risk incentives in an interbank network - banque de france · bank 1 (z) bank 2 (0) bank 3 (z)...

Introduction Example General Framework Numerical Example Policy Conclusion

Risk Incentives in an Interbank Network

Miguel Faria-e-CastroNew York University

Endogenous Financial Networks and Equilibrium DynamicsBanque de France

July 10, 2015

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Financial Crises and Bank Herding

I Non-anonymous markets

I Limited liability

I Risk shifting

Incentives to

I Correlate failures

I Endogenous intermediation?

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This Model

Main features:

I Non-anonymous market for debt contracts

I Liquidity shocks motivate risky debt contracts

I Direct (counterparty) and indirect (systemic) risk

I Endogenous network formation

I Arbitrary number of heterogeneous banks

Main results:

I “Risk-based” intermediation

I Bank herding, risk-shifting behavior

I Network structure amplifies risk

I Policy: caps on lending, LOLR, creditor bailouts

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Literature

I OTC Financial Markets: Duffie et al. (2005), Lagos andRocheteau (2009), Afonso and Lagos (2012), Acharya andBisin (2014).

I Financial Networks: Allen and Gale (2000), Freixas et al.(2000), Atkeson et al. (2013), Fique and Page (2013),Acemoglu et al. (2015).

I Strategic Formation of Financial Networks: Cohen-Coleet al. (2011), Zawadowski (2013), Farboodi (2014), Glodeand Opp (2014).

I Bank Herding: Acharya (2009), Acharya and Yorulmazer(2008).

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Example

I 2 periods, 3 banks

I Banks own long-term asset a, senior claims v

bad : a = v − ε < v

good : a = v + ε > v

I Payoffs are non-pledgeable ⇒ debt contracts

I Banks endowed with net liquidity position z

I Liquidity technology at rate r , or borrow from bank withliquidity surplus

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Example continuedAssume the following payoff structure

State Bank 1 Bank 2 Bank 3

s1 v − ε v + ε v + εs23 v + ε v − ε v − εs3 v + ε v + ε v − ε

I 1 and 3 are negatively correlated

I 2 is (partially) correlated with both

Endowments:

z1 = −zz2 = 0

z3 = z

Default is socially costly, with zero recovery rate.6 / 23


Autarky

Bank 1(−z)

Bank 2(0)

Bank 3(z)

I Bank 1 borrows from outside at rate r

I Bank 3 willing to offer a cheaper contract

I Gains from trade, avoid costly liquidity technology

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3 lends to 1

Bank 1(−z)

Bank 2(0)

Bank 3(z)

L3→1

I Dominates autarky as long as r > 1

I Socially optimal

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May not be stable...

Bank 1(−z)

Bank 2(0)

Bank 3(z)

L3→2L2→1

Under certain conditions,I Bank 2 willing to lend to 1 at a lower rateI Bank 3 willing to lend to 2 at a lower rateI Pairwise stable, but inefficient equilibriumI Bank 3 now defaults in state s23!

Limited Liability + Non-Anonymous Market = Systemic Risk9 / 23


General Model - Environment

I Two dates t = 0, 1

I One good, cash (liquidity)

I N islands, each populated by a representative bank anddepositors

I Banks indexed by long-term assets and initial deposits (ai , di )

I t = 0: liquidity shock zi , banks may engage in debt contracts.

I t = 1: Illiquid long-term assets pay Ri ∼ g(R), contracts aresettled

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Interbank Lending

I Banks lend to each other to clear liquidity positionsI Lending from bank i to bank j

ìj

I Interbank lending forms a credit network

L =

0 `12 . . . `1N

`21 0 . . . `2N...

.... . .

...`N1 `N2 . . . 0

This network is

I directedìj 6= `ji

I weightedìj ∈ R+

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Flow of Funds and Payoffs

Banks must clear liquidity positions at t = 0∑j 6=i

`ij + ci︸︷︷︸outflows

=∑j 6=i

`ji + vi + zi︸︷︷︸inflows

Profits at t = 1

π+i (Ri ,θ) =

Riai + ci +∑j 6=i

θj rij`ij −∑j 6=i

rji`ji − f (vi )− di

+

wheref ′ > 0 f ′′ > 0

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Networked Markets

I Bank i borrows from a competitive market i

I Other banks can participate to lend to i

Bank 1Market 1, rB1

Bank 2Market 2, rB2

Bank 3Market 3, rB3

`12

`32

`23

`13`21

`31

I One interest rate in each market rBiI Identity of the lender is irrelevant (converse not true!)

Bi ≡∑j 6=i

`ji

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Equilibrium at t = 1

At t = 1, banks have chosen their portfolios L, c.

Limited liability ⇒ bank unable to repay debts for low values of Ri

1. Available assets: Riai + ci − f (vi ) + repayments

2. Senior debt: di

3. Junior debt: rBi Bi

Costs of Default: Fraction δ of total assets is recovered, δRiai

I π+i (Ri ,θ) depends on counterparty defaults and repayments

I ...which in turn may depend on πi !

I Fixed point problem as in Eisenberg and Noe (2001), withcosts of default.

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Banks’ Problem at t = 0

Take θ(R; L, c) and prices as given.

maxci ,vi ,{ìj}j 6=i ,Bi

ER[π+i (R,θ)]

subject to

FF :∑j 6=i

ìj + ci = Bi + vi + zi

Optimality implies rBi = f ′(vi ) ≡ ri ,

ci : 1 ≤ ri

ìj : E[θj |πi ≥ 0]rj ≤ ri , ∀j 6= i

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EquilibriumA Competitive Equilibrium consists of a collection of portfolioallocations L ∈ RN×N , c ∈ RN ; prices r ∈ RN ; and a repaymentrule θ(R; L, c) ∈ RN such that:

1. Given expected repayments E[θ(R; L, c)] and prices r,portfolio allocations L, c solve the bank’s problem for eachrepresentative bank i ∈ N.

2. Prices are such that all N interbank credit markets clear

Bi =∑j 6=i

`ji , ∀i ∈ N

and defined as ri = f ′(vi ) whenever Bi = 0.

3. For each joint realization of long-term asset payoffsR ∼ G (R), θ(R; L, c) constitutes a repayment equilibrium.

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Results

I Perceived Repayments

θ̃i→j ≡ E[θj |πi ≥ 0]

I Interest Rate Dispersion: if i → j

rj ≥ rj θ̃i→j = ri ≥ ri

I Herding Behavior: if i → j

rj θ̃i→j = ri

I Risk-Based Intermediation: i → j , j → k, but i 9 k if

θ̃j→k × θ̃i→j ≥ θ̃i→k17 / 23


Six Bank Example

Parameter Value

N 6ai 1di 0.8

f (v)(ψ1 + 1

2ψ2v)· v

ψ1 1ψ2 5δ 0.7

Distributions

Gi ≡ U [0.8, 1.2]

zi ∼ U [−0.15, 0.15],N∑i=1

zi = 0

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Six Bank Example

Bank si ri Pr[default] Risk Shifting Risk Sharing

1 0.05 2.46% 2.40% 2.40% −2.40%2 0.15 0.00% 0.00% 0.00% 0.00%3 −0.10 34.23% 25.50% 2.05% 74.50%4 0.01 2.14% 2.10% 2.10% −2.10%5 −0.05 11.54% 10.35% −0.85% 19.20%6 −0.06 17.72% 15.05% −0.55% 37.10%

1

2

3

4

5

6

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Policy - Cap on Lending∑i 6=j

`ij ≤ f (ai , di , zi ;φ)

Equivalent to a leverage constraint

0.02 0.04 0.06 0.08 0.1 0.12

0.15

0.2

0.25

Volume

φ0.02 0.04 0.06 0.08 0.1 0.12

29

30

31

32

33

34

No. Connect ions

φ

0.02 0.04 0.06 0.08 0.1 0.12

9.84

9.85

9.86

9.87

Welfare

φ0.02 0.04 0.06 0.08 0.1 0.12

0

1

2

3

4

5x 10

−3

φ

Risk Measures

Sharing

Shifting

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Policy - Discount Window Lending

I Captures riskier borrowersI Reduces both direct and indirect risks in the systemI May destroy risk-sharing links and induce formation of

risk-shifting ones

1 1.1 1.2 1.3 1.4 1.5

0.05

0.1

0.15

0.2

0.25Volume

r̄

1 1.1 1.2 1.3 1.4 1.5

20

22

24

26

28

30

32

No. Connect ions

r̄

1 1.1 1.2 1.3 1.4 1.59.9

10

10.1

10.2

10.3

Welfare

r̄

1 1.1 1.2 1.3 1.4 1.50

0.005

0.01

0.015

0.02

0.025

0.03

r̄

Risk Measures

Sharing

Shifting

1 1.1 1.2 1.3 1.4 1.5

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Discount Borrowing

r̄

1 1.1 1.2 1.3 1.4 1.5

0.01

0.02

0.03

0.04

0.05

Expected Costs

r̄ 21 / 23


Policy - Creditor Bailouts

I Expectation of bailout lowers rates and curbs risk shifting

I No time consistency issues

I Systemic insurance, as in Dell’Ariccia and Ratnovski (2013)

Policy E[W ] E[r ] E[vol ] E[costs]

No Policy 10.0619 1.68% 0.357 0Unexpected 10.0721 1.68% 0.357 0.0347

Expected 10.1090 0.00% 0.487 0.0334

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Conclusion

I Key ingredients:

1. Idiosyncratic shocks2. Limited liability3. Non-anonymous markets

I Results:

1. Risk-based intermediation2. Bank herding3. Endogenous amplification of risk4. Role for policy

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risk incentives in an interbank network - banque de france · bank 1 (z) bank 2 (0) bank 3 (z)...

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