How Should Central Counterparty Clearing Reduce Risk? Collateral Requirements and Entry Restrictions.∗ Jean S´ebastien Fontaine, Hector Perez-Saiz and Joshua Slive Bank of Canada November 2014
Abstract We analyze collateral requirements and entry restrictions by a Central Counterparty (CCP) in over-the-counter (OTC) markets where a few dealers trade with hedgers. More dealers increase competition but competition increases default risk. Collateral reduces the default probability but collateral is costly. Based on total welfare, a CCP should typically use collateral and avoid entry restrictions. Collateral requirements are conservative only if dealers transfer risk efficiently: the required level of collateral is too costly otherwise. Restricting entry and lowering collateral is desirable only if collateral is expensive and dealers are efficient. Dealers favor entry restrictions and lower collateral to raise their mark-up.
∗
The opinions and conclusions expressed herein are those of the authors and do not necessarily represent the views of the Bank of Canada or its staff. We thank for comments and suggestions Toni Ahnert, Jason Allen, Narayan Bulusu, Santiago Carbo, James Chapman, Alexander David, Darrell Duffie, Rod Garratt, Corey Garriott, Hanna Halaburda, Fr´ed´eric Hervo, Thorsten Koeppl, Yaron Leitner, Albert Menkveld, David Martinez Miera, Cyril Monnet, James Moser, Adolfo de Motta, Thomas Nellen, Tiago Pinheiro, Lukasz Pomorski, Enchuan Shao, Joseph Schroth, Haoxiang Zhu and participants at the Bank of Canada Workshop on Financial Intermediation and Market Dynamics, the Bank of Spain, the CEA 2012 conference, the Dutch National Bank conference on financial infrastructures, ESEM 2012, IBEFA 2012, the Federal Reserve Bank of Chicago, the Joint Federal Reserve Board-Bank of Canada Economics of Payments V Conference, the joint Bank of England-Banque de France-ECB conference on OTC derivatives and Mcgill University. We also thank Glen Keenleyside for editorial assistance. An earlier version of this paper was titled “Dealers’ Competition and Control of a Central Counterparty ”.
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Introduction Imperfect competition and default risk are two salient features of Over-the-Counter (OTC)
markets. First, a small number of large global dealers offer customized and differentiated intermediation services. Second, counterparty default risk is prevalent for swaps and derivatives contracts. This concentration of default risk in the hands of few dealers exacerbated the 2007–09 financial crisis (Duffie, 2010a), and led to an increased use of central counterparties (CCPs).1 The CCP uses membership restrictions and collateral rules to reduce default risk but at the costs of reduced competition in the OTC market. We provide an equilibrium framework to analyze the trade-off between default risk and competition in the OTC market when choosing CCP rules. A CCP clears and novates all trades, becoming a party to every trade – acting as a buyer to every seller, as a seller to every buyer – and absorbing counterparty risk. CCPs have far-reaching effects. For instance, CCPs enhance multilateral netting (Duffie and Zhu, 2011), increase the diversification and mutualization of counterparty risk (Koeppl and Monnet, 2010; Biais, Heider, and Hoerova, 2012) and can mitigate systemic liquidity risk (Menkveld, 2013). Nonetheless, the extant literature on CCPs has neglected the trade-off between default risk and competition in the OTC market. This issue has been highlighted informally by Duffie (2010b) and Pirrong (2011).2 It has also been addressed by recent regulatory reforms: the U.S. Commodity Futures Trading Commission has set limits on entry barriers as part of the Dodd-Frank Act implementation. Since then, the largest OTC interest rate derivative clearer (LCH.Clearnet’s SwapClear) reduced the capital requirement for its members from $5 billion to $50 million and eliminated the derivatives book size requirement (previously $1 trillion). Similarly, SwapClear’s founding members gave up control of the clearing service in response to new regulation in Europe.3 We consider an OTC market with (i) imperfect competition, (ii) endogenous dealers’ default 1
The G-20 stated in September 2009 that all standardized OTC derivatives contracts should be cleared by a CCP. Duffie (2010b):“Large [clearing members (CMs)], which tend to be dealers, may have a conflict of interest in claiming that smaller CMs are unsafe merely because of being small. [...] allowing smaller CMs introduces more competition between large dealers and smaller market participants.” Pirrong (2011), page 28: “One way to exercise this market power for the benefit of members is to limit membership to an inefficiently small number through the imposition of unduly restrictive membership requirements.” 3 CCPs were for long perceived to use membership criteria and risk management rules to exclude new entrants. For instance: “Bank of New York Mellon has been trying to become a so-called clearing member since early this year. But three of the four main clearing houses told the bank that its derivatives operation has too little capital, and thus potentially poses too much risk to the overall market.” A Secretive Banking Elite Rules Trading in Derivatives, New York Times, 2010. 2
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probability and (iii) where a CCP clears all trades. We model imperfect competition between a small number of differentiated dealers, but where differentiation decreases as more dealers enter the market. Dealers meet a continuum of risk-averse hedgers, offering a swap contract to transfer hedgers’ aggregate endowment risk. Dealers are specialist with access to hedging strategies allowing them to transfer this risk away. However, dealers may default since they use imperfect hedging strategies. For instance, the broker-dealer arm of a bank holding company may default due to losses in other business lines. As a result, the swap contract transforms hedgers’ aggregate risk into idiosyncratic dealers’ default risk (hedging strategies are uncorrelated across dealers). The probability of a default is endogenous. Due to dealers’ limited liability, the contract’s price that maximizes dealers’ utility may imply a positive probability of default. This probability is determined in equilibrium and depends on dealers’ market power, the quantity of collateral they pledge and the efficiency of their hedging strategies. Dealers are the CCP’s clearing members. The CPP pools margins posted by dealers (we use margins and collateral interchangeably in the following) to fulfill its obligations and reaps diversification benefits, as in Koeppl and Monnet (2010). Note that hedgers cannot insure against default – markets are incomplete – but dealers’ idiosyncratic default risk can be diversified within the CCP. Nonetheless, a CCP does not provide a guarantee (unlike deposits insurance) and the losses due to default are borne equally by all hedgers. This is analog to a fiscal authority absorbing the losses using tax resources. The CCP sets membership and collateral requirements ex-ante to reduce the risk and the costs of default. These rules also affect competition in the OTC market. We study the optimal CCP rules using the sum of hedgers’ and dealers’ welfare.4 First, the CCP should typically favor free entry and use collateral requirements to manage default risk. Consider the effects of free entry. On the one hand, a higher number of dealers reduces hedgers’ trading costs and increases the diversification benefits within the CCP. This increases welfare. On the other hand, competition eventually leads to a lower contract’s price, lower revenues and may also lead to a higher probability of a dealer defaulting. But margin requirements can counteract this effect. Margins transfer wealth from dealers to hedgers in the event of default. In addition, collateral reduces the probability of default since it mitigates the effects of limited liability 4 We take for given that only one CCP operates in the market. We also take as given that the CCP clears all trades; either because hedgers’ preferences or due to regulations.
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on incentives: each dealer has more skin in the game. Second, since collateral is costly, the CCP should set conservative collateral requirements – targeting a lower default probability – only if dealers transfer risk sufficiently efficiently. Hedging efficiency plays a key role in our results. If dealers are inefficient, a conservative rule requires too much collateral relative to the welfare improvements achieved by lowering default risk. In this case – if dealers are inefficient, the collateral rule should be less conservative and the optimal default probability is higher. Third, restricting entry is a substitute to margins and may be optimal if the opportunity costs of collateral are too high. Restricting entry raises the equilibrium price because of market power, raising dealers’ profit and lowering the risk of a default directly instead of via dealers’ incentives. However, restricting entry increases trading costs due to differentiation. Again, hedging efficiency plays a key role. We find that a CCP should restrict entry only if collateral is too costly and if dealers transfer risk efficiently. One implication is that restricting entry may be optimal if the increased collateral requirements due to regulatory reforms raises the opportunity costs of safe collateral.5 Hedgers and dealers prefer different CCP rules. The hedgers’ preferred rules are very similar to the optimal rules based on total welfare because most of dealers’ costs and benefits are exactly passed through via the price. The only difference between total and hedgers’ welfare is given by the extent of differentiation and determined by the trading costs and the number of dealers. This creates a rent borne by hedgers and earned by dealers. Therefore, from the point of view of hedgers, the costs of collateral must be higher and dealers must be more efficient to justify entry restrictions. In contrast, total dealers’ profits are maximized with tight entry restrictions that reduces competition and increases the mark-up. Dealers also prefer a lower level of collateral. Importantly, the costs of collateral are not the reason why dealers prefer lower margins: these costs are passedthrough to hedgers in equilibrium. Indeed, lowering the level of collateral reduces the cost component of the price. But dealers can captures this benefit. Lowering the capital requirements allow dealer to further restricting entry and to further raise their mark-up, but while keeping the 5
The increased collateral requirements follow from several distinct regulatory reforms. See, e.g., Duffie, Scheicher, and Vuillemey (2014); Lopez, Mendes, and Vikstedt (2013) and the references therein
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contract’s price below the hedgers’ reservation value. Hence, our framework provides a novel explanation for dealers’ and banks’ desire to use less collateral (or equity) and to limit entry in the CCP and support recent CCP regulations limiting privileged access and control by CCP members. Instead, current regulation emphasizes collateral requirements and mutualization to mitigate default risk.6 We consider two extensions to the model. First, we explore the interaction with other rules used to reduce risk, typically summarized in a CCP’s “rule-book”. Indeed, the CCP is in a unique position to directly monitor the quantity of risk borne by its member, since it stands on one side of every trade. We allow for monitoring by the CCP, but at a cost, introducing trade-offs between margins, entry restrictions and monitoring. Each rule acts via a different mechanism. Tight monitoring act as a constraint on dealers’ behavior: they must choose a price that is consistent with the risk-taking. In contrast, collateral affects dealers’ incentives while entry restrictions act on the mark-up. If collateral is cheap relatively to monitoring costs, the efficient CCP rules corresponds to the benchmark case above, otherwise monitoring is the privileged risk-management tool. As above for entry restrictions, we find that monitoring rules should be conservative when collateral is costly and if dealers are efficient. We also study an extension where dealers choose endogenously between hedging strategies that differ in risk. Endogenous hedging choice does not remove the fundamental trade-off between competition and risk and our benchmark results remain qualitatively unchanged. Allowing for riskier hedging strategies increases the incentive effect of limited liability on dealers, raising the bar against using conservative collateral requirement. It also raises the bar against entry restrictions since the mark-up must be higher. However, monitoring costs are constant and the threshold for using conservative monitoring rules is unchanged. The microstructure literature studies the industrial organization of OTC markets (Stoll, 1978; Kyle, 1985, 1989). Dealers’ market power is often linked to search frictions in this context (Duffie, Gˆarleanu, and Pedersen, 2005) but the implications for dealers’ default are ignored. A trade-off between competition and default risk arises in the context of banking (Keeley, 1990; Hellmann et al., 2000; Boyd and De Nicol` o, 2005; Martinez Miera and Repullo, 2010; Vives, 2010). While banks’ 6
See Lazarow (2011) for details on the governance structure of some large CCPs. See Duffie et al. (2010), and the discussion in Fontaine et al. (2012) for policy perspectives on OTC market infrastructure.
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deposits are fully insured in these models – and depositors do no account for the possibility of default – we differ in that a CCP is not a guarantee and that hedgers take dealers’ default into account. Therefore, while dealers can undercut each others to gain market share, the effect of competition on the probability of default is endogenous and reflected in the (closed-form) equilibrium price. We also differ in that dealers’ risk transfer efficiency plays a crucial role in our results; a feature that is absent from the banks’ borrowing and lending activities. Other papers also study how changes in the clearing infrastructure affect trading on the OTC market. CCPs reduce the counterparty-risk externality (Leitner, 2012; Acharya and Bisin, 2014; Thompson, 2010) and reduce an asset’s sensitivity to information (Carapella and Mills, 2012). Monnet and Nellen (2012) study trading and clearing arrangements on OTC markets with twosided limited commitment. Koeppl (2012) studies investments in financial market infrastructure and their governance structure. More recently, Atkeson, Eisfeldt, and Weill (2013) consider the effect of the entry decision on the trading pattern in the OTC market. Stephens and Thompson (2011) study the interaction between competition and risk in a credit default swap market with heterogenous insurers. They discuss how mutualization of losses within the CCP eliminates the comparative advantage of stable insurers. Another strand of the literature studies different CCP configurations in partial equilibrium settings (Renault, 2010; Haene and Sturm, 2009; Rausser et al., 2010; Cruz Lopez et al., 2012). Finally, the empirical literature remains sparse, in large part due to limited access to data. Jones and Perignon (2012) argue that a substantial share of the systemic risk faced by the CME Clearing House is due to proprietary trading on the part of its members: membership restrictions may not address the most salient sources of risk. Duffie et al. (2014) use an extensive data set of bilateral credit default swap positions to estimate the impact on collateral demand of new clearing and margin regulations. The content of the paper is organized as follows. Section 2 introduces the model and discusses our main assumptions about hedgers, dealers and the structure of the CCP. Section 3 derives the equilibrium and discusses its main properties. Section 4 contains the main results as well as extensions to the model. All proofs of the main results are included in the appendix, while proofs of results in the extensions are included in the online appendix.
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2
Model We consider a trading environment similar in spirit to Duffie, Gˆarleanu, and Pedersen (2005) and
Lagos and Rocheteau (2009), where hedgers and dealers trade an OTC contract. Our model has two key distinct features. First, dealers can hedge risk only imperfectly and they may (endogenously) default. Duffie, Gˆ arleanu, and Pedersen (2005) have dealers hedge their risk perfectly. Second, we model imperfect competition between a small number of dealers directly via differentiation and not via search frictions. This approach preserves market power for dealers while allowing us to compute explicitly the probability that any number of dealers will default. Together, these features allow us to analyze how dealers’ default risk changes with competition as more dealers enter the OTC market.
2.1
Assumptions
Trade occurs between two types of agents: a unit measure of risk-averse hedgers and a discrete number n ∈ N of specialized risk-neutral dealers. Hedgers are uniformly dispersed on a circle with circumference 1 while the n dealers are equally spaced along the circle. Figure 1 illustrates this economic environment based on Salop (1979) but to which we will add the possibility of a default. There are two periods illustrated in Figure 2. Dealers meet hedgers to trade at the beginning of the first period and offer them to trade a swap contract. The CCP clears all contracts and receives the collateral pledged by dealers at the end of the first period. During the second period, all shocks are realized and some dealers may default. The CCP settles all contracts at the end of the second period. Hedgers Each hedger owns m units of a numeraire and one unit of an asset producing a random endowment of the consumption good e˜:
ee =
e > 0 with probability
0
q
(1)
with probability 1 − q,
where a tilde indicates a random variable. Therefore, hedgers are exposed to a common endowment shock. Following, e.g., Lagos and Rocheteau (2009) and Koeppl and Monnet (2010), each hedger
6
has quasi-linear preferences u(e e) + m,
(2)
where m is the numeraire, u(·) is concave, u0 (0) = ∞, u0 (∞) = 0, and u(0) = 0. We introduce market power directly via horizontal differentiation (and not via search frictions). The location of hedgers and dealers in the economy captures this effect (see Figure 1). Hedger i’s costs of trading with dealer j located at a distance of di,j is given by ci,j = di,j · t where the trading cost parameter t > 0 is in terms of the numeraire. Dealers
Dealers have linear utility but limited liability. During the first period, dealers offer a
swap contract contingent on the realization of ee to hedgers. The contractual terms of the swap are given exogenously. Figure 3 illustrates its payoffs. The contract exchanges the endowment e] = qe against the random ee. The hedger’s payoff is se ≡ e¯ − ee. Dealer’s offer a mean e ≡ E[e price pj , in units of the numeraire, to be paid by hedgers when the contract settles. The price is determined in equilibrium. Dealers have the special ability to trade and hedge the risk in other e j transferring some markets. Specifically, dealer j has access to a hedging strategy e hj ≡ se − ∆ e j is a residual or basis risk that is uncorrelated across dealers. but not all the contract’s risk: ∆ Therefore, the swap contract transforms hedgers’ aggregate risk into idiosyncratic dealers’ default risk. A trader’s position is illustrated in Figure 3. A dealer trading yj contracts must pledge yj K unit of capital with opportunity costs r per unit of collateral and its period-1 expected utility is given by e j ), −yj K)]. E[udj ] = E[max(yj (pj − rK − ∆ Default
(3)
Default is endogenous and occurs if the pledged capital and the price of the swap are
e j are less than the loss incurred from trading and the cost of capital rK. The realizations of ∆ independent across dealers and independent of the endowment ee with distribution given by
ej = ∆
(q − 1)σe with probability
qσe
q
(4)
with probability 1 − q,
e j ] = 0 with variance var[∆ e j] = so the hedged position has zero expected payoff E[−e s+e h] = E[−∆ σ 2 var[e s]. The parameter σ plays a key role in our analysis, measuring how efficiently dealers’ 7
activities transfer the risk borne by hedgers. If σ = 0, dealers never default and hedgers face no e j represents risk that cannot be traded away by dealers. risk in equilibrium. Otherwise, the basis ∆ If σ = 1, there is not variance reduction but dealers’ default remain uncorrelated. This contrasts with perfectly correlated default in the absence of hedging. e j + rK − K) lies in one Depending on the price, the probability of a default D(pj ) ≡ Pr(pj < ∆ of three regions: Safe region: D(pj ) = 0
if
p ≤ pj
Risky region: D(pj ) = 1 − q if p ≤ pj < p Default region: D(pj ) = 1
if
(5)
pj < p,
with boundarires p ≡ qσe − K + rK and p ≡ (q − 1)σe − K + rK. The Default region is irrelevant in equilibrium and we ignore it in the following. Central counterparty clearing
The CCP novates all trades at the end of the first period.
Novation means that the CCP stands between hedgers and dealers, absorbing counterparty risk. Each trade between a hedger and a dealers is broken in two trades: one between the hedger and the CCP and one between the dealers and the CCP. Every dealer must pledge K units of collateral to the CCP for each contract. As in Koeppl and Monnet (2010), novation is not a guarantee. The CCP can only redistribute the proceeds from contracts with surviving dealers and the collateral pledged by defaulting dealers at the end of the second period (see below for details). Hedgers pay the swap’s price only if the contractual terms are met in full. Consider an event where eb ≤ n dealers default on their contracts. Figure 4 illustrate the flows between dealers, the CCP and the hedgers. The CCP uses the proceeds from contracts with the surviving dealers and the collateral from the defaulting dealers to fulfill a pro-rata share of its obligations to hedgers. In other words, hedgers receive a share of e¯ against ee from the n−eb surviving dealers, as well as of the collateral K from the eb defaulting dealers. A hedger’s consumption of the e is given by: consumption good C eb e e K) ≡ ee + n − b (¯ C(n, e − ee) + K. n n
(6)
e = e˜ + K and if no dealer defaults (eb = 0) we For instance, if all dealers default (eb = n) we have C
8
h i e = e¯. A hedger’s period-1 expected utility from consumption is υ(n, K) ≡ E u(C(n, e K)) , have C where the expectation is taken over the number of defaulting dealers ˜b and the endowment e˜. A hedger’s trading surplus is the difference between its expected utility from consumption given that he trades and the expected utility from consumption given that he does not trade. For convenience, we define the surplus as (1 − D(p))Π(n, K) with Π(n, K) given by
Π(n, K) ≡
υ(n, K) − qu(e) > 0, q
(7)
yielding the surplus (1 − D(p))Π(n, K) = υ(n, K) − qu(e). Importantly, this surplus increases monotonically with the number of members or as the collateral rises: ∂Π(n, K) ≥0 ∂n
and
∂Π(n, K) ≥ 0. ∂K
(8)
In a Safe equilibrium, we have that D(p) = 0 and υ(n, K) = u(¯ e) and the surplus is given by
Π ≡ u(¯ e) − qu(e) > 0.
(9)
Finally, hedger i trading with dealer j has total expected utility given by (neglecting the numeraire m): E[uhi (pj )] =
Π + qu(e) − pj − di,j t
if
qΠ(n, K) + q(u(e) − pj ) − di,j t if
2.2
pj ≥ p
(10)
pj ∈ [p, p).
Discussion
We interpret the exchange between dealers and hedgers as an OTC market where customization plays a role. We use the circular city model of Salop (1979) to model price competition with horizontal differentiation. Dealers offer differentiated ancillary services to their clients: cash accounts, accounting services and other customized services. Alternatively, dealers may have private knowledge about different hedgers and establishing a relationship may be costly. Hence, differentiation costs can be seen as an approximation to search frictions. Formally, the trading costs ci,j are incurred directly by each hedger and represent the real costs of participating in the market. These costs decrease as more dealers enter the market – the distance 9
to the nearest dealer is shorter – reflecting the higher diversity of customized services or the lower cost to establish a relationship. Importantly, ci,j is not available to dealers in the second period to meet its commitment and it cannot be recovered by hedgers in the case of a default. Central Counterparty
We take as exogenous the existence of the CCP. Alternatively, the CCP
may be mandated by the regulatory authority.7 In any case, a dealer’s default is idiosyncratic and reallocating the losses across hedgers improves welfare: the CCP diversifies dealers’ default risk. Hence, increasing the number of dealers or increasing collateral reduces the wedge between the surplus in the Safe and Risky region Π−(1−D(p))Π(n, K) = u(¯ e)−υ(n, K), since υ(n, K) increases with n and K (see Equation 8). Novation by a CCP implements this reallocation, improving hedgers’ welfare without reducing dealers’ expected profits (Koeppl and Monnet, 2010). Dealers
A dealer is not a bank or an insurance company but a trading specialist in financial
markets. Note that there is no point for dealers to trade the hedgers’ endowment risk between ˜ j . In the each other. In addition, we take for given that dealers use some hedging strategy h absence of hedging, hedgers pay the contract’s price when the endowment is high ee = e but dealers e j between the hedging strategy and the default when the endowment is low ee = 0. The basis ∆ swap contract is given exogenously. For instance, this basis may arise because of losses incurred by dealers in other business lines. e is key to derive clear analytical results related to Using a simple distribution for the shock ∆ the joint probability distribution of defaults when some, many or all dealers may default. This e j or extending the length of is crucial for our purpose. Adding more points to the support of ∆ the support of ee changes the default regions but much of the tractability would be lost without changing the thrust of the paper. Finally, we take as given that the hedging strategy is not perfectly correlated with the swap’s payoff. There is no diversification benefits within the CCP otherwise. The difference between the swap and its hedge may arise because hedging markets are incomplete, or because of gains or losses incurred by the dealer in other business lines. Note that trading away the endowment risk may be beneficial to hedgers even if σ is high (e.g., σ > 1), since the hedging strategy offers diversification benefits via novation by the CCP. 7
The G-20 have agreed that markets for OTC derivatives should be centrally cleared.
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Hedgers We assume m > p and that hedgers can commit to pay the contract’s price (the endowment is contractible). Hence, hedgers credibly promise to deliver their endowment and they do not default in equilibrium. Hedgers are not clearing members but their trades are nonetheless novated by the CCP. We think of hedgers as indirect clearers whose access to the CCP is included in the customized services provided by a dealer. Any cost incurred by a dealer to monitor the hedgers can be included in the trading costs t. Nonetheless, clients have a segregated account with the CCP as a protection against a default: the CCP has the ability to assess and enforce payment in the event of default by the clearing member. Finally, hedgers only trade with the CCP members, either because of regulation or because non-members’ risk cannot be diversified.
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Equilibrium This section describes the equilibrium in this market for different numbers of dealers n. As
in Duffie, Gˆ arleanu, and Pedersen (2005), dealers are homogeneous ex ante and it is natural to consider an equilibrium where a unique price prevails across the OTC market. Therefore, we follow Salop (1979) and focus on symmetric Nash equilibrium where every dealer j chooses its price pj to maximize its expected utility E[udj ,c ]:
E[udj ,c ] =
yc (pj , pˆ)(pj − rK)
yc (pj , pˆ)(pj − rK − (q − 1)σe −
1−q q K)q
if
pj , pˆ ≥ p
if
pj , pˆ ∈ [p, p)
,
(11)
taking into account the hedgers’ demand yc (pj , pˆ) and the price pˆ set by other dealers. The hedgers’ demand yc can easily be derived from Equation 10. Details of this and other derivations are shown in the appendix. Proposition 1 summarizes the equilibrium properties as the number of dealers n vary. We use the subscript c and u to designate a covered or uncovered equilibrium, respectively; and we use the superscript s and r to designate an equilibrium in the Safe and the Risky region, respectively, which we discuss in details below.
Proposition 1 Existence of a Unique Symmetric Equilibrium Consider cases where n > 1, r < 2, 0 ≤ σ ≤ σ ¯≡
Π 2qe
11
¯ and 0 ≤ K < K(σ) ≡ qσe.
• Safe Uncovered Equilibrium: for any 1 ≤ n ≤ nsu , where nsu =
t Π−rK ,
the unique sym-
metric equilibrium is uncovered (dealers are local monopolies) and in the Safe region (default risk is low). The price is psu =
Π+rK 2 .
• Safe Covered Equilibrium: for any nsc ≤ n ≤ nsc , where nsc =
3t 2(Π−rK)
and nsc =
t qσe−K
with nsc ≤ nsc , the unique symmetric equilibrium is covered (dealers compete) and in the Safe region (default risk is low). The price is psc =
t n
+ rK.
• Risky Covered Equilibrium: for any n > nrc where nrc is given in the appendix the unique symmetric equilibrium is covered (dealers compete) and in the Risky region (default risk is high). The price is prc = (q − 1)σe +
1−(1−r)q K q
+
t qn .
We consider cases where 0 ≤ σ ≤ σ ¯ because the equilibrium lies in the Risky region for all n if σ >σ ¯ , in which case dealers’ risk is too high relative to the hedgers’ surplus in the Safe region. ¯ We consider cases with 0 ≤ K < K(σ) because the equilibrium lies in the Safe region for all n e j states (or check that nsc → +∞). Both the otherwise: the pledged collateral is sufficient in all ∆ ¯ Risky and Safe regions exist when σ ≤ σ ¯ and 0 ≤ K < K(σ). Figure 5 shows the different equilibrium region across different values of n and K. For given K, if n < nsu the equilibrium does not “cover” the market (i.e., yj < 1/n) and dealers are local monopolists. We label this case the uncovered equilibrium. Under one interpretation, the uncovered equilibrium is an illiquid OTC market where some hedgers (the more distant ones to the dealers) do not participate, and the trading volume is low. This is the left-most area in Figure 5. As n increases, additional hedgers choose to trade and the equilibrium eventually covers the market (i.e., yj = 1/n). We label this case the covered equilibrium. The threshold between the two equilibrium types arises when the trading costs of the hedgers located the furthest from any given dealers, given t n,
by
equal the hedgers’ trading surplus minus the cost of collateral Π − rK.8 At this point, the
marginal hedger is indifferent between trading adjacent dealers. Therefore, the covered equilibrium features imperfect competition. The covered equilibrium can be in the Safe or the Risky region. The equilibrium is Safe if competition is low enough, nsc ≤ n ≤ nsc , so that the price and dealers’ revenue are high enough. 8
Clearly this case exists only if nsu > 1, otherwise every equilibrium covers the market
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Otherwise, if n > nsc , competition is intense, the markup t/n does not cover losses in all states e j (net of collateral) and the equilibrium lies in the Risky region. The threshold between the ∆ Safe and Risky regions nsc also depends on dealers’ efficiency. If dealers are very efficient (σ is small) then a large increase in the number of dealers is required before competitive pressure pushes the equilibrium in the Risky region (i.e., nsc nsc ). Capital also plays a role in determining the threshold nsc . Higher capital shifts nsc the right of in Figure 5. In other words, higher capital mitigate the effect increased competition on the default probability. Equilibrium Price The contract’s price decreases as the number of dealers increases. First, the covered equilibrium introduces competition. It is easy to check that the price decreases at the boundary between the uncovered and the covered equilibrium; psu > psc when n = nsu . Second, the price in the covered region psc decreases further as n increases. The price paid by hedgers exceeds the expected payoff due to market power – which yields the mark-up t/n – to the cost of capital rK passed-through to hedgers: psc =
t + rK > E[˜ s] = 0. n
(12)
The equilibrium price corresponds to the expected payoff only if the market is competitive (t = 0) and collateral is not costly (r = 0). The price decreases further in the Risky region and accounts for the effect of default: prc =
1 t D + rK − (qσe − K) 1−Dn 1−D
(13)
where D is a dealer’s default probability (see Section 2). As above the cost of capital is passed through to the hedgers. However, default has two additional effects. First, it reduces the effect that competition has on the mark-up. Second, it reduces the price by a term proportional to (qσe − K). This term arises because the limited liability benefits to dealers are passed-through in equilibrium. To see this, rewrite the dealer’s expected utility as
E[urd,c ] = ycr (p, pˆ) [(1 − D)(p − rK) + D(qσe − K)] ,
(14)
which combines its profits p − rK if there is not default with a positive term qσe− K ≥ 0 combining the benefits of limited liability and the costs of the foregone collateral in a default. The net effect of reaching the Risky region reduces the price. 13
We briefly describe the effects of novation. Novation increases the risk transfer efficiency, reducing the volatility of hedgers’ consumption. These welfare gains increase with n and K, mitigating the effect of competition on default risk. However, novation does not affect the equilibrium price within each regions. Competition between dealers implies that the price does not rise beyond a level consistent with the extent of differentiation, trading costs and default risk.
4
Optimal Entry Restrictions and Collateral Requirements CCPs must define the required margins that members must pledge to establish a trading posi-
tion. Abruzzo and Park (2014) show that margin changes are infrequent: the average time between changes may be as high as one year, depending on the type of financial contracts. CPPs are also defined by the number and characteristic of its clearing members. Changes to a CPP membership are even less frequent; the changes driven by regulatory actions provides one case. Together, the collateral requirements and membership structure are two key aspects of the economic environment under the control of the so-called “risk committee” or of the regulator.
4.1
Total Welfare
Within the context of our model, the CCP sets the number of members n and the margin requirements K ex ante.9 Since the rules that determine margins and membership change very infrequently, we can study how the market equilibrium changes between different choices of rules. Proposition 2 describes the optimal choices if the CCP (or its regulator) uses the sum of hedgers’ and dealers’ welfare as a criteria. The optimum changes whether the cost of collateral is low or expensive. We consider each case in turn (Cases A and B). Dealers’ efficiency plays a key role in the determination of the optimal CCP rule, with threshold σl and σe in Case A and B, respectively. Proposition 2 Optimal CCP Rules (K and n) Define the thresholds σl and σe with σe < σl and values given in the Appendix. The choices of K and n that maximize the sum of dealers’ and hedgers’ welfare is given by the following two cases: • Case A: the cost of capital is low (r < 1/4). 1. Free-entry is efficient. 9
Entry may be restricted by stringent operational or size requirements amounting to large fixed costs to entry.
14
¯ 2. If σ < σl , the efficient margin is K(σ) and the equilibrium is in the Safe region. ∗ ¯ 3. If σ ≥ σl , the efficient margin is K < K(σ) and the equilibrium is in the Risky region. • Case B: the cost of capital is high (r ≥ 1/4). 1. Free-entry may not be efficient. 2. If σ < σe , restricting entry to nsc is efficient and the equilibrium is in the Safe region. The efficient margin is K = 0. 3. If σ ≥ σe , free entry is efficient and the equilibrium is in the Risky region. The efficient ¯ margin is K ∗ < K(σ). The sum of dealers’ and hedgers’ expected utility in a covered equilibrium is given by:
Wcs = Π + qu(e) −
t − rK, 4n
Wcr = (1 − D) (Π(n, K) + u(e) − rK) + D (qσe − K) −
(15) t , 4n
(16)
in the Safe region and Risky region, respectively.10 The effect of n and K is unambiguous in the Safe region. Total welfare decreases with the collateral requirement K due its opportunity costs but total welfare increases with the number of dealers n due to lower differentiation and lower ¯ maximizes Wcs . The same effects are trading costs. If r < 1/4, then free entry and collateral K present in the Risky region but K and n also changes the trading surplus Π(n, K). Increasing n raises the diversification benefits, lowering the hedgers’ consumption volatility. Hence, the effect of n is unambiguous in the Risky region as well. In contrast, the effect of collateral is ambiguous in the Risky region. Margin requirements are costly but margins increase the mean and reduce the volatility of consumption. The level of collateral K ∗ is precisely that level maximizing Wcr ; K ∗ decreases with the costs r. For our purpose, the important comparison is across welfare between equilibrium regions: between Wcr and Wcs . Raising collateral requirements (or restricting entry) to leave the Risky region and reach the Safe region is costly. On the other hand, reaching the equilibrium to the Safe region increases hedgers’ surplus. Figure 6 illustrates numerically the optimal CCP rules as we vary σ and r. It shows that the optimal CCP rules are conservative – the equilibrium is in the Safe region – only if dealers are sufficiently efficient (i.e., σ is sufficiently low). This holds in general, whether in Case A or B. In Case A, the opportunity of collateral cost is low and the CCP should 10
The uncovered equilibrium is never optimal.
15
use collateral to control risk. In this case, lower efficiency raises the collateral required to reach the ¯ ¯ Safe regionK(σ) = qσe, increasing the costs rK(σ) of the conservative CCP rule directly. In Case B, the opportunity costs of collateral is high and it may be optimal for the CCP to restrict entry to reach the Safe region. Lower efficiency increases the required severity of the entry restrictions (it decreases n ¯ sc ), increasing the welfare loss due to trading costs t/¯ nsc . Whether in cases A or B, the higher costs of the conservative eventually surpass the hedgers’ benefits from reaching the Safe region. The conservative rule eventually becomes suboptimal. ¯ Note that the choice of K ∗ instead of the conservative rule K(σ) for low values of σ does not imply that the level of collateral requirements decreases. The collateral requirement K ∗ is ¯ given from the maximization of welfare in the Risky region (Equation 16). The requirement K(σ) ¯ increases with σ but we have that K ∗ = K(σ) at the boundary σ = σl (or σ = σe as in case B). Comparing Cases A and B reveals that entry restrictions and collateral requirements are substitutes. Figure 7 shows how the collateral requirement K ∗ changes as its opportunity costs r increase in a case where σ is greater than both σe and σl so that K ∗ is always optimal. The results illustrate that the collateral requirements decrease monotonically as the the opportunity cost r increases. In the opposite case, where σ is less than both σe and σl and dealers are efficient, the optimal CCP rule uses a different mechanism depending on the cost of collateral. In Case A, the opportunity cost is low, the CCP should use collateral to control risk and free entry is efficient irrespective of dealers’ efficiency. Free entry maximizes CCP diversification and minimizes trading costs. Entry restrictions eventually become optimal when the cost of collateral is high (Case B). The mechanisms used to reach reach the Safe region differ between Cases A and B. Note the term qσe − K in Equation 16. This term reflects positive impact of limited liability on dealers’ welfare following a default. Hence, lower efficiency increases total welfare in the Risky region via the ¯ dealers’ welfare. In Case A, the optimal choice of collateral K(σ) exactly counteracts the benefits of limited liability. Indeed, this mechanism guarantees that dealers find it optimal to select a price ¯ in the Safe region: the net benefits from a default disappears when collateral is set to K(σ). In Case B, the optimal number of CCP members raises the markup to p, again guaranteeing that the dealers’ choose a price in the Safe region (to see this, substitute the value for n ¯ sc in the price psc ).
16
Proposition 2 has meaningful implications about rules used by a CCP to control the risk and efficiency of its members. In our simple model, all dealers have the same level of efficiency. Nonetheless, consider the case where regulations can alter the composition of dealers’ efficiency. If entry is restricted to efficient dealers, then it is optimal to set conservative rules. As discussed above, the welfare costs of the requirements are lower when dealers are efficient. In fact, restricting entry may be optimal to a few efficient dealers (and excluding others) and if the costs of collateral are too high. In contrast, if the CCP chooses to give access to inefficient entrants, it is too costly to set conservative requirements in this case. The dependence on the costs of collateral also has meaningful implications. These costs are exogenous in our model and given by r. Nonetheless, widespread adoptions of stringent collateral rules could raise this rate. Our results suggest that the combination of entry restrictions and margins could be optimal in this case.
4.2
Dealers’ vs Hedgers’ Welfare
The CCP rules in Proposition 2 trade off hedgers’ and dealers’ welfare. Several CCPs are private entities, they are often owned by their members, and their members decide or influence the rules via the risk committee. Proposition 3 shows that the optimal rules are very different from the perspective of the dealers’ welfare. Proposition 3 Dealers’ Optimal CCP Rules Restricting entry to nsc and using low collateral requirement K = 0 maximize the sum of dealers’ welfare and the equilibrium is in the Safe region. A dealer’s welfare in the covered equilibrium is driven by its market power and given by
E[usd,c ] = E[urd,c ] =
t , n2
(17)
both in the Safe and Risky region. A dealer’s welfare is unaffected by the cost of collateral or default. The former is passed-through to hedgers while dealers do not internalize the effects of default on hedgers. The expected benefits of limited liability are also passed through to hedgers via a lower price. 17
A dealer’s welfare is slightly different in the uncovered equilibrium:
E[usd,u ]
2 = t
Π − rK 2
2 ,
(18)
which is decreasing in the cost of collateral rK due to imperfect pass through. Default is irrelevant because the uncovered equilibrium is Safe. The number of dealers n is also irrelevant since dealers are local monopolists in this case. It is easy to see that any dealer prefers to be a monopolist. If dealers compete, then reducing n increases every dealer’s profit. However, increasing the number of local monopolists increases total dealers’ welfare (as a group). Therefore, using the welfare of all dealers as a criteria, we find that dealers choose to cover the market but only barely. Then, the markup is high for each dealer and total profits are high. In addition, dealers prefer a lower level of collateral. High costs of collateral are often cited as a reason why banks and dealer prefer to keep capitalization or collateral low (Admati and Hellwig, 2014). This is not the case here since these costs of are passed-through to hedgers in equilibrium. Instead, lowering margins decreases nsc , allowing dealers to increase the mark-up t/n and profits, while maintaining the price p = t/n + rK below the hedgers’ outside option. Otherwise, a higher price is not optimal for dealers since it would lead to an uncovered equilibrium. Proposition 3 is consistent with well known industrial organization results where firms collude or cooperate, promoting regulations that create ex-ante entry barriers and help decreasing competition ex-post. The results are also consistent with several recent regulatory reviews of possible anticompetitive practice in CCPs.11 Note that we do not discuss how dealers can coordinate their actions and agree to an ex-ante set of rules that maximize total profits.12 Perhaps the CCP manager is appointed by its members. The manager will maximize the utility of the dealers if its incentives are aligned (e.g., the manager gets a fraction of members’ profits). This possibility of rents points to potential competition between CCPs. In the long-run, a coalition of competing dealers 11
The U.S. Department of Justice has recently extended its probe of derivative markets to investigate “the possibility of anticompetitive practices in credit derivatives clearing” among others (Reuters, 8 June 2012). In addition, the Financial Stability Board (FSB) identified fair and open access to CCPs as one of four safeguards for clearing OTC derivatives. The Principles for Financial Markets Infrastructures (FMI), which defines the international standards for FMI, states that “participation requirements should be justified in terms of the safety and efficiency of the FMI and [...] have the least-restrictive impact on access that circumstances permit.” 12 See, among others, Donsimoni et al. (1986); Bloch (1996); Thoron (1998) on the stability of coalitions.
18
could eventually launch a new CCP, eroding the monopolistic rents associated with membership in the incumbent CCP. Heuristically, competition between CCPs may have implications for the rules employed in equilibrium (see e.g., Santos and Scheinkman 2001) but we leave a formal analysis of these issues for future research. For our purpose, we note that CCPs face significant returns to scale due to increasing netting efficiencies (Duffie and Zhu, 2011) and that most OTC markets currently have only CCP in operation. Note that Hedgers are necessarily worse off if the CCP implements the dealers’ rule. First, the benefits of CCP diversification are not exhausted, reducing the surplus from trading. Second, trading costs remain high since dealers are more distant on average: the equilibrium price remains high due to the lack of competition. What if the CCP is managed (or regulated) for the hedgers’ benefits exclusively? The answer is given in Proposition 4.
Proposition 4 Hedgers’ Optimal CCP Rules Define the thresholds σh,l and σh,e with σh,e < σh,e . The choices of K and n that maximize the sum of hedgers’ welfare are exactly as in Proposition 2 except that: 1. The threshold cost of collateral that justifies entry restrictions is higher: r ≥ 5/4 instead of ¯ rK(σ)1/4. 2. The threshold dealers efficiency level that justifies conservative CCP rules is the same when the cost of collateral is low, σh,l = σl . 3. The threshold dealers efficiency level that justifies conservative CCP rules is lower when the cost of collateral is high, σh,e < σe .
The hedgers’ welfare in the Safe and Risky covered equilibrium are respectively given by
E[Ucs ] =Π + qu(e) −
5t − rK 4n
E[Ucr ] =(1 − D) (Π(n, K) + u(e) − rK) + D (qσe − K) −
19
(19) 5t , 4n
(20)
which differ from total welfare in Equations 15-16 by a term t/n.13 The difference between total and hedgers’ welfare is given by the dealers’ welfare. The hedgers’ welfare includes costs and benefits that are passed through from dealers to hedgers via the price. These terms disappear in the total welfare. Since the collateral costs rK and dealers’ benefits from limited liability qσe − K are exactly passed through via the price, the only difference between total and hedgers’ welfare is given by the trading costs t/n. Therefore, hedgers’ preferred CCP rules are qualitatively similar to results in Proposition 2, based on total welfare. We find that hedgers’ welfare is reduced by the higher mark-up t/n in the Safe region so that the thresholds for r and σ to to justify entry restrictions are tighter. Indeed, Proposition 4 shows that free entry is preferable as long as r < 5/4, in contrast with Proposition 2 where the threshold is r < 1/4. Similarly, for r ≥ 5/4, the efficiency threshold required to justify entry restriction is tighter, σh,e < σe .
4.3
Margins and Monitoring
The CCP sees all of its members’ exposure since it novates all trades. This gives the CCP a unique position to directly monitor the quantity of risk borne by its members. Indeed, real-world CCPs combine margins and membership requirements with a host of other risk controls that are summarized in the “rule-book”. This monitoring activity acts very much like a Value-at-Risk (VaR) constraint, limiting the risk of each dealer. Formally, we assume monitoring costs M > 0 per unit of trade (in units of the numeraire) and allow the CCP to directly limit how much risk brought by each of its member, e j + M ≤ α. D(pj ) = Pr pj < ∆
(21)
The relevant decision by the CCP is to choose α between α < 1 − q, whereas the equilibrium is constrained in the Safe region, and α > 1 − q, whereas the equilibrium is unconstrained. The monitoring fee M is passed-through from the CCP to its members in the second period.14 The 13
Again, the uncovered equilibrium is never optimal. The sum of hedgers’ utility in the uncovered equilibrium is given by Π − rK Π − rK E[Uus ] = n + qu(e) , t 4 where E[Uus ] integrates each hedger’s utility E[uhi (pj )] along the circle. 14 But timing of payment is irrelevant since dealers do not default in the Safe region.
20
utility of dealers when using monitoring is
e j ), 0)]. E[ud,j ] = E[max(yj (pj − M − ∆
(22)
It is never optimal to combine monitoring with collateral requirements in our model but we discuss possible extensions in the conclusion. The equilibrium with monitoring is derived in the online appendix, where we modify the definition of the equilibrium to account for the constraint given by Equation 21. Proposition 5 shows that monitoring and collateral requirement act as substitutes when r < 1/4. There is no essential difference when r ≥ 1/4 and we do not report this case to preserve space. Proposition 5 Optimal CCP Rules (K, n and monitoring) Consider cases with r < 1/4. Define σl as in Proposition 2 as well as σ1 (M, r) and σ3 (M, r) (defined in the Appendix). The choices of n and K that maximize the sum of dealers’ and hedgers’ welfare are given by: • Free entry is optimal. • If the cost of capital is low (r < 1/4). 1. If σ ≤ σl and σ ≤ σ1 (M, r), then the optimal margin is K(σ), the CCP does not monitor risk, and the equilibrium is in the Safe region. 2. If σ ≥ σl and σ ≥ σ3 (M, r), then the optimal margin is K = K ∗ , the CCP does not monitor risk, and the equilibrium is in the Risky region. 3. If σ > σ1 (M, r) and σ < σ3 (M, r), then the optimal margin is K = 0, the CCP monitors risk, and the equilibrium is in the Safe region. Importantly, monitoring affects the equilibrium via a different mechanism than margins or entry restrictions. While margins affect the dealers’ incentives and while entry restrictions act on the mark-up, monitoring acts as a constraint: it forces dealers to choose a price that satisfies Equation 21. Figure 8 summarizes the results, showing the thresholds σl , σ1 (M, r) and σ3 (M, r) as a function of the monitoring costs M . First note that the three possibilities listed in Proposition 5 are exhaustive. Second, free entry is never optimal (as in Case A above). Finally, for high values of M (to the right in the figure), monitoring is never efficient and we are essentially back to the results in Proposition 2. 21
For relatively low monitoring costs M , new cases arise in the left of the figure in an triangle given by σ > σ1 (M, r) and σ < σ3 (M, r). On the edge of this area, monitoring and margins become substitutes. Suppose that σ < σ1 (M, r) < σl , then margins are optimal: dealers are efficient enough ¯ while monitoring is relatively expensive. The efficient collateral requirement is K = K(σ) and there is no monitoring. However, as M decreases we may eventually reach a point σ < σl < σ1 (M, r) where monitoring is optimal and the collateral requirement is low.15 However, monitoring may not be optimal even if M = 0 since the price that dealers must offer to reach the Safe region is too high (dealers do not transfer risk sufficiently efficiently). Alternatively, suppose that σ ≥ σl and σ ≥ σ3 (M, r). Then both collateral and monitoring are expensive relative to dealers’ efficiency. The efficient collateral requirement is K = K ∗ and there is no monitoring. The new case arises when we decrease monitoring costs so that σl < σ < σ3 (M, r) (moving from right to left in Figure 8). In this case, dealers are sufficiently efficient for the conservative monitoring rule to be optimal. In other words, monitoring is a substitute to collateral requirements. The effect of monitoring is very similar in the case r > 1/4 (unreported). Dealers’ efficiency affects the substitution between monitoring and margin requirement. Keeping M fixed this time, when σ < σl , a decline in dealers’ efficiency (an increase in σ) reduces the advantages of margin requirements in controlling risk. This arises because margins work through the incentives of dealers. As dealers’ efficiency falls, the effect of limited liability on their incentive ¯ must increase to reach the safe equilibrium. increases, and margin requirements K Finally, the Appendix shows that monitoring costs are passed through to hedgers and leave dealers’ equilibrium profits unchanged in the covered equilibriums. Therefore, differentiating between the optimal rules from the point of view of hedgers or from the point of view of dealers lead to results that are qualitatively similar to Propositions 3-4.
4.4
Endogenous hedging strategies
A risker hedging strategy (higher σ) increases dealers’ utility because of limited liability. It seems plausible that dealers would choose riskier strategies if they could, potentially changing the nature of the optimal CCP rules. We find that dealers choose low-risk strategies when competition is 15
Collateral and monitoring are never used together because the CCP cannot choose different degree of monitoring “intensity” which would control the probability that a dealers violates the constraint 21
22
low and the equilibrium is in the Safe region. Conversely, dealers choose high-risk strategies when competition is intense and the equilibrium is in the Risky region. Hence, allowing dealers to choose the risk of the hedging strategy do not eliminate the trade-off between default and competition, but it increases the costs of leaving the Safe region. To check this, we consider the more general case where dealers can choose between σ and a riskier strategy σδ ≡ δσ with δ ≥ 1. The risk transfer is less efficient and hedgers’ consumption is more volatile when dealers choose σδ . For simplicity, we consider the case of no monitoring costs M = 0. We also maintain that σδ < σ ¯ , as above. Each dealers now simultaneously choose a price and a hedging strategy to maximize their expected profits in Equation (11), given the hedgers’ demand and given the choices of other dealers. As above, we focus on the symmetric equilibrium. Nonetheless, endogenous hedging strategies affect the characterization of the equilibrium only marginally. Proposition 6 Equilibrium with Endogenous Hedging Strategies Define σδ ≡ δσ ≤ σ ¯ with δ ≥ 1, then under the conditions of Proposition 1, • If nsc ≤ n ≤ nsc (δ) the symmetric equilibrium is covered and in the Safe region with price as in Proposition 1. Dealers choose either σ or σδ with no effect on the probability of default. • If nsc (δ) < n ≤ nsc , there are two covered symmetric equilibria. One symmetric equilibrium in the Risky region where dealers choose σδ and one symmetric in the Safe region equilibrium where dealers choose σ. This case maximizes social welfare. • If n > nsc , the symmetric equilibrium is covered and in the Risky region, with price as in Proposition 1. Dealers choose σδ . A detailed exposition of the equilibrium with endogenous hedging choice is given in the appendix. Heuristically, start with a covered equilibrium in the Safe region where all dealers choose σ. As above, this arises when trading costs t are high enough given the number of dealers n and the level of collateral K. Dealers have limited liability and, for a given price, they prefer hedging strategies that are more volatile. However, the contract’s equilibrium price accounts for dealers’ risk. Indeed, choosing σδ and a price in the Risky region is not profitable for any of the dealers, and 23
the symmetric equilibrium remains in the Safe region. Conversely, start from an equilibrium where every dealer chooses σδ > σ and a price in the Risky region. Then, it is not profitable for any dealer to choose σ a price in the Safe region if trading costs are low. In a sense, this is simply expanding the proof that a symmetric equilibrium in Proposition 1 is robust to deviation strategy. However, introducing σδ open an intermediate range for n where two covered symmetric equilibria coexist. In one case, the low-risk strategy is selected and the equilibrium price is in the Safe region. In the other case, the high-risk strategy is selected and the equilibrium price is in the Risky region. The equilibrium is indeterminate because dealers do not internalize default risk and they are indifferent between these cases. Hedgers prefer the safe equilibrium. Therefore, the main effect of δ > 1 is to increase volatility in the Risky region. Since dealers tend to choose riskier strategies, then the introduction of endogenous strategies increases the costs of conservative CCP rules. If anything, this pushes the trade-off away from entry restrictions and toward the use of collateral requirements. In the next propositions we show this result for r < 1/4. Proposition 7 Optimal CCP Rules (K and n) with Endogenous Hedging Strategies Define the threshold σl,δ with σl,δ < σl with values given in the Appendix. When r < 1/4, the choices of K and n that maximize the sum of dealers’ and hedgers’ welfare are: • Free-entry is efficient. ¯ δ (σ) > K(σ) ¯ • If σ < σl,δ , the efficient margin is K and the equilibrium is in the Safe region. ¯ δ (σ) and the equilibrium is in the Risky region. • If σ ≥ σl,δ , the efficient margin is K ∗ < K As above, free entry is efficient when the costs of collateral is low. The only changes are the required collateral requirement to reach the Safe region as well as the efficiency threshold beyond with this conservative requirement is worth the costs. The requirement increases because the effect of limited liability on dealers’ incentives is higher for σδ > σ. In turns, this implies that the opportunity costs ¯ δ are higher. Since the hedgers’ utility in the Safe region is unchanged, then the conservative rK requirements are optimal for a narrower range of dealers’ efficiency (smaller values of σ). When r ≥ 1/4, riskier strategies tilts the optimal away from entry restrictions (unreported). This arises because dealers must be offered a greater rent (n must be lower) for the Safe equilibrium 24
with limited competition to be optimal. Finally, a riskier hedging strategy tilts the optimal rules toward the use of monitoring. This arises because margin requirements act on the dealers’ incentive while monitoring acts directly on their behavior.
5
Conclusion Clearing houses and CCPs have long played a key role in securities markets and are becoming
central to OTC markets. In order to manage risk, a CCP typically imposes stringent membership requirements and other risk controls. Our analysis emphasizes that these restrictions affect the OTC market’s structure. In particular, if members of the CCP control its rules and regulations, they use tighter risk controls to restrict entry and commit to a lower degree of competition. This increases their expected profits and decreases the probability of a default, but imposes costs to hedgers that are too high relative to the social optimum. In contrast, maximizing total welfare, a CCP typically allows for free entry. This minimizes rent and maximizes the diversification benefits stemming from the CCP. Our approach open several avenues for future research. First, default funds and mutualization may also affect the trade-off between competition and default risk. Our main results will likely remain unaltered: a CCP that uses these additional mechanisms to reduce default risk is likely to tilt the trade-off toward more competition. On the other hand, dealers could use these risk controls to further limit the effect of competition and achieve higher profits. Second, considering fixed costs for dealers entrants may reduce the extent of free entry but would not affect the underlying tradeoffs. Third, potential externalities from a dealer’s default would introduce an important tradeoff. For instance, the negative externality from default of too-big-to-fail dealers or the positive externality from having a robust pool of diverse dealers (with free entry). Fourth, a formal gametheoretic analysis could shed light on coordination between dealers; for instance, incumbent dealers may allow access to new entrants but at a price; or a coalition of incumbents and new entrant dealers may form a competing CCP. Finally, we briefly considered monitoring at the extensive margin (whether or not to monitor) but ignored the case where monitoring must be fine-tuned at the intensive margin (how intensively you should monitor).
25
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Figure 1: Salop’s Circular Economy This figure illustrate the location of hedgers and dealers on the circular economy. The circle represents the continuum of hedgers with n dealers located at equal interval represented by the yellow dot. The distance between every pair of dealers is 1/n so that any hedger with distance dij to the nearest dealer j is also located at the distance of 1/n − dij from a competing dealer.
Figure 2: Timeline This figure illustrate the model’s timeline. Dealers and hedgers agree on a price and trade the swap contract in the beginning of the first period. Clearing by the CCP occurs at the end of the first period. Shocks realized and some dealers may default in the beginning of the second period. The CCP settles all contracts at the end of the second period. 29
Figure 3: Dealers’ swap contract and hedging This figure illustrates the exposures of dealer j from the swap contract and from the hedging strategy when dealer does not default.
Figure 4: Settlement by the CCP at the end of the Second Period This figure illustrates the role of the CCP in the event that some dealers default (with probability D = 1 − q). The surviving dealers settle the swap contract with the CCP, exchanging e˜ − e¯ and the price p. The CCP also takes possession of the collateral K from any defaulting dealer and settles its obligation to hedgers.
30
Figure 5: Equilibrium Types This figure illustrates the different equilibrium in Proposition 1 as the number of dealers n and the level of collateral requirement K change.
Figure 6: Margin Requirement and Entry Restriction This figures illustrates the choices of K and n in Proposition 2 that maximize total welfare as the opportunity costs of collateral r and the dealers’ efficiency σ changes. Case A is shows to the left and Case B is shown to the right of the vertical line given by r = 1/4, respectively.
31
Figure 7: Optimal Collateral Requirement This figure illustrates the optimal CCP rules for different values of dealers’ efficiency σ and monitoring costs M in the case with where the opportunity costs of collateral are low r < 1/4. The case with r ≥ 1/4 is very similar but entry restrictions become optimal in case where the conservative collateral requirement were optimal.
Figure 8: CCP Rules – Margins or Monitoring
32
A A.1
Preliminary results Dealers’ utility and default
e j + rK − K with distribution given by Default occurs when pj < ∆ p ≡ qσe − K + rK with probability 1 − q e ∆j − K + rK = p ≡ (q − 1)σe − K + rK with probability q.
(23)
e j − K + rK) is given by Therefore, the default probability distribution D(pj ) = Pr(pj < ∆ Safe region: D(pj ) = 0 if p ≤ pj Risky region: D(pj ) = 1 − q if p ≤ pj < p Default region: D(pj ) = 1 if pj < p,
(24)
and from Eq. (3) a dealer j expected utility is given by e j )/pj − rK − ∆ e j ≥ −K] E[udj ] = E[yj (pj − rK − ∆ e j ≥ −K) − yj K · Pr(pj − rK − ∆ e j < −K), · Pr(pj − rK − ∆
(25)
which simplifies to
yj (pj − rK) yj (q(pj − rK) + (1 − q)(qσe − K)) E[udj ] = −yj K
A.2
if if if
p ≤ pj pj ∈ [p, p) pj < p.
(26)
Novation and hedger’s utility
The random number of defaulting dealers eb follows a binomial distribution with parameters D(p) and n, with mean nD(p) and variance nD(p)(1 − D(p)). Hedger i trading with dealer j has utility given by e K)) − (1 − D(pj ))pj − di,j t, uhi (pj ) = u(C(n,
(27)
e K) is the hedger’s consumption. For an equilibrium in the Risky region (p ≤ pj < p), the expected where C(n, utility of hedger i trading with dealer j is given by E[uhi (pj ; n, K)] = υ(n, K) − qpj − di,j t, where the first term, υ(n, K), corresponds to the expected utility from the consumption good: h i h i e K))] = qE u(C) e | ee = e + (1 − q)E u(C) e | ee = 0 . υ(n, K) ≡ E[u(C(n,
(28)
(29)
This expectation increases with n because of risk-aversion. The consumption volatility decreases with n but the e are given by mean consumption does not change. The conditional means of C e | ee = e) = qe + e(1 − q)2 + (1 − q)K E(C e | ee = 0) = qe − eq(1 − q) + (1 − q)K, E(C
33
(30)
which do not change with n but increase with K, and their conditional variances are given by e | ee = e) = (1 − q)q (K + e − qe)2 V ar(C n (1 − q)q e | ee = 0) = V ar(C (K − qe)2 , n
(31)
which decrease strictly with n, increase with K, and reach zero when n → +∞. The distribution of consumption when the number of dealers is n and the distribution when the number of dealers is n+1 differ by a mean-preserving spread (the dispersion decreases with n). Then, υ(n, K) is increasing in n because u is concave (see Rothschild and Stiglitz 1970 and, e.g., the review in Levy 1992). The benefits from diversification, Π(n, K) ≡ υ(n,K)−qu(e) , increase q with the number of members since υ(n, k) increases with n, therefore the hedger’s expected utility increases with n (and with K) in the Risky region.
A.3
Further properties
e increases with K, u0 (0) = +∞ and u(+∞) = 0, the derivative of υ(n, K) with respect to K has the Since C following properties: ∂υ(n, K) ∂υ(n, K) ∂υ(n, K) ≥ 0, lim = 0, lim = +∞. (32) K→+∞ K→0 ∂K ∂K ∂K Also,
∂Π(n,K) K→+∞ ∂K
lim
∂Π(n,K) K→0 ∂K
= 0, lim
= +∞, and by concavity of the utility function, Π(n, K) is concave in K.
Also, Π(n, K) > Π(n, 0) ≡ Π(n) > Π
for K > 0 and n > 1.
(33)
Finally, note that qΠ(+∞, K = 0) ≤ Π, with equality only if q = 1 (when there is default).
A.4
Demand in the covered equilibrium
Dealers compete in a covered equilibrium. The demand schedule for dealer j, yc (pj , pb), is the quantity of contracts sold by dealer j as function of its price pj and the price pˆ set by other dealers (we focus on symmetric equilibria). To derive hedgers’ demand, compare the utility obtained from trading with alternative dealers. The expected utility of a hedger i from trading with the nearest dealer j located at a distance di,j is (neglecting the constant m) p ≤ pj Π + qu(e) − pj − di,j t if E[uhi (pj )] = (34) q(Π(n, K) + u(e) − pj ) − di,j t if p ≤ pj < p. Consider the second nearest dealer, j − 1 say, located at distance n1 − di,j on the other side of the hedger. The expected utility from trading with dealer j − 1 setting price pˆ is Π + qu(e) − pˆ − ( n1 − di,j )t if p ≤ pˆ E[uhi (ˆ p)] = (35) 1 q(Π(n, K) + u(e) − pˆ) − ( n − di,j )t if p ≤ pˆ < p, and the hedger will prefer to trade with dealer j if ( E[uhi (pj )] ≥ E[uhi (ˆ p)] ⇔
1 −pj +ˆ p+ n t ≥ di,j 2t 1 −qpj +q pˆ+ n t ≥ di,j 2t
34
if
p ≤ pj , pˆ
if pj , pˆ ∈ [p, p).
(36)
We obtain a similar condition for those hedgers between dealer j and j + 1. Then the demand schedule for dealer j is given by, −p +bp+ 1 t j n if p ≤ pj , pˆ t 1 −qpj +qb p+ n t (37) yc (pj , pb) = if pj , pˆ ∈ [p, p) t 0 if pj , pˆ < p. Note that hedgers’ demand is zero in the default region confirming the irrelevance of the case p < p. In that case, all dealers default. Therefore hedgers would incur costs ci,j but receive nothing in exchange: there is no trade.
A.5
Demand in the uncovered equilibrium
A hedger does not trade with any dealer if its expected utility is less than its outside value, which is equal to the utility from not trading, u(e)q + u(0)(1 − q) = qu(e). (38) Comparing Eq. (34) and Eq. (38) shows that some hedgers choose not to trade if the price or the trading costs t are large enough. In this case, each dealer is a local monopoly with demand given by 2 p ≤ pj if t max (Π − pj , 0) 2 q max (Π(n, K) − pj , 0) if p ≤ pj < p yu (pj ) = (39) t 0 if pj < p. The demand schedule has a kink at the point between the uncovered and the covered cases. This kink reflects the effect of competition on the price elasticity of hedgers (compare the slope of the hedgers’ demand between the Eq. 37 and 39).
B
Main Propositions
B.1 B.1.1
Proposition 1: Existence of a unique Symmetric Equilibrium Uncovered equilibrium
From first-order conditions calculated to maximize Eq. (26) with respect to pj , the equilibrium price in the Safe uncovered region, p ≤ pj ≤ Π, is given by: Π + rK psu = , (40) 2 and the demand in equilibrium for every dealer equal to 2 Π + rK 2 Π − rK Π− = . (41) yu (psu ) = t 2 t 2 This price cannot be too high, psu ≤ Π, otherwise a dealer does not trade with any hedger: psu =
Π + rK ¯u ≡ Π . ≤Π⇔K≤K 2 r
(42)
¯ ¯ ¯ u . We verify that this price is in We assume that K < K(σ), which is defined below, and we show that K(σ)
35
which can be rewritten as (r − 2)K ≤ Π − 2qσe.
(44)
By assumption, Π − 2qσe ≥ 0, K ≥ 0 and r < 2, and therefore condition (44) is always satisfied. Consider next the price for an uncovered equilibrium in the Risky region. Since hedgers’ expected utility is quadratic, the price pru satisfies the necessary and sufficient first-order condition: ∂E[urd,u (pj = pru )] ∂pj
=−
2q 2q (qpru − q(q − 1)σe − (1 − q + qr)K) + (Π(n, K) − pru )q = 0, t t
(45)
implying that pru
=
Π(n, K) + (q − 1)σe +
1−q+qr K q
2
.
(46)
From Eq. (39) demand for dealers j is given by yu (pru )
2 =q t
Π(n, K) − (q − 1)σe −
1−q+qr K q
! ,
2
(47)
and its expected utility is given by E[usd,u ] = yu (psu )(psu − rK) =
2 t
Π − rK 2
2 .
(48)
The price pru must satisfy p ≤ pru < p, yielding two conditions on σ. First we must have that Π(n, K) + (q − 1)σe +
1−q+qr K q
2 Π(n, K) +
1+q−qr K q
⇔σ>
(1 + q)e
< qσe − K + rK >
Π , (1 + q)e
(49)
and second, that Π(n, K) + (q − 1)σe +
⇔
1−q+qr K q
2 Π(n, K) + (1 − q)σe +
≥ (q − 1)σe − K + rK 1+(1−r)q K q
2
≥ 0,
(50) (51)
which is always satisfied, since σ ≥ 0, q ≤ 1, r ≤ 2 and K ≥ 0. We show next that under certain conditions, the Risky uncovered equilibrium can be excluded. Equation (43) and the fact that Π(n, K) ≥ Π require that Π(n, K) ≥ (r − 2)K + 2qσe.
(52)
On the other hand, Eq. (49) requires that Π(n, K) < (1 + q)σe +
36
(1 + r)q − 1 − 2 K. q
(53)
These two conditions in Eq. (52) and (53) do not overlap: the uncovered equilibrium is either Safe or Risky. (1 + r)q − 1 (r − 2)K + 2qσe − (1 + q)σe − −2 K q 1−q (K − qσe) ≤ 0. (54) = q ¯ ≡ qσe by assumption. Therefore, from Eq. (43), if r ≤ 2 we can The last inequality follows because K ≤ K exclude the Risky uncovered equilibrium. We calculate the utility of hedgers in the Safe and Risky region: Π−ps u t
s E[Uh,u ]
Z
psu
[Π + qu(e) −
= 2n
Π − rK − zt] dz = n t
Π − rK + qu(e) 4
(55)
0
q r E[Uh,u ]
Π(n,K)−pr u t
Z
[qΠ(n, K) + qu(e) − qpru − zt] dz
= 2n 0
= 2nq
Π(n, K) − pru t
1 Π(n, K) − pru q t + qu(e) . 2 t
(56)
To calculate total welfare in the Safe region, we add Eq. (48) and Eq. (55). The price paid is a transfer between hedgers and dealers and cancels out. We obtain: Wus = n B.1.2
2 t
Π − rK 2
2 +n
Π − rK t
Π − rK + qu(e) . 4
(57)
Covered equilibrium in the Safe region
Definition 1 In a symmetric price (Nash) equilibrium in the covered region, every dealer chooses optimally the same price taking as given the other dealers’ prices. Since we focus on a symmetric equilibrium, from Eq.(26) the equilibrium price in the Safe region psc is given by the first-order condition: ∂E[ud,c (pj = psc , pb = psc )] ∂yc s = (p − rK) + yc = 0, (58) ∂pj ∂pj c and, since yc = 1/n in a symmetric covered equilibrium, ∂E[ud,c (pj = psc , pb = psc )] 1 1 t = − (psc − rK) + = 0 ⇔ psc = + rK. ∂pj t n n
(59)
The second order condition is trivially satisfied. This equilibrium is in the Safe region if p ≤ psc , qσe − K + rK ≤
t t + rK ⇔ n ≤ ≡ nsc . n qσe − K
37
(60)
This expression can be rewritten as K ≥ qσe −
t . n
(61)
Also, from Eq. (60), when ¯ K → K(σ) ≡ qσe ⇒ nsc → +∞, and the covered equilibrium is always in the Safe region. Also, since σ ≤ σ ≡
(62) Π 2qe
and since r ≤ 2, then
Π ¯u ≡ Π , ¯ ≤K K(σ) ≡ qσe ≤ 2 r
(63)
¯ u has been defined in Eq. (42). where K An equilibrium is covered if the marginal hedger’s utility (located at a distance d = value. For the Safe region we have u(qe) − psc −
1 2n )
is greater than its outside
1 3t t ≥ qu(e) ⇔ n ≥ ≡ nsc . 2n 2(Π − rK)
(64)
¯ ¯ u ≡ Π . We also verify that the Safe covered equilibrium is Note that the term nsc is positive if K ≤ K(σ) ≤K r s s non-empty, nc < nc : t 2Π + (3 + r)K 3t < ⇔σ< , (65) 2(Π − rK) qσe − (r + 1)K 3qe which is satisfied since σ ≤ σ ≡
Π 2qe .
¯ Therefore, if K < K(σ), σ ≤ σ and r < 2, the equilibrium transitions from the Safe uncovered region , to the Safe covered region to the Risky covered region as n increases. Deviation from a symmetric equilibrium We verify that no dealer deviates by setting a price in the Risky region given that other dealers set the price psc = nt + rK in the Safe region. First, we compute the corresponding demand function. For the case where a dealer j deviates to set a price pj in the Risky region given that other dealers set a price pb in the Safe region, hedgers’ demand to trade with dealer j is given by the inequality: q(Π(1, K) + u(e) − pj ) − dj t ≥ Π(1, K) + qu(e) − pb − (
1 − dj )t ⇔ n
(q − 1)Π(1, K) − qpj + pb + n1 t ≥ dj . 2t
(66)
The demand for dealer j, ycr (pj , pb), is given by ycr (pj , pb) =
(q − 1)ΠK − qpj + pb + n1 t , t
where ΠK ≡ Π(1, K) and the best deviation price prj
s
(67)
is given by the first order condition,
∂E[ud,c (pj , pb = nt )] ∂y r = c (qpj − q(q − 1)σe − (1 − q + qr)K) + ycr q = 0 ⇔ ∂pj ∂pj (q − 1)ΠK − qpj + nt + rK + q − (qpj − q(q − 1)σe − (1 − q + qr)K) + t t (q−1)ΠK +(1−q+qr)K rK 2t (q − 1)σe + + q q + qn r s pj ≡ , 2 38
t n
q=0⇔ (68)
which yields the following demand: ycr (prj s , pb) = =
(q − 1)ΠK − q
(q−1)σe+
(q−1)ΠK +(1−q+qr)K 2t + rK + qn q q
2
+ pb + n1 t
t −q (q−1)σe 2
+
(q−1)ΠK −(1−q+qr)K 2
+
rK 2
+ n1 t
t
,
(69)
and the following expected utility: E[ud,c ] = ycr (prj s , pb)(pjr s − (q − 1)σe − =
−q (q−1)σe + 2
(1 − q + qr)K )q q
(q−1)ΠK −(1−q+qr)K 2
+
rK 2
+ n1 t
t (q − 1)σe +
(q−1)ΠK +(1−q+qr)K q
+
rK q
+
2t qn
·
(1 − q + qr)K q q (q − 1)σe (q − 1)ΠK − (1 − q + qr)K rK t 2 1 −q + + + . = t 2 2 2 n
2
− (q − 1)σe −
(70)
The deviation is not optimal if the expected profits from Eq. (70) are lower than the no-deviation case in Eq. (84), −q(q − 1)σe + (q − 1)ΠK − (1 − q + qr)K + rK ≤ 0 ⇔ (q − 1)(−qσe + ΠK + (1 − r)K) ≤ 0 ⇔ ΠK + (1 − r)K σ≤ qe which is satisfied:
ΠK + (1 − r)K Π−K Π − qσe Π Π ≥ ≥ ≥ −σ ≥ , qe qe qe qe 2qe
¯ since r < 2, K ≤ K(σ) ≡ qσe and σ ≤ σ ≡ B.1.3
(71)
(72)
Π 2qe .
Covered equilibrium in the Risky region
The equilibrium price prc in the Risky region is given by the first-order condition, ∂E[ud,c (pj = prc , pb = prc )] ∂yc = (qprc − q(q − 1)σe − (1 − q + qr)K) + yc q = 0 ∂pj ∂pj 1 − q + qr t ⇔ prc = (q − 1)σe + K+ , q qn
(73)
and the second-order condition is satisfied. This price yields an equilibrium in the Risky region if prc < qσe − K + rK ⇔ n > nsc ≡
39
t . qσe − K
(74)
This is a covered equilibrium if the expected utility of the marginal hedger (i.e., d = value. The threshold nrc solves the following condition
1 2n )
is larger than its outside
q(Π(nrc , K) − prc ) + qu(e) − dt = qu(e) ⇔ qΠ(nrc , K) − q(q − 1)σe − (1 − q + qr)K =
3t . 2nrc
(75)
We define nrc ≡ max(nrc , nsc ). For some values of the parameters, we have that nrc = nsc and, therefore, the transition between a covered equilibrium in the Safe region toward a covered equilibrium in the Risky region is smooth. This is the case when K = 0 and q ≥ 12 . To show this, we define n brc as qΠ − q(q − 1)σe =
3t 3t ⇔n brc ≡ , r 2b nc 2[qΠ − q(q − 1)σe]
(76)
where Eq. (76) is just as Eq. (75) with K = 0 and Π instead of Π(n, K). Since Π(n, K) increases with n, then nrc (K = 0) < n brc . The condition for n brc < nsc (K = 0) to be satisfied is n brc ≡
3t 2Π t ≤ nsc (K = 0) ≡ ⇐⇒ σ ≤ , 2[qΠ − q(q − 1)σe] qσe (1 + 2q)e
(77)
2Π Π and if q ≥ 21 then (1+2q)e . Since we have shown that nrc,K=0 < n brc < nsc,K=0 when q ≥ 12 , K = 0 and ≥ σ = 2qe σ ≤ σ, then the transition between a covered equilibrium in the Safe region, and a covered equilibrium in the Risky region is smooth.
Deviation from a symmetric equilibrium We verify that a dealer does not want to deviate and set a price t K + qn in the Risky region. For the in the Safe region given that other dealers set a price prc = (q − 1)σe + 1−q+qr q case where a dealer j deviates to set a price pj in the Safe region given that the rest of the dealers set a price prc in the Risky region, the hedgers that trade with dealer j are given by the following inequality: 1 b b Π(n, K) + qu(e) − pj − dj t ≥ q(Π(n, K) + u(e) − prc ) − ( − dj )t ⇔ n 1 r b (1 − q)Π(n, K) − pj + qpc + n t ≥ dj , 2t
(78)
b where Π(n, K) is the surplus when n − 1 dealers default with probability 1 − q, and one dealer defaults with b probability 0. Note that Π(n, K) > Π(n, K) > Π. The demand for dealer j in this case (denoted as ycs (pj , prc )) is given by: b (1 − q)Π(n, K) − pj + qprc + n1 t s r , (79) yc (pj , pc ) = t where pj ≥ p for this deviation to be in the Safe region. From first order conditions, dealer j sets an optimal price in the safe region,
pj =
rK + 2
∂ycs (pj , prc ) (pj − rK) + yc = 0 ⇔ ∂pj b (1 − q)Π(n, K) + q(q − 1)σe + (1 − q + qr)K + 2
40
2t n
.
(80)
This price must be greater or equal to p. Let assume that this condition is not satisfied: b (1 − q)Π(n, K) + q(q − 1)σe + (1 − q + qr)K + 2t rK n + ≤ qσe − K + rK ⇔ 2 2 2t b (1 − q)Π(n, K) + q(q − 3)σe + (3 − q + (q − 1)r)K + ≤ 0. n Since we are in the risky region, n ≥
t qσe−K
⇔
t n
(81)
≤ qσe − K. Therefore,
2t b ≤ (1 − q)Π(n, K) + q(q − 3)σe + (3 − q + (q − 1)r)K + n b (1 − q)Π(n, K) + q(q − 3)σe + (3 − q + (q − 1)r)K + 2(qσe − K) = b (1 − q)Π(n, K) + q(q − 1)σe + (1 − q + (q − 1)r)K.
(82)
b is a continuous function, there exists q ∗ < 1 such For any value of K, if q = 1, Eq. (82) is equal to zero. Since Π 1 that for any q ≥ q ∗ the optimal price of firm j is lower or equal than p. Since the utility of the deviating dealer j is a concave function, optimal price in the safe region for dealer j must be equal to p. Then, the utility of the deviating dealer is equal to ycs (p, prc )(p − rK). A condition for this deviation not to be profitable is that the utility of dealer j is lower than the utility of not deviating, ycs (p, prc )(p − rK) = 1 2t t b (1 − q)Π(n, K) + q(q − 3)σe + (3 − q + (q − 1)r)K + + qσe − K (qσe − K) ≤ 2 . t n n
(83)
We show numerically that there exists q2∗ < 1 such that, for any q ≥ q2∗ , Eq. (83) is satisfied. Figure 9 show several numerical examples of q ∗ = max(q1∗ , q2∗ ) that guarantee that the deviation is not profitable for any q ≥ q ∗ . We show q ∗ as a function of σ and t for several cases of K/e and e. B.1.4
Utility of dealers and hedgers in the covered equilibrium
The utility of every dealer in a symmetric covered equilibrium in the Safe region is E[usd,c ] = yc (psc )(psc − rK) =
t . n2
(84)
In addition, each dealer trades with 1/n hedgers that are located at a distance of 1/(2n) on each of its side. The utility of these hedgers in a Safe equilibrium is given by 1/(2n)
Z 2 0
1 [u(qe) − zt − pc ] dz = n
5t u(qe) − − rK . 4n
(85)
The expected utility of dealers in the Risky equilibrium is given by E[urd,c ] = yc (prc )(qprc − q(q − 1)σe − (1 − q + qr)K) =
t , n2
(86)
and the utility for hedgers covered by one dealer is Z
1/(2n)
[q(Π(n, K) + u(e) −
2 0
prc )
1 − zt] dz = n
5t q(Π(n, K) − (q − 1)σe) − (1 − q + qr)K + qu(e) − 4n 41
.
(87)
(a) e = 15 and
(c) e = 5 and
K e
K e
= 0.0004
(b) e = 15 and
= 0.0012
(d) e = 5 and
K e
K e
= 0.0002
= 0.0006
Figure 9: Sufficient Conditions for Existence of the Equilibrium Numerical illustration of the inequality given in Equation 83. Values for q above the plane satisfy q > q ∗ , guaranteeing the existence of a covered equilibrium in the Risky region under the conditions given in Proposition 1. The total utility of hedgers is n times the values in Eq. (85) and Eq. (87): s E[Uh,c ] = Π + qu(e) −
5t − rK, 4n
r E[Uh,c ] = q(Π(n, K) + u(e) − (q − 1)σe) −
42
(88) 5t − (1 − q + qr)K, 4n
(89)
which is strictly increasing in n and decreasing in K within each region. Finally, total welfare follows from Eq. (88) or Eq. (89) but adding the utility of dealers: Wcs = Π + qu(e) −
t − rK, 4n
(90)
Wcr = q(Π(n, K) + u(e) − (q − 1)σe) −
B.2
t − (1 − q + qr)K. 4n
(91)
Proposition 2: Optimal CCP Rules (K and n)
Using Eqs. (90), (91) and Proposition 1, t 3t t = u(qe) − − rK for n ∈ , 4n 2(Π − rK) qσe − K t Wcr = q(Π(n, K) − (q − 1)σe) + qu(e) − − (1 − q + qr)K 4n Wcs
(92) for n ∈ (max(nrc , nsc ), +∞).
(93)
Note that welfare is increasing in n in each region. Therefore, it is sufficient to compare regions using the highest value for n in each case (we treat n here as a real number). Note that we always have that Wus ≤ Wcs because t some hedgers are not trading and the effect on prices cancel out in the total welfare. Evaluate Wcs at n = qσe−K : Wcs = Π + qu(e) −
qσe − K qσe (1 − 4r)K − rK = Π + qu(e) − + . 4 4 4
(94)
The choice of K that maximizes Wcs in Eq. (94) depends on r. We consider cases when r < 1/4 and when r ≥ 1/4 ¯ separately. First, if r < 1/4 then K(σ) maximizes Wcs and there is free entry. Welfare is Wcs = Π + qu(e) −
qσe (1 − 4r)qσe + = Π + qu(e) − rqσe. 4 4
(95)
Evaluate Wcr when n → +∞, Wcr = qΠ(+∞, K) + qu(e) + q(1 − q)σe − (1 − q + qr)K. Because lim
K→0
∂Π(+∞,K) ∂K
(96)
¯ that maximizes Eq. (96). From the = +∞, there exists a level of collateral 0 < K ∗ < K
implicit function theorem, and because Π(n, K) is concave in K, we have tends to the infinity.
∂K ∗ ∂r
< 0. Note that K ∗ tends to 0 as r
With these optimal levels of collateral, we study the inequality Wcs ≤ Wcr , Wcs ≤ Wcr ⇔ Π + qu(e) − rqσe ≤ q(Π(+∞, K ∗ ) − (q − 1)σe) + qu(e) − (1 − q + qr)K ∗ Π − qΠ(+∞, K ∗ ) + (1 − q + qr)K ∗ ≡ σl . ⇔σ≥ qe(1 − q + r)
(97)
If r ≥ 1/4, then K = 0 maximizes Wcs and access is constrained to n = nsc : Wcs = Π + qu(e) −
43
qσe . 4
(98)
We study the inequality Wcs ≤ Wcr , qσe ≤ q(Π(+∞, K ∗ ) − (q − 1)σe) + qu(e) − (1 − q + qr)K ∗ 4 Π − qΠ(+∞, K ∗ ) + (1 − q + qr)K ∗ ≡ σe . ⇔σ≥ qe( 54 − q)
Wcs ≤ Wcr ⇔ Π + qu(e) −
(99)
Note that σe increases with r, that σe < σl when r < 1/4 and that σe = σl when r = 1/4.
B.3
Proposition 3: Dealers’ Optimal CCP Rules
The sum of utilities of all dealers in the uncovered Safe equilibrium is nE[usd,u ] n is set as high as possible, n = nsu ≡
=
nyu (psu )(psu
t Π−rK ,
nE[usd,u ] =
2 − rK) = n t
Π − rK 2
2 .
(100)
to maximize Eq. (100) . This gives t 2 Π − rK t
Π − rK 2
2 =
Π − rK , 2
(101)
which has a maximum at K = 0. In a covered Safe equilibrium, dealers’ utility does not depend on collateral and decreases with n, t t nE[usd,c ] = n 2 = , (102) n n which has a maximum at n = n ≡ nsc and K = 0, we obtain
3t 2(Π−rK)
≡ nsc with K = 0. By comparing Eq. (101) at K = 0,and Eq. (102) at t 3t 2Π
≥
Π 2Π Π 2 1 ⇔ ≥ ⇔ ≥ , 2 3 2 3 2
(103)
which is always satisfied. Therefore, dealers prefer n = nsc and K = 0. When n is fixed, collateral can be used strategically by dealers to maximize profits. There are three possible cases ¯ depending on the value of n. First, if n ≤ nsu for any value of K ∈ [0, K(σ)), then from Eq. (101) it is optimal to s set K = 0 because collateral is costly. Second, if n ≤ nu for high-enough values of K and n ≥ nsc for low-enough of K, then there are two possible cases depending on the value of K. Since Eq. (101) decreases with K, K can be chosen as the minimum value such that n ≤ nsu , i.e., t 1 t n≤ ⇔K≥ Π− , (104) Π − rK r n and in that case, from Eq. (100) K = 1r Π − nt and nE[usd,u ]
2 =n t
!2 Π − r 1r Π − nt t = . 2 2n
(105)
Also, K can be chosen low-enough such that n ≥ nsc , n≥
nsc
3t 1 3t ≡ ⇔K≤ Π− , 2(Π − rK) r 2n
44
(106)
and from Eq. (102), utility of dealers is equal to 1 3t optimal for dealers to set K = r Π − 2n .
t n.
Since utility in Eq. (105) is always lower than
t n,
then it is
Third, and finally, if n ≥ nsc for any value of K, then since utility in Eq. (102) does not depend on K, the value of K does not affect the utility of the dealers.
B.4
Proposition 4: Hedgers’ Optimal CCP Rules
Using Eqs. (88), (89), and Proposition 1: Π − rK Π − rK t + qu(e) for n ∈ [1, ] t 4 Π − rK 3t t 5t s − rK for n ∈ [ , ] E[Uh,c ] = u(qe) − 4n 2(Π − rK) qσe − K 5t r E[Uh,c ] = q(Π(n, K) − (q − 1)σe) + qu(e) − − (1 − q + qr)K, 4n for n ∈ (max(nrc , nsc ), +∞).
s E[Uh,u ]=n
(107) (108) (109)
Hedger’s utility increases with n in each region. Therefore, it is sufficient to compare regions using the highest value for n in each case. First, we show that the uncovered equilibrium can never be optimal because the hedgers are better off in a covered equilibrium. In the uncovered Safe region we have t Π − rK Π − rK Π − rK s ]= E[Uh,u + qu(e) = + qu(e). (110) Π − rK t 4 4 Therefore, the maximum is reached with K = 0: Π4 + qu(e) and the price is Π2 . When K = 0, the price in the t uncovered Safe region is greater than the price in the Safe covered region at the highest n = qσe if: Π t Π ≥ ⇐⇒ σ ≤ , 2 n 2qe
(111)
which is satisfied by assumption. Hedgers pay lower transaction costs and a lower price in the covered case and s ]. If entry is restricted to the uncovered Safe equilibrium cannot be optimal. Consider now the case of E[Uh,c t s n = nc ≡ qσe−K , the utility is s E[Uh,c ] = Π + qu(e) − 5
qσe − K 5qσe (5 − 4r)K − rK = Π + qu(e) − + . 4 4 4
(112)
The choice of K to maximize Eq. (112) depends on the cost of collateral r. First, if r ≤ 54 then it is optimal to ¯ choose K(σ) and with free entry open access (n → +∞).16 In this case the utility of hedgers is s E[Uh,c ] = Π + qu(e) −
Second, if r >
5 4
5qσe (5 − 4r)K + = Π + qu(e) − rqσe. 4 4
then it is optimal to choose K = 0, access is constrained to n = s E[Uh,c ] = Π + qu(e) − 5
16
t qσe
and utility is
qσe . 4
¯ Choosing a higher level of collateral is not optimal because higher K increases nsc only when K < K(σ).
45
(113)
(114)
r ] is optimal when n → +∞, Finally E[Uh,c r E[Uh,c ] = qΠ(+∞, K) + qu(e) + q(1 − q)σe − (1 − q + qr)K.
Because lim
K→0
∂Π(+∞,K) ∂K
(115)
= +∞, there exists a level of collateral K ∗ > 0 that maximizes Eq. (115). This is the same
optimum found to maximize Eq. (96). Also, K ∗ decreases with r and tends to 0 as r tends to the infinity. s ] ≥ E[U r ] when r ≤ 5 : With these optimal levels of collateral, we study the inequality E[Uh,c h,c 4 s r E[Uh,c ] ≥ E[Uh,c ] ⇔ Π + qu(e) − rqσe ≥ q(Π(+∞, K ∗ ) − (q − 1)σe) + qu(e) − (1 − q + qr)K ∗
⇔σ≤
Π − qΠ(+∞, K ∗ ) + (1 − q + qr)K ∗ ≡ σh,l . qe(1 − q + r)
(116)
s ] ≥ E[U r ] when r > 5 : We also study the inequality E[Uh,c h,c 4
qσe ≥ q(Π(+∞, K ∗ ) − (q − 1)σe) + qu(e) − (1 − q + qr)K ∗ 4 Π − qΠ(+∞, K ∗ ) + (1 − q + qr)K ∗ ⇔σ≤ ≡ σh,e . qe(1 − q + 54 )
s r ] ≥ E[Uh,c ] ⇔ Π + qu(e) − 5 E[Uh,c
(117)
It is easy to see that σh,e is increasing with r, and that σh,l > σh,e Also, note that K ∗ decreases with r and tend ¯ to zero as r → +∞ and K ∗ ≤ K(σ).
46
Online Appendix to “How Should a Central Clearing Counterparty Reduce Risk? Collateral Requirements and Entry Restrictions” Jean-S´ebastien Fontaine Hector Perez Bank of Canada
Joshua Slive
This online appendix contains proofs for results in Section 4.3 and 4.4 of the main text.
A
Margins and monitoring
We can derive a dealer’s expected utility when the CCP uses monitoring by substituting rK for M and setting K = 0 in Eq. (25) and other equations: e j ), 0)] = yj pj − E[∆ e j + M/pj ≥ ∆ e j + M ] · Pr(pj ≥ ∆ e j + M ), E[ud,j ] = E[max(yj (pj − M − ∆ (118) which simplifies to E[ud,j ] =
yj (pj − M ) yj (q(pj − M ) + (1 − q)qσe)
if if
pM ≤ pj , pj ∈ [pM , pM )
(119)
where pM ≡ qσe + M and pM ≡ (q − 1)σe + M . Monitoring implies that we are in the Safe region. We use a modified definition of equilibrium to account for the default probability constraint: Definition 2 In a constrained symmetric price (Nash) equilibrium, every dealer chooses the same optimal price e j + M ) ≤ α. given the other dealers’ prices, subject to the default probability constraint Pr(pj < ∆ We say that dealers are not constrained when the constrained equilibrium price is the same as the equilibrium price without monitoring (the unconstrained equilibrium in Proposition 1). Otherwise, we say that dealers are constrained (in equilibrium). The following Lemma characterizes the unique constrained equilibrium. Lemma 1 A constrained symmetric price equilibrium exists and is unique. If α ≥ 1 − q, from Proposition 1 the equilibrium trivially satisfies the default constraint. If α < 1 − q then we have that: • if n ≤ nsc (K = 0) then dealers are not constrained, and the price in the Safe region is psc =
t n
+ M.
• if n ≥ nrc (K = 0) then dealers are constrained, they choose p = pM ≡ qσe + M and the utility of every dealer is n1 (pM − M ).
Proof. Suppose the equilibrium is unconstrained in the Safe covered region. Following identical steps from Proposition 1, the equilibrium price is t psc = + M, (120) n
47
and the equilibrium is in the Safe region if pM ≤ psc , n≤
t ≡ nsc (K = 0). qσe
(121)
Therefore, if n ≤ nsc (K = 0), dealers set the equilibrium price in the Safe region and the constraint Pr(pj < e j + (1 − r)K) ≤ α is satisfied for any α ≥ 0. Similarly, the equilibrium in the Risky region is obtained as in Eq. ∆ (73), t prc = (q − 1)σe + M + . (122) qn It is possible to find a threshold nrc (K = 0) ≡ max(nrc (K = 0), nsc (K = 0)) as in Proposition 1 where nrc (K = 0) is defined similarly to Eq. (75): q(Π(nrc , 0) − prc ) + qu(e) − dt = qu(e) ⇔ qΠ(nrc , 0) − q(q − 1)σe − qM =
3t . 2nrc
(123)
Therefore, if n > nrc (K = 0), the unconstrained equilibrium is in the Risky region and doesnot satisfy the default e j + M ≤ α only if pj ≥ pM . probability constraint if α < 1 − q. The constrained equilibrium satisfies Pr pj < ∆ If this is an uncovered equilibrium, then we need to solve the following optimization problem max yu (pj )(pj − M ) = 2 pj
Π − pj (pj − M ) t
s.t. pj ≥ pM ,
(124)
and the maximum is reached with pj = Π+M ≥ pM . However, this equilibrium is not uncovered since n ≥ nrc (K = 2 0): every hedger would trade. Next, consider the case where the constrained equilibrium is covered. We show that p = pM is a constrained symmetric Nash equilibrium. If every other dealer chooses p = pM , then any dealer j’s optimal choice solves, max yc (pj , pb = pM )(pj − M ) = pj
−pj + pM + n1 t (pj − M ) t
s.t.
pj ≥ pM .
(125)
The profit function is quadratic in the price and the (unconstrained) maximum is reached with p∗j = 21 (pM + nt +M ), but n ≥ nrc (K = 0) implies that pM > nt +M and that p∗j < pM . By concavity of the utility function the constrained optimum for dealer j must be pM . At this equilibrium every dealer has utility 1 (p − M ). n M
(126)
This constrained symmetric equilibrium is unique. Any other candidate equilibrium where all dealers set pb > pM cannot be a constrained symmetric equilibrium. If every other dealer chooses pb > pM , then, from Eq. (125), any p + nt + M ) if 12 (b p + nt + M ) ≥ pM , or p∗j = pM if 12 (b p + nt + M ) < pM . In any case p∗j < pb dealer j chooses p∗j = 21 (b because if p∗j = 21 (b p + nt + M ), then 1 t t (b p + + M ) < pb ⇔ + M < pb, 2 n n
(127)
and this is true because n ≥ nrc (K = 0) implies that nt + M < pM , and pM < pb by assumption. Also, if p∗j = pM then p∗j < pb by assumption, implying that there cannot be a constrained symmetric equilibrium.
48
A.1
Proposition 5: Optimal CCP Rules (K, n and monitoring)
We compare monitoring with margin requirements. The uncovered equilibrium is not considered since a covered equilibrium is welfare improving. Recall that prices do not show up in total welfare. From Eq. (92), total welfare s , is given by in the covered region with monitoring (α < 1 − q), Wc,M s Wc,M = Π + qu(e) −
t − M. 4n
(128)
Total welfare increases with n and therefore it is optimal to choose n → +∞: s Wc,M = Π + qu(e) − M.
(129)
When using collateral, consider a covered equilibrium in the Safe region: Wcs = Π + qu(e) −
t − rK for nsc ≤ n ≤ nsc . 4n
(130)
¯ is optimal when r < 1/4, and there is free entry (i.e., It is optimal to choose n = nsc . From Eq. (94), K(σ) s s nc → +∞). Wc is then: Wcs = Π + qu(e) − rqσe. (131) If r ≥ 1/4, then K = 0 is optimal and access is constrained to n = nsc (see Eq. 98). Total welfare is Wcs = Π + qu(e) −
qσe . 4
(132)
Note that if α ≥ 1 − q (no monitoring), then K ∗ (as in Eq. 96) and n → +∞ are optimal in the Risky covered region: Wcr = qΠ(+∞, K ∗ ) + q(u(e) + (1 − q)σe) − (1 − q + qr)K ∗ . (133) To find the optimum, we need to compare Eq. (129) and Eq. (131) (or Eq. 132 and Eq. 133, depending on the value of r). If r < 1/4, we compare Eq. (129) and Eq. (131): s Wcs ≥ Wc,M ⇔ Π + qu(e) − rqσe ≥ Π + qu(e) − M ⇔ M σ ≤ σ1 (M, r) ≡ . rqe
(134)
Then, we compare Eq. (131) and Eq. (133): Wcs ≥ Wcr ⇔ Π + qu(e) − rqσe ≥ qΠ(+∞, K ∗ ) + q(u(e) + (1 − q)σe) − (1 − q + qr)K ∗ ⇔ Π − qΠ(+∞, K ∗ ) + (1 − q + qr)K ∗ σ ≤ σl ≡ . qe (1 − q + r)
(135)
If r ≥ 1/4, we compare Eq. (129) and Eq. (132): qσe ≥ Π + qu(e) − M ⇔ 4 4M σ ≤ σ2 (M, r) ≡ . qe
s Wcs ≥ Wc,M ⇔ Π + qu(e) −
49
(136) (137)
Then, we compare Eq. (132) and Eq. (133): Wcs ≥ Wcr ⇔ Π + qu(e) −
qσe ≥ qΠ(+∞, K ∗ ) + q(u(e) + (1 − q)σe) − (1 − q + qr)K ∗ ⇔ 4 Π − qΠ(+∞, K ∗ ) + (1 − q + qr)K ∗ σ ≤ σe ≡ . qe 54 − q
(138)
For any value of r, we compare Eq. (129) and Eq. (133): Π + qu(e) − M ≥ qΠ(+∞, K ∗ ) + q(u(e) + (1 − q)σe) − (1 − q + qr)K ∗ ⇔ Π − qΠ(+∞, K ∗ ) + (1 − q + qr)K ∗ − M σ ≤ σ3 (M, r) ≡ , qe (1 − q)
(139)
where σ3 (M, r) = σl when r = 0 and M = 0. Assuming that M is not too high (M ≤ M ), so σ3 (M, r) ≥ 0. To summarize: • If r < 1/4 (collateral is not too costly) – If σ ≤ σ1 (M, r) and σ ≤ σl then the optimum is in the Safe region, free entry (n → +∞), no risk ¯ controls, and K = K(σ). – If σ ≥ σ1 (M, r) and σ ≤ σ3 (M, r) then the optimum is in the Safe region, free entry (n → +∞), risk controls, and K = 0. – If σ ≥ σ3 (M, r) and σ ≥ σl then the optimum is in the Risky region, free entry (n → +∞), no risk controls, and K = K ∗ . • If r ≥ 1/4 (collateral is costly) – If σ ≤ σ2 (M, r) and σ ≤ σe then the optimum is in the Safe region, access constrained to n = nsc , no ¯ risk controls, and K = K(σ). – If σ ≥ σ2 (M, r) and σ ≤ σ3 (M, r) then the optimum is in the Safe region, free entry (n → +∞), risk controls, and K = 0. – If σ ≥ σe and σ ≥ σ3 (M, r) then the optimum is in the Risky region, free entry (n → +∞), no risk controls, and K = K ∗ .
B B.1
Endogenous hedging Proposition 6: Equilibrium with Endogenous Hedging Strategies
Dealers choose a price and one of two hedging strategies, σ and σδ . We use a similar definition of equilibrium as in Proposition 1, with dealers choosing simultaneously two strategies: Definition 3 In a symmetric price (Nash) equilibrium in the covered region with endogenous hedging, every dealer chooses optimally the same price and hedging strategy taking as given the other dealers’ prices and hedging strategies. t Note that nsc (δ) = δσqe−K becomes the upper threshold for the Safe covered region when σδ = δσ is selected (and equal to the lower threshold for the Risky covered region). To probe this proposition, we first derive three intermediary results.
50
Lemma 2 Assume that nsc ≤ n ≤ nsc (δ). Then, • If all dealers choose σδ , when dealer j deviates and chooses σ, the optimal price for dealer j is still in the Safe region. • If all dealers choose σ, when dealer j deviates and chooses σδ , the optimal price for dealer j is still in the Safe region.
Proof. By assumption on n, when all dealers choose σδ , the symmetric equilibrium must be in the Safe region (see Proposition 1). Assume dealer j deviates and chooses σ. If the optimal price is in the Safe region, then from first order conditions in Eq. (58), dealer j also chooses price psc = nt + rK (as in Eq. 59). This price is above p for dealer j because condition in Eq. (60) is satisfied by assumption: n ≤ nsc (δ) < nsc =
t . qσe − K
(140)
Also, dealer j is not better off setting a price in the Risky region. We can apply the same arguments as in the “Deviation from a symmetric equilibrium” section in Proposition 1. From Eq. (70) in Proposition 1, the utility of a deviating dealer increases with σ. Therefore, he is better off setting a price in the Risky region when σδ is Π chosen. But because σδ ≤ 2qe , then Eq. (71) is satisfied and this is not a profitable deviation. For the second part of the Lemma, we use similar arguments. By assumption about n, when all dealers choose σ, the symmetric equilibrium must be in the Safe region (because n < nsc (δ) < nsc ). If the optimum price is in the Safe region, then dealer j chooses price psc = nt + rK. This price is above qσδ e for dealer j because condition in Eq. (60) is satisfied. Also, dealer j is better off in the Safe region. If dealer j deviates to the Risky region we can apply the same condition from Eq. (71) which is satisfied because σδ ≤ σ. Therefore, the price equilibrium is in the Safe region. Lemma 3 Assume that nsc (δ) < n ≤ nsc . Then, • If all dealers choose σ, it cannot be a profitable deviation for dealer j to choose σδ and set the price in the Safe region. In a profitable deviation, price must be in the Risky region. • If all dealers choose σδ , it cannot be a profitable deviation for dealer j to choose σ and set the price in the Risky region. In a profitable deviation, price must be in the Safe region.
Proof. Assume that all dealers choose σ. From Proposition 1, they choose a price psc = nt + rK in the Safe region. If dealer j deviates and chooses σδ then dealer j could choose a price in the Safe region or in the Risky region. If dealer j chooses a price in the Safe region, then psc = nt + rK cannot be the optimal price because t nsc (δ) = δqσe−K < n and t t < n ⇔ + rK < p ≡ δqσe − K + rK, (141) δqσe − K n and therefore nt + rK is not in the Safe region. By concavity of the utility function, dealer j chooses the minimium price in the Safe region, pj = δqσe − K + rK. This cannot be a profitable deviation for dealer j because dealer j is better off choosing σ and setting psc = nt + rK in the Safe region. Given that other dealers set a price nt + rK, by concavity of the profit function dealer j is better off choosing σ and psc = nt + rK. Therefore, in a profitable deviation, dealer j must set a price in the Risky region. For the second part of the lemma, assume that all dealers 51
t K + qn in the Risky region. choose σδ . By assumption on n, they set an equilibrium price prc = (q − 1)σδ e + 1−q+qr q If dealer j deviates and chooses σ then dealer j could optimally choose a price in the Safe region, or in the Risky region. Following Proposition 1, dealer j’s optimal choice in the Risky region is:
E[uj,c ] =
−qpj + q(q − 1)σδ e + (1 − q + qr)K + t
t n
+ n1 t
(pj − (q − 1)σe −
(1 − q + qr)K )q. q
(142)
.
(143)
The best deviating price is pj =
(q − 1)σδ e +
(1−q+qr)K q
+
2t qn
+ (q − 1)σe +
(1−q+qr)K q
2
The demand at this price is yc (pj , pi =
q(q − 1)σδ e q(q − 1)σe 1 − + . 2t 2t n
(144)
Therefore, the deviating profit is 1 E[uj,c ] = t
q(q − 1)σδ e q(q − 1)σe t − + 2 2 n
2
1 = t
q(1 − q) t (σ − σδ ) + 2 n
2 .
(145)
Because σ < σδ , then the expected utility from Eq. (145) is smaller than the utility obtained in Proposition 1 (see Eq. 86). Therefore, this is not a profitable deviation. Hence, in a profitable deviation, dealer j must set a price in the Safe region. Lemma 4 Assume that n > nsc . Then, • If all dealers choose σδ , when dealer j deviates and chooses σ, the optimal price for dealer j is still in the Risky region. • If all dealers choose σ, when dealer j deviates and chooses σδ , the optimal price for dealer j is still in the Risky region.
Proof. By assumption, when all dealers choose σδ the symmetric equilibrium must be in the Risky region because t nrc (σδ ) < nrc (σ) < n and the price must be (q − 1)σδ e + 1−q+qr K + qn . When a dealer j deviates and chooses σ, q dealer j cannot set a profitable price in the Safe region. We apply the sufficient conditions given in Eq. (82) and (83) in Proposition 1. For the second part of the Lemma, we use similar arguments. By assumption about n, the symmetric equilibrium must be in the Risky region. When a dealer j deviates, chooses σδ , and sets a price in the Safe region, then the dealer is worse off because we can apply same sufficient conditions as in Eq. (82) and (83). Therefore, the equilibrium price is in the Risky region. B.1.1
Proof of first part of Proposition 6
Assume all dealers choose σδ . From Lemma 2, if a dealer j deviates and chooses σ, the optimal price for dealer j is still in the Safe region. Therefore, since there is no profitable deviation, the symmetric equilibrium is such that all dealers set a price (59) in the Safe region and choose σδ . Similar arguments are used to find the symmetric equilibrium where all dealers choose σ and set a price Eq. (59) in the Safe region.
52
B.1.2
Proof of second part of Proposition 6
First, we show that σ and p = nt + rK is a symmetric equilibrium (in the Safe region ). From Proposition 1 and by assumption on n, dealers set a symmetric equilibrium price psc = nt + rK in the Safe region. Let assume that dealer j deviates and chooses σδ . From Lemma 3 it cannot be a profitable deviation to set a price in the Safe region. Therefore, the deviating price must be in the Risky region. From Eq. (70) the utility of the deviating dealer is (q − 1)σδ e (q − 1)ΠK − (1 − q + qr)K rK t 2 1 −q + + + . (146) E[uj,c ] = t 2 2 2 n This deviation is not profitable as long as −q(q − 1)σδ e + (q − 1)ΠK − (1 − q + qr)K + rK ≤ 0. which is satisfied using Eq (71) and the fact that σδ ≤ σ. Hence, σ and p =
t n
(147)
+ rK is a symmetric equilibrium.
Second, we show that σδ and a price in the Risky region is a symmetric equilibrium. From Proposition 1 and by t K + qn in the Risky region. If dealer j deviates assumption on n, all dealers set a price prc = (q − 1)σe + 1−q+qr q and chooses σ then, by Lemma 3 it is not a profitable deviation for dealer j to set a price in the Risky region. Therefore, dealer j chooses a price in the Safe region. We apply the sufficient conditions given in Eqs. (82) and (83) t in Proposition 1. Therefore deviating is not profitable for the dealer j. Hence, σδ and psc = (q−1)σδ e+ 1−q+qr K + qn q is a symmetric equilibrium. B.1.3
Proof of third part of Proposition 6
First, we show that selecting σ cannot be a symmetric equilibrium. In this case, by Proposition 1, dealers set a t price prc = (q − 1)σe + 1−q+qr K + qn in the Risky region. If one dealer j deviates and chooses σδ , then by Lemma q 4, dealer j sets a price in the Risky region: E[uj,c ] =
−qpj + q(q − 1)σe + (1 − q + qr)K + t
t n
+ n1 t
(pj − (q − 1)σδ e −
(1 − q + qr)K )q. q
(148)
.
(149)
The optimal price is pj = By assumption, Eq. (74) implies
(q − 1)σδ e +
(1−q+qr)K q
+ (q − 1)σe +
(1−q+qr)K q
2 2t qn
< 2σe −
pj =
2K q
+
2t qn
and this price is in the Risky region:
(q − 1)σδ e + (q − 1)σe +
2(1−q+qr)K q
+
2t qn
2 σδ + σ K (1 − q + qr)K < (q − 1) e + σe − + 2 q q (q − 1)σδ e (q + 1)σe = + + (r − 1)K 2 2 2qσδ e < − K + rK < qσδ e − K + rK. 2
53
(150)
The demand at this price is 2(1−q+qr)K
q (q − 1)σδ e + (q − 1)σe + q − t 2 q(q − 1)σδ e q(q − 1)σe 1 =− + + . 2t 2t n
+
2t qn
+
q(q − 1)σe 1 − q + qr 2 + K+ t t n (151)
Therefore, the deviating profit is 1 E[uj,c ] = t
q(1 − q) t (σδ − σ) + 2 n
2 .
(152)
Since σδ > σ, Eq. (152) is greater than utility obtained in Proposition 1 (Eq. 86) and equal to nt2 . Therefore, σ cannot be a symmetric equilibrium because any dealer can increase its expected utility by deviating from the other dealers and setting σδ . Second, we show that σδ is a symmetric equilibrium. Again, by Proposition 1, dealers set price prc = (q − 1)σδ e + 1−q+qr t K + qn in the Risky region. If one dealer j deviates and chooses σ, then by Lemma 4, dealer j sets a price q in the Risky region. The deviating profit of dealer j is 1 E[uj,c ] = t
q(1 − q) t (σ − σδ ) + 2t n
2 .
(153)
Since σ < σδ , Eq. (153) is smaller than utility obtained in Proposition 1 in Eq. (86), equal to nt2 . Therefore, this deviation is not profitable. Hence, dealers choose the highest σδ and a price in the Risky region.
B.2
Proposition 7: Optimal CCP rules (K and n) with endogenous hedging strategies
From Proposition 6, depending on the value of n, we can find the symmetric equilibrium in prices and in the choice of σ or σδ . In the case where nsc (δ) < n ≤ nsc , there are two possible symmetric Nash equilibria that could be played by dealers. We assume that with probability λ the Safe equilibrium is played, and with probability 1 − λ the Risky equilibrium is played. Using Eqs. (90), Eq. (91), and Proposition 1, we can find the following welfare values for each region: Wcs = u(qe) −
t − rK for nsc ≤ n ≤ nsc (δ) 4n
(154)
t − λrK − (1 − λ)(1 − q + qr)K 4n + λu(qe) + (1 − λ)qu(e) + (1 − λ)q(Π(n, K) − (q − 1)σδ e)
Wcr (λ) = − for
nsc (δ)
(155)
nsc .
Wcr = q(Π(n, K) − (q − 1)σδ e) + qu(e) −
t 4n
for n > nsc ,
(156)
where Wcr (λ) is the expected welfare assuming that the Safe equilibrium is played with probability λ and the Risky equilibrium with probability 1 − λ. For any value of K, welfare is increasing in n in each region. Therefore, it is sufficient to compare regions using the highest value for n in each case. Also, we do not consider the uncovered region because we always have that Wus ≤ Wcs because some hedgers are not trading in the uncovered region and t the effect on prices cancel out when calculating total welfare. Consider the case of Wcs when n = qσδ e−K : Wcs = u(qe) −
qσδ e − K qσδ e (1 − 4r)K − rK = u(qe) − + . 4 4 4
54
(157)
The optimal choice of K to maximize Wcs in Eq. (157) depends on the value of r. Also, the optimal choice of K to maximize Wcr (λ) in Eq. (155) depends on the value of r : qσe − K − λrK − (1 − λ)(1 − q + qr)K 4 + λu(qe) + (1 − λ)qu(e) + (1 − λ)q(Π(n, K) − (q − 1)σδ e) qσe (1 − 4λr − 4(1 − λ)(1 − q + qr))K =− + 4 4 + λu(qe) + (1 − λ)qu(e) + (1 − λ)q(Π(n, K) − (q − 1)σδ e)
Wcr (λ) = −
where
(1 − 4λr − 4(1 − λ)(1 − q + qr))K 4 is increasing in λ. If r < 1/4 then then the highest value of K is optimal to maximize Eq. (157), therefore ¯ δ ≡ qσδ e is optimal and there is free entry. In this case, the region nsc (δ) < n ≤ nsc collapses to 0 because K nsc (δ) → +∞. Therefore, we compare welfare in the Safe and Risky regions using the following inequality: r(λ) =
Wcs ≤ Wcr ⇔ Π + qu(e) − rqσδ e ≤ q(Π(+∞, K ∗ ) − (q − 1)σe) + qu(e) − (1 − q + qr)K ∗ ⇔ Π − qΠ(+∞, K ∗ ) + (1 − q + qr)K ∗ ≤ qσe(1 − q + δr) ⇔ Π − qΠ(+∞, K ∗ ) + (1 − q + qr)K ∗ σ≥ ≡ σl,δ < σl . qe(1 − q + δr)
55
(158)
