Self-Fulfilling Debt Crises, Revisited: TheArt of the Desperate Deal

Mark Aguiar Satyajit Chatterjee

Harold L. Cole Zachary Stangebye

September 2, 2016

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Desperate Deals

I What does a sovereign do when faced with the prospect of afailed auction?

I In Cole-Kehoe they default

I In practice, they look for alternative financing

I Often tolerating high spreads.

I Occasionally ending up defaulting.

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Desperate Deals

I What does a sovereign do when faced with the prospect of afailed auction?

I In Cole-Kehoe they default

I In practice, they look for alternative financing

I Often tolerating high spreads.

I Occasionally ending up defaulting.

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Portugal

I Difficulty in raising funds through auctions starting in 2011

I Private placement of bonds in January 2011 (reported aspurchased by China)

I Official ¤78 billion package in May 2011I ¤34.2 billion dispersed in 2011 and ¤28.5 in 2012

I Dual auction in October 2012I Bought “September 2013” bonds

I Sold “October 2015” bonds

I Goal: Clear “space” for auctions in 2013/2014

I Launched new issue in early 2013

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Self-Fulfilling Debt Crises, Revisited

I What is a self-fulfilling debt crisis?

I Cole-Kehoe: Failed auction today generates default today

I Zero price for any amount of bonds issued

I Choose to default - hence self-fulfilling.

I Our model generates less extreme rollover crises than in C-K.

I Keeps C-K’s “static” rollover-crisis multiplicity but considers aricher notion of “failed” auctions.

I Generate spikes in spreads without defaults, like the data.

I Our model has some surprising welfare implications.(Buybacks not all bad.)

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Self-Fulfilling Debt Crises, Revisited

I What is a self-fulfilling debt crisis?

I Cole-Kehoe: Failed auction today generates default today

I Zero price for any amount of bonds issued

I Choose to default - hence self-fulfilling.

I Our model generates less extreme rollover crises than in C-K.

I Keeps C-K’s “static” rollover-crisis multiplicity but considers aricher notion of “failed” auctions.

I Generate spikes in spreads without defaults, like the data.

I Our model has some surprising welfare implications.(Buybacks not all bad.)

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Self-Fulfilling Debt Crises, Revisited

I What is a self-fulfilling debt crisis?

I Cole-Kehoe: Failed auction today generates default today

I Zero price for any amount of bonds issued

I Choose to default - hence self-fulfilling.

I Our model generates less extreme rollover crises than in C-K.

I Keeps C-K’s “static” rollover-crisis multiplicity but considers aricher notion of “failed” auctions.

I Generate spikes in spreads without defaults, like the data.

I Our model has some surprising welfare implications.(Buybacks not all bad.)

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FrameworkEndowment: Stochastic Growth as in Aguiar-Gopinath

I Small open economy

I Discrete time

I Markov process for endowment growth

yt ≡ lnYt =t∑

s=0

gs + zt

= yt−1 + gt + zt − zt−1

I gt follows an AR(1) process and zt is iid

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FrameworkBonds: Random Maturity as in Chatterjee-Eyigungor

I Sovereign issues non-contingent “random-maturity” bonds

I Bonds mature with Poisson probability λ

I Assume that in a non-degenerate portfolio of bonds, a fractionλ matures with probability 1

I Perpetual-youth bonds allow for tractably incorporatingmaturity without adding separate state variables for eachcohort of bond issuances

I Bonds pay coupon r∗ each period up to and including maturity

I Payments due in period t: (r∗ + λ)Bt

I New issuances: Bt+1 − (1− λ)Bt

I Bt+1

Yt≤ b̄ prevents Ponzi schemes.

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FrameworkLenders

I Risk averse OLG lenders (risk aversion not conceptuallyimportant)

I Financial markets are segmented: Finite wealth available toparticipate in bond market

I Tractability: Period t’s set of investors hold bonds for oneperiod and then sell them to a new cohort of investors at startof t + 1

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TimingModification of Cole-Kehoe

Initial State:s

AuctionB ′ − (1− λ)B

at priceq(s,B ′)

Settlement

No Default

Default

V R(s,B ′)

VD(s)

Next Pe-riod: s ′

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Settlement

I Auction Revenue:

x(s,B ′) ≡ max{q(s,B ′)(B ′ − (1− λ)B), 0

}I Proceeds from auction are held in escrow until government

makes repayment decision

I If government repays, can draw on settlement funds forrepayment and consumption

I If government defaults, auction revenue disbursed tobondholders in proportion to face value of claims:

RD(s,B ′) =x(s,B ′)

B ′ + (r∗ + λ)B

I If B ′ < (1− λ)B: Buyback funds are paid out and gone.

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Settlement

I Auction Revenue:

x(s,B ′) ≡ max{q(s,B ′)(B ′ − (1− λ)B), 0

}I Proceeds from auction are held in escrow until government

makes repayment decision

I If government repays, can draw on settlement funds forrepayment and consumption

I If government defaults, auction revenue disbursed tobondholders in proportion to face value of claims:

RD(s,B ′) =x(s,B ′)

B ′ + (r∗ + λ)B

I If B ′ < (1− λ)B: Buyback funds are paid out and gone.

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Settlement

I No Default:

I Old lenders receive (r∗ + λ)B

I New lenders hold B ′ into next period

I Default:

I Old lenders receive RD(s,B ′)(r∗ + λ)B

I New lenders receive RD(s,B ′)B ′

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The Government’s ProblemPreferences

I Sovereign government makes all consumption-savings-defaultdecisions

I Sovereign’s preferences over sequence of aggregateconsumption {Ct}∞t=0:

E∞∑t=0

βtu(Ct)

with

u(C ) =C 1−σ

1− σ

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Value Functions

I V (s) denotes start-of-period value of government

I V R(s,B ′) denotes value if having auctioned B ′ − (1− λ)Bthe government decides to repay (r∗ + λ)B at settlement

I VD(s) denotes the value of defaulting at settlement(independent of amount auctioned) ⇒ lose fraction φ ofendowment until “redemption” from default status

I Strategic default implies:

V (s) = max

⟨max

B′≤b̄YV R(s,B ′),VD(s)

⟩

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Bellman Equations

I If repay...

V R(s,B ′) = u(C ) + βE[V (s ′)|s,B ′

],

with

C = Y + q(s, b′)(B ′ − (1− λ)B)− (r∗ + λ)B.

I If default...

VD(s) = u(Y D) + βEV E (s ′)

V E (s) = u((1− φ)Y ) + β(1− ξ)E[V E (s ′)

∣∣∣∣s]+ βξE

[V (s ′)

∣∣∣∣s,B ′ = 0

]

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Equilibrium

I Markov Equilibrium

I States s ∈ S elements of s are:I Endowment: (Y , g , z)

I Bonds: B

I Beliefs: ρ

I Policy Functions:

I Bond-issuance: B(s)

I Default: D(s, b′) ∈ [0, 1]

I Bond-demand: µ∗(s, b′)

I Price function: q(s,B ′) ∈ [0, 1]

I Market clearing: µ∗(s,B ′)W = q(s,B ′)B ′.

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Multiplicity of Equilibria

I There is a “static” multiplicity in a given period

I Arises because of timing convention: Failed auction even forsmall levels of bond issuances can be supported in equilibrium

I Suppose the continuation equilibrium is held constant and weconsider alternative price schedules for the current period’sauction

I Consider two scenarios for today’s auction, holding constantequilibrium behavior going forward

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Cole-Kehoe Crisis

I Zero price for any B ′ ≥ (1− λ)B:

V R(s, (1− λ)B)

= u (Y − (r∗ + λ)B) + βE[V (s ′)|s,B ′ = (1− λ)B

]< VD(s).

Non-Crisis

I A pair (q̃, B̃) such that:

V R(s,B ′) =

u(Y − (r∗ + λ)B + q̃[B̃ − (1− λ)B]

)+ βE

[V (s ′)|s, B̃

]> VD(s).

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Constructing Equilibria

I For a given state s and equilibrium policy functions, we canprice a bond conditional on no-default at settlement thisperiod

I Such within-period commitment is the assumption ofEaton-Gersovitz models

I Our exercise assumes only commit for the current period, andthen no commitment going forward

I Let qEG (s,B ′) denote this price

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Crisis Zone

I Define a “Crisis Zone” by evaluating V R(s,B ′) under qEG :

C ≡{s ∈ S

∣∣∣∣ maxB′≤(1−λ)B

V R(s,B ′) ≤ VD(s) &

maxB′≥(1−λ)B

V R(s,B ′) ≥ VD(s)

}.

I This set identifies states in which:I Faced with qEG , the government would have no reason to

default

I Faced with q = 0 for B ′ > (1− λ)B, it will default

I Crisis zone combination of high B and low (Y , g , z)

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Self-Fulfilling Crises

I In Cole-Kehoe equilibrium, a rollover crisis is an equilibrium inwhich prices are zero for any positive amount of debt issuance

I We relax this and consider a broader set of crisis equilibria

I Build on the mixed strategy equilibria of Aguiar and Amador(2014)

I That model had potential buybacks and randomization off theequilibrium path

I We now bring this onto the equilibrium path and consider crisisissuances

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Desperate DealsRethinking Failed Auctions

Desperate Deal Price Schedule: An Indifference Condition

qD(s,B ′) =

{q̃

∣∣∣∣VD(s) = u(Y − (r∗ + λ)B + q̃[B ′ − (1− λ)B]

)+ βE

[V (s ′)|s,B ′

]}.

I Support as mixed-strategy equilibrium with appropriate choiceof D(s,B ′) ∈ [0, 1]

I Feasible for 0 ≤ qD(s,B ′) ≤ qEG (s,B ′)

I Government indifferent over feasible B ′ so make selection.

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Crisis Price Schedule

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

B0

Y

qD

qEG

I Desperate deals (indifference) prices in black. EG (good)prices in red. CK price is 0 for issuances.

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Evolution of Beliefs

I iid probability of crisis if s ∈ CI q = qD

I Assume B(s) = B ′/Y = B ′−1/Y−1 (prior debt-output ratioequals new ratio)

I Contrast with “No Desperate Deals” modelI Just original Cole-Kehoe Eq.

I q = 0 for any positive issuance in a crisis

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CalibrationPre-Set Parameters

I Calibrate endowment to Mexico 1980Q1-2001Q4

I Set risk aversion coefficient for sovereign and lenders at 2

I Set quarterly risk free rate to 1%

I Set average maturity to 8 quarters

I Set average exclusion to 8 quarters

I Probability ρ = rC is 14

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The Role of Desperate Deals

Target Moment Data Benchmark No Deals

BY 65.6% 66.6% 64.5%r − r∗ 3.4% 3.4% 3.4%σ(r − r∗) 2.5% 2.5% 0.1 %Default Freq 2.0% 2.0% 2.1%

I Massive increase in volatility relative to No Deals(and without deterministic growth, nonlinear default costs andvolatility output).

I Lower debt level with No Deals reduces difference in defaults.(Desperate deals reduces borrowing discipline.)

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Crises and Default

I Crises:I Fraction of quarters in crisis zone: 8.0%

I Rollover crises occur 2% of the time

I Defaults:I Default rate 2% per annum (targeted)

I 97% of defaults coincide with negative growth

I 70% of defaults coincide with rollover crisis

I Conditional on rollover crisis, default on average 15% of time

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Distribution of r − r ∗

Conditional on CrisisFr

eque

ncy

0 .1 .2 .3 .4Spread

Default Unrealized Default Realized

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Equilibrium Price ScheduleWith and Without Crisis Issuances

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

B0

Y

q

BenchmarkNo Deals

I Better price schedule from lenders’ anticipation of bettertreatment with desperate deals.

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Welfare Resultssome surprises

I Desperate Deals equilibrium slightly dominates No DealsI Prices better conditional on crisis

I Government captures this through ex ante price schedule

I Off-setting effect: Higher debt and more defaults due to morefavorable spreads

I Buybacks during rollover crises raises welfareI Counters Bulow-Rogoff’s Buyback Boondoggle

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Welfare Resultssome surprises

I Desperate Deals equilibrium slightly dominates No DealsI Prices better conditional on crisis

I Government captures this through ex ante price schedule

I Off-setting effect: Higher debt and more defaults due to morefavorable spreads

I Buybacks during rollover crises raises welfareI Counters Bulow-Rogoff’s Buyback Boondoggle

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Welfare Resultssome surprises

I Desperate Deals equilibrium slightly dominates No DealsI Prices better conditional on crisis

I Government captures this through ex ante price schedule

I Off-setting effect: Higher debt and more defaults due to morefavorable spreads

I Buybacks during rollover crises raises welfareI Counters Bulow-Rogoff’s Buyback Boondoggle

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Conclusion

I Models based upon Eaton-Gersovitz environment strugglewith matching spread volatility.

I Cole-Kehoe environment also generates limited volatility inspreads and extreme outcome conditional on a crisis

I In our approach, self-fulfilling crises generate a mixture offundamental and belief-driven defaults, and desperate deals

I Crises now look more like what we see in the data.

I Extreme spreads look like we see in the data.

I Interesting welfare implications (relative to classicBulow-Rogoff).

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