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Why are collateral levels so extreme?
Agostino Capponi∗ W. Allen Cheng†
Abstract
We develop a game-theoretic model of centralized clearing to analyze a clearing-house’s choice of transaction fee and collateral requirements. The clearinghouse’srequirements affect not only the size and riskiness of her participating client base,but also the transaction fees charged to clients by her clearing member. We showthat empirically observed extreme collateral levels (compared to aggregate fees) canbe explained as the equilibrium arising from strategic interactions between profitmaximizing agents.
We analytically characterize the equilibrium fee-to-collateral ratio and find thatit depends on the relative riskiness of the contract and the relative costliness ofoperating a client clearing business (both relative to the depth of clients’ privatetrading benefits.) We observe that the clearinghouse tends to impose a very highcollateral requirement when the contract is very risky, so that participating clientsare mostly speculators as opposed to fundamental value traders, and impose lowerrequirements when the operational cost of client clearing is high, so to incentivizethe member’s participation.
1 Introduction
Clearinghouses use collateral, essentially a security that pays off only when there is adefault, to protect themselves from counterparty risk. In comparison, the transactionfees that clearinghouses charge are guaranteed upfront payments in all states of nature,and are their main source of revenue. Since the default probability of each individualclient is usually small, it might be tempting to conclude that the clearinghouse shouldbe indifferent between increasing collateral levels by a large amount and increasing feesby a small amount.
However, this stands in contrast with empirical evidence. High collateral levels (rel-ative to fees) consistently prevail in many derivatives markets, while low collateral levelsare common among primary asset markets. For example, for the year 2014, a conserva-tive estimate of Intercontinental Exchange’s (ICE) collateral holdings pledged for clearedCDS trades was around 20 billion USD, while the company’s revenue from CDS trans-actions totaled 161 million USD. Assuming an average five-year maturity of the CDStrades, this gives a fee-to-collateral ratio of around 161×5
20,000 ≈ 4%. This indicates that ICE
∗Department of Industrial Engineering and Operations Research, Columbia University, New York,NY 10027, USA, [email protected]†Department of Industrial Engineering and Operations Research, Columbia University, New York,
NY 10027, USA, [email protected]
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preferred collateral a lot more than fees. On the other hand, equity exchange-associatedclearinghouses often charge almost negligible collateral compared to aggregate brokeragefees.
Intuitively, a clearinghouse desires high levels of collateral either if she deems theclient to have high probability of default, or if she is very risk averse, i.e. she placessignificant emotional weight on the states of nature where a default occurs. Since a clear-inghouse could decide to not clear contracts when clients are deemed likely to default,the more plausible explanation seems to be that high collateral clearinghouses are veryrisk averse.
This paper shows that this needs not be the case. The empirically observed extremefee-to-collateral ratios can be explained as an equilibrium phenomenon arising from thestrategic interactions between agents. We design a game where the risk-neutral profitmaximizing clearinghouse clears client trades submitted via a clearing member bank. Shesets her requirements (transaction fee and collateral) for cleared trades which directlyaffect not only the size and riskiness of the clearinghouse’s participating client base, butalso the transaction fees charged to clients by her clearing member. Clients have theoption to default, and may do so when the contract value moves against them. The bankmay choose to leave the clearinghouse if the client clearing business is not profitable.We show that the resulting equilibrium can only involve very low or very high collaterallevels, depending on the riskiness of the traded contract and the benefits that clients cancapture from trading.
The clearinghouse’s requirements obviously reduce the clients’ profit from trading:the transaction fee directly lowers the clients’ overall surplus from trading, and collaterallowers the value of the client’s default option. Since participating clients are those whocan realize sufficient benefits from trading, increasing these requirements reduces theclearinghouse’s participating client base. In addition, we show that the profit of theclearing member bank decrease when requirements are higher, so that the clearinghousemay lose profitable client business if the bank chooses to not become a clearing member.
The requirements also act as a screening device. When they are high, participatingclients have smaller incentives to default either because they can capture large benefitsfrom maintaining the trade or because they would lose large amounts of collateral ifthey default. This means that the fraction of participating clients who default, given afixed realization of the contract value, is thus lower. The need for using such screeningdevices comes from the fact that the bank and clearinghouse either cannot or chooseto not discriminate between clients. In practice, while clearinghouses often have accessto the identities of the clients submitting trades, there are many clients for the clear-inghouse to keep track of every one of them. Indeed, transaction fees and collateralare usually charged based on some portfolio specific rule rather than tailored to clientspecific characteristics.1 In particular, firm specific characteristics such as credit quality
1ICE Clear Credit charges every clearing member with the same CDS portfolio the same amount ofinitial margin. Every client currently pays the transaction fee of $6 per million notional cleared for CDSindex contracts. For more detail see ICE Clear Credit’s schedule of fees and online documentation ofmargining rules.
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and asset size often do not factor into the fee and collateral calculations.Our model economy consists of three groups of risk-neutral agents: the clearing-
house, a potential clearing member bank, and a continuum of clients who may trade asingle, mandatorily cleared, contract. The economy has two periods, ex-ante and ex-postthe realization of the contract value. Prior to the realization, agents decide whether ornot to participate in the clearing process. The clearinghouse sets her fee and collat-eral requirements. The bank then sets his own transaction fee and participates if thebusiness is deemed profitable; that is, if his revenue can cover his operational costs forparticipation. High fee and collateral requirements both disincentivize the clients fromtrading. After the realization, clients default strategically, resulting in losses which areborne by the clearinghouse.2 Client’s trades are motivated by their private benefits andthe value of their default option. Non-defaulting clients receive (heterogenous) privatebenefits when they carry out cleared trades. In (subgame perfect) equilibrium all agentschoose strategies that maximize their profits, taking into account the reaction of others.We classify all possible equilibria of the model.
We find that when the operational cost of becoming a clearing member is low, so thatthe bank’s individual rationality constraint is non-binding, the prevailing equilibrium iseither one involving infinite collateral (when the contract is very risky) or one thatinvolves zero collateral (when the contract is not risky). We find that it is the relativeriskiness of the contract, measured by the volatility of the contract value over the depthof the private benefits to be realized, that determines which equilibrium prevails. Thismeans that infinite collateral equilibria prevails when many clients trade for speculativepurposes (when the default option value is high) and zero collateral equilibrium prevailswhen clients trade more for fundamental value (private benefits are large).
When the operational cost of becoming a clearing member is high, so that the bank’sindividual rationality constraint is binding, the clearinghouse can opt to set reducedrequirements to increase the bank’s profit, and incentivize him to participate as a clearingmember. In this case the extreme collateral phenomenon still prevails, as long as it isindividually rational for the clearinghouse to set up the clearing channel. The fact thatthe bank’s individual rationality constraint may be binding can be observed from therecent exit of many clearing members. For instance, in May 2014, the Royal Bank ofScotland announced the wind down of its clearing business due to increasing operationalcosts. This was followed by State Street, BNY Mellon, and more recently, Nomura;each shutting down part or all of their OTC clearing business. This “constrained”equilibrium has certain welfare implications, as we demonstrate the possible existenceof Pareto improvements in the model economy when this equilibrium prevails.
Our baseline model considers the case where collateral comes at no cost. The result-ing equilibria can involve infinite collateral levels which, while serving well our purposesof explanation and demonstrating the intuition, is far from realistic. We consider alsothe case where collateral comes at a small but positive cost. We show that our quali-tative results from the baseline model is naturally inherited while quantitatively we can
2We are implicitly assuming that the clearinghouse is acting as a true central counterparty. Allcounterparty risk and collateral management duties are borne by the clearinghouse.
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calibrate the model to match empirically observed fee and collateral levels.To the best of our knowledge, our paper is the first theoretical study to explain collat-
eral levels in a centralized clearing setting. Previous work in the context of collateralizedtrading often assume that either margining rules are exogenously given (Garleanu andPedersen (2011)), or that margins are set following some mix of the expected short-fall, VaR, and maximum shortfall measures (Anderson and Joeveer (2014), Duffie et al.(2015)). Johannes and Sundaresan (2007) and Capponi (2013) assume that (variation)margin payments on the valuation of swaps are set exactly to track mark to marketvalue changes. In contrast, our work looks for a micro-founded collateral setting rulevia solving a profit maximization problem. Our modeling approach is closely related tothose of Stiglitz and Weiss (1981) in the modeling of strategic defaults, and Holmstromand Tirole (1997) in the inclusion of private benefits.
On the other hand, the determination of optimal levels of collateral has been exten-sively analyzed in the corporate finance literature. For instance, Stiglitz and Weiss (1981)and Besanko and Thakor (1987) both find collateral as a useful screening device eitherthrough adverse selection or incentive effects, and that collateral levels arise from profit-maximizing Nash equilibria. Geanakoplos (1997) analyzes (general) collateral equilibriafor the case when assets are used to collateralize security trades, implicitly assuming the“smallness” of each agent. The centralized clearing setting is quite different, however.First, all agents trade publicly available contracts and are exposed to the same marketrisks, rather than bringing independent individual borrower risk to the table (Diamond(1984)), thus there is less asymmetric information about the risks clients are taking on.Second, the direction of future exposure is uncertain as either counterparty could beout of the money in the future. Third, collateral posting is not between clients, but isoften unilateral from the clients to the clearinghouse. While initial margins is postedby clients to the clearinghouse, the clearinghouse usually does not post initial marginsto the clients (Pirrong (2011)). Fourth, the clearinghouse often has significant marketpower.3
The rest of the paper is organized as follows. Section 2 explains the set up of thebaseline model. Section 3 solves for and characterizes all possible equilibria. Section4discusses welfare implications of the model. Section 5 analyzes the extended modelwhere collateral comes at a small but nonzero cost. Section 6 concludes. All proofs aredelegated to the appendix.
2 The model
Our model consists of three groups of risk neutral agents: the clearinghouse (CH), apotential clearing member bank (CM), and a continuum of clients trading a single,mandatorily cleared, derivative contract. There are two periods, separated by the real-ization of the contract value. Agents choose their participation in the trading/clearingprocess ex-ante; ex-post, clients default strategically and the clearinghouse bears the
3For instance, while ICE, CME, and LCH all clear CDS, the bulk of CDS are still cleared throughICE both in the US and Europe.
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losses.4 The clearinghouse has a large endowment of equity and does not default onher obligations. The bank only serves as an intermediary in the model so he does notdefault.
Before the contract value is realized, the clearinghouse can set her collateral C andfee δc requirements per contract cleared. The bank is the prime broker of a continuumof clients with unit mass, whose “private benefit parameters” B are described by thedistribution F , i.e. F (t) is the fraction of clients whose private benefits that do not exceedt. In view of C and δc, the bank can set his fee per contract cleared δb. After δc, C andδb have been declared, each client, characterized by his private benefit parameter B, hasthe choice between trading long one contract (L), trading short one contract (S), andnot-trading (NT ). These actions provide private benefit B, −B, and 0, respectively.Since each individual client is small, he cannot become a clearing member himself andcannot afford to trade more than one contract.5
Client choices are submitted to the bank. After receiving the trade orders, the bankhas a choice between joining (J) and not joining (NJ) as a clearing member. The bankincurs an operational cost G for joining.6 The bank’s payoff is given by:
Bank’s Payoff = δb ×mass of clients who trade−G. (2.1)
If the bank were to become a clearing member, a clearing channel is set up. Clientswho trade pay the total fee of δ := δb + δc where δc goes to the clearinghouse andδb to the bank, before the contract value is realized. They also post collateral to theclearinghouse.
After this is done, the contract value is realized with value ε ∼ H. The distributionsF and H are assumed to be Laplace with parameters (0, λ) and (0, γ), respectively. Thatis, the cumulative distribution functions are
F (t) ={
1− 12e−λt , t ≥ 0
12eλt , t < 0
,
H(t) ={
1− 12e−γt , t ≥ 0
12eγt , t < 0
.
We use f and h to denote the associated density functions. Notice that while H is aprobability distribution describing the random realization of a random variable ε, F isnot; it is used to describe the distribution of deterministic private benefits among a unitmass of clients.
Before proceeding further, we discuss the economic interpretation of the parametersγ and λ. The mean absolute deviation of a Laplace (0, γ) distribution is 1
γ . γ serves as4Since there are only two periods, it suffices to consider only initial margins and not variation margin
posting.5In practice, clients usually are asset management funds or money market funds and are very small
compared to clearing members, who are usually large broker-dealers (Pirrong (2011)).6As a clearing member, the bank needs to meet certain capital requirements, contribute to a default
fund, set up operational channels for client clearing, and, in extreme circumstances, bear large losses ofthe clearinghouse.
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CH
δc, C
CM
δb
CT
{L, S,NT}
CM
{J,NJ}
Fee collection
ε realized
CT
{D,ND}
CH suffers losses
Figure 1: The extensive form game timeline
a measure of the volatility of the random contract value, with the contract value beingmore volatile when γ is small. It thus represents the extent to which clients trade dueto speculation, generating surplus primarily from the default option. λ, on the otherhand, serves as a measure of the depth of deteministic private benefits to be realized fromfinancial trading. When λ is small, there are many clients with large private benefits. Itthus represents the extent to which clients trade due to fundamental value, generatingsurplus primarily from capturing private benefits.
After the contract value is realized, clients default strategically. If a long client chooseto not default (ND), he receives his private benefit B and the contract realization ε; ifhe chooses to default (D), he does not receive the private benefit and loses his collateralC. The short case is analogous: he either receives −B and −ε or loses C. They defaultwhenever it is more profitable to do so.
Last, the clearinghouse collects payments from clients who are out of the money andis obligated to pay the clients who are in the money. When out-of-the-money clientsdefault, the clearinghouse experiences a shortfall in payments and must make up for thedifference using her own equity thereby incurring a loss. The clearinghouse’s payoff isthus
δc × client trades− loss per contract× clients who trade and default. (2.2)
A graphical illustration of the timeline of the game is given in Figure 1.
3 Equilibrium collateral levels
In this section we solve for the subgame perfect Nash equilibria using backward induction.Throughout the paper a tilde sign denotes an implicitly defined function version of thevariable, now as a function of the parameters of an underlying optimization problem ordefining equation.
We first solve for clients’ choice of defaults, assuming that the clearing channel hasbeen set up. Since clients default whenever it is more profitable, the long client’s payofffunction is:7
max(B − δ + ε,−δ − C). (3.1)
The short client’s payoff function is
max(−B − δ − ε,−δ − C) (3.2)7In Section 5 and the majority of the proofs, we consider the generalized case where collateral comes
at a cost α ≥ 0. In that case the long client’s payoff is given by max(B − δ − αC + ε,−δ − (1 + α)C).
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Notice that clients default only when the market moves against them. In particular, ifε < 0, all long buyers with insufficient private benefit B + ε ≤ −C will default; if ε > 0,all short buyers with insufficient (negative) private benefit −B − ε ≤ −C will default.
Next, the bank joins (J) as a clearing member if the client clearing business isprofitable:
δb ×mass of clients who trade−G ≥ 0.
Clients trade when their ex-ante expected payoff is positive. Our first theorem showsthat there is a unique private benefit threshold governing the trading decisions of theclients.
Theorem 1. Fix δ, C ≥ 0, then there exists a unique trading threshold B = B(δ, C)such that a client wants to trade long if B ≥ B, and wants to trade short if B ≤ −B.
A straightforward calculation (included in the appendix) shows that B is the solutionto
δ = B +∫ ∞B+C
(1−H(x)) dx.
The above expression indicates that this “trading threshold” does not depend on thedistribution of private benefits F . Indeed, after δ and C are declared, each client needsonly evaluate the profitability of his own potential trade, disregarding the trading actionsof his fellow clients. From Theorem 1 we see immediately that clearinghouse requirementscreate a “no-trade” region.
The clearinghouse may want the clients with private benefits B > B > −B to trade.However, the requirements are such that this would not be profitable for these clients.In our model, the clearinghouse and bank only have information about the distributionof client private benefits and cannot distinguish between clients before the trades aresubmitted. Therefore they cannot incentivize more clients to trade by offering themreduced fees or lowered collateral levels. This phenomenon is illustrated in figure 2. Weremark that since F is symmetric, the number of positions traded by each bank’s clientsnet to zero (the market always clears).
Notice that a client with private benefits B such that B ≤ B ≤ −B would want tobe both long and short.8 However, multiple contracts are usually governed by a masteragreement that dictates that when a client defaults on one position, he must defaulton all positions governed by the agreement (Hull (2012)). Thus, in the case whereB ≤ B ≤ −B, the client’s payoff from trading both long and short a contract is
max(B − δ + ε−B − δ − ε,−2δ − 2C) = −2δ − 2C.
Hence, he would never choose such a position when fees or collateral are positive. Inthis case the client chooses the option that gives higher expected payoff: long if B > 0and short if B < 0. Thus the level of private benefits beyond which clients will trade
8Notice that this case exists if and only if B ≤ 0.
7
B
No Trade Region
LongShort
~B! ~B 0j
f(B)
Figure 2: Each client trades depending on his level of private benefits. Only clients with high (low)benefits trade long (short).
long is bounded below by zero.9 It will thus be convenient to define the effective tradingthreshold B(δ, C) = max(B(δ, C), 0).
We can now express the bank’s payoff function given in Eq. (2.1) as
R(δb; δc, C,G) := 2δb(1− F (B))−G, (3.3)
where 1− F (B) is the fraction of clients who are long the contract.The clearinghouse’s payoff function is
X(δc, C) :=2δc(1− F (B)) + (ε+ C)(F (−C − ε)− F (B))+ + (−ε+ C)(F (−C + ε)− F (B))+.
The first term corresponds to the clearinghouse’s income from transaction fees. Thesecond term is the aggregate loss from long clients who default: the loss per default−ε−C multiplied by the mass of clients who traded long ex-ante and defaulted ex-post(F (−C−ε)−F (B))+. The last term is the aggregate loss from short clients who default.The case of long clients defaulting on a negative realization of ε is illustrated in Figure3. When a low contract value εL is realized all clients with private benefits between Band −C − ε have traded long the contract and choose to default, creating losses to theclearinghouse. When a high value εH is realized, there are no defaults, since the clientswho would have defaulted did not trade in the first place.
The clearinghouse’s expected payoff can be explicitly computed and is given by
E[X(δc, C)] = δce−λB − λ
2(λ+ γ)e−λB−γ(B+C)
(1γ
+ 1λ+ γ
+ B
). (3.4)
We can now define our subgame perfect Nash equilibria.9As we will see later, this effect of the master agreement induces a discontinuity in the rate at which
the payoff of the bank changes with respect to the clearing requirements.
8
B
Long
Everyone below this line defaults
!C ! "L!C ! "H0
f(B)
~B
No defaults from participating clients
Figure 3: After the contract value is realized, clients with insufficient benefits default. When a lowvalue εL is realized, clients who trade and default create losses for the clearinghouse. When a high valueεH is realized, there are no defaults.
Definition 1. An equilibrium is a triple (δc, C, δb(·, ·)) such that
(δc, C) ∈ argmax(δc,C)∈R2
+
E[X(δc, C)]; (3.5)
δb(δc, C) ∈ argmaxδb∈R+
R(δb; δc, C,G) for all (δc, C) ∈ R2+; (3.6)
R(δb; δc, C, G) ≥ 0 (3.7)E[X(δc, C)] ≥ 0. (3.8)
Notice that we only focus on the clearinghouse and the bank’s choice in equilibriumto reduce clutter. In equilibrium, the bank, in response to δc and C, chooses a fee levelδb(δc, C) that maximize his expected payoff. The clearinghouse takes this into accountand sets δc = δc and C = C to maximize her profit. The last two constraints are theindividual rationality (IR) constraints of the bank and the clearinghouse, respectively.We assume both the bank and the clearinghouse have a reservation value of 0: The bankonly joins when his revenue covers his costs, and the clearinghouse will only initiate thebusiness if her expected profit is nonnegative.10
We next show that the bank always has a unique choice of fees that maximizes hispayoff:
Theorem 2. Fix δc, C ≥ 0, then R(δb; δc, C,G) has a unique local maximum at δb =δb(δc, C). In addition, δb(δc, C) is continuous.
10Although not setting up the clearing channel is not an explicit choice of strategy our model set up,the clearinghouse can effectively “kill the market” by choosing δc =∞.
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Theorem 2 indicates there will not be a sudden “jump” in bank fees when the clear-inghouse changes her requirements. The unambiguity of the bank’s choice will play animportant role in the analysis of the clearinghouse’s action.
Interestingly, although δb(δc, C) is continuous, depending on δc and C, the bank’sresponse to increases in the clearinghouse’s requirements fees may be increasing his fees(augmenting fees) or decreasing his fees (complementing fees). It will be convenient todefine the “regime switching” function:
ξ(δc, C) := 1 + λδc −12e−γC
(1 + λ
γ
)(3.9)
Our next result gives the precise conditions identifying the two regimes.
Theorem 3. The following statements hold:
1. (Augmenting fees.) If ξ ≥ 0, δb is given as the unique solution to
γ(δc + C) + log 2− 1 = −(λ+ γ)δb − log(1− λδb), (3.10)
greater than or equal to γλ+γ
1λ . In this case ∂δb
∂δc= ∂δb
∂C ≥ 0, and B ≥ 0.
2. (Complementing fees.) If ξ < 0,
δb = 12γ e
−γC − δc. (3.11)
In this case ∂δb∂δc
< 0, ∂δb∂C < 0, and B = 0.
We refer to the first regime as the bank imposing augmenting fees. In this regime,when the clearinghouse increases requirements, the participating clients base shrinks; inresponse, the bank increases his fee, which further decreases the participating client base.However, the gain from increased fee outweighs the loss in the mass of participatingclients. We refer to the second regime as the bank imposing complementing fees. Inthis regime, the entire client base is participating; when the clearinghouse increasesrequirements, the participating clients base again shrinks. Since the clearinghouse’srequirements are low, the bank decreases his fee so that again the full client base isparticipating. In this case, the loss from the decreased fee is outweighed by the gain inthe mass of participating clients. Some examples of the revenue functions (income as afunction of the bank’s fee) the banks face is given in Figure 4.
Additionally, Theorem 3 shows that the clearinghouse’s strategy space is separatedinto two distinct regions. Which regime the bank adapts to depends on her choice ofclearing requirements. Figure 5 illustrates this phenomenon. Notice that the comple-menting fee regime is feasible only when γ
λ < 1, i.e. when the contract is sufficientlyrisky.
10
/b0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
R
Optimal choice
(a) Augmenting fees. (δc, C) = (2, 1)
/b0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0
0.5
1
1.5
R
Optimal choice
(b) Complementing fees.(δc, C) = (1, 1)
Figure 4: Revenue as a function of the bank’s fees for γ = 0.2 and λ = 1. The bank’s choice of fees isalways unique.
After characterizing the bank’s response, we can solve for the clearinghouse’s action.For future purposes, it will be convenient to work with the normalized quantities:
u := λδb,
v := γδc
w := γC,
θ := γ
λG := λG
Normalized bank’s profit := λR(δb; δc, C,G),Normalized clearinghouse’s expected profit := γE[X(δc, C)].
(3.12)
From this point on when we refer to profits, we refer to normalized profits unless specifiedotherwise.
When the clearinghouse sets her requirements, she must take into account the twopossible regimes that can result from her actions and the bank’s IR constraint. Thegeometry of the constraint is given by the following theorem:
Theorem 4. The graph L := {(δc, C)|R(δb(δc, C); δc, C,G) = 0} is a decreasing, convex,C1 curve. Increasing G shifts L downwards. In addition, L crosses the curve ξ = 0 atmost once.
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C
9 < 0
9 > 0
Augmenting Fees
/c
0
/c =1
2.!
1
26
Complementing FeesC =
1
.
3log
31 +
6
.
4! log 2
4
Figure 5: The clearinghouse’s strategy space is separated into two sections. Depending on her choiceof requirements, the bank reacts in different regimes. The complementing regime is feasible only whenγλ< 1.
This geometry is demonstrated in Figure 6. As G increases, the feasible region inthe δc − C plane is reduced.
We are now positioned to solve for the clearinghouse’s equilibrium action.
Theorem 5. The clearinghouses equilibrium action is one of five types.
(i) (Unconstrained infinite collateral equilibrium) (u, v, w) = (1, θ,∞). The clear-inghouse’s (normalized, expected) profit is e−2. The bank’s (normalized) profit ise−2 − G. This can occur only when G ≤ e−2.
(ii) (Constrained infinite collateral equilibrium) (u, v, w) = (1,−θ(log G+ 1),∞). Theclearinghouse’s profit is −θG(log G + 1). The bank’s profit is 0. This can occuronly when e−1 ≥ G ≥ e−2.
(iii) (Unconstrained zero collateral equilibrium) u is the unique solution to the equation
− log 2 + 1− log(1− u) + θ2
u− (1 + θ)2 = 0
which is larger than −θ2+θ√θ2+4
2 . (v, w) =(−θ2
u + (1 + θ)2 − (1 + θ)u, 0)
. The
clearinghouse’s profit is θu2+(θ4+2θ3+2θ2)u−θ4−θ3
(1+θ)2u (2(1 − u))1θ . The bank’s profit is
u(2(1− u))1θ − G.
(iv) (Constrained zero collateral equilibrium) u is the unique solution to the equation
u(2(1− u))1θ = G
12
C
/c
0
GH
GL
9 = 0
IR constraintsFee regime switch
Figure 6: Bank’s IR constraint. The bank’s individual rationality constraint limits the feasible regionof the clearinghouse’s fee/collateral choices. Higher operational costs (GH > GL) reduce the feasibleregion.
which is larger than θ1+θ . (v, w) = (1− log 2− (1 + θ)u− log(1− u), 0). The clear-
inghouse’s profit is Gu
(−(2+θ)(θ+(1+θ+θ2)u)
(1+θ)2 + (θ+u)(log 2−log(1−u))1+θ
). The bank’s profit
is 0.
(v) (No clearing) The clearinghouse does not set up the clearing channel. This canoccur only when G ≥ e−1.
In equilibria where the clearing channel is set up, the bank must be imposing augmentingfees. Which equilibrium prevails depends only on the value of θ and G.
Theorem 5 shows that the empirically observed low fee-to-collateral ratio can beexplained as a result of risk neutral agents’ strategic actions. What is most compellingis that observable equilibria either involves very high levels of collateral or very low levelsof collateral, and which equilibrium prevails depends only on the relative riskiness of thecontract and the operational cost of joining.
Consider first the case where G is very low so that the banks’ IR constraint isnot-binding. In this case the prevailing equilibrium is either the unconstrained infinitecollateral equilibrium or the unconstrained zero collateral equilibrium. We can eas-ily demonstrate numerically that there exists situations in which either equilibrium isachieved. Table 1 gives some numerical values of the expected payoff function whenλ = 1.
By plotting the optimized expected payoff as a function of γ, fixing λ = 1, we seean even stronger phenomenon. As illustrated in Figure 7, there is a critical value γ∗beyond which the clearinghouse chooses zero collateral, and below which the clearing-house chooses infinite collateral. When γ is small, the contract value is very volatile
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γ||(δc,C,E[X(δc, C)]) Infinite Collateral Zero Collateral0.2 (1,∞, .1353)∗ (2.9493, 0, .1119)0.59 (1,∞, .1353) (1.3261, 0, .1356)∗1 (1,∞, .1353) (1.0681, 0, .1380)∗∞ (1,∞, .1353)∗ (1, 0, .1353)∗
Table 1: Expected payoffs at possible extrema for λ = 1. Asterisks describe which equilibrium is chosenby the clearinghouse.
and the expected loss of the clearinghouse from client defaults is high. The clearing-house thus chooses to eliminate defaults by setting high levels of collateral. When γ islarge, the contract value is not as volatile and the expected loss of the clearinghousefrom client defaults is low. The clearinghouse thus chooses to incentivize more clientsto trade by setting collateral at very low levels. As γ approaches infinity, the contractvalue converges to zero. In this case the decisions of clients become independent of thecollateral requirements, so that the zero collateral equilibrium and the infinite collateralequilibrium coincide and give the same optimized value.
The phenomenon observed in Figure 7 is quite general. By Theorem 5 we know thatwhich equilibrium prevails when the bank’s IR constraint is not binding depends onlyon the relative riskiness (relative to the depth of private benefits) of the contract value,θ. Since from Figure 7 the threshold γ∗ ≈ 0.58, we know that θ∗ = γ∗/1 also serves as athreshold for all pairs (γ, λ): if θ > θ∗, the clearinghouse chooses zero collateral (infinitefee-to-collateral ratio), and infinite collateral (zero fee-to-collateral ratio) otherwise.
Next we consider general case when the cost of joining, G, is high, so that the bank’sIR constraint may be binding. Due to the algebraic complexity of the expressions inTheorem 5, we numerically solve for the clearinghouse’s expected profit for when shedemands infinite or zero collateral, subject to the bank’s IR constraint being satisfied.We record the equilibrium chosen for various values of θ and G and present the resultsin Figure 8. Again, Theorem 5 implies that the phenomenon observed in Figure 8 isgeneral for all triples (γ, λ,G).
We remark that all five equilibria outlined in Theorem 5 are observed in Figure 8. Inaddition, we observe even stronger phoenomena: (i) infinite collateral equilibria prevailwhen the contract is risky (θ is low), (ii)infinite collateral equilibria prevail when thebank’s normalized operational cost is high, and that (iii) there exists situations wherethere is no equilibrium when the operational cost is too high. Indeed, when G is too high,the clearinghouse must set requirements very low for the bank to receive enough revenuefrom his client clearing business; such requirements, however, mean the clearinghouse istaking losses, and it is therefore not individually rational for her to set up the clearingbusiness. No clearing equilibria exist in this case.
14
.0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Exp
ecte
d P
ayof
f
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
Chooses In-nite Collateral Chooses Zero Collateral
Infinite CollateralZero Collateral
Figure 7: Maximum clearinghouse expected payoff as a function of γ for λ = 1. The clearinghouseschooses infinite collateral is below the threshold value γ∗. Beyond this threshold value, the clearinghousechooses zero collateral.
4 Pareto improvements
One of the most intriguing phenomena we observe in Figure 8 is the “taking over” of zerocollateral equilibria by infinite collateral equilibria when the operational costs becomehigher. In this section we discuss how a social planner can make Pareto improvementsto the model economy by increasing the operational cost G via a lump sum tax whenthe equilibrium is constrained. The primary benefit of such a tax is to lower the feesimposed by the clearinghouse and stimulate trading activity in the model economy.When there are more trades, the captured additional surplus more than compensatesfor the decreased revenue.
Theorem 6. Suppose the prevailing equilibrium is a constrained infinite collateral equi-librium. Assume that if the operational cost increases to G + τ a constrained infinitecollateral equilibrium still prevails. Then social planner can improve social welfare byimposing a lump-sum tax τ on the bank, increasing the operational cost from G to G+τ ,and transferring the tax to the clearinghouse.
When the bank’s operational cost increases, the clearinghouse must decrease theirfee’s so that the bank still participates. The clearinghouse, receiving the lump sumtransfer τ now has profit:
P (τ) := τ − (G+ τ)(log λ(G+ τ) + 1).
15
~G0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
3
C1, Unrestricted
C1, Restricted
C0, Unrestricted
C0, Restricted
No Equilibrium
Figure 8: Prevailing equilibrium for various θ and G. Both high riskiness and high operational costspush the clearinghouse towards demanding infinite collateral (C∞). There is no equilibrium when G istoo high.
Since the prevailing equilibrium is an infinite collateral equilibrium, we must have λ(G+τ) < e−1. This implies
∂P
∂τ= −(log λ(G+ τ) + 1) > 0, for λ(G+ τ) < e−1.
Thus, while the fees per client traded decrease, this is more than offset by the increasedparticipation client base and lump sum transfer. The clearinghouse is better off. Thebank is kept at zero profit and is indifferent. Since in any infinite collateral equilibriumbank’s set u = 1, more clients trade and old clients pay a lower fee and are better off.
5 Costly Collateral
So far we have focused our analysis on the case where collateral is freely available toall clients. While this provides us with clear intuition about the arising equilibria aspresented in Figure 8, the infinite collateral equilibrium clearly does not arise in practice.In this section we consider the case where collateral comes at a cost of α per unit.We show that the qualitative results presented in Section 3 remain the same: thereexists equilibria that parallel the notions of zero and infinite collateral equilibria, whereequilibria involve collateral levels that are “large but finite.” For this section we restrictour attention to the cases where the bank’s IR constraint is non-binding.
16
The payoff function for the long client for costly collateral is:
max(B − δ + ε− αC,−δ − (1 + α)C). (5.1)
Since our proofs in Appendix A take into account collateral costs, one can easily see thatthe following analogies of Theorems 1–4 hold true:
1′. There is a unique trading threshold B for (δ, C) ∈ R2+. The market always clears.
2′. The banks always has a unique choice of fees δb which is continuous in the clearing-house’s requirements.
3′. Define ξα := ξ + λαC. The bank imposes augmenting fees when ξα ≥ 0, given as thethe unique solution to
γ(δc + (1 + α)C) + log 2− 1 = −(λ+ γ)δb − log(1− λδb)
greater than or equal to γγ+λ
1λ . Otherwise δb = 1
2γ e−γC − δc − αC.
4′. The bank’s IR constraint is decreasing and convex.
In what follows we will restrict our attention to the case when the bank’s IR constraintis non-binding (e.g. G = 0). Our first result characterizes the possible equilibrium forcostly collateral
Theorem 7. Let(δc, C, δb(·, ·)
)=:
(vγ ,
wγ ,
uλ
)be an equilibrium. Assume G = 0 and
α ≥ 0. Then one of the following hold:
(i) u solvesη(u; θ, α) = 0,
where
η(u) := −2θ − θ2 + θ2
u− log(2− 2u),
and
v = −θ2 + (1 + θ)2u− (1 + θ)u2
u.
w = 0.
(ii) u solvesκ(u; θ, α) = 0,
where
κ(u; θ, α) := α2(1 + θ)2 + α(θ3 + 3θ2 + θ + (1 + θ − 2θ2 − θ3)u)− θ(θ + θ2 − u)(u− 1)− α(1 + θ)(u− 1) log(2− 2u).
17
and
v = α− θ2(1 + α) + ((1 + θ)2 + (−1 + θ + θ2)α)u− (1 + θ)u2 − α(1− u) log(2− 2u)α+ u
,
w = −(1 + θ)u− log(2− 2u) + 1− v1 + α
= θ2 − 2θu− θ2u− u log(2− 2u)α+ u
.
The first case in Theorem 7 shows that the zero collateral equilibrium is naturallyinherited into the case of costly collateral. In fact, since zero collateral equilibria maxi-mizes clearinghouse profit when the clearinghouse’s requirements are restricted to C = 0,the clearinghouse’s requirements are independent of α for zero collateral equilibria.
The second case parallels the case of unconstrained infinite collateral as given inTheorem 5. We perform a analysis of the equilibria arising when collateral costs are“local to zero” in Theorem 8, in which we demonstrate that collateral levels that prevailin equilibrium can be large and approach infinity as α go to zero.
Theorem 8. For each θ ∈[0, −1+
√5
2
], there exists α(θ) such that for A := [0, α(θ)],
there exists a differentiable function u∞θ : A→[
θ1+θ , 1
]such that u∞θ (α) solves κ(u; θ, α) =
0. In addition,
1. limα−>0 u∞θ (α) = 1,
2. ∂u+θ
∂α < 0.
3. The associated equilibrium fee v = v(α, u∞θ (α)) increases with α, while the equilib-rium collateral level w = w(α, u∞θ (α)) decreases with α.
In what follows we will refer to u∞θ (α) as the infinite collateral path. Theorem 8 im-plies that for small α and θ < 0.618,11 when α increases the clearinghouse’s requirementschange in such a way that the bank’s normalized fees move along the infinite collateralpath. This corresponds to decreases in the collateral and fee requirements. Intuitively,as collateral becomes costlier to come by, less clients are willing to trade. In view of this,the clearinghouse shifts away from collateral requirements and asks for more fees. Thisin turn reduces the bank’s fees.
To see how general are the local results presented in Theorem 8, we numerically solvefor the fees and (unnormalized) collateral levels arising along the infinite collateral pathand present them in Figure 9 for γ = 0.2 and λ = 1.
Given γ and λ, one can easily calibrate our extended model to explain observed feeand collateral levels. For instance, keeping γ = 0.2 and λ = 1, a collateral cost of two tothree basis points (corresponding to ratios of 3.9 and 4.25 percent, respectively) matchesthe empirically observed four percent fee-to-collateral ratio of ICE Clear Credit.
In Figure 10 we present collateral levels for various values of θ and α along the infinitecollateral path. We observe that collateral levels generally decrease as α increases,
11Notice that from our numerical results in Figure 7, all values of θ where the unconstrained infinitecollateral equilibrium prevails are under 0.6.
18
,0 0.005 0.01 0.015 0.02 0.025 0.03
0
5
10
15
20
25
30
/c=CC
Figure 9: Fee to collateral ratios and collateral levels along the branch of solutions u∞(α) for γ = 0.2and λ = 1. Collateral levels decrease with cost.
following the results of Theorem 8. At the same time, we observe another interestingphenomenon: for fixed α collateral levels are not a monotonic function of θ. Indeed, forcertain levels of alpha, we see that collateral levels, both normalized and unnormalized,increase when we move from θ = 0.1 to θ = 0.2, but then decrease when we move furtherto θ = 0.3.
We offer an explanation to this phenomenon as follows. There are two counteractingeffects when θ increases. When the contract is riskier, the clearinghouse would like tocharge more collateral to protect herself against default losses. At the same time, moreclients are willing to trade since the value of the default option is higher. In the case ofour baseline model when collateral comes at no cost, the clearinghouse can effectivelylower the portion of “speculating” clients by setting collateral to infinity.
When collateral comes at a cost, clients essentially pay an upfront payment to cred-itors to obtain collateral. This creates a leak of value out of the the system since theupfront payment does not increase the clearinghouse’s profit. When the contract isriskier, more clients are willing to borrow collateral and speculate on the high value ofthe default option. The lowered aggregate surplus actually lowers the clearinghouse’sprofit and thus she would rather lower her collateral requirement.
Our numerical results show that for certain parameter values, this second effectcan dominate and result in the clearinghouse lowering collateral requirements when thecontract becomes riskier.
19
,0 0.005 0.01 0.015 0.02 0.025 0.03
0
5
10
15
20
25
30
C
. = 0:1
. = 0:2
. = 0:3
. = 0:4
. = 0:5
(a) Collateral levels
,0 0.005 0.01 0.015 0.02 0.025 0.03
0
1
2
3
4
5
6
.C
. = 0:1
. = 0:2
. = 0:3
. = 0:4
. = 0:5
(b) Normalized collateral levels
Figure 10: Collateral levels along the infinite collateral path for various γ and λ = 1. While collaterallevels generally decrease with costs, they do not necessarily increase with riskiness.
6 Concluding remarks
Our model of centralized clearing is designed to explain the extreme collateral levelsobserved in the market. Our results indicate that fee and collateral choices of the clear-inghouse affect her payoffs in two different ways: they influence the payoff per clientcleared and the mass of clients who trade. We have analyzed the equilibrium that re-sults from the profit maximization process of all agents in the model. We demonstratedthe possible existence of Pareto improvements in equilibrium, and provided a calibrat-able framework with collateral costs. It should be noted that we are not asserting thatthe low fee-to-collateral ratios must result from strategic expected profit maximization,but rather propose it as a plausible alternative to assigning agents some (unobservable)high level of risk aversion and utility constraints.
Our choice of the Laplace distribution is due to the straightforward economic inter-pretation of its parameters and the analytical tractability. However, it is also desirable toassess the robustness of our results with respect to different distributional specificationsof private benefits and contract value.
Our results can be easily extended to the case of multiple clearing members whenthe clearinghouse sets requirements separately for each clearing member’s client base. Amore interesting but difficult extension is to consider multiple clearing members wherethe clearinghouse is restricted to set uniform requirements for clients clearing via all
20
member banks. In this case one must take into account all of the private benefit distri-butions and the relative sizes of the potential client bases that can be cleared throughthe clearing member. We expect that when one clearing member has many more clientsthan others, the clearinghouse would choose requirements close to the case of one membertreated in this paper.
One important part of the default waterfall we do not address is the clearinghouse’schoice of guaranty fund requirements. In our framework, there is no need for guarantyfunds since the clearing member does not default. One future line of research would beto analyze how clients would endogenize potential clearing member defaults and howthis affects the equilibrium collateral levels.
A Proofs
In this section we restate and prove the generalized versions of the theorems in section3, taking into account the cost of collateral (see section 5) . The results presented insection 3 can be obtained by setting α = 0.
Theorem 1a. Fix δ, C, α ≥ 0, then there exists a unique trading threshold B = B(δ, C;α)such that a client wants to trade long if B ≥ B, and wants to trade short if B ≤ −B.
Proof of Theorem 1a. We start with the long case. Assume B is a solution to
δ + αC = E[(B + ε)1ε>−B−C − C1ε≤−B−C ] (A.1)
That is, B is a level of private benefit at which expected profit for the client is zero. Set
φ1(B) := E[(B + ε)1ε>−B−C − C1ε≤−B−C ],
we see that
limB→∞
φ1(B) = limB→∞
E[B1ε>−B−C ] + E[ε1ε>−B−C ]− E[C1ε≤−B−C ]
=∞+ 0− C =∞.
by using the fact that E[ε] = 0 and applying the monotone convergence theorem. Inaddition,
limB→−∞
φ1(B) = limB→−∞
−(−B − C)E[1ε>−B−C ] + E[ε1ε>−B−C ]− C
= −C,
where we have used the fact that E|ε| < ∞ implies that limx→∞ xP (ε > x) = 0. Thus,if δ, C, α ≥ 0, there must exist B = B such that Eq. (A.1) is satisfied.
Next, notice that φ1(B) is a strictly increasing function of B, since
φ1(B) =∫ ∞−B−C
(B + x) dH(x) +∫ −B−C−∞
−C dH(x),
21
φ′1(B) = 1−H(−B − C) > 0.
So the solution must be unique. Since H is symmetric, the short case follows by asymmetry argument.
Theorem 2a. Fix δc, C, α ≥ 0, then R(δb; δc, C,G) has a unique local maximum atδb = δb(δc, C;α). In addition, δb(δc, C;α) is continuous in δc and C.
Before the proof of Theorem 2a, we introduce two auxiliary propositions:
Proposition 1. Let δ, C, α ≥ 0. Then B solves
δ + αC = B + 12γ e
−γ(B+C). (A.2)
In addition B ≥ 0 if and only if
δ + αC ≥ 12γ e
−γC .
Proof of Proposition. By definition
δ + αC =∫ ∞−B−C
(B + x) dH(x) +∫ −B−C−∞
−C dH(x)
= B − (B + C)H(−B − C) +∫ 0
−B−Cx dH(x) +
∫ ∞0
x dH(x)
= B − (B + C)H(−B − C) + xH(x)∣∣∣0−B−C
−∫ 0
−B−CH(x) dx+
∫ ∞0
(1−H(x)) dx
= B −∫ B+C
0(1−H(y)) dy +
∫ ∞0
(1−H(x)) dx
= B +∫ ∞B+C
(1−H(x)) dx
Here we used the layer cake representation of expectation to derive the third equalityand the fact that H is a symmetric distribution to derive the fourth equality. Now plugB = 0 into Eq. (A.2), we have
δ + αC =∫ ∞C
(1−H(x)) dx = 12γ e
−γC .
Notice that
∂
∂B
(B +
∫ ∞B+C
(1−H(x)) dx)
= H(B + C) > 0,
so B(δ, C;α) ≥ 0 if and only if δ ≥ 12γ e−γC .
The next proposition follows immediately from differentiating Eq. (A.2):
22
Proposition 2.
∂B
∂δ= 1H(B + C)
> 0
∂B
∂C= 1 + α
H(B + C)− 1 ≥ 0
Proof of Theorem 2a. We first show the existence of a maximizer. Define the bank’srevenue function as
φ2(δb) := δb(1− F (B(δb + δc, C;α))).
By Proposition 1, for large enough δb, we have B(δb + δc, C;α) = B(δb + δc, C;α). ByProposition 2, ∂B
∂δ > 1 since H is a distribution function. In addition, B is continuoussince B is continuous. Thus, it follows that
limδb→∞
B(δb + δc, C;α) =∞ (A.3)
limδb→∞
δb
B(δb + δc, C;α)<∞. (A.4)
Since∫∞
0 tdF (t) = 12λ <∞, it follows by dominated convergence:
limB→∞
B(1− F (B)) = 0,
which implies,
limδb→∞
φ2(δb) = limδb→∞
δb
B(δb + δc, C;α)limB→∞
B(1− F (B)) = 0.
Where we used Eq. (A.3) and (A.4) to derive the second limit. Since φ2(δb) ≥ 0 andφ2(0) = φ2(∞) = 0, and φ2 is continuous, there exists an interior maximizer of φ2(δb)on (0,∞).
Next we show uniqueness. Fix δc, C, α ≥ 0. By Proposition 1, the bank’s payofffunction can be written as
2δb(1− F (B))−G =
δbe−λB(δb+δc,C) −G, δb + δc + αC ≥ 12γ e−γC
δb −G, δb + δc + αC < 12γ e−γC (A.5)
Suppose δb ∈{δb∣∣∣B(δb + δc, C;α) < 0
}={δb∣∣∣δb + δc + αC < 1
2γ e−γC
}is a local
maximizer of φ2, then φ2 can be always increased by choosing δb slightly larger thanδb, thus δb cannot be a local maximizer. Therefore, a local maximizer must be in theregion {δb|B(δb + δc, C;α) ≥ 0}. This implies that a local maximizer δb either solves theequation
0 = B(δb + δc, C;α), (A.6)
23
or is a critical point of the differentiable function
φ3(δb) := δbe−λB(δb+δc,C;α).
Obviously, if Eq. (A.6) has a nonnegative solution, it must be δ0b := 1
2γ e−γC − δc − αC,
in this case the bank’s payoff is exactly δ0b −G by Eq (A.5). Also by Eq (A.5), if a local
maximizer is a critical point of φ3, it must be larger than δ0b .
We now analyze the critical points of the function φ3 strictly larger than δ0b . We will
show there is at most one such critical point.The first order condition is
0 = 1− λδb
1− 12e−γ(B(δb+δc,C;α)+C)
(A.7)
By Eq. (1), we also have
δb + δc + αC = B + 12γ e
−γ(B+C) (A.8)
Thus, combining Eq. (A.7) and (A.8) any critical point of φ3(δb) must satisfy
γ(δc + (1 + α)C) + log 2− 1 = −λδb − γδb − log(1− λδb) (A.9)
Defineφ4(δb) := −λδb − γδb − log(1− λδb),
andφ5(γ) := γ(δc + C) + log 2− 1.
We see that
φ4(0) = 0lim
δb→1/λφ4(δb) =∞
φ′′4(δb) = λ2
(1− λδb)2 ≥ 0, for δb ∈ (0, λ−1).
Now, the unique minimum of φ4(δb) is given by the first order condition:
φ′4(δb) = −λ− γ + λ
1− λδb= 0.
δ∗b = γ
λ(λ+ γ) <1λ
(A.10)
φ4(δ∗b ) = −γλ
+ log(
1 + γ
λ
)< 0
The above analysis shows that on {δb|0 < λδb < 1}, φ3(δb) has exactly one criticalpoint when φ5(γ) > 0 or φ5(γ) = φ4(δ∗b ) and exactly two critical points when φ4(δ∗b ) <φ5(γ) < 0.
24
We will now show that when there are two critical points, the smaller critical pointis always less than δ0
b . Define
φ7 := γ(δc + (1 + α)C),u := λδb,
θ := γ
λ,
then we can rewrite Eq. (A.9) as:
φ7 + log 2− 1 = −(1 + θ)u− log(1− u). (A.11)
Fixing φ7 ≥ 0, we see that
min{(δc,C)∈R2
+|γ(δc+(1+α)C)=φ7}δ0b = 1
γ
(12 − φ7
)=: m(φ7)
argmin{(δc,C)∈R2
+|γ(δc+(1+α)C)=φ7}δ0b =
(φ7γ, 0)
It is clear that there exists, for some choice of λ, γ, δc, C, two critical points of φ3(x)greater than δ0
b if and only if for some φ7 ∈ [0, 1−log 2], the smaller solution to Eq.(A.11)is larger than λm(φ7).
Denote the smaller branch of solutions to Eq. (A.11) as u−(φ7). It holds that,
λm(1− log 2) =log 2− 1
2θ
> u−(1− log 2) = 0.
By Eq. (A.10), we see that u−(φ7) < θ1+θ . Since −(1 + θ) + 1
1−u > 0 for all u ∈ [0, θ1+θ )
we have:
∂
∂φ7
(λm(φ7)− u−(φ7)
)= −1
θ− 1−(1 + θ) + 1
1−u−= −
u−
1−u−
θ(−(1 + θ) + 1
1−u−) < 0
Thus m(φ7) > u−(φ7) for Y ∈ [0, 1− log 2]. Thus there is at most one critical point, δ†b ,greater than δ0
b .Since the payoff function is differentiable everywhere but at δ0
b , the function cannotbe maximized simultaneously at δ†b or δ0
b , unless they are equal. Indeed, using the MeanValue Theorem we have that if δ†b 6= δ0
b , either φ(δ†b) > φ(δ0b ) or φ(δ†b) < φ(δ0
b ). Againsince there is no critical point in between, only one of them can be a local maximizer.Thisallows us to conclude there is a unique maximizer: the function is either maximized atδ†b or δ0
b .The second statement of the theorem, i.e., the continuity of the function δb(δc, C),
follows from Berge’s Maximum Theorem.
25
Theorem 3a. Define
ξα := 1 + λδc + αλC − 12e−γC
(1 + λ
γ
). (A.12)
Then the following statements hold:
1. (Augmenting fees.) If ξα ≥ 0, δb is given as the unique solution to
γ(δc + (1 + α)C) + log 2− 1 = −(λ+ γ)δb − log(1− λδb),
greater than or equal to γλ+γ
1λ . In this case ∂δb
∂δc= 1
1+α∂δb∂C ≥ 0, and B ≥ 0.
2. (Complementing fees.) If ξα < 0,
δb = 12γ e
−γC − δc − αC.
In this case ∂δb∂δc
< 0, ∂δb∂C < 0, and B = 0.
Proof of Theorem 3a. Define δ0b := 1
2γ e−γC − δc − αC.
Suppose δ0b ≤ 0. By Proposition 1 we have B(δb + δc, C;α) > 0 for all δb ≥ 0, thus δb
must be the larger solution to Eq. (3.10) by Eq. (A.9). Notice that ξα < 0 implies thatδ0b ≥ 0, so ξα ≥ 0 when δ0
b ≤ 0.Now suppose δ0
b > 0. we see the right derivative of the bank’s payoff function Eq. (3.3)at δ0
b is given by
limδb→δ0
b
1− λδb
1− 12e−γ(B(δb+δc,C;α)+C)
= 1−λ( 1
2γ e−γC − δc − αC)1− 1
2e−γC
∝ 1− 12e−γC − λ( 1
2γ e−γC − δc − αC) = 1 + λδc + λαC − 1
2e−γC(1 + λ
γ).
Since the local maximum of the payoff function is unique by Theorem 2, δb must bethe larger solution to Eq. (3.10) when the right derivative is positive, and equal to δ0
b
when the right derivative is negative. The sign of the derivatives follow from directdifferentiation.
Before the proof of Theorem 4, we summarize the relations between the implicitlydefined functions so far. All relations come from rearranging previous results or directdifferentiation. Recall the definitions of normalized quantities from Eq. (3.12).
Corollary 1. Define Kα := {(δc, C)|δc ≥ 0, C ≥ 0, ξα ≥ 0}. Then in the interior of Kα,the following relations hold:
1− u = 12e−γB−w (A.13)
θu+ v + αw = γB + 12e−γB−w (A.14)
26
γB = (1 + θ)u+ v − 1 + αw (A.15)v + (1 + α)w + log 2− 1 = −(1 + θ)u− log(1− u). (A.16)
γB = 1 + θ
1 + αu+ v
1 + α− 1
1 + α− α
1 + αlog(2− 2u) (A.17)
∂δb∂δc
= 11 + α
∂δb∂C
= θ
−(1 + θ) + 11−u
. (A.18)
Moreover, u ∈ [ θ1+θ , 1].
Theorem 4a. Define the IR correspondences as
δLc (C,G) := {δc|R(δb(δc, C;α); δc, C,G) = 0},CL(δc, G) := {C|R(δb(δc, C;α); δc, C,G) = 0}.
Then the following statements hold
1. Fix C ≥ 0. Then δLc (·, G) is a singleton for all G ≥ 0. The function δLc (C, ·) isdecreasing.
2. Fix δc ≥ 0. Then CL(δc, ·) is a singleton for all δc ≥ 0. The function CL(δc, ·) isdecreasing.
3. Fix G ≥ 0. Then δLc (·, G) is a singleton for all C ≥ 0. The function δLc (·, G) is C1,decreasing, and convex.
4. For any G ≥ 0, the IR constraint crosses the regime switching line ξ = 0 line atmost once.
To prove Theorem 4a, we will make use of the following proposition:
Proposition 3. Along L = {(δc, C)|R = 0}, the following statements hold:
1. When the bank is imposing augmenting fees ∂δc(C)∂C = u− 1− α < 0.
2. When the bank is imposing complementing fees ∂δc(C)∂C = −1
2e−w − α < 0.
Proof of Proposition. We first consider the case when banks are imposing augmentingfees, then by the Envelope Theorem, we can write
0 =(∂δc∂C
∂
∂δc+ ∂
∂C
)δbe−λB(δb+δc,C;α),
∂δc∂C
= H(B + C)− 1− α = u− 1− α.
Here we have used Eq. (A.13) from Corollary 1.When the bank is imposing complementing fees, we have G = δb = 1
2γ e−γC−δc−αC.
Thus∂δc∂C
= −12e−γC − α.
27
Proof of Theorem 4a. Holding C fixed, we see that
∂R(δb; δc, C,G)∂G
+ δc(G)∂G
∂R(δb; δc, C,G)∂δc
= 0,
By the Envelope Theorem and Proposition 3, we have
∂δc(G)∂G
= 1u− 1 < 0.
Thus for each C and G there can be only one δLc satisfying the IR constraint. FromProposition 3 we can see that δLc is decreasing. Analogous arguments show that δLc (C, ·)and CL(G, ·) are all singleton-valued correspondences, decreasing in their arguments.
Further differentiation of the results in Proposition 3 gives
1. (Augmenting fees) ∂2δLc (·,G)∂C2 = λ
(∂δb∂δc
∂δc∂C + ∂δb
∂C
)= λ∂δb∂δc
(H(B + C)− 1− α+ 1 + α) >0,
2. (Complementing fees) ∂2δLc (·,G)∂C2 = γ
2e−γC > 0.
δLc (·, G) is thus convex.To show the “single crossing” property, observe that along the curve ξα = 0 we have
∂δc∂C
∣∣∣ξα=0
= −α− 12e−γC (1 + γ) < −α− 1
2e−γC = ∂δc
∂C
∣∣∣complem.
≤ −α− 12e−γ(B+C) = ∂δc
∂C
∣∣∣aug.
.
Thus the curves can cross at most once if they cross at all.For the C1 statement in Theorem 4, we see that to check for continuity of the deriva-
tive, we only need to check along the regime switching line ξα = 0 for α = 0, whenaugmenting fees and complementing fees agree. We have:
λδb − 1 = λ
2γ e−γC − (1 + λδc) = −1
2e−γC .
Thus δLc (·, G) is C1 when α = 0.
Before we prove Theorem 5, we present three useful propositions.
Proposition 4. The first order condition ∂E[X]∂δc
= 0 implies
v = α− θ2(1 + α) + ((1 + θ)2 + (−1 + θ + θ2)α)u− (1 + θ)u2 − α(1− u) log(2− 2u)α+ u
.
(A.19)
This in conjunction with the first order condition ∂E[X]∂C = 0 implies
v = θ + (1 + θ + αθ)(θ2 + α(1 + θ)2)(1 + θ)(α(1 + θ) + u+ θ) + (1 + θ)2 + α(θ2 − θ − 1)
(1 + θ)(α(1 + θ) + u+ θ) u
28
− (1 + θ)α(1 + θ) + u+ θ
u2 + α(u− 1) log(2− 2u)α(1 + θ) + u+ θ
(A.20)
and that u solves
0 = κ(u; θ, α), (A.21)
where
κ(u; θ, α) := α2(1 + θ)2 + α(θ3 + 3θ2 + θ + (1 + θ − 2θ2 − θ3)u)− θ(θ + θ2 − u)(u− 1)− α(1 + θ)(u− 1) log(2− 2u).
Proof of Proposition 4. Define Λ as
Λ(δc, C,B) := δce−λB − λ
2(λ+ γ)e−λB−γ(B+C)
(1γ
+ 1λ+ γ
+B
).
Then
∂Λ
∂δc= e−λB
∂Λ
∂C= e−λB
(e−γ(B+C) λγ
2(λ+ γ)
(B + λ+ 2γ
γ(λ+ γ)
))∂Λ
∂B= e−λB
(−λδc −
λ
2(λ+ γ)e−γ(B+C) + λ
2 e−γ(B+C)
(B + λ+ 2γ
γ(λ+ γ)
))Evaluating each derivative at B = B, we obtain
eλB∂Λ
∂δc= 1. (A.22)
eλB∂Λ
∂C= 1− u
1 + θ
((1 + θ)u1 + α
+ v
1 + α+ θ
1 + θ+ α
1 + α− α
1 + αlog(2− 2u)
)(A.23)
eλB∂Λ
∂B= 1θ
(−v + (1− u)(γB + 1)
)(A.24)
= 1θ
((1 + θ)(1− u)1 + α
u+ −α− u1 + αv + (1− u)α
1 + α(1− log(2− 2u))
)(A.25)
1 + ∂δb∂δc
= λ2δbλ− (1− u)(λ+ γ) = u
1− (1− u)(1 + θ) (A.26)
The first order condition with respect to δc is
0 = ∂Λ
∂δc+ ∂Λ
∂B
∂B
∂δ
(1 + ∂δb
∂δc
)(A.27)
After some algebra, this gives Eq. (A.19).
29
The first order condition with respect to C is
0 = ∂Λ
∂C+ ∂Λ
∂B
(∂B
∂δ
∂δb∂C
+ ∂B
∂C
)(A.28)
= ∂Λ
∂C+ ∂Λ
∂B(1 + α)
(∂B
∂δ
∂δb∂δc
+ (1 + α)∂B∂δ− 1
)(A.29)
= ∂Λ
∂C− (1 + α) ∂Λ
∂δc− ∂Λ
∂B(A.30)
After some algebra, this gives Eq. (A.20).Since at a critical point both first order conditions must hold, we have that the left
hand sides of the two first order conditions are equal. Rearranging, we find that u solvesEq. (A.21).
Proposition 5. Define φ8(δc) := E[X(δc, 0)]. Then there exists a unique δ∗c at whichφ8 is maximized and ξα > 0. This maximized value is positive. In addition, the bank’sfees u := λδb(δ∗c , 0) is the unique solution to η(u) = 0 greater than θ
1+θ , where
η(u) := −2θ − θ2 + θ2
u− log(2− 2u). (A.31)
Proof of Proposition 5. We start with existence. Set C = 0. By Theorem 3, the bankimposes augmenting fees when
ξα(δc, 0) = 1 + 1θv − 1
2
(1 + 1
θ
)≥ 0,
where we are using the definition of ξα given in Eq. (A.12). This is equivalent to
v ≥ 1− θ2 .
We first consider the point v = 1−θ2 , i.e. the threshold above which the bank switches
from imposing complementing to augmenting fees. Notice this is only relevant whenθ < 1. At this point, we have from Eq. (A.14), u = 1
θ
(12 −
1−θ2
)= 1
2 and B = 0. Theright derivative of φ8 is given by:
e−λB(
1 + 1θ
((1 + θ)(1− u)− v) u
1− (1− u)(1 + θ)
). (A.32)
and is equal to
e−λB(
1 + 1θ
((1 + θ)(1− u)− v) u
1− (1− u)(1 + θ)
)∣∣∣∣u= 1
2 ,v= 1−θ2
= 1 + 11− θ > 0.
at v = 1−θ2 . Thus the clearinghouse’s profit is increasing at this point. Next, consider a
point v > 1−θ2 , then by Eq. (A.16), we have
v + log 2− 1 = −(1 + θ)u− log(1− u), (A.33)
30
thus when v increases, u approaches 1 monotonically. Thus there exists some v∗(θ) suchthat for v > v∗(θ) we have
∂E [X(δc, 0)]∂δc
< 0
by Eq. (A.32). Combined with the previous analysis this means that expected profit ismaximized at a finite point where banks are strictly imposing augmenting fees. SinceE[X(∞, C)] = 0, this maximized value is positive.
The condition for optimality is then given by Eq. (A.19) and (A.33). Rearranging,this means u solves η(u) = 0. Notice that this function always has exactly one zero onu ∈ [ θ
1+θ , 1).
η
(θ
1 + θ
)= −θ − log 2 + log(1 + θ) < 0
η(1−) =∞,
η′(u) = 0 only at u∗ = θ2 + θ√θ2 + 4
2
η(u∗) = 2θ2
θ√θ2 + 4− θ2
− log(
1− −θ2 + θ
√θ2 + 4
2
)− (θ + 1)2 + 1− log 2
= −θ2 + θ
√θ2 + 4
2 − log(
1− −θ2 + θ
√θ2 + 4
2
)− 2θ − log 2 < 0,
the last inequality follows from the following two inequalities
θ ≥ − log(
1− −θ2 + θ
√θ2 + 4
2
)≥ −θ
2 + θ√θ2 + 4
2 (A.34)
which holds true for all θ > 0. To show the first inequality, we notice that θ =− log
(1− −θ2+θ
√θ2+4
2
)at θ = 0. In addition, a cumbersome but straightforward com-
putation shows that
∂
∂θ
(θ + log
(1− −θ
2 + θ√θ2 + 4
2
))= 1− 2
4 + θ2 ≥ 0
Thus the first inequality in (A.34) holds. The second inequality follows from basiccalculus since − log(1 − x) ≥ x for all x ∈ [0, 1). Since φ8 has at exactly one criticalpoint, and we have shown that local maxima do not occur on the boundary, it must bethe global maximum.
Proposition 6. Fix θ > 0, α ≥ 0, then no local maxima of E[X(δc, C)] can occur on (i)δc = 0, (ii) C = 0, and (iii) the regime switching line ξα = 0.
31
Proof of Proposition 6. Assume the contrary. Along ξα = 0, we can consider the maxi-mization problem:
maxδc,C
E[X(δc, C)]
subject to ξα = 1 + λδc + λαC − 12e−γC(1 + λ
γ) = 0.
Notice that when ξ = 0, B = 0 by Theorem 3 and Proposition 1. Introducing theLagrange multiplier µ, we obtain that the following must hold at such an optimum:
0 = 1 + µλ
0 = γ2
2 e−γC
(1γ− 1λ+ γ
)(1γ
+ 1λ+ γ
)+ µ
(αλ+ γ
2 e−γC(1 + λ
γ)).
Rearranging
α = 12e−γC−θ3 − 3θ2 − θ
(1 + θ)3 .
Since this system of equations has no solution, a local maxima cannot occur along theboundary ξ = 0 unless δc = 0 or C = 0. Yet Proposition 5 shows that there cannot bea local maximum at the point where ξα = 0 and C = 0 intersect. Proposition 5 alsoshows that there is a choice of requirements where the clearinghouse is making positiveexpected profit, and thus it would never choose δc = 0 since in that case expected profitis nonpositive.
For Theorem 5 we consider only the case when α = 0.
Proof of Theorem 5. Fix a triple (γ, θ,G) ∈ R3. We first define a pair of normalizedrequirements (v, w) to be: a zero collateral equilibrium if w = 0; an interior collateralequilibrium if 0 < w < ∞; and an infinite collateral equilibrium if w = ∞. The equi-librium is constrained if the bank’s profit at that pair of requirements is zero, and isunconstrained otherwise. Since the bank is can be imposing augmenting fees or comple-menting fees, this leads to 12 cases to be considered. The proof is broken down into foursteps.
[Step 1.] We first rule out all unconstrained equilibria with complementing fees (3cases). By Theorem 3, when the bank is imposing complementing fees we must haveξ ≤ 0. We have B = 0 and the clearinghouse’s expected payoff function Eq. (3.4) istherefore given as
E[X(δc, C)] = δc −γ
2 e−γC
(1γ− 1λ+ γ
)(1γ
+ 1λ+ γ
)Thus, when ξ < 0, the clearinghouse can increase profit by increasing δc and C, and thusin unconstrained equilibria with complementing fees they must choose δc, C such that
32
ξ = 0. which is ruled out by Proposition 6. This rules out all unconstrained equilibriawith strictly complementing fees.
[Step 2.] Next we show the possible existence of unconstrained infinite and zerocollateral equilibria with strictly augmenting fees (cases (i) and (iii)), while ruling outunconstrained interior collateral equilibria with augmenting fees (3 cases). We searchfor all possible maxima over the space K0 := {(δc, C)|δc ≥ 0, C ≥ 0, ξ ≥ 0}. The payofffunction is obviously continuous, and is continuously differentiable on K0\{ξ(δc, C) = 0},where ξ is defined in Eq. (3.9). Notice that maxima along the boundary of K0 are ruledout by Proposition 6, thus we need only consider maxima in the interior of K0.
Using the expressions given by Proposition 4, we see that at critical point:
0 = −θ2(u− 1)(u− θ2 − θ)u(1 + θ)(θ + u)
The above equation implies that critical points can occur only at (u∞, v∞) := (1, θ) and(uint, vint) := (θ+θ2, 1−θ3− θ3
1+θ ). (u, v) = (1, θ) directly implies C =∞ by Eq. (A.16).These results, along with Eq. (A.16), give what we refer to as the (unconstrained) infiniteand interior collateral equilibrium, respectively.
When C = ∞, the expression for B simplifies to B = δ and E [X(δc,∞)] =δce−λ(δb+δc). In addition, u = 1. This fully characterizes the infinite collateral equi-
librium in case (i).We can then rule out the interior collateral equilibrium with uint = θ2 + θ, vint =(
1− θ3 − θ3
1+θ
). Notice that the equilibrium is only well defined when uint ≤ 1, and
coincides with infinite collateral equilibrium at θ2 +θ = 1. At (uint, vint) we can evaluate
λB = 1 + 3θ + θ2
1 + θ
γE[X(uint, vint)] = vinte−λB − 1
1 + θe−λB(1− uint)
(1 + θ
1 + θ+ γB
)= e−
1+3θ+θ21+θ
θ2(2 + θ)1 + θ
∂
∂θe−
1+3θ+θ21+θ
θ2(2 + θ)1 + θ
= −e− θ2θ+1−
3θθ+1−
1θ+1 θ
(θ4 + 2θ3 − θ2 − 5θ − 4
)(θ + 1)3
∝ −θ4 − 2θ3 + θ2 + 5θ + 4 > 0
The last inequality follows from the fact that for uint < 1 we must have θ ≤ 1. Sincethis derivative is positive, the maximum expected profit (maximized over all θ) of theclearinghouse at the interior collateral equilibrium is θ2 + θ = 1. For every other θ,the clearinghouse’s profit is strictly less than the infinite collateral equilibrium. It isthus never a global maximum unless it coincides with the infinite collateral equilibrium,ruling out the interior collateral equilibrium as the prevailing equilibrium.
To characterize zero collateral equilibria, we use Proposition 5. This fully character-izes the unconstrained zero collateral equilibrium (case (iii)).
33
[Step 3.] Now we rule out all constrained equilibria with complementing fees (3cases). We start with the zero collateral equilibrium. Assume the bank is imposingcomplementing fees , then we must have G ≥ 1
2 by Theorem 4 (see Figures 5 and 6). Inaddition, since B = 0, u = G. The clearinghouse’s normalized expected profit is givenby
γE[X(δc, C)] = v − θ + 22(1 + θ)2 = 1
2 − θG−θ + 2
2(1 + θ)2 ≤12 −
θ
2 −θ + 2
2(1 + θ)2
= −θ3 + θ2 + 12(θ + 1)2 < 0
so the clearinghouse would not participate in clearing, hence ruling out this equilibrium.When collateral equals infinity we must have ξ ≥ 0 so the bank is imposing augment-
ing fees, ruling out the infinite collateral equilibrium.Last we show a constrained interior collateral equilibrium with complementing fees
cannot exist. Assume the contrary, since the constraint is binding and the bank isimposing complementing fees, we must have B = 0, v = 1
2e−w − θu, and u = G by
Corollary 1 and Theorem 3. The clearinghouse’s normalized expected profit is given by
γE[X(δc, C)] = v − 12e−w θ + 2
2(1 + θ)2 = 12e−w − θG− 1
2e−w θ + 2
2(1 + θ)2
= 12e−w θ
2 + θ − 1(1 + θ)2 − θG ≤ max{1
2θ2 + θ − 1(1 + θ)2 −
θ
2 ,−θ
2} < 0
This rules out constrained interior collateral equilibrium with complementing fees.[Step 4.] Finally, we show the possible existence of constrained infinite and zero
collateral equilibria with strictly augmenting fees (cases (ii) and (iv)), the possible non-existence of equilibria, whiling ruling out constrained interior collateral equilibrium withaugmenting fees.
The Lagrangian associated with the clearinghouse’s maximization problem is:
L(δc, C, µ) := E[X(δc, C)] + µ(δbe−λB −G
)We assume the multiplier is positive at the critical points. In the case when it is zero,the results are the same as that given in step 2. When this “strictly complementarity”holds, the binding condition µ > 0 implies, we must have that ∂δc
∂C is the same along thelevel curve of the clearinghouse’s expected profit and the bank’s IR constraint.
Since the bank is imposing augmenting fees, along E[X] = const., we have
eλB∂E[X]∂δc
= 1 + u((1 + θ)(1− u)− v)θ(1− (1− u)(1 + θ)) ,
eλB∂E[X]∂C
= (1− u)(u+ v
1 + θ+ θ
(1 + θ)2 + u(1 + θ)((1 + θ)(1− u)− v)θ(1− (1− u)(1 + θ))
).
Combining this with Proposition 3, we see that critical points of the Lagrangian mustsatisfy:
34
−(1− u)
(u+ v
1+θ + θ(1+θ)2 + u(1+θ)((1+θ)(1−u)−v)
θ(1−(1−u)(1+θ))
)1 + u((1+θ)(1−u)−v)
θ(1−(1−u)(1+θ))
= u− 1 (A.35)
When u = 1, we must have v <∞ otherwise no clients trade. This means C =∞ andwe can compute the constrained infinite collateral equilbirum (u, v, w) = (1,−θ(log G+1),∞, (case (ii)).At this point the bank’s profit is zero and the clearinghouse’s (nor-malized) expected profit is −θG(log G + 1). Thus, for the clearinghouse to participate,G ≤ e−1. In addition, from step 2 we see that an unconstrained infinite collateral equi-librium gives the bank’s revenue 1
λe−2, so we must have G ≥ e−2 for the IR constraint
to be binding. We can also compute the shadow value associated with the IR constraint:µ = 2 + log(Gλ).
If u 6= 1, it seems that there may be a constrained “interior collateral equilibrium”:
(u+ v
1+θ + θ(1+θ)2 + u(1+θ)((1+θ)(1−u)−v)
θ(1−(1−u)(1+θ))
)1 + u((1+θ)(1−u)−v)
θ(1−(1−u)(1+θ))
= 1
or
u+ v
1 + θ+ θ
(1 + θ)2 − 1 + u((1 + θ)(1− u)− v)1− (1− u)(1 + θ) = 0
v
1 + θ+ θ
(1 + θ)2 − 1 + u(1− v)1− (1− u)(1 + θ) = 0
v = θ(1− u) + 11 + θ
We will now rule out this possibility. From Corollary 1 we have:
2(1− u) = e−γB−w,
w = −u− log(1− u)− log 2− θ2
1 + θ,
− γB = θ log Gu,
so we arrive at
2(1− u) = eθ log Gu
+u+log(1−u)+log 2+ θ21+θ ,
1 = eθ log Gu
+u+ θ21+θ
0 = log Gu
+ u
θ+ θ
1 + θ(A.36)
Again by Corollary 1 we see this imposes a restriction on G:
0 = log Gu
+ u
θ+ θ
1 + θ≥ log G
u+
θ1+θθ
+ θ
1 + θ,
35
e−1 ≥ G
u≥ G.
which in turn imposes a restriction on θ:
e−1 ≥ G ≥ u ≥ θ
1 + θ.
Let θH ≈ 0.58 be the solution to e−1 = θ1+θ , we must have θ ≤ θH .
Eq. (A.36) also imposes another restriction on θ. Since log u − uθ is maximized at
u = θ:θ
1 + θ= − log G+ log u− u
θ≤ 1 + log θ − 1 = log θ
Let θL ≈ 1.93 be the unique solution to log θ = θ1+θ , we must have θ ≥ θL. Altogether
this means there is no θ that allows for a constrained interior collateral equilibrium withaugmenting fees.
Next we turn to the constrained zero collateral equilibrium. From Corollary 1 wecan write
(2(1− u))−1θ = eλB
thus when the constraint is binding, u must solve
u(2(1− u))1θ = G.
Eq. (A.16) then characterizes the equilibrium (case ( iv)). To see uniqueness of the
solution, notice that ∂u(2(1−u))1θ
∂u = 0 only at u = θ1+θ , so u(2(1− u))
1θ can equal G only
at one u on[
θ1+θ , 1
]Last, we see that when G is very large, none of the equilibria presented are individ-
ually rational (case (v)). Since 1 ≥ u ≥ ue−λB, the bank’s normalized revenue is alwaysbounded above by 1. A G greater than 1 will result in the nonexistence of any feasibleequilibria. Since when G ≤ e−1 some infinite collateral equilibrium must exist, G mustbe greater than e−1 for this to occur.
Notice that all profit functions and IR constraints can be written in a fashion thatonly depends on θ and G, thus which equilibrium prevails only depends on θ and G.
Proof of Theorem 7. This is given directly by combining Propositions 4 and 5.
Proof of Theorem 8. We see that κ(u; θ, α) = 0 has a solution at u = 1 when α = 0.
∂κ(1; θ, 0)∂α
= (1 + θ)2 > 0.
By the Implicit Function theorem, there exists α1(θ) such that we can a continuouslydifferentiable branch of solutions u∞(α) such that
u∞(α) = 1.
36
For θ < −1+√
52 , we have θ2 + θ < 1 and
limα→0
∂κ(u; θ, α)∂u
= θ(1− θ2 − θ)− limα→0
α(1 + θ) log(2(1− u)) > 0.
Thus ∂u∞(0)∂α = −
∂κ(1;θ,0)∂α
∂κ(1;θ,0)∂u
< 0. By continuity of the derivative, it remains negative in aregion [0, α2(θ)].
Next, from Theorem 7 we see that we can the associated fee and collateral by pluggingin u = u∞(α) :
v(u, α) = α− θ2(1 + α) + ((1 + θ)2 + (−1 + θ + θ2))u− (1 + θ)u2 − α(1− u) log 2(1− u)α+ u
w(u, α) = θ2 − 2θu− θ2u− u log 2(1− u)α+ u
Since we see
limα→0,u→1
∂v(u, α)∂α
= limα→0,u→1
θ2u2 − 2θ2u+ θ2 + 2θu2 − 2θu+ u2 log(2(1− u))− u log(2(1− u))(α+ u)2 = 0.
limα→0,u→1
∂v(u, α)∂u
= limα→0,u→1
α2θ2 + α2θ + 2αθ2 − 2αθu+ 2αθ − αu+ θ2 − θu2 − u2
(α+ u)2
+ limα→0,u→1
α2 log(2(1− u)) + α log 2(1− u)(α+ u)2
= −1− θ + θ2 + limα→0,u→1
α2 log(2(1− u)) + α log 2(1− u)(α+ u)2 < 0.
So there is a region [0, α3(θ)], in which dvdα = ∂v
∂α + ∂v∂u
∂u∞
∂α > 0 Similarly, we can write:
limα→0,u→1
∂w(u, α)∂u
= limα→0,u→1
αθ2u− αθ2 + 2αθu− 2αθ + αu− α log(2(1− u)) + αu log(2(1− u)) + θ2u− θ2 + u2
(1− u)(α+ u)2
= limα→0,u→1
−α(1− u) log(1− u) + 1(1− u)(α+ u)2 ≥ 0.
Since w ≥ 0 at α = 0, obviously limα→0,u→1∂w(u,α)∂α < 0. Together this means there is
a region [0, α4(θ)], in which dwdα = ∂w
∂α + ∂w∂u
∂u∞
∂α < 0. To finish the proof, take α(θ) =mini=1,2,3,4 αi(θ).
37
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