er th
an
the
risk ca
pita
l of th
e fi
rm ta
ken
as a
wh
ole
. Th
at is, th
ere
is a d
eclin
e in
tota
l costs to
be
exp
ecte
d b
y p
oolin
g th
e
activ
ities
of th
e fi
rm, a
nd
this
ad
van
tag
e n
eed
s to
be
share
d fa
irly b
etw
een
the
con
stituen
ts. We
stress
fairn
ess, a
s a
ll
con
stituen
ts are
from
the
sam
e fi
rm, a
nd
non
e sh
ou
ld re
ceiv
e p
refe
ren
tial tre
atm
en
t for th
e p
urp
ose
of th
is allo
catio
n e
xercise
.
In th
at se
nse
, the
risk ca
pita
l of a
con
stituen
t, min
us its a
lloca
ted
share
of th
e d
iversifi
catio
n a
dvan
tag
e, is e
ffectiv
ely
a fi
rm-
inte
rnal risk m
easu
re.
Th
e a
lloca
tion
exe
rcise is b
asica
lly p
erfo
rmed
for co
mp
ariso
n p
urp
ose
s: know
ing
the p
rofit g
en
era
ted
and
the risk ta
ken
by
the co
mp
on
en
ts of th
e fi
rm, a
llow
s for a
mu
ch w
iser co
mp
ariso
n th
an
know
ing
on
ly o
f pro
fits. T
his id
ea o
f a rich
er in
form
atio
n
set u
nd
erlie
s the p
op
ula
r con
cep
ts of risk-a
dju
sted
perfo
rmance
measu
res (ra
pm
) and
retu
rn o
n risk-a
dju
sted
cap
ital (ro
rac).
Our a
pp
roach
of th
e a
lloca
tion
pro
ble
m is
axio
matic, in
a se
nse
that is
very
simila
r to th
e a
pp
roach
take
n b
y A
rtzner,
Delb
aen
, Eb
er a
nd
Heath
[3]. Ju
st as th
ey
defin
ed
a se
t of n
ece
ssary
“good
qu
alitie
s” of a
risk m
easu
re, w
e su
gg
est a
set o
f
pro
pertie
s to b
e fu
lfille
d b
y a
fair risk ca
pita
l allo
catio
n p
rincip
le. T
heir se
t of a
xio
ms d
efin
es th
e co
here
nce
of risk m
easu
res, o
ur
set o
f axio
ms d
efines th
e co
here
nce
of risk
cap
ital a
lloca
tion
prin
ciple
s. (Incid
en
tally
, the sta
rting
poin
t of o
ur d
evelo
pm
en
t, the
risk cap
itals o
f the fi
rm a
nd
its con
stituents, is a
cohere
nt risk m
easu
re)
We m
ake
, thro
ug
hou
t this a
rticle, lib
era
l use
of th
e co
nce
pts a
nd
resu
lts of g
am
e th
eory. A
s we h
op
e to
con
vin
ce th
e re
ad
er,
gam
e th
eory
pro
vid
es a
n e
xcelle
nt fra
mew
ork
on
which
to ca
st the
allo
catio
n p
rob
lem
, an
d a
elo
qu
en
t lan
gu
ag
e to
discu
ss it.
Th
ere
is an
imp
ressiv
e a
mou
nt o
f litera
ture
on
the
allo
catio
n p
rob
lem
with
in th
e a
rea
of g
am
e th
eory
, with
ap
plica
tion
s ran
gin
g
from
tele
ph
on
e b
illing
to a
irport la
nd
ing
fees a
nd
to w
ate
r treatm
en
t costs. T
he
main
sou
rces fo
r this a
rticle a
re th
e se
min
al
article
s of S
hap
ley [2
8] a
nd
[30
] on
on
e h
an
d; a
nd
the
book o
f Au
bin
[5], th
e a
rticles o
f Bille
ra a
nd
Heath
[9]), a
nd
Mirm
an
an
d
Taum
an
[18
], on
the o
ther h
and
.
At a
more
gen
era
l level, th
e in
tere
sted
read
er m
ay co
nsu
lt a g
am
e th
eory
refe
ren
ce a
s the n
ice O
sborn
e a
nd
Ru
bin
stein
[21
],
the
ed
ited
book o
f Roth
[24
] (inclu
din
g th
e su
rvey o
f Tau
man
[32
]), or th
e su
rvey a
rticle o
f You
ng
[33
], wh
ich co
nta
in le
gio
ns o
f
refe
ren
ces o
n th
e su
bje
ct.
Th
e a
rticle is d
ivid
ed
as fo
llow
s. We
reca
ll the
conce
pt o
f coh
ere
nt risk
measu
re in
the
next se
ction
. Sectio
n 3
pre
sen
ts the
idea o
f the co
here
nce
in a
lloca
tion
. Gam
e th
eory
con
cep
ts are
intro
du
ced
in se
ction
4, w
here
the risk
cap
ital a
lloca
tion
pro
ble
m
is mod
elle
d a
s a g
am
e b
etw
een
portfo
lios. W
e tu
rn in
sectio
n 5
to fu
zzy g
am
es, a
nd
the
coh
ere
nce
of a
lloca
tion
is exte
nd
ed
to
that se
tting
. This is w
here
the A
um
an
n-S
hap
ley v
alu
e e
merg
es a
s a m
ost a
ttractiv
e a
lloca
tion
prin
ciple
. We tre
at th
e q
uestio
n o
f
the
non
-neg
ativ
ity o
f allo
catio
ns in
sectio
n 6
. Th
e fi
nal se
ction
is devote
d to
a “to
y e
xam
ple
” of a
coh
ere
nt risk
measu
re b
ase
d
on
the m
arg
in ru
les o
f the S
EC
, and
to a
lloca
tion
s that a
rise w
hile
usin
g th
at m
easu
re.
Rem
ark: B
ew
are
that tw
o co
nce
pts o
f coh
ere
nce
are
discu
ssed
in th
is pap
er: th
e co
here
nce
of risk
measu
res w
as in
trod
uce
d
in [3
], bu
t is use
d it h
ere
as w
ell; th
e co
here
nce
of a
lloca
tion
s is intro
du
ced
here
.
Risk m
easu
re a
nd risk ca
pita
l
In th
is pap
er, w
e fo
llow
Artzn
er, D
elb
aen
, Eb
er a
nd
Heath
[3] in
rela
ting
the
risk o
f a fi
rm to
the
un
certa
inty
of its fu
ture
worth
.
Th
e d
an
ger, in
here
nt to
the id
ea o
f risk, is that th
e fi
rm’s w
orth
reach
such
a lo
w n
et w
orth
at a
poin
t in th
e fu
ture
, that it m
ust
stop
its activ
ities. R
isk is then
defined
as a
ran
dom
varia
ble
X re
pre
sen
ting
a fi
rm’s n
et w
orth
at a
specifi
ed
poin
t of th
e fu
ture
.
A risk
measu
re ρ
qu
an
tifies th
e le
vel o
f risk. Sp
ecifi
cally
, it is a m
ap
pin
g fro
m a
set o
f ran
dom
varia
ble
s (risks) to th
e re
al
nu
mb
ers: ρ
(X) is th
e a
mou
nt o
f a n
um
´era
ire (e
.g. ca
sh d
olla
rs) wh
ich, a
dd
ed
to th
e fi
rm’s a
ssets, e
nsu
res th
at its fu
ture
worth
be a
ccep
tab
le to
the re
gu
lato
r, the ch
ief risk o
ffice
r or o
thers. (Fo
r a d
iscussio
n o
f acce
pta
ble
worth
s, see [3
]) Cle
arly
, the h
eftie
r
the
req
uire
d sa
fety
net is, th
e riskie
r the
firm
is perce
ived
. We
call ρ
(X) th
e risk
cap
ital o
f the
firm
. Th
e risk
cap
ital a
lloca
tion
pro
ble
m is to
allo
cate
the a
mou
nt o
f risk ρ(X
) betw
een th
e p
ortfo
lios o
f the fi
rm.
We
will a
ssum
e th
at a
ll rand
om
varia
ble
s a
re d
efin
ed
on
a fi
xed
pro
bab
ility sp
ace
(Ω, A
, P). B
y L
∞(Ω, A
, P), w
e m
ean
the
space
of b
ou
nd
ed
rand
om
varia
ble
s; we
assu
me
that ρ
is o
nly
defin
ed
on
that sp
ace
. Th
e re
ad
er w
ho
wish
es to
do
so ca
n
gen
era
lize th
e re
sults a
lon
g th
e lin
es o
f [12
].
In th
eir p
ap
ers, A
rtzner, D
elb
aen
, Eb
er a
nd
Heath
([3], [2
]) have
sug
geste
d a
set o
f pro
pertie
s th
at risk
measu
res sh
ou
ld
satisfy
, thu
s defin
ing
the co
nce
pt o
f coh
ere
nt m
easu
res o
f risk1:
Defin
ition
1 A
risk measu
re ρ
: L
∞ → R
is coh
ere
nt if it sa
tisfies th
e fo
llow
ing
pro
pertie
s:
Su
bad
ditiv
ity Fo
r all b
ou
nd
ed
ran
dom
varia
ble
s X a
nd
Y , ρ
(X +
Y ) ≤
ρ(X
)+ ρ
(Y )
Mon
oto
nicity
For a
ll bou
nd
ed
ran
dom
varia
ble
s X, Y
such
that X
≤ Y
2 , ρ
(X) ≥
ρ(Y
)
Positiv
e h
om
og
en
eity
For a
ll λ ≥
0 a
nd
bou
nd
ed
ran
dom
varia
ble
X, ρ
(λX
)= λ
ρ(X
)
1On th
e to
pic, se
e a
lso A
rtzner’s [1
], and D
elb
aen
’s [12
] an
d [1
3]
2The re
latio
n X
≤ Y
betw
een tw
o ra
nd
om
varia
ble
s is take
n to
mean
X(ω
) ≤ Y
(ω) fo
r alm
ost a
ll ω ∈
Ω, in
a p
rob
ability
space
(Ω, F ,P
).
Transla
tion
invaria
nce
For a
ll α ∈
R a
nd
bou
nd
ed
ran
dom
varia
ble
X,
ρ(X
+ α
rf )=
ρ(X
) −α
where
rf is th
e p
rice, a
t som
e p
oin
t in th
e fu
ture
, of a
refe
ren
ce, riskle
ss
investm
en
t whose
price
is 1 to
day.
Th
e p
rop
ertie
s th
at d
efin
e co
here
nt risk
measu
res a
re to
be
un
dersto
od
as n
ece
ssary
con
ditio
ns fo
r a risk
measu
re to
be
reaso
nab
le. Le
t us b
riefly ju
stify th
em
. Su
bad
ditiv
ity re
flects th
e d
iversifi
catio
n o
f portfo
lios, o
r that “a
merg
er d
oes n
ot cre
ate
extra
risk” [3
, p.2
09
]. Mon
oto
nicity
says th
at if a
portfo
lio Y
is a
lways
worth
more
than
X, th
en
Y ca
nn
ot b
e riskie
r than
X.
Hom
og
en
eity
is a
limit ca
se o
f sub
ad
ditiv
ity, re
pre
sen
ting
what h
ap
pen
s w
hen
there
is p
recise
ly n
o d
iversifi
catio
n e
ffect.
Transla
tion
invaria
nce
is a n
atu
ral re
qu
irem
en
t, giv
en
the m
ean
ing
of th
e risk m
easu
re g
iven
ab
ove a
nd
its rela
tion
to th
e n
um
´
era
ire.
In th
is pap
er, w
e w
ill not b
e co
nce
rned
with
specifi
c risk
measu
res, u
ntil o
ur e
xam
ple
of se
ction
7; w
e h
ow
ever a
ssum
e a
ll
risk measu
res to
be co
here
nt.
Cohere
nce
of th
e a
lloca
tion
prin
ciple
An
allo
catio
n p
rincip
le is a
solu
tion
to th
e risk
cap
ital a
lloca
tion
pro
ble
m. W
e su
gg
est in
this se
ction
a se
t of a
xio
ms, w
hich
we
arg
ue
are
nece
ssary
pro
pertie
s of a
“reaso
nab
le” a
lloca
tion
prin
ciple
. We
will ca
ll coh
ere
nt a
n a
lloca
tion
prin
ciple
that sa
tisfies
the se
t of a
xio
ms. T
he fo
llow
ing
defin
ition
s are
use
d:
•X
i, i ∈
1,2
,...,n
, is a b
oun
ded
ran
dom
varia
ble
rep
rese
ntin
g th
e n
et w
orth
at tim
e T
of th
e i th
portfo
lio o
f a fi
rm. W
e a
ssum
e
that th
e n
th p
ortfo
lio
is a riskle
ss instru
men
t with
net w
orth
at tim
e T
eq
ual to
Xn
= α
rf , w
here
rf th
e tim
e T
price
of a
riskless in
strum
en
t with
price
1
tod
ay.
•X
, the b
ou
nd
ed
ran
dom
varia
ble
rep
rese
ntin
g th
e fi
rm’s n
et w
orth
at so
me
n
poin
t in th
e fu
ture
T, is d
efin
ed
as X
Xi.
i=1
•N
is the se
t of a
ll portfo
lios o
f the fi
rm.
•A
is the se
t of risk ca
pita
l allo
catio
n p
rob
lem
s: pairs (N
,ρ) co
mp
ose
d o
f a se
t of n
portfo
lios a
nd
a co
here
nt risk m
easu
re ρ
.
•K
= ρ
(X) is th
e risk ca
pita
l of th
e fi
rm.
We ca
n n
ow
define:
Defin
ition
2 A
n a
lloca
tion
prin
ciple
is a fu
nctio
n Π
: A R
n th
at m
ap
s
→
each
allo
catio
n p
rob
lem
(N, ρ
) into
a u
niq
ue a
lloca
tion
:
Π1(N
, ρ) K
1
Π:(N
, ρ)
−→
Π2(N
, ρ)
. .
=
K2
. .
such
that K
i = ρ
(X).
..
i∈N
Πn(N
, ρ) K
n
Th
e co
nd
ition
en
sure
s th
at th
e risk
cap
ital is
fully
allo
cate
d. T
he
Ki–n
ota
tion
is u
sed
wh
en
the
arg
um
en
ts a
re cle
ar fro
m th
e
con
text.
Defin
ition
3 A
n a
lloca
tion
prin
ciple
Π is
coh
ere
nt if fo
r every
allo
catio
n p
rob
lem
(N, ρ
), the
allo
catio
n Π
(N, ρ
)satisfi
es th
e th
ree
pro
pertie
s:
1) N
o u
nd
ercu
t
∀ M
⊆ N
,
Ki ≤
ρX
i i∈M
i∈M
2) S
ym
metry
If by jo
inin
g a
ny su
bse
t M ⊆
N\
i, j, p
ortfo
lios i a
nd
j both
make
the sa
me co
ntrib
utio
n to
the risk ca
pita
l, then
Ki =
Kj .
3) R
iskless a
lloca
tion
Kn
= ρ
(αrf )=
−α
Reca
ll that th
e n
th p
ortfo
lio is a
riskless in
strum
ent.
Furth
erm
ore
, we ca
ll non
-neg
ativ
e co
here
nt a
lloca
tion
a co
here
nt a
lloca
tion
wh
ich sa
tisfies K
i ≥ 0
, ∀i ∈
N.
It is o
ur p
rop
ositio
n th
at th
e th
ree
axio
ms o
f Definitio
n 3
are
nece
ssary
con
ditio
ns o
f the
fairn
ess, a
nd
thus cre
dib
ility, o
f
allo
catio
n p
rincip
les. In
that se
nse
, coh
ere
nce
is a y
ard
stick by w
hich
allo
catio
n p
rincip
les ca
n b
e e
valu
ate
d.
Th
e p
rop
ertie
s can
be
justifi
ed
as fo
llow
s. Th
e “n
o u
nd
ercu
t” pro
perty
en
sure
s that n
o p
ortfo
lio ca
n u
nd
ercu
t the
pro
pose
d
allo
catio
n: a
n u
nd
ercu
t occu
rs wh
en
a p
ortfo
lio’s a
lloca
tion
is hig
her th
an
the
am
ou
nt o
f risk ca
pita
l it wou
ld fa
ce a
s an
entity
sep
ara
te fro
m th
e fi
rm. G
iven
sub
ad
ditiv
ity, th
e ra
tion
ale
is simp
le. U
pon
a p
ortfo
lio jo
inin
g th
e fi
rm (o
r any su
bse
t there
of), th
e
tota
l risk ca
pita
l incre
ase
s b
y n
o m
ore
than
the
portfo
lio’s
ow
n risk
cap
ital: in
all fa
irness, th
at p
ortfo
lio ca
nn
ot ju
stifiab
ly b
e
allo
cate
d m
ore
risk ca
pita
l than
it can
possib
ly h
ave
bro
ug
ht to
the
firm
. Th
e p
rop
erty
also
en
sure
s that co
alitio
ns o
f portfo
lios
cann
ot u
nd
ercu
t, with
the
sam
e ra
tion
ale
. Th
e sy
mm
etry
pro
perty
en
sure
s th
at a
portfo
lio’s
allo
catio
n d
ep
en
ds o
nly
on
its
con
tribu
tion
to risk
with
in th
e fi
rm, a
nd
noth
ing
else
. Acco
rdin
g to
the
riskless a
lloca
tion
axio
m, a
riskless p
ortfo
lio sh
ou
ld b
e
allo
cate
d e
xactly
its risk m
easu
re, w
hich
incid
enta
lly w
ill be n
eg
ativ
e. It a
lso m
ean
s that, a
ll oth
er th
ing
s bein
g e
qual, a
portfo
lio
that in
crease
s its cash
positio
n, sh
ould
see its a
lloca
ted
cap
ital d
ecre
ase
by th
e sa
me a
mou
nt.
Gam
e th
eory
an
d a
lloca
tion
to a
tom
ic pla
yers
Gam
e th
eory
is th
e stu
dy
of situ
atio
ns w
here
pla
yers
ad
op
t vario
us stra
teg
ies to
best a
ttain
their in
div
idu
al g
oals. Fo
r now
,
pla
yers w
ill be a
tom
ic, mean
ing
that fra
ction
s of p
layers a
re co
nsid
ere
d se
nse
less. W
e w
ill focu
s here
on co
alitio
nal g
am
es:
Defin
ition
4 A
coalitio
nal g
am
e (N
, c) con
sists of:
%•
a fi
nite
set N
of n
pla
yers, a
nd
%•
a co
st fun
ction
c that a
ssocia
tes a
real n
um
ber c(S
) to e
ach
sub
set S
of N
(calle
d a
coalitio
n).
We d
en
ote
by G
the se
t of g
am
es w
ith n
pla
yers.
Th
e g
oal o
f each
pla
yer is to
min
imize
the co
st she in
curs, a
nd
her stra
teg
ies co
nsist o
f acce
ptin
g o
r not to
take
part in
coalitio
ns
(inclu
din
g th
e co
alitio
n o
f all p
layers).
In th
e lite
ratu
re, th
e co
st fun
ction
is usu
ally
assu
med
to b
e su
bad
ditiv
e: c(S
∪ T
) ≤ c(S
)+ c(T
) for a
ll sub
sets
S a
nd
T o
f N
with
em
pty
inte
rsectio
n; a
n a
ssum
ptio
n w
hich
we m
ake
as w
ell.
One o
f the m
ain
qu
estio
ns ta
ckled
in co
alitio
nal g
am
es, is th
e a
lloca
tion
of th
e co
st c(N) b
etw
een
all p
layers; th
is qu
estio
n is
form
alize
d b
y th
e co
nce
pt o
f valu
e:
Defin
ition
5 A
valu
e is a
fun
ction Φ
: G R
n th
at m
ap
s each
gam
e (N
, c)
→
into
a u
niq
ue a
lloca
tion
:
Φ1(N
, c) K1
Φ:(N
, c)
−→
Φ2(N
, c) . . .
=
K2
. . .
where
Ki =
c(N)
i∈N
Φn(N
, c) Kn
Ag
ain
, the
Ki–n
ota
tion
can
be
use
d w
hen
the
arg
um
en
ts are
clear fro
m th
e co
nte
xt, a
nd
when
it is also
clear w
heth
er w
e m
ean
Πi(N
, ρ)o
rΦi(N
, c).
4.1
The co
re o
f a g
am
e
Giv
en
the
sub
ad
ditiv
ity o
f c, the
pla
yers
of a
gam
e h
ave
an
ince
ntiv
e to
form
the
larg
est co
alitio
n N
, since
this
brin
gs a
n
imp
rovem
en
t of th
e to
tal co
st, wh
en
com
pare
d w
ith th
e su
m o
f their in
div
idu
al co
sts. Th
ey n
eed
on
ly fi
nd
a w
ay to
allo
cate
the
cost c(N
) of th
e fu
ll coalitio
n N
, betw
een
them
selv
es; b
ut in
doin
g so
, pla
yers still try
to m
inim
ize th
eir o
wn
share
of th
e b
urd
en
.
Pla
yer i w
ill even
thre
ate
n to
leave th
e co
alitio
n N
if she is a
lloca
ted
a sh
are
Ki o
f the to
tal co
st
that is h
igh
er th
at h
er o
wn
ind
ivid
ual co
st c( i
). Sim
ilar th
reats m
ay co
me fro
m co
alitio
ns S
⊆ N
:if Ki e
xceed
s c(S) th
en
every
pla
yer i in
S co
uld
i∈S
carry
an
allo
cate
d co
st low
er th
an
his cu
rren
t Ki,if S
sep
ara
ted
from
N.
Th
e se
t of a
lloca
tions th
at d
o n
ot a
llow
such
thre
at fro
m a
ny p
layer n
or co
alitio
n is ca
lled
the co
re:
Defin
ition
6 T
he co
re o
f a co
alitio
nal g
am
e (N
, c) is the se
t of a
lloca
tion
s
K ∈
Rn
for w
hich
i∈S
Ki ≤
c(S) fo
r all co
alitio
ns S
⊆ N
.
A co
nd
ition
for th
e co
re to
be
non-e
mp
ty is
the
Bond
are
va-S
hap
ley
theore
m. Le
t C b
e th
e se
t of a
ll coalitio
ns o
f N, le
t us
den
ote
by 1
S ∈
Rn
the ch
ara
cteristic v
ecto
r of th
e co
alitio
n S
:
(1S
) i = 1
if i ∈ S
0 o
therw
ise
A b
ala
nce
d co
llectio
n o
f weig
hts is a
colle
ction
of |C
that S
∈C
λS
1S
=1
N . A
gam
e is b
ala
nce
d if S
∈C
λS
c(S) ≥
c(N) fo
r all b
ala
nce
d
colle
ction
s of w
eig
hts. T
hen
:
| nu
mb
ers λ
S in
[0, 1
] such
Th
eore
m 1
(Bon
dare
va-S
hap
ley, [1
1], [2
9]) A
coalitio
nal g
am
e h
as a
non
-em
pty
core
if an
d o
nly
if it is bala
nce
d.
Pro
of: se
e e
.g. [2
1].
4.2
The S
hap
ley v
alu
e
Th
e S
hap
ley v
alu
e w
as in
trod
uce
d b
y L. S
hap
ley [2
8] a
nd
has e
ver sin
ce re
ceiv
ed
a co
nsid
era
ble
am
ou
nt o
f inte
rest (se
e [2
4]).
We u
se th
e a
bb
revia
tion
∆i(S
)= c(S
∪ i) −
c(S) fo
r an
y se
t S ⊂
N, i
∈
S. Tw
o p
layers i a
nd
j are
inte
rchan
geab
le in
(N, c) if e
ither o
ne
make
s the
sam
e co
ntrib
utio
n to
an
y co
alitio
n S
it may jo
in, th
at
con
tain
s neith
er i n
or j:∆
i(S)=
∆j (S
) for e
ach
S ⊂
N a
nd
i, j ∈ S
. A p
layer i is a
du
mm
y if it
brin
gs th
e co
ntrib
utio
n c(i) to
an
y co
alitio
n S
that d
oes n
ot co
nta
in it a
lread
y: ∆i(S
)= c(i). W
e n
eed
to d
efin
e th
e th
ree p
rop
ertie
s:
Sym
metry
If pla
yers i a
nd
j are
inte
rchan
geab
le, th
en
Φ(N
, c) i =Φ
(N, c) j
Du
mm
y p
layer Fo
r a d
um
my p
layer, Φ
(N, c) i =
c(i)
Ad
ditiv
ity o
ver g
am
es Fo
r two g
am
es (N
, c1) a
nd
(N, c
2), Φ(N
, c1
+ c
2)=
Φ(N
, c1)+
Φ(N
, c2), w
here
the g
am
e (N
, c1+
c2) is d
efin
ed
by (c
1+c
2)(S)=
c1(S
)+ c
2(S) fo
r all S
⊆ N
.
Th
e ra
tionale
of th
ese
pro
pertie
s will b
e d
iscusse
d in
the n
ext se
ction
. Th
e a
xio
matic d
efin
ition
of th
e S
hap
ley v
alu
e is th
en
:
Defin
ition
7 ([2
8]) T
he
Sh
ap
ley v
alu
e is th
e o
nly
valu
e th
at sa
tisfies th
e p
rop
ertie
s of sy
mm
etry, d
um
my
pla
yer, a
nd
ad
ditiv
ity
over g
am
es.
Let u
s now
brin
g to
geth
er th
e co
re a
nd
the S
hap
ley v
alu
e: w
hen
does th
e S
hap
ley v
alu
e y
ield
allo
catio
ns th
at a
re in
the co
re
of th
e g
am
e ?
Th
e o
nly
perta
inin
g re
sults
to o
ur kn
ow
led
ge
are
that o
f Sh
ap
ley
[30
] an
d A
ub
in [5
]. Th
e fo
rmer in
volv
es th
e
pro
perty
of stro
ng
sub
ad
ditiv
ity:
Defin
ition
8 A
coalitio
nal g
am
e is stro
ng
ly su
bad
ditiv
e if it is b
ase
d o
n a
stron
gly
sub
ad
ditiv
e3
cost fu
nctio
n:
c(S)+
c(T) ≥
c(S ∪
T )+
c(S ∩
T )
for a
ll coalitio
ns S
⊆N
and
T ⊆
N.
Th
eore
m 2
([30
]) If a g
am
e (N
, c) is stron
gly
sub
ad
ditiv
e, its co
re co
nta
ins th
e S
hap
ley v
alu
e.
Th
e se
con
d co
nd
ition
that e
nsu
res th
at th
e S
hap
ley v
alu
e is in
the co
re, is:
Th
eore
m 3
([5]) If fo
r all co
alitio
ns S
, | S|≥
2,
(−1
) | S|−
| T | c(T
) ≤ 0
T ⊆
S
then
the co
re co
nta
ins th
e S
hap
ley v
alu
e.
Th
e im
plica
tion
s of th
ese
two re
sults a
re d
iscusse
d in
the n
ext se
ction
.
Let u
s en
d th
is sectio
n w
ith th
e a
lgeb
raic d
efin
ition
of th
e S
hap
ley v
alu
e, w
hich
pro
vid
es b
oth
an
inte
rpre
tatio
n (se
e [2
8] o
r
[24
]), an
d a
n e
xp
licit com
puta
tion
al a
pp
roach
.
Defin
ition
9 T
he S
hap
ley v
alu
e K
Sh
for th
e g
am
e (N
, c) is defi
ned
as:
KS
h ( s −
1)!(
n
−
s )!
,i ∈N
=
i
n! c (S
) −c(S
\i
)
S∈
Ci
where
s = S
, and
Ci re
pre
sents a
ll coalitio
ns o
f N th
at co
nta
in i.
||
Note
that th
is req
uire
s the e
valu
atio
n o
f c fo
r each
of th
e 2
n p
ossib
le co
alitio
ns, u
nle
ss the p
rob
lem
has so
me sp
ecifi
c structu
re.
Dep
en
din
g o
n w
hat c is, th
is task m
ay b
eco
me im
possib
ly lo
ng
, even
for m
od
era
te n
.
3By d
efin
ition, a
strong
ly su
bad
ditiv
e se
t functio
n is su
bad
ditiv
e. W
e fo
llow
Shap
ley [3
0] in
ou
r term
inolo
gy; n
ote
that h
e ca
lls con
vex, a
functio
n sa
tisfyin
g
the re
verse
rela
tion
of stro
ng su
bad
ditiv
ity.
4.3
Risk ca
pita
l allo
catio
ns a
nd g
am
es
Cle
arly
, we
inte
nd
to m
od
el risk
cap
ital a
lloca
tion
pro
ble
ms a
s coalitio
nal g
am
es. W
e ca
n a
ssocia
te th
e p
ortfo
lios o
f a fi
rm w
ith
the p
layers o
f a g
am
e, a
nd
the risk m
easu
re ρ
with
the co
st fun
ction c :
c(S)
ρX
i for S
⊆N
(1)
i∈S
Allo
catio
n p
rincip
les n
atu
rally
beco
me v
alu
es.
Note
that g
iven
(1), ρ
bein
g co
here
nt a
nd
thu
s sub
ad
ditiv
e in
the se
nse
ρ(X
+ Y
) ≤ ρ
(X)+
ρ(Y
) of D
efin
ition
1, im
plie
s that c
is sub
ad
ditiv
e in
the se
nse
c(S ∪
T ) ≤
c(S)+
c(T ) g
iven
ab
ove.
Th
e co
re A
lloca
tion
s satisfy
ing
the “n
o u
nd
ercu
t” pro
perty
lie in
the co
re o
f the g
am
e, a
nd
if non
e d
oes, th
e co
re is e
mp
ty. There
is only
a in
terp
reta
tion
al d
istinctio
n b
etw
een
the tw
o co
nce
pts: w
hile
a “re
al” p
layer ca
n th
reate
n to
leave th
e fu
ll coalitio
n N
, a
portfo
lio ca
nn
ot w
alk a
way fro
m a
bank. H
ow
ever, if th
e a
lloca
tion
is to b
e fa
ir, un
dercu
tting
shou
ld b
e a
void
ed
. Ag
ain
, this h
old
s
also
for co
alitio
ns o
f ind
ivid
ual p
layers/p
ortfo
lios.
Th
e n
on
-em
ptin
ess
of th
e co
re is
there
fore
crucia
l to th
e e
xiste
nce
of co
here
nt a
lloca
tion
prin
ciple
s. From
Theore
m 1
, we
have:
Th
eore
m 4
If a risk
cap
ital a
lloca
tion
pro
ble
m is m
od
elle
d a
s a co
alitio
nal g
am
e w
hose
cost fu
nctio
n c is d
efi
ned
with
a co
here
nt
risk measu
re ρ
thro
ug
h (1
), then its co
re is n
on
-em
pty.
Pro
of: Le
t 0 ≤
λS
≤1
for S
∈C
, and
λS1
S =
1N. T
hen
S
∈C
λS
c(S)=
ρλ
SXi i∈
S
S∈
C S
∈C
λSX
i
≥ ρ
i∈S
S∈
C
= ρ
λS X
i
i∈N
S∈
C,S
i
= c(N
)
By T
heore
m 1
, the
core
of th
e g
am
e is n
on
-em
pty.
Th
e S
hap
ley v
alu
e W
ith th
e a
lloca
tion
pro
ble
m m
od
elle
d a
s a g
am
e, th
e
Sh
ap
ley v
alu
e y
ield
s a risk
cap
ital a
lloca
tion
prin
ciple
. Much
more
, it is a co
here
nt a
lloca
tion
prin
ciple
, bu
t for th
e “n
o u
nd
ercu
t”
axio
m. S
ym
metry
is satisfi
ed
by d
efin
ition
. Th
e riskle
ss allo
catio
n a
xio
m o
f Definitio
n 3
is imp
lied
by th
e d
um
my p
layer a
xio
m:
from
ou
r defin
ition
s of se
ction 3
, the re
fere
nce
, riskless in
strum
ent (ca
sh a
nd
eq
uiv
ale
nts) is a
du
mm
y p
layer.
Note
that a
dd
itivity
over g
am
es is a
pro
perty
that th
e S
hap
ley v
alu
e p
osse
sses b
ut th
at is n
ot re
qu
ired
of co
here
nt a
lloca
tion
prin
ciple
. As d
iscusse
d in
sectio
n 5
.3.2
, ad
ditiv
ity co
nflicts w
ith th
e co
here
nce
of th
e risk m
easu
res.
Th
e S
hap
ley
valu
e a
s co
here
nt a
lloca
tion
prin
ciple
From
the
ab
ove, th
e S
hap
ley
valu
e p
rovid
es u
s w
ith a
cohere
nt a
lloca
tion
prin
ciple
if it map
s gam
es to
ele
men
ts of th
e co
re. It is th
e ca
se w
hen
the co
nd
ition
s of e
ither T
heore
ms 2
or 3
are
satisfi
ed
. Th
e
case
of T
heore
m 2
is perh
ap
s disa
pp
oin
ting
, as th
e stro
ng
sub
ad
ditiv
ity o
f c imp
lies a
n o
verly
string
en
t con
ditio
n o
n ρ
:
Th
eore
m 5
Let ρ
be
a p
ositiv
ely
hom
og
eneou
s risk
measu
re, su
ch th
at ρ
(0) =
0. Le
t c b
e d
efi
ned
over th
e se
t of su
bse
ts o
f
ran
dom
varia
ble
s in L
∞, thro
ug
h
c(S)
ρ ( i∈
S X
i). Th
en
if c is stron
gly
sub
ad
ditiv
e, ρ
is linear.
Pro
of: C
on
sider a
ny ra
nd
om
varia
ble
s X, Y, Z
in L
∞. Th
e stro
ng
sub
ad
ditiv
ity o
f c imp
lies
ρ(X
+ Z
)+ ρ
(Y +
Z) ≥
ρ(X
+ Y
+ Z
)+ ρ
(Z) b
ut a
lso
ρ(X
+ Z
)+ ρ
(Y +
Z)=
ρ(X
+(Y
+ Z
) − Y
)+ ρ
(Y +
Z)
ρ(X
+(Y
+ Z
)) + ρ
((Y +
Z) −
Y )
≤
= ρ
(X +
Y +
Z)+
ρ(Z
)
so th
at
ρ(X
+ Z
)+ ρ
(Y +
Z)=
ρ(X
+ Y
+ Z
)+ ρ
(Z)
By ta
king
Z =
0, w
e o
bta
in th
e a
dd
itivity
of ρ
. Th
en
, com
bin
ing
ρ(−
X)=
ρ(X
− X
) − ρ
(X)=
−ρ
(X)
12
with
the p
ositiv
e h
om
og
en
eity
of ρ
, we o
bta
in th
at ρ
is hom
og
eneou
s, an
d th
us lin
ear.
Th
at risks b
e p
lain
ly a
dd
itive is d
ifficu
lt to a
ccep
t, since
it elim
inate
s all p
ossib
ility o
f div
ersifi
catio
n e
ffects.
Un
fortu
nate
ly, th
e co
nd
ition
of T
heore
m 3
is a
lso a
strong
on
e, a
t least in
no
way
imp
lied
by
the
coh
ere
nce
of th
e risk
measu
re ρ
.
We
thu
s fall sh
ort o
f a co
nvin
cing
pro
of o
f the
existe
nce
of co
here
nt a
lloca
tion
s. How
ever, w
e co
nsid
er n
ext a
n o
ther ty
pe
of
coalitio
nal g
am
es, w
here
an
sligh
tly d
iffere
nt d
efin
ition
of co
here
nce
yie
lds m
uch
stron
ger e
xiste
nce
resu
lts.
5 A
lloca
tion
to fra
ction
al p
layers
In th
e p
revio
us se
ction
, portfo
lios w
ere
mod
elle
d a
s pla
yers o
f a g
am
e, e
ach
of th
em
ind
ivisib
le. T
his in
div
isibility
assu
mp
tion
is
not a
natu
ral o
ne, a
s we co
uld
con
sider fra
ction
s of p
ortfo
lios, a
s well a
s coalitio
ns in
volv
ing
fractio
ns o
f portfo
lios. T
he p
urp
ose
of th
is sectio
n is to
exam
ine a
varia
nt o
f the a
lloca
tion g
am
e w
hich
allo
ws d
ivisib
le p
layers.
Th
is tim
e, w
e d
ispen
se w
ith th
e in
itial se
para
tion
of risk-ca
pita
l allo
catio
n p
rob
lem
s a
nd
gam
es, a
nd
intro
duce
the
two
simulta
neou
sly. As b
efo
re, p
layers a
nd
cost fu
nctio
ns a
re u
sed
to m
od
el re
spectiv
ely
portfo
lios a
nd
risk m
easu
res, a
nd
valu
es
giv
e u
s allo
catio
n p
rincip
les.
5.1
Gam
es w
ith fra
ction
al p
layers
Th
e th
eory
of co
alitio
nal g
am
es h
as b
een
exte
nd
ed
to co
ntin
uou
s pla
yers w
ho n
eed
neith
er b
e in
nor o
ut o
f a co
alitio
n, b
ut w
ho
have
a “sca
lab
le” p
rese
nce
. Th
is poin
t of v
iew
seem
s mu
ch le
ss inco
ng
ruou
s if the
pla
yers in
qu
estio
n re
pre
sen
t portfo
lios: a
coalitio
n co
uld
con
sist of six
ty p
erce
nt o
f portfo
lio A
, an
d fi
fty p
erce
nt o
f portfo
lio B
. Of co
urse
, this m
eans “x
perce
nt o
f each
instru
men
t in th
e p
ortfo
lio”.
Au
man
n a
nd
Sh
ap
ley’s b
ook “V
alu
es o
f Non
-Ato
mic G
am
es” [7
] was th
e se
min
al w
ork o
n th
e g
am
e co
nce
pts d
iscusse
d in
this
sectio
n. T
here
, the
inte
rval [0
, 1] re
pre
sents th
e se
t of a
ll pla
yers, a
nd
coalitio
ns a
re m
easu
rab
le su
bin
terv
als (in
fact, e
lem
en
ts
of a
σ-a
lgeb
ra). A
ny su
bin
terv
al co
nta
ins o
ne o
f smalle
r measu
re, so
that th
ere
are
no
ato
ms, i.e
. smalle
st en
tities th
at co
uld
be
calle
d p
layers; h
en
ce th
e n
am
e “n
on
-ato
mic g
am
es”.
Som
e o
f the n
on
-ato
mic g
am
e th
eory
was la
ter re
cast in
a m
ore
intu
itive se
tting
: an
n-d
imensio
nal v
ecto
r λ ∈
Rn
+ re
pre
sen
ts the “le
vel o
f pre
sen
ce” o
f the e
ach
of n
pla
yers in
a co
alitio
n. T
he o
rigin
al p
ap
ers
on
the to
pic a
re A
ub
in’s
[5] a
nd
[6], B
illera
an
d R
aan
an
’s [10
], Bille
ra a
nd
Heath
’s [9], a
nd
Mirm
an a
nd
Tau
man
’s [18
].
Au
bin
calle
d su
ch g
am
es fu
zzy; w
e ca
ll them
(coalitio
nal) g
am
es w
ith fra
ction
al p
layers:
Defin
ition
10
A co
alitio
nal g
am
e w
ith fra
ctional p
layers (N
, Λ, r) co
nsists o
f
• a
fin
ite se
t N o
f pla
yers, w
ith | N
= n
;
|
1.
• a
positiv
e v
ecto
r Λ ∈
Rn
2.
+, e
ach
com
pon
en
t rep
rese
ntin
g fo
r on
e o
f the n
pla
yers h
is full in
volv
em
ent.
%• a
real-v
alu
ed
cost fu
nctio
n r : R
n R
, r : λr(λ
) such
that r(0
) = 0
.
→
→
Pla
yers a
re p
ortfo
lios; th
e v
ecto
r Λ re
pre
sents, fo
r each
portfo
lio, th
e “size
” of th
e p
ortfo
lio, in
a re
fere
nce
unit. (T
he Λ
could
also
rep
rese
nt th
e b
usin
ess v
olu
mes o
f the b
usin
ess u
nits). T
he ra
tio λ
i then
den
ote
s a p
rese
nce
or a
ctivity
Λi
level, fo
r pla
yer/p
ortfo
lio i, so
that a
vecto
r λ ∈
Rn
+ ca
n b
e u
sed
to re
pre
sen
t
a “co
alitio
n o
f parts o
f pla
yers”. W
e still d
en
ote
by X
i the ra
nd
om
varia
ble
of th
e n
et w
orth
of p
ortfo
lio i a
t a fu
ture
time
T ), a
nd
Xn
keep
s its riskless in
strum
en
t defin
ition
of se
ction
3, w
ith tim
e T
net w
orth
eq
ual to
αrf , w
ith α
som
e co
nsta
nt. T
hen
the
cost
fun
ction
r can b
e id
en
tified
with
a risk m
easu
re ρ
thro
ug
h
λi r(λ
)
ρ Λ
i Xi
i∈N
so th
at r(Λ
) = ρ
(N ). B
y e
xte
nsio
n, w
e a
lso ca
ll r(λ) a
risk measu
re. T
he e
xp
ressio
n X
i is the p
er-u
nit fu
ture
net w
orth
of p
ortfo
lio i.
Λi
Th
e d
efinitio
n o
f coh
ere
nt risk m
easu
re (D
efin
ition
1) is a
dap
ted
as:
Defin
ition
11
A risk m
easu
re r is co
here
nt if it sa
tisfies th
e fo
ur p
rop
ertie
s: Su
bad
ditiv
ity4
For a
ll λ∗
and
λ∗
∗ in
Rn
, r(λ∗
+ λ
∗∗) ≤
r(λ∗)+
r(λ∗
∗) M
on
oto
nicity
For a
ll λ∗
and
λ∗
∗ in
Rn
,
λ∗
λ∗
∗
Λii X
i i
Λi X
i
≤⇒
i∈
Ni∈
N
r(λ∗) ≥
r(λ∗
∗)
where
the le
ft-han
d sid
e in
eq
uality
is ag
ain
un
dersto
od
as in
footn
ote
2. D
eg
ree o
ne h
om
og
en
eity
For a
ll λ ∈
Rn, a
nd
for a
ll
γ ∈
R+,
r(γλ)=
γr(λ
)
Transla
tion
invaria
nce
For a
ll λ ∈
Rn
,
r(λ)=
r
λ1
λ2
. . .
λn
−1
0
−
λn
Λn
α
One ca
n ch
eck th
at r is co
here
nt if a
nd
only
if ρ is.
5.2
Coh
ere
nt co
st allo
catio
n to
fractio
nal p
layers
Th
e p
ortfo
lio size
s g
iven
by
Λ a
llow
us
to tre
at a
lloca
tion
s o
n a
per-u
nit
basis. W
e th
us
intro
du
ce a
vecto
r k
∈ R
n, each
com
ponen
t of w
hich
rep
rese
nts th
e p
er u
nit a
lloca
tion
of risk
cap
ital to
each
portfo
lio. T
he
cap
ital a
lloca
ted
to e
ach
portfo
lio is
ob
tain
ed
by a
simp
le H
ad
am
ard
(i.e. co
mp
on
en
t-wise
) pro
du
ct
Λ .∗
k = K
(2)
Let u
s also
defin
e, in
a m
an
ner e
qu
ivale
nt to
the co
nce
pts o
f sectio
n 4
:
4Note
that u
nder d
eg
ree o
ne h
om
og
en
eity
, sub
add
itivity
is eq
uiv
ale
nt to
convexity
r(αλ
∗ +
(1 −
α)λ
∗∗) ≤
αr(λ
∗)+
(1 −
α)r(λ
∗∗)
Defin
ition
12
A fu
zzy v
alu
e is a
map
pin
g a
ssign
ing
to e
ach
coalitio
nal g
am
e w
ith fra
ction
al p
layers (N
, Λ, r) a
un
iqu
e p
er-u
nit
allo
catio
n v
ecto
r
φ1(N
, Λ,r) k
1
φ :(N
, Λ,r)
−→
φ2(N
, Λ,r)
. . .
=
k2
. . .
φn(N
, Λ,r) k
n
with
Λtk =
r(Λ)
(3)
Ag
ain
, we u
se th
e k-n
ota
tion
when
the a
rgu
men
ts are
clear fro
m th
e co
nte
xt.
Cle
arly
, a fu
zzy v
alu
e p
rovid
es u
s with
an
allo
catio
n p
rincip
le, if w
e g
en
era
lize th
e la
tter to
the co
nte
xt o
f div
isible
portfo
lios.
We ca
n n
ow
define th
e co
here
nce
of fu
zzy v
alu
es:
Defin
ition
13
Let r b
e a
coh
ere
nt risk m
easu
re. A
fuzzy
valu
e φ
:(N, Λ
,r) −→
k ∈ R
n
is coh
ere
nt if it sa
tisfies th
e p
rop
ertie
s defi
ned
belo
w, a
nd
if k is an
ele
men
t of th
e fu
zzy co
re:
•A
gg
reg
atio
n in
varia
nce
Su
pp
ose
the risk m
easu
res r a
nd
r¯satisfy
r(λ)=
¯
r(Γλ) fo
r som
e m
× n m
atrix
Γ and
all λ
such
that 0
≤ λ
≤ Λ
then
φ(N
, Λ,r)=
Γtφ
(N, ΓΛ
,r¯)(4
)
1.
•C
on
tinu
ity T
he m
ap
pin
g φ
is contin
uou
s over th
e n
orm
ed
vecto
r space
Mn
of co
ntin
uou
sly d
iffere
ntia
ble
fun
ction
s r
: Rn
2.
+ −
→ R
that v
an
ish a
t the o
rigin
.
%•
Non
-neg
ativ
ity u
nd
er r n
on
-decre
asin
g5
If r is non
-decre
asin
g, in
the se
nse
that r(λ
) ≤ r(λ
∗) w
hen
ever 0
≤ λ
≤ λ
∗
≤ Λ
, then
φ(N
, Λ,r) ≥
0(5
)
5Calle
d m
on
oto
nicity
by so
me a
uth
ors.
• D
um
my p
layer a
lloca
tion
If i is a d
um
my p
layer, in
the se
nse
that
r(λ) −
r(λ∗)=
(λi −
λ∗
i ) ρ(X
i) Λi
when
ever 0
≤ λ
≤ Λ
and
λ∗
= λ
exce
pt in
the i th
com
ponen
t, then
ki =
ρ(X
i) (6)Λ
i
•Fu
zzy co
re T
he a
lloca
tion
φ(N
, Λ,r) b
elo
ng
s to th
e fu
zzy co
re o
f the g
am
e (N
, Λ,r) if fo
r all λ
such
that 0
≤ λ
≤ Λ
,
λtφ
(N, Λ
,r) ≤ r(λ
)(7
)
as w
ell a
s Λtφ
(N, Λ
,r)= r(Λ
).
Th
e p
rop
ertie
s req
uire
d o
f a co
here
nt fu
zzy v
alu
e ca
n b
e ju
stified
esse
ntia
lly in
the sa
me m
ann
er a
s was d
on
e in
Defin
ition
3.
Ag
gre
gatio
n in
varia
nce
is a
kin to
the
sym
metry
pro
perty: e
qu
ivale
nt risks
shou
ld re
ceiv
e e
qu
ivale
nt a
lloca
tion
s. Con
tinu
ity is
desira
ble
to e
nsu
re th
at sim
ilar risk
measu
res y
ield
simila
r allo
catio
ns. N
on-n
eg
ativ
ity u
nd
er n
on-d
ecre
asin
g risk
measu
res is a
natu
ral re
qu
irem
en
t to e
nfo
rce th
at “m
ore
risk” imp
ly “m
ore
allo
catio
n”. Th
e d
um
my
pla
yer p
rop
erty
is the
eq
uiv
ale
nt o
f the
riskless a
lloca
tion
of D
efin
ition
3, a
nd
is nece
ssary
to g
ive “risk
cap
ital” th
e se
nse
we g
ave it in
sectio
n 2
: an
am
ou
nt o
f riskless
instru
men
t nece
ssary
to m
ake
a p
ortfo
lio a
ccep
tab
le, riskw
ise. Fin
ally
, note
that th
e fu
zzy co
re is
a sim
ple
exte
nsio
n o
f the
con
cep
t of co
re: a
lloca
tion
s ob
tain
ed
from
the fu
zzy co
re th
roug
h (2
) allo
w n
o u
nd
ercu
t from
any p
layer, co
alitio
n o
f pla
yers, n
or
coalitio
n w
ith fra
ction
al p
layers. S
uch
allo
catio
ns a
re fa
ir, in th
e sa
me
sense
that co
re e
lem
en
t were
con
sidere
d fa
ir in se
ction
4.3
. Much
less
is kn
ow
n a
bou
t this
allo
catio
n p
rob
lem
than
is kn
ow
n a
bou
t the
simila
r pro
ble
m d
escrib
ed
in se
ction
4. O
n th
e
oth
er h
an
d, o
ne so
lutio
n co
nce
pt h
as b
een
well in
vestig
ate
d: th
e A
um
an
n-S
hap
ley p
ricing
prin
ciple
.
5.3
The A
um
ann-S
haple
y V
alu
e
Au
man
n a
nd
Sh
ap
ley e
xte
nd
ed
the co
nce
pt o
f Shap
ley v
alu
e to
non
-ato
mic g
am
es, in
their o
rigin
al b
ook [7
]. The re
sult w
as
calle
d th
e A
um
an
n-S
hap
ley v
alu
e, a
nd
was la
ter re
cast in
the co
nte
xt o
f fractio
nal p
layers g
am
es, w
here
it is defin
ed
as:
1
φA
S (N
, Λ,r)=
kA
S =
∂r (γΛ
) dγ (8
)
ii
∂λi fo
r pla
yer i o
f N. T
he p
er-u
nit co
st kA
S is th
us a
n a
vera
ge o
f the m
arg
inal
0
i costs o
f the i th
portfo
lio, a
s the le
vel o
f activ
ity o
r volu
me in
crease
s un
iform
ly fo
r all p
ortfo
lios
from
0 to
Λ. T
he v
alu
e h
as a
simp
ler e
xp
ressio
n, g
iven
ou
r assu
med
coh
ere
nce
of th
e risk m
easu
re r; in
deed
, consid
er th
e re
sult
from
stan
dard
calcu
lus:
Lem
ma 1
If f is a k–h
om
og
en
eous fu
nctio
n, i.e
. f(γx)=
γk f(x
), then
∂f (x)
∂xi
is (k − 1
)-hom
og
eneou
s.
As a
resu
lt, since
r is 1–h
om
og
en
eou
s,
φA
S
(N, Λ
,r)= k
AS
= ∂r(Λ
) (9)
ii
∂λi a
nd
the p
er-u
nit a
lloca
tion
vecto
r is the g
rad
ient o
f the m
ap
pin
g r e
valu
ate
d a
t the
full p
rese
nce
level Λ
: φ(N
, Λ,r) A
S =
kA
S =
∇r(Λ
) (10
)
We ca
ll this g
rad
ien
t “Au
man
n-S
hap
ley p
er-u
nit a
lloca
tion”, o
r simp
ly “A
u-m
an
n-S
hap
ley p
rices”. T
he a
mou
nt o
f risk cap
ital
allo
cate
d to
each
portfo
lio is th
en g
iven
by th
e co
mp
on
en
ts of th
e v
ecto
r
KA
S =
kA
S .(1
1)
∗ Λ
5.3
.1 A
xio
matic ch
ara
cteriza
tion
s of th
e A
um
an
n-S
hap
ley v
alu
e
As in
the S
hap
ley v
alu
e ca
se, a
chara
cteriza
tion
con
sists of a
set o
f pro
pertie
s, wh
ich u
niq
uely
defin
e th
e A
um
an
n-S
hap
ley
valu
e. M
an
y ch
ara
cteriza
tion
s exist (se
e Ta
um
an
[32
]); we co
nce
ntra
te h
ere
on
that o
f Aub
in [5
] an
d [6
], an
d B
illera
an
d H
eath
[9]. B
oth
chara
cteriza
tions a
re fo
r valu
es o
f gam
es w
ith fra
ction
al p
layers a
s defin
ed
ab
ove; o
nly
their a
ssum
ptio
ns o
n r d
iffer
from
ou
r assu
mp
tion
s: their co
st fun
ctions a
re ta
ken to
van
ish a
t zero
and
to b
e co
ntin
uou
sly d
iffere
ntia
ble
, bu
t are
not a
ssum
ed
coh
ere
nt. A
ub
in a
lso im
plicitly
assu
mes r to
be h
om
og
en
eou
s of d
eg
ree o
ne. Le
t us d
efin
e:
A fu
zzy v
alu
e φ
is linear if fo
r an
y tw
o g
am
es (N
,Λ, r
1) an
d (N
,Λ, r
2) an
d sca
lars
γ1
and
γ2, it is a
dd
itive
an
d 1
-hom
og
en
eou
s in
the risk m
easu
re: φ
(N, Λ
,γ1r
1 +
γ2r
2)= γ
1 φ
(N, Λ
,r1)+
γ2
φ(N
, Λ,r
2)
Th
en
, the fo
llow
ing
pro
pertie
s of a
fuzzy
valu
e a
re su
fficie
nt to
un
iqu
ely
defi
ne th
e A
um
an
n-S
hap
ley v
alu
e (8
):
Au
bin
’s B
illera
& H
eath
’s • lin
earity
• lin
earity
• a
gg
reg
atio
n
invaria
nce
• a
gg
reg
atio
n in
varia
nce
• co
ntin
uity
• n
on
-neg
ativ
ity u
nd
er r n
on
d
ecre
asin
g
In fa
ct, both
Aubin
, and
Bille
ra a
nd
Heath
pro
ve
that th
e A
um
an
n-S
haple
y v
alu
e sa
tisfies
all fo
ur
pro
pertie
s in th
e ta
ble
above.
So, is th
e A
um
an
n-S
haple
y v
alu
e a
cohere
nt fu
zzy v
alu
e w
hen
r is a co
here
nt risk
measu
re ? N
ote
first
that th
e co
here
nce
of r im
plie
s its hom
ogeneity
, as w
ell a
s r(0) =
0. B
ein
g co
ntin
uously
diff
ere
ntia
ble
is not
auto
matic
how
ever;
let u
s a
ssum
e fo
r n
ow
that
r d
oes
have
contin
uou
s d
eriv
ativ
es. T
he
eventu
al
nond
iffere
ntia
bility
will b
e d
iscusse
d la
ter.
Cle
arly
, two
pro
pertie
s are
missin
g fro
m th
e se
t ab
ove
for φ
to q
ualify
as co
here
nt: th
e d
um
my
pla
yer
pro
perty
and
the
fuzzy
core
pro
perty. T
he
form
er ca
use
s n
o p
roble
m: g
iven
(9), th
e v
ery
meanin
g o
f a
du
mm
y p
layer in
Definitio
n 1
3 im
plie
s:
Lem
ma
2 W
hen
the
allo
catio
n p
roce
ss is base
d o
n a
coh
ere
nt risk
measu
re r, th
e A
um
ann
-Shaple
y p
rices
(9) sa
tisfy th
e d
um
my p
layer p
rop
erty.
Con
cern
ing th
e fu
zzy co
re p
rop
erty
, one v
ery
inte
restin
g re
sult o
f Aub
in is th
e fo
llow
ing:
Theore
m 6
([5]) T
he
fuzzy
core
(7) o
f a fu
zzy g
am
e (N
, r, Λ) w
ith p
ositiv
ely
hom
ogeneou
s r is equal to
the
sub
diff
ere
ntia
l ∂r(Λ) o
f r at Λ
.
As A
ubin
note
d, th
e th
eore
m h
as tw
o v
ery
importa
nt co
nse
quen
ces:
Theore
m 7
([5]) If th
e co
st functio
n r is co
nvex (a
s well a
s positiv
ely
hom
ogeneous), th
en
the fu
zzy co
re is
non-e
mp
ty, convex, a
nd
com
pact.
If furth
erm
ore
r is diff
ere
ntia
ble
at Λ
, then th
e co
re co
nsists o
f a sin
gle
vecto
r, the g
rad
ient ∇
r(Λ).
The d
irect co
nse
qu
ence
of th
is is the A
um
ann-S
hap
ley v
alu
e is in
deed a
cohere
nt fu
zzy v
alu
e, g
iven th
at
5.3
.3 O
n th
e d
iffere
ntia
bility
req
uire
men
t
Con
cern
ing
the d
iffere
ntia
bility
of th
e risk
measu
res/co
st fun
ction
s, rece
nt re
sults a
re e
nco
ura
gin
g. Ta
sche [3
1] a
nd
Sca
illet [2
6]
giv
e co
nd
ition
s un
der w
hich
a co
here
nt risk
measu
re, th
e e
xp
ecte
d sh
ortfa
ll, is diff
ere
ntia
ble
. Th
e co
nd
ition
s are
rela
tively
mild
,
esp
ecia
lly in
com
pariso
n w
ith th
e te
mera
riou
s assu
mp
tion
s com
mon
in th
e a
rea
of risk
man
ag
em
en
t. Exp
licit first d
eriv
ativ
es
are
pro
vid
ed
, wh
ich h
ave
the
follo
win
g in
terp
reta
tion: th
ey
are
exp
ecta
tion
s o
f the
risk fa
cto
rs, cond
ition
ed
on
the
portfo
lio
valu
e b
ein
g b
elo
w a
certa
in q
uan
tile o
f its d
istribu
tion
. Th
is is
very
inte
restin
g: it sh
ow
s th
at w
hen
Au
man
n-S
hap
ley
valu
e is
use
d w
ith a
shortfa
ll risk measu
re, th
e re
sultin
g (co
here
nt) a
lloca
tion
is ag
ain
of a
shortfa
ll typ
e:
Ki =
E −
Xi | i X
i ≤ q
α
21
w
here
qα
is a q
uantile
of th
e d
istribu
tion
of X
i.
i
Even
when
r is not d
iffere
ntia
ble
, som
eth
ing
can
ofte
n b
e sa
ved
. Ind
eed
, sup
pose
that r is n
ot d
iffere
ntia
ble
at Λ
, bu
t is the
sup
rem
um
of a
set o
f para
mete
rized
fun
ctions th
at a
re th
em
selv
es co
nvex, p
ositiv
ely
hom
og
en
eou
s an
d d
iffere
ntia
ble
at Λ
:
r(λ) =
sup
w(λ
,p) (1
2)
p∈
P
where
P is a
com
pact se
t of p
ara
mete
rs of th
e fu
nctio
ns w
, and
w(λ
,p) is u
pp
er se
mico
ntin
uou
s in p
. Th
en
Au
bin
[5] p
roved
:
Th
e fu
zzy co
re is th
e clo
sed
con
vex h
ull o
f all th
e v
alu
es φ
AS
(N,Λ
,w(Λ
,p)) o
f the fu
nctio
ns w
that a
re “a
ctive” a
t Λ, i.e
. that a
re
eq
ual to
r(Λ).
Th
us, sh
ou
ld (1
2) a
rise, —
wh
ich is n
ot u
nlike
ly, th
ink o
f Lag
ran
gia
n re
laxatio
n w
hen
r is defin
ed
by a
n o
ptim
izatio
n p
rob
lem
—, th
e a
bove re
sult p
rovid
es a
set o
f cohere
nt v
alu
es to
choose
from
.
5.3
.4 A
ltern
ativ
e p
ath
s to th
e A
um
an
n-S
hap
ley v
alu
e
It is v
ery
inte
restin
g th
at th
e re
cen
t rep
ort o
f Tasch
e [3
1] co
mes fu
nd
am
enta
lly to
the
sam
e re
sult o
bta
ined
in th
is se
ction
,
nam
ely
that g
iven
som
e d
iffere
ntia
bility
con
ditio
ns o
n th
e risk
measu
re ρ
, the
corre
ct way
of a
lloca
ting
risk ca
pita
l is thro
ug
h
the A
um
an
n-S
hap
ley p
rices (9
). Tasch
e’s ju
stifica
tion
of th
is con
tentio
n is h
ow
ever co
mp
lete
ly d
iffere
nt; h
e d
efin
es a
s “suita
ble
”,
cap
ital a
lloca
tion
s su
ch th
at if th
e risk-a
dju
sted
retu
rn o
f a p
ortfo
lio is
“ab
ove
avera
ge”, th
en
, at le
ast lo
cally
, incre
asin
g th
e
share
of th
is portfo
lio im
pro
ves th
e o
vera
ll retu
rn o
f the fi
rm. N
ote
that th
e w
ork o
f Sch
mock a
nd
Stra
um
ann
[27
] poin
ts ag
ain
to
the
sam
e co
nclu
sion
. In th
e a
pp
roach
of [3
1] a
nd
[27
], the
Au
man
n-S
hap
ley p
rices a
re in
fact th
e u
niq
ue
satisfa
ctory
allo
catio
n
prin
ciple
.
Oth
ers im
porta
nt re
sults o
n th
e to
pic
inclu
de
that o
f Artzn
er a
nd
Ostro
y [4
], wh
o, w
orkin
g in
a n
on
-ato
mic
measu
re se
tting
,
pro
vid
e a
ltern
ativ
e ch
ara
cteriza
tion
s o
f diff
ere
ntia
bility
an
d su
bd
iffere
ntia
bility
, with
the
goal o
f esta
blish
ing
the
existe
nce
of
allo
catio
ns th
rou
gh, b
asica
lly, E
ule
r’s theore
m. S
ee a
lso th
e fo
rthco
min
g D
elb
aen
[13
].
On
Eu
ler’s
theore
m6, n
ote
also
that th
e fe
asib
ility (3
) of th
e a
lloca
tion
vecto
r follo
ws
dire
ctly fro
m it, a
nd
that o
ut o
f
con
sidera
tion
for th
is, som
e a
uth
ors h
ave
calle
d th
e a
lloca
tion
prin
ciple
(9) th
e E
ule
r prin
ciple
. See
for e
xam
ple
the
atta
chm
ent
to th
e re
port o
f Patrik, B
ern
eg
ger, a
nd
R¨
ueg
g [2
3], w
hich
pro
vid
es so
me p
rop
ertie
s of th
is prin
ciple
.
We
shall e
nd
this se
ction
by
dra
win
g th
e a
tten
tion
of th
e re
ad
er to
the
imp
orta
nce
of th
e co
here
nce
of th
e risk
measu
re ρ
(and
the r d
eriv
ed
from
it) for th
e a
lloca
tion
.
Th
e su
bad
ditiv
ity o
f the
risk m
easu
re: is a
nece
ssary
con
ditio
n fo
r the
existe
nce
of a
n a
lloca
tion
with
no
un
dercu
t, in b
oth
the
ato
mic a
nd
fractio
nal p
layers co
nte
xts.
Th
e h
om
og
en
eity
of th
e risk m
easu
re: e
nsu
res th
e sim
ple
form
(9) o
f the A
um
an
n-S
hap
ley p
rices.
Both
sub
ad
ditiv
ity a
nd
hom
og
en
eity: a
re u
sed
to p
rove
that th
e co
re in
non
-em
pty
(Th
eore
m 4
), in th
e a
tom
ic gam
e se
tting
. In
the
fuzzy
gam
e se
tting
, the
two
pro
pertie
s are
use
d to
show
that th
e A
um
ann
-Sh
ap
ley
valu
e is in
the
fuzzy
core
(un
der
diff
ere
ntia
bility
). Th
ey a
re a
lso u
sed
in th
e n
on
-neg
ativ
ity p
roof o
f the a
pp
en
dix
.
Th
e riskle
ss pro
perty: is ce
ntra
l to th
e d
efin
ition
of th
e riskle
ss allo
catio
n (d
um
my p
layer) p
rop
erty.
Th
e n
on-n
eg
ativ
ity o
f the a
lloca
tion
Giv
en
our d
efin
ition
of risk
measu
re, a
portfo
lio m
ay w
ell h
ave a
neg
ativ
e risk
measu
re, w
ith th
e in
terp
reta
tion
that th
e p
ortfo
lio
is then sa
fer th
an
deem
ed
nece
ssary.
Sim
ilarly
, there
is no
justifi
catio
n p
er se
to e
nfo
rce th
at th
e risk
cap
ital a
lloca
ted
to a
portfo
lio b
e n
on
-neg
ativ
e; th
at is, th
e
allo
catio
n o
f a n
eg
ativ
e a
mou
nt d
oes n
ot p
ose
a co
nce
ptu
al p
rob
lem
. Un
fortu
nate
ly, in
the a
pp
licatio
n
6Wh
ich sta
tes th
at if F is a
real, n
–varia
ble
s, hom
og
en
eous fu
nctio
n o
f degre
e k, th
en
∂F ( x ) ∂F ( x ) ∂F ( x )
++
xn ∂x
n =
kF (x)
x1 ∂x
1 +
x2 ∂x
2
··· w
e w
ou
ld like
to m
ake
of th
e a
lloca
ted
cap
ital, n
on
-neg
ativ
ity is a
pro
ble
m. If
retu
rn
the a
moun
t is to b
e u
sed
in a
RA
PM
-typ
e q
uotie
nt
allo
cate
d ca
pita
l , n
eg
ativ
ity h
as
a ra
ther n
asty
dra
wb
ack, a
s a
portfo
lio w
ith a
n
allo
cate
d ca
pita
l slightly
belo
w ze
ro e
nd
s up
with
a n
eg
ativ
e risk-a
dju
sted
measu
re o
f larg
e m
ag
nitu
de, w
hose
inte
rpre
tatio
n is
less th
an
ob
vio
us. A
neg
ativ
e a
lloca
tion
is there
fore
not so
mu
ch a
con
cern
with
the a
lloca
tion
itself, th
an
with
the u
se w
e w
ou
ld
like to
make
of it.
A cro
ssed
-fin
gers, a
nd
perh
ap
s m
ost p
rag
matic
ap
pro
ach
, is to
assu
me
that th
e co
here
nt a
lloca
tion
is in
here
ntly
non-
neg
ativ
e. In
fact, o
ne
cou
ld re
aso
nab
ly e
xp
ect n
on
-neg
ativ
e a
lloca
tions to
be
the
norm
in re
al-life
situatio
ns. Fo
r ex
am
ple
,
pro
vid
ed
no p
ortfo
lio o
f the fi
rm e
ver d
ecre
ase
s the risk m
easu
re w
hen
ad
ded
to a
ny su
bse
t of p
ortfo
lios o
f the fi
rm:
c(S ∪
i
) ≥c(S
) ∀S
⊆N
, ∀i ∈
N \ S
then
the
Shap
ley
valu
e is n
ece
ssarily
non
-neg
ativ
e. T
he
eq
uiv
ale
nt co
nd
ition
for th
e A
um
an
n-S
hap
ley p
rices is th
e p
rop
erty
of
non
-neg
ativ
ity u
nd
er n
on
-decre
asin
g r (e
qu
atio
n (5
)): if the a
nte
ced
en
t alw
ays h
old
s, the p
er-u
nit a
lloca
tion
s are
non
-neg
ativ
e.
An
oth
er a
pp
roach
wou
ld b
e to
en
force
non
-neg
ativ
ity b
y re
qu
iring
more
of th
e risk
measu
re. Fo
r exam
ple
, the
core
an
d th
e
non
-neg
ativ
ity o
f the
Ki’s fo
rm a
set o
f linear in
eq
ualitie
s (an
d o
ne
linear e
qu
ality
), so th
at th
e e
xiste
nce
of a
non-n
eg
ativ
e co
re
solu
tion
is eq
uiv
ale
nt to
the
existe
nce
of a
solu
tion
to a
linear sy
stem
. Sp
ecifi
cally
, a h
yp
erp
lan
e se
para
tion
arg
um
ent p
roves
that su
ch a
solu
tion
will e
xist if th
e fo
llow
ing
cond
ition
on
ρ h
old
s:
+,ρ
Xi m
in λ
iXi (1
3)∀
λ ∈
Rn
i∈N
λ
i≤
ρ i∈
N
i∈N
Th
e p
roof is g
iven
in a
dd
end
um
. Th
e co
nd
ition co
uld
be in
terp
rete
d a
s fol
low
s. First assu
me th
at ρ
i∈N
Xi >
0, w
hich
is reaso
nab
le, if w
e a
re in
deed
to a
lloca
te so
me risk
cap
ital. T
hen
(13
) says th
at th
ere
is no
positiv
e lin
ear co
mb
inatio
n o
f (each
an
d e
very
) portfo
lios, th
at ru
ns n
o risk. In
oth
er w
ord
s, a p
erfe
ctly h
ed
ged
portfo
lio
cann
ot b
e a
ttain
ed
by sim
ply
re-w
eig
htin
g th
e p
ortfo
lios, if a
ll portfo
lios a
re to
have
a p
ositiv
e w
eig
ht. H
ow
ever, u
nle
ss on
e is
willin
g to
imp
ose
such
a co
nd
ition
on
the
risk m
easu
re, th
e fa
ct is that th
e issu
e o
f the
non
-neg
ativ
ity re
main
s un
satisfa
ctorily
reso
lved
for th
e m
om
en
t.
Allo
catio
n w
ith a
n S
EC
-like risk m
easu
re
In th
is sectio
n, w
e p
rovid
e so
me e
xam
ple
s of a
pp
licatio
ns o
f the S
hap
ley a
nd
Au
man
n-S
hap
ley co
nce
pts to
a p
rob
lem
of m
arg
in
(i.e. risk ca
pita
l) allo
catio
n.
Th
e risk
measu
re w
e u
se is
deriv
ed
from
the
Secu
rities a
nd
Exch
an
ge
Com
missio
n (S
EC
) rule
s fo
r marg
in re
quire
ments
(Reg
ula
tion
T), a
s describ
ed
in th
e N
atio
nal A
ssocia
tion
of S
ecu
rities D
eale
rs (NA
SD
) docu
men
t [19
]. Th
ese
rule
s are
use
d b
y
stock
exch
an
ges to
esta
blish
the m
arg
ins re
qu
ired
of th
eir m
em
bers, a
s gu
ara
nte
e a
gain
st the risk
that th
e m
em
bers’ p
ortfo
lios
involv
e (th
e C
hica
go B
oard
of O
ptio
ns E
xchan
ge
is one
such
exch
an
ge). T
he
rule
s them
selv
es a
re n
ot co
nstru
ctive, in
that th
ey
do n
ot sp
ecify
how
the m
arg
in sh
ould
be co
mp
ute
d; th
is com
pu
tatio
n is le
ft to e
ach
mem
ber o
f the e
xchan
ge, w
ho m
ust fi
nd
the
smalle
st marg
in co
mp
lyin
g w
ith th
e ru
les. R
ud
d a
nd
Sch
roed
er [2
5] p
roved
in 1
98
2 th
at a
linear o
ptim
izatio
n p
rob
lem
(L.P.)
mod
elle
d th
e ru
les a
deq
uate
ly, a
nd
was su
fficie
nt to
esta
blish
the
min
imu
m m
arg
in o
f a p
ortfo
lio, th
at is, to
evalu
ate
its risk
measu
re. It is
worth
men
tion
ing
that g
iven
this
L.P.-base
d risk
measu
re, th
e co
rresp
on
din
g co
alitio
nal g
am
e h
as b
een
calle
d
linear p
rod
uctio
n g
am
e b
y O
wen
[22
], see a
lso [1
0].
For th
e p
urp
ose
of th
e a
rticle, w
e re
strict the
risk m
easu
re to
simp
listic portfo
lios o
f calls o
n th
e sa
me
und
erly
ing
stock, a
nd
with
the sa
me e
xp
iratio
n d
ate
. This re
striction
of th
e S
EC
rule
s is take
n fro
m A
rtzner, D
elb
aen, E
ber a
nd
Heath
[3] w
ho u
se it a
s
an
exam
ple
of a
non
-cohere
nt risk
measu
re. In
the ca
se o
f a p
ortfo
lio o
f calls, th
e m
arg
in is ca
lcula
ted
thro
ug
h a
rep
rese
nta
tion
of th
e ca
lls by
a se
t of sp
read
op
tion
s, each
of w
hich
carry
ing
a fi
xed
marg
in. To
ob
tain
a co
here
nt m
easu
re o
f risk, we
pro
ve
late
r that it is su
fficie
nt to
rep
rese
nt th
e ca
lls by a
set o
f spre
ad
s and
bu
tterfl
y o
ptio
ns. N
ote
that su
ch a
chan
ge to
the m
arg
ins
rule
s was p
rop
ose
d b
y th
e N
AS
D a
nd
very
rece
ntly
acce
pte
d b
y th
e S
EC
, see [2
0].
7.1
Coh
ere
nt, S
EC
-like m
arg
in ca
lcula
tion
We co
nsid
er a
portfo
lio co
nsistin
g o
f CP ca
lls at strike
price
P, w
here
P b
elo
ng
s to a
set o
f strike p
rices P
=
Pm
in,Pm
in +
10
,...,Pm
ax
−1
0,P
max
. Th
is assu
mp
tion
ab
ou
t the fo
rmat o
f the strike
price
s set P
, inclu
din
g th
e in
terv
als o
f 10
, make
s the n
ota
tion
more
pala
tab
le, w
ithou
t loss o
f gen
era
lity. For co
nven
ien
ce, w
e d
en
ote
the se
t P\
Pm
in,Pm
ax
by P
−. W
e a
lso m
ake
the sim
plify
ing
assu
mp
tion
that th
ere
are
as m
an
y lo
ng
calls a
s short ca
lls in th
e p
ortfo
lio, i.e
. C
P =
0. B
oth
assu
mp
tion
s rem
ain
valid
thro
ug
hou
t sectio
n 7
.P ∈
P
We w
ill den
ote
by C
P th
e v
ecto
r of th
e C
P p
ara
mete
rs, P ∈
P. W
hile
CP fu
lly d
escrib
es th
e p
ortfo
lio, it ce
rtain
ly d
oes n
ot
describ
e th
e fu
ture
valu
e o
f the p
ortfo
lio, w
hich
dep
en
ds o
n th
e p
rice o
f the u
nd
erly
ing
stock a
t a fu
ture
date
. Alth
oug
h risk
measu
res w
ere
defin
ed
as a
map
pin
gs o
n ra
nd
om
varia
ble
s, we n
everth
ele
ss write
ρ(C
P ) sin
ce th
e ρ
con
sidere
d h
ere
can
be
defin
ed
by u
sing
on
ly C
P . O
n th
e o
ther h
an
d, th
ere
is a sim
ple
linear re
latio
nsh
ip b
etw
een C
P a
nd
the fu
ture
worth
s (un
der a
n
ap
pro
pria
te d
iscretiza
tion
of th
e sto
ck price
sp
ace
), so th
at a
n e
xp
ressio
n su
ch a
s ρ C
∗ +
C∗
∗ is a
lso ju
stified
. On
ly in
the
PP
case
of th
e p
rop
erty
“mon
oto
nicity
” need
we tre
at w
ith m
ore
care
the d
istinctio
n b
etw
een
nu
mb
er o
f calls a
nd
futu
re w
orth
.
We ca
n n
ow
define o
ur S
EC
-like m
arg
in re
qu
irem
en
t. To e
valu
ate
the m
arg
in (o
r risk measu
re) ρ
of th
e p
ortfo
lio C
P , w
e fi
rst
rep
licate
its calls w
ith sp
read
s and
bu
tterfl
ies, d
efin
ed
as fo
llow
s:
Varia
ble
Instru
men
t Calls e
qu
ivale
nt
SH
,K
Sp
read
, lon
g in
H, sh
ort in
K
O
ne lo
ng
call a
t price
H
, one
short ca
ll at strike
K
Blo
ng
H
Lon
g b
utte
rfly, ce
nte
red
at
H
One lo
ng
call a
t H−
10
, two
short ca
lls at H
, on
e lo
ng
ca
ll at H
+1
0
Bsh
ort
H
Sh
ort b
utte
rfly, ce
nte
red
at
H
One sh
ort ca
ll at H
−1
0, tw
o
lon
g ca
lls at H
, on
e sh
ort
call a
t H +
1
0
The
varia
ble
s shall re
pre
sent th
e n
um
ber o
f each
specifi
c in
strum
ent. A
ll H a
nd
K a
re u
ndersto
od
to b
e in
P,o
r P−
for th
e b
utte
rflie
s; H =
K fo
r the sp
read
s.
As
in th
e S
EC
rule
s, fixe
d m
arg
ins
are
attrib
ute
d to
the
instru
men
ts u
sed
for th
e re
plica
ting
portfo
lio, i.e
. spre
ad
s a
nd
bu
tterfl
ies in
ou
r case
. Sp
read
s carry
a m
arg
in o
f (H−
K) +
= m
ax(0
,H−
K); sh
ort b
utte
rflie
s are
giv
en
a m
arg
in o
f 10
, wh
ile lo
ng
bu
tterfl
ies re
qu
ire n
o m
arg
in. In
simp
le la
ng
uag
e, e
ach
instru
men
t req
uire
s a
marg
in e
qu
al to
the
worst p
ote
ntia
l loss, o
r
neg
ativ
e p
ayoff
, it cou
ld y
ield
.
By d
efin
ition
, the m
arg
in o
f a p
ortfo
lio o
f spre
ad
s an
d b
utte
rflie
s is the su
m o
f the m
arg
ins o
f its com
pon
en
ts.
On th
e b
asis o
f [25
], the m
arg
in ρ
(CP ) o
f the p
ortfo
lio ca
n b
e e
valu
ate
d w
ith th
e lin
ear o
ptim
izatio
n p
rob
lem
(SEC
-LP):
min
imize
f tY
subje
ct to
AY =
CP
(SEC
−LP
)
Y
≥ 0
S
where
: Y sta
nd
s for Y
=
Blo
ng
w
here
S is a
colu
mn
vecto
r of a
ll spre
ad
s
Bsh
ort
varia
ble
s co
nsid
ere
d (a
pp
rop
riate
ly o
rdere
d: b
ull sp
read
s, then
bear sp
read
s), an
d B
long
and
Bsh
ort
are
ap
pro
pria
tely
ord
ere
d
colu
mn v
ecto
rs of b
utte
rflie
s varia
ble
s; f tY is sh
orth
and
nota
tion
for
10
Bsh
ort
f tY =
(H −
K) +
SH
,K +
;
H
H,K
∈P
H∈
P−
and
A is
11 0
···
00
−1
···
0
0 −
1
···
00 0
···
−1
−
1
1 0
··· ···
0 0 0
1
··· 0
0
0···
0
1
0···
0
−2
1
1
−2
0
1
··· ··· ···
0 0 0
00
−1
···
0
2 −
1 ···
20
−1
···
0
0 −
1 ···
A =
. .
.
. .
.
. .
.
0 0
··· 0
0 0
··· 1
0 0
···
−1
. .
. .
. .
. .
. .
. .
. .
. .
. .
0 0
··· 0
0
0
··· 1
0 0
0 0
··· ···
−1
1
0
0
0 0
··· ···
−2 1
. .
.
. .
.
. .
.
0 0
0 0
··· ···
−1 2 0
0···
−1
Th
e o
bje
ctive
fun
ction
rep
rese
nt th
e m
arg
in; th
e e
qu
ality
con
strain
ts en
sure
that th
e p
ortfo
lio is e
xactly
rep
licate
d. T
he
risk
measu
re th
us d
efined
is coh
ere
nt; th
e p
roof is g
iven
next.
7.2
Pro
of o
f the co
here
nce
of th
e m
easu
re
We p
rove h
ere
that th
e risk
measu
re ρ
ob
tain
ed
thro
ug
h (S
EC
-LP) is co
here
nt, in
the se
nse
of D
efinitio
n 1
. We p
rove e
ach
of th
e
fou
r pro
perty
in tu
rn, b
elo
w.
1) S
ub
ad
ditiv
ity:
and
∗∗C
P
For a
ny tw
o p
ortfo
lios ∗
CP
, ρ
∗∗
+ C
P
∗C
P
≤
)+( ∗
∗
ρC
P( ∗
ρC
P
) Pro
of:
If solv
ing
(SEC
-LP) w
ith ∗
CP
as rig
ht-h
an
d sid
e o
f the e
qu
ality
con
strain
ts yie
lds a
solu
tion
, and
solv
ing
with
∗
∗∗
S
CP
yie
lds a
solu
tion
S∗
∗,
∗∗
+ C
P
then
is a fe
asib
le so
lutio
n fo
r the (S
EC
-LP) w
ith ∗
∗∗
∗
+S
S
CP
as rig
ht-
han
d sid
e. S
ub
ad
ditiv
ity fo
llow
s dire
ctly, g
iven
the lin
earity
of th
e o
bje
ctive fu
nctio
n.
2) D
eg
ree o
ne h
om
og
en
eity: Fo
r an
y γ
≥ 0
an
d a
ny p
ortfo
lio C
P ,
ρ(γ
CP )=
γρ
(CP )
Pro
of: T
his is a
gain
a d
irect co
nse
qu
en
ce o
f the
linear o
ptim
izatio
n n
atu
re o
f ρ,a
s γS
is a so
lutio
n o
f the
(SEC
-LP) w
ith γ
CP
as
righ
t-hand
side
of th
e co
nstra
ints, w
hen
S is a
solu
tion
of th
e (S
EC
-LP) w
ith C
P a
s righ
t-han
d sid
e. O
f cou
rse, th
e v
ery
defin
ition
of h
om
og
eneity
imp
lies th
at w
e a
llow
fractio
ns o
f calls to
be so
ld a
nd
boug
ht.
3) Tra
nsla
tion
invaria
nce
: 7 A
dd
ing
to a
ny
portfo
lio o
f calls
CP
an
am
oun
t of riskle
ss in
strum
en
t worth
α to
day, d
ecre
ase
s th
e
marg
in o
f CP b
y α
.
Pro
of: T
here
is little to
pro
ve
here
; we
rath
er n
eed
to d
efin
e th
e b
eh
avio
ur o
f ρ in
the
pre
sen
ce o
f a riskle
ss instru
men
t, an
d
natu
rally
choose
the tra
nsla
tion
invaria
nce
pro
perty
to d
o so
. Th
is pro
perty
simp
ly a
nch
ors th
e m
ean
ing
of “m
arg
in”.
4) M
onoto
nicity:
and
∗∗C
P
For a
ny tw
o p
ortfo
lios ∗
CP
such
that th
e fu
ture
worth
of ∗
CP
is alw
ays le
ss than
or e
qu
al to
that o
f ∗∗C
P
,
) ( ∗∗
ρC
≥
P
( ∗ρ
CP
)
7Note
that in
the in
tere
st of a
tidie
r nota
tion, w
e d
eparte
d fro
m o
ur p
revio
us u
sag
e a
nd
did
not u
se th
e la
st com
pon
en
t of C
P to
den
ote
the riskle
ss instru
ment
Befo
re p
rovin
g m
on
oto
nicity
, let u
s first in
trod
uce
the
valu
es V
P , fo
r P ∈
P
min
+1
0,...,P
max,P
max
+1
0
, which
rep
rese
nt th
e
futu
re p
ayoff
s, or w
orth
s, of th
e p
ortfo
lio fo
r the fu
ture
price
s P o
f the u
nd
erly
ing
. (Ob
vio
usly
, the la
tter se
t of p
rices m
ay b
e to
o
coarse
a re
pre
senta
tion
of p
ossib
le fu
ture
price
s, and
is use
d to
keep
the n
ota
tion
com
pact; sta
rting
with
a fi
ner P
wou
ld re
lieve
this
pro
ble
m) A
gain
, we
write
VP
to d
en
ote
the
vecto
r of a
ll VP
’s. Th
e co
mp
on
ents
of V
P a
re co
mp
lete
ly d
ete
rmin
ed
by
the
nu
mb
er o
f calls in
the p
ortfo
lio:
P −
10V
P =
Cp(P
−p
) ∀P ∈
P
min
+1
0,...,P
max,P
max +
10
p
=P
min
which
is alte
rnativ
ely
writte
n V
P
MC
P , w
ith th
e sq
uare
, invertib
le m
atrix
M:
M =
10
0 0
···
20
10
0
···
30
20
10
···
. . ..
....
... .
Th
e a
nte
ced
en
t of th
e m
on
oto
nicity
pro
perty
is, of co
urse
, the co
mp
on
en
twise
V ∗
≤ V
∗∗.
PP W
e w
ill also
use
the fo
llow
ing
lem
ma:
Lem
ma 3
Un
der su
bad
ditiv
ity, two e
qu
ivale
nt fo
rmu
latio
ns o
f mon
oto
nicity
are
, for a
ny th
ree p
ortfo
lios C
P , C
∗ a
nd
C∗
∗:
PP
V ∗
≤ V
∗∗
= ⇒
ρ(M
−1V
P ∗
) ≥ ρ
(M−
1V ∗
∗)
PP P
and
0 ≤
VP
= ⇒ ρ
(M−
1VP ) ≤
0
Pro
of: T
he u
pp
er co
nd
ition
is suffi
cien
t, as it im
plie
s
ρ(0
) ≥ ρ
(M−
1VP ),
and
ρ(0
) = 0
from
the v
ery
structu
re o
f (SE
C-LP
). Th
e u
pp
er co
nd
ition
is nece
ssary
, as
ρ(M
−1V
∗∗
P
)= M
−1(V
P ∗
+(V
∗∗
−V
P ∗
)
ρ P
ρ(M
−1V
P ∗
)+ ρ
M−
1(V ∗
∗
P
−V
P ∗
)
≤
≤ ρ
(M−
1VP ∗
). 29
Pro
of o
f mon
oto
nicity
: As a
con
seq
uen
ce o
f the
ab
ove
lem
ma, it is su
fficie
nt to
pro
ve
that if a
portfo
lio o
f calls
alw
ays h
as
non
-neg
ativ
e fu
ture
payoff
, then its a
ssocia
ted
marg
in is n
on
-positiv
e.
A lo
ok
at (S
EC
-LP) sh
ow
s that th
e m
arg
in a
ssigned
to th
e p
ortfo
lio w
ill be
non
-positiv
e (in
fact, ze
ro), if a
nd
on
ly if a
non
-
neg
ativ
e, fe
asib
le so
lutio
n o
f (SEC
-LP) e
xists in
wh
ich a
ll spre
ad
s varia
ble
s SH
,K w
ith H
>K
and
all sh
ort b
utte
rflie
s varia
ble
s Bsh
ort
have v
alu
e ze
ro. T
his m
ean
s that th
ere
exists a
solu
tion
to th
e lin
ear sy
stem
:
AY
= C
P 0
(1
4)
Y
≥
(15
)
where
Ais m
ad
e o
f the fi
rst an
d th
ird p
arts o
f the A
that w
as d
efined
for (S
EC
-LP):
A =
11 0
10
0
··· ···
00 −
21 0
−1
··· ···
0 −
1 0
1 −
20
··· ···
00 0
01
0
··· ···
.. ... .
.. ... .
.. ... . 00
00
0 1
··· ···
00 1
00
−2
··· ···
00 −
100
1
··· ···
,
and
Y is th
e a
pp
rop
riate
vecto
r of sp
read
s and
bu
tterfl
ies v
aria
ble
s. We o
bta
in a
new
, eq
uiv
ale
nt sy
stem
of e
qu
atio
ns M
AY =
MC
P =
VP b
y p
re-m
ultip
lyin
g b
y th
e in
vertib
le m
atrix
Min
trod
uce
d a
bove. R
eca
ll now
that
we h
ave m
ad
e th
e a
ssum
ptio
n th
at th
e p
ortfo
lio co
nta
ins a
s man
y sh
ort ca
lls as lo
ng
calls,
i.e. C
P =
0. T
hu
s, we n
eed
on
ly p
rove th
at th
ere
exists a
non
-neg
ativ
e
P ∈
P
solu
tion
to th
e sy
stem
M
AY =
VP w
hen
ever V
P ≥
0 a
nd
e tM
−1V
P =
0 (e
is a ro
w v
ecto
r of 1
’s). A sim
ple
ob
serv
atio
n o
f M
t Ash
ow
s that its co
lum
ns sp
an
the sa
me su
bsp
ace
as th
e se
t of co
lum
ns
10 0
0
0
0 . . . 0
0
,
1
0 . .
. 0
0
, ··· ,
0
0 . . . 1
0
,
0
0 . . . 0
1
00 0
1
30
O
bse
rvin
g fu
rtherm
ore
that
t
0 0
.
. . 0
−1
1
e tM
−1
=
so th
at a
ny V
P sa
tisfyin
g e
tM−
1VP =
0 h
as id
en
tical la
st two co
mp
on
en
ts, the rig
ht-h
an
d sid
e o
f M
A Y
= V
P ca
n a
lways b
e e
xp
resse
d a
s a n
on
-neg
ativ
e lin
ear co
mb
inatio
n o
f the co
lum
ns o
f M
A.
7.3
Com
puta
tion o
f the a
lloca
tions
Giv
en
this
risk m
easu
re a
s a
linear o
ptim
izatio
n p
rob
lem
, the
Sh
ap
ley
valu
e is
easy
to co
mp
ute
wh
en
the
“tota
l portfo
lio” is
div
ided
in a
small n
um
ber o
f sub
portfo
lios. First, th
e m
arg
in o
f every
possib
le co
alitio
n o
f sub
portfo
lios is ca
lcula
ted
. Th
en
, the
marg
in a
lloca
ted
to e
ach
sub
portfo
lio is co
mp
ute
d, u
sing
the fo
rmu
la o
f the S
hap
ley v
alu
e g
iven
in D
efin
ition
9.
Th
e co
mp
uta
tion
of th
e A
um
an
n-S
hap
ley v
alu
e is e
ven
simp
ler. N
ote
that b
y w
orkin
g in
the fra
ction
al p
layers fra
mew
ork, w
e
imp
licitly a
ssum
e th
at fra
ction
s of p
ortfo
lios a
re se
nsib
le in
strum
ents. W
e ch
oose
the
vecto
r of fu
ll pre
sen
ce o
f all p
layers Λ
to
be th
e v
ecto
r of o
nes e
. Reca
ll that th
e A
um
an
n-S
hap
ley p
er u
nit m
arg
in a
lloca
ted
to th
e i th
sub
portfo
lio is
∂r ( e )
kA
S =
(16
)
i
∂λi
where
r(λ) is th
e m
arg
in re
qu
ired
of th
e su
m o
f all th
e su
bp
ortfo
lios i, e
ach
scale
d b
y a
scala
r λi, so
that r(e
)= ρ
(CP
). In v
ecto
r
nota
tion, k
AS
= ∇
r(e).
Let u
s first d
efin
e th
e lin
ear o
pera
tor L w
hich
map
s the le
vel o
f pre
sen
ce o
f the su
bp
ortfo
lios to
nu
mb
ers o
f calls in
the g
lob
al
portfo
lio. If th
ere
are
|P
| d
iffere
nt ca
lls (e
qu
ivale
ntly
here
, |P
strike p
rices), a
nd
n
sub
portfo
lios, th
en
| L is an
|P
× n m
atrix
, such
that Le
= C
P . E
xam
ple
s are
the th
ree-b
y-five
| m
atrice
s at th
e to
p o
f the ta
ble
s giv
en
in se
ction 7
.4.
Now
, the o
ptim
al d
ual so
lutio
n δ
∗ o
f the
linear p
rog
ram
(SEC
-LP), o
bta
ined
auto
matica
lly w
hen
com
putin
g th
e m
arg
in o
f the
tota
l portfo
lio, p
rovid
es th
e ra
tes o
f
chan
ge o
f the m
arg
in, w
hen
the p
rese
nce
of e
ach
specifi
c call v
arie
s. How
ever, u
sing
the co
mp
lem
enta
rity co
nd
ition
satisfi
ed
at
the o
ptim
al so
lutio
n p
air (Y
∗,δ
∗), w
e ca
n w
rite
f tY ∗
= −
(δ∗) tC
P (1
7)
=(−
(δ∗) tL)e
(18
)
so th
at th
e co
mp
on
en
ts of L
tδ∗
giv
e th
e m
arg
inal ra
tes o
f chan
ge o
f the o
bje
ctive v
alu
e o
f (SE
C-LP
), as a
fun
ction
of su
bp
ortfo
lio
pre
sen
ce, e
valu
ate
d a
t the p
oin
t of fu
ll pre
sen
ce o
f all su
bp
ortfo
lios.
Pu
t in o
ne se
nte
nce
, the m
ost in
tere
sting
resu
lt of th
is sectio
n is th
at th
e A
um
an
n-S
hap
ley a
lloca
tion
is on
ly a
matrix
pro
du
ct
aw
ay fro
m th
e lo
ne e
valu
atio
n o
f the m
arg
in fo
r the to
tal p
ortfo
lio.
Finally
, con
cern
ing
the
un
iqu
en
ess o
f the
allo
catio
n a
nd
the
diff
ere
ntia
bility
of th
e risk
measu
re, w
e ca
n o
nly
say
that th
ey
dep
en
d d
irectly
on
the
un
iqu
en
ess o
f the
op
timal so
lutio
n o
f the
du
al p
rob
lem
of (S
EC
-LP). A
lthou
gh
there
is not sp
ecia
l reaso
n
for m
ultip
le o
ptim
al d
ual so
lutio
ns to
occu
r here
, it can
well h
ap
pen
, in w
hich
case
we
have
ob
tain
ed
on
e o
f man
y a
ccep
tab
le
allo
catio
ns, p
er se
ction
5.3
.3.
7.4
Num
erica
l exam
ple
s of co
here
nt a
lloca
tion
We ca
n o
bta
in a
som
ew
hat m
ore
pra
ctical fe
elin
g o
f Sh
ap
ley a
nd
Aum
an
n-S
hap
ley a
lloca
tion
s by lo
okin
g a
t exam
ple
s.
For a
ll allo
catio
n e
xam
ple
s belo
w, th
e re
fere
nce
“tota
l” portfo
lio is th
e sa
me; its v
alu
es o
f CP ,P
∈
10
,20
,30
,40
,50
are
:
C1
0 C
20
C3
0 C
40
C5
0
Tota
l
−1
−2
8 −
72
mean
ing
on
e sh
ort ca
ll at strike
10
, two sh
ort ca
lls at strike
20
, eig
ht lo
ng
calls a
t strike 3
0, e
tc. It carrie
s a m
arg
in o
f 40
: ρ(C
P )
= 4
0.
In e
ach
of th
e ta
ble
s belo
w, w
e g
ive
the
div
ision
of th
e to
tal p
ortfo
lio in
thre
e su
bp
ortfo
lios su
ch th
at C
P1
+ C
P2
+ C
P3
= C
P ,
follo
wed
by th
e S
hap
ley a
lloca
tion
an
d th
e A
um
an
n-S
hap
ley a
lloca
tion
. Th
e ta
ble
s are
follo
wed
by so
me
ob
serv
atio
ns. C
on
sider fi
rst the d
ivisio
n:
C10
C20
C30
C40
C50
Sh
ap
ley
Au
man
n −
S
hap
ley
CP
1
−1
0
6
−6
1
15
2
0
CP
2
0
−2
2
0
0
20
2
0
CP
3
0
0
0
−1
1
5
0
Total
−1
−2
8
−7
2
40
4
0
In th
is exam
ple
, coalitio
ns o
f subportfo
lios in
cur m
arg
ins a
s follo
ws: ρ
(CP
1 +
CP
2 )=
40
ρ(C
P1
+ C
P3
)= 2
0 ρ
(CP
2
+ C
P3
)= 3
0 ρ
(CP
1 )=
20
ρ(C
P2
)= 2
0 ρ
(CP
3 )=
10
Con
sider a
seco
nd
exam
ple
:
C10
C20
C30
C40
C50
Sh
ap
ley
Au
man
n −
S
hap
ley
CP
1
−1
0
2
−2
1
20
2
0
CP
2
0
−1
6
−5
0
0
10
CP
3
0
−1
0
0
1
20
1
0
Total
−1
−2
8
−7
2
40
4
0
Here
, coalitio
ns o
f subp
ortfo
lios p
ortfo
lios in
cur th
e m
arg
ins:
ρ(C
P1
+
CP
2 )=
30
ρ
(CP
1 +
CP
3 )=
50
ρ(C
P2
+ C
P3
)= 2
0
ρ(C
P1
)= 2
0
ρ(C
P2
)= 1
0
ρ(C
P3
)= 3
0
Finally
, the th
ird e
xam
ple
is:
C10
C20
C30
C40
C50
Sh
ap
ley
Au
man
n −
S
hap
ley
CP
1
−1
−1
4
−2
0
26
.66
3
0
CP
2
0
−1
4
−3
0
6.6
6
10
CP
3
0
0
0
−2
2
6.6
6
0
Total
−1
−2
8
−7
2
40
4
0
wh
ere
the co
alitio
ns o
f subp
ortfo
lios in
cur:
ρ(C
P1
+ C
P2
)= 4
0 ρ
(CP
1 +
CP
3 )=
30
ρ(C
P2
+ C
P3
)= 1
0 ρ
(CP
1 )=
30
ρ(C
P2
)= 1
0 ρ
(CP
3 )=
20
On th
ese
exam
ple
s, we n
ote
that:
%•
Th
e S
hap
ley a
nd
Au
man
n-S
hap
ley a
lloca
tion
s do n
ot a
gre
e. A
n e
qu
al a
lloca
tion
in th
is settin
g w
ou
ld h
ave b
een
fortu
itous.
%•
Nu
ll allo
catio
ns d
o o
ccur. N
ull (o
r neg
ativ
e) a
lloca
tion
s can
not b
e ru
led
ou
t, and
do h
ap
pen
here
, both
for th
e
Conclu
sion
In th
is article
, we
have
discu
ssed
the
allo
catio
n o
f risk ca
pita
l from
an
axio
matic
persp
ectiv
e, d
efinin
g in
the
pro
cess w
hat w
e
call co
here
nt a
lloca
tion
prin
ciple
s.
Our o
rigin
al g
oal is
to e
stab
lish a
fram
ew
ork
with
in w
hich
fin
an
cial risk
allo
catio
n p
rincip
les
cou
ld b
e co
mp
are
d a
s
mean
ing
fully
as p
ossib
le. O
ur sta
nd
is th
at th
is ca
n b
e a
chie
ved
by
bin
din
g th
e co
nce
pt o
f coh
ere
nt risk
measu
res
to th
e
existin
g g
am
e th
eory
resu
lts on
allo
catio
n.
We
sug
gest tw
o se
ts o
f axio
ms, e
ach
definin
g th
e co
here
nce
of risk
cap
ital a
lloca
tion
in a
specifi
c se
tting
: eith
er th
e
con
stituen
ts o
f the
firm
are
con
sidere
d in
div
isible
en
tities
(in th
e co
alitio
nal g
am
e se
tting
), or, to
the
con
trary
, they
are
con
sidere
d to
be d
ivisib
le (in
the co
nte
xt o
f gam
es w
ith fra
ctional p
layers). In
the fo
rmer ca
se, w
e fi
nd
that th
e S
hap
ley v
alu
e is
a co
here
nt a
lloca
tion p
rincip
le, th
ou
gh
only
und
er ra
ther re
strictive co
nd
ition
s on th
e risk m
easu
re u
sed
.
In th
e fra
ctional p
layers
settin
g, th
e A
um
an
n-S
hap
ley
valu
e is
a co
here
nt a
lloca
tion
prin
ciple
, un
der a
mu
ch la
xer
diff
ere
ntia
bility
con
ditio
n o
n th
e risk
measu
re; u
nd
er lin
earity
, it is a
lso th
e u
niq
ue
coh
ere
nt p
rincip
le. In
fact, g
iven
that th
e
allo
catio
n p
roce
ss starts w
ith a
coh
ere
nt risk
measu
re, th
is coh
ere
nt a
lloca
tion
simp
ly co
rresp
on
ds to
the
gra
die
nt o
f the
risk
measu
re w
ith re
spect to
the
pre
sen
ce le
vel o
f the
con
stituents o
f the
firm
. As a
con
seq
uen
ce, th
e A
um
an
n-S
hap
ley
ap
pro
ach
,
beyon
d its th
eore
tical so
un
dn
ess, fu
rther h
as a
com
pu
tatio
nal fe
, in th
at it is a
s easy
to e
valu
ate
, as th
e risk itse
lf is.
Refe
rence
s
[1] P
h. A
rtzner, “A
pp
licatio
n o
f Coh
ere
nt R
isk M
easu
res to
Cap
ital R
eq
uire
men
ts in In
sura
nce
”, North
Am
erica
n A
ctuaria
l Jou
rnal
(19
99
)
[2] P
h. A
rtzner, F. D
elb
aen
, J.-M. E
ber, D
. Heath
, “Th
inkin
g co
here
ntly
”, RIS
K, v
ol. 1
0, n
o. 1
1 (1
99
7).
[3] P
h. A
rtzner, F. D
elb
aen
, J.-M. E
ber, D
. Heath
, “Cohere
nt m
easu
res o
f risk”, Math
em
atica
l Finan
ce, v
ol. 9
, no. 3
(19
99
), 20
3–
22
8.
[4] P
h. A
rtzner, J. M
. Ostro
y, “G
rad
ien
ts, Su
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Ap
pend
ix
A P
roof o
f the n
on
-neg
ativ
ity co
nd
ition
Th
e fo
llow
ing
resu
lt from
sectio
n 6
is pro
ved
here
.
Th
eore
m 8
A su
fficie
nt co
nd
ition fo
r a n
on
-neg
ativ
e, “n
o u
nd
ercu
t” allo
catio
n
to e
xist is:
+,ρ
Xi m
in λ
iXi∀
λ ∈
Rn
i∈N
λ
i≤
ρ i∈
N
i∈N
Pro
of: Le
t us re
call th
at w
e d
en
ote
d b
y 1
S ∈
Rn
the ch
ara
cteristic v
ecto
r of
the co
alitio
n S
:
(1S
)i =1
if i ∈S
0 o
therw
ise
A n
on-n
eg
ativ
e, “n
o u
nd
ercu
t” allo
catio
n K
exists w
hen
∃K
∈R
nsuch
that
Kt1
S ≤
ρX
i ∀S
N
i∈S
(19
)
Kt1
N =
ρX
i
i∈N
K ≥
0
Usin
g Fa
rkas’s le
mm
a, th
is is eq
uiv
ale
nt to
1N
yN
+1
S y
S ≥
0, ∀
yN
∈R
, ∀y
S ∈
R+,S
N
S
N
= ⇒
ρX
i yN
+ ρ
Xi y
S +
≥ 0
(20
) i∈N
S
Ni∈
S
which
in tu
rn is e
quiv
ale
nt to
yS
, ∀y
S ≥
0,S
N, a
nd
∀i ∈
N,
yN
≥−
Si
= ⇒
ρ( X
i)yS
≥−
ρX
i yN
(21
) S
Ni∈
Si∈
N
Now
, usin
g th
e h
om
og
eneity
an
d th
e su
bad
ditiv
ity o
f ρ,
ρX
i yS
= ρ
yS
Xi
S
Ni∈
SS
Ni∈
S
≥ ρ
y
S X
i
S
Ni∈
S
= ρ
Xi y
S
i∈N
Si
Th
ere
fore
, a su
fficie
nt co
nd
ition
for (1
9) (o
r (20
) or (2
1)) to
hold
, is
yS
, ∀y
S ≥
0,S
N
, ∀i ∈
N,
yN
≥−
S
i
= ρ
Xi y
S ≥
ρX
i (−y
N )
⇒
i∈N
Sii∈
N
Finally
, usin
g th
e d
efin
ition
λi
Si y
S , w
e ca
n w
rite th
e su
fficie
nt co
nd
ition
for (1
9)
∀λ
i ≥0
,i ∈N
,ρ λ
iXi ≥
ρX
i min
λi i∈
Ni∈
Ni∈
N
Note
that in
the la
st step
, we a
lso u
sed
ρ i∈
N X
i ≥0
, a n
ece
ssary
con
ditio
n fo
r the e
xiste
nce
of a
non
-neg
ativ
e, “n
o u
nd
ercu
t” allo
catio
n; ch
eckin
g
yN
=1
,yS
=0
∀S
N in
(20
) show
s this p
oin
t.
