Bail-ins and Bailouts: Incentives, Connectivity, and ... · Bail-ins and Bailouts: Incentives, Connectivity, and Systemic Stability Benjamin Bernard1 Agostino Capponi 2Joseph E. Stiglitz

Bail-ins and Bailouts: Incentives,Connectivity, and Systemic Stability

Benjamin Bernard1 Agostino Capponi2 Joseph E. Stiglitz2

1UCLA2Columbia University

Task Force Meeting, New York

August 29, 2018

Introduction Model of Financial Network Model of Intervention Credibility of the Threat Numerical Example

Introduction


Systemic risk in an interbank network

shock

a

Financial institutions are connected through bilateral contracts (Lij)ij .

Shock hits banks’ outside assets, leading to fundamental defaults F .

Shock propagates through network, leading to contagious defaults C.

Bail-ins and Bailouts Benjamin Bernard 1 / 21



shock

Fa

Financial institutions are connected through bilateral contracts (Lij)ij :






shock

Fa







shock

Fa






Research Objectives

Existing studies take the banks as inactive agents, and measure the cas-cading contagion effects:

there is either no intervention to stop a default cascade,

intervention follows exogenously specified protocols.

2-step agenda:

Endogenize intervention after the occurrence of defaults,

Endogenize network formation.



Research Objectives

Existing studies take the banks as inactive agents, and measure the cas-cading contagion effects:

there is either no intervention to stop a default cascade,

intervention follows exogenously specified protocols.

2-step agenda:

Endogenize intervention after the occurrence of defaults,

Endogenize network formation.



Methods of intervention

Bailout: Government provides liquidity through taxpayer money.

Among others, Citigroup, AIG Insurance, and UBS were bailed out bytheir respective government during the recent crisis.

Bail-in: Creditors voluntarily forgive part of the debt in exchange for equityin the reorganized company.

The hedge fund Long Term Capital Management was bailed-in in 1998.Under the supervision of the Federal Reserve Bank of New York, a totalof 14 banks agreed to participate in a recapitalization plan.

Subsidized/assisted bail-in: Contributions from regulator and banks.

Bear Stearns was sold to JP Morgan Chase for $1.2 billion with agovernment injection of $30 billion.



Model of Financial Network


Financial Network

Financial network is described by:

Banks have bilateral exposures Lij , denoting j ’s liability to i . Denote

Lj =n∑

i=1

Lij , πij =Lij

Lj.

Each bank i has investment e i in assets outside the interbank network.

Each bank i has net cash holdings c i .

Welfare losses: Liquidation of assets / bankruptcy costs:

Liquidation losses are equal to (1− α)× liquidated assets.

Bankruptcy cost is (1− β)× recovered assets.



Financial Network

Financial network is described by:

Banks have bilateral exposures Lij , denoting j ’s liability to i . Denote

Lj =n∑

i=1

Lij , πij =Lij

Lj.

Each bank i has investment e i in assets outside the interbank network.

Each bank i has net cash holdings c i .

Welfare losses: Liquidation of assets / bankruptcy costs:

Liquidation losses are equal to (1− α)× liquidated assets.

Bankruptcy cost is (1− β)× recovered assets.



Model of Intervention


Stages of the game

The game has the following stages:

1. The government proposes an assisted bail-in (b, s).

2. Each bank i chooses ai , whether or not it accepts the proposal.

3. If some banks reject, government has three options:

(a) Proceed with the rescue, but make up for the contributions of defectingbanks. Resulting welfare losses are wλ(b, s, a).

(b) Resort to public bailout (0, sP) with welfare losses wP = wλ(0, sP).

(c) Abandon the rescue, which leads to welfare losses wN = wλ(0, 0).

An SPE σ is weakly renegotiation proof (WRP) if after every history ht ,there exists no continuation SPE, which Pareto-dominates σ|ht .



Stages of the game











Stages of the game











Stages of the game











Why weak-renegotiation?

Suitable equilibrium selection criterion because the coordination of abail-in is precisely a negotiation

During the bail-in of LCTM, Peter Fisher of the FRBNY sat downwith representatives of LTCM’s creditors

Implausible that they would have ever agreed on a bail-in plan thatis Pareto-dominated.



Accepting Equilibrium

Lemma

Let (b, s) be a feasible proposal of a complete bail-in. In an acceptingequilibrium a, bank i with bi > 0 accepts if and only if

1. wλ(b, s, (0, a−i )) ≥ min (wN ,wP), and

2. bi − s i ≤

∑n

j=1 πij(Lj − pjN) if wN ≤ wP ,

0 if wP < wN .

Bank i is willing to contribute only if

there is no accepting equilibrium without bank i (no free-riding).

share of benefits accruing to bank i outweighs cost of contribution.



Complete rescues

In a complete rescue (b, s) with accepting equilibrium a, p̄(b, s, a) = L.Maximal incentive-compatible net contribution is

ηi :=

min

(∑nj=1 π

ij(Lj − pjN), (c i + αe i + (πL)i − Li )+)

if λα ≥ 1− α,

min(∑n

j=1 πij(Lj − pjN), (c i + (πL)i − Li )+

)if λα < 1− α.

If wP < wN , then p̄(sP) = L, hence no bank is willing to contribute.

⇒ No-intervention threat is not credible.

If wP ≥ wN , the threat is credible and regulator will aim to include bankswhose incentive-compatible contributions have the largest welfare-impact.



Complete rescues


ηi :=

min

(∑nj=1 π


if λα ≥ 1− α,

min(∑n


)if λα < 1− α.






Complete rescues


ηi :=

min

(∑nj=1 π


if λα ≥ 1− α,

min(∑n


)if λα < 1− α.






Equilibrium bail-in plan

Theorem

Let ν i denote the welfare-impact of largest incentive-compatible contri-bution of bank i . Let i1, . . . , in−|F| be a non-increasing ordering of ν i .

1. If wP < wN , the unique SPE outcome is a public bailout.

2. If wN ≤ wP , the unique WRPE outcome is a subsidized bail-in with

w∗ = min

(wP −

m∑j=1

ν ij , wN − ν im+1

),

where m := min(k∣∣∣ wP −

∑kj=1 ν

ij < wN

).

w∗ wPw

wN

w{i1,i2,i3}



Equilibrium bail-in plan

Theorem

Let ν i denote the welfare-impact of largest incentive-compatible contri-bution of bank i . Let i1, . . . , in−|F| be a non-increasing ordering of ν i .

1. If wP < wN , the unique SPE outcome is a public bailout.

2. If wN ≤ wP , the unique WRPE outcome is a subsidized bail-in with

w∗ = min

(wP −

m∑j=1

ν ij , wN − ν im+1

),

where m := min(k∣∣∣ wP −

∑kj=1 ν

ij < wN

).

w∗ wPw

wN

w{i1,i2,i3}



Credibility of the Regulator’s Threat


Amplification of the shock

Lemma

Let χ0 and χN denote the aggregate losses accruing to the financial sectorimmediately after the initial shock and after a default cascade withoutintervention, respectively. The threat is credible if and only if

χN − χ0 ≤ λχ0 + min (λα, 1− α)n∑

i=1

ì (p̄(sP)).

Larger weight λ to tax-dollars improves credibility of the threat.

From literature without intervention: dense interconnections, low re-covery rates, and a large initial shock amplify the initial shock.




Lemma


χN − χ0 ≤ λχ0 + min (λα, 1− α)n∑

i=1

ì (p̄(sP)).






Lemma


χN − χ0 ≤ λχ0 + min (λα, 1− α)n∑

i=1

ì (p̄(sP)).






Lemma


χN − χ0 ≤ λχ0 + min (λα, 1− α)n∑

i=1

ì (p̄(sP)).





Credibility and Welfare Losses

Lemma

For fixed L, c , e, α, β, the equilibrium welfare losses after intervention aresmaller in network π1 than in network π2 if the regulator’s threat iscredible in network π1 but not in network π2.



Strongly Concentrated Welfare Losses

Definition

Consider two financial systems (L, π, c , e), (L, π′, c , e). For any m < n,interbank losses are strongly m more concentrated in network π than inπ′ if

(i) η(i)(π) ≥ η(i)(π′) for every i ≤ m + 1, where x (i) denotes thei th-largest entry of vector x ,

(ii) wN(π) ≥ wN(π′).

Welfare losses in π equal the sum of the banks’ incentive compatiblecontributions: wN(π) =

∑ni=1 η

i (π).

Largest m + 1 losses in network π are higher than the correspondinglosses in π′.



Welfare Losses and Concentration

Proposition

Consider two financial systems (L, π, c , e), (L, π′, c , e) with homogeneouscash holdings such that L = πL. Suppose

wN(π′) ≤ wN(π) ≤ wN(π′) + λ(ηim(π)+1 (π)− ηim(π)+1 (π′)).

If interbank losses are strongly m(π) more concentrated in network πthan in π′, then equilibrium welfare losses are lower in π than in π′.

Even if welfare losses are higher in the least concentrated network inthe absence of intervention, they become lower in the presenceof intervention.



Takeaway

A model for financial intervention, where

the structure of the intervention plan arises endogeneously as theresult of the strategic interaction between regulator and banks,

the regulator cannot commit to an ex-post suboptimal resolution policy.

Equilibrium intervention plan:

Prior studies have shown that, if shocks are not too high, dense in-terconnections are preferable to sparser interconnections

The presence of intervention enlarges the range of shocks for whichsparser networks are socially preferrable:

Bail-ins can be tailored in a way that the benefits accrue morestrongly to the contributorLowers the banks’ incentive to free-ride and increases the sizes of thebank’s incentive compatible contribution.As the amplification of a shock is smaller in a more concentratednetwork, the government can more credibly stand by idly



Takeaway

A model for financial intervention, where

the structure of the intervention plan arises endogeneously as theresult of the strategic interaction between regulator and banks,

the regulator cannot commit to an ex-post suboptimal resolution policy.

Equilibrium intervention plan:

Prior studies have shown that, if shocks are not too high, dense in-terconnections are preferable to sparser interconnections

The presence of intervention enlarges the range of shocks for whichsparser networks are socially preferrable:

Bail-ins can be tailored in a way that the benefits accrue morestrongly to the contributorLowers the banks’ incentive to free-ride and increases the sizes of thebank’s incentive compatible contribution.As the amplification of a shock is smaller in a more concentratednetwork, the government can more credibly stand by idly



Thank you!



Assisted Bail-ins

Each bank i receives a subsidy s i to purchase a part bi of debt fromfundamentally defaulting banks.

This is essentially a set of centralized transfers between banks:

bi1{ai=1} − s i is bank i ’s net contribution to the bail-in.∑i (s

i − bi1{ai=1}) is the government’s contribution.

Includes public bailouts and privately backed bail-ins as special cases.

After transfers:

Liabilities are cleared as before with c i (b, s, a) = c i + s i − bi1{ai=1}.

Bank i ’s value is V i (b, s, a) := V i(p̄(b, s, a)

)and welfare losses are

wλ(b, s, a) := wλ(p̄(b, s, a)

)+ λ

n∑i=1

(s i − bi1{ai=1}).



General rescues

Regulator minimizes welfare losses over subsets B of banks to be rescued.

For a fixed B, coordination of bail-in works analogously:

Only banks contribute, for which the threat is credible,

Largest contributors are added first.



Numerical Example


Specific choice of network topologies

(a) The complete network. (b) The ring network.

We compare the credibility in the ring network πR and the complete net-work πC in a financial system with Li = L and c i = c for every bank i .

A shock hits the financial system such that

there is 1 fundamentally defaulting bank with shortfall χ0,

nl banks are lowly capitalized with value of outside asset el ,

nh banks are highly capitalized with eh > el .



Specific choice of network topologies

(a) The complete network.

FL

L

L

L

LH

H

H

H

H

(b) The ring network.

We compare the credibility in the ring network πR and the complete net-work πC in a financial system with Li = L and c i = c for every bank i .

A shock hits the financial system such that

there is 1 fundamentally defaulting bank with shortfall χ0,

nl banks are lowly capitalized with value of outside asset el ,

nh banks are highly capitalized with eh > el .



Comparison of welfare losses

Welfare losses of optimal bail-in is of the form

wP − ηi1 − ηi2 − . . .− ηim ,

where ηi is the maximal incentive-compatible contribution of bank i , re-lated to i ’s losses in absence of intervention.

In densely connected network: shock distributed among many banks.Incentive to free-ride is large → small contributions.

In sparsely connected network: few creditors who would suffer largelosses in case of no intervention → large contributions.

1

00 χ∗ χ∗

β

shock sizes

Lower in ringUndecided



Comparison of welfare losses

Welfare losses of optimal bail-in is of the form

wP − ηi1 − ηi2 − . . .− ηim ,

where ηi is the maximal incentive-compatible contribution of bank i , re-lated to i ’s losses in absence of intervention.

In densely connected network: shock distributed among many banks.Incentive to free-ride is large → small contributions.

In sparsely connected network: few creditors who would suffer largelosses in case of no intervention → large contributions.

1

00 χ∗ χ∗

β

shock sizes

Lower in ringUndecided



Without intervention: clearing payments

A clearing payment vector p is a solution to

pi =

Li if c i + αe i +∑

j πijpj ≥ Li ,

β(c i + αe i +

∑j π

ijpj)+

otherwise.

Banks in D(p) :={i∣∣ pi < Li

}default.

Bank i has to liquidate ì (p) = min(

1α

(Li − c i −

∑j π

ijpj)+, e i

).

Senior creditors (depositors) of i lose δi (p) =(c i + αe i +

∑j π

ijpj)−

.




A clearing payment vector p is a solution to

pi =

Li if c i + αe i +∑

j πijpj ≥ Li ,

β(c i + αe i +

∑j π

ijpj)+

otherwise.

Banks in D(p) :={i∣∣ pi < Li

}default.

Bank i has to liquidate ì (p) = min(

1α

(Li − c i −

∑j π

ijpj)+, e i

).

Senior creditors (depositors) of i lose δi (p) =(c i + αe i +

∑j π

ijpj)−

.




With clearing payment vector p, the value of each bank i is

V i (p) =

0 i ∈ D(p),

c i + e i − (1− α)ì (p) + (πp)i − pi otherwise.

Welfare losses are equal to

wλ(p) = (1− α)n∑

i=1

ì (p) +1− ββ

∑i∈D(p)

pi + λ∑

i∈D(p)

δi (p).

︸︷︷︸deadweight losses

︸︷︷︸depositor’s losses

Lemma

Set of clearing payments forms a (non-empty) complete lattice. Greatestclearing payment p̄ Pareto-dominates all other clearing payments.





V i (p) =

0 i ∈ D(p),



wλ(p) = (1− α)n∑

i=1

ì (p) +1− ββ

∑i∈D(p)

pi + λ∑

i∈D(p)

δi (p).



Lemma






V i (p) =

0 i ∈ D(p),



wλ(p) = (1− α)n∑

i=1

ì (p) +1− ββ

∑i∈D(p)

pi + λ∑

i∈D(p)

δi (p).



Lemma




Weak renegotiation proofness

Government makes a proposal such that at least 5 banks need to acceptfor the government to proceed.

Any response where at most 3 banks accept is a trivial SPE.

Outcome is identical to all banks rejecting the proposal.

For given proposal (b, s), a continuation equilibrium a = (a0, a1, . . . , an)is called an accepting equilibrium if a0(b, s, a1, . . . , an) = “proceed”.

Lemma

All accepting equilibria are WRP. Rejecting equilibria are WRP if and onlyif there exists no accepting equilibrium.



Weak renegotiation proofness

Government makes a proposal such that at least 5 banks need to acceptfor the government to proceed.

Any response where at most 3 banks accept is a trivial SPE.

Outcome is identical to all banks rejecting the proposal.

For given proposal (b, s), a continuation equilibrium a = (a0, a1, . . . , an)is called an accepting equilibrium if a0(b, s, a1, . . . , an) = “proceed”.

Lemma

All accepting equilibria are WRP. Rejecting equilibria are WRP if and onlyif there exists no accepting equilibrium.



Comparison of credibility in different networks

If threat is credible in π and π′, two counteracting forces:

incentive-compatible contributions are larger in π than in π′,

since wN(π) ≥ wN(π′), the no free-riding condition implies fewerbanks included in bail-in in π.

Interbank losses are strongly more concentrated in π than in π′ if incentivecompatible contributions are larger in π than in π′

Lemma

Let (L, π, c , e) and (L, π′, c , e) be regular with wN(π) = wN(π′). If lossesare strongly more concentrated in π than in π′, then w∗(π) ≤ w∗(π

′).



Comparison of credibility in different networks

Prior studies have shown that dense interconnections may amplifyrather than absorb initial losses (Acemoglu et al. (2015), Haldane(2009))

Our results suggest that this phenomenon is strengthened in thepresence of intervention:

In a more concentrated network, bail-ins can be tailored in a waythat the benefits accrue more strongly to the contributorLowers the banks’ incentive to free-ride and increases the sizes of thebank’s incentive compatible contribution.As the amplification of a shock is smaller in a more concentratednetwork, the government can more credibly stand by idlyEnlarge the region of shock sizes, for which a bail-in strategy iscredible.



Literature

Franklin Allen and Douglas Gale: Systemic risk, interbank relations and liquidityprovision by the central bank, Journal of Political Economy, 108 (2000), 1–33

Daron Acemoglu, Asuman Ozdaglar, and Alireza Tahbaz-Salehi: Systemic risk andfinancial stability in financial networks, American Economic Review, 105 (2015),564–608

Larry Eisenberg and Thomas H. Noe: Systemic risk in financial systems, Manage-ment Science, 47(2) (2001), 236–249

Darrell Duffie and Chaojun Wang: Efficient Contracting in Network FinancialMarkets, Graduate School of Business, Stanford University

Matt Elliott, Ben Golub, and Matthew Jackson: Financial Networks and Conta-gion, American Economic Review, 104(10) (2014), 3115–3153

Leonard C. G. Rogers and Luitgard A. M. Veraart: Failure and rescue in an inter-bank network, Management Science, 59(4) (2013), 882–898


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Bail-ins and Bailouts: Incentives, Connectivity, and ... · Bail-ins and Bailouts: Incentives, Connectivity, and Systemic Stability Benjamin Bernard1 Agostino Capponi 2Joseph E. Stiglitz