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Mathematics and Financial Economics

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01.07.2020

Systemic credit freezes in financial lending networks

verfasst von: Daron Acemoglu, Asuman Ozdaglar, James Siderius, Alireza Tahbaz-Salehi

Erschienen in: Mathematics and Financial Economics | Ausgabe 1/2021

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Abstract

This paper develops a network model of interbank lending, in which banks decide to extend credit to their potential borrowers. Borrowers are subject to shocks that may force them to default on their loans. In contrast to much of the previous literature on financial networks, we focus on how anticipation of future defaults may result in ex ante “credit freezes,” whereby banks refuse to extend credit to one another. We first characterize the terms of the interbank contracts and the patterns of interbank lending that emerge in equilibrium. We then study how shifts in the distribution of shocks can result in complex credit freezes that travel throughout the network. We use this framework to analyze the effects of various policy interventions on systemic credit freezes.

Vorheriger Artikel Dual representations for systemic risk measures based on acceptance sets

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For example, see Acemoglu et al. [2], Cabrales et al. [17], Elliott et al. [25], Gai and Kapadia [29], Jorion and Zhang [34], and Allen and Gale [9].

Brunnermeier [14] and Duffie [22] for general discussions.

Also see Zawadowski [40], Farboodi [27], and Erol [26], who study how endogenous formation of financial networks can shape systemic risk.

This stage is introduced to rule out equilibria that may arise due to coordination failures: banks may refuse to extend credit to others if they worry that no bank will subsequently extend them a credit line with sufficiently favorable terms. The withdrawal stage in the model rules out the possibility of such miscoordinations. See Di Maggio and Tahbaz-Salehi [20] for a discussion.

This assumption thus rules out the possibility of “fractional defaults” as in Eisenberg and Noe [23] and Acemoglu et al. [2], whereby banks may only default on a fraction of their obligations to their creditors.

This statement assumes that, given financial network $({{\,\mathrm{{\mathbf {R}}}\,}},{{\,\mathrm{{\mathbf {x}}}\,}})$, the repayment equilibrium at $t=2$ is unique for all realizations of ${{\,\mathrm{{\mathbf {z}}}\,}}$. We show in the Appendix that, for all ${{\,\mathrm{{\mathbf {z}}}\,}}$, the repayment equilibrium is indeed unique for any financial network emerging in equilibrium.

This restriction is introduced in order to ensure that the best response of banks when offering interest rates to their potential borrowers converge to the equilibrium point in question as we take the limit of the trembles towards zero.

See Appendix A for a formal definition of strong equilibrium and more details on its implications for equilibrium refinement. This concept is closely related to “trembling-hand perfect equilibrium” in extensive-form games.

Because the standard Lebesgue measure is not well-defined over the space of continuous probability distributions, we use the notion of generic probability distribution from [39]. This notion is based on the use of “probes,” such as polynomial functions of order k as approximations to smooth probability distributions. Generic properties are those that hold for almost all order k polynomials. See Appendix C for more details.

Throughout we refer to credit freezes in order to emphasize that following a change in the distribution of shocks ${\mathcal {Q}}$, the decision not to lend by some banks leads to stoppages in credit flows.

This result only holds for economies with a single entrepreneur. As we show in Sect. 5, the impact of increased competition on lending is ambiguous when there are multiple entrepreneurs in the network.

Note, however, that the chain subnetwork along which lending takes place is endogenously determined, as it depends on the structure of ${{\,\mathrm{{\mathbf {G}}}\,}}$ and the shock distribution ${\mathcal {Q}}$. Hence, limiting attention to arbitrary chain networks is not without loss of generality.

Formally, there is always a strong equilibrium where either (i) $1 flows from the depositor to the entrepreneur or (ii) there is a systemic credit freeze. However, there may be other equivalent strong equilibria, for instance, where bank 1 offers a prohibitively large interest rate to bank 2, but with no flow of funds anywhere in the chain.

Notice the contrast with Corollary 1: while the corollary considers the addition of a link to a network (with a given set of banks), this theorem considers adding a new bank to a chain network (which thus removes a link and adds two new links to the new bank).

We restrict $\zeta $ to be in (0, 1) so that no bank can fully absorb a counterparty loss. This assumption guarantees that any default cascade that begins at some agent j propagates upstream to all its direct and indirect lenders.

See Chateauneuf et al. [18].

These observations also imply that if we allow banks to choose the riskiness of their outside investments, limited liability may push them towards riskier assets, but with significant negative systemic implications.

A high bankruptcy cost F encourages banks to diversify in order to avoid costly default. Our assumption that $F < {\underline{F}}$ ensures that the lack of diversification as shocks become more correlated does not dominate the increase in expected profits from making the loans.

Note that, in this proposition, we assume large values of F to control for risk attitudes, as in Proposition 2 (see the discussion in Sect. 4.2).

Recall from Theorem 3 that while, in equilibrium, interbank borrowing and lending always occurs in a tree structure, the opportunity network ${\mathbf {G}}$ need not be a tree. We now separately consider the implications of a tree opportunity network.

We require sufficiently large F for the same reason as in Propositions 2 and 5: bank $\Omega $ with high risk-bearing capacity is more risk-averse to lending; this guarantees that there is no shift in risk attitudes by channeling funds through the additional intermediary.

In the context of Example 1, inserting a risk-bearing bank resets the compensating interest rate differential between the borrower and lender back to 0. Hence, if a fraction of the banks have risk-bearing capacity, then these differentials do not grow unboundedly as the chain gets longer.

Note that this effect is distinct from the one emphasized by Farboodi [27] . In Farboodi [27], core banks have higher-return but riskier projects, allowing peripheral banks to obtain intermediation rents using their own source of funds, which in turn creates inefficient levels of systemic risk. In our case, we obtain essentially the opposite result: voluntary intermediation comes from the fact that peripheral banks can insulate themselves and reduce potential default cascades by channeling funds through larger “safer” intermediaries who are unlikely to default.

For discussions of optimal policies in models based on ex post contagion, see Bernard et al. [12] and Kanik [35].

For simplicity, we are modeling this policy intervention as a direct liquidity injection or transfer. It is equivalent to a subsidized loan from the central bank. In particular, if the bank has to repay the central bank an amount $r_i \epsilon _i$ (where $r_i$ is the discount interest rate from the central bank) at time $t=2$, provided that doing so does not put the bank in default, then all of our results apply identically.

While in reality asset purchases do not target a single bank, we think of such a policy as targeting the distressed assets composing this bank’s balance sheet. For instance, the Fed’s policy to purchase mortgage-backed securities (MBS) during the crisis was in-part designed to target large dealer banks whose balance sheets comprised of sizable MBS positions.

Providing the depositor with liquidity does not change her incentives for lending, so the central bank must condition these funds on their use for interbank lending. The policy is equivalent to one where the central bank acts as a “depositor” itself, and directly lends to banks connected to the depositor in ${\mathbf {G}}$ (but not others).

This adverse shift corresponds to a leftward shift of the distribution function ${\mathcal {Q}}(z_j)$. The amount of the shift, $\delta $, is the anticipated liquidity shock bank j now faces.

This result is in the same spirit as Jackson and Pernoud [33], but relates to ex ante rescue policies (before the realization of liquidity shocks) to ensure lending markets continue to function when future solvencies are in-question.

See https://www.bis.org/publ/bcbs283.pdf.

Note that we cite Theorem 2 in the proof of Theorem 3 to show it is a directed tree, but we are leveraging only uniqueness of the interest rate and borrowing stages, and not the unique repayment equilibrium, which is the only time we use Theorem 3 in this proof.

Note that these expectations depend on the offer order ${\mathcal {O}}$, but are simply integrals over realizations of liquidity shocks, as is the form in Ott and Yorke [39], given that banks are not indifferent between making multiple offers for $R_{i \rightarrow j}$ when $x_{i \rightarrow j} > 0$ as shown in the following paragraph.

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Titel: Systemic credit freezes in financial lending networks
verfasst von: Daron Acemoglu
Asuman Ozdaglar
James Siderius
Alireza Tahbaz-Salehi
Publikationsdatum: 01.07.2020
Verlag: Springer Berlin Heidelberg
Erschienen in: Mathematics and Financial Economics / Ausgabe 1/2021
Print ISSN: 1862-9679
Elektronische ISSN: 1862-9660
DOI: https://doi.org/10.1007/s11579-020-00272-z

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