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Collateral, rehypothecation, and efficiency
Accepted Manuscript
Collateral, Rehypothecation, and Efficiency Hyejin Park, Charles M. Kahn PII: DOI: Reference:
S1042-9573(18)30037-8 10.1016/j.jfi.2018.06.001 YJFIN 793
To appear in:
Journal of Financial Intermediation
Please cite this article as: Hyejin Park, Charles M. Kahn, Collateral, Rehypothecation, and Efficiency, Journal of Financial Intermediation (2018), doi: 10.1016/j.jfi.2018.06.001
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Collateral, Rehypothecation, and Efficiency∗ June 9, 2018
Abstract
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Hyejin Park† and Charles M. Kahn‡
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This paper studies rehypothecation, a practice in which financial institutions re-pledge collateral pledged to them by their clients. Rehypothecation enhances provision of funding liquidity to the economy, but it also incurs deadweight cost by misallocating the asset among the agents when counterparties fail. We examine the possibility of a conflict between the intermediary and its borrower on rehypothecation arrangements. The direction of this conflict depends on haircuts of the contract between them: if the contract involves over-collateralization, there tends to be an excessive use of rehypothecation, and if the contract involves under-collateralization, there tends to be an insufficient use of rehypothecation. This offers an empirical prediction of a link between the size of haircut in collateralized financial contracts, borrower information, and the likelihood of rehypothecation.
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Keywords: collateral, rehypothecation, moral hazard, misallocation JEL Classification: D53, D62, G21, G28
∗ We are grateful to the Editors and an anonymous referee for their comments which helped to improve this paper. We also thank Dan Bernhardt, Vincent Maurin, Giorgia Piacentino, Jason Donaldson, Jan Willem van den End, Gerold Willershausen, Phil Prince, Manmohan Singh, Mina Lee, and David Andolfatto, as well as seminar participants at the University of Illinois at Urbana-Champaign, the IBEFA San Francisco Fed conference, the St. Louis Fed Summer Workshop, the MFA Annual Meeting, the Atlanta Federal Reserve Conference in Amelia Island, Florida, and Cambridge Corporate Finance Theory Symposium for their valuable comments. Any remaining errors are ours. † Department of Economics, University of Illinois at Urbana-Champaign. E-mail: [email protected] ‡ Department of Finance, University of Illinois at Urbana-Champaign. E-mail: [email protected]
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Introduction
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Financial contracts are regularly enforced through collateral.1 Assets most desirable as collateral are scarce in the economy and the cost of generating these assets is nonnegligible. With the increased volume of financial transactions, market participants find that it is important to economize on the limited amount of available collateral.2 One way economize on collateral is through collateral re-use, allowing a single unit of collateral to support multiple transactions. Re-use reduces the cost of holding collateral; it also allows the borrower access to funds at a cheaper rate. An important example of collateral re-use by financial intermediaries is “rehypothecation,” the practice of repledging a customer’s collateral for its own trading or borrowing. This paper addresses some basic open questions about this practice: under what circumstances rehypothecation arises and what benefits and costs it produces, and whether decentralized decisions to participate in rehypothecation achieve a socially efficient outcome or not. In particular, our main question is whether there is alignment between the objectives of the borrower and intermediary: are there cases where the intermediary overor under-uses the borrower’s collateral? Rehypothecation was one of the most popular devices for many dealer banks to serve their own funding liquidity needs before the recent financial crisis.3 After the failure of Lehman Brothers in 2008, however, rehypothecation significantly dropped as hedge funds restricted re-use of their collateral.4 At the same time, regulation on rehypothecation has also been advocated by legislators and policy-makers.5 Nonetheless, understanding of the 1
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Collateral occurs in a large variety of settings, from residential houses in mortage to T-bills in repo transactions. Holmstr¨ om (2015) notes that the history of collateralized lending can be traced back to pawning in China during the 7th century. For another insightful discussion on the origin of collateralized lending, see Geanakoplos (1996). 2 Singh (2010, 2011) emphasizes the scarcity and varying quality of collateralizable assets and their efficient use. Krishnamurthy and Vissing-Jorgensen (2012) estimate the liquidity and safety premium on Treasuries paid by the investors on average from 1926 to 2008 was 72 basis points per year. In a similar vein, Greenwood, Hanson, and Stein (2012) point out the monetary premium embedded in short-term Treasury bills, the highest quality collateral. 3 While we model rehypothecation, with suitable modifications our model can be applied to other forms of collateral re-use as well. In a companion paper (Kahn and Park, 2016), we also consider alternative institutional arrangement which can achieve the benefits of rehypothecation using a simple, but less standard, model of incentives as a motivating example. 4 According to Singh (2010, 2011) the value of collateral held in 2007 by the six largest U.S. investment banks that was permitted to be rehypothecated was around $ 4.5 trillion. Post the crisis, in 2009, however, this value sharply dropped to $2 trillion. 5 For example, the Dodd-Frank Act requires that collateral in most swap contracts to be deposited in
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economics underlying this practice is still incomplete, and there are still debates on how to regulate rehypothecation, as evidenced by wide variation in the rules on rehypothecation across countries.6 We start with a framework in which a borrower is subject to a moral hazard problem and must post collateral to prevent him from engaging in risk-taking actions. Previous models of this problem, however, have not considered the possibility that lenders re-use collateral for their own purposes and the risk that they might fail to return the collateral. We incorporate the possibility of collateral re-use by the counterparty into a contracting model, offering the first analytical framework that investigates individuals’ incentives behind this practice. In our model the borrower transfers collateral to the intermediary at the beginning of the contract. In other words, collateral in our model is like a repurchase agreement; the borrower transfers his asset to the lender at the time a contract is initiated and buys it back at a later point.7 This early transfer of collateral introduces the risk that the lender may not be able to return collateral at the time when the borrower wants it, a risk exacerbated by rehypothecation.8 Indeed, as observed from the failure of Lehman Brothers in 2008 or MF Global in 2011, this is not simply a theoretical possibility.9 The efficiency of rehypothecation is determined by the relative size of the following two
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a segregated account of a central counterparty, precluding re-use. 6 For example, in the U.S., SEC rule 15c3-3, a prime broker may rehypothecate assets to the value of 140% of the client’s liability to the prime broker. In the U.K., there is no limit on the amount that can be rehypothecated. See Monnet (2011) for more detailed explanation on various regulatory regimes for rehypothecation. 7 Early transfer of collateral is observed frequently in shadow banking sectors where loans are mostly short-term and of large value and seizing collateral after bankruptcy of borrowers is difficult and time consuming. In contrast, for long-term loans between a bank and a consumer such as a mortgage loan, it is relatively easy to seize collateral from the borrower should the need arise, and collateral mostly stays with the borrower. 8 The traditional situation was described as follows, in the case when a broker’s client purchased on margin and the broker rehypothecates to a bank to finance the call loan: “Numerous court decisions have established that the bank’s rights are strong, generally superior even to those of the customer. Although the situation is infrequent, if the broker became unsecured as a result of price activity and was not able to meet the bank’s loan terms, the bank has the legal right to sell the customer’s pledged securities to protect its loan. Naturally, the customer in this situation will have a difficult time salvaging much equity.” (Teweles et al., 1998). 9 With regard to Lehman, for example, Fleming and Sarkar (2014) state, “Counterparties that had allowed Lehman to hypothecate their collateral to unrelated third parties in connection with securities transactions that could not be unwound found that their collateral had become unrecoverable. . . [they] did not know when their collateral would be returned to them, nor did they know how much they would recover given the deliberateness and unpredictability of the bankruptcy process.”
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effects: Rehypothecation lowers the cost of holding collateral and makes illiquid collateral more liquid, thereby providing more funding liquidity into the market. On the other hand, rehypothecation failure leads to the misallocation of collateral. In our model, this misallocation of the asset arises from the following two frictions: (i) we assume that the asset is ‘illiquid’ in the sense that the asset is likely to be more valuable to the initial owner than to the other agents (for example, an asset in the borrower’s portfolio might be more useful for that portfolio than for others)10 ; (ii) we assume that agents can trade only through the intermediary (for example, large dealer banks have better access to potential clients than those clients have to one another). This, in turn, implies that if the intermediary fails, the asset might end up in the wrong hands: the asset cannot be returned to the initial owner who values it the most, but instead it is seized by the third party for whom the collateral may be less valuable. We consider the possibility that the intermediary can choose whether to participate in rehypothecation or not, by comparing the payoff when he rehypothecates and the payoff when he does not rehypothecate. We characterize the conditions under which a conflict arises between the intermediary and the borrower. A key factor that determines the direction of the conflict is the wedge between the borrower’s valuation on his collateral and the payment promised to the lender for retrieving it. Whether a loan contract involves overor under-collateralization depends in turn on the incentive problem that the contract is designed to mitigate. If the contract involves over-collateralization, in the sense that the value of collateral to the borrower exceeds the payment for retrieving it, the intermediary is likely to overuse the borrower’s collateral. The reason is that the intermediary does not internalize this positive externality of returing collateral to the borrower and sometimes might want to engage in rehypothecation even when rehypothecation is costly from the borrower’s viewpoint. Similarly, if the contract involves under-collateralization, the intermediary might be tempted to underuse the borrower’s collateral for the opposite reason: he values returning it to the borrower more than the borrower actually values receiving it back. This result also has the additional implication that if the initial borrower has the right to permit or restrict rehypothecation, her choice will depend on whether the contract involves under- or over-collateralization; if the contract involves under-collateralization, she tends to permit rehypothecation more often; if the contract involves over-collateralization, 10
Park (2017) shows how this wedge between valuations on the same asset arises endogenously in an infinite-horizon dynamic setting. We discuss other possibilities in section 6.
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she tends to be reluctant to permit rehypothecation. Thus, the inefficiency due to the intermediary’s excessive rehypothecation might be mitigated relatively easily if the borrower could prohibit rehypothecation in advance, while the inefficiency due to the insufficient rehypothecation will remain without other devices to induce rehypothecation.
Related Literature
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A number of theoretical and empirical papers consider collateral as a device to reduce contracting frictions in the presence of moral hazard, among them, Aghion and Bolton (1997), Boot and Thakor (1994), and Holmstr¨om and Tirole (1998, 2011).11 Boot, Thakor, and Udell (1991) offer empiricial evidence as well as a theory that collateral mitigates moral hazard and provides a screening mechanism. Along these lines, Berger, Frame, and Ioannidou (2016) test a variety of theories of collateral and find empirical dominance of the theories in which collateral mitigates losses and incentivizes the borrowers to take less risk. They also find that liquid collateral is associated with especially low risk premia, and offer an explanation for the conflicting results in the literature about the empirical relation between collateral and loan risk. Agarwal, Green, Rosenblatt, and Yao (2015) document that even after controlling for mark-to-market asset valuation, initial collateral remains an important predictor of mortgage default, and those who pledge higher collateral have a lower default likelihood. In the context of derivatives trading, Biais, Heider, and Hoerova (2012) discuss how collateral mitigates the moral hazard problem of the derivatives providers. In particular they assume that collateral is deliverd to the lender before the borrower’s final payoffs are realized. One of the advantages of depositing collateral in advance is that it can protect the collateral from the potential borrower’s risky action as new information arrives after the contract begins. In a similar framework, Bolton and Oehmke (2014) examine
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Mills and Reed (2012) describe the two fundamental roles of collateral: first, it incentivizes the borrower to repay, second, it provides a lender with some compensation through its liquidation if the borrower does not fulfill her payment obligations. However, in the absence of additional frictions, there is no difference between a collateralized loan and raising funds by a sale of the asset (and a subsequent repurchase in some states of the world). Thus various accounts consider additional frictions which collateral specifically addresses. Besides moral hazard, the possibilities include signaling/adverse selection, changing bargaining power under renegotiation, or protection against dilution of claim at the hands of other creditors (for this last, see Donaldson, Gromb and Piacentino, 2017). Each of these explanations, to the extent that they serve to provide a wedge between the value of the collateral and the size of the loan, can serve as an alternative basis for our results.
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the implications of the recent privileged bankruptcy treatment of derivatives and argue that the seniority of derivatives can be inefficient since it transfers default risk to the debt market although the risk can be born more efficiently in the derivative market. A contribution of our paper is that we consider a default risk associated with the receiver of collateral which was absent in those models. The source of this counterparty risk in our model is rehypothecation. If the lender defaults having re-used the borrower’s collateral, the collateral may not be returned to its most efficient use. Our paper is also related with the literature on two-sided credit risk. Mills and Reed (2012) develop a repo contract model in which both a borrower and a lender are subject to risks of default. Their model is in turn based on the approach of Kocherlokota (1996) that considers a problem of risk sharing in an environment without commitment to future actions. Kehoe and Levine (2001) and Alvarez and Jermann (1995) investigate asset trade and pricing with multiparty credit risk in a general setting. Derivatives pricing under bilateral credit risks is considered by Franzen (2001) and Huge and Lando (1999). Our paper is also related to works that build on the scarcity of safe and liquid assets. Gorton and Pennacchi (1990) show that in the presence of informational frictions, financial intermediaries can create safe assets, providing liquidity to the economy. Krishnamurthy and Vissing-Jorgensen (2013) show the existence of a persistent demand for safe and liquid assets, providing evidence that safe assets are scarce. Gorton, Lewellen, and Metrick (2012) and Singh (2010, 2011) discuss the shortage of safe and liquid assts in the shadow banking system. More recently, Bonaccorsi di Patti and Sette (2016) shows that the freeze in the securitization market led to a reduced supply of credit by banks. In our paper, a decrease in rehypothecation increases the cost of financing to the borrower and the cost of holding collateral to the lender, thereby reducing supply of liquidity to the system. Since the financial crisis in 2007, there has been growing interest in rehypothecation from both policy groups and academic researchers. Monnet (2011) discusses policy implication of rehypothecation. Kahn (2009) shows how an analogous concern arises in the context of international payments arrangements. Bottazi, Luque, and Pascoa (2012) examine how asset prices increase by the leverage built up in the process of rehypothecation in a general equilibrium model. Andolfatto, Martin, and Zhang (2014) analyze the welfare implications of rehypothecation in a monetary search model. Maurin (2014) extends the general equilibrium model with collateral constraint by Geanakoplos (2010) and compares rehypothecation with other alternative trading techniques such as tranching and pyramiding. He shows that rehypothecation has no effect on trading outcomes in 6
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complete markets, and thus the effectiveness of rehypothecation depends on the market structure. In the finance literature, Lee (2015) discusses a tradeoff of rehypothecation in terms of economic efficiency and financial stability. She emphasizes that a sudden decline of rehypothecation can lead to an inefficient repo run by creating a positive feedback loop between the repo spread and fire-sale discounts. Eren (2014) and Infante (2014) consider a situation in which a dealer bank earns additional liquidity by setting different margins to their clients on the opposite sides. Park (2017) extends the model in the current paper into an infinite-horizon dynamic model to examine how shocks associated with the intermediary affect collateral allocation in the economy and initiate fluctuations in aggregate output. Our paper is organized as follows. The baseline model is presented in Section 3. In Section 4, the model is extended to a three-player model of rehypothecation and the contracting outcome is derived. Section 5 provides examples where a conflict between the parties about rehypothecation arises. Section 6 discusses institutional alternatives to rehypothecation.
The Baseline Model
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We start from a simple two-player model in which a borrower is subject to a moral hazard problem tempting him to engage in a risky action that is not desirable from the lender’s perspective.12 The moral hazard problem leads to a shortage of pledgeable income of the borrower, in the sense that the future pledgeable return of the borrower’s investment is not enough to repay the initial cost of the investment. For this reason, uncollaterlized borrowing may not be possible. In such a scenario, requiring the borrower to post his valuable asset as collateral can mitigate the borrower’s incentive to engage in the risky action, and makes it possible for capital to be transferred from the lender to the borrower. 12 While our baseline model is similar to those of Tirole (2006) and Bolton and Oehmke (2014) as well as a simplified version of Boot, Thakor, and Udell (1991), in detail it differs from each. An important difference in our model is that collateral works as an incentive for the borrower to take less risk so that collateral relaxes the borrower’s incentive constraint, while in Tirole’s model collateral (net worth) works as an insurance for the lender so that collateral relaxes the lender’s participation constraint. Thus, in Tirole’s model, the borrowing capacity is determined by the outside value of collateral (riskless price that the collateral can be resold at in case of the borrower’s default), while in our model, it is determined by the inside value of collateral (the borrower’s private valuation of collateral). Key differences from Bolton and Oehmke (2014) are noted below.
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We incorporate in this basic model the additional risk that the lender might be unable to return the borrower’s collateral, and show that if the risk is too high, the borrower will be reluctant post his asset as collateral at all, and borrowing will not occur. The subsequent section extends the two-player into a three-player model in which we examine rehypothecation.
A Two-Agent Model
aRA + (1 − a)(pA RA + b)
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There are two dates, (the “beginning date” and the “end date”)13 and two representative agents in the economy, a firm A and an outside investor B. Agents are risk-neutral and consume at the end date. For simplicity, the price of consumption good is normalized to be 1. Firm A is endowed with a (non-transferable) investment opportunity which requires an input at the beginning date to produce an output at the end date. We assume that the outcome of the investment is uncertain and takes one of two values; if the investment succeeds, the investment produces RA > 1 units of good per unit of inputs and if it fails, it produces zero units. The outcomes are costlessly observable by the outside investors. The probability of the success of the borrower’s investment depends on his hidden action. We assume that A can choose either a safe action or a risky action. The safe action leads to success with certainty and the risky action leads to success with probability pA < 1. On the other hand, the risky action gives A a private benefit b > 0 (measured per unit of inputs). We let a be the indicator for choice of action by A (1 for safe action, 0 for risky action)14 thus the expected return of the investment is given by
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The investment needs input at the beginning date, but A is endowed with no input. The outside investor B is endowed with capital at the beginning date that can be used for funding A’s investment. For simplicity, we assume that A tries to borrow for his investment from B by issuing simple debt; A receives funding I from B at the beginning date by promising to pay a certain amount of his investment outcome X to B at the end 13
When we get to the model with rehypothecation we will denote the beginning date as “date 0” and the end date as “date 2” and add in an intermediate date 1. The intermediate date is redundant in the two-player case. 14 It can be shown that mixed strategies for a are never optimal.
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date (a promise that can only be carried out if the investment is successful). If he does not deal with A, B’s outside option is assumed to be worth J ≥ 0. We assume that the parameters satisfy the following two conditions:15 Assumption 1. min{RA , pA RA + b} > 1 > pA RA .
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The first inequality implies that A’s project is worth carrying out regardless of his action. The second inequality implies that, from the perspective of B, lending to A is profitable only if A takes the safe action. Assumption 2. RA − 1 < pA (RA − 1) + b.
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This assumption implies that when A invests with the funding from B, A will always find it profitable to take the risky action rather than the safe action for any given contract such that X ≥ I (or, equivalently, for any contract with a positive interest rate). To see this, notice that the left side represents A’s expected net surplus (per unit of the inputs) after paying out the investment cost to B if he takes the safe action and the right side represents A’s expected net surplus if he takes the risky action. Assumptions 1 and 2 together imply that in the absense of collateralization, A cannot borrow funds for his investment from B. Intuitively, this is because A will prefer to take the risky action after receiving funds from B, and with foreknowledge of this, B will then be unwilling to provide funds. Lemma 1. Under Assumptions 1 and 2, uncollateralized borrowing is not feasible.
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Thus we need to consider alternative arrangements. We assume that A is also endowed with a transferable, indivisible asset which is illiquid in the following two senses: First, it yields the consumption goods only at the end date. Second, it produces more output when it is in the hands of the initial owner, firm A, than in the hands of the outside investor, B. Specifically we assume that the asset yields Z units of the good if it is held by A at the end date, while it yields less than that, Z0 (< Z) units of the good if it is held by B at that time.16 15
For every b > 0 there is an open set of pairs (pA , RA ) satisfying these parametric restrictions strictly. In our model, we assume that the borrower has an indivisible asset whose value is not affected by the borrower’s action. In comparison, Bolton and Oehmke (2014) consider the case in which the borrower has a certain amount of assets which may yield high or low return depending on whether the borrower takes a safe or risky action, and depositing a part of these assets into the lender’s margin account (i.e. posting these assets as collateral) keeps the value of these assets stable and mitigates the borrower’s incentive to engage in risky action by lowering the amount of assets under his management. 16
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Consider an arrangement where A posts his asset to B as collateral. Where others investigating the usefulness of collateral have assumed the B is a perfectly reliable counterparty, we extend our analysis by including the possibility that B is also less than perfectly reliable and might lose A’s collateral. Let pB be the probability that B does not lose A’s collateral (this probability will be endogenized in the full model below). For simplicity, A is assumed to have all the bargaining power and makes a take-it-orleave-it offer to B. At the beginning date, A borrows funding I from B in exchange for delivering the collateral, and then A chooses his action a. If the investment succeeds, A makes the promised payment X to B, and if the investment fails, A defaults and B seizes the collateral (if B loses the collateral, A does not repay the loan). The appendix describes A’s choice of contract in general. In the case where B is perfectly reliable (pB = 1), the contract (I, X, a) offered by A solves the following maximization problem: max a(RA I − X) + (1 − a)[pA (RA I − X) + bI − (1 − pA )Z] I,X,a
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a ∈ arg max a(RA I − X) + (1 − a)[pA (RA I − X) + bI − (1 − pA )Z] a∈{0,1}
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(IC) (P )
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aX + (1 − a)[pA X + (1 − pA )Z0 ] − I ≥ J I, X ≥ 0
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The objective function is A’s expected utility. Constraint (IC) is A’s incentive constraint which implies that a is ex-post incentive compatible given the choice of the other terms of the contract. Constraint (P ) is B’s participation constraint. Lastly, constraint (R) is the resource constraint which implies that A cannot promise to pay more than the return of the project. Note that the term −(1 − pA )Z in the objective function and on the right side of the incentive constraint (IC) indicates the additional loss to the borrower from the potential loss of his collateral. The participation constraint (P ) also includes a term representing the compensation that the lender receives from the collateral in the event of a default. Finally, if Z = Z0 = 0, the problem reduces to borrowing in the absence of collateral.
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If Z0 and J and are sufficiently small, it is optimal to take the safe action and both (IC) and (P ) bind. For convenience we define B ≡ RA −
b 1 − pA
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by Assumption 2, B < 1. When B is positive, it can be interpreted as the pledgeable income of the borrower, the maximum amount that the lender can take out of the borrower’s investment (see Holmstr¨om and Tirole, 1998). B increases with increasing the return from the investment and with decreasing the (expected) gain from choosing the risky investment over the safe investment. First we consider the case where B is perfectly reliable. Proposition 1 (Contract Choice with Collateralized Borrowing). Suppose pA < 1, J ≤ Z J , and pB = 1. A chooses a collateralized contract in which and Z0 ≤ 1−p A (I, X, a) =
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In particular, A takes the safe action.
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Note that in this problem, as the value of the collateral to the borrower increases, the constraints are relaxed and the amount of investment increases. In this respect, the relevant value is the value to the borrower, not the lender, because the collateral is being used as an incentive, not a repayment. Roughly speaking, the need to maintain the incentives for safe behavior increases the collateral needed to back the borrowing. Note that depending on the details of the incentive problem, the value of the collateral to the borrower can be greater than or less than the promised repayment from the borrower.17 We will call the loan “overcollateralized” if Z > X and “undercollateralized” if the inequality is reversed. For small enough values of J, this depends solely on the sign of the quantity B: Corollary 1. Under the conditions of the previous theorem the chosen contract is overcollateralized (undercollateralized) if B is negative (positive). 17
If the purpose of collateral were to force the borrower to make a repayment ex post, then it would always be necessary to have the loan overcollateralized. Here, the incentive problem is not for ex post repayment (we assume that if the borrower has the funds he will make the promised repayment) but over the ex ante risk choices which determine whether the funds will be available.
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When B is less than perfectly reliable, then, the expected payoff to A becomes a[RA I − pB X − (1 − pB )Z] + (1 − a)[pA (RA I − pB X) + bI − (1 − pA pB )Z].
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The expected payment when successful is reduced from X to pB X, and these is now a cost associated with the loss of collateral due to B’s failure. The incentive constraint for A is modified accordingly. The payoff for B now takes into account the fact that he only receives payment from A if he does not lose A’s collateral. Thus the participation constraint becomes:
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pB [aX + (1 − a)(pA X + (1 − pA )Z0 )] − I ≥ J.
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The other constraints to the problem are unchanged. If B is sufficiently reliable, the contract approximates the case of pB = 1 described above. On the other hand if pB is low, in other words, if B loses collateral too frequently, A is reluctant to post his asset as collateral due to the concern of not getting it back. Again, A’s investment will not be undertaken even when collateralized borrowing is feasible. For example, suppose Z0 = 0 and J = 0. Then, the solution (I, X, a) takes the following form. Z pB Z , X= , a=1 (4) I= 1−B 1−B
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b where B = RA − 1−p < 1 as before. A Substituting this into A’s utility function, we have
[pB RA + (1 − pB )B − 1]Z . 1−B
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Note that as pB decreases, A’s utility decreases and for sufficiently low pB A will prefer not to borrow at all.
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Collateral Chain: Rehypothecation Model
In this section, we extend the previous two-player model into a three-player model in which B repledges A’s collateral to the third party, C. This extended model describes how the same unit of collateral can be rehypothecated to support multiple transactions. As a preliminary result, using this model, we confirm that rehypothecation generates a 12
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trade-off. On the one hand, rehypothecation improves the provision of funding liquidity in the economy so that additional productive investments can be undertaken. To be specific, in our model, the intermediary B can raise funds for another productive investment by repledging A’s collateral to the third party C, even in the case in which B’s pledgeable income is not enough to back up the debt that B issues for funding his investment. On the other hand, rehypothecation may incur a deadweight cost when it fails, that is, when B (the intermediary who sits in the middle of the collateral chain) goes bankrupt having re-pledged A’s collateral to the third party, C. In that case, the collateral will be seized by C and cannot be returned to the initial owner, A, who puts a higher value on the asset than does C, leading to the misallocation of the asset. In our model, this misallocation of the asset originates from the following two frictions in the market. The first friction is a wedge between the agents’ valuations on the posted asset. In particular, we assume that the initial owner of the asset values his asset more highly than do the other agents in the economy. The second friction is a trade friction: the initial collateral owner, A, and the final cash lender, C, cannot trade on their own, but only through the intermediary, B. Therefore, in case of rehypothecation failure, these two asssumptions together lead to the misallocation of A’s asset to C.18 To facilitate analysis, throughout this section, we assume that B’s ex-post decision whether to rehypothecate or not is given. Under this assumption, we then calculate the total surplus in the case where B rehypothecates and in the case where B does not rehypothecate, and compare the welfare in these two cases to examine what benefits and costs rehypothecation produces in the economy. In section 5, we relax the assumption that B’s decision to rehypothecate is exogenously given, and see whether his decision aligns or conflicts with the ex-ante socially efficient decision. In other words, we ask if there are circumstances where B does not want to rehypothecate ex post when it is socially efficient to rehypothecate, or B prefers to rehypothecate ex post when it is inefficient to rehypothecate.
18
In general, C might be able to sell the asset and A may repurchase it from the variety of sources. However, as long as there are some trading frictions between A and C, for example, if there exist non-zero transaction costs, there will be some deadweight loss when B fails. Kahn and Park (2016) consider how changes in these transaction costs affect the desirability of rehypothecation.
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4.1 4.1.1
The Model Players and Endowments
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We add an additional, risk neutral player C who also only consumes at the end date. We also add an additional, intermediate date to the model; now the dates are denoted 0, 1, 2. As before, we take the price of date 2 consumption good to be 1. As before A has an opportunity of an investment which requires funding at the beginning date 0 and produces outcome at the end date 2. As before, A is also endowed with a single unit of indivisible illiquid asset, which only yields goods at date 2, and B is endowed with a large amount of funding which is available at date 0 for input for A’s investment. At the intermediate date 1, B also has an investment opportunity which requires an immediate input at that time to produce an outcome at date 2. However, at date 1 B does not have capital that can be spent as an input for his project nor any other pledgeable assets.19 On the other hand, C does not have access to B’s investment, but has a large amount of capital available at the intermediate date that can be spent as an input for B’s investment. Thus, in order that B’s investment is undertaken, capital must be transferred from C to B. As in the previous section, we assume that B issues simple debt to borrow funds for his investment; B borrows funds from C at date 1 by promising to pay a part of his investment outcome to C at date 2. We make the following assumptions regarding B’s investment opportunity. With probability pB the project yields positive output RB > 0 (measured per unit of the investment cost) and with probability 1 − pB it yields no output. We assume that B’s investment is productive in the sense that the expected return of B’s investment is greater than the investment cost (both are measured per unit of inputs). Assumption 3. pB RB > 1 .
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However, the outcome of B’s investment is not verifiable to its creditor, C. For example, even when the project succeeds, B can falsely report that his investment fails, and can avoid paying the loan to C. This implies that debt financing solely backed by the future return of the investment is not feasible for B.20 19
B’s endowment at date 0 might not be storable until the next period when he has an investment opportunity, that is, there is a liquidity mismatch problem faced by B. 20 Note, therefore that we are not assuming that the purpose of the collateral in the BC loan is to affect B’s ex ante incentives, but only his ex post incentive to repay the loan. This simplifies the determination
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4.1.2
Sequential Contracts: Collateral Chain
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Suppose B is now allowed to repledge A’s collateral deposited in his account to borrow funds from C. This can help to raise funds for B’s investment, which otherwise cannot be funded on its own. By posting A’s collateral, B can assure C that he will make the payment to recover the collateral – if B does not recover the collateral, B cannot receive the payment promised by A. On the other hand, collateral provides an insurance to C against B’s default by allowing C to liquidate it to collect losses, if any.21 Formally, under rehypothecation, the same single unit of collateral is used to support more than one transaction, the contract between A and B at date 0 and that between B and C at date 1, thereby creating collateral chain in the economy. In this three period model, these two contracts arise sequentially, and the timing of the model proceeds as follows (also see Figure 1).
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• At date 0, A borrows funds for his investment, denoted by I1 , from B by pledging his asset, which is worth Z to himself and Z0 (< Z) to the others, and promises to pay a part of the investment outcome, denoted by X1 , to B, conditional B’s returning collateral to A. After entering the contract, A then chooses a ∈ [0, 1].
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• At date 1, B has an investment opportunity, and decides whether to repledge A’s collateral and undertake the investment, or keep A’s collateral to return it safely to A in the next period. If B decides to rehypothecate, B borrows funds for his investment, denoted by I2 , from C by repledging A’s collateral, and promises to pay a part of the investment outcome, denoted by X2 , to C for recovering A’s collateral from C.
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• At date 2, both A’s and B’s investment outcomes are realized and the contracts are executed according to the prearranged terms. If B pays X2 he repurchases A’s collateral from C, and then returns it to A to receive X2 . However, if B defaults, C seizes A’s collateral repledged by B, and it cannot be returned to A, while at the same time, A is also exempt from paying the loan X1 to B.
of the choice of amount of collateral compared to loan size, allowing us to focus on the conflict between A and B. A more general treatment would consider both effects of collateral on both loans. 21 We assume that B is unable to receive funding back from A unless the collateral is returned at the same time. Thus B will be unable to repay C unless the project succeeds. In a more general setting (see previous footnote) it will be worthwhile to consider the extent to which B could arrange “bridge financing” to use A’s repayment to finance the collateral redemption.
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B borrows I2 from C collateral
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A pays X1 to B
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B pays X2 to C collateral
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A borrows I1 from B
Figure 1: Timing of Contracts
4.2
Model Solution
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Next, we solve for the contracting problem for this model. Throughout this section, we focus on the case in which B’s decision whether to rehypothecate A’s collateral at date 1 is exogenously given – we will endogenize this decision in the subsequent section. First, consider the case in which B does not rehypothecate A’s collateral at date 1. In that case, the model involves only one contract made between A and B, and effectively, it boils down to the previous two-player model. Next, consider the case in which B rehypothecates at date 1. In that case, the model has a sequence of two contracts, the date 0 contract between A and B and the date 1 contract between B and C. Formally, the date 0 contract between A and B is defined as a profile (I1 , X1 , a) where I1 represents the investment cost that A borrows from B by pledging his asset as collateral at date 0, X1 represents the payment promised by A to pay for getting back his asset from B at date 2, and a denotes the action taken by A. Similarly, the date 1 contract between B and C is defined by a profile (I2 , X2 ) where I2 represents the investment cost that B borrows from C by repledging A’s collateral at date 1 and X2 represents the payment promised by B to repurchase A’s collateral from C at date 2. First, we solve for the contracting problem between B and C at date 1 by taking the date 0 contract between A and B as given, and then we solve for the contracting problem between A and B at date 0. 16
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4.2.1
Contracting Problem between B and C at date 1
For any given contract between A and B at date 0, (I1 , X1 , a), the contracting problem between B and C is to choose (I2 , X2 ) which solves the following problem.
I2 ,X2
subject to I1 ≤ pB X2 + (1 − pB )Z0
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max pB [RB I2 − X2 + aX1 + (1 − a)(pA X1 + (1 − pA )Z0 )]
X2 ≤ aX1 + (1 − a)(pA X1 + (1 − pA )Z0 )
(P1 )
(R1 )
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The objective function is B’s expected utility when he makes the investment by repledging A’s collateral. With probability pB , the investment returns RB I2 , and B pays a part of the return X2 to repurchase A’s collateral from C, and then delivers this collateral to receive X1 from A. The participation constraint (P1 ) states that C’s utility must be at least the reservation value, 0. The right side of (P1 ) is the expected revenue from lending: C receives X2 from B with probability pB and Z0 by seizing the collateral if B defaults with probability 1 − pB . The left side of (P1 ) is the cost I2 that C provides to B’s investment. The last constraint (R1 ) is the resource constraint which implies that B’s promise, X2 , cannot be greater than what B is going to earn by recovering the collateral from C, the expected payment that B would receive from A by returning the collateral: B receives X1 with probability a + (1 − a)pA and Z0 by liquidating the collateral if A defaults with probability (1 − a)(1 − pA ). Then, Assumption 3 and the linearity of the problem ensure that in both cases, all the constraints are binding in optimum. Solving these constraints simultaneously, we can write the contract at date 1 as a function of the date 0 contract.
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Lemma 2. Suppose Assumption 3 holds. Then, at optimum, the contract at date 1 satisfies I2 = apB X1 + (1 − a)(pA pB X1 + (1 − pA pB )Z0 ) X2 = aX1 + (1 − a)(pA X1 + (1 − pA )Z0 ) In other words, B passes the collateral along to C. If B does not fail, it pays X1 , the expected value of the payment by A. If B fails, C retains the collateral valued at Z0 . The 17
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weighted average of these two quantities is the amount that C lends to B. 4.2.2
Contracting Problem between A and B at date 0
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Moving backward, we consider the contracting problem between A and B at date 0. First, we consider the case in which A takes the safe action. To facilitate analysis, we substitute the results in Lemma 2 into the objective function of the date 1 problem, so that B’s expected utility can be written as a function of (I2 , X2 ) as follows: pB (RB I2 −X2 +X1 )−I1 = pB RB [pB (a+(1−a)pA )X1 +{1−pB (a+(1−a)pA )}Z0 ]−I1 (6)
max
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Again, we assume that A has all the bargaining power and makes a take-it-or-leave-it offer to B, and if B rejects the offer, he receives the reservation value, J. The contracting problem at date 0 becomes the following: a(RA I1 − pB X1 − (1 − pB )Z) + (1 − a)[pA (RA I1 − pB X1 ) + bI1 − (1 − pA pB )Z] (7)
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a ∈ arg max a(RA I1 − pB X1 − (1 − pB )Z) + (1 − a)[pA (RA I1 − pB X1 ) + bI1 − (1 − pA pB )Z] (ICA )
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a∈{0,1}
pB RB [pB (a + (1 − a)pA )X1 + {1 − pB (a + (1 − a)pA )}Z0 ] − I1 ≥ J
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(P0 ) (R0 )
I1 , X1 ≥ 0
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The objective function represents A’s expected utility. The incentive constraint for A, (ICA ), the participation constraint for B, (P0 ), and the resource constraint, (R0 ), are modified taking into account the fact that B re-uses collateral at date 1. We make the following two parametric assumptions on pB and RB : Assumption 4. max{pB 2 RA RB , pB RB (pA RA + b)} > 1 > pA pB 2 RA RB . Assumption 4 corresponds to Assumption 1. It implies that it is efficient to undertake the project regardless of A’s action, but from the perspective of B, providing funds for A’s project is profitable only if A takes the safe action. For example, suppose A’s collateral 18
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Assumption 5. RA − pB RA < pA (RA − pB RA ) + b.
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is worthless to B and C, that is, Z0 = 0. In such case, the maximum amount that B can promise to C by repledging A’s collateral is pB RA if A takes the safe action, and pA pB RA if A takes the risky action (both measured per unit of inputs). Then, the total revenue to B is pB RA × pB RB > 1 if A takes the safe action, and pA pB RA × pB RB < 1 if A takes the risky action, implying that B provides funding to A only if A takes the safe action.
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Assumption 5 is the same as Assumption 2, except that the quantity 1 on both sides of Assumption 2 is now replaced by pB RA .22 As before, pB RA is the maximum level of A’s payment (measured per unit of inputs) if A takes the safe action. Assumption 5 can be interpreted by considering that if the value of the asset to A were to approach zero,23 A would want to take the risky action rather than the safe action whenever he invests with the borrowed money. Then, using the same technique as in the previous section, we show that if Z0 and J are sufficiently small, it is optimal to take the safe action and both the incentive constraint (ICA ) and the participation constraint (P0 ) bind.
pB RB [pB Z + (1 − pB )Z0 ] − J Z + (1 − pB )BRB Z0 − (pB /B)J , ,1 1 − BpB RB 1 − BpB RB
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where
and J ≤ min{J 0 , J 00 }. Then, at
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Lemma 3. Suppose pA < 1 and Z0 ≤ optimum, the date 0 contract satisfies
h p 2R R − 1 i 1 B A B pB Z + (1 − pB )pB RB Z0 1 − BpB RB pB RA − B −1 h i 1 1 pB Z 00 J ≡ pB R B + + (1 − p )Z B 0 pB 2 RA RB − 1 1 − BpB RB 1 − BpB RB
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J0 ≡
Taken all the results together, the equilibrium outcome is characterized as follows:
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Note that Assumption 5 is implied by Assumption 2 if pB RA > 1. For example, think of the collateral posted by A as the letter of claim of the debt, which B must hold in order to be able to collect the debt from A, but the letter itself is unlikely to generate other private values for A. 23
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Proposition 2. When pairs of agents choose contracts sequentially and rehypothecation is permitted, the equilibrium contracts are p R [p Z + (1 − p )Z ] − J Z + (1 − p )BR Z − (p /B)J B B 0 B B B B B 0 , ,1 , (I1 , X1 , a) = 1 − BpB RB 1 − BpB RB
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(I2 , X2 ) = (pB X1 + (1 − pB )Z0 , X1 ).
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Note that the scale of both the borrower’s and the intermediary’s investments, I1 and I2 increase as collateral is more valuable – that is, Z or Z0 increases. Also those quantities are increasing in the productivity of the intermediary’s project, RB , and the probability of success of the intermediary’s investment, pB .
Welfare Analysis of Rehypothecation
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In this section, we evaluate social welfare with and without rehypothecation, and compare them to examine the benefits and costs of rehypothecation in the economy. Rehypothecation has the following fundamental trade-offs. On one hand, it helps to provide more funding liquidity to the economy so that additional productive investment can be undertaken, by enabling the limited amount of collateral to back multiple transactions. On the other hand, it introduces the risk of “rehypothecation failure,” the risk that the intermediary might default having repledged his borrower’s collateral to a third party. One potential cost associated with this failure of the intermediary is the deadweight cost of misallocating the asset. This misallocation of the asset arises from illiquidity and trading frictions; the borrower is likely to put a higher value on his assets than his creditor, and they need the intermediary to transfer collateral between them. Thus, in case that the intermediary fails collateral cannot be returned to its highest use. Formally, the social welfare with rehypothecation, denoted by WR , is defined by the sum of A’s, B’s and C’s utility. Using the notation in the previous sections, it can be written as follows. WR ≡ [RA I1 −pB X1 +pB Z−Z]+[pB RB I2 −pB X2 +pB X1 −I1 ]+[pB X2 +(1−pB )Z0 −I2 ]. (8)
where (I1 , X1 , I2 , X2 ) are from Proposition 2.
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Simplifying, WR = (RA − 1)I1 + (pB RB − 1)I2 − (1 − pB )(Z − Z0 )
(9)
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Note that the social welfare consists of the following three components: (i) the surplus generated from A’s investment, (RA − 1)I1 (ii) the surplus generated from B’s investment, (pB RB − 1)I2 , and (iii) the cost generated from the misallocation of the asset in case of rehypothecation failure, (1 − pB )(Z − Z0 ). On the other hand, the social welfare without rehypothecation, denoted by W0 , consists of A’s utility and B’s utility only (since the transaction between B and C cannot happen). Since in this case B never fails to return the collateral, the social welfare without rehypothecation is simply the surplus from A’s investment, W0 = (RA − 1)I
(10)
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where I is from Proposition 1. Suppose A allows B to re-use collateral whenever B wants to do so.24 Then, comparison of equations (9) and (10) shows that rehypothecation allows B’s project to be funded, (pB RB − 1)I2 , but it incurs the cost of misallocating collateral, (1 − pB )(Z − Z0 ), due to the risk that B might not return collateral with probability of 1 − pB .25 To summarize, the efficiency of rehypothecation is determined by the relative size of these trade-off effects. Rehypothecation enhances the provision of funding liquidity to the system so that additional productive investment can be undertaken, but it introduces the counterparty risk which incurs the cost of misallocating collateral.
5
Conflict between Agents’ Incentives to Rehypoth-
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We have so far assumed that the agents’ decisions on the rehypothecation arrangement are exogenously determined. Now we consider the possibility that B can choose whether 24
In the following section, we will consider the possibility that A has the option of prohibiting B from re-pledging collateral. 25 The size of A’s project may also vary; A’s project may be either larger or smaller when rehypothecation is allowed.
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Over-Collateralization and Excessive Rehypothecation
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to rehypothecate A’s collateral or not: B compares the payoff if he rehypothecates A’s collateral to the payoff if he simply holds on to it until when A makes the payment for it. Then we examine whether this decision made by B is aligned with A’s objectives, that is, whether B’s decision is also efficient ex ante. In what follows, we build examples where A’s and B’s objectives are in conflict. In the first example, B prefers to rehypothecate when rehypothecation is inefficient from A’s perspective, and thus the social welfare will be improved if A can preclude rehypothecation in advance (provided that A has complete information about B’s project). In the second example, B does not rehypothecate even when rehypothecation is efficient from A’s perspective, and thus the welfare will be improved if B can commit to rehypothecate in advance. The direction of this conflict between A and B depends on whether the contract between them involves over-collateralization or under-collateralization, that is, whether A’s private value of collateral, Z, is greater or smaller than the payment for recovering it from B, X1 . With over-collateralization, B tends to be excessively eager to rehypothecate. Intuitively, this is because collateral is worth only up to the payment from A, X1 , from B’s perspective, while it is actually worth Z(> X1 ) to A. However, B does not internalize A’s value on his asset, and thus B tends to want to rehypothecate too frequently. With under-collateralization, B tends to be excessively cautious to rehypothecate for the opposite reason. Intuitively, the collateral is worth X1 from B’s perspective, which is more than its value to A, and thus B does not want to rehypothecate even though rehypothecation may be a desirable gamble from A’s perspective.
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In this subsection, we build an example where over-collateralization induces B to excessively participate in rehypothecation from A’s perspective. The setting in this example is the same as in the previous analysis. For simplicity, we set B’s reservation value to be J = 0 and the external value of collateral to be Z0 = 0. We make the following additional assumptions. Assumption 6. B ≡ RA −
b 1−p
< 0 and 1 − B < (1 − B)pB 2 RB < 1 − BpB RB .
The following result shows that if this assumption holds, a conflict arises between the interests of A and B. 22
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Proposition 3. Suppose Assumption 6 holds and set Z0 = J = 0. Then, B prefers to rehypothecate although it is undesirable from A’s perspective.
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b < 0, combined To see this, note that the first inequality of Assumption 6, B ≡ RA − 1−p with the binding incentive constraint (ICA ), implies that the equilibrium contract involves over-collateralization. The second inequality of Assumption 6, 1 − B < (1 − B)pB 2 RB , implies that B wants to rehypothecate. Finally, the last inequality, (1 − B)pB 2 RB < 1 − BpB RB implies that rehypothecation is undesirable for A.
Under-Collateralization and Insufficient Rehypothecation
Assumption 7. B ≡ RA −
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Now we build an example where under-collateralization leads B to be unwilling to rehypothecate, although rehypothecation is desirable from A’s perspective. Again, the setting in this example is the same as in the previous analysis and we set Z0 = J = 0. For this example we make the following assumption regarding parameters: > 0. and 1 − BpB RB < (1 − B)pB 2 RB < 1 − B.
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Compared to the over-collateralization example of the previous subsection, this assumption changes the direction of the conflict between A’s and B’s interests as follows.
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Proposition 4. Suppose Assumption 7 holds and set Z0 = J = 0. Then, B does not want to rehypothecate though rehypothecation is efficient from A’s perspective.
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This result can be interpreted in a similar way as in the case of over-collateralization. b > 0, combined with the binding inThe first inequality of Assumption 7, B ≡ RA − 1−p A centive constraint (ICA ) implies that the optimal contract involves under-collateralization. The second inequality of Assumption 7, (1 − B)pB 2 RB < 1 − B, implies that B does not want to rehypothecate. Finally, the last inequality, 1 − BpB RB < (1 − B)pB 2 RB , implies that rehypothecation is not desirable for A. To summarize, these examples show that if the contract involves over-collateralization, the intermediary tends to be too eager to rehypothecate, while the contract involves undercollateralization, it tends to be too cautious to rehypothecate. The inefficiency due to the intermediary’s over-eagerness to rehypothecate might be mitigated relatively easily if the initial borrower has the right to preclude rehypothecation in advance. However, the inefficiency due to excessive reluctance to rehypothecate will still remain without an institutional device that induces rehypothecation. In any event, if 23
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6
Comments
6.1
Heterogeneity of Valuations
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the borrower’s information about the intermediary’s investment opportunity is imperfect, the borrower’s decision on the rehypothecation arrangement may not achieve an optimal outcome in either case. For example, if the return of B’s project, RB , or, the probability of B’s default, 1 − pB , are private information to B, and it is costly for A to learn these, then A might prevent socially efficient rehypothecation from taking place unnecessarily, due to the concern of losing collateral.
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Rehypothecation has costs and benefits. The costs we have addressed here stem from the fact that rehypothecation increases the chances that the financial asset ends up in the “wrong hands”—in other words, that the agent who ends up with the asset after counterparty failure actually values it less than the initial borrower. In the case of financial assets there are several possible sources for this difference in valuations: First, different individuals may start from different non-tradable portfolios,26 or hold different, irreconcilable views about the future values of assets.27 Differences in the transactions costs of resale faced by the various parties can play an equivalent role; this role is considered explicitly in Kahn and Park (2016).28 For otherwise highly liquid assets another possible source of diffence in valuations is legal uncertainty—the title to the asset may be encumbered by uncertainty in the process of hypothecation and rehypothecation across jurisdictions. Finally, the difference in valuation may ultimately be attributed to differing liquidity needs: the initial borrower may be constrained in her ability to acquire additional pledgeable assets and therefore place an above-market premium on their return. Not surprisingly, if the agents’ valuations on collateral are less heterogeneous, the cost of misallocation of collateral will decrease, and rehypothecation will be more efficient. On the other hand, our main result, a conflict of interests between the contracting parties 26
See, for example, Brunnermeier and Pedersen (2009). Geanakoplos (2010) analyzes the leverage cycles in a market in which agents have different beliefs about the future returns of collateral asset. Based on this framework, Simsek (2013) develops a model in which lenders do not value the collateral as much as borrowers do, and show how the nature of the borrower’s optimism affects the tightness of the borrowing constraint. Finally, He and Xiong (2012) investigate how the dynamics of agents’ heterogenous beliefs generate a fluctuation of the asset price and prices of the contract. 28 Boot, Thakor, and Udell (1991) emphasized costs of dealing with collateral. 27
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Segregated Accounts
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about rehypothecation, does not rely on heterogeneous valuations; rather it arises from the incentive role of collateral. For example, as long as the size of the haircut is set in order to solve the borrower’s incentive problem, the promised payment will differ from the collateral value, even if the intrinsic value of collateral is the same across all the agents. If the amount promised to be paid to the intermediary is greater than the value of collateral, then the intermediary is likely to be too cautious about rehypothecation from the borrower’s perspective. If the lender could somehow commit to a more aggressive rehypothecation policy, a superior loan contract could be achieved between him and the borrower.29
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As a protection against the loss of collateral in a counterparty failure, collateral is often sequestered into a separate protected account with the intermediary, into which account assets can be replaced by other assets in the event that they are used for rehypothecation. However this practice varies across markets and countries. For example, in the US, assets segregated under SEC Rule 153-3 can be deposited at either (1) an affiliated clearing entity of the prime broker or (2) a third-party custodian bank. The choice has the following trade-offs. The custodian bank does not lend nor reuse the assets, but requires a custody fee for keeping the assets. In contrast, the prime broker does not charge a custody fee, but keeps the clients’ assets in an omnibus account (pooled with other clients’ assets) and they can use these assets with the written consent of the clients.30 If the prime broker uses these assets from the omnibus account, it must maintain a number of assets in the omnibus account with the sum of fully paid securities by SEC Customer Protection Rule. Hedge funds now increasingly demand their prime brokers provide more protections on their assets by requesting segregation at other custodian banks or limiting the amount of their assets permitted to be reused.31 However customer protection for rehypothecated assets has been much less clear in the United Kingdom and continental Europe (Singh and Aitken, 2010). It is for this reason that Lehman transfered many customer assets to its UK affiliate in the years leading up to the financial crisis. 29
If the haircut has the opposite sign, the direction of the conflict will go into the other way. If clients want complete segregation of their assets, they need to request their prime broker to open an individual account. But, this requires some fees, and was not used by most clients. 31 See Carlsson-Sweeny (2010) for more information on the recent changes in prime brokerage agreements. 30
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While segregated accounts considerably mitigate the problem of misallocation, they do not eliminate it. First of all, as we have seen, financial institutions find legal workarounds and sometimes have failed to honor the requirements for segregation—a fact only discovered after failure. Secondly, in situations where the substitution of one asset for another is possible within the segregated account, in principle the substitutions may not be regarded as equivalent by the initial borrower, reestablishing the problem in miniature. For example, there are accounts of financial institutions routing different qualities of asset for rehypothecation in bespoke or triparty markets.32
Implications for Stability and Systemic Risk
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Throughout this paper, we have assumed that the lender’s default risk is constant. However in the real world default risk varies over time for individual firms, and varies systematically with the business cycle. Borrowers will wish to prohibit rehypothecation when the lender’s credit rating drops, in turn making borrowing more expensive. Thus rehypothecation can act as another accelerator in financial crises: when a borrower blocks the lender’s ability to rehypothecate because of doubts about its credit rating, he makes it more difficult for the lender to borrow funds, further damaging the lender’s credit rating. In our model, collateral chains are short, containing at most two links. This mirrors the institutional arrangements often seen with formal rehypothecation, in which the middleman is in a specialized position as both lender of cash and lender of securities to less-specialized clients. However, as noted at the beginning, other forms of collateral reuse are also common in the financial system, and these can entail longer collateral chains. As the chains become longer, the reinforcement effects noted are likely to become more severe. While the cessation of rehypothecation can reinforce a financial downturn, it can also lead to a faster recovery. Park (2017) addresses this issue in a dynamic extension of the model, examining how borrowers’ responses to shocks affect the size of the downturn and the speed of recovery after the shock. In that model, if borrowers prohibit rehypothecation after shocks, output drops more severely, but further potential misallocations of capital are reduced. For this reason, the distribution of collateral is improved and recovery of the 32
“Think of the bilateral repo market as analogous with the old-clothing trade...Big lots come in from hedge funds and security lenders, and the collateral desks at large banks will paw through it, searching for gems. Those gems go out bilateral to customers who will pay a premium,. The remainder goes to the guys in the back of the line: tri-party repo.” (Singh, 2014, p. 9)
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economy occurs more quickly. Conversely, if rehypothecation continues after the shock, the immediate output drop is less severe, but the recovery is slower. Park’s extension also provides further systemic implications of rehypothecation. In her framework, prolonged periods of good state (boom) lead to too much rehypothecation, increasing the output in the short run, but accumulating misallocated collateral and concomitant reductions in long-run output.
Alternative Solutions
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In this section we briefly consider alternative arrangements to solve the problem of guaranteeing a sequence of loans with limited collateral available. For financial collateral, such as used in repo transactions, it is typical that a borrower hands it over to a lender at the inception of the contract. Probably the most important reason for this is to save costs of the collection process in case of the borrower’s bankruptcy. For example, in mortgage transactions, upon the debtor’s declaration of bankruptcy, collateral is frozen under the bankruptcy law (an automatic stay goes into effect). This exposes the lenders to the risk of sale at uncertain prices since they have to wait for a longer time for liquidating collateral. In this respect, transfering collateral to the lenders earlier can shorten this time of the collection process. The repo arrangement has the legal effect of pulling collateral out of the bankruptcy proceedings by giving legal ownership to the creditor. This makes the transfer in case of debtor default much easier, but puts the debtor at greater risk in the case of creditor default. One possibility would be for the initial borrower to issue her own transferable debt. Then when the intermediary needs to raise funds he can sell this debt to a third party.33 In practice, resale of debt exists in some financial markets and not in others: bonds are resellable, repos are not. Using transferable debt requires that third parties know the creditworthiness of the initial borrower, information that is likely to be part of the expertise of intermediaries but not of the typical participant in these markets (often a nonfinancial corporation lending its temporary excess funds); nor would it be in the interest of the intermediary to give an accurate description of the initial borrower’s creditworthiness to anyone to whom he was attempting to sell the debt. Using financial assets as collateral 33
Such an arrangement is similar to using transferable debt as “inside money” (Kahn and Roberds, 2007).
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Conclusion
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reduces (although does not eliminate) the difficulties of determining values.34 The inefficiencies in the model arise from the inability of the initial borrower to redeem the collateral from the final creditor. In the model we have simply assumed that the two parties never meet; in Kahn and Park (2016) we assume that they can meet in an open market but are subject to a transactions cost in that market. This assumption is motivated by observations that there are trading frictions in the markets, such as over-the-counter (OTC) markets in which repo and security lending transactions take place.35 Nonetheless, it would seem to be the case that arrangements which facilitate meetings between individuals attempting to reallocate the collateral would be obvious ways to increase the efficiency of these markets. One natural institutional arrangement to consider in this context is the establishment of centralized counterparties (CCPs) to serve as clearinghouses for these sorts of trades. CCP’s are an important part of the proposals for reform of the financial system post crisis. The Dodd-Frank Act mandates central clearing of many swaps; in this case collateral will be held by a central counterparty. On the one hand, this precludes direct re-use of collateral. On the other hand, the arrangement makes it easier to transfer positions to third parties. Whether a CCP represents a superior solution depends on the degree to which terms of the contract between parties can be made independent of private information about counterparties. Information on counterparties, including the collateral terms that are optimal for particular trades with them, is likely to be part of the comparative advantage of prime brokers. Thus a move from OTC to centralized arrangements is likely to generate both losers and winners among customers.36
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We develop a model of the economic underpinnings of rehypothecation. Rehypothecation is an example of the re-use of the same collateral to support multiple transactions. Rehypothecation helps to provide more funding liquidity into the system by allowing the 34 See Donaldson and Micheler (2016) for a further discussion of the consequences of the differences in asset resellability. 35 Notable papers that build on frictions in the OTC markets are Duffie, Gˆ arleanu and Perdersen (2005) and Lagos and Rocheteau (2009). Edwards et al. (2007) provide empirical evidence of the existence of transaction costs in OTC markets for corporate bonds. 36 If agents A and C cannot meet directly, a second important alternative to rehypothecation is asset securitization. Roughly speaking, securitization lowers the problem of misallocation but reduces C’s protection. See Maurin (2014) for further discussion.
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lender to use her borrower’s collateral, which would otherwise sit idle in the lender’s account, for another productive investment. However, it can incur a deadweight cost due to misallocating the asset when the lender fails having repledged the borrower’s collateral to the third party to whom the collateral is less likely to be as valuable to the initial owner. We show that the individuals’ incentives to participate in rehypothecation may not be aligned with economic efficiency. In other words, cases arise in which agents choose socially inefficient levels of rehypothecation. Importantly, the direction of the inefficiency depends on terms of the agents’ contracts and ultimately as the incentive problems that those contracts are designed to solve. If the contract involves over-collateralization, the intermediary tends to be overly eager to rehypothecate; if the contract involves undercollateralization, the intermediary tends to be overly cautious to rehypothecate. As noted above, Park (2017) investigates the output dynamics in an extension of this model in which rehypothecation endogenously arises and varies with macroeconomic conditions. In this model, she derives initial implications regarding the systemic effects, in terms of magnification of shocks to creditworthiness or speed of recovery of output. However other natural extensions are worth consideration. Thus far we have not incorporated insurance motives, random fluctuation in the value of the collateral, and more general effects of aggregate and idiosyncratic shocks. These important practical considerations are left for future research.
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Huge, B., Lando, D., et al. (1999). Swap pricing with two-sided default risk in a ratingbased model. European Finance Review, 3(3):239–268. Infante, S. (2014). Money for nothing: The consequences of repo rehypothecation. Available at SSRN 2460979. Kahn, C. M. (2009). Collateral policy in a world of round-the-clock payment. University of Illinois College of Business Working Paper, pages 10–0100.
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Kahn, C. M. and Park, H. (2016). Collateral chains and incentives. The Journal of Financial Market Infrastructures, 5(1). Kahn, C. M. and Roberds, W. (2007). Transferability, finality, and debt settlement. Journal of Monetary Economics, 54(4):955–978.
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Park, H. (2017). Rehypothecation, Collateral Mismatch and Financial crises. PhD thesis, University of Illinois. Simsek, A. (2013). Belief disagreements and collateral constraints. Econometrica, 81(1):1– 53.
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Singh, M. (2010). The sizable role of rehypothecation in the shadow banking system. IMF Working Papers.
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Singh, M. (2011). Velocity of pledged collateral: analysis and implications. IMF Working Papers.
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Singh, M. (2016). Collateral and financial plumbing. International Monetary Fund Washington, DC.
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Teweles, R. J. and Bradley, E. S. (1998). The stock market, volume 64. John Wiley & Sons.
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Tirole, J. (2010). The theory of corporate finance. Princeton University Press.
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Proofs
By the revelation principle the optimal contract is a triple (I, X, a) solving the following maximization problem: max a[RA I − pB X − (1 − pB )Z] + (1 − a)[pA (RA I − pB X) + bI − (1 − pA pB )Z] (A.11)
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Is ,Xs ,a
subject to
a ∈ arg max a[RA I − pB X − (1 − pB )Z] + (1 − a)[pA (RA I − pB X)+bI − (1 − pA pB )Z] a∈{0,1}
(IC) (P )
RA I ≥ X
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pB [aX + (1 − a)(pA X + (1 − pA )Z0 )] − I ≥ J I, X, J ≥ 0
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The objective function is A’s expected utility. The moral hazard problem restricts a to values which are ex-post incentive compatible given the choice of the other terms of the contract, this is restriction (IC). Constraint (P ) ensures that B’s expected profit from lending be at least equal to his reservation value J. Lastly, constraint (R) says that A cannot pay more than the return from the investment, RA I (if the investment fails, A makes no payment to B).
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Proof of Lemma 1 . In the case where pB = 1, (IC) is equivalent to the following three conditions: a ∈ {0, 1}
(A.12)
and if a > 0 then (1 − pA )(RA I − X) ≥ bI
(A.13)
and if a < 1 then (1 − pA )(RA I − X) ≤ bI
(A.14)
First, consider the case where a = 0. Then [1 + (1 − a)pA ]X ≤ I, so that (P ) can only be satisfied when X = I = J = 0. By (A.13),
b I RA − 1 − pA 34
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and by Assumption 2 RA − therefore
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On the other hand, if a = 0, the requirement reduces to pA X ≥ I but I ≥ X/RA ≥ pA X
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I ≥ X.
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Proof of Proposition 1 . There is not feasible contract in which a = 0 and borrowing takes place. If it did then it must be the case that pA X ≥ I. But by Assumption 1 this is possible only if I = X = J = 0. That leaves only the case a = 1 to examine. In this case we can simplify the problem as follows: max RA I − X I,X
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where the letters to the right of the constraints denote the Lagrange multipliers, all non negative. Taking the first order conditions we have µ − Bλ = RA (ν + 1) µ−λ=ν+1
From the latter condition we conclude µ is positive, and since RA > 1, combining the
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conditions we conclude that (1 − B)λ > 0
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or λ > 0. Since 1 − B > 0 by Assumption 2, both conditions (λ) and (µ) hold with equality at an optimum. Solving these conditions for X and I yields the formulas in the statement of the proposition. The restrictions on the relation between Z0 and J in the statement of the proposition guarantee that X and I satisfy the remaining inequalities in the problem.
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Proof of Lemma 3 . If a = 0, there is no feasible contract in which borrowing takes place. If it did then it must be the case that pB RB [pA pB X1 + (1 − pA pB )Z0 ] − J ≥ I, and by the conditions on Z0 , this implies that it must be the case that pA pB 2 RA RB X ≥ I. But, by Assumption 4, this is possible only if I1 = X1 = J = 0. Now we examine the case a = 1. There are two possible cases. (Case 1) If RA > pA RA + b, (P0 ) must bind in any optimum. Suppose not. Increasing I1 until this is the case; it improves the objective without violating any constraints. Thus we can simplify the problem as follows:
pB RB [pB Z + (1 − pB )Z0 ] − J 1 − BpB RB J − pB RB (1 − pB )Z0 I1 ≥ pB 2 RA RB − 1 I1 ≤
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where BpB RB < 1 by Assumption 4. −1 Since J ≤ pB RB pB 2 RA1RB −1 + 1−Bp1B RB
pB Z 1−BpB RB
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pB RB [pB Z + (1 − pB )Z0 ] − J Z + (1 − pB )BRB Z0 − (pB /B)J , 1 − BpB RB 1 − BpB RB
(Case 2) If RA < pA RA + b, either (P0 ) or (ICA ) binds in optimum. If (P0 ) binds, this
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reduces to (Case 1). If (ICA ) binds, we can rewrite the problem as follows. max (RA − B)I1 − Z I1
subject to
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pB RB [pB Z + (1 − pB )Z0 ] − J 1 − BpB RB pB Z I1 ≥ pB RA − B I1 ≤
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i Since J ≤ + (1 − pB )pB RB Z0 , the set of I1 that satisfy these constraints is non-empty and (PB ) binds. Thus, the solution is the same as in (Case 1) above. pB 2 RA RB −1 pB Z pB RA −B
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h
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Proof of Proposition 3 . Recall that the incentive constraint (ICA ) binds at optimum, RA I1 − pB X1 − (1 − pB )Z = pA (RA I1 − pB X1 ) + bI1 − (1 − pA pB )Z
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which reduces to,
X1 = pB −1 BI1 + Z.
pB 2 RB X1 > X1 .
(A.15)
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b Since B ≡ RA − 1−p < 0 by Assumption 6, X1 < Z, i.e., the contract involves overA collateralization. Now we shows that B wants to rehypothecate at date 1 if and only if
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where the left side is B’s payoff with rehypothecation and the right side is the payoff without rehypothecation when Z0 = 0. This inequality holds by the second inequality of Assumption 6. On the other hand, A does not want to rehypothecate. If Assumption 6 holds, the following must hold, RA I − X > RA I1 − pB X1 − (1 − pB )Z (A.16) where I, X are from Proposition 1 and I1 , X1 are from Proposition 2. The left side is A’s payoff without rehypothecation and the right side is the payoff with rehypothecation.
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By substituting X and X1 with a function of I and I1 , respectively, we can rewrite it as follows. (RA − B)I − Z > (RA − B)I1 − Z.
(A.17)
pB 2 RB Z Z > 1−B 1 − BpB RB by the last inequality of Assumption 6.
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Substituting I in Proposition 1 and I1 in Proposition 2 into this inquality, we obtain, (A.18)
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Proof of Proposition 4 . Note that the second inequality of Assumption 7 is equivalent to pB 2 RB < 1.
Since X1 is positive and Z0 = 0, this is equivalent to
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The left side, B’s payoff with rehypothecation, is smaller than the right side, the payoff without rehypothecation. On the other hand, the second inequality of Assumption 7 implies that pB 2 R B Z Z . < 1−B 1 − BpB RB
(A.20)
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Using the results in Proposition 1 and Proposition 2, this can be rewritten as follows. RA I − X < RA I1 − pB X1 − (1 − pB )Z
(A.21)
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The left side, A’s payoff without rehypothecation, is smaller than the right side, the payoff with rehypothecation.
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Collateral, Rehypothecation, and Efficiency Hyejin Park, Charles M. Kahn PII: DOI: Reference:
S1042-9573(18)30037-8 10.1016/j.jfi.2018.06.001 YJFIN 793
To appear in:
Journal of Financial Intermediation
Please cite this article as: Hyejin Park, Charles M. Kahn, Collateral, Rehypothecation, and Efficiency, Journal of Financial Intermediation (2018), doi: 10.1016/j.jfi.2018.06.001
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Collateral, Rehypothecation, and Efficiency∗ June 9, 2018
Abstract
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Hyejin Park† and Charles M. Kahn‡
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This paper studies rehypothecation, a practice in which financial institutions re-pledge collateral pledged to them by their clients. Rehypothecation enhances provision of funding liquidity to the economy, but it also incurs deadweight cost by misallocating the asset among the agents when counterparties fail. We examine the possibility of a conflict between the intermediary and its borrower on rehypothecation arrangements. The direction of this conflict depends on haircuts of the contract between them: if the contract involves over-collateralization, there tends to be an excessive use of rehypothecation, and if the contract involves under-collateralization, there tends to be an insufficient use of rehypothecation. This offers an empirical prediction of a link between the size of haircut in collateralized financial contracts, borrower information, and the likelihood of rehypothecation.
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Keywords: collateral, rehypothecation, moral hazard, misallocation JEL Classification: D53, D62, G21, G28
∗ We are grateful to the Editors and an anonymous referee for their comments which helped to improve this paper. We also thank Dan Bernhardt, Vincent Maurin, Giorgia Piacentino, Jason Donaldson, Jan Willem van den End, Gerold Willershausen, Phil Prince, Manmohan Singh, Mina Lee, and David Andolfatto, as well as seminar participants at the University of Illinois at Urbana-Champaign, the IBEFA San Francisco Fed conference, the St. Louis Fed Summer Workshop, the MFA Annual Meeting, the Atlanta Federal Reserve Conference in Amelia Island, Florida, and Cambridge Corporate Finance Theory Symposium for their valuable comments. Any remaining errors are ours. † Department of Economics, University of Illinois at Urbana-Champaign. E-mail: [email protected] ‡ Department of Finance, University of Illinois at Urbana-Champaign. E-mail: [email protected]
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Introduction
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Financial contracts are regularly enforced through collateral.1 Assets most desirable as collateral are scarce in the economy and the cost of generating these assets is nonnegligible. With the increased volume of financial transactions, market participants find that it is important to economize on the limited amount of available collateral.2 One way economize on collateral is through collateral re-use, allowing a single unit of collateral to support multiple transactions. Re-use reduces the cost of holding collateral; it also allows the borrower access to funds at a cheaper rate. An important example of collateral re-use by financial intermediaries is “rehypothecation,” the practice of repledging a customer’s collateral for its own trading or borrowing. This paper addresses some basic open questions about this practice: under what circumstances rehypothecation arises and what benefits and costs it produces, and whether decentralized decisions to participate in rehypothecation achieve a socially efficient outcome or not. In particular, our main question is whether there is alignment between the objectives of the borrower and intermediary: are there cases where the intermediary overor under-uses the borrower’s collateral? Rehypothecation was one of the most popular devices for many dealer banks to serve their own funding liquidity needs before the recent financial crisis.3 After the failure of Lehman Brothers in 2008, however, rehypothecation significantly dropped as hedge funds restricted re-use of their collateral.4 At the same time, regulation on rehypothecation has also been advocated by legislators and policy-makers.5 Nonetheless, understanding of the 1
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Collateral occurs in a large variety of settings, from residential houses in mortage to T-bills in repo transactions. Holmstr¨ om (2015) notes that the history of collateralized lending can be traced back to pawning in China during the 7th century. For another insightful discussion on the origin of collateralized lending, see Geanakoplos (1996). 2 Singh (2010, 2011) emphasizes the scarcity and varying quality of collateralizable assets and their efficient use. Krishnamurthy and Vissing-Jorgensen (2012) estimate the liquidity and safety premium on Treasuries paid by the investors on average from 1926 to 2008 was 72 basis points per year. In a similar vein, Greenwood, Hanson, and Stein (2012) point out the monetary premium embedded in short-term Treasury bills, the highest quality collateral. 3 While we model rehypothecation, with suitable modifications our model can be applied to other forms of collateral re-use as well. In a companion paper (Kahn and Park, 2016), we also consider alternative institutional arrangement which can achieve the benefits of rehypothecation using a simple, but less standard, model of incentives as a motivating example. 4 According to Singh (2010, 2011) the value of collateral held in 2007 by the six largest U.S. investment banks that was permitted to be rehypothecated was around $ 4.5 trillion. Post the crisis, in 2009, however, this value sharply dropped to $2 trillion. 5 For example, the Dodd-Frank Act requires that collateral in most swap contracts to be deposited in
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economics underlying this practice is still incomplete, and there are still debates on how to regulate rehypothecation, as evidenced by wide variation in the rules on rehypothecation across countries.6 We start with a framework in which a borrower is subject to a moral hazard problem and must post collateral to prevent him from engaging in risk-taking actions. Previous models of this problem, however, have not considered the possibility that lenders re-use collateral for their own purposes and the risk that they might fail to return the collateral. We incorporate the possibility of collateral re-use by the counterparty into a contracting model, offering the first analytical framework that investigates individuals’ incentives behind this practice. In our model the borrower transfers collateral to the intermediary at the beginning of the contract. In other words, collateral in our model is like a repurchase agreement; the borrower transfers his asset to the lender at the time a contract is initiated and buys it back at a later point.7 This early transfer of collateral introduces the risk that the lender may not be able to return collateral at the time when the borrower wants it, a risk exacerbated by rehypothecation.8 Indeed, as observed from the failure of Lehman Brothers in 2008 or MF Global in 2011, this is not simply a theoretical possibility.9 The efficiency of rehypothecation is determined by the relative size of the following two
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a segregated account of a central counterparty, precluding re-use. 6 For example, in the U.S., SEC rule 15c3-3, a prime broker may rehypothecate assets to the value of 140% of the client’s liability to the prime broker. In the U.K., there is no limit on the amount that can be rehypothecated. See Monnet (2011) for more detailed explanation on various regulatory regimes for rehypothecation. 7 Early transfer of collateral is observed frequently in shadow banking sectors where loans are mostly short-term and of large value and seizing collateral after bankruptcy of borrowers is difficult and time consuming. In contrast, for long-term loans between a bank and a consumer such as a mortgage loan, it is relatively easy to seize collateral from the borrower should the need arise, and collateral mostly stays with the borrower. 8 The traditional situation was described as follows, in the case when a broker’s client purchased on margin and the broker rehypothecates to a bank to finance the call loan: “Numerous court decisions have established that the bank’s rights are strong, generally superior even to those of the customer. Although the situation is infrequent, if the broker became unsecured as a result of price activity and was not able to meet the bank’s loan terms, the bank has the legal right to sell the customer’s pledged securities to protect its loan. Naturally, the customer in this situation will have a difficult time salvaging much equity.” (Teweles et al., 1998). 9 With regard to Lehman, for example, Fleming and Sarkar (2014) state, “Counterparties that had allowed Lehman to hypothecate their collateral to unrelated third parties in connection with securities transactions that could not be unwound found that their collateral had become unrecoverable. . . [they] did not know when their collateral would be returned to them, nor did they know how much they would recover given the deliberateness and unpredictability of the bankruptcy process.”
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effects: Rehypothecation lowers the cost of holding collateral and makes illiquid collateral more liquid, thereby providing more funding liquidity into the market. On the other hand, rehypothecation failure leads to the misallocation of collateral. In our model, this misallocation of the asset arises from the following two frictions: (i) we assume that the asset is ‘illiquid’ in the sense that the asset is likely to be more valuable to the initial owner than to the other agents (for example, an asset in the borrower’s portfolio might be more useful for that portfolio than for others)10 ; (ii) we assume that agents can trade only through the intermediary (for example, large dealer banks have better access to potential clients than those clients have to one another). This, in turn, implies that if the intermediary fails, the asset might end up in the wrong hands: the asset cannot be returned to the initial owner who values it the most, but instead it is seized by the third party for whom the collateral may be less valuable. We consider the possibility that the intermediary can choose whether to participate in rehypothecation or not, by comparing the payoff when he rehypothecates and the payoff when he does not rehypothecate. We characterize the conditions under which a conflict arises between the intermediary and the borrower. A key factor that determines the direction of the conflict is the wedge between the borrower’s valuation on his collateral and the payment promised to the lender for retrieving it. Whether a loan contract involves overor under-collateralization depends in turn on the incentive problem that the contract is designed to mitigate. If the contract involves over-collateralization, in the sense that the value of collateral to the borrower exceeds the payment for retrieving it, the intermediary is likely to overuse the borrower’s collateral. The reason is that the intermediary does not internalize this positive externality of returing collateral to the borrower and sometimes might want to engage in rehypothecation even when rehypothecation is costly from the borrower’s viewpoint. Similarly, if the contract involves under-collateralization, the intermediary might be tempted to underuse the borrower’s collateral for the opposite reason: he values returning it to the borrower more than the borrower actually values receiving it back. This result also has the additional implication that if the initial borrower has the right to permit or restrict rehypothecation, her choice will depend on whether the contract involves under- or over-collateralization; if the contract involves under-collateralization, she tends to permit rehypothecation more often; if the contract involves over-collateralization, 10
Park (2017) shows how this wedge between valuations on the same asset arises endogenously in an infinite-horizon dynamic setting. We discuss other possibilities in section 6.
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she tends to be reluctant to permit rehypothecation. Thus, the inefficiency due to the intermediary’s excessive rehypothecation might be mitigated relatively easily if the borrower could prohibit rehypothecation in advance, while the inefficiency due to the insufficient rehypothecation will remain without other devices to induce rehypothecation.
Related Literature
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A number of theoretical and empirical papers consider collateral as a device to reduce contracting frictions in the presence of moral hazard, among them, Aghion and Bolton (1997), Boot and Thakor (1994), and Holmstr¨om and Tirole (1998, 2011).11 Boot, Thakor, and Udell (1991) offer empiricial evidence as well as a theory that collateral mitigates moral hazard and provides a screening mechanism. Along these lines, Berger, Frame, and Ioannidou (2016) test a variety of theories of collateral and find empirical dominance of the theories in which collateral mitigates losses and incentivizes the borrowers to take less risk. They also find that liquid collateral is associated with especially low risk premia, and offer an explanation for the conflicting results in the literature about the empirical relation between collateral and loan risk. Agarwal, Green, Rosenblatt, and Yao (2015) document that even after controlling for mark-to-market asset valuation, initial collateral remains an important predictor of mortgage default, and those who pledge higher collateral have a lower default likelihood. In the context of derivatives trading, Biais, Heider, and Hoerova (2012) discuss how collateral mitigates the moral hazard problem of the derivatives providers. In particular they assume that collateral is deliverd to the lender before the borrower’s final payoffs are realized. One of the advantages of depositing collateral in advance is that it can protect the collateral from the potential borrower’s risky action as new information arrives after the contract begins. In a similar framework, Bolton and Oehmke (2014) examine
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Mills and Reed (2012) describe the two fundamental roles of collateral: first, it incentivizes the borrower to repay, second, it provides a lender with some compensation through its liquidation if the borrower does not fulfill her payment obligations. However, in the absence of additional frictions, there is no difference between a collateralized loan and raising funds by a sale of the asset (and a subsequent repurchase in some states of the world). Thus various accounts consider additional frictions which collateral specifically addresses. Besides moral hazard, the possibilities include signaling/adverse selection, changing bargaining power under renegotiation, or protection against dilution of claim at the hands of other creditors (for this last, see Donaldson, Gromb and Piacentino, 2017). Each of these explanations, to the extent that they serve to provide a wedge between the value of the collateral and the size of the loan, can serve as an alternative basis for our results.
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the implications of the recent privileged bankruptcy treatment of derivatives and argue that the seniority of derivatives can be inefficient since it transfers default risk to the debt market although the risk can be born more efficiently in the derivative market. A contribution of our paper is that we consider a default risk associated with the receiver of collateral which was absent in those models. The source of this counterparty risk in our model is rehypothecation. If the lender defaults having re-used the borrower’s collateral, the collateral may not be returned to its most efficient use. Our paper is also related with the literature on two-sided credit risk. Mills and Reed (2012) develop a repo contract model in which both a borrower and a lender are subject to risks of default. Their model is in turn based on the approach of Kocherlokota (1996) that considers a problem of risk sharing in an environment without commitment to future actions. Kehoe and Levine (2001) and Alvarez and Jermann (1995) investigate asset trade and pricing with multiparty credit risk in a general setting. Derivatives pricing under bilateral credit risks is considered by Franzen (2001) and Huge and Lando (1999). Our paper is also related to works that build on the scarcity of safe and liquid assets. Gorton and Pennacchi (1990) show that in the presence of informational frictions, financial intermediaries can create safe assets, providing liquidity to the economy. Krishnamurthy and Vissing-Jorgensen (2013) show the existence of a persistent demand for safe and liquid assets, providing evidence that safe assets are scarce. Gorton, Lewellen, and Metrick (2012) and Singh (2010, 2011) discuss the shortage of safe and liquid assts in the shadow banking system. More recently, Bonaccorsi di Patti and Sette (2016) shows that the freeze in the securitization market led to a reduced supply of credit by banks. In our paper, a decrease in rehypothecation increases the cost of financing to the borrower and the cost of holding collateral to the lender, thereby reducing supply of liquidity to the system. Since the financial crisis in 2007, there has been growing interest in rehypothecation from both policy groups and academic researchers. Monnet (2011) discusses policy implication of rehypothecation. Kahn (2009) shows how an analogous concern arises in the context of international payments arrangements. Bottazi, Luque, and Pascoa (2012) examine how asset prices increase by the leverage built up in the process of rehypothecation in a general equilibrium model. Andolfatto, Martin, and Zhang (2014) analyze the welfare implications of rehypothecation in a monetary search model. Maurin (2014) extends the general equilibrium model with collateral constraint by Geanakoplos (2010) and compares rehypothecation with other alternative trading techniques such as tranching and pyramiding. He shows that rehypothecation has no effect on trading outcomes in 6
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complete markets, and thus the effectiveness of rehypothecation depends on the market structure. In the finance literature, Lee (2015) discusses a tradeoff of rehypothecation in terms of economic efficiency and financial stability. She emphasizes that a sudden decline of rehypothecation can lead to an inefficient repo run by creating a positive feedback loop between the repo spread and fire-sale discounts. Eren (2014) and Infante (2014) consider a situation in which a dealer bank earns additional liquidity by setting different margins to their clients on the opposite sides. Park (2017) extends the model in the current paper into an infinite-horizon dynamic model to examine how shocks associated with the intermediary affect collateral allocation in the economy and initiate fluctuations in aggregate output. Our paper is organized as follows. The baseline model is presented in Section 3. In Section 4, the model is extended to a three-player model of rehypothecation and the contracting outcome is derived. Section 5 provides examples where a conflict between the parties about rehypothecation arises. Section 6 discusses institutional alternatives to rehypothecation.
The Baseline Model
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We start from a simple two-player model in which a borrower is subject to a moral hazard problem tempting him to engage in a risky action that is not desirable from the lender’s perspective.12 The moral hazard problem leads to a shortage of pledgeable income of the borrower, in the sense that the future pledgeable return of the borrower’s investment is not enough to repay the initial cost of the investment. For this reason, uncollaterlized borrowing may not be possible. In such a scenario, requiring the borrower to post his valuable asset as collateral can mitigate the borrower’s incentive to engage in the risky action, and makes it possible for capital to be transferred from the lender to the borrower. 12 While our baseline model is similar to those of Tirole (2006) and Bolton and Oehmke (2014) as well as a simplified version of Boot, Thakor, and Udell (1991), in detail it differs from each. An important difference in our model is that collateral works as an incentive for the borrower to take less risk so that collateral relaxes the borrower’s incentive constraint, while in Tirole’s model collateral (net worth) works as an insurance for the lender so that collateral relaxes the lender’s participation constraint. Thus, in Tirole’s model, the borrowing capacity is determined by the outside value of collateral (riskless price that the collateral can be resold at in case of the borrower’s default), while in our model, it is determined by the inside value of collateral (the borrower’s private valuation of collateral). Key differences from Bolton and Oehmke (2014) are noted below.
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3.1
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We incorporate in this basic model the additional risk that the lender might be unable to return the borrower’s collateral, and show that if the risk is too high, the borrower will be reluctant post his asset as collateral at all, and borrowing will not occur. The subsequent section extends the two-player into a three-player model in which we examine rehypothecation.
A Two-Agent Model
aRA + (1 − a)(pA RA + b)
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There are two dates, (the “beginning date” and the “end date”)13 and two representative agents in the economy, a firm A and an outside investor B. Agents are risk-neutral and consume at the end date. For simplicity, the price of consumption good is normalized to be 1. Firm A is endowed with a (non-transferable) investment opportunity which requires an input at the beginning date to produce an output at the end date. We assume that the outcome of the investment is uncertain and takes one of two values; if the investment succeeds, the investment produces RA > 1 units of good per unit of inputs and if it fails, it produces zero units. The outcomes are costlessly observable by the outside investors. The probability of the success of the borrower’s investment depends on his hidden action. We assume that A can choose either a safe action or a risky action. The safe action leads to success with certainty and the risky action leads to success with probability pA < 1. On the other hand, the risky action gives A a private benefit b > 0 (measured per unit of inputs). We let a be the indicator for choice of action by A (1 for safe action, 0 for risky action)14 thus the expected return of the investment is given by
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The investment needs input at the beginning date, but A is endowed with no input. The outside investor B is endowed with capital at the beginning date that can be used for funding A’s investment. For simplicity, we assume that A tries to borrow for his investment from B by issuing simple debt; A receives funding I from B at the beginning date by promising to pay a certain amount of his investment outcome X to B at the end 13
When we get to the model with rehypothecation we will denote the beginning date as “date 0” and the end date as “date 2” and add in an intermediate date 1. The intermediate date is redundant in the two-player case. 14 It can be shown that mixed strategies for a are never optimal.
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date (a promise that can only be carried out if the investment is successful). If he does not deal with A, B’s outside option is assumed to be worth J ≥ 0. We assume that the parameters satisfy the following two conditions:15 Assumption 1. min{RA , pA RA + b} > 1 > pA RA .
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The first inequality implies that A’s project is worth carrying out regardless of his action. The second inequality implies that, from the perspective of B, lending to A is profitable only if A takes the safe action. Assumption 2. RA − 1 < pA (RA − 1) + b.
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This assumption implies that when A invests with the funding from B, A will always find it profitable to take the risky action rather than the safe action for any given contract such that X ≥ I (or, equivalently, for any contract with a positive interest rate). To see this, notice that the left side represents A’s expected net surplus (per unit of the inputs) after paying out the investment cost to B if he takes the safe action and the right side represents A’s expected net surplus if he takes the risky action. Assumptions 1 and 2 together imply that in the absense of collateralization, A cannot borrow funds for his investment from B. Intuitively, this is because A will prefer to take the risky action after receiving funds from B, and with foreknowledge of this, B will then be unwilling to provide funds. Lemma 1. Under Assumptions 1 and 2, uncollateralized borrowing is not feasible.
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Thus we need to consider alternative arrangements. We assume that A is also endowed with a transferable, indivisible asset which is illiquid in the following two senses: First, it yields the consumption goods only at the end date. Second, it produces more output when it is in the hands of the initial owner, firm A, than in the hands of the outside investor, B. Specifically we assume that the asset yields Z units of the good if it is held by A at the end date, while it yields less than that, Z0 (< Z) units of the good if it is held by B at that time.16 15
For every b > 0 there is an open set of pairs (pA , RA ) satisfying these parametric restrictions strictly. In our model, we assume that the borrower has an indivisible asset whose value is not affected by the borrower’s action. In comparison, Bolton and Oehmke (2014) consider the case in which the borrower has a certain amount of assets which may yield high or low return depending on whether the borrower takes a safe or risky action, and depositing a part of these assets into the lender’s margin account (i.e. posting these assets as collateral) keeps the value of these assets stable and mitigates the borrower’s incentive to engage in risky action by lowering the amount of assets under his management. 16
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Consider an arrangement where A posts his asset to B as collateral. Where others investigating the usefulness of collateral have assumed the B is a perfectly reliable counterparty, we extend our analysis by including the possibility that B is also less than perfectly reliable and might lose A’s collateral. Let pB be the probability that B does not lose A’s collateral (this probability will be endogenized in the full model below). For simplicity, A is assumed to have all the bargaining power and makes a take-it-orleave-it offer to B. At the beginning date, A borrows funding I from B in exchange for delivering the collateral, and then A chooses his action a. If the investment succeeds, A makes the promised payment X to B, and if the investment fails, A defaults and B seizes the collateral (if B loses the collateral, A does not repay the loan). The appendix describes A’s choice of contract in general. In the case where B is perfectly reliable (pB = 1), the contract (I, X, a) offered by A solves the following maximization problem: max a(RA I − X) + (1 − a)[pA (RA I − X) + bI − (1 − pA )Z] I,X,a
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aX + (1 − a)[pA X + (1 − pA )Z0 ] − I ≥ J I, X ≥ 0
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The objective function is A’s expected utility. Constraint (IC) is A’s incentive constraint which implies that a is ex-post incentive compatible given the choice of the other terms of the contract. Constraint (P ) is B’s participation constraint. Lastly, constraint (R) is the resource constraint which implies that A cannot promise to pay more than the return of the project. Note that the term −(1 − pA )Z in the objective function and on the right side of the incentive constraint (IC) indicates the additional loss to the borrower from the potential loss of his collateral. The participation constraint (P ) also includes a term representing the compensation that the lender receives from the collateral in the event of a default. Finally, if Z = Z0 = 0, the problem reduces to borrowing in the absence of collateral.
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If Z0 and J and are sufficiently small, it is optimal to take the safe action and both (IC) and (P ) bind. For convenience we define B ≡ RA −
b 1 − pA
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by Assumption 2, B < 1. When B is positive, it can be interpreted as the pledgeable income of the borrower, the maximum amount that the lender can take out of the borrower’s investment (see Holmstr¨om and Tirole, 1998). B increases with increasing the return from the investment and with decreasing the (expected) gain from choosing the risky investment over the safe investment. First we consider the case where B is perfectly reliable. Proposition 1 (Contract Choice with Collateralized Borrowing). Suppose pA < 1, J ≤ Z J , and pB = 1. A chooses a collateralized contract in which and Z0 ≤ 1−p A (I, X, a) =
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Note that in this problem, as the value of the collateral to the borrower increases, the constraints are relaxed and the amount of investment increases. In this respect, the relevant value is the value to the borrower, not the lender, because the collateral is being used as an incentive, not a repayment. Roughly speaking, the need to maintain the incentives for safe behavior increases the collateral needed to back the borrowing. Note that depending on the details of the incentive problem, the value of the collateral to the borrower can be greater than or less than the promised repayment from the borrower.17 We will call the loan “overcollateralized” if Z > X and “undercollateralized” if the inequality is reversed. For small enough values of J, this depends solely on the sign of the quantity B: Corollary 1. Under the conditions of the previous theorem the chosen contract is overcollateralized (undercollateralized) if B is negative (positive). 17
If the purpose of collateral were to force the borrower to make a repayment ex post, then it would always be necessary to have the loan overcollateralized. Here, the incentive problem is not for ex post repayment (we assume that if the borrower has the funds he will make the promised repayment) but over the ex ante risk choices which determine whether the funds will be available.
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When B is less than perfectly reliable, then, the expected payoff to A becomes a[RA I − pB X − (1 − pB )Z] + (1 − a)[pA (RA I − pB X) + bI − (1 − pA pB )Z].
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The expected payment when successful is reduced from X to pB X, and these is now a cost associated with the loss of collateral due to B’s failure. The incentive constraint for A is modified accordingly. The payoff for B now takes into account the fact that he only receives payment from A if he does not lose A’s collateral. Thus the participation constraint becomes:
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The other constraints to the problem are unchanged. If B is sufficiently reliable, the contract approximates the case of pB = 1 described above. On the other hand if pB is low, in other words, if B loses collateral too frequently, A is reluctant to post his asset as collateral due to the concern of not getting it back. Again, A’s investment will not be undertaken even when collateralized borrowing is feasible. For example, suppose Z0 = 0 and J = 0. Then, the solution (I, X, a) takes the following form. Z pB Z , X= , a=1 (4) I= 1−B 1−B
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Collateral Chain: Rehypothecation Model
In this section, we extend the previous two-player model into a three-player model in which B repledges A’s collateral to the third party, C. This extended model describes how the same unit of collateral can be rehypothecated to support multiple transactions. As a preliminary result, using this model, we confirm that rehypothecation generates a 12
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trade-off. On the one hand, rehypothecation improves the provision of funding liquidity in the economy so that additional productive investments can be undertaken. To be specific, in our model, the intermediary B can raise funds for another productive investment by repledging A’s collateral to the third party C, even in the case in which B’s pledgeable income is not enough to back up the debt that B issues for funding his investment. On the other hand, rehypothecation may incur a deadweight cost when it fails, that is, when B (the intermediary who sits in the middle of the collateral chain) goes bankrupt having re-pledged A’s collateral to the third party, C. In that case, the collateral will be seized by C and cannot be returned to the initial owner, A, who puts a higher value on the asset than does C, leading to the misallocation of the asset. In our model, this misallocation of the asset originates from the following two frictions in the market. The first friction is a wedge between the agents’ valuations on the posted asset. In particular, we assume that the initial owner of the asset values his asset more highly than do the other agents in the economy. The second friction is a trade friction: the initial collateral owner, A, and the final cash lender, C, cannot trade on their own, but only through the intermediary, B. Therefore, in case of rehypothecation failure, these two asssumptions together lead to the misallocation of A’s asset to C.18 To facilitate analysis, throughout this section, we assume that B’s ex-post decision whether to rehypothecate or not is given. Under this assumption, we then calculate the total surplus in the case where B rehypothecates and in the case where B does not rehypothecate, and compare the welfare in these two cases to examine what benefits and costs rehypothecation produces in the economy. In section 5, we relax the assumption that B’s decision to rehypothecate is exogenously given, and see whether his decision aligns or conflicts with the ex-ante socially efficient decision. In other words, we ask if there are circumstances where B does not want to rehypothecate ex post when it is socially efficient to rehypothecate, or B prefers to rehypothecate ex post when it is inefficient to rehypothecate.
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4.1 4.1.1
The Model Players and Endowments
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We add an additional, risk neutral player C who also only consumes at the end date. We also add an additional, intermediate date to the model; now the dates are denoted 0, 1, 2. As before, we take the price of date 2 consumption good to be 1. As before A has an opportunity of an investment which requires funding at the beginning date 0 and produces outcome at the end date 2. As before, A is also endowed with a single unit of indivisible illiquid asset, which only yields goods at date 2, and B is endowed with a large amount of funding which is available at date 0 for input for A’s investment. At the intermediate date 1, B also has an investment opportunity which requires an immediate input at that time to produce an outcome at date 2. However, at date 1 B does not have capital that can be spent as an input for his project nor any other pledgeable assets.19 On the other hand, C does not have access to B’s investment, but has a large amount of capital available at the intermediate date that can be spent as an input for B’s investment. Thus, in order that B’s investment is undertaken, capital must be transferred from C to B. As in the previous section, we assume that B issues simple debt to borrow funds for his investment; B borrows funds from C at date 1 by promising to pay a part of his investment outcome to C at date 2. We make the following assumptions regarding B’s investment opportunity. With probability pB the project yields positive output RB > 0 (measured per unit of the investment cost) and with probability 1 − pB it yields no output. We assume that B’s investment is productive in the sense that the expected return of B’s investment is greater than the investment cost (both are measured per unit of inputs). Assumption 3. pB RB > 1 .
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B’s endowment at date 0 might not be storable until the next period when he has an investment opportunity, that is, there is a liquidity mismatch problem faced by B. 20 Note, therefore that we are not assuming that the purpose of the collateral in the BC loan is to affect B’s ex ante incentives, but only his ex post incentive to repay the loan. This simplifies the determination
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4.1.2
Sequential Contracts: Collateral Chain
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Suppose B is now allowed to repledge A’s collateral deposited in his account to borrow funds from C. This can help to raise funds for B’s investment, which otherwise cannot be funded on its own. By posting A’s collateral, B can assure C that he will make the payment to recover the collateral – if B does not recover the collateral, B cannot receive the payment promised by A. On the other hand, collateral provides an insurance to C against B’s default by allowing C to liquidate it to collect losses, if any.21 Formally, under rehypothecation, the same single unit of collateral is used to support more than one transaction, the contract between A and B at date 0 and that between B and C at date 1, thereby creating collateral chain in the economy. In this three period model, these two contracts arise sequentially, and the timing of the model proceeds as follows (also see Figure 1).
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• At date 0, A borrows funds for his investment, denoted by I1 , from B by pledging his asset, which is worth Z to himself and Z0 (< Z) to the others, and promises to pay a part of the investment outcome, denoted by X1 , to B, conditional B’s returning collateral to A. After entering the contract, A then chooses a ∈ [0, 1].
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• At date 1, B has an investment opportunity, and decides whether to repledge A’s collateral and undertake the investment, or keep A’s collateral to return it safely to A in the next period. If B decides to rehypothecate, B borrows funds for his investment, denoted by I2 , from C by repledging A’s collateral, and promises to pay a part of the investment outcome, denoted by X2 , to C for recovering A’s collateral from C.
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• At date 2, both A’s and B’s investment outcomes are realized and the contracts are executed according to the prearranged terms. If B pays X2 he repurchases A’s collateral from C, and then returns it to A to receive X2 . However, if B defaults, C seizes A’s collateral repledged by B, and it cannot be returned to A, while at the same time, A is also exempt from paying the loan X1 to B.
of the choice of amount of collateral compared to loan size, allowing us to focus on the conflict between A and B. A more general treatment would consider both effects of collateral on both loans. 21 We assume that B is unable to receive funding back from A unless the collateral is returned at the same time. Thus B will be unable to repay C unless the project succeeds. In a more general setting (see previous footnote) it will be worthwhile to consider the extent to which B could arrange “bridge financing” to use A’s repayment to finance the collateral redemption.
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B borrows I2 from C collateral
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Figure 1: Timing of Contracts
4.2
Model Solution
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Next, we solve for the contracting problem for this model. Throughout this section, we focus on the case in which B’s decision whether to rehypothecate A’s collateral at date 1 is exogenously given – we will endogenize this decision in the subsequent section. First, consider the case in which B does not rehypothecate A’s collateral at date 1. In that case, the model involves only one contract made between A and B, and effectively, it boils down to the previous two-player model. Next, consider the case in which B rehypothecates at date 1. In that case, the model has a sequence of two contracts, the date 0 contract between A and B and the date 1 contract between B and C. Formally, the date 0 contract between A and B is defined as a profile (I1 , X1 , a) where I1 represents the investment cost that A borrows from B by pledging his asset as collateral at date 0, X1 represents the payment promised by A to pay for getting back his asset from B at date 2, and a denotes the action taken by A. Similarly, the date 1 contract between B and C is defined by a profile (I2 , X2 ) where I2 represents the investment cost that B borrows from C by repledging A’s collateral at date 1 and X2 represents the payment promised by B to repurchase A’s collateral from C at date 2. First, we solve for the contracting problem between B and C at date 1 by taking the date 0 contract between A and B as given, and then we solve for the contracting problem between A and B at date 0. 16
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4.2.1
Contracting Problem between B and C at date 1
For any given contract between A and B at date 0, (I1 , X1 , a), the contracting problem between B and C is to choose (I2 , X2 ) which solves the following problem.
I2 ,X2
subject to I1 ≤ pB X2 + (1 − pB )Z0
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max pB [RB I2 − X2 + aX1 + (1 − a)(pA X1 + (1 − pA )Z0 )]
X2 ≤ aX1 + (1 − a)(pA X1 + (1 − pA )Z0 )
(P1 )
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The objective function is B’s expected utility when he makes the investment by repledging A’s collateral. With probability pB , the investment returns RB I2 , and B pays a part of the return X2 to repurchase A’s collateral from C, and then delivers this collateral to receive X1 from A. The participation constraint (P1 ) states that C’s utility must be at least the reservation value, 0. The right side of (P1 ) is the expected revenue from lending: C receives X2 from B with probability pB and Z0 by seizing the collateral if B defaults with probability 1 − pB . The left side of (P1 ) is the cost I2 that C provides to B’s investment. The last constraint (R1 ) is the resource constraint which implies that B’s promise, X2 , cannot be greater than what B is going to earn by recovering the collateral from C, the expected payment that B would receive from A by returning the collateral: B receives X1 with probability a + (1 − a)pA and Z0 by liquidating the collateral if A defaults with probability (1 − a)(1 − pA ). Then, Assumption 3 and the linearity of the problem ensure that in both cases, all the constraints are binding in optimum. Solving these constraints simultaneously, we can write the contract at date 1 as a function of the date 0 contract.
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Lemma 2. Suppose Assumption 3 holds. Then, at optimum, the contract at date 1 satisfies I2 = apB X1 + (1 − a)(pA pB X1 + (1 − pA pB )Z0 ) X2 = aX1 + (1 − a)(pA X1 + (1 − pA )Z0 ) In other words, B passes the collateral along to C. If B does not fail, it pays X1 , the expected value of the payment by A. If B fails, C retains the collateral valued at Z0 . The 17
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weighted average of these two quantities is the amount that C lends to B. 4.2.2
Contracting Problem between A and B at date 0
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Moving backward, we consider the contracting problem between A and B at date 0. First, we consider the case in which A takes the safe action. To facilitate analysis, we substitute the results in Lemma 2 into the objective function of the date 1 problem, so that B’s expected utility can be written as a function of (I2 , X2 ) as follows: pB (RB I2 −X2 +X1 )−I1 = pB RB [pB (a+(1−a)pA )X1 +{1−pB (a+(1−a)pA )}Z0 ]−I1 (6)
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Again, we assume that A has all the bargaining power and makes a take-it-or-leave-it offer to B, and if B rejects the offer, he receives the reservation value, J. The contracting problem at date 0 becomes the following: a(RA I1 − pB X1 − (1 − pB )Z) + (1 − a)[pA (RA I1 − pB X1 ) + bI1 − (1 − pA pB )Z] (7)
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a ∈ arg max a(RA I1 − pB X1 − (1 − pB )Z) + (1 − a)[pA (RA I1 − pB X1 ) + bI1 − (1 − pA pB )Z] (ICA )
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pB RB [pB (a + (1 − a)pA )X1 + {1 − pB (a + (1 − a)pA )}Z0 ] − I1 ≥ J
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(P0 ) (R0 )
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The objective function represents A’s expected utility. The incentive constraint for A, (ICA ), the participation constraint for B, (P0 ), and the resource constraint, (R0 ), are modified taking into account the fact that B re-uses collateral at date 1. We make the following two parametric assumptions on pB and RB : Assumption 4. max{pB 2 RA RB , pB RB (pA RA + b)} > 1 > pA pB 2 RA RB . Assumption 4 corresponds to Assumption 1. It implies that it is efficient to undertake the project regardless of A’s action, but from the perspective of B, providing funds for A’s project is profitable only if A takes the safe action. For example, suppose A’s collateral 18
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Assumption 5. RA − pB RA < pA (RA − pB RA ) + b.
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is worthless to B and C, that is, Z0 = 0. In such case, the maximum amount that B can promise to C by repledging A’s collateral is pB RA if A takes the safe action, and pA pB RA if A takes the risky action (both measured per unit of inputs). Then, the total revenue to B is pB RA × pB RB > 1 if A takes the safe action, and pA pB RA × pB RB < 1 if A takes the risky action, implying that B provides funding to A only if A takes the safe action.
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Assumption 5 is the same as Assumption 2, except that the quantity 1 on both sides of Assumption 2 is now replaced by pB RA .22 As before, pB RA is the maximum level of A’s payment (measured per unit of inputs) if A takes the safe action. Assumption 5 can be interpreted by considering that if the value of the asset to A were to approach zero,23 A would want to take the risky action rather than the safe action whenever he invests with the borrowed money. Then, using the same technique as in the previous section, we show that if Z0 and J are sufficiently small, it is optimal to take the safe action and both the incentive constraint (ICA ) and the participation constraint (P0 ) bind.
pB RB [pB Z + (1 − pB )Z0 ] − J Z + (1 − pB )BRB Z0 − (pB /B)J , ,1 1 − BpB RB 1 − BpB RB
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and J ≤ min{J 0 , J 00 }. Then, at
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Lemma 3. Suppose pA < 1 and Z0 ≤ optimum, the date 0 contract satisfies
h p 2R R − 1 i 1 B A B pB Z + (1 − pB )pB RB Z0 1 − BpB RB pB RA − B −1 h i 1 1 pB Z 00 J ≡ pB R B + + (1 − p )Z B 0 pB 2 RA RB − 1 1 − BpB RB 1 − BpB RB
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J0 ≡
Taken all the results together, the equilibrium outcome is characterized as follows:
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Note that Assumption 5 is implied by Assumption 2 if pB RA > 1. For example, think of the collateral posted by A as the letter of claim of the debt, which B must hold in order to be able to collect the debt from A, but the letter itself is unlikely to generate other private values for A. 23
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Proposition 2. When pairs of agents choose contracts sequentially and rehypothecation is permitted, the equilibrium contracts are p R [p Z + (1 − p )Z ] − J Z + (1 − p )BR Z − (p /B)J B B 0 B B B B B 0 , ,1 , (I1 , X1 , a) = 1 − BpB RB 1 − BpB RB
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(I2 , X2 ) = (pB X1 + (1 − pB )Z0 , X1 ).
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Note that the scale of both the borrower’s and the intermediary’s investments, I1 and I2 increase as collateral is more valuable – that is, Z or Z0 increases. Also those quantities are increasing in the productivity of the intermediary’s project, RB , and the probability of success of the intermediary’s investment, pB .
Welfare Analysis of Rehypothecation
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In this section, we evaluate social welfare with and without rehypothecation, and compare them to examine the benefits and costs of rehypothecation in the economy. Rehypothecation has the following fundamental trade-offs. On one hand, it helps to provide more funding liquidity to the economy so that additional productive investment can be undertaken, by enabling the limited amount of collateral to back multiple transactions. On the other hand, it introduces the risk of “rehypothecation failure,” the risk that the intermediary might default having repledged his borrower’s collateral to a third party. One potential cost associated with this failure of the intermediary is the deadweight cost of misallocating the asset. This misallocation of the asset arises from illiquidity and trading frictions; the borrower is likely to put a higher value on his assets than his creditor, and they need the intermediary to transfer collateral between them. Thus, in case that the intermediary fails collateral cannot be returned to its highest use. Formally, the social welfare with rehypothecation, denoted by WR , is defined by the sum of A’s, B’s and C’s utility. Using the notation in the previous sections, it can be written as follows. WR ≡ [RA I1 −pB X1 +pB Z−Z]+[pB RB I2 −pB X2 +pB X1 −I1 ]+[pB X2 +(1−pB )Z0 −I2 ]. (8)
where (I1 , X1 , I2 , X2 ) are from Proposition 2.
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Simplifying, WR = (RA − 1)I1 + (pB RB − 1)I2 − (1 − pB )(Z − Z0 )
(9)
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Note that the social welfare consists of the following three components: (i) the surplus generated from A’s investment, (RA − 1)I1 (ii) the surplus generated from B’s investment, (pB RB − 1)I2 , and (iii) the cost generated from the misallocation of the asset in case of rehypothecation failure, (1 − pB )(Z − Z0 ). On the other hand, the social welfare without rehypothecation, denoted by W0 , consists of A’s utility and B’s utility only (since the transaction between B and C cannot happen). Since in this case B never fails to return the collateral, the social welfare without rehypothecation is simply the surplus from A’s investment, W0 = (RA − 1)I
(10)
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where I is from Proposition 1. Suppose A allows B to re-use collateral whenever B wants to do so.24 Then, comparison of equations (9) and (10) shows that rehypothecation allows B’s project to be funded, (pB RB − 1)I2 , but it incurs the cost of misallocating collateral, (1 − pB )(Z − Z0 ), due to the risk that B might not return collateral with probability of 1 − pB .25 To summarize, the efficiency of rehypothecation is determined by the relative size of these trade-off effects. Rehypothecation enhances the provision of funding liquidity to the system so that additional productive investment can be undertaken, but it introduces the counterparty risk which incurs the cost of misallocating collateral.
5
Conflict between Agents’ Incentives to Rehypoth-
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We have so far assumed that the agents’ decisions on the rehypothecation arrangement are exogenously determined. Now we consider the possibility that B can choose whether 24
In the following section, we will consider the possibility that A has the option of prohibiting B from re-pledging collateral. 25 The size of A’s project may also vary; A’s project may be either larger or smaller when rehypothecation is allowed.
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Over-Collateralization and Excessive Rehypothecation
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to rehypothecate A’s collateral or not: B compares the payoff if he rehypothecates A’s collateral to the payoff if he simply holds on to it until when A makes the payment for it. Then we examine whether this decision made by B is aligned with A’s objectives, that is, whether B’s decision is also efficient ex ante. In what follows, we build examples where A’s and B’s objectives are in conflict. In the first example, B prefers to rehypothecate when rehypothecation is inefficient from A’s perspective, and thus the social welfare will be improved if A can preclude rehypothecation in advance (provided that A has complete information about B’s project). In the second example, B does not rehypothecate even when rehypothecation is efficient from A’s perspective, and thus the welfare will be improved if B can commit to rehypothecate in advance. The direction of this conflict between A and B depends on whether the contract between them involves over-collateralization or under-collateralization, that is, whether A’s private value of collateral, Z, is greater or smaller than the payment for recovering it from B, X1 . With over-collateralization, B tends to be excessively eager to rehypothecate. Intuitively, this is because collateral is worth only up to the payment from A, X1 , from B’s perspective, while it is actually worth Z(> X1 ) to A. However, B does not internalize A’s value on his asset, and thus B tends to want to rehypothecate too frequently. With under-collateralization, B tends to be excessively cautious to rehypothecate for the opposite reason. Intuitively, the collateral is worth X1 from B’s perspective, which is more than its value to A, and thus B does not want to rehypothecate even though rehypothecation may be a desirable gamble from A’s perspective.
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In this subsection, we build an example where over-collateralization induces B to excessively participate in rehypothecation from A’s perspective. The setting in this example is the same as in the previous analysis. For simplicity, we set B’s reservation value to be J = 0 and the external value of collateral to be Z0 = 0. We make the following additional assumptions. Assumption 6. B ≡ RA −
b 1−p
< 0 and 1 − B < (1 − B)pB 2 RB < 1 − BpB RB .
The following result shows that if this assumption holds, a conflict arises between the interests of A and B. 22
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Proposition 3. Suppose Assumption 6 holds and set Z0 = J = 0. Then, B prefers to rehypothecate although it is undesirable from A’s perspective.
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b < 0, combined To see this, note that the first inequality of Assumption 6, B ≡ RA − 1−p with the binding incentive constraint (ICA ), implies that the equilibrium contract involves over-collateralization. The second inequality of Assumption 6, 1 − B < (1 − B)pB 2 RB , implies that B wants to rehypothecate. Finally, the last inequality, (1 − B)pB 2 RB < 1 − BpB RB implies that rehypothecation is undesirable for A.
Under-Collateralization and Insufficient Rehypothecation
Assumption 7. B ≡ RA −
b 1−pA
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Now we build an example where under-collateralization leads B to be unwilling to rehypothecate, although rehypothecation is desirable from A’s perspective. Again, the setting in this example is the same as in the previous analysis and we set Z0 = J = 0. For this example we make the following assumption regarding parameters: > 0. and 1 − BpB RB < (1 − B)pB 2 RB < 1 − B.
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Compared to the over-collateralization example of the previous subsection, this assumption changes the direction of the conflict between A’s and B’s interests as follows.
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Proposition 4. Suppose Assumption 7 holds and set Z0 = J = 0. Then, B does not want to rehypothecate though rehypothecation is efficient from A’s perspective.
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This result can be interpreted in a similar way as in the case of over-collateralization. b > 0, combined with the binding inThe first inequality of Assumption 7, B ≡ RA − 1−p A centive constraint (ICA ) implies that the optimal contract involves under-collateralization. The second inequality of Assumption 7, (1 − B)pB 2 RB < 1 − B, implies that B does not want to rehypothecate. Finally, the last inequality, 1 − BpB RB < (1 − B)pB 2 RB , implies that rehypothecation is not desirable for A. To summarize, these examples show that if the contract involves over-collateralization, the intermediary tends to be too eager to rehypothecate, while the contract involves undercollateralization, it tends to be too cautious to rehypothecate. The inefficiency due to the intermediary’s over-eagerness to rehypothecate might be mitigated relatively easily if the initial borrower has the right to preclude rehypothecation in advance. However, the inefficiency due to excessive reluctance to rehypothecate will still remain without an institutional device that induces rehypothecation. In any event, if 23
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6
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6.1
Heterogeneity of Valuations
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the borrower’s information about the intermediary’s investment opportunity is imperfect, the borrower’s decision on the rehypothecation arrangement may not achieve an optimal outcome in either case. For example, if the return of B’s project, RB , or, the probability of B’s default, 1 − pB , are private information to B, and it is costly for A to learn these, then A might prevent socially efficient rehypothecation from taking place unnecessarily, due to the concern of losing collateral.
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Rehypothecation has costs and benefits. The costs we have addressed here stem from the fact that rehypothecation increases the chances that the financial asset ends up in the “wrong hands”—in other words, that the agent who ends up with the asset after counterparty failure actually values it less than the initial borrower. In the case of financial assets there are several possible sources for this difference in valuations: First, different individuals may start from different non-tradable portfolios,26 or hold different, irreconcilable views about the future values of assets.27 Differences in the transactions costs of resale faced by the various parties can play an equivalent role; this role is considered explicitly in Kahn and Park (2016).28 For otherwise highly liquid assets another possible source of diffence in valuations is legal uncertainty—the title to the asset may be encumbered by uncertainty in the process of hypothecation and rehypothecation across jurisdictions. Finally, the difference in valuation may ultimately be attributed to differing liquidity needs: the initial borrower may be constrained in her ability to acquire additional pledgeable assets and therefore place an above-market premium on their return. Not surprisingly, if the agents’ valuations on collateral are less heterogeneous, the cost of misallocation of collateral will decrease, and rehypothecation will be more efficient. On the other hand, our main result, a conflict of interests between the contracting parties 26
See, for example, Brunnermeier and Pedersen (2009). Geanakoplos (2010) analyzes the leverage cycles in a market in which agents have different beliefs about the future returns of collateral asset. Based on this framework, Simsek (2013) develops a model in which lenders do not value the collateral as much as borrowers do, and show how the nature of the borrower’s optimism affects the tightness of the borrowing constraint. Finally, He and Xiong (2012) investigate how the dynamics of agents’ heterogenous beliefs generate a fluctuation of the asset price and prices of the contract. 28 Boot, Thakor, and Udell (1991) emphasized costs of dealing with collateral. 27
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Segregated Accounts
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about rehypothecation, does not rely on heterogeneous valuations; rather it arises from the incentive role of collateral. For example, as long as the size of the haircut is set in order to solve the borrower’s incentive problem, the promised payment will differ from the collateral value, even if the intrinsic value of collateral is the same across all the agents. If the amount promised to be paid to the intermediary is greater than the value of collateral, then the intermediary is likely to be too cautious about rehypothecation from the borrower’s perspective. If the lender could somehow commit to a more aggressive rehypothecation policy, a superior loan contract could be achieved between him and the borrower.29
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As a protection against the loss of collateral in a counterparty failure, collateral is often sequestered into a separate protected account with the intermediary, into which account assets can be replaced by other assets in the event that they are used for rehypothecation. However this practice varies across markets and countries. For example, in the US, assets segregated under SEC Rule 153-3 can be deposited at either (1) an affiliated clearing entity of the prime broker or (2) a third-party custodian bank. The choice has the following trade-offs. The custodian bank does not lend nor reuse the assets, but requires a custody fee for keeping the assets. In contrast, the prime broker does not charge a custody fee, but keeps the clients’ assets in an omnibus account (pooled with other clients’ assets) and they can use these assets with the written consent of the clients.30 If the prime broker uses these assets from the omnibus account, it must maintain a number of assets in the omnibus account with the sum of fully paid securities by SEC Customer Protection Rule. Hedge funds now increasingly demand their prime brokers provide more protections on their assets by requesting segregation at other custodian banks or limiting the amount of their assets permitted to be reused.31 However customer protection for rehypothecated assets has been much less clear in the United Kingdom and continental Europe (Singh and Aitken, 2010). It is for this reason that Lehman transfered many customer assets to its UK affiliate in the years leading up to the financial crisis. 29
If the haircut has the opposite sign, the direction of the conflict will go into the other way. If clients want complete segregation of their assets, they need to request their prime broker to open an individual account. But, this requires some fees, and was not used by most clients. 31 See Carlsson-Sweeny (2010) for more information on the recent changes in prime brokerage agreements. 30
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While segregated accounts considerably mitigate the problem of misallocation, they do not eliminate it. First of all, as we have seen, financial institutions find legal workarounds and sometimes have failed to honor the requirements for segregation—a fact only discovered after failure. Secondly, in situations where the substitution of one asset for another is possible within the segregated account, in principle the substitutions may not be regarded as equivalent by the initial borrower, reestablishing the problem in miniature. For example, there are accounts of financial institutions routing different qualities of asset for rehypothecation in bespoke or triparty markets.32
Implications for Stability and Systemic Risk
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Throughout this paper, we have assumed that the lender’s default risk is constant. However in the real world default risk varies over time for individual firms, and varies systematically with the business cycle. Borrowers will wish to prohibit rehypothecation when the lender’s credit rating drops, in turn making borrowing more expensive. Thus rehypothecation can act as another accelerator in financial crises: when a borrower blocks the lender’s ability to rehypothecate because of doubts about its credit rating, he makes it more difficult for the lender to borrow funds, further damaging the lender’s credit rating. In our model, collateral chains are short, containing at most two links. This mirrors the institutional arrangements often seen with formal rehypothecation, in which the middleman is in a specialized position as both lender of cash and lender of securities to less-specialized clients. However, as noted at the beginning, other forms of collateral reuse are also common in the financial system, and these can entail longer collateral chains. As the chains become longer, the reinforcement effects noted are likely to become more severe. While the cessation of rehypothecation can reinforce a financial downturn, it can also lead to a faster recovery. Park (2017) addresses this issue in a dynamic extension of the model, examining how borrowers’ responses to shocks affect the size of the downturn and the speed of recovery after the shock. In that model, if borrowers prohibit rehypothecation after shocks, output drops more severely, but further potential misallocations of capital are reduced. For this reason, the distribution of collateral is improved and recovery of the 32
“Think of the bilateral repo market as analogous with the old-clothing trade...Big lots come in from hedge funds and security lenders, and the collateral desks at large banks will paw through it, searching for gems. Those gems go out bilateral to customers who will pay a premium,. The remainder goes to the guys in the back of the line: tri-party repo.” (Singh, 2014, p. 9)
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economy occurs more quickly. Conversely, if rehypothecation continues after the shock, the immediate output drop is less severe, but the recovery is slower. Park’s extension also provides further systemic implications of rehypothecation. In her framework, prolonged periods of good state (boom) lead to too much rehypothecation, increasing the output in the short run, but accumulating misallocated collateral and concomitant reductions in long-run output.
Alternative Solutions
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In this section we briefly consider alternative arrangements to solve the problem of guaranteeing a sequence of loans with limited collateral available. For financial collateral, such as used in repo transactions, it is typical that a borrower hands it over to a lender at the inception of the contract. Probably the most important reason for this is to save costs of the collection process in case of the borrower’s bankruptcy. For example, in mortgage transactions, upon the debtor’s declaration of bankruptcy, collateral is frozen under the bankruptcy law (an automatic stay goes into effect). This exposes the lenders to the risk of sale at uncertain prices since they have to wait for a longer time for liquidating collateral. In this respect, transfering collateral to the lenders earlier can shorten this time of the collection process. The repo arrangement has the legal effect of pulling collateral out of the bankruptcy proceedings by giving legal ownership to the creditor. This makes the transfer in case of debtor default much easier, but puts the debtor at greater risk in the case of creditor default. One possibility would be for the initial borrower to issue her own transferable debt. Then when the intermediary needs to raise funds he can sell this debt to a third party.33 In practice, resale of debt exists in some financial markets and not in others: bonds are resellable, repos are not. Using transferable debt requires that third parties know the creditworthiness of the initial borrower, information that is likely to be part of the expertise of intermediaries but not of the typical participant in these markets (often a nonfinancial corporation lending its temporary excess funds); nor would it be in the interest of the intermediary to give an accurate description of the initial borrower’s creditworthiness to anyone to whom he was attempting to sell the debt. Using financial assets as collateral 33
Such an arrangement is similar to using transferable debt as “inside money” (Kahn and Roberds, 2007).
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Conclusion
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reduces (although does not eliminate) the difficulties of determining values.34 The inefficiencies in the model arise from the inability of the initial borrower to redeem the collateral from the final creditor. In the model we have simply assumed that the two parties never meet; in Kahn and Park (2016) we assume that they can meet in an open market but are subject to a transactions cost in that market. This assumption is motivated by observations that there are trading frictions in the markets, such as over-the-counter (OTC) markets in which repo and security lending transactions take place.35 Nonetheless, it would seem to be the case that arrangements which facilitate meetings between individuals attempting to reallocate the collateral would be obvious ways to increase the efficiency of these markets. One natural institutional arrangement to consider in this context is the establishment of centralized counterparties (CCPs) to serve as clearinghouses for these sorts of trades. CCP’s are an important part of the proposals for reform of the financial system post crisis. The Dodd-Frank Act mandates central clearing of many swaps; in this case collateral will be held by a central counterparty. On the one hand, this precludes direct re-use of collateral. On the other hand, the arrangement makes it easier to transfer positions to third parties. Whether a CCP represents a superior solution depends on the degree to which terms of the contract between parties can be made independent of private information about counterparties. Information on counterparties, including the collateral terms that are optimal for particular trades with them, is likely to be part of the comparative advantage of prime brokers. Thus a move from OTC to centralized arrangements is likely to generate both losers and winners among customers.36
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We develop a model of the economic underpinnings of rehypothecation. Rehypothecation is an example of the re-use of the same collateral to support multiple transactions. Rehypothecation helps to provide more funding liquidity into the system by allowing the 34 See Donaldson and Micheler (2016) for a further discussion of the consequences of the differences in asset resellability. 35 Notable papers that build on frictions in the OTC markets are Duffie, Gˆ arleanu and Perdersen (2005) and Lagos and Rocheteau (2009). Edwards et al. (2007) provide empirical evidence of the existence of transaction costs in OTC markets for corporate bonds. 36 If agents A and C cannot meet directly, a second important alternative to rehypothecation is asset securitization. Roughly speaking, securitization lowers the problem of misallocation but reduces C’s protection. See Maurin (2014) for further discussion.
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lender to use her borrower’s collateral, which would otherwise sit idle in the lender’s account, for another productive investment. However, it can incur a deadweight cost due to misallocating the asset when the lender fails having repledged the borrower’s collateral to the third party to whom the collateral is less likely to be as valuable to the initial owner. We show that the individuals’ incentives to participate in rehypothecation may not be aligned with economic efficiency. In other words, cases arise in which agents choose socially inefficient levels of rehypothecation. Importantly, the direction of the inefficiency depends on terms of the agents’ contracts and ultimately as the incentive problems that those contracts are designed to solve. If the contract involves over-collateralization, the intermediary tends to be overly eager to rehypothecate; if the contract involves undercollateralization, the intermediary tends to be overly cautious to rehypothecate. As noted above, Park (2017) investigates the output dynamics in an extension of this model in which rehypothecation endogenously arises and varies with macroeconomic conditions. In this model, she derives initial implications regarding the systemic effects, in terms of magnification of shocks to creditworthiness or speed of recovery of output. However other natural extensions are worth consideration. Thus far we have not incorporated insurance motives, random fluctuation in the value of the collateral, and more general effects of aggregate and idiosyncratic shocks. These important practical considerations are left for future research.
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References Agarwal, S., Green, R. K., Rosenblatt, E., and Yao, V. (2015). Collateral pledge, sunkcost fallacy and mortgage default. Journal of Financial Intermediation, 24(4):636–652.
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A
Proofs
By the revelation principle the optimal contract is a triple (I, X, a) solving the following maximization problem: max a[RA I − pB X − (1 − pB )Z] + (1 − a)[pA (RA I − pB X) + bI − (1 − pA pB )Z] (A.11)
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Is ,Xs ,a
subject to
a ∈ arg max a[RA I − pB X − (1 − pB )Z] + (1 − a)[pA (RA I − pB X)+bI − (1 − pA pB )Z] a∈{0,1}
(IC) (P )
RA I ≥ X
(R)
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pB [aX + (1 − a)(pA X + (1 − pA )Z0 )] − I ≥ J I, X, J ≥ 0
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The objective function is A’s expected utility. The moral hazard problem restricts a to values which are ex-post incentive compatible given the choice of the other terms of the contract, this is restriction (IC). Constraint (P ) ensures that B’s expected profit from lending be at least equal to his reservation value J. Lastly, constraint (R) says that A cannot pay more than the return from the investment, RA I (if the investment fails, A makes no payment to B).
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CE
PT
Proof of Lemma 1 . In the case where pB = 1, (IC) is equivalent to the following three conditions: a ∈ {0, 1}
(A.12)
and if a > 0 then (1 − pA )(RA I − X) ≥ bI
(A.13)
and if a < 1 then (1 − pA )(RA I − X) ≤ bI
(A.14)
First, consider the case where a = 0. Then [1 + (1 − a)pA ]X ≤ I, so that (P ) can only be satisfied when X = I = J = 0. By (A.13),
b I RA − 1 − pA 34
≥X
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and by Assumption 2 RA − therefore
b <1 1 − pA
On the other hand, if a = 0, the requirement reduces to pA X ≥ I but I ≥ X/RA ≥ pA X
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so again X = I = J = 0.
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I ≥ X.
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Proof of Proposition 1 . There is not feasible contract in which a = 0 and borrowing takes place. If it did then it must be the case that pA X ≥ I. But by Assumption 1 this is possible only if I = X = J = 0. That leaves only the case a = 1 to examine. In this case we can simplify the problem as follows: max RA I − X I,X
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PT
ED
subject to
BI ≥ X − Z
(λ)
X ≥J +I
(µ)
RA I ≥ X
(ν)
I, X ≥ 0
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where the letters to the right of the constraints denote the Lagrange multipliers, all non negative. Taking the first order conditions we have µ − Bλ = RA (ν + 1) µ−λ=ν+1
From the latter condition we conclude µ is positive, and since RA > 1, combining the
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conditions we conclude that (1 − B)λ > 0
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or λ > 0. Since 1 − B > 0 by Assumption 2, both conditions (λ) and (µ) hold with equality at an optimum. Solving these conditions for X and I yields the formulas in the statement of the proposition. The restrictions on the relation between Z0 and J in the statement of the proposition guarantee that X and I satisfy the remaining inequalities in the problem.
I1
ED
subject to
J 1 (pB RA RB − 1)I1 − (1 − pB )(Z − Z0 ) − pB RB pB RB
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max
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Proof of Lemma 3 . If a = 0, there is no feasible contract in which borrowing takes place. If it did then it must be the case that pB RB [pA pB X1 + (1 − pA pB )Z0 ] − J ≥ I, and by the conditions on Z0 , this implies that it must be the case that pA pB 2 RA RB X ≥ I. But, by Assumption 4, this is possible only if I1 = X1 = J = 0. Now we examine the case a = 1. There are two possible cases. (Case 1) If RA > pA RA + b, (P0 ) must bind in any optimum. Suppose not. Increasing I1 until this is the case; it improves the objective without violating any constraints. Thus we can simplify the problem as follows:
pB RB [pB Z + (1 − pB )Z0 ] − J 1 − BpB RB J − pB RB (1 − pB )Z0 I1 ≥ pB 2 RA RB − 1 I1 ≤
PT
(ICA )
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where BpB RB < 1 by Assumption 4. −1 Since J ≤ pB RB pB 2 RA1RB −1 + 1−Bp1B RB
pB Z 1−BpB RB
(R)
+ (1 − pB )Z0 , the set of I1 that
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satisfy these constraints is non-empty and (ICA ) binds. Thus, in this solution, (I1 , X1 ) =
pB RB [pB Z + (1 − pB )Z0 ] − J Z + (1 − pB )BRB Z0 − (pB /B)J , 1 − BpB RB 1 − BpB RB
(Case 2) If RA < pA RA + b, either (P0 ) or (ICA ) binds in optimum. If (P0 ) binds, this
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reduces to (Case 1). If (ICA ) binds, we can rewrite the problem as follows. max (RA − B)I1 − Z I1
subject to
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pB RB [pB Z + (1 − pB )Z0 ] − J 1 − BpB RB pB Z I1 ≥ pB RA − B I1 ≤
(R)
i Since J ≤ + (1 − pB )pB RB Z0 , the set of I1 that satisfy these constraints is non-empty and (PB ) binds. Thus, the solution is the same as in (Case 1) above. pB 2 RA RB −1 pB Z pB RA −B
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1 1−BpB RB
h
(PB )
Proof of Proposition 3 . Recall that the incentive constraint (ICA ) binds at optimum, RA I1 − pB X1 − (1 − pB )Z = pA (RA I1 − pB X1 ) + bI1 − (1 − pA pB )Z
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which reduces to,
X1 = pB −1 BI1 + Z.
pB 2 RB X1 > X1 .
(A.15)
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PT
ED
b Since B ≡ RA − 1−p < 0 by Assumption 6, X1 < Z, i.e., the contract involves overA collateralization. Now we shows that B wants to rehypothecate at date 1 if and only if
AC
where the left side is B’s payoff with rehypothecation and the right side is the payoff without rehypothecation when Z0 = 0. This inequality holds by the second inequality of Assumption 6. On the other hand, A does not want to rehypothecate. If Assumption 6 holds, the following must hold, RA I − X > RA I1 − pB X1 − (1 − pB )Z (A.16) where I, X are from Proposition 1 and I1 , X1 are from Proposition 2. The left side is A’s payoff without rehypothecation and the right side is the payoff with rehypothecation.
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By substituting X and X1 with a function of I and I1 , respectively, we can rewrite it as follows. (RA − B)I − Z > (RA − B)I1 − Z.
(A.17)
pB 2 RB Z Z > 1−B 1 − BpB RB by the last inequality of Assumption 6.
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Substituting I in Proposition 1 and I1 in Proposition 2 into this inquality, we obtain, (A.18)
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Proof of Proposition 4 . Note that the second inequality of Assumption 7 is equivalent to pB 2 RB < 1.
Since X1 is positive and Z0 = 0, this is equivalent to
(A.19)
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pB RB (pB X1 + (1 − pB )Z0 ) < X1 .
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ED
The left side, B’s payoff with rehypothecation, is smaller than the right side, the payoff without rehypothecation. On the other hand, the second inequality of Assumption 7 implies that pB 2 R B Z Z . < 1−B 1 − BpB RB
(A.20)
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Using the results in Proposition 1 and Proposition 2, this can be rewritten as follows. RA I − X < RA I1 − pB X1 − (1 − pB )Z
(A.21)
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The left side, A’s payoff without rehypothecation, is smaller than the right side, the payoff with rehypothecation.
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