Do secu i y p ices ise
o all when ma gins
a e aised?
Jean-Ma c Bo azzi
Má io R. Páscoa
Guille mo Ramí ez
Wo king Pape
# 616
2017
Do Secu i y P ices Rise o Fall When
Ma gins A e Raised?
By Jean-Ma c Bo azzi, M´
a io R. P´
ascoa and Guille mo Ram´
ı ez1
Pa is S. o Economics and Capula; Uni e si y o Su ey; No a S. o Business
and Economics
Abs ac
When epo ma gins a e aised by a cen al clea ing coun e pa y (CCP),
he impac on secu i y p ices depends on whe he he ma ke is ‘long’ o
‘sho ’. As bo h he long and he sho mus pos ma gins, he p ice impac
depends on which side is mo e le e aged: ade s long in he secu i y o
hose sho -selling i . I a aised ma gin o ces mo e posi ion unwind om
he long han om he sho , he p ice will go down o clea he ma ke .
Howe e , i sho posi ions a e mo e hi hen he long ones, a aised hai cu
leads o a highe secu i y p ice!
1We a e hank ul o commen s om he audience a he SAET 2017 con e ence (June
2017). G. Ram´ı ez was suppo ed by a doc o al g an gi en by FCT, Po ugal (g an
BD 74704/2010). P´ascoa and Ram´ı ez acknowledge suppo om p ojec PTDC/IIM-
ECO/5360/2012 (FCT, Po ugal).
1
1 INTRODUCTION 2
1 In oduc ion
Wi hou ehypo heca ion, aised ma gin ypically dec eases he p ice o he colla -
e al (houses p ices o mo gage, o he p ice o o he colla e al used o asse back
secu i ies). In ac , he only le e age p o ided is o he long side o he colla e al
ma ke , who is unding colla e al pu chases. Reduced le e age, e e y hing else
being equal, means less buying and hence a lowe colla e al p ice.
The si ua ion is less clea , howe e , in he p esence o ehypo heca ion, o
example in he secu i ies ma ke s. Secu i ies se e as colla e al when inancing
hei own pu chase and hen cash lende s manage o epledge o sho sell2 he
colla e al hey ha e accep ed. Tha is, in hese ma ke s, sel -colla e aliza ion com-
bined wi h colla e al euse allow o sho posi ions. Since le e age is p o ided o
bo h he long and he sho side, he p ice impac o a le e age educ ion can also
go ei he way. Wi h cen al coun e pa y clea ing houses (CCPs), bo h he long
side and he sho side need o pos ma gin and a e hus a ec ed by a ia ion o
hai cu 3: i is well documen ed how highe ola ili y (o jus highe sp eads on
yields o so e eign bonds ela i e o a AAA benchma k) make CCPs (and u u es
exchanges) inc ease he ini ial ma gins cha ged in epo ades o bo h sides o he
ma ke 4, agen s ha pledge secu i ies and agen s ha accep hese as colla e al
o cash loans. posi ions
I has been a gued, in mos o he li e a u e we e iew below, ha he impac
o highe hai cu s is always o lowe he p ice o he colla e al. I his we e ue his
would mean ha aising ma gin is always p ocyclical : i he p ice o he colla e al
2The coun e pa ies o hese sho s can also epledge he secu i ies (an indi ec ehypo he-
ca ion) and a new i e a ion s a s.
3S icly speaking, he hai cu indica es, in pe cen age e ms, how he loan alls below he
colla e al alue, whe eas he ini ial ma gin exp esses he colla e al as a pe cen age o he
loan (i is he e o e he in e se o one minus he hai cu ).
4I should be clea ha a simila analysis can be done eplacing secu i y wi h deli e able
u u es and hai cu wi h u u es ini ial ma gin.
1 INTRODUCTION 3
is al eady unde p essu e, a highe hai cu pushes i u he down in some icious
cycle ha in he wo s case can eed on i sel .
Howe e , o ake jus an example, i we obse e wha happened in he Eu opean
so e eign deb c isis, we see yields d opping a e some ma gins we e aised. This
is mo e no o ious o he 10 yea go e nmen bonds o he ollowing coun ies and
o he ollowing episodes: I eland, 12 and 26 No embe 2010 and again o he
six consecu i e ma gin inc eases om Ap il h ough June 2011; I aly, 9 No 2011
and July 2012; Po ugal, 11 May 2011 and 29 June 2011; Spain, 20 June 2012.
The e a e ne e heless also some down spi als in p ices5, bu his is ce ainly no
he mos equen ma ke eac ion a e ma gins a e aised. La e , in pa icula o
I ish and Spanish bonds, as he c isis s a ed o ade away, specula ion dec eased
and he all in hai cu s was accompanied by a all in yields.
The e may be o he ac o s behind such yield mo emen , bu i does sugges
ha he impac o aising ma gin is no always o push he p ice o he colla e al
down. Ou iew is ha in secu i ies ma ke s i depends on posi ioning. Fo his
eason, as posi ioning changes quickly o e ime, one can expec he impac o
ma gins o be mo e isible in sho e m ma ke eac ions a he han in long
e m ends. I is well-known ha a a ious imes specula i e posi ions we e
qui e signi ican . This aises he possibili y ha aising hai cu could a imes
ha e had mo e impac on he sho side. And a con ibu ing ac o o yields going
down is sho s being he le e aged ade s being o ced o dele e age when ma gin
is aised. In ma ke pa lance, a secu i y ma ke is long when he longs a e he mo e
le e aged agen s and sho when i is he sho s ha a e mo e le e aged ins ead.
We will gi e a mo e p ecise de ini ion o a e minology ha supe icially seems
o clash wi h ma ke clea ing (which equi es longs and sho s o be ma ched).
The emphasis is on whe e he le e age is when he hai cu ises. I he sho side
519 No embe 2010 and 25 Ma ch 2011 o I eland; h oughou Ap il 2011 and on 14 June
2011 o Po ugal
2 RELATION WITH THE LITERATURE 4
ge s o be educed mo e han he long side (which happens when ‘ he ma ke is
sho ’), hen he p ice o he secu i y going up, e e y hing else being equal, enables
ma ke s o clea .
2 Rela ion wi h he Li e a u e
Mos o he li e a u e emphasized he p ocyclical ole o epo hai cu s and he e-
sul ing p opaga ing down-spi al. Ad ian and Shin (2009) obse e a s ong posi i e
co ela ion be ween le e age and balance shee size o US inancial in e media ies,
in e ing om i a p ocyclical ole o le e age. A heo e ical wo k by B unne meie
and Pede sen (2009) no ices ha sho selling also equi es capi al in he o m
o ma gin and p o ides an in e es ing esul poin ing ou ha a ma gins spi al
a ises when long ma gins a e dec easing in p ices o sho ma gins a e inc easing
in p ices. I seems o be implici ha p ices migh inc ease wi h sho ma gins and
ha a coun e -cyclical pa e n could happen i he sho ma gins we e dec easing
in p ices, jus like he long ma gins.
The obse ed, no so in equen , escapes o wha jus be o e looked like an
implacable icious ci cle logic mo i a ed us. When, in esponse o a p ice d op
and ola ili y, ma gins a e inc eased and he p ice subsequen ly ises, is he aised
ma gin a con ibu ing ac o o his ise o does he ise in p ice happen in spi e
o a headwind ma gin e ec ? We sugges ha he ma gin e ec can some imes be
suppo i e: highe ma gin also can nudge p ices highe .
In he empi ical li e a u e, he analysis ha Go on and Me ick (2012) did
o he 2008 inancial c isis a gued ha he spi al in e ac ion o epo hai cu s and
p ices o s uc u ed secu i ies led o a un on epo and insol ency o he US banking
sys em. While ha p o-cyclicali y may ha e ac ually occu ed in wha was hen
a long ma ke o hose secu i ies, K ishnamu hy e al. (2014) downplayed i s

2 RELATION WITH THE LITERATURE 5
magni ude and impac on he whole inancial sys em. These di e en assessmen s
may be due, as In an e (2015) poin ed ou , o Go on and Me ick (2012) ha ing
ocused on bila e al ades, while in he la ge US i-pa y ma ke hai cu s may
ha e been changed much less. We a gue o a balanced iew: in a ma ke wi h
bo h longs and sho s, le e age by i sel does no ha e o be p ocyclical, i depends
whe e le e age is mo e p e alen .
In any case, he p esumed p ocyclical na u e o ma gins se by CCPs (mo e
sys ema ic han in o e - he-coun e (OTC) ma ke ) has e en led o he iew ha
CCPs migh be ac o o ins abili y. This iew was ne e heless me wi h some
scep icism by Eu opean policy-make s ha alue CCPs’ ole as a i ewall agains
he p opaga ion o de aul shocks and ind an inc easingly cen ally clea ed ma ke
easie o moni o (see Cons ˆancio (2012)).
In he case o he Eu opean so e eign deb c isis, a lo o posi ioning was done
by banks (hedges o illiquid posi ion) and hedge unds. We ind acking CCP
ma gin a good way o gauge e olu ion o ma gins ele an o hose agen s. No
jus because Banks end o use CCPs bu all ele an ma gins ha e be mo ing
b oadly oge he : (1) hedge unds use p ime b oke s who ac as clea e s o hem
(also cha ging ma gin ega dless o whe he he ade is sho o long) (2) Fu u es
con ac s (whe e he exchange asks o ini ial ma gin ega dless o posi ion di ec-
ion). Ou ocus on cen al clea ing was no chosen only o analy ical pu poses
bu also seems o i be e wi h wha happened in ha c isis we chose o illus a e
ou poin .
The e is an impo an li e a u e ha has s udied colla e al in he gene al equi-
lib ium (GE) heo y adi ion and his is he li e a u e ha is closes ela ed o ou
wo k. The pape s by Geanakoplos (1997) and Geanakoplos and Zame (1997) se
he s age: adically changing he way bo owing is ea ed in GE models, eplacing
abs ac nega i e asse posi ions (subjec o exogenous nai e cons ain s) by loans
2 RELATION WITH THE LITERATURE 6
secu ed by asse s. Subsequen ly, many con ibu ions explo ed implica ions o how
colla e al ma gins a e se (like absence o Ponzi schemes6, wel a e p ope ies7o
he ola ili y o colla e al p ices8) bu he mos ele an o ou heme a e he pa-
pe s on how le e age impac s asse p ices, s emming om Geanakoplos (2003).
In pa icula , he pape s by Geanakoplos (2010), Fos el and Geanakoplos (2012)
and Fos el and Geanakoplos (2015), con empla e a con inuum o ade s, wi h di -
e en belie s on binominal asse e u ns, and compa e unsecu ed bo owing wi h
bo owing agains a colla e al which is ei he a du able good o a inancial asse
ha canno be sho -sold.
One o he main insigh s o hese h ee seminal pape s is ha le e age makes
he p ice o he asse ha se es as colla e al ise. To s a , le e age equilib ium
p ices we e shown o be g ea e han no-le e age p ices. I was no iced ha p ices
inc ease wi h exogenous loan- o- alue a ios, bu he emphasis was hen placed
on he endogenous de e mina ion o le e age ( uling ou de aul on non- ecou se
loans). The endogenous dele e age always ein o ced a all in p ices caused by bad
news abou he wo se case payo .
We use Fos el and Geanakoplos’ (2012) and (2015) binomial model wi h a
con inuum o deg ees o agen s’ op imism on asse payo s. We ind i su p isingly
e sa ile and ap o illus a e ich and sub le phenomena o gene al ele ance. We
use i o show ha lowe ma gin migh no always aise asse p ices, i asse s can be
sho sold. The key di e ence - mo e impo an han ecou se - is i he colla e al
may be eused by he c edi o on whose hands i was pledged. In a nu shell: Asse
p ices inc ease wi h le e age in a long ma ke and dec ease in a sho ma ke . In
a long ma ke he op imis s a e le e aged, while i is he pessimis s who a e i i is
sho . To make he compa ison easie , ou examples a e as close as possible o he
6See A aujo, P´ascoa and To es-Ma inez (2002) and Kuble and Schmede s (2003).
7See A aujo, Kuble and Schomme (2012) and Geanakoplos and Zame (2014).
8See B umm, G ill, Kuble and Schmede s (2015).
2 RELATION WITH THE LITERATURE 7
amewo k used in hose pape s, in pa icula o he benchma k model in Fos el
and Geanakoplos (2012).
As in Fos el and Geanakoplos (2012), we con empla e a binomial economy ( he
ini ial node is ollowed by wo nodes) and wo asse s, one ha can be pledged as
colla e al and ano he one ha canno . The o me has s ochas ic e u ns and he
la e is iskless. We can in e p e he o me as a isky secu i y and he la e
as cash. Also as in Fos el and Geanakoplos (2012) and Fos el and Geanakoplos
(2015), he e is a con inuum o agen s, wi h he same ini ial holdings o asse s a
he i s da e bu di e ing in he weigh s a ached o he wo s a es o na u e in
hei linea u ili y unc ions.
We depa om Fos el and Geanakoplos (2012) and Fos el and Geanakoplos
(2015) in h ee aspec s. Fi s and o emos he colla e al can be eused (and sho
sold). Reuse o inancial colla e al has always been done bu had been igno ed in
he li e a u e o a long ime. Recen wo k includes he pape by Bo azzi, Luque
and P´ascoa (2012), add essing le e age and exis ence o euse equilib ia in OTC
ma ke s, and he wo king pape by B umm, G ill, Kuble and Schmede s (2017),
which s udies ola ili y and wel a e implica ions o euse.
Second, loans a e ecou se, as is ypically he case in epo (wi h an excep ion
allowed by he Fed du ing he 2008 c isis). De aul is a bank up cy igge 9and,
he e o e, no so common in epo ma ke s. Fo his eason, and also o simplici y
and ac abili y, we ocus on asse p ice luc ua ions and look o ull epaymen
ou comes10.
Thi d, agen s ha e endowmen s (o he iskless asse ) also a he wo nodes
9The colla e al alue can be below he p omised epu chase p ice and ye de aul migh
no occu . On he o he hand, epo colla e al is exemp by he US bank up cy code om
au oma ic s ay and he cash lende can liquida e he colla e al immedia ely, bu i he
liquida ion alue exceeds (o is below) he epu chase p ice, he di e ence is deemed p ope y
o he insol en es a e (o can be claimed agains ha es a e, espec i ely).
10 Wo k in p og ess by Bo azzi, P´ascoa and Rami ez (2017) discusses how bank up cy can be
modelled bu he e we lea e ha in ica e non-con ex p oblem aside.
2 RELATION WITH THE LITERATURE 8
o he second da e. Such endowmen s played no ole in ensu ing sol ency in he
non- ecou se case, bu we can now choose hem high enough so ha he sho s
can con empla e maximum le e age o a gi en le el o hai cu , wi hou going
bank up a he second da e.
Ou model is used o cons uc wo benchma k cases o a epo ma ke clea ed
h ough a CCP. One whe e he ma ke is sho and aising hai cu s/ini ial ma gin
leads o highe secu i y p ices (as in he so e eign deb s episodes men ioned abo e).
Ano he , whe e he ma ke is long and aising he hai cu makes ha he secu i y
p ice go down. Wha de e mines whe he he ma ke will be long o sho is how
secu i y e u ns a y ac oss nodes by compa ison wi h endowmen s o he iskless
asse . The la e migh be oo low (in he s a e whe e he secu i y has highe
e u n) o allow o he sho s o le e up o wha he hai cu allows o .
The pape shows how ehypo hecac ion allows o coun e -examples o well-
known esul s ha had been es ablished in he absense o such euse o he col-
la e al. In he nex sec ion we p o ide a e y b ie mo i a ion on he basis o
obse ed hai cu . In sec ion 4 we desc ibe he model o cen ally clea ed epo
and in he ollowing sec ions we p esen he examples, discussing in mo e de ail a
sho ma ke whe e p ices ise in esponse o highe ma gin. Tha being said we
canno claim ha hai cu s ha e a s abilizing e ec by hemsel es. Fi s he e a e
many ma ke s whe e le e age is only o e ed o he long side. Also when sho s a e
possible, le e aged posi ioning could s abilise he p ice when ma gin is aised only
i he ma ke would be well equipped, in he sense o ha ing plen y o le e aged
sho s willing o ake p o i in any excessi e down p ice mo e.
4 THE MODEL 15
asse Xand secu i y Y, espec i ely, and a he same ime can pledge ψiuni s
o he secu i y in epo (in exchange o ob aining a loan wi h he same alue
qψi) o accep θiuni s o he secu i y as colla e al (in exchange o gi ing a loan
wi h he same alue qθi). We assumed epo con ac s a e se led a he secu i y
ma u i y da e12 . In his simple case, in he second leg, he loan is epaid (a he
p ede e mined epu chase p ice) bu wha is ac ually gi en back o he bo owe
is jus he asse payo s.
A non-nega i i y cons ain applies o X, epo long and epo sho posi ions:
(2) xi, θi, ψi≥0
Secu i y posi ions a e bounded by he in e ac ion wi h epo posi ions. Wha can
be pledged as colla e al canno exceed he secu i y long posi ion. So a , his is
would be a adi ional colla e al cons ain . Bu we go beyond, as he colla e al
can be eused by he c edi o : can be pledged in ano he epo ade o can be
sold. Selling he colla e al ha is empo a ily in he c edi o ’s hands is wha a
sho sale is. Naked sho sales, un ela ed o he amoun o he secu i y ha was
bo owed (accep ed as colla e al), a e no allowed in eali y and a e uled ou in
ou model. Hence, secu i y posi ion can be nega i e yibu sho sales a e bounded
by he ne epo posi ion. The box cons ain cap u es he wo-sided in e ac ion
be ween secu i y and epo posi ions13:
(3) yi+θi−ψi≥0
In o he wo ds, he ne secu i y i le balance held by each agen (in each agen ’s
12 also known as epo o ma u i y...al e na i ely, a 3 da es a ian o he model could be wo ked
ou easily.
13 See Bo azzi, Luque and P´ascoa (2012) on his cons ain and also Du ie (1996) o p e-
cu so s o his cons ain .

4 THE MODEL 16
“box”) mus be non nega i e. I he euse o he colla e al we e no allowed (as
in mo gages), his cons ain would collapse o a plain colla e al cons ain o he
o m yi−ψi≥0.
Repo ades a e cen ally clea ed by an exchange (also known as cen al clea -
ing coun e pa y, CCP), ha cha ges a ma gin o bo h sides o he ma ke , in a
p opo ion 1 −ho he alue o he epo ade14. Then, agen ibudge cons ain
a da e 0 is as ollows:
(xi−1) + q(yi−1) + q(θi−ψi) + (1 −h)q(θi+ψi)≤0(4)
O equi alen ly,
(xi−1) + q(yi−1) + q(2 −h)θi−qhψi≤0(5)
A he second da e, agen s se le hei epo obliga ions, using hei endowmen s
and he e u ns om hei i s da e secu i y (long o sho ) posi ions. Repo loans
a e epaid a he epo a e ρ. The CCP gi es back o he agen s he ma gins i
had collec ed, acc ued o in e es , a he same a e, he epo a e
The budge cons ain o agen iin s a e sis as ollows:
Ci
s=ωs+xi+yiDs+ (1 + ρ)q(θi−ψi) + (1 + ρ)q(1 −h)(θi+ψi)≥0(6)
Tha is,
Ci
s=ωs+xi+yiDs+ (1 + ρ)q[(2 −h)θi−hψi]≥0(7)
A plan (xi, yi, θi, ψi) is said o be easible o agen ia p ices (q, ρ) i i sa is ies
14 See Bo azzi, Luque and P´ascoa (2012) on he OTC case and how hai cu s a e di e en in
ha case.
4 THE MODEL 17
he budge cons ain s (5) and (7), he box cons ain (3) and he non-nega i i y
es ic ions (2). A plan ha maximizes (1) among all plans ha a e easible o i
a (q, ρ) is said o op imal o igi en (q, ρ).
4.2 Equilib ium
The CCP collec s ma gins om bo h sides o he epo ma ke and is supposed o
pay back hese ma gins, a he epo in e es a e. To accomplish his, i should
in es he ma gins and he e is no be e way han o in es also in epo, ha is,
o be also a epo long (accep he secu i y as colla e al and p o ide a loan, on i s
own), wi h a posi ion ΘEX ≥0.
The ma ke clea ing condi ion o he epo ma ke is:
Z1
0
θidi + ΘEX =Z1
0
ψidi,(8)
Wha he CCP in es s in epo is he ma gins i collec s om bo h sides o he
ma ke , ha is,
ΘEX = (1 −h)·Z1
0
θidi +Z1
0
ψidi
Hence, epo ma ke s clea i
(2 −h)Z1
0
θidi =hZ1
0
ψidi(9)
The ma ke clea ing condi ions o he wo asse ma ke s a e:
Z1
0
xidi = 1 and Z1
0
yidi = 1(10)
An equilib ium consis s in p ices (q, ρ) and an alloca ion o agen s’ plans (xi, yi, θi, ψi)i∈(0,1)
5 SHORT MARKETS VERSUS LONG MARKETS 18
such ha (i) each agen ’s plan is op imal o him a hese p ices and (ii) asse
and epo ma ke s clea 15.
5 Sho ma ke s e sus long ma ke s
5.1 Le e age
Le e age in secu i ies ma ke is usually measu ed by he a io o he alue o
posi ion o he down paymen needed o build ha posi ion. In he case o a
long posi ion, he pu chase o he secu i y can be inanced in he epo ma ke -
pledging he whole long posi ion and ob aining a cash loan wi h he same alue,
bu incu ing he cos o paying he ma gin o he CCP, ha is, spending jus a
ac ion 1 −ho he secu i y alue. Le e age is gi en by 1/(1 −h). Analogously,
in he case o a sho posi ion, i he agen is sho selling he whole colla e al ha
he has accep ed, i p esumes ha he ga e a cash loan o he same alue and paid
a ma gin o he CCP. Again, he ne cos is jus he ac ion 1 −ho he secu i y
alue being sho ed. So, le e age is once mo e gi en by 1/(1 −h). Howe e , in
bo h cases, his is wha le e age is when he agen has decided o le e up o ull
po en ial gi en he hai cu . Lowe le e age would esul i he long posi ion we e
no en i ely pledged o i he sho posi ion we e lowe han he whole colla e al
ha was accep ed.
Colla e al euse may play also a ole in allowing o he longs o le e up o ull
hai cu po en ial. The agg ega e ( ully) le e aged long posi ion, which is in e sely
p opo ional o he hai cu 1 −h, may exceed wha he agg ega e ini ial holdings
we e o his secu i y. Such ou come can only sa is y ma ke clea ing i sho sales
a e done in equilib ium. Howe e , how much le e age can be achie ed depends
15 Second da e ma ke clea ing ollows om asse ma ke clea ing and he ul illmen o second
da e budge cons ain s as equali ies (due o mono onici y in p e e ences).
5 SHORT MARKETS VERSUS LONG MARKETS 19
bo h on how weal hy agen s a e and how low he hai cu is.
In equilib ium and o he ma ke a la ge we will say ha he ma ke is long i
ne ade o longs is mo e sensi i e o le e age han sho s’. The ma ke is sho i
highe sensi i i y is wi h sho s. In he examples ha we p esen in his pape we
exhibi equilib ia whe e he longs o he sho s a e le e ed up o ull po en ial bu
no bo h a he same ime, hence p o iding clea examples o ‘long’ and ‘sho ’
ma ke s. The side hi ing ha ull le e age cons ain - only one side does- will
ell us whe he he ma ke is long o sho , espec i ely. In bo h cases, being
ully le e aged means ha he secu i y posi ion (long o sho , espec i ely) can
be buil o he mos ha he ma gin downpaymen allows o (by combining he
i s da e budge and box binding equali ies)16. Second da e endowmen s should
no be an obs acle o such le e age. Tha is, in he case o a sho ma ke , s a e D
endowmen s should be high enough (o ma gins no oo low) o allow o he sho s
o ully le e while s ill ha ing non-nega i e income (no becoming insol en ) a
ha s a e, whe e secu i y e u ns ou weigh epo epaymen s. Simila ly, in he case
o a long ma ke , s a e Uendowmen s should be high enough o allow o he longs
o ully le e wi hou becoming insol en in he s a e whe e secu i y e u ns all
below he epo epaymen ha is due.
Finally, we no e ha in bo h examples, he colla e al ge s eused. In he
‘sho ’ case his is ob ious bu we obse e also ha in he ‘long’ case sho sales
a e always done in equilib ium. The longs’ weal h ( hei ini ial holdings o he wo
asse s, e alua ed a equilib ium p ices) is high enough (by compa ison wi h wha
he hai cu is), so ha agg ega e long posi ions exceed he gi en agg ega e ini ial
holdings.
We will show nex ha , in ou economy, whe he he ma ke is long o sho is
wha de e mines how inc eases in le e age ( educ ions in ma gin) a ec he p ice
16 Mo eo e , he e can no be an equilib ium whe e bo h he longs and he sho s a e ully
le e aged, as shown in Appendix A.4.
5 SHORT MARKETS VERSUS LONG MARKETS 20
o he secu i y.
We also iden i y condi ions ha gi e ise o sho and long ma ke s. Sho
ma ke s a e likely o occu when he endowmen in he s a e ha sho selle s
alue less (ωDin his economy) is high ela i e o he endowmen in he o he
s a e. When ωUis high enough ela i e o ωD, a long ma ke is mo e likely o
occu .
5.2 Le e age and p ices
Mo e o mally, le us ocus on equilib ia whe e all sho s ake he same sho
posi ion and all longs ake he same long posi ion, as will be he case in he
examples p esen ed below. We e e o such equilib ia as ypical equilib ia. The e
will be a ma ginal agen mwho is indi e en be ween buying o selling secu i y
Yand we deno e a ypical agen in he se (m, 1) by Hand a ypical agen in
he se (0, m) by L. The o me is a pessimis and will be a sho . The la e is
an op imis and will be a long. I will be shown om i s o de condi ions wi h
espec o y ha when minc eases qdec eases.
De ini ion 1. We say ha a ypical equilib ium (q.ρ, x, y, θ, ψ)is a sho (long)
ma ke equilib ium i he ne ade o a sho is mo e (less, espec i ely) elas ic
wi h espec o he loan- o- alue a io h han he ne ade o a long.
In o he wo ds, in a sho ma ke i is he sho s who a e mo e sensi i e o
hai cu s, o changes in le e age condi ions. They hey beha e as being he ones
ha a e mo e le e aged. The elas ici y o he ne ade o a sho wi h espec
o hmeasu es wha is he p opo ional change in he absolu e alue o he equi-
lib ium ne ade (yH−1) o a sho in esponse o some p opo ional change in
h, by allowing all equilib ium a iables o adjus o his change ( ha is, mu a is
mu andis). I is gi en by dln(1−yH)
dln h, as yH<0. This should no be con used wi h

5 SHORT MARKETS VERSUS LONG MARKETS 21
he pa ial elas ici y ∂ln(1−yH)
∂ln h, which assumes he secu i y p ice and he epo a e
o emain cons an (a ce e is pa ibus elas ici y). Simila ly, he elas ici y o he ne
ade o a long wi h espec o hmeasu es how sensi i e is he absolu e alue o
he ne ade o a long o changes in hand is gi en by dln(yL−1)
dln h, o yL>1.
Ou i s esul es ablishes ha he way how secu i y p ices espond o ma gins,
in ypical equilib ia, depend on whe he ma ke s a e long o sho .
P oposi ion 1. The secu i y p ice qinc eases wi h he loan- o- alue a io hin a
long ma ke bu dec eases wi h hin a sho ma ke .
P oo . Ma ke clea ing equi es myL+ (1 −m)yH= 1. As he loan- o- alue a io
hchanges, mhas o change so ha 1 emains a con ex combina ion o he new
posi ions yLand yH. Mo e o mally,
(11) m=1−yH
yL−yH=1
1 + yL−1
1−yH
We see ha mis inc easing wi h he a io 1−yH
yL−1o he absolu e alue o he
ne ade o a sho (1 −yH) o he ne ade o a long (yL−1). Hence, m
inc eases (and, he e o e, q alls) wi h hi and only i he loga i hm o his a io
inc eases wi h h, ha is, i and only i dln(1−yH)
dh >dln(yL−1)
dh , o equi alen ly, i
and only i dln(1−yH)
dln h>dln(yL−1)
dln h, as claimed. And qinc eases wi h hi and only i
dln(1−yH)
dln h<dln(yL−1)
dln h.
In he nex subsec ions we examine how sho o long ma ke s may a ise and
see how secu i y p ices ge a ec ed by ma gins in bo h cases.
5.3 Sho ma ke
We cons uc an equilib ium wi h a sho ma ke . This ype o equilib ium only
occu s when ωDis la ge enough so ha wha ends up cons aining he sho
5 SHORT MARKETS VERSUS LONG MARKETS 22
posi ion o agen s i∈(m, 1) is no he non-nega i i y o Ci
Dbu ins ead he
binding box cons ain . The sho s sell all hei endowmen s o asse s Xand Y o
build up he la ges possible sho posi ion in Y. We ind ha he posi ions o a
pessimis agen Has ollows:
yH=−θH=−1 + q
q·1
1−h
(12)
ψH=xH= 0(13)
CH
U, CH
D>0
Equa ion (12) shows ha when i is he box cons ain ha bounds sho -selle s’
posi ions, hei posi ions a e in e sely ela ed o he hai cu , ha is gi en by he
le e age 1/(1−h). We will see ha his is a sho ma ke as when ma gin is aised
sho s need o co e mo e posi ion han longs.
Deno e a ypical agen in he se (0, m) by L. This is a buye o bo h Xand
Y. F om ma ke clea ing in he secu i y we see ha he secu i y long posi ion o
his agen mus be he ollowing:
yL=1
m1 + (1 −m)1 + q
q·1
1−h
(14)
I he agen s ha a e long in Ya e pledging i (ψL>0) and no doing any e e se
epo (θL= 0), he epo ma ke clea ing condi ion (9) equi es
(15) ψL= (2−h
h)(1−m
m)1 + q
q(1 −h)
The box cons ain equi es yL−ψL≥0, which holds i and only i
(16) h/2≥(1 −m)1 + q
q
5 SHORT MARKETS VERSUS LONG MARKETS 23
The e will be a slack in he box o he non-le e ed agen Li (16) holds as a s ic
inequali y. The o he posi ions o agen Lsa is y
(17) CL
U= 0, CL
D>0, xL=1
m
Agen s in (0, m) a e mo ing all hei consump ion om s a e U o s a e D,
ha is, CL
U= 0.
To be su e ha hese plans a e op imal o each agen , we w i e agen i’s
op imiza ion p oblem in e ms o h ee choice a iables, yi,θiand ψi. We deno e
EiD=γiDU+ (1 −γi)DD, which is agen i’s expec ed paymen o secu i y Y. On
he igh o each es ic ion we w i e he co esponding Lag ange mul iplie .
max
Eiω+ (1 + q)+(EiD−q)yi+q(2 −h)ρθi−qhρψi
s. .
xi= (1 + q)−qyi−q(2 −h)θi+qhψi≥0 (µi
x)
yi+θi−ψi≥0 (µi)
Ci
s=ωs+ (1 + q)+(Ds−q)yi+q(2 −h)ρθi−qhρψi≥0 (λi
s)
θi≥0 (νi
θ), ψi≥0 (νi
ψ)
The i s o de condi ions (FOC) o he p oblem o agen i∈(0,1) a e:
yi:EiD−q=qµi
x−µi−X
s
λi
s(Ds−q)
θi:q(2 −h)ρ=q(2 −h)µi
x−µi−νi
θ−q(2 −h)ρX
s
λi
s
ψi:qhρ =qhµi
x−µi+νi
ψ−qhρ X
s
λi
s,
5 SHORT MARKETS VERSUS LONG MARKETS 24
No ice ha an agen may ha e bo h θiand ψiposi i e only i he mul iplie µi o
he box cons ain is ze o.
Gi en ha Xpays a null in e es a e, we look o equilib ia whe e he epo
a e is ze o as well17. Fo agen s i, wi h i > m, he i s o de condi ions a e
sa is ied o ρ= 0 wi h he ollowing mul iplie s:
µH
x=q−EHD
q(1 −h)>0
µH=q(2 −h)µH
x>0
νH
θ= 0
νH
ψ= 2q(1 −h)µH
x
λH
U=λH
D= 0
When i<m, he mul iplie s ha sa is y he i s o de condi ions o ρ= 0 a e:
µL
x=µL= 0
νL
θ=νL
ψ= 0
λL
U=ELD−q
q−DU
>0
λL
D= 0
Sho s ha e posi i e shadow alues o bo h he box cons ain (a possession alue
o he colla e al, due o he desi e o sho sell) and he non-nega i i y cons ain
on X(exp essing a wish o do a ”‘naked sho sale”’ in X). I he o me we e
posi i e and he la e ze o, he epo a e would ha e o be nega i e, below he
ze o in e es a e on X.
17 In Fos el and Geanakoplos (2012) he secu ed in e es a e was also ze o. I Xhad s ochas ic
e u ns, he epo a e could all in be ween he implied in e es a es on X, in e p e able as
in e es on ese es (IOR).
A APPENDIX 31
(a) yi(b) θiand ψi
Figu e 6: Equilib ium co esponding o hai cu o 1% (h= 0.99): m= 0.881,
q= 0.754, xi= 1/m o i≤mand xi= 0 o i>m(assuming ωD≥107.6).
(a) yi(b) θiand ψi
Figu e 7: Equilib ium co esponding o hai cu o 3% (h= 0.97): m= 0.801,
q= 0.789, xi= 1/m o i≤mand xi= 0 o i>m(assuming ωD≥34.7).
An agen abo e he ma ginal agen (i>m) is sho selling he whole colla e al
ha was pledged o him. An agen below he ma ginal agen (i < m) is long in
he secu i y bu pledging less han he long posi ion.
Ma ke clea ing equi es myL+(1−m)yH= 1. As he hai cu inc eases, mhas
o change so ha 1 emains a con ex combina ion o he new posi ions yLand yH.
Now, he g aphs on he le -hand-side o Figu es 6 and 7 show ha , as he hai cu

A APPENDIX 32
is aised, he ully le e aged (indi idual) sho posi ion |yH| alls much mo e han
he (indi idual) long posi ion yL. The sho posi ion yHchanges om −232.7 o
−75.6 (is 32% o wha i was be o e), while he long posi ion yLchanges om 32.7
o 20.0 (is s ill 61% o wha i was be o e). Bo h yHand yLgo close o 1, bu
he o me much mo e han he la e . The weigh s on he con ex combina ion
mus e lec ha : m alls and, he e o e, q=EmD ises.
We saw in he p oo o P oposi ion 1 ha mis inc easing wi h he a io 1−yH
yL−1
o he absolu e alue o he ne ade o a sho (1 −yH) o he ne ade o a
long (yL−1). In a sho ma ke , his a io alls as he hai cu is aised. Figu e
8 illus a es how mand q a y as his a io changes. In he i s g aph we see
ha his a io is he slope o a ay joining he o igin o he poin ep esen ing ne
ade posi ions o he wo ypical agen s. The second g aph akes an ampli ied
look a wha happens close o he o igin, whe e mcan be ound as he e ical
coo dina e o he poin whe e he ay c osses he simplex. Such minduces q=
DD−(DD−DU)m, ead on he ho izon al axis.
A APPENDIX 33
(a) Ne ades o longs and sho s (b) mand qde e mined om ne ades a io
Figu e 8
2) On o he epo posi ions o he non-le e ed agen s: as he non-le e ed longs
ha e null shadow alues o he box cons ain , he abo e secu i y ades could
be implemen ed wi h o he epo posi ions o he longs (and he CCP). Each long
could ake posi ions ˜
θLand ˜
ψLon bo h sides o he epo ma ke and ha e no slack
in he box (˜
θL=˜
ψL−yL). This is no he same as pledging he long posi ion in
exchange o he ne epo loan. Being on bo h sides o he ma ke en ails an ex a
ma gin, which will be gi en back by he CCP in he second leg. Tha is, he long
is a he same ime lending cash o he CCP (a he epo a e), he eby undoing
pa ially his secu ed bo owing. Wha ma ks he di e ence be ween le e ed and
non-le e ed agen s is no so much whe he he e is no slack in he box bu a he
how ied up is he secu i y posi ion (long o sho ) o he opposi e epo posi ion
(sho o long, espec i ely).
A APPENDIX 34
Such al e na i e epo ades would no change how q ela es o h. Ma ke
clea ing equi es bo h (2 −h)(1 −m)θH+ (2 −h)m˜
θL=hm ˜
ψLand myL=
1 + (1 −m)|yH|, whe e |yH|=θH. We ge ˜
ψL=2−h
1−h·1
2m. I is easy o see ha
˜
θL≥0 (and ˜
ψL≥yL≥ψL) i and only i (16) holds and, he e o e, equilib ium
p ices qa e s ill ela ed o hacco ding o he solid black cu e o Figu e 5.
3) On ne ing o clien s epo posi ions: we ha e assumed ha he epo sho
and long posi ions o each agen do no ge ne ed o he pu poses o assessing
ma ginable posi ions. Howe e , ou esul s a e obus o ull o pa ial ne ing18.
As shown in sec ion A.5, condi ions (16) and (19) s ill hold unde ull o pa ial
ne ing (and, he e o e, also Figu e 5). The only in e es ing di e ence is ha unde
ull ne ing, epo po olios o non-le e ed agen s will be uniquely de e mined, o
each hai cu : non-le e ed agen s can no longe ha e al e na i e epo po olios
wi h posi ions on bo h sides o he ma ke ( hose discussed in Commen 2)).
A.2 Long ma ke
A long ma ke is mo e likely o occu when ωUis la ge ela i e o ωD. The
equilib ium is again cha ac e ized by a ma ginal agen m∈(0,1). The se (m, 1)
o agen s has a ypical agen deno ed by Hand he se (0, m) has a ypical agen
deno ed by L. Now, he agen s wi h high alua ions ( he op imis s) o he expec ed
e u n o secu i y Yha e long posi ions on he secu i y and a e le e aged, leading
18 LCH.Clea ne does pa ial ne ing (see LCH.Clea ne SA (2011)): he ma ginable posi ion
is a epo long o sho posi ion dec emen ed by he ollowing: he minimum o he wo
mul iplied by a ac o signi ican ly less han one.
A APPENDIX 35
na u ally o a long ma ke as can be seen below.
yL=ψL=1 + q
q·1
1−h
(20)
θL=xL= 0(21)
CL
U, CL
D>0
while agen s wi h low alua ions o secu i y e u ns a e sho and no le e aged,
yH=1
1−m1−m·1 + q
q·1
1−h
(22)
θH=h
2−h
m
1−m
1 + q
q(1 −h)
(23)
ψH= 0(24)
xH=1
1−m
(25)
CH
U>0CH
D= 0
The box cons ain o he non-le e ed sho s holds i and only i yH+θH≥0, o
equi alen ly, o
(26) h/2≤1−m1 + q
q
The p oblem o he agen s is he same as in he sho ma ke example. Fo
he posi ions p esen ed abo e, he i s o de condi ions a e sa is ied when agen s’
posi i e mul iplie s a e he ollowing: µL
x=ELD−q
q(1−h),µL=qhµL
x,νL
θ= 2q(1 −h)µL
x
and λH
D=q−EHD
DD−q.
We will conside he case when he pa ame e s o he economy a e all as be o e
wi h only one excep ion: we in e change he alues o he endowmen s ωUand ωD.
Le DU= 0.70, DD= 1.15. We assume now
A APPENDIX 36
Assump ion 2:ωD= 0 and ωU≥(1 + DD)DD−(2−h)DU
DU(1−h)( o ensu e ha
CL
U≥0).
Agen s’ belie s a e again gi en by γi=i. Now, CH
D= 0 implies ha le e age
should be such ha
(27) 1
1−h=(1 −m)(1 + q)
DD−q+ 1q
(1 + q)m
whe e m=DD−q
DD−DU.
In Figu e 9, condi ion (26) holds he egion abo e he g ay cu e. Combining
condi ions (27) and (26) we ge a lowe bound on hcompa ible wi h ψH≥0. Fo
habo e ha lowe bound (h≥0.96189), equa ion (27) desc ibes how qe ol es
wi h h. The black cu e in Figu e 9 illus a es how he secu i y p ice inc eases
wi h he loan- o- alue a io h(and, he e o e, wi h le e age) in he long ma ke 19.
This ela ionship is gene alized in he ollowing p oposi ion, p o en in sec ion A.3:
P oposi ion 3. I commodi y endowmen s sa is y Assump ion 2, hen he ma -
ke is long and he secu i y p ice inc eases as le e age goes up (and he e o e as
ma gins dec ease).
Obse e ha sho sales a e always done in equilib ium. In ac , yH<0 i and
only i m1+q
q
1
1−h≥1. By (27), we ge yH<0 i and only i (1−m)(1+q)
DD−q+ 1 ≥1,
which holds i ially.
19 As in he sho ma ke examples, we could ha e allowed now each non-le e ed sho o be
jus p edominan ly epo long, a he han exclusi ely epo long, wi h no slack in he box,
bu s ill wi h a (sho ) secu i y posi ion less ied up he (long) epo posi ion, han is he
case o he long coun e pa ies.

A APPENDIX 37
Figu e 9: q s. h- Long ma ke
A.3 P oo o P oposi ions 2 and 3
We show ha unde Assump ions 1 o 2, he secu i y p ice dec eases o inc eases,
espec i ely, wi h h, which also implies ha he ma ke is sho o long, espec-
i ely, by he a gumen in he p oo o P oposi ion 1.
Fo gi en a alue o h, he equilib ium p ice o he secu i y mus sa is y he
ollowing nonlinea sys em o wo equa ions:
H≡1
1−h=m(1 + q)
q−DU
−1q
(1 + q)(1 −m)
(19)
q=EmD(18)
F om equa ion (19) we see ha i we keep qcons an and inc ease m, le e age (H)
A APPENDIX 38
inc eases:
∂H
∂m =1 + q
q−DUq
(1 + q)(1 −m)+m(1 + q)
q−DU
−1q
(1 + q)(1 −m)2
=q
(1 + q)(1 −m)2(q−DU)[(1 −m)(1 + q) + m(1 + q)−q+DU]
=q(1 + DU)
(1 + q)(1 −m)2(q−DU)>0
whe e he las inequali y ollows om q > DU, due o (18). Now le ¯q= 1/q and
no e ha :
∂H
∂¯q=1
(1 −m)(1 + ¯q)2(1 −¯qDU)2(1 −¯qDU)2+m(1 + ¯q)2DU>0(28)
This implies ha ∂H
∂q <0 and ha along any isole e age cu e (ob ained by ixing
a alue o Hin (19)) we ha e:
dq
dm =−
∂H
∂m
∂H
∂q
>0
F om ∂H
∂m >0 we ge ha , o a ixed alue o q, inc easing mimplies mo ing o
a isole e age cu e co esponding o a highe alue o H. The new secu i y p ice
ha sol es he wo-equa ion sys em is hen ound a he in e sec ion o his cu e
and he s aigh line gi en by (18) (q=DD+m(Du−DD)). This implies ha as
le e age Hinc eases, he p ice o he secu i y qmus dec ease as shown in Figu e
10.
A APPENDIX 39
Figu e 10: Impac o le e age on equilib ium when he ma ke is sho
The e is ano he condi ion ha mus hold in equilib ium. I is condi ion (16),
ensu ing ha he box cons ain s is sa is ied o he non-le e ed longs. Elimina ing
hin he sys em o med by equa ion (18) and he equali y e sion o (16), we ge
a quad a ic equa ion in q, o each m, whose wo oo s a e desc ibed by cu es c1
and c2in Figu e 10. The g ey egion in Figu e 10 is he se o poin s (m, q) whe e
q∈[DU, DD] and (16) holds. Equilib ium pai s (m, q) a e ound wi hin his egion
whe e each isole e age ( o some h) c osses he dashed line ep esen ing (18). The
isole e ages po ayed in he igu e a e o he hai cu s o 1% and 3%, as in he
nume ical example o sec ion 5.3 (and o i s pa ame e alues o Dand ω).
An analogous a gumen p o es ha , in a long ma ke , p ices inc ease wi h
inc eases in le e age, as in P oposi ion 3. To see his simply no e ha now equi-
lib ium p ices mus sa is y bo h (18) and
(27) 1
1−h=(1 −m)(1 + q)
DD−q+ 1q
(1 + q)m
A APPENDIX 40
De ining ¯m= 1 −mwe ha e:
∂H
∂¯m=q(1 + DD)
(1 + q)(1 −¯m)2(DD−q)>0
Which in u n implies ha now ∂H
∂m <0. Le ing again ¯q= 1/q we ha e:
∂H
∂¯q=−1
m(1 + ¯q)2(DD¯q−1)2(DD¯q−1)2+ (1 −m)(1 + ¯q)2DD<0(29)
This implies ha ∂H
∂q >0 and, he e o e, along each isole e age we ha e dq
dm >0.
Now highe le e age implies mo ing o a isole e age o he le . This implies a
highe p ice q o he secu i y and a lowe alue o m. This is illus a ed in Figu e
11.
Figu e 11: Impac o le e age on equilib ium when he ma ke is long
Equilib ium equi es in addi ion ha (26) should hold in a long ma ke . This
ensu es ha he box cons ain o he non-le e ed sho s is sa is ied. Fo q∈
[DU, DD], such condi ion holds in he g ey egion o Figu e 11 (cons uc ed om