Re-using the collateral of others
A General Equilibrium Model of Rehypothecation ∗
Vincent Maurin†
European University Institute
[Preliminary and Incomplete - Please do not circulate]
March 30, 2014
Abstract
Rehypothecation rights allow financial institutions to re-use the collateral they received as lenders
for their own secured borrowing. In this paper, I introduce rehypothecation into a competitive economy where agents face collateral constraints due to limited commitment. On the one hand, rehypothecation facilitates collateral circulation as pledged assets become available to support additional
borrowing. On the other hand, limited commitment generates risk along the chain of repledges.
I show that an efficient financial structure without re-use rights can substitute for rehypothecation. With such a structure, the economy realizes an efficient use of durable assets as collateral and
rehypothecation is not warranted to alleviate collateral scarcity issues. This paper thus mitigates
claims that the recent set of regulations on collateral management will constrain financial markets
by lowering collateral velocity. Remarkably, this equivalence result holds with decentralized trading,
an important feature of Over The Counter (OTC) markets for swaps, derivatives or repos.
However, with incomplete markets, rehypothecation may increase efficiency by freeing up encumbered collateral. These gains are typically larger in OTC markets.
Keywords : Limited Commitment, Collateral Constraints, Rehypothecation, GE.
JEL codes : D41, D53.
∗ I am particularly indebted to my supervisor Piero Gottardi, for his guidance throughout this project, and Douglas
Gale for his advice during my stay at NYU in the Fall of 2013 where I started this work. I would also like to thank Alberto
Bisin, Cyril Monnet and David Sanches for insightful discussions and the audience at the EUI Micro Working Group for
useful comments.
† Email : [email protected]
1
1
Introduction
In credit markets, lenders frequently require borrowers to post collateral as a protection against default. Households pledge their house as collateral for a mortgage while banks use financial assets such
as government securities. While households remain the owners of their house during the transaction,
financial intermediaries can arrange for the legal transfer of the asset used as collateral. This practice,
known as rehypothecation allow lenders to sell or re-use collateral to carry additional transactions. While
rehypothecation may seem bizarre, the ISDA1 reports that borrowers grant such rehypothecation rights
in 73.7% of trades surveyed. To fix ideas, suppose bank A obtains cash from bank B against Treasury
Bills it agrees to buy back at a latter date (a repurchase or repo agreement). With rehypothecation,
bank B can re-use these T-Bills as collateral when it enters a swap agreement with bank C. Re-use
may occur over many similar transactions and ultimately creates “collateral chains” from the initial
borrower (bank A) to the ultimate lender (bank C). These chains allow (potentially scarce) good quality
collateral to circulate in financial markets. They also generate risk as borrowers may be wary of losing
access to their re-used asset. On account of those risks, Canadian law prohibits rehypothecation while
several pending reforms in the US2 effectively prevent re-use. Whether the decrease in risk warrants
the associated reduction in collateral circulation3 has not yet received a clear answer. The mere size
of collateralized markets (derivatives weigh $700 trillions in notional amounts) thus calls for a formal
analysis of rehypothecation.
In this paper, I provide a theoretical model to analyze the trade-off between circulation of collateral
and collateral risk. To the best of my knowledge, this paper is the first to address these issues in a
general equilibrium framework. The main contribution of this work is to show that rehypothecation can
ultimately substitute for missing financial securities in facilitating the circulation of collateral. However, rehypothecation proves redundant if agents may trade these collateral efficient securities. Indeed,
the whole financial structure (and not only rehypothecation rights) determines the velocity of the asset
used as collateral. With incomplete markets however, rehypothecation can strictly increase welfare by
enhancing collateral circulation. These gains appear to be larger in segmented markets where trading is
decentralized. This last effect arises because intermediaries in decentralized markets typically hold both
short and long positions and the former require posting collateral. As far as the “anti-rehypothecation”
reforms are concerned, their impact thus hinges crucially on the degree of market incompleteness.
The model builds on the seminal contribution by Geanakoplos (1996) where durable assets serve as
collateral, or promise-keeping devices4 . I extend this framework to allow lenders to re-use the collateral
they receive. Segregation of collateral from the lender’s other asset - the implicit contract provision in
1 International
Swaps and Derivatives Association, 2013 Margin Survey
VII of the Dodd-Frank Act on central clearing of swaps implies that collateral cannot be rehypothecated.
Similarly, dealer banks now have to obligation to notify their clients that their collateral can be segregated. The EU’s own
EMIR regulation introduces similar requirements for cleared swaps.
3 Singh (2011) measures collateral velocity, i.e. the number of times a piece of source collateral ends up being used,
around 3. Taken at face value, this number suggests that a general ban on rehypothecation would thus provoke a threefold
contraction in collateralized financial markets.
4 Some important models of default use a different technology than collateral to enforce promises. In Kehoe and Levine
(1993), defaulting agents are excluded from financial markets. Dubey et al. (2005) uses non-pecuniary penalties for default.
However, my work focuses precisely on the utilization of collateral, hence this modeling approach
2 Section
2
the literature - thus appears as a special case. My work contributes to the debate about asset scarcity
in collateralized markets5 as rehypothecation may provide an alternative to tranching or pyramiding
studied in Fostel and Geanakoplos (2012) or Gottardi and Kubler (2014). While rehypothecation generically underperforms these techniques, it enhances collateral circulation in incomplete markets. I also
emphasize the importance of collateral circulation and hence the higher value of rehypothecation when
trading is decentralized, a dimension that the recent literature on OTC markets mostly overlooked.
In my model, a buyer of a security receives an asset as a protection against default from the seller.
When the security grants rehypothecation rights, he may re-use the collateral to secure his own short
positions in other securities. The possibility of re-use clashes with limited commitment on the lender’s
side as he may fail to return the asset initially pledged as collateral. Still, borrowing can be enforced
in equilibrium with partial segregation of collateral at every round of pledging. I show how a simple
contract provision, the possibility to substitute cash for collateral at the delivery date (Assumption A),
allows to accommodate rehypothecation in the framework of Geanakoplos (1996) and others. Appendix
A develops a companion model to rationalize this assumption at a minor cost.
Rehypothecation received much attention from policymaking circles as the 2007 financial crisis exposed risky collateral management practices at some dealer banks6 . Monnet (2011), Singh (2011) or
Kirk et al. (2014) describe the trade-off between collateral circulation and collateral risk and provide
rough measures of circulation at the aggregate or bank level. However, the economic literature has yet to
provide a theory for rehypothecation. Bottazzi et al. (2012) account for circulation of securities through
repurchase agreements but sidestep the commitment issue attached to returning securities. Monnet and
Nellen (2012) do analyze these frictions but their model cannot capture circulation with one-shot bilateral trades.
Rehypothecation also emphasizes the similarities between collateral and money. As means of exchange, they determine the economy’s ability to trade efficiently in an environment with limited commitment. The concept of money velocity extends to collateral with rehypothecation as the same piece
of asset can back several transactions. When it comes to regulations, a restriction on rehypothecation
resembles the minimum reserve requirements in fractional reserve banking. The analogy also proves
compelling7 when comparing borrowers in my model to households in Lucas and Stokey (1987). The
collateral constraint replaces the Cash In Advance constraint of their monetary model. The commitment
problem put aside, rehypothecation would allow a Lucas agent to front some of his to finance his cashgood consumption.
Finally, my work also connects to a broader literature on leverage. Financial institutions typically
finance their asset with little equity or small haircuts in the repo market. With rehypothecation, dealer
banks may similarly intermediate significant volumes with a small amount of proprietary collateral. Several contributions including prominently Admati et al. (2010) fueled a hot debate on the social costs
and private benefits of high leverage. The literature generally recognizes that leverage mainly amplifies
5 See
CGFS (2013) for a recent take on the practical aspects of this question
(in)famous is the 2011 failure of MF Global, a broker dealer who took bets on the European debt market re-using
the collateral of its clients
7 I thank Cyril Monnet for pointing out this analogy
6 Most
3
shocks in the presence of frictions. For instance, Allen and Gale (2000a) discuss the role of risk-shifting
while Shleifer and Vishny (1992) stress liquidity mismatch through fire sales and Allen and Gale (2000b)
highlight financial network externalities. In my framework, the leverage induced by collateral re-use
cannot be disentangled from the limited commitment friction which generates collateral risk.
The paper is organized as follows. Section 2 introduces the model and discusses its novel features.
The main equivalence result on rehypothecation appears on Section 3 while Section 4 shows how rehypothecation can restore efficiency in an incomplete market environment. Finally, Section 5 concludes.
2
The Model
2.1
Set-up
This section presents the set-up of a general equilibrium model of financial markets with rehypothecation. The additional features imposed over the standard one-sided limited commitment environment
(henceforth 1SLC) require a comprehensive exposition. In order to streamline the presentation, I delay
the discussion of these novel assumptions to section 2.4. Section 2.2 formulates the agent’s optimization problem and defines the equilibrium concept. Section 2.3 discusses the implications of absence of
arbitrage in a rehypothecation economy and shows the existence of equilibrium with rehypothecation.
2.1.1
Physical Environment
Consider a 1 good endowment economy with two periods t = 0, 1 and several states of the world s = 1, ...S
in period 1. The economy is characterized by a set of agents I = {1, 2, ..., I} with endowments {ω i (s)}i∈I
s∈S .
Each type i represents a continuum of agents of mass 1. Preferences are Von Neumann-Morgenstern
over consumption streams. Function ui denotes instantaneous utility so that agent i’s preferences over
consumption bundle (c0 , c1 ) are given by
U i (c0 , c1 ) = ui (c0 ) +
S
X
π(s)ui (c1 (s))
s=1
where π(s) denotes the probability that state s realizes in period 1. Utility functions ui : R++ → R are
strictly monotone, C 2 , strictly concave and verify the Inada condition : limc→0 uic (c) = 0.
The economy is endowed with a tradable Lucas tree an asset which delivers a quantity x(s) of
consumption good in each state s ∈ {1, ...S} of period 1 but no dividend in period 0. Initial holdings of
P i
i θ0 .
the tree are {θ0i }i∈I ∈ RI+ while θ0 stands for the total quantity of tree in the economy, i.e. θ0 :=
Each agent has two different accounts called a segregated and a non-segregated account to handle
his holdings of the tree. This distinction becomes effective when I describe securities in the financial
environment. Finally, agents can renege at no cost on promises to deliver goods in the future.
4
2.1.2
Financial Environment
Agents cannot pledge their non-tree endowment as they suffer no utility loss from reneging on their
promises. However, they can still pledge the tree as collateral to borrow.. Finally, I allow lenders to
re-use the collateral they receive. A financial security (or simply security) is thus defined as follows :
Definition 1: A security j is a triplet (R̄j , k̄j , αj ) ∈ RS+ × R+ × [0, 1] where. R̄j (s) is the face value
in state s, k̄j is the amount of tree to be posted as collateral and αj is the fraction of this collateral to
be held in the buyer’s segregated account.
My definition with rehypothecation collapses to the usual concept of a collateralized security (see
(Geanakoplos, 1996) and others) when αj = 1. When αj < 1, the security grants rehypothecation rights
to the buyer. The set of all tradable securities is J . In a nutshell, posting collateral gives credibility to
the borrower’s promise while segregation protects the latter from lender’s wrongdoings.
The transfers involved by the sale of asset j are illustrated on Figure 1. When he sells a promise
j, an agent must transfer k̄j physical units of the tree, a fraction αj of which is segregated8 . On the
contrary, the lender may use the 1 − αj fraction of non-segregated collateral to carry his own transactions. Hence, he implicitly issues a promise to return the non-segregated collateral. Since reneging on
Promise R̄j
Borrower
Lender
Segregated
Non-Segregated
Sell
αj k̄j
(1 − αj )k̄j
Re-pledge
Market
Collateral k̄j
Figure 1: Sale of security j
promises is costless in our environment, this promise cannot be taken at face value and lenders may fail
to return the non-segregated collateral. Ultimately, rehypothecation introduces a double sided limited
commitment (henceforth 2SLC) problem. To make a distinction, I call Failure the action of a lender
not turning back collateral as in Johnson (1997). The terminology Default is saved for the decision of
the borrower not to repay. In the following, I analyze default and failure pattern for securities in J . The
following assumption will prove helpful for this purpose.
Assumption A : In period 1, a delivery of m units of collateral is equivalent to delivering mx(s)
units of consumption good, for any m ≥ 0.
For now, it is enough to say that Assumption A equalizes for any agent, the consumption value
8 Practically, the segregated collateral could be stored in a third party’s dedicated account. In the tri-party repo market,
BNY Mellon and JP Morgan provide this collateral storage facility.
5
of the tree to the implicit cost of returning collateral. Hence, an agent fails or defaults independently
of his non-segregated collateral and cash balances when he enters period 1. This ultimately ensures
that securities payoffs are homogeneous across lender/borrower pair and allows to consider anonymous
markets. The next subsection offers a discussion of Assumption A which is microfounded in Appendix
A thanks to an extended model of trading.
2.1.3
Default Resolution Mechanism : DRM
I now characterize securities’ payoff provided that lenders may also fail to return collateral. To avoid
uninteresting complications, I restrict the analysis to symmetric equilibria which require agents of the
same type to follow the same default and fail pattern across securities. Furthermore, an agent either
defaults or fails but cannot randomize between the two strategies. Practically, this rules out partial
default or failure.
I now state the provisions of this DRM explicit. Although their precise identity is immaterial here,
let (i, h) be a type-pair of borrower and lender for promise j. Denote dij (s) ∈ {0, 1} (resp. fjh (s))
decisions to default (resp. fail) by each trader in state s. The DRM maps (dij (s), fjh (s)) to trader’s
payoffs. It exhibits substitutability because agents ultimately pay or deliver only when the reciprocal
promise is kept on the other side of the trade 9 . Hence, borrower i’s payoff Πi (dij (s)) is
Πi (0) = fjh (s)αj k̄j x(s) + (1 − fjh (s))(k̄j x(s) − R̄j (s))
Πi (1) = 0
where fjh (s) ∈ {0, 1} is taken as given. Importantly if i does not default (dij (s) = 0) he effectively pays
only if counterparties h do not fail, fjh (s) = 0. In the legal parlance, collateral pledgors have a right to
set-off when the receiver fails to comply with his obligation to return collateral.
Suppose - without loss of generality given Assumption A - that long trader h actually holds the collateral
on his balance sheet. He then faces the following trade-off:
Πh (0) = dij (s)k̄j x(s) + (1 − dij (s))R̄j (s)
Πh (1) = (1 − αj )k̄j x(s)
where dij (s) ∈ {0, 1} is taken as given. If h announces that he does not fail (fjh (s) = 0), he gets the
repayment when type i pays and the collateral if type i defaults. When failing (fjh (s) = 1), he can only
walk away with the non-segregated part of collateral.
Observe that when an agent defaults or fails on one side of the trade, failure or default is a strictly
9 This feature obviously requires some coordination in the payment/delivery system. A plausible mechanism for implementation relies on two stages for settlement. In the first stage, short (resp. long) agents announce decision to default
(resp. fail) on their positions. In the second stage, payments and collateral transfers are implemented through the DRM
based on these decisions by an automaton.
6
dominated option for the other agent10 . Hence, optimal decisions are given by :
dij (s) = 1{R̄j (s)>k̄j x(s)}
fjh (s) = 1{R̄j (s)<(1−αj )k̄j x(s)}
Default occurs when the face value exceeds the collateral value (as in the 1SLC model). Failure occurs
when the face value falls short of the non-segregated fraction of collateral. In other contingencies,
payment and delivery follow contractual obligations.
Combining the results above, we can write the actual payoff of a security Rj = (Rj (1), ..., Rj (S)) as



(1 − αj )k̄j x(s) if R̄j (s) < (1 − αj )k̄j x(s)


Rj (s) = R̄j (s)
if R̄j (s) ∈ [(1 − αj )k̄j x(s), k̄j x(s)]



k̄ x(s)
if R̄j (s) > k̄j x(s)
j
(1)
Observe that all three components of a security matter to determine the actual payoff but not the identity
of the traders. Expression (1) differs from payoffs Geanakoplos (1996) and others only to the extent that
the failure option is activated. Indeed, equation (1) boils down to Rj (s) = min{R̄j (s), k̄j d(s)} when
αj = 1. Alternatively, when αj = 0 (no segregation), the payoff is given by Rj (s) = k̄j x(s). The
transaction is exactly11 a sale of the tree put up as “collateral”.
I denote E(J ) the economy where the set of agents I, preferences U := {ui }i∈I and endowments
(W, θ 0 ) are fixed. Securities in J are traded competitively in a centralized market. The matrix R ∈
M(S)×J+1 collects the payoffs of securities J .
2.2
Agent’s Optimization Problem and Equilibrium
Consumer Problem
Denote θi agent i’s position in the market for the tree. Since short security positions need to be
i−
backed up by collateral, it is useful to distinguish purchases φi+
j ≥ 0 and sales φj ≥ 0 of security j ∈ J .
Prices of collateral and securities are respectively p ∈ R+ and q ∈ RJ+ .
10 The DRM thus avoids undesirable outcomes whereby either (i) payments remain un-allocated or (ii) default and failure
occur simultaneously. To put it otherwise, this DRM uniquely determines the allocation of the segregated collateral (the
physical asset).
11 This points out to the fact that credit markets in such an economy ultimately require a segregation technology for the
durable asset. Otherwise, credit markets are equivalent to spot markets for the tree.
7
The problem of agent i given asset prices and returns - defined by equation (1) - can be written
max
{c,θ,φ}
s. to
ui (ci0 ) +
X
π(s)ui (ci1 (s))
(2)
s∈S
ci0 +
X
qj φji+ + pθi ≤ ω0i + pθ0i +
j∈J
ci1 (s) +
X
qj φi−
j
X
i
i
φi−
j Rj (s) ≤ ω1 (s) + θ x(s) +
j∈J
θi +
X
X
φi+
j Rj (s)
(4)
j∈J
(1 − αj )k̄j φi+
j −
j∈J
|
(3)
j∈J
X
k̄j φi−
j ≥0
(5)
j∈J
{z
}
Re-use
Equations (3)-(4) are standard pieces of a financial market budget constraint. Besides asset returns,
rehypothecation affects collateral constraints (5) at the individual level. Collateral pledged needs not
be acquired only in the market for the tree but can also consists of non-segregated collateral received.
In my specification, collateral transfers are simultaneous. In particular, agents may effectively hold a
negative position in the collateral market θi < 0 if they receive collateral through long positions. This
feature reminds of Bottazzi et al. (2012) where agents cannot issue securities but are able to sell them
after they borrowed it. The collateral constraint (5) is similar in spirit to their “Box constraint”.
Equilibrium Concept
A symmetric REE of economy E(J ) is a feasible allocation (c0 , c1 ) ∈ R(S+1)×I , a price vector (p, q) ∈
RJ+1
such that ∀i ∈ I, (ci0 , ci1 ) solves agent i’s optimization problem (2)-(5) and securities market clear,
+
i.e.
(Feasibility)
I
X
i=1
(Market Clearing)
I
X
ci0 ≤
I
X
θi = K,
ci1 (s) ≤
I
X
ω1i (s) + Kx(s) ∀s = 1..S
(6)
i=1
i=1
i=1
i=1
2.3
I
X
ω0i ,
I
X
i−
(φi+
j − φj ) = 0,
∀j = 1..J
(7)
i=1
No arbitrage and Equilibrium Existence
An arbitrage strategy is a self-financing portfolio with a strictly positive payoff in at least one state.
With collateral constraints, feasibility also matters. Let us introduce the following matrix

−p

W =x
1
−q T
+q T
R
(1 − α) ◦ k̄
T


−R 
−k̄
T
Definition 2: There is an opportunity of arbitrage iff there exists ψ = (θ, φ+ , φ− ) ∈ R × R2J
+ such
that W ψ > 0.
8
The modification to the standard formulation features in the last line of W . A self-financing portfolio
which delivers strictly positive returns does not qualify as an arbitrage strategy unless it verifies collateral
constraint (5). Because promises are all positive, absence of arbitrage (AOA) implies the following
inequalities relating collateral to security prices
∀j ∈ J ,
−pk̄j + qj ≤ 0
and
− qj + (1 − αj )k̄j p ≤ 0
The first inequality states that it cannot be profitable to buy collateral to finance short sales since
securities pay less than the collateral backing them. To put it differently, haircuts are positive. The
second inequality (specific to rehypothecation) ensures that buying a security to sell the non segregated
collateral cannot yield profit. Hence, the price qj of security j must lie in the following range
∀j ∈ J ,
(1 − α)k̄j p ≤ qj ≤ pk̄j
(8)
In this context, an equilibrium of our economy with rehypothecation always exists :
Proposition 1 : Economy E(J ) admits a competitive equilibrium.
Proof : B Given the regularity conditions imposed on utility functions, the existence of equilibrium
follows as in the standard 1SLC environment. A good reference for a detailed proof is Theorem 3 of
Kilenthong (2011).C
The existence result would hold in most 2 period financial market models with any linear constraint
(this is the case of collateral constraint 5). Hence, the contribution here consists in making a general
equilibrium model of rehypothecation amenable to this larger class of models to exploit known results.
Existence of equilibrium does not require any new technique, hence the outside reference for the proof.
2.4
Discussion of the Set-up
Default and Fail in Equilibrium
In this set-up, default by borrowers and failure by lenders are compatible with the orderly function of
markets (cf. Dubey et al. (2005)) because collateral requirements are well specified and agents correctly
anticipate default patterns. However, even for non-recourse loans, outright default is considered as a rare
event. This discrepancy arises as my stylized environment cannot incorporate margins of adjustments
available in real-life financial transactions. In the repo market for instance, haircuts would typically
increase when liquidity or solvency risk matters. In contrast, the collateral requirements are exogenous
and fixed in the model. For swap transactions, daily margin calls for collateral precisely aim at avoiding
default by the party with a negative Mark to Market position. This feature obviously clashes with a
static 2 period model where uncertainty resolution is not gradual.
Informal evidence on default and failure in collateralized lending markets also suggest that the latter
9
event is far less frequent. Failing lenders would thus carry a more severe stigma than defaulting borrowers. In my model, default and failure are natural consequences of the fundamental assumption that
agents cannot pledge their whole endowment. As such, it cannot accommodate the asymmetry between
default and failure suggested above.
However, the analysis will make clear that any security may be substituted by another without default or failure. The option to fail for lender ultimately captures some notion of collateral risk along the
chain of repledge.
Rehypothecation Rights
In my model, rehypothecation rights are security-specific and part of the financial structure. The
2013 ISDA Margin Survey shows that rehypothecation rights indeed vary across securities, from 48.3%
of collateral posted on commodity trades to 83% for credit derivatives, suggesting that rehypothecation
appears more desirable for some transactions.
It is interesting to see how the parameter α relates to Singh (2011)’s macro estimate for collateral
velocity. If αj = α for all securities j ∈ J , (1 − α) of every unit of collateral can be re-used. In turn
(1 − α) of this fraction can be re-used and so on. Hence a quantity K of source collateral can ultimately
generate up to Kα units of pledged collateral where Kα =
K
α.
To match Singh’s estimate of collateral
velocity around 3, α should be no more than 1/3 in this model.
The regulatory stance towards collateral re-use varies significantly across jurisdictions. Rehypothecation is banned in Canada. In the US, brokers/dealers may re-use an amount up to 140% of the debit
balance of their client under Regulation T (cf SEC Rule 15c3-3). The debit balance is an equilibrium
object in our model hence the different formulation as a fraction of total collateral posted. Under EU
rules, counterparties actually bargain over the amount eligible for re-use. Save for the possibility to
bargain, my framework can fit these provisions with a lower bound ᾱ on αj .
Assumption A : Cash for collateral
Rehypothecation rights allow a lender to re-use the non-segregated collateral. At settlement date in
period 1, the total collateral pledged might exceed the physical amount of tree available. Assumption
A neutralizes the concerns raised by simultaneous settlement as agents may substitute the consumption
good for collateral in period 1. Indeed, Assumption A would not be warranted if agents were to trade
and settle over several successive rounds with a market for the tree as settlement takes place. Appendix
A formalizes this intuition to show that A appears as a mild restriction in making such a sequential
trading environment amenable to the model of the main text. Proposition 4 ultimately shows that every
equilibrium outcome of the main model with A is an equilibrium of the extended sequential model when
agents trade the tree at the settlement stage. To put it otherwise, there always are equilibria of this
extended model where the liquidity constraints that would hinder the reverse circulation of collateral
do not bind. Even if Appendix A gives a satisfying micro-foundation, practical relevance of A is still
worth mentioning. The cash for tree provision retains the flavor of collateral arrangements12 which
12 This
is the case of the English Credit Support Annex to the ISDA Master Agreement. See Monnet (2011) for a
10
only bind receivers to return “equivalent” property. Counterparties may thus agree on the delivery of a
different security than that pledged initially. Assumption A offers a similar flexibility to trading partners.
The next section presents the main Proposition of the paper which states that rehypothecation is
redundant when markets are complete in a sense made precise below. The result extends to decentralized
trading.
3
Rehypothecation and complete markets
Singh (2011)’s criticism of recent reforms against rehypothecation rests on the observation that they will
likely reduce collateral velocity. Circulation of collateral would improve market efficiency by allowing
traders to borrow against the collateral they receive as lenders. In my framework, this claim suggests
that rehypothecation allows to save on the tree, the only source of pledgeable income in this economy.
In this section however, I prove that there exists a financial structure without rehypothecation rights
which allows to reach the same allocations as an unconstrained one. This result (stated in Proposition 2)
uses first the spanning argument of Lemma 1. Then I show that substituting securities by their spanning
portfolio does not hinder collateral circulation.
The reader might think that this result is straightforward. Indeed, without collateral constraints,
Arrow Securities suffice to complete the market and no additional security can bring any improvement.
To illustrate why this is not the case with collateral constraints, I borrow the following example from
Gottardi and Kubler (2014).
Example 1
Suppose there are 3 states of the world with equal probabilities, 3 agents with identical preferences and
identical period 0 endowment, including the tree endowment θ0i = 4. The tree pays off 1 in any state.
Endowments in period 1 are given by
e1 = (0, 6, 9),
e2 = (6, 9, 0),
e3 = (9, 0, 6)
There is a Pareto optimal allocation ci1 = c1 = (9, 9, 9). Agents can trade Arrow securities collateralized
by one unit of the tree but cannot re-use collateral. The payoff matrix is thus given by R = I3 .
Then c cannot be implemented. Indeed, to reach c1 , agent 1 would need to keep his tree endowment,
buy 5 units of security 1, short 1 units of security 2 and 4 of security 3. Such a pattern of trades obviously
clashes with the collateral constraint. The next subsection should convince the reader that no other set
of securities and trade patterns without rehypothecation allow agents to achieve c1 .
With pyramiding - a technology that allow securities to be used as collateral for other securities Gottardi and Kubler (2014) show that c can be attained. The later holds with the fundamental limited
pledgeability constraint unchanged. We may thus rephrase our question as follows. Can rehypothecation
comparison of the different Credit Support Annexes. Unfortunately, the ISDA does not provide detail numbers on the use
of each of this Annex.
11
bring about the same benefits than pyramiding? Proposition 1 shows that this is not the case.
3.1
Complete Financial Structure
In this section, I introduce a set J1 of securities without rehypothecation rights (i.e. αj = 1 for any
j ∈ J1 ) that I use to state the main result. Consider first the unrestricted set of securities J0 where:
J0 =
(R̄j , k̄j , αj ) ∈
RS+
× R++ × [0, 1]
For the rest of this Section, I consider economy E(J0 ). Without loss of generality, I normalize the payoff
of the tree to be x(s) = 1 in any state s. I now introduce a subset of J0 called J1 of securities without
rehypothecation.
J1 =
S
(R̄j , k̄j , αj ) ∈ {0, 1} × {1} × {1}
Observe that J1 contains but is not equal to the set of (collateralized) Arrow Securities with securityspecific collateral constraints13 . The first step in showing that restricting financial trades to J1 entails
no loss of generality relies on a completeness argument. For this, I introduce the following definition.
Definition 3 : A financial structure J is complete iff
∀j0 ∈ J0 ,
∃J (j0 ) ⊂ J ,
(θ, φ) ∈ R × R|J (j)|
such that

θ1 S + P
R
j∈J (j0 ) φj Rj = Rj0
P

j∈J (j0 ) αj |φj |k̄j ≤ αj0 k̄j0
(θ, φ) is called a replicating portfolio
In words, for every security, there must be a replicating portfolio made of the tree and securities in
the set J that (i) delivers the same payoff (spanning aspect) and (ii) does not require more collateral
segregation (collateral use aspect). As usual, to verify the first criterion (i) it is enough to exhibit a
S × S submatrix of [1S , R] with rank S. The second criterion (ii) acknowledges the role of the tree as
collateral and recognizes the importance of tree availability or unencumbered collateral to carry trades.
By definition, criterion (ii) would be redundant with portfolio margins instead of security specific margins. At a fundamental level, observe that one cannot separate this definition from the restrictions put
on the collateral technology. First, only the tree can back a security (no pyramiding). Second, a piece
of tree can only back one security (i.e. security specific margins or no tranching).
The next Lemma proves that J1 is indeed a complete financial structure according to Definition 3.
13 A simple discussion clarifies this point. Suppose indeed that S = 4. An agent wants to short (0, 0, 1, 1). With Arrow
securities only, he needs to short (0, 0, 1, 0) and (0, 0, 0, 1) hence requiring 2 units of collateral. Only 1 is necessary if the
security is available. It is clear that with ex-post collateral constraints as in Chien et al. (2011), the argument does not
hold and Arrow Securities do suffice. Security-specific collateral constraints are closer to market practice however.
12
Lemma 1 : J1 is a complete financial structure.
The proof borrowed from Kilenthong (2011) is in Appendix B. For every security j in J0 , the replicating portfolio is unique given the convention for ordering states in the proof. Hence, we can introduce
the well-defined function
ψ : J0 7→ J1S−1 × RS+
j → (J1 (j), ψ(j))
which maps j into a set of S − 1 securities of J and quantities of these securities ψ(j). I refer loosely to
ψ(j) as the replicating portfolio for security j in the following. It contains 1 − αj units of the tree and a
total of αj units (of the tree) is required to short securities in J1 (j)
Lemma 1 thus shows that for spanning purposes, a smaller set of securities without rehypothecation
suffices. This is a natural result since we focus only on payoffs here. To reach our conclusion, we need
to show that shutting down collateral circulation does not restrict feasible allocations through collateral
constraints. This is the object of the next subsection.
3.2
An equivalence result with complete markets
This section features the main result of this paper. It states that the complete financial structure without
rehypothecation rights J1 can deliver the same outcomes as the unconstrained structure J0 .
Proposition 2 Any equilibrium allocation of E(J0 ) is an equilibrium allocation of E(J1 )
The proof is presented in Appendix B. The reader will realize that the results holds any other complete financial structure. The structure J1 is special because it prevents rehypothecation.
Proposition 2 reads as a negative statement. What can be done with rehypothecation can be done
without, if enough securities collateralized by the tree are available to trade. In particular, the circulation
of pledged collateral through rehypothecation can be mimicked by a no-rehypothecation portfolio. To
put it otherwise, rehypothecation may at best substitute for market completeness. On the normative
side, this result mitigates claims that banning rehypothecation would take a huge toll on collateralized
financial markets. Symmetrically, an equivalence result does not call for a ban either. Proposition 2
provides restrictions on the financial environments where rehypothecation has a genuine economic value.
This result ultimately points to the fact that collateral circulation depends endogenously on the financial structure, not only through rehypothecation rights. Face values and collateral requirements also
condition the efficient use of collateral in a given financial structure. Because they are set in an optimal
way in J1 , the fundamental limit on velocity imposed by the collateral technology itself already binds.
Coming back to Example 1, rehypothecation thus fails to alleviate the collateral scarcity problem
to implement the efficient allocation. More generally, as a financial innovation, it under-performs both
tranching and pyramiding to economize on collateral. Fundamentally, rehypothecation does not intro-
13
duce any technological change to the financial environment. On the contrary, pyramiding allows financial
securities to be used as collateral, i.e. gives a pledging power to any form of financial income, not only
that from the durable asset. Similarly, tranching, by drawing different claims on the same piece of
durable asset exhausts the pledging power of the tree.
As indicated in the introduction, most secured markets (swaps, derivatives, repos) have an OTC
structure. One might suspect that the segmented nature of OTC markets might give economic value to
the circulation of collateral thanks to rehypothecation. In particular, by allowing for collateral circulations between agents that are not direct trading partners, intermediaries could improve efficiency. I show
in the next subsection that centralized trading does not actually drive the equivalence result as long as
securities in the spanning set J1 may be (competitively) traded.
3.3
Robustness to Decentralized Trading
This section shows that the form of market segmentation particular to OTC markets may not itself break
down the irrelevance result. For the sake of comparison, I maintain the competitive nature of the model.
My analysis thus misses bargaining, non-competitive price setting or search frictions of OTC markets.
Still, decentralized trading appears as an important departure which justifies a dedicated analysis.
Again I suppose that all agents can trade every securities in J1 but some agents may not trade
with others. Hence, traders are now disposed on a connected graph G. A symmetric adjacency matrix
A ∈ MI×I indicates which agents are connected to each other. If A(i, h) = A(h, i) = 1, agents i and
h can trade together. Hence the set of agents I and the adjacency matrix A ultimately characterize
the graph G := {I, A}. Let AI0 denote the adjacency matrix restricted to a subset I0 ⊆ I. Market
segmentation arises because agents must trade on Local Markets which correspond to subgraphs of G.
My definitions of Local Markets (LM) reads as follows.
Definition 4 A set of LM is a finite set of subgraphs H = {Hn := (In , AIn )}n=1..N such that:
i) I = ∪N
n=1 In
ii) ∀n = 1..N, Hn is complete.
iii) ∀n 6= n0 , Hn 6⊆ Hn0
Point i) states that every agent participates in at least one LM and hence rules out exclusion from financial markets. Point ii) requires traders in a LM to be able to trade with each other. Finally, with iii), I
impose that no local market is contained in another LM. This last assumption is not crucial but facilitates the interpretation of the LM concept. However, local markets may overlap and one may describe
agents at the intersection of several local markets as intermediaries. Even in a competitive setting, local
markets matter with collateral constraints because positions need to be cleared at the local level. Hence,
I refer to the trading structure studied in the previous section as one with centralized clearing 14 . When
14 Observe that centralized clearing occurs on a security basis in our model. Similarly, most Central Clearing Counterparties to date clear a single class of assets. Hence my centralized trading model ultimately provides the closest counterpart
14
agents trade on LM, there is decentralized clearing. The economy with decentralized trading is denoted
by E(H, J ). To gain some intuition on trading in LM, consider the following example.
Example 2 : Local Markets with Multilateral Clearing
Multilateral clearing in our model represents the situation where a group of agents subscribe to a
Central Clearing Counterparty. With this arrangement, the trades of more than two agents may be
cleared together. To see the link with Definition 5, consider the set of Largest Complete Subgraphs. A
Largest Complete Subgraph is a complete subgraph of G which is not contained in another complete
subgraph of G. It is easy to see that the requirements of Definition 5 are verified. Figure 2 divides a
graph into its largest complete subgraphs where a color stands for a Local Market.
4
3
5
2
1
6
7
Figure 2: Largest Complete Subgraphs
The picture makes clear that large complete subgraph accomodates for both multilateral and pure
bilateral trading. With the terminology introduced above, agent 2, 3, 5 and 6 qualify as intermediaries.
The LM framework also encompasses tiered structures with a small number of institutions clear with a
Central Clearing Counterparty (CCP) and provide clearing services to their clients in turn.
Let us now come back to the general case. For every agent i defines H(i) = {H ∈ H |i ∈ H} as the set
of LM where agent i can trade. The set H(i) 6= ∅ since G is connected. Agents trade competitively the
tree and securities on every local market they belong to. Because of segmentation, there are theoretically
as much security prices as there are such LM. Arbitrage between Local Markets with different prices for
securities may be prevented by collateral constraints.
i−
iB
Every agent i posts long/short demands (φi+
Hj , φHj ) for every security j and buys collateral θH in
every LM H ∈ H(i) he belongs to given prices {pH , q H }H∈H(i) . The same notations without subscript H
indicates total demands. Agent i must post collateral in every local market to cover his local trades but
he can freely allocate collateral acquired in any local market he trades in either through direct purchases
P
iB
or re-use. Let us call θi = θ0i + H∈H(i) θH
the total collateral position of agent i. Then, it is easy to
realize that the local collateral constraints collapse to the following aggregate collateral constraint for
to the collateral netting benefits provided by a CCP, save the fact that terms of trade are usually not determined in a
competitive manner in actual markets
15
agent i
θi + (1 − α)
X X
X X
k̄j φi+
Hj ≥
H∈H(i) j∈J
k̄j φi−
Hj
(9)
H∈H(i) j∈J
With segregated markets, market clearing is required at the local level, i.e.
∀H ∈ H,
X
iB
θHj
=0
(10)
i−
φi+
Hj − φHj = 0
(11)
i∈H
∀H ∈ H,
X
i∈H
Fundamentally, in a competitive environment, the difference between decentralized and centralized trading hinges on the local clearing conditions. Indeed, for given net positions, local clearing typically requires
agents (and in particular intermediaries) to hold larger gross positions but the same net positions. With
collateral constraints, short and long positions do not enter in a symmetric fashion as the former require
collateral. The issue of collateral allocation and collateral velocity appears all the more relevant in this
setting.
We have seen in Proposition 2 that securities in J1 without rehypothecation substitute for the complete set of securities J0 . One might think that the equivalence could break down in a decentralized
setting where intermediaries would improve collateral circulation with rehypothecation. The next proposition shows that this is not the case.
Corollary to Proposition 2: Every equilibrium allocation of (decentralized) economy E(H, J0 ) is
an equilibrium allocation of (decentralized) economy E(H, J1 )
Proof B The Corollary holds since the proof for Proposition 2 itself did not use that trading was
centralized. In particular, the argument can be reproduced for every local market in H without any
other modification than to the notation. C
As long as markets are competitive and the complete set of securities J1 is available, the result carries
on with decentralized trading. Decentralized clearing does not affect the ability of financial structure J1
to support trades with an efficient use of collateral.
In this analysis, decentralized trading generates market segmentation. The companion model of appendix A provides a different interpretation of market segmentation as (sequential) limited participation.
In this case, agents would not be allowed to trade in all trading rounds. Replacing local markets by
trading rounds, the reader might realize that our result also holds in this alternative setting.
Still, as I highlighted in the introduction, collateral arrangements are quite common in secured financial markets, suggesting that there should be economic value to rehypothecation. I show in the next
section that, with incomplete markets, rehypothecation can strictly increase welfare and that the gains
are typically larger with decentralized trading.
16
4
The value of rehypothecation
The previous section showed that rehypothecation appears redundant under complete markets. Allocations implementable with rehypothecation may be reached with a financial structure prohibiting re-use,
that is J1 . There exists a number of reasons why such a complete set of securities might not be available to trade15 . This section discusses how rehypothecation may increase the collateral efficiency of an
incomplete financial structure. To do so, I introduce two economies where agents may only trade in a
non-contingent bond (besides the asset itself). These examples illustrate more generally how circulation
of collateral improve upon an inefficient financial structure and why these gains should be larger in a
decentralized economy.
4.1
Inefficient Financial Structure
There are two states in period 1 with probability 1/2 each and 2 types of agents with identical utility
functions. The tree pays off x(s) = s in state s = 1, 2. Agents’ endowments are as follows :


ω02 = ω


θ02 = 1



ω 2 (s) = ω + (−1)s a − s



ω1 = ω

 0
θ01 = 1



ω 1 (s) = ω − (−1)s a − s
1
1
It is clear that the efficient allocation c∗ is given by cit = ω for i = 1, 2 and t = 0, 1. To achieve this in
period 1, agent i = 1, 2 needs to forgo a units of consumption in state s = i.
Observe that agents holdings of the tree exactly compensates their exposure to aggregate risk built
in their non-tree endowment. Hence, agents’ endowment risk is purely idiosyncratic. Such risk cannot
be hedged when it requires strategic disclosure of private information. Still, when agents can only trade
a bond, rehypothecation delivers the efficient allocation attainable with Arrow Securities.
Arrow Securities
Suppose the Arrow security j = 1 with payoff R̄1 (s) = 1{s=1} is available to trade. The (unique)
portfolio for agent 1 in the tree and j to finance c∗ is (θ1 , φ1j ) = (1+a/2, −3a/2). The collateral constraint
for agent 1 writes:
θ1 ≥ −φ1j
⇒
1+
a
3a
≥
2
2
⇒
a≤1
(12)
The last inequality actually coincides with the fundamental constraint16 that an agent consumption
cit = ω cannot fall below his non-tree endowment in period 1 which is ωti (s). Otherwise, an agent could
15 Market incompleteness is taken as exogeneous since my model abstracts from the supply side for financial securities.
Private information about aggregate or idiosyncratic outcomes can explain why some securities are not traded. In the
first case, DeMarzo and Duffie (1999) and more recently Dang et al. (2012) show that debt-like securities are desirable
because of their low information sensitivity. For the second case, there exists a long tradition of Bewley models where
idiosyncratic shocks are private information and hence uninsurable. Even without such frictions, Carvajal et al. (2012)
prove that security designers might find it optimal not to complete the market.
16 Gottardi and Kubler (2014) formalize this condition within their concept of Arrow Debreu equilibrium with Limited
Pledgeability
17
renege on every promise he signed and consume his non-tree endowment. Applying this inequality for
agent 1 in period 1 delivers indeed a ≤ 1
Bond economy
Suppose now idiosyncratic risk cannot be hedged so that only a non-contingent bond j 0 with face
value R̄j 0 (s) = 1 can be traded together with the tree. Shorting the bond requires one unit of the tree
as collateral. This alternative financial implements allocation c∗ at a higher collateral cost. Indeed, the
(unique) portfolio in asset and bond to deliver c∗ is (θ1 , φ1j 0 ) = (1 + 2a, −3a). I show below that collateral
efficiency in a bond economy can only be restored through rehypothecation.
Claim : In a bond economy, when 0.5 < a ≤ 1 allocation c∗ can be implemented if and only if
rehypothecation is possible.
Proof :
B To prove necessity, observe that portfolio (θ1 , φ1j 0 ) requires agent 1 to hold 1 + 2a units of tree.
This is possible only if this amount does not exceed the supply in the economy since agent 2 must hold
a positive amount of the tree. Hence, c∗ can be implemented if 1 + 2a ≤ θ02 + θ01 = 2, i.e. if a ≤ 0.5.
To prove sufficiency, suppose agents may re-use a share 1 − α = 1/2 of the tree for every unit of the
bond purchased. Lenders do not fail as (1−α)x(s) = s/2 ≤ 1 for s = 1, 2, i.e. the non-segregated value of
collateral always lies below the face value of the bond. Hence, the bond pays off its face value Rj 0 (s) = 1
in any state s = 1, 2. The market clearing and collateral constraints for agent 1 and 2 necessary to
implement portfolio (θ1 , φ1j 0 ) = (1 + 2a, 3a) now write:
1 + 2a + θ2 = 2
1 + 2a − 3a ≥ 0
1
θ2 + 3a ≥ 0
2
As we already observed, rehypothecation allows to take (virtual) negative positions in the tree. To verify
the market clearing constraint, agent 2 needs to hold θ2 = 1 − 2a which can be negative. This is possible
when re-using the tree he receives as collateral for lending to agent 1. Hence, if a ≤ 1, agent 2’s collateral
constraint holds because he can re-use a fraction 1/2 of the collateral acquired on the long position in
the bond. The optimal allocation c∗ can be financed with rehypothecation. C
While formally correct, the reference in the proof to “short positions” of a physical asset may disconcert the reader. What rehypothecation does is to create virtual collateral out of fixed amount of
physical asset. To gain further intuition about this feature, let us use the sequential trading structure,
borrowed from the model of Appendix A. The steps below, illustrated on Figure 3 describe how portfolio
(θ1 , φ1j 0 ) = (1+2a, −3a) can be implemented. For each agent, the left column records the cumulated position in the bond while the right column records holdings of the tree, with the Segregated/Non-Segregated
distinction when relevant.
1 Trading Round 1 : Agent 1 purchases a ≤ 1 = θ02 units of the tree from agent 2. He uses 2a ≤ 1 + a
18
Agent 1
Bond
Tree
−2a
1−a
Agent 2
Pledge 2a
Bond
Sell a
2a
Tree
S
NS
a
1
(a) Step 1
Agent 1
Bond
Tree
−3a
1−a
Agent 2
Pledge a
Bond
Sell a
3a
Tree
S
NS
3a/2
1 − a/2
(b) Step 2
Figure 3: Trading Rounds with re-use
units of the tree to secure a short position of −2a in the bond. After this transaction, agent 1 holds
1 − a units of tree. Agent 2 has a (resp. 1) units in his segregated (resp. non segregated) account.
2 Trading Round 2 : Agent 2 sells a units of the tree to agent 1 which uses it to finance an additional
short position of −a in the bond. After this transaction, agent 1 holds 1 − a units of tree. Agent
2 has 3a/2 (resp. 1 − a/2) units in his segregated (resp. non segregated) account.
Combining these trades, agent 1 finally bought 2a units of the tree from 2 and shorted 3a units of the
bond. To do this, agent 2 re-sold the non-segregated tree obtained during the first round of trading.
How can agent 2 ultimately accept a short position of 3a in the bond that can exceed the 2 units of
physical collateral available? Sequential settlement help explain this:
1. Settlement Round 1 : To settle the a short position of Trading Round 2, agent 1 pays a while agent
2 transfers a/2 units of tree from his non segregated account (added to the a/2 segregated units).
After this first settlement, agent 1 holds 1 unit of the tree. Agent 2 has a (resp. 1 − a) units in his
segregated (resp. non segregated) account.
2. Settlement Round 2 : To settle the 2a long position traded in Trading Round 2 agent 2 needs to
return a units of collateral (besides the a segregated units). While he only has 1 − a units on his
account, he can buy a units of the asset from 1 at price x(s) = s and turn it back to make good
on the bond position.
The argument thus relies on the possibility to buy the tree for agent 2 during the second round of
settlement. In the model of the main text, there is only one simultaneous settlement stage and Assumption A implicitly replaces the market for the tree. For the interested reader, Appendix A formalizes this
connection.
By allowing for circulation of pledged collateral, rehypothecation allows agent 1 to finance a large
19
short position of −3a in the bond. The need to take large short positions stems for the collateral inefficiency of the financial structure. Indeed, the tree+bond financial structure fails the second requirement
of Definition 3. Rehypothecation allows for circulation which mitigates this inefficiency.
With decentralized trading, some agents naturally emerge as intermediaries between other agents.
While intermediaries typically end up having small net positions, they usually hold large gross positions.
This pattern is documented by Atkeson et al. (2013) for CDS markets for instance. The discrepancy
between net and gross positions typical of OTC markets comes at a cost with collateral constraints. Indeed, the example of Section 4.2 shows that collateral inefficiency can be aggravated with decentralized
trading.
4.2
Centralized vs. Bilateral Trading
In this example, agents simply desire to trade endowment across time. A bond efficiently realizes this
transfer. However, in a decentralized economy, collateral scarcity may bite when agents (endogenously)
need to play the role of intermediaries.
To account for decentralized trading, at least 3 agents are needed. There are S states of the world.
The asset pays off x(s) = s for s = 1, .., S. Agents have the same utility function and do not discount
period 1 payoffs. Let a and b such that 0 < a < b. Endowments are as follows :

1


ω0 = ω + a − b

ω11 (s) = ω − s + b − a



θ 1 = 1
0

2


ω0 = ω + b

ω12 (s) = ω − s − b



θ 2 = 1
0

3


ω0 = ω − a

ω13 (s) = ω + a − s



θ 3 = 1
0
Given convex preferences and the absence of discounting, it is easy to see that the optimal allocation is
cit = c∗ = ω for all i = 1, 2, 3 and all t = 0, 1. Agents 3 and 1 wish to borrow while agent 2 desires to
lend. To put it otherwise, agents want to trade endowment across time and not across states as in 4.1.
Hence, a non-contingent bond with face value R(s) = 1 appears as the perfect instrument. For borrowers
not to default, it needs to be collateralized by one unit of asset so k̄ = 1. Finally, for lenders not to fail,
rehypothecation should be limited by α ≥
S−1
S
Centralized Trading
With centralized trading, the bond suffices to reach the efficient allocation. In this case, agent 2 lends
respectively b − a and a to agent 1 and 3. These trades verifies collateral constraints if b − a ≤ θ01 = 1
and a ≤ θ03 = 1. The allocation c∗ is supported by price q = 1 for the bond given perfect risk sharing
and absence of discounting. The inequalities above are also equivalent to the fundamental constraints
on the limited pledgeability of income and no other trade pattern can improve upon this one.
Decentralized Trading
Suppose now that trading is bilateral as shown in figure 4 so that agent 1 qualifies as an intermediary.
Precisely, to implement c∗ agent 1 needs to lend a to agent 3 and borrows b from agent 2. Hence, the
20
net trade is the same but agent 1 needs to hold both long and short positions in the bond.
b
2
a
1
3
Figure 4: Bilateral Trading - Period 0 transfers
With bilateral trading, the conditions to attain c∗ are more stringent, even a complete financial
structure as b ≤ 1 + a/S must hold17 . With a bond, it is clear that in the absence of rehypothecation,
the new constraint is b ≤ 1 for agent 1 to finance his short position with agent 2.
With rehypothecation now, agent 1 can re-use a fraction 1/S of the a units of collateral pledged by
3. This allows him to borrow as much as 1 + a/S from agent 2. Hence, with bilateral trading and a bond
economy, rehypothecation delivers the same outcome as a complete financial structure.
4.3
Rehypothecation and financial innovation
An important remark is in order here. Remember that a security was defined as a triplet (R̄j , k̄j , αj ),
i.e. face value, collateral requirement and rehypothecation rights. Hence, modifying J to allow for rehypothecation ultimately constitutes a financial innovation. We have seen in Section 3 that as a financial
innovation, rehypothecation merely competes against introducing the securities in J1 without rehypothecation rights. Hence, for our purpose, it seems more appealing and not less legitimate to consider
this change as a lift on a rehypothecation ban. The regulatory interpretation gives a different flavor to
rehypothecation rights as exogenous limitations on feasible transactions.
At the technological level however, this view takes an implicit stand on the dimensions along which
financial innovation proves easier : rehypothecation rights rather than face value and collateral requirement. Without a theory of security design, the debate might prove shaky. However, on the basis of legal
costs, one can argue that granting rehypothecation rights just requires a clause to an existing contract
while the other forms of innovation implies drafting whole new contracts. Furthermore, special accounts
must be set or collateral depository institutions created, to implement collateral segregation. These
institutional provisions should also come at a cost.
Geanakoplos (1996) already suggested that innovation should be naturally directed towards the securities that (i) provide greater risk-sharing and (ii) make the most efficient use of collateral. Rehypothecation ultimately appears as a relatively cheap channel along which these adjustments can be made.
Indeed, it introduces (i) state-contingent decisions by lenders that may better accommodate the trading
needs of an economy and (ii) circulation by re-use of dormant collateral with minimal adjustments to
existing securities.
17 Here, one would need to introduce security j with payoff R (s) = (S − s)/S collateralized by (S − 1)/S units of the
j
tree. Agent 3 would sell a units of this security and a/S units of asset to agent 1. Agent 1 can then borrow up to his asset
holdings which are 1 + a/S. Hence the new inequality.
21
5
Conclusion
This paper introduced rehypothecation in a competitive economy where agents post collateral to short
securities. Re-use may facilitate the circulation of collateral but limited commitment creates collateral
risk along the chain of repledges. I show that rehypothecation can be replaced by an efficient financial
structure without rehypothecation which delivers the same velocity of collateral. This result is robust
to market segmentation whereby agents trade on decentralized markets with local clearing constraints.
Hence, my results mitigate the claim from bankers and some academics that a ban on rehypothecation
would severely affect secured financial markets. In the presence of market incompleteness however, the
irrelevance result may break down as rehypothecation allows to free up (inefficiently) encumbered collateral. Decentralized trading magnifies these gains as intermediaries typically need to take long and short
positions simultaneously, which are costly in terms of collateral use.
The ultimate role of rehypothecation is to help match the supply of good quality asset (mainly
government securities) with the increasing demand for collateral in secured financial markets. A clear
consensus about the empirical relevance of such asset scarcity has yet to emerge (cf CGFS (2013) for
a recent analysis). Still, many recent developments in financial markets can be analyzed through this
prism. For instance, the new regulations on central clearing practices motivated a number of studies to
assess their impact on collateral demand (see for example Duffie and Zhu (2011) for an early contribution). However, a comprehensive analysis of central clearing efficiency is still missing. On the supply
side, Gorton and Metrick (2009) argues that pre-crisis securitization activity was driven by the growing
demand for high quality collateral. Tranching and pyramiding, the main techniques attached to securitization, have been shown by Gottardi and Kubler (2014) to realize an efficient use of durable assets as
collateral. Empirically, the collapse in securitization activity or the prominent use of plain vanilla assets
as collateral both point at liquidity issues that my framework could not capture. The vast questions of
what constitutes good collateral and how best to supply it in the presence of such frictions still rank
high on the research agenda for collateral markets.
References
Admati, A. R., P. M. DeMarzo, M. F. Hellwig, and P. Pfleiderer (2010). Fallacies, irrelevant facts, and
myths in the discussion of capital regulation: Why bank equity is not expensive. Research Papers
2065, Stanford University, Graduate School of Business.
Allen, F. and D. Gale (2000a). Bubbles and Crises. Economic Journal 110 (460), 236–55.
Allen, F. and D. Gale (2000b). Financial Contagion. Journal of Political Economy 108 (1), 1–33.
Atkeson, A. G., A. L. Eisfeldt, and P.-O. Weill (2013). The Market for OTC Derivatives. NBER Working
Papers 18912, National Bureau of Economic Research, Inc.
22
Bottazzi, J.-M., J. Luque, and M. R. Páscoa (2012). Securities market theory: Possession, repo and
rehypothecation. Journal of Economic Theory 147 (2), 477–500.
Carvajal, A., M. Rostek, and M. Weretka (2012). Competition in Financial Innovation. Econometrica 80 (5), 1895–1936.
CGFS (2013). Asset encumbrance, financial reform and the demand for collateral assets. Cgfs papers,
BIS.
Chien, Y., H. Cole, and H. Lustig (2011). A multiplier approach to understanding the macro implications
of household finance. Review of Economic Studies 78 (1), 199–234.
Dang, T. V., G. B. Gorton, and B. Holmström (2012). Ignorance, debt and financial crises. February
2012 Draft.
DeMarzo, P. and D. Duffie (1999). A liquidity-based model of security design. Econometrica 67 (1),
65–100.
Dubey, P., J. Geanakoplos, and M. Shubik (2005). Default and punishment in general equilibrium.
Econometrica 73 (1), 1–37.
Duffie, D. and H. Zhu (2011). Does a central clearing counterparty reduce counterparty risk? Review of
Asset Pricing Studies 1 (1), 74–95.
Fostel, A. and J. Geanakoplos (2012). Tranching, cds, and asset prices: How financial innovation can
cause bubbles and crashes. American Economic Journal: Macroeconomics 4 (1), 190–225.
Geanakoplos, J. (1996). Promises promises. Cowles Foundation Discussion Papers 1143, Cowles Foundation for Research in Economics, Yale University.
Gorton, G. B. and A. Metrick (2009). Securitized banking and the run on repo. NBER Working Papers
15223, National Bureau of Economic Research, Inc.
Gottardi, P. and F. Kubler (2014). Dynamic competitive economies with complete markets and collateral
constraints. Technical report.
Johnson, C. A. (1997). Derivatives and rehypothecation failure: It’s 3:00 p.m., do you know where your
collateral is? Arizone Law Review 30 (949).
Kehoe, T. J. and D. K. Levine (1993). Debt-constrained asset markets. Review of Economic Studies 60 (4), 865–88.
Kilenthong, W. (2011). Collateral premia and risk sharing under limited commitment. Economic Theory 46 (3), 475–501.
Kirk, A., J. McAndrews, P. Sastry, and P. Weed (2014). Matching collateral supply and financing
demands in dealer banks. Economic policy review, Federal Reserve Bank of New York.
23
Lucas, Robert E., J. and N. L. Stokey (1987). Money and interest in a cash-in-advance economy.
Econometrica 55 (3), pp. 491–513.
Monnet, C. (2011). Rehypothecation. Business Review (Q4), 18–25.
Monnet, C. and T. Nellen (2012). Clearing with two-sided limited commitment.
Shleifer, A. and R. W. Vishny (1992). Liquidation values and debt capacity: A market equilibrium
approach. The Journal of Finance 47 (4), 1343–1366.
Singh, M. (2011). Velocity of pledged collateral: Analysis and implications. IMF Working Papers 11/256,
International Monetary Fund.
Appendices
A
The role of Assumption A
Under Assumption A, receivers may return either cash (i.e. consumption good) or collateral to pledgors
in order to settle positions. Hence, if collateral he re-pledged is not returned to a trader, he may substitute
it with cash for settling a long position. While trading securities in period t = 0 require the transfer of
physical collateral, assumption A ultimately allows traders to “create” collateral in the settlement period
t = 1. This property dramatically improves the tractability of the model as a single simultaneous budget
constraint may account for the sequential circulation of collateral. However, rehypothecation allows the
economy as a whole to pledge more collateral than physically present. Hence, the actual necessity to
deliver physical units might have interesting implications for the settlement of claims and the circulation
of collateral that Assumption A allowed us to overlook so far. Actually, I show that assumption A.2
functions as an equilibrium selection device in the sequential model described below.
A.1
A model with sequential re-use
The sequential model keeps the physical set-up and the limited pledgeability friction of the main model.
However, financial transactions are now sequential rather than simultaneous.
Let us indeed divide period 0 (resp. period 1) into T trading (resp. settlement) rounds. In each
trading round τ ∈ {1, .., T }, agents may trade the collateral and the securities in J in a competitive
market at respective prices p0,τ and q τ . Now, a security identifier should include the round in which it
is traded18 . For simplicity, I assume that in any round τ , Jτ = J . Although payoffs might ultimately
differ, securities traded (promises, collateral requirements and rehypothecation rights) are identical across
trading rounds.
In period 1, uncertainty is resolved and the settlement stage takes place in a backward fashion.
18 In
the general case, payoffs may also depend on the identity of the counterparties as discussed below
24
Securities traded in trading round τ ∈ {1, .., T } are settled in settlement round T − τ + 1. The tree,
which pays dividends after the last settlement round, can be traded in every round τ at price p1,τ .
Allowing agents to trade the tree after uncertainty is resolved is crucial for the delivery of physical
collateral. Consumption in period 0 and 1 take place at the last round of the trading and settlement
stages respectively. For every variable, subscript 0 (resp. 1) denotes the trading (resp. settlement) stage
as before. The second subscript τ denotes the round. Figure 5 presents the time-line of the sequential
trading model.
A.1.1
Period 0 : Trading stage
Agent i enters trading round 1 < τ ≤ T with (i) consumption good (sometimes referred to as cash) in
i,N S
i,S
i
quantity ω0,τ
−1 , (ii) a quantity of segregated (resp. non-segregated) collateral θ0,τ −1 (resp. θ0,τ −1 ) and
(iii) a record of securities traded during the τ − 1 previous rounds Φiτ −1 = (φi1 , .., φiτ −1 ) ∈ R2J×(τ −1) .
i,B
and secuDuring round τ , given respective prices (p0,τ , q τ ), he chooses new purchases of collateral θ0,τ
rities φiτ .
i,S
i,N S
i
= 0. Initial
:= θ0i and θ0,0
The initial values of the variables introduced above are ω0,0
:= ω0i , θ0,0
holdings of the collateral are entirely pledgeable and hence located in the non-segregated account. For
i
.
period t = 0, the agent consumes what consumption good he has once trades are over, i.e. c0i := ω0,T
i,B
Finally, given optimal choices (θ0,τ
, φiτ ), the following equations govern the evolution of the cash and
collateral accounts :
i,B
i,+
i,−
i
i
ω0,τ
= ω0,τ
−1 − p0,τ θ0,τ − q τ .φτ + q τ .φτ
X
i,S
i,S
θ0,τ
= θ0,τ
k̄j φi,+
−1 + α
τ,j
(13)
(14)
j
i,N S
i,N S
i,B
θ0,τ
= θ0,τ
−1 + θ0,τ −
X
k̄j φi,−
τ,j + (1 − α)
j
X
k̄j φi,+
τ,j
(15)
j
Furthermore, cash and collateral account balances must verify the following constraints in any period τ
i,N S
i,B
θ0,τ
−1 + θ0,τ ≥
X
k̄j φi,−
τ,j
(16)
j
i
ω0,τ
≥0
(17)
Inequality (16) is the sequential collateral constraint equivalent to (5) in the simultaneous model of the
main text. Together with accumulation equation (15), it accounts for circulation of collateral implicit to
rehypothecation. Indeed, the tree received from long positions may only be pledged (or sold) one trading
round after it was acquired. In the model of the main text, these transfers take place simultaneously.
Constraint (17) requires agents to hold positive cash holdings at the end of every trading round. In
other words, agents cannot engage in inter-round credit, a natural feature in an environment with limited
commitment. One of the reason why the sequential model differs from the simultaneous one is because
these two liquidity constraints may bind.
25
Consumption
t=0
Settlement
Claims
T −τ +1
Trading Round
(ωτi −1 ,θτi −1 ,
1
2
Φ1τ −1 )
τ
s Realized
T
1
Trading Stage
2
τ
Consumption
t=1
T
Settlement Stage
Figure 5: Timeline of the sequential trading model
Two limit cases are worth mentioning. Setting αj to 1 bans collateral re-use de jure and trading
rounds after the first are redundant. Similarly, allowing for only one round T = 1 suppresses de facto the
possibility to re-use collateral with sequential constraints. Note however that in this case, 2SLC would
still matter as receivers retain control over a fraction 1 − αj of the collateral pledged for security j.
A.1.2
Period 1 : Settlement stage
Collateral pledged after the trading stage might exceed collateral physically available, hence the importance of sequential settlement. In settlement round 1 ≤ τ ≤ T , only promises traded at round T − τ + 1
are settled. As uncertainty is resolved before settlement takes place, I abstract from indexing variables
by the realized state s in the following.
Symmetrically to the trading stage, I introduce the variables summarizing an agent’s position upon
entering settlement round τ , i.e. (i) consumption good (sometimes referred to as cash) in quantity
i,N S
i
i
ω1,τ
−1 , (ii) a quantity of non-segregated collateral θ1,τ −1 and (iii) securities yet to be settled ΦT −τ +1 =
(φi1 , .., φiT −τ −1 ) ∈ R2J×(T −τ −1) . At every settlement stage τ , an agent must decide whether to default
on securities he shorted at trading stage T − τ + 1 and whether to deliver collateral on his long positions.
i,B
To accomplish the latter, he may either dip in his segregated account or buy collateral (quantity θ1,τ
)
in the collateral market of round τ .
In every round τ , payments and deliveries follow the same Default Resolution Mechanism as in the
main model. That is, sellers who pay always get the segregated collateral back (and the non-segregated
if buyers deliver) while buyers get the segregated collateral only if sellers default.
i,S
i,N S
i,S
i
i
:= θ0,T
and θ1,0
= θ0,T
.
The initial values of the variables introduced above are ω1,0
:= ω1i , θ1,0
For period t = 1, the agent consumes what consumption good he has once settlement is over plus the
i,N S
i
+ θ1,T
x.
dividends from his final tree holdings, i.e ci1 = ω1,T
i,N S
i
Definition 5: Given initial positions (ω1,0
, θ1,0
, Φi1T )i∈I , a settlement equilibrium is a price sei
quence p1 = (p1,1 , .., p1,T ), defaults diτ,j ∈ {0, 1} and fail fτ,j
∈ {0, 1} decisions, and collateral purchases
26
i,B
θ1,τ
such that
1. p1,τ clear the collateral market in round τ
i,B
i
2. (diτ,j , fτ,j
, θ1,τ
)τ =1..T maximize19 ci1 given p1
Because an agent can renege on all his promises (to pay and to deliver), cash endowments and
non-segregated collateral at the beginning of the settlement stage are not seizable. Hence period 1 coni,N S
sumption must verify. ci1 ≥ ω1i + θ1,1
x ≥ ω1i
Identifying equilibria of the sequential model proves difficult. With non-trivial price sequences for
the tree, default and failure decisions depend on a trader’s collateral and cash balance. This makes settlement equilibrium determination extremely cumbersome and as a result, the whole sequential trading
model becomes intractable. However, I show in the following that there exists a class of equilibrium
which is considerably easier to analyze and bridges the gap between the simultaneous model of the main
text and this sequential model.
Definition 6 : An equilibrium of the settlement stage is called a fundamental equilibrium if ∀τ = 1..T ,
p1,τ = x.
In a fundamental equilibrium, the collateral price equals the dividend value. In particular holding
cash and collateral at the onset of the settlement stage is equivalent and delivering collateral to buy it
later is a neutral operation. Non-fundamental settlement equilibria can arise since securities payments
can exceed the fundamental value of the non-segregated collateral which a receiver must buy to deliver.
A.2
Relationship between the models
As suggested before, characterizing the sequential model entirely is beyond the scope of this paper. However, this section shows that the sequential model encompasses the simultaneous one as a special case.
For this, we need to establish the following lemma first.
Lemma 2
There always exists a fundamental equilibrium of the settlement game. In such an equilibrium, the
following property holds : ∀τ = 1..T , a security (j, τ ) has a payoff given by equation (1).
Proof
B Consider the price sequence p1 = x.1RT . At this price, there is no possibility of arbitrage and
holding collateral and cash is equivalent. I want to show that agent’s optimal decisions under p1 are
compatible with it being an equilibrium price. For this, I use backward intuition starting from the last
trading round.
In settlement round T , each agent must settle holdings i, φi1 . Now observe that having a collateral
19 In
the general case, it proves cumbersome to write the sequential constraints at the settlement stage
27
price p1,T = x implies that a receiver which does not hold collateral in his non-segregated account can
“produce” it at cost x. This is exactly the crucial feature of Assumption A of the simultaneous model.
Hence, optimal default and failure decisions yield the same outcomes in settlement round T .
Now, consider an earlier trading round τ ans suppose that payments and deliveries have been implemented as described above. Since p1,T −1 = .. = p1,τ = x, the same applies to settlement round τ because
any receiver in this round can buy back collateral returned later through a revenue-neutral operation.C
Since two j securities traded at different round yield the same payoff given by (1), we can write the
missing piece of the budget constraint relating period 1 consumption to portfolio :
ci1 (s) = ωsi +
θ̄0i +
T
X
!
i,B
θ0,τ
x(s) +
τ =1
X
Rα,j (s)
j
T
X
!
φi+
τ,j
−
φi−
τ,j
(18)
τ =1
where Rα,j (s) is defined as in (1). With a fundamental settlement equilibrium, the sequential portfolios
are payoff equivalent to a simultaneous portfolio consisting of the sum of all sequential trades.
Building on the properties of the fundamental settlement equilibrium, I can now state the equivalence
result between the sequential model Seq presented in this appendix and the simultaneous model Sim of
the main text.
Proposition 3
Every equilibrium allocation of model Sim is an equilibrium allocation of model Seq.
Proof
B Let (c, p, q) be an equilibrium of the Sim model. By definition, for every trader i, there exists a
i
portfolio (θi , φi ) ∈ R × R2J
+ which finances c under (p, q).
As the reader will have guessed, the settlement equilibrium of the Seq will be a fundamental one with
p1,τ = x for all τ . In addition, the relevant prices for the Seq model are p0,τ = p and q τ = q for any τ ,
i.e. constant collateral and security prices. Formally this translates the fact that constraints (16)-(17)
of the Seq model will not bind.
For simplicity, I note SimBC(p, q, ω0 , θ0 ) (resp. SeqBC T (p, q, ω0 , θ0 )) the budget constraint in the
simultaneous model (resp. the sequential model with T trading rounds). For the sequential budget
constraint, remember that the first two arguments are formally sequences and that we omitted the price
sequence for the tree at the settlement stage. The result obtains if budget feasible allocations are the
same in the following sense.
i) For every T , SeqBC T (p, q, ω0 , θ0 ) ⊂ SimBC(p, q, ω0 , θ0 )
ii) SimBC(p, q, ω0 , θ0 ) ⊆ ∪∞
T =1 SeqBC T (p, q, ω0 , θ0 )
Point i) is easy to show as an agent faces the same prices but additional constraints with the sequential
budget set, namely, liquidity constraint (17) and collateral constraint (16).
28
Point ii) is slightly more involved. Let c ∈ SimBC(p, q, ω0 , θ0 ). By definition, there exists a portfolio
ψ = (θ, φ) ∈ R × R2J
+ delivering c under (p, q). Let us consider the following trades over two rounds :
−
B
1. Buy θ0,1
= θ − θ0 + p1 q.φ+ units of tree and short φ−
1 =φ
B
2. Buy θ0,2
= −q.φ+ units of tree and buy φ+
It is easy to see that as a result of the two operations, the agent effectively holds total portfolio (θ, φ)
which finances c under prices (p, q). We are left to verify whether constraints in trading round 1 are
verified. For this, I use the fact that (p, q) are equilibrium prices of the simultaneous model and hence
verify absence of arbitrage. In particular, AOA equation (19) implies that
1
q − (1 − α)k̄ .φ+ ≥ 0
p
(19)
Thus, for the collateral constraint observe that
1
NS
B
θ0,0
+ θ0,1
= θ0 + θ − θ0 + q.φ+ ≥ θ + (1 − α)k̄.φ+ ≥ k̄.φ− = k̄.φ−
1
p
where the first inequality uses (19), the second is the collateral constraint (5) of the simultaneous model.
As for the liquidity constraint (17), note that
B
ω0,1 = ω0 − pθ0,1
+ q.φ− = ω0 − p(θ − θ0 ) − q.φ+ + q.φ− = c0 ≥ 0
where the inequality follows from c ∈ SimBC(p, q, ω0 , θ0 )
Liquidity and collateral constraint are trivially verified in trading round 2 since the transaction is selffinancing and no security is shorted.
Finally, since budget feasible plans are the same and c is optimal in the simultaneous model under
prices (p, q), this is also the consumer’s choice in the sequential model under (p, q). C
The proof for Proposition 3 relies on a budget set argument. One should not infer from the analysis
above that we characterized the actual equilibrium trades in the sequential model leading to c under
(p, q), let alone that two trading rounds of the sequential model would suffice to reach the allocation in
equilibrium. Indeed, while they are budget feasible on an individual basis, the trades described in the
proof are not mutually consistent with the equilibrium requirement of market clearing. This is precisely
because it is hard to assess how many rounds of trading will be necessary in equilibrium that I used this
alternative budget set argument.
29
B
B.1
Proofs
Proof of Lemma 1
B Let j = (R̄j , k̄j , αj ) ∈ J0 and consider security j 0 = (Rj /k̄j , 1, αj ) the face value of which is proportional to the actual payoff of j. Since R̄j 0 (s) ∈ [(1 − αj ), 1], we have Rj 0 (s) = R̄j 0 (s) = (1/k̄j )Rj . Hence,
security j can be replicated by k̄j units of security j 0 . It is thus enough to find a replicating portfolio for
j 0 . We can then restrict our attention to the following set :
J˜0 =
(R̄j , k̄j , αj ) | R̄j (s) ∈ [(1 − αj ), 1], k̄j = 1, αj ∈ [0, 1]
⊂ J0
which is exactly the set of no-default/no-fail securities collateralized by one unit of the tree. Let now
j = (R̄, 1, α) ∈ J˜0 and re-order the states s = 1, .., S so that
1 ≥ R̄(1) ≥ R̄(2) ≥ · · · ≥ R̄(S) ≥ (1 − α)
(20)
Whenever R̄(s) = R̄(s0 ), let the initial ordering prevail. Next consider the S−1 securities {j1 , j2 , .., jS−1 } :=
J1 (j) ⊂ J1 which verify
Rjl (s) = 1, if 1 ≤ s ≤ S − l
Rjl (s) = 0, otherwise
Contract jl has the same payoff as one unit of collateral in the first S − l state. I now derive the portfolio
ψ(j) = (θ, φj1 , .., φjS−1 ) ∈ RS+ of tree and securities of J1 (j) to replicate j. To this effect, set
∀l ≥ 1,
φjl = R(S − l) − R(S − l + 1)
θ = R(S)
By construction, this portfolio replicates security j’s payoff. The collateral segregated by ψ(j) is
k S (ψ(j)) =
S−1
X
αjl φjl = R(1) − R(S) ≤ 1 − (1 − α) = α = k S (j)
l=1
and hence less than the collateral segregated when shorting j. We thus proved that any contract in J0
can be replicated by contracts in J1 according to Definition 3. C
B.2
Proof of Proposition 2
B Let E0 := (c, p0 , q 0 ) be an equilibrium in economy E(J0 ). By definition, for every agent i, there
|J |
|J |
i−
0
exists a portfolio (θ0i , φi+
× R+ 0 of collateral and securities of J0 which finances the
0 , φ0 ) ∈ R × R+
allocation ci under prices (p0 , q 0 ) according to budget constraint (3)-(5). In what follows, I show that
c is an equilibrium allocation of economy E(J1 ) where only securities in J1 are available for trade. Let
30
E1 = (c, p1 , q 1 ) this equilibrium to be constructed.
Every security is priced in E0 , even if it is not traded. Let us set prices (p1 , q 1 ) for securities of J1
in E1 to their value in E0 . Since J1 ⊂ J0 budget feasible allocations in E1 are feasible in E0 .
Suppose now a security j = (R̄j , k̄j , αj ) ∈ J0 \J1 is traded in equilibrium E0 . Using the proof of
Lemma 1, one can set k̄j = 1 and R̄j (s) ∈ [(1 − αj ), 1] without loss of generality. Let ψ(j) be the
replicating portfolio where the first element is a quantity of the tree and the second argument relevant
i+
quantities of securities in J1 . For any agent i, replace every long position φi+
j (resp. short position φj )
i+
in security j by φi+
j (resp. −φj ) units of portfolio ψ(j). The following points prove that the substitution
achieves our goal
a) Using market clearing for security j in E0 , the securities in ψ(j) verify market clearing in E1 . To
put it otherwise, the substitution is resource neutral.
b) Second, any agent’s payoff in period 1 stays identical by definition of ψ(j). Furthermore, the replicating portfolio’s price in E0 equals that of the security. If the former were lower, by monotonicity
of preferences long agents would have bought ψ(j) instead of j. If it were higher, short agents
would have sold ψ(j) instead of j. Hence, the substitution is also cost neutral.
c) Finally, substituting ψ(j) for j does not violate the collateral constraint. The reason is that,
by construction the replicating portfolio leads to the same collateral segregation as the original
security. To see this, consider a long agent first. The net variation in the collateral constraint from
this substitution is
∆θ+ = −(1 − αj ) + (1 − αj ) = 0
The term to enter negatively is the quantity of collateral that can be repledged out of 1 unit of
security j. The positive term is the quantity of the tree in the replicating portfolio. For a short
agent in j, the substitution yields
∆θ− = +1 − (1 − αj ) − αj = 0
The term to enter positively is the collateral requirement for j. The first negative term accounts
for the sale of 1 − αj units of tree while the second one represents the collateral requirement to
short the securities in ψ(j).
Hence, we have shown that c is budget feasible with securities in E1 . Since this is the optimal choice
of agents under a larger budget set, c is the optimal choice of agents in E(J1 ) and thus constitutes an
equilibrium allocation. C
END
31