Default and Interest Rate Shocks: Renegotiation Matters*
Victor Almeida†

Carlos Esquivel‡

Timothy J. Kehoe§

Juan Pablo Nicolini¶
February 7, 2023

[Most recent version]
Abstract
We develop a sovereign default model with endogenous re-entry to financial markets via
debt renegotiation. We use this model to evaluate how shocks to risk-free interest rates trigger
default episodes through two channels: borrowing costs and expected renegotiation terms after
default. The first channel makes repayment less attractive when risk-free interest rates are highdue to higher borrowing costs. The second channel works through the expected subsequent
renegotiation process: when risk-free rates are high, lenders are willing to accept a higher
haircut in exchange for resuming payments. Thus, high risk-free rates imply better renegotiation terms for a borrower, making default more attractive ex-ante. We calibrate the model
to study the 1982 Mexican default, which was preceded by a drastic increase in federal funds
rates in the US. We find that the renegotiation process is key for reconciling the model to thewidespread narrative that the increase in US interest rates triggered the 1982 default episode.
Keywords: Sovereign default, Renegotiation, Interest rate shocks.
JEL Codes: F34, F41* ACKNOWLEDGEMENTS† Carleton College‡ Rutgers University; Email: carlos.esquivel@rutgers.edu§ University of Minnesota, Federal Reserve Bank of Minneapolis, and National Bureau of Economic Research;

Email: tkehoe@umn.edu¶ Federal Reserve Bank of Minneapolis and Universidad Torcuato Di Tella; Email: juanpa@utdt.edu

1Introduction

The 1980s featured the most wide-spread sovereign debt crisis in history. The left panel of Figure 1
shows how it compares with historically difficult periods such as the Napoleonic Wars, World Wars
I and II, and the Great Depression. Starting in 1982 with the Mexican default, the crisis reacheda peak of 25 countries suspending some or all debt payments by 1985. There were on average 19
countries in default in each year during the decade, out of which 11 were from Latin America. Dueto its depth and length, the economic collapse that accompanied the debt crisis during the 1980s isreferred to as the “lost decade” in Latin America.
Figure 1: Debt crises and interest rates, 1980s case
25

25

20

20

Countries in default

15

10

5

6
15

4
2

10

0
5

US real interest rate
0
1800

0
1830

1860

1890

1920

1950

1980

8

Percentage points

Number of countries in default

Number of countries in default(left axis)

10(right axis)-2-41959 1964 1969 1974 1979 1984 1989 1994 1999 2004

The data of countries in default are from ?. The US real interest rate is the annual yield on 1-year US treasury bonds
minus observed inflation.

The crisis was preceded by aggressive interest rate increases in the U.S. by the then Chairmanof the Federal Reserve Paul Volcker, which were intended to tame rising inflation. The right panelof Figure 1 shows how real interest rates in the U.S. were a leading indicator of the number ofcountries in default during the 1980s, with the initial rate-hike being shortly followed by a cascadeof sovereign defaults.
This “Volcker shock” is often credited for being the main trigger of the crisis. The usual
narrative focuses on the direct impact that higher interest rates had on debt service, since mostdebt had been contracted at floating rates in foreign currency (see for instance Ocampo (2014)
and Tourre (2017)). We argue that such an interest rate shock has an additional indirect effect on1

default incentives through the terms that emerge from an eventual debt renegotiation.1
We develop a stylized model of sovereign debt renegotiation and show that governments get
more beneficial outcomes when interest rates are high.2 The intuition is simple: lenders’ opportunity cost of holding delinquent debt is higher when interest rates are high, so they are willing toaccept a smaller recovery in order for payments to resume. We compare two mechanisms throughwhich risk-free interest rates affect default incentives ex-ante. The first—which we call the standard mechanism—works through lenders discounting future debt payments at a higher rate. This
directly lowers the market price of government debt, which in turn makes servicing current debt
more expensive. The second—which we call the renegotiation mechanism—refers to how a highrisk-free rate improves the expected terms for the government of a future renegotiation, which inturn increases the value of defaulting in the present. In addition, these expectations also decrease
the market price of government debt in good standing, since lower payments from future recovery after default are also priced in. Thus, while the standard mechanism only decreases the valueof repaying the debt, the renegotiation mechanism both increases the value of defaulting and decreases the value of repaying. This intuition suggests that renegotiation considerations play a more
important role when interest rate shocks trigger default events, which is what we explore in ourquantitative exercise.
We build on our stylized environment to develop a quantitative sovereign default model withendogenous debt renegotiation and persistent shocks to the risk-free interest rate.3 We calibrate
the model to the 1982 Mexican default. We use data prior to 1982 to calibrate the parametersthat govern the income process, borrowing, and default probabilities. Importantly, we calibrate theparameters that govern the bargaining game so that the average haircut generated endogenously by
the model equals the haircut to Mexican debt under the Brady plan. We find that, in the ergodic
distribution, 22 percent of interest rate hikes trigger a default event. In order to compare therelevance of both mechanisms we consider two counterfactual economies that feature a fixed debt
haircut instead of endogenous renegotiation. For the first, we consider a haircut of 100 percent,
1 As documented by Sturzenegger and Zettelmeyer (2008) and Benjamin and Wright (2009), among others, de-faulting countries and their lenders negotiate reductions on the defaulted debt.
2 Our result is consistent with Guimaraes (2011), who proves for in a similar simple environment that shocks torisk-free interest rates have a larger impact on the incentive compatible level of debt, which directly affects the extent
of debt relief.
3 Guimaraes (2011) shows that the effect of interest rate shocks on default incentives is increasing in the persistenceof the shocks, which stresses the importance of dynamic considerations during the process of debt renegotiation.

2

which would be akin to the canonical models in the literature in which governments are readmitted
to financial markets with no debt (see Arellano (2008), Aguiar and Gopinath (2006), Chatterjee
and Eyigungor (2012)).4 For the second, we consider a fixed haircut equal to the average targetedin the benchmark calibration. In these two counterfactual economies the fraction of interest rate
hikes that trigger a default event are 0.06 and 0.13, respectively. We draw two conclusions fromthese exercises. The first is that, absent the possibility of some debt recovery, interest rate hikes
have a small effect on default incentives. This implies that the usual narrative of the “Volcker
shock” triggering the Mexican default in 1982 solely through higher interest costs is unlikely. Our
second conclusion is that, in the presence of some debt recovery after default, the renegotiation
mechanism described above (i.e. the government expecting favorable renegotiation terms if interest
rates remain high) accounts for roughly half of the default risk generated by interest rate hikes.
Related literature.—This paper is closely related to the literature that studies debt renegotiation in quantitative sovereign default models. Yue (2010) develops a model in which debt renegotiation happens after a default has occurred. An important difference between her environmentand ours is the set of assumptions regarding the outside options in the renegotiation game. She assumes that if a renegotiation attempt were unsuccessful then the government would face indefinite
autarky and lenders would recover nothing. We relax this by, instead, assuming that the alternative
to a present renegotiation outcome is to wait for a future one. This increases the outside valueof lenders as long as they have some bargaining power. Moreover, this outside value is directly
affected by the level of the risk-free interest rate in the renegotiation period, which is crucial to ourrenegotiation mechanism. In related papers, Benjamin and Wright (2009), Pitchford and Wright(2011), Bai and Zhang (2012), and Benjamin and Wright (2018) study delays in sovereign debt
renegotiation. They develop environments in which delays arise endogenously as strategies in thebargaining game to restructure debt. Similarly, Dvorkin, Sanchez, Sapriza, and Yurdagul (2023)
study sovereign debt restructurings in a model in which governments and lenders make alternatingoffers and endogenous delays are possible through taste shocks that are realized after these offers
have been made. A key difference between our model and theirs is that our assumptions prevent
4 In a recent paper, Arellano, Mateos-Planas, and Rios-Rull (2022) study an environment in which a government

can choose to default on a fraction of its debt. The main difference between their model and the existing literature on
debt renegotiation that we review below is that they allow for the government to continue to borrow while in partial
default.

3

delays from happening in equilibrium. What is essential for our results is the threat of delay. Even
if neither player chooses to delay in equilibrium, the fact that they could affects the outcome of thebargaining game. These assumptions provide a great deal of simplification and allow us to focus
on the role of risk-free interest rates. Hatchondo, Martinez, and Sosa-Padilla (2014) develop amodel of voluntary debt exchanges in which the government and lenders can choose to reduce theface value of the debt before a default occurs. These exchanges are mutually beneficial and happen
in equilibrium when the stock of debt is to the right side of the Laffer curve. Unlike them, we do
not allow for debt renegotiation to prevent a default in our model. Mihalache (2020) documentsthat debt relief programs are mostly implemented through maturity extensions, rather than throughreductions to the face value of debt. The essence of our results would not change if maturity extensions were included in the renegotiation game. Whether it is through a higher debt haircut or
through a more convenient maturity extension, our main result about the government getting amore favorable outcome with high rates would still hold.
This paper also contributes to the literature that studies the effect of shocks to risk-free interest
rates on default risk. Our work is closely related to Guimaraes (2011), who develops a stylized
model of debt renegotiation similar to ours and uses it to compare the role that income and interest
rate shocks have on debt relief. One of his findings is that the increase in world interest rates atthe beginning of the 1980’s can solely account for over half of the debt forgiveness obtained by
the main Latin American countries through the Brady plan. Our quantitative findings echo his andhighlight the ex-ante role of these expected outcomes on the initial default decisions. Moreover, we
disentangle the different channels through which the interest rate shock affected default incentives.
Singh (2020) develops a model to study clustered defaults and uses it to analyze the 1980’s debt
crisis. He finds that the Volcker shock was not a decisive factor for the clustered defaults and,instead, global shocks that jointly affected income in the defaulting countries played a major role.
However, his model does not feature debt renegotiation and, thus, his findings with such a model
are consistent with our results. Johri, Khan, and Sosa-Padilla (2022) incorporate an estimatedtime-varying process for the risk-free interest rate to a model of sovereign default. They findthat shocks to the interest rate have a sizable impact on sovereign spreads, even in the absenceof renegotiation. An important feature of their model is the time-varying variance of the risk-freerate since the large quantitative effect is mostly driven by shocks during periods of high volatility.
4In our model the volatility of the risk-free rate is fixed. However, as long as risk-free rates are
persistent, the renegotiation mechanism that we identify is an amplifier of the effect of interest rate
shocks, regardless of their variance.
Layout.—Section 2 presents a one-shot model of sovereign default and renegotiation whichhighlights the intuition behind our main result. Section 3 presents the general model and thequantitative exercises. Section 4 concludes.

5

2

One-shot modelThis model presents a one-shot game of sovereign default. An impatient government faces astochastic stream of income and issues short-term defaultable debt. The government is able tosustain positive debt levels due to the threat of financial autarky for a stochastic number of periods
and a dead-weight income loss during exclusion. After exclusion, the government and the lenders
negotiate over a flow of income (without the dead-weight loss) that remains constant from that
period on and the game ends.
Figure 2: Income throughout the game

Figure 2 illustrates the evolution of income over time throughout the game. We use this game tohighlight how debt renegotiation affects default incentives ex-ante and, crucially, how the level ofthe risk-free interest rate—which scales the lenders’ outside option—affects the negotiated terms.
Environment.—Time is discrete and runs forever. There is a small-open economy popu
lated by a government with preferences for streams of consumption represented by 𝑈𝑡 {𝑐 𝑠 }∞
𝑠=𝑡 =
Í

𝑠−𝑡 𝑢 (𝑐 ) where 𝛽 ∈ (0, 1) is a discount factor, and 𝑢 is continuously differentiable,
E𝑡 ∞
𝛽
𝑠
𝑠=𝑡
strictly increasing, striclty concave, lim𝑐→0 𝑢 (𝑐) = −∞ and lim𝑐→0 𝑢0 (𝑐) = ∞. Each period, thegovernment receives a stochastic endowment 𝑦 𝑡 ∈ (0, 2) which is iid over time with E [𝑦 𝑡 ] = 1 and
CDF 𝐹 (𝑦). The sovereign can issue one-period non-contingent debt 𝑏 𝑡+1 in international financial
markets. Debt is purchased by a measure 1 of identical risk-neutral lenders with deep pockets who
have access to a risk-free bond that pays a fixed interest rate 𝑟. At the beginning of each period, thegovernment observes the realization of 𝑦 𝑡 and decides whether to repay its outstanding debt. If thegovernment chooses to default, then it is immediately excluded from financial markets and income
is 𝜆 < 1. After default, the government remains in financial autarky and income continues to be 𝜆6until an opportunity to renegotiate arises, which happens with probability 𝜃. After renegotiation,
the government receives a constant stream of income E [𝑦 𝑡 ] = 1 forever, out of which it consumes
1 − 𝜌 ∗ in each period. The value 𝜌 ∗ is captured by the lenders (i.e. it cannot be defaulted on) andis determined in a negotiation period through Nash bargaining. We assume that either party canchoose to reject a proposed 𝜌 ∗ and wait for a new renegotiation opportunity. In equilibrium, thisnever happens because the assumption of 𝜆 < 1 implies that there is a strictly positive surplus thatcan be split in the negotiation period. However, it is the possibility to reject and wait that allows
the interest rate to have an important role in the determination of 𝜌 ∗ .5
In order to define and characterize the equilibrium and its properties we proceed backwards.
We first characterize the outcome of the renegotiation game 𝜌 ∗ by guessing, and then verifying,that it is unique. Then, we use this characterization to define the equilibrium of the model.
Renegotiation.—Given a negotiated payment 𝜌 ∗ , the value of the government in autarky after∗)∗
renegotiation is 𝑉 𝐴 (𝜌 ∗ ) = 𝑢(1−𝜌
1−𝛽 . Thus, the value of the government in default, for a given 𝜌 , is:𝑉 𝐷 (𝜌 ∗ ) =𝑢 (𝜆)𝛽𝜃𝑉 𝐴 (𝜌 ∗ )+1 − 𝛽 (1 − 𝜃) 1 − 𝛽 (1 − 𝜃)(1)

. Similarly, the value of a representative lender who holds defaulted bonds is:
𝑄 𝐷 (𝜌 ∗ ) =1−𝜃 𝐷 ∗𝜃
𝑄 𝐴 (𝜌 ∗ ) +𝑄 (𝜌 )
1+𝑟
1+𝑟

(2)

∗
∗
𝐴
where 𝑄 𝐴 (𝜌 ∗ ) = 1+𝑟𝑟 𝜌 is the value that lenders get after negotiating 𝜌 . Plugging 𝑄 into theabove equation we get that the value of holding defaulted bonds is
𝑄 𝐷 (𝜌 ∗ ) =𝜃 1+𝑟 ∗
𝜌𝑟 𝜃 +𝑟(3)

which is strictly decreasing in 𝑟. When an opportunity to renegotiate arises, lenders and the gov5 In previous work, ?

assumes that renegotiation can take place only once after default, which implies that theoutside value of lenders is to receive nothing. In such an environment, the renegotiation terms would not be affectedby the level of the risk-free interest rate.

7

ernment engage in Nash bargaining. We define 𝜌 ∗ as
 𝛼  𝐺𝑂𝑉
 1−𝛼
( 𝜌)
𝜌 ∗ = arg max 𝑆 𝐿𝐸 𝑁 ( 𝜌)
˜
𝑆
˜
𝜌˜

(4)

𝑆 𝐺𝑂𝑉 ( 𝜌)˜ = 𝑉 𝐴 ( 𝜌)˜ −𝑉 𝐷 (𝜌 ∗ ) ≥ 0

𝑠.𝑡.

𝑆 𝐿𝐸 𝑁 ( 𝜌)˜ = 𝑄 𝐴 ( 𝜌)˜ − 𝑄 𝐷 (𝜌 ∗ ) ≥ 0
where 𝛼 ∈ (0, 1) is the lenders’ bargaining power, and 𝑆 𝐺𝑂𝑉 and 𝑆 𝐿𝐸 𝑁 are the surpluses of thegovernment and the lenders, respectively. Note that both participation constraints consider the
option to wait for a future renegotiation with outcome 𝜌 ∗ . Assuming an interior solution, thefirst-order condition of the problem in (4) is(1 − 𝛼) 𝑟 𝑢0 (1 − 𝜌 ∗ ) ∗𝑢 (1 − 𝜌 ∗ ) − 𝑢 (𝜆)𝛼𝜌=
1 − 𝛽 (1 − 𝜃)
𝜃 +𝑟
1− 𝛽


(5)

where we have used the definitions of 𝑉 𝐴 , 𝑉 𝐷 , 𝑄 𝐴 , and 𝑄 𝐷 above.
Lemma 1. If 𝛼 ∈ (0, 1), then there is a unique 𝜌 ∗ ∈ (0, 1) that solves the bargaining problem in(4).
Proof : See Appendix A.
The proof of Lemma 1 shows that, as long as both parties have some bargaining power, equation(5) has a unique solution in the interior of (0, 1). Intuitively, the assumption that 𝜆 < 1 implies
that there is a strictly positive surplus to split. The fact that both parties can choose to delay
renegotiation implies that, as long as 𝛼 ∈ (0, 1), both should get a strictly positive value out of
the game. If the lenders have all the bargaining power, then they would offer 𝜌 ∗ = 1 − 𝜆, which
would leave the government indifferent between accepting and delaying forever. Similarly, if thegovernment has all the bargaining power, then it will offer the lenders 𝜌 ∗ = 0, which would leavethem indifferent between accepting and delaying forever.
Given the above characterization of the renegotiation game, the value of the government ingood financial standing is
𝑉 (𝑏, 𝑦) = max𝑑𝑉 𝐷 (𝜌 ∗ ) + (1 − 𝑑) 𝑉 𝑃 (𝑏, 𝑦)𝑑∈{0,1}8(6)

where 𝑑 is the default decision. The value of repaying the debt is{𝑢 (𝑐) + 𝛽E [𝑉 (𝑏0, 𝑦0)]}
𝑉 𝑃 (𝑏, 𝑦) = max
0

(7)

𝑏

𝑠.𝑡.

𝑐 + 𝑏 ≤ 𝑦 + 𝑞 (𝑏0) 𝑏0

where 𝑞 is the price schedule for government bonds. Note that 𝑉 𝑃 is strictly increasing in 𝑦 for any
given 𝑏, so the default set D (𝑏) = 𝑦 ∈ (0, 2) |𝑉 𝑃 (𝑏, 𝑦) < 𝑉 𝐷 (𝜌 ∗ ) is characterized by a cutoff
value 𝑦 ∗ (𝑏) such that 𝑉 𝑃 (𝑏, 𝑦 ∗ (𝑏)) = 𝑉 𝐷 (𝜌 ∗ ).
Equilibrium.—An equilibrium is value functions 𝑉 and 𝑉 𝑃 , policy functions 𝑑 and 𝑏0, a priceschedule 𝑞, and a renegotiation outcome 𝜌 ∗ such that: (i) 𝜌 ∗ solves the bargaining problem in (4);(ii) given 𝜌 ∗ and 𝑞, the value and policy functions solve the functional equations (6) and (7); and(iii) given 𝜌 ∗ and 𝑑, the price schedule is actuarially fair:𝑞 (𝑏0) =1 − 𝐹 (𝑦 ∗ (𝑏0)) 𝐹 (𝑦 ∗ (𝑏0)) 𝑄 𝐷 (𝜌 ∗ )
+
1+𝑟
1+𝑟
𝑏0

(8)

where 𝑦 ∗ is the cutoff value implied by the policy function 𝑑.

2.1

Renegotiation matters

Here we discuss how the risk-free rate 𝑟 affects default incentives and borrowing choices ex-antethrough two mechanisms. The first, which we call the standard mechanism, refers to how 𝑟 affects
the budget constraint of the government in repayment through its direct effect on how lenders
discount (i.e. the denominators in equation (8)). The second, which we call the renegotiation
mechanism, refers to how 𝑟 affects the renegotiation outcome 𝜌 ∗ and how, through it, it affects
both the price schedule 𝑞 in repayment and the value of defaulting.
Proposition 1. If 𝛼 ∈ (0, 1), then 𝜌 ∗ is decreasing in 𝑟.
Proof : First, note that for 𝑟 to have a meaningful role in renegotiation it is crucial that bothparties have something to gain from the renegotiation process. As mentioned above, if 𝛼 = 1then 𝜌 ∗ = 1 − 𝜆 and if 𝛼 = 0 then 𝜌 ∗ = 0, regardless of the value of 𝑟. Rewrite equation (5) thatcharacterizes 𝜌 ∗ as𝑢 (1 − 𝜌 ∗ ) − 𝑢 (𝜆) 1 − 𝛼 1 − 𝛽 (1 − 𝜃) 𝑟=
𝑢0 (1 − 𝜌 ∗ ) 𝜌 ∗
𝛼
1− 𝛽
𝜃 +𝑟
9

and note that the left-hand-side is decreasing in 𝜌 ∗ (this follows from 𝑢 being increasing andconcave). Also note that the right-hand-side is strictly increasing in 𝑟. If the interest rate increases
then the right-hand-side increases, so 𝜌 ∗ must decrease for the left-hand-side to increase too andthe equality to hold.
Recall that from equation (3) we get that the lenders’ outside option is decreasing in 𝑟. Intuitively, a larger interest rate reduces the lenders’ outside option because their opportunity costof delaying payment increases. This makes them more willing to accept a lower 𝜌 ∗ which improves the government’s outcome after renegotiation. Hereafter we adopt the notation 𝜌 ∗ (𝑟) and
use Proposition 1 to analyze how the level of 𝑟 affects default incentives ex-ante.
We can use (3) to write the value of repaying the debt as{𝑢 (𝑐) + 𝛽E [𝑉 (𝑏0, 𝑦0, 𝑟)]}𝑉 𝑃 (𝑏, 𝑦, 𝑟) = max
0

(9)

𝑏

𝑠.𝑡.

𝑐+𝑏 ≤ 𝑦+1 − 𝐹 (𝑦 ∗ (𝑏0, 𝑟)) 0 𝜃𝐹 (𝑦 ∗ (𝑏0, 𝑟))𝑏 +(𝜃 + 𝑟) 𝑟
1+𝑟|{z}| {z }
standard mechanism𝜌 ∗ (𝑟)|{z}
renegotiation mechanism

standard mechanism

where 𝑟 affects the budget constraint by discounting the value of future debt payments (the standard
mechanism) and by changing the value of 𝜌 ∗ (the renegotiation mechanism). The cutoff 𝑦 ∗ isdefined by
𝑉 𝑃 (𝑏, 𝑦 ∗ (𝑏, 𝑟) , 𝑟) = 𝑉 𝐷 (𝜌 ∗ (𝑟))(10)and depends on 𝑟 directly through its effect on the budget constraint in 𝑉 𝑃 and indirectly throughits effect on 𝜌 ∗ .
Proposition 2. For any given 𝑏 such that the repayment set is not empty, the default set isexpanding in 𝑟.
Proof : Proposition 1 implies that 𝑉 𝐷 (defined in equation (1)) is increasing in 𝑟. Also, it isclear from equation (9) that 𝑉 𝑃 is decreasing in 𝑟. Then, for equation (10) to hold 𝑦 ∗ must also
increase as 𝑟 increases.
Proposition 2 provides the main result of this Section: default incentives are increasing in𝑟. The intuition is that high interest rates improve the government’s value of defaulting because
they improve the terms that it would get out of an eventual renegotiation. Absent endogenous
renegotiation, the interest rate would still affect default incentives, but only through the standard
10

mechanism. Suppose a counterfactual economy in which debt recovery is 𝜌 ∗ = 𝜅, where 𝜅 ∈ (0, 1)
is some fixed number. Then, equation (9) becomes{𝑢 (𝑐) + 𝛽E [𝑉 (𝑏0, 𝑦0, 𝑟)]}𝑉 𝑃 (𝑏, 𝑦, 𝑟) = max
0

(11)

𝑏

𝑠.𝑡.

𝑐+𝑏 ≤ 𝑦+1 − 𝐹 (𝑦 ∗ (𝑏0, 𝑟)) 0 𝜃𝐹 (𝑦 ∗ (𝑏0, 𝑟))𝑏 +𝜅(𝜃 + 𝑟) 𝑟
1+𝑟|{z}| {z }
standard mechanism

standard mechanism

and the cutoff 𝑦 ∗ is defined by 𝑉 𝑃 (𝑏, 𝑦 ∗ (𝑏, 𝑟) , 𝑟) = 𝑉 𝐷 (𝜅). In this case, the risk-free rate is irrelevant for the payoffs after default. The limiting case of 𝜅 = 0 corresponds to a model in which
lenders recover nothing after default and the government remains in autarky forever. This limitingcase would further undermine the role of the standard mechanism by reducing the second term in
the budget constraint from equation (11).
In the following section we present a quantitative model of sovereign default that featuresendogenous renegotiation in a similar fashion as above and show that, for a calibration to the
Mexican default in 1982, the quantitative relevance of the standard mechanism is dwarfed by thatof the renegotiation mechanism. In order to make the latter point, we use two counterfactual caseswith a fixed recovery rate considering the case of no recovery 𝜅 = 0—which would be akin to
the standard in the literature—and with some recovery 𝜅 > 0—which would allow the standard
mechanism to have a larger effect.

3

Quantitative model

We now extend the above one-shot model into a quantitative sovereign default model with renegotiation. The key additions are shocks to the real interest rate and readmission to financial markets
after debt renegotiation. We also allow for long-term debt and persistent income shocks. Given
these assumptions, results similar to Propositions 1 and 2 cannot be proved, but, as will be clear
later, the intuition behind both results persists.
Shocks and preferences.—The risk-free interest rate can take two values 𝑟 𝑡 ∈ {𝑟 𝐿 , 𝑟 𝐻 }, with𝑟 𝐿 < 𝑟 𝐻 , and follows a Markov chain, where 𝜋𝑖 𝑗 with 𝑗 ∈ {𝐿, 𝐻} are the transition probabilities.
Each period, the economy receives a stochastic endowment of a tradable good 𝑦 𝑡 that follows a11
log-normal AR(1) process log (𝑦 𝑡 ) = 𝜌 log (𝑦 𝑡−1 ) + 𝜖𝑡 , with |𝜌| < 1 and 𝜖𝑡 ∼ 𝑁 0, 𝜎𝜖2 . Unlike inthe one-shot model, the endowment follows this stochastic process regardless of the government’s
financial standing. The government has preferences for consumption in each period represented
1−𝜎by 𝑢 (𝑐) = 𝑐1−𝜎 and discounts the future at a rate 𝛽.
Debt and default.—The government can issue long-term non-contingent debt in international
financial markets. Similar to Hatchondo and Martinez (2009), a bond consists of a perpetuity withgeometrically declining payments: a bond issued in period 𝑡 promises to pay 𝛾 (1 − 𝛾) 𝑗−1 unitsof the tradable good in period 𝑡 + 𝑗, ∀ 𝑗 ≥ 1. The law of motion for bonds is given by 𝑏 𝑡+1 =(1 − 𝛾) 𝑏 𝑡 + 𝑥𝑡 where 𝑏 𝑡 is the amount of bonds due at the beginning of period 𝑡, 𝛾 is the fractionof bonds that matures each period, and 𝑥𝑡 is the issuance of new bonds. Debt is purchased by ameasure 1 of identical risk-neutral competitive lenders with deep pockets who discount the future
1
at the current risk-free rate 1+𝑟. At the beginning of each period, the government observes the𝑡realization of the shocks and 𝑏 𝑡 and decides whether to repay or default. If it chooses to default
then it gets immediately excluded from financial markets. While in default, there is an asymmetric
cost to output cost of the form 𝜙 (𝑦 𝑡 ) = max 0, 𝜙0 𝑦 𝑡 + 𝜙1 𝑦 𝑡2 , where 𝜙0 < 0 < 𝜙1 .6 Also, at the
beginning of each period after default, an opportunity to renegotiate the outstanding debt arises
with probability 𝜃.7
Renegotiation.—When an opportunity to renegotiate arises, lenders and the government engage in Nash bargaining to determine a new debt level 𝑏 𝑅 for the government to re-enter financial
markets with. In the renegotiation period, after 𝑏 𝑅 has been determined, the government pays 𝛾𝑏 𝑅
and is allowed to issue new debt. Readmission with 𝑏 = 𝑏 𝑅 must be mutually beneficial and wecontinue to assume that both the government and the lenders can choose to reject an offer anddelay renegotiation for a later opportunity. As in the one-shot model, a delay does not happen
in equilibrium because the dead-weight cost to output in default (both in the present and future
periods) implies that there is always a positive surplus to split. Unlike the one-shot model, however, the surplus is not constant, but rather dependent on both the current level of output—which
determines real resources to be split—and on the level of the risk-free interest rate—which affects
6 We take this functional form from Chatterjee and Eyigungor (2012), which implies that the cost is zero for0 ≤ 𝑦 𝑡 ≤ − 𝜙𝜙01 and more than proportionally increasing for 𝑦 𝑡 > − 𝜙𝜙01 . This asymmetry allows the model to gener-

ate countercyclical spreads and default, as is observed in the data.
7 Hatchondo, Martinez, and Sosa-Padilla (2014) consider the option to renegotiate immediately as an alternative todefault. Our results would not change if we considered this option, which we exclude for simplicity.

12the lenders’ outside option and the value of new debt that the government could issue. Also, notethat the renegotiated debt level 𝑏 𝑅 is only a function of the present and future surplus to be split,
and does not depend on how much debt was defaulted on.8

3.1

Recursive formulation

The state of the economy is (𝑏, 𝑦, 𝑟). The value function of the government in good standing is:
𝑉 (𝑏, 𝑦, 𝑟) = max(1 − 𝑑) 𝑉 𝑃 (𝑏, 𝑦, 𝑟) + 𝑑𝑉 𝐷 (𝑦, 𝑟)(12)𝑑∈{0,1}

where 𝑑 is the default decision. If the government decides to repay it makes coupon payments 𝛾𝑏and gets to issue new bonds. The value of the government in repayment is:
𝑉 𝑃 (𝑏, 𝑦, 𝑟) = max0 {𝑢 (𝑐) + 𝛽E [𝑉 (𝑏0, 𝑦0, 𝑟 0)]}

(13)

𝑐,𝑏

𝑠.𝑡.

𝑐 + 𝛾𝑏 ≤ 𝑦 + 𝑞 𝑃 (𝑏0, 𝑦, 𝑟) [𝑏0 − (1 − 𝛾) 𝑏]

where 𝑞 𝑃 is the price schedule of newly issued bonds. The value of the government if it chooses
to default is:ih  𝐷 0 0 o
0 0
0 0
𝑅
𝑃
𝑉 (𝑦, 𝑟) = 𝑢 (ℎ (𝑦)) + 𝛽 𝜃E 𝑉 𝑏 (𝑦 , 𝑟 ) , 𝑦 , 𝑟 + (1 − 𝜃) E 𝑉 (𝑦 , 𝑟 )
𝐷

n

(14)

where ℎ (𝑦) = 𝑦 − 𝜙 (𝑦) is the output net of default costs, and 𝑏 𝑅 (𝑦0, 𝑟 0) is the value of renegotiateddebt when the state is (𝑦0, 𝑟 0). When an opportunity to renegotiate arises, 𝑏 𝑅 is determined asn 𝛼  𝐺𝑂𝑉
 1−𝛼 o(𝑦, 𝑟)𝑏 𝑅 (𝑦, 𝑟) = arg max 𝑆 𝐿𝐸 𝑁 (𝑦, 𝑟)
𝑆
𝑏˜
h

i

˜ 𝑦, 𝑟 , 𝑦, 𝑟 𝑏˜ − 𝑄 𝐷 (𝑦, 𝑟) ≥ 0
𝑠.𝑡. 𝑆 𝐿𝐸 𝑁 (𝑦, 𝑟) = 𝛾 + (1 − 𝛾) 𝑞 𝑃 𝑏 𝑃 𝑏,˜ 𝑦, 𝑟 −𝑉 𝐷 (𝑦, 𝑟) ≥ 0
𝑆 𝐺𝑂𝑉 (𝑦, 𝑟) = 𝑉 𝑃 𝑏,

(15)

8 This is an important difference between our model and the one in Hatchondo, Martinez, and Sosa-Padilla (2014).

In their environment, the exchanged debt depends on outstanding debt because the outside option of the lenders is thecurrent market value of it. This is because they model voluntary debt exchanges that happen instead of default, rather
than after default. They assume that if lenders reject the exchange they can collect the current market value of thedebt, while we assume that if they reject then renegotiation is delayed to a future period.

13

where 𝑏 𝑃 is the policy function of the government’s problem in repayment (13) and 𝑄 𝐷 (𝑦, 𝑟) is
the value of a representative lender holding delinquent bonds:𝜃
E 𝛾 + (1 − 𝛾) 𝑞 𝑃 (𝑏00, 𝑦0, 𝑟 0) 𝑏 𝑅 (𝑦0, 𝑟 0)
1+𝑟1−𝜃  𝐷 0 0 +
E 𝑄 (𝑦 , 𝑟 )1+𝑟

𝑄 𝐷 (𝑦, 𝑟) =(16)

with 𝑏00 = 𝑏 𝑃 𝑏 𝑅 (𝑦0, 𝑟 0) , 𝑦0, 𝑟 0 . The participation constraints in (15) capture how both the government and the lenders have the option to delay renegotiation for a future period.
Note that, while the relation between the renegotiated debt 𝑏 𝑅 and the risk-free rate 𝑟 is not
as transparent as in the one-shot model, the same intuition laid out in the latter persists. Equation(16) shows that when the risk-free rate is high lenders discount the future at a higher rate, which
directly lowers their outside option 𝑄 𝐷 . Thus, with high 𝑟 lenders are more willing to accept
a lower 𝑏 𝑅 since they value immediate payments more than potentially higher future ones. The
government understands that it will get better terms if renegotiation happens when the risk-free rate
is high. So, if interest rates are expected to remain high (e.g. the process for 𝑟 is highly persistent),
then high interest rates in the present make default more attractive through expectations of better
renegotiation terms (i.e. low 𝑏 𝑅 ).
The price of debt in good financial standing 𝑞 𝑃 reflects the actuarially fair value of newly issued
bonds 𝑏0:
𝑞 𝑃 (𝑏0, 𝑦, 𝑟) =

1
1+𝑟
|{z}




E {1 − 𝑑 (𝑏0, 𝑦0, 𝑟 0)} 𝛾 + (1 − 𝛾) 𝑞 𝑃 (𝑏00, 𝑦0, 𝑟 0)

(17)

standard mechanism






𝐷 (𝑦 0, 𝑟 0) 1𝑄E 𝑑 (𝑏0, 𝑦0, 𝑟 0)
+

0


1+𝑟
𝑏
|{z}
|
{z
}




standard mechanism  renegotiation mechanism 
where 𝑏00 = 𝑏 𝑃 (𝑏0, 𝑦0, 𝑟 0) is the government’s debt issuance if it repays in the next period. Equation(17) shows how the risk-free rate affects the price of debt through both mechanisms. Throughthe standard mechanism, an increase in 𝑟 lowers the market value of debt because it increases
the rate at which lenders discount the future. This decreases the real amount of resources that14

the government can raise from a new debt issuance, which in turn makes default more attractive.
Through the renegotiation mechanism, 𝑄 𝐷 decreases when the interest rate is high if it is expected
to remain high when an opportunity to renegotiate arrives. This further decreases 𝑞 𝑃 by making
the second term in equation (17) lower.
Equilibrium.—An equilibrium is value and policy functions for the government, a price schedule 𝑞 𝑃 , a value of holding defaulted debt 𝑄 𝐷 , and a function for renegotiated bonds 𝑏 𝑅 such that:(i) given 𝑞 𝑃 , 𝑄 𝐷 and 𝑏 𝑅 , the value and policy functions of the government satisfy equations (12),(13) and (14); (ii) given 𝑏 𝑅 and the government’s policy functions, the value 𝑄 𝐷 satisfies the functional equation (16); (iii) given the value and policy functions of the government and given 𝑄 𝐷 ,𝑏 𝑅 solves the bargaining problem in (15); and (iv) given the policy functions and 𝑄 𝐷 , the price 𝑞 𝑃
satisfies equation (17).

3.2

Calibration

We consider the 1982 Mexican debt crisis, which was preceded by a sizable increase in US interest
rates. We use our model to assess the extent to which this increase in US interest rates triggered
the Mexican default decision and whether renegotiation dynamics played an essential role.
Table 1 presents all parameter values that we calibrate directly. Each period in the model
corresponds to 1 year. The risk aversion parameter is set to a standard value, 𝜎 = 2. The AR(1)income process estimation uses HP-filtered logged Mexican GDP data from 1921 to 1983, whichyields an auto-correlation parameter 𝜌 = 0.705 and a standard deviation of innovations of 𝜎𝜖 =0.040. We set 𝛾 = 0.75 so that the average bond duration equals 16 months, which was the average
maturity of the outstanding syndicated loans Mexico had by 1982 (see Negrete Cardenas (1999)).
Table 1: Externally calibrated parameters
Parameter
low rhigh rPr(low to high r)
Pr(high to low r)
Pr(renegotiation)
maturity raterisk aversion
income process

𝑟𝐿
𝑟𝐻
𝜋 𝐿,𝐻
𝜋 𝐻 ,𝐿
𝜃
𝛾
𝜎
𝜌
𝜎𝜖

Value

Details

0.012
0.062
0.01
0.20
0.19
0.75
2
0.705
0.040

1955 - 1980
1981 - 1985
Duration of 100 years
Duration of 5 years
5.2 years exclusion (Gelos, Sahay, and Sandleris (2011))
Sixteen-month bonds
Standard
AR(1) estimation
annual data 1933-1983

15

The probability of switching from the high risk-free interest rate regime to the low one is set
to 𝜋 𝐻,𝐿 = 0.20 so that it generates an expected duration of 5 years for the high regime. This is
the time it took interest rates in the U.S. to start decreasing, as can be seen in Figure 3. Hence,
implicit in our analysis is the assumption the Mexican government had the correct expectation forthe duration of high world interest rates. We set the probability of switching from the low to thehigh risk-free interest rate regime to 𝜋 𝐿,𝐻 = 1% so that shocks like the one we are studying are very
infrequent events.
Figure 3: Real risk-free interest rate

Figure 3 also displays the average interest rate during the Volcker shock (1980-1985) and theaverage interest rate before that (1955-1980).9 Therefore, we set the risk-free interest rate in the
low regime to 𝑟 𝐿 = 0.012, and to 𝑟 𝐻 = 0.062 in the high regime.
We set the lenders’ bargaining power parameter 𝛼, the discount factor 𝛽, and the output cost
parameters 𝜙0 and 𝜙1 to jointly match four moments of the Mexican economy: a haircut of 0.24
following the Brady plan, an average debt-to-GDP ratio of 0.19, a default probability of 0.03,
and an average spread of 0.03.10 The first column in Table 2 reports the parameter values for the
9 For simplicity, we assume only two possible states for the risk-free interest rate. This highlights the mechanics ofthe model as well as the role of the Volcker shock in triggering default episodes.
10 Due to limited data availability, our target for the average spread considers the spreads implied in the EMBI Index

16benchmark calibration.
Table 2: Parameters chosen to match data moments
Parameters
BenchmarkBargaining power𝛼

Targets

Full exogenous haircut

Partial exogenous haircut

0.11

from data
Haircut in 1990

0.24

Discount factor

𝛽

0.82

0.77

0.89

Debt-to-GDP ratio

0.19

Quadratic income

𝜙0

-0.20

-0.62

-0.46

Default probability

0.03

cost function

𝜙1

0.23

0.69

0.49

Average spreads

0.03

In order to quantify the relevance of the renegotiation mechanism we consider two counterfactual cases in which the haircut to defaulted debt is determined exogenously. That is, we assume
that, once a renegotiation opportunity arrives, the government is readmitted to financial markets
with a debt level equal to 𝑏 𝑅 = (1 − 𝜅) 𝑏 where 𝜅 ∈ [0, 1] is the exogenous haircut. We also assume
that the government can choose to reject this offer, in which case defaulted debt remains at 𝑏 andthe government continues to be in autarky until a new opportunity arrives. The value in default is
now a function of the level of debt that the government defaulted on 𝑏:𝑉 𝐷 (𝑏, 𝑦, 𝑟) = 𝑢 (ℎ (𝑦)) + 𝛽 𝜃E [𝑉 (𝜅𝑏, 𝑦0, 𝑟 0)] + (1 − 𝜃) E 𝑉 𝐷 (𝑏, 𝑦0, 𝑟 0)

(18)


where the continuation value 𝑉 (𝑏, 𝑦, 𝑟) = max𝑑∈{0,1} (1 − 𝑑) 𝑉 𝑃 (𝑏, 𝑦, 𝑟) + 𝑑𝑉 𝐷 (𝑏, 𝑦, 𝑟) considers
the government’s ability to choose to remain in default. We consider two cases: the case of full
exogenous haircut with 𝜅 = 1 and the case of partial exogenous haircut with 𝜅 = 0.24.11 For eachof these two cases we recalibrate the model to match the same moments as in the benchmark. The
second and third columns of Table 2 report these values.

3.3Interest rate shocks and default

In order to analyze how renegotiation affects default incentives and, more importantly, the abilityof interest rate hikes to induce defaults, we divide the state space into three regions for pairs offor Mexico, which is available from 1997 onward. Our results are not sensitive to using the volatility of the tradebalance during the relevant period as an alternative target.
11 Note that the case of 𝜅 = 1 is nested by the benchmark model by setting 𝛼 = 0. This gives the government all
the bargaining power and allows it to make take-it-or-leave-it offers to the lenders. The lenders are still allowed todelay, in which they would wait for a future opportunity to receive a similar deal. In equilibrium, the value of any
future renegotiation is erased by the government having all the bargaining power at all times, which pushes the lender’soutside option to 0.

17

income and debt (𝑦, 𝑏): (i) one in which the government defaults for any risk-free interest rate, (ii)
one in which it repays for any risk-free interest rate, and (iii) the region in which the government
defaults only when the risk-free interest rate is high.
The left panel of Figure 4 presents these regions for the case in which there is no renegotiationand no debt recovery. This corresponds to the calibration in the second column of Table 2, which isthe case where 𝜅 = 1 (or 𝛼 = 0, as discussed in footnote 11). The right panel presents these regions
for the same calibration but setting 𝛼 = 0.20.
Figure 4: Default regions, effect of renegotiation(b) Renegotiation with 𝛼 = 0.2(a) No renegotiation, no recovery

Introducing renegotiation has two important implications in the model. First, it allows thegovernment to sustain higher levels of debt. This is because lenders expect some positive recovery
after a potential default so, for any given default probability implied by some state, the market
value of debt is higher. Second, it expands the region in which default happens only with highinterest rates but now with low.
Figure 5 presents the same regions for the benchmark model with renegotiation and the counterfactual case with an exogenous fixed haircut of 𝜅 = 0.24.

18

Figure 5: Default regions(b) Fixed exogenous haircut, 𝜅 = 0.24(a) Renegotiation, benchmark

Note that the black regions are thicker than their counterpart in the left panel of Figure 4. Thishighlights the role of renegotiation—as the comparison of both cases in Figure 4 did—but it alsostresses the role of debt recovery after default, even if it is exogenous. However, note that the black
region is more vertical in the case of endogenous renegotiation. This implies that, for a given levelof debt, there is a larger range of income shocks that would be consistent with a default triggered
solely by an interest rate hike.
The above analysis of default sets is akin to comparing policy functions of different models,
which allows to understand how endogenous decisions drive simulated outcomes. We now analyze
default events in the ergodic distribution of each case. For each model, we simulate 100,500
periods and drop the first 500 to avoid results being driven by initial conditions. We use thesetime series to compute the probability of an interest-rate hike triggering a default event, that is
Pr (𝑑𝑡 = 1|𝑑𝑡−1 = 0, 𝑟 𝑡 = 𝑟 𝐻 , 𝑟 𝑡−1 = 𝑟 𝐿 ). Table 3 report this statistic for all three cases.
Table 3: Probability of interest-rate hikes triggering a default

Pr (default event|interest-rate hike)No renegotiation,
no recovery
0.06

Fixed exogenous Endogenous
haircut
renegotiation
0.13
0.22

In the model with no renegotiation and no debt recovery only six percent of interest rate hikes
trigger a default, while in our benchmark model this number is 22 percent. This makes the usual19narrative of the 1982 Mexican default being triggered only by higher interest costs unlikely. The
sole expectation of some debt recovery, even if it were independent of the interest rate, more thandoubles the likelihood of interest rate shocks triggering a default (from 0.06 to 0.13). This is almost
doubled again from 0.13 to 0.22 if this recovery endogenously depends on the level of the interest
rate, which is our renegotiation mechanism.
We now compare default episodes in the benchmark model with those in the model with nodebt recovery. Figure 6 displays the average paths around default episodes of income shocks, therisk-free interest rate, and a hypothetical haircut:
haircut𝑡 = 1 −𝑏 𝑅 (𝑦 𝑡 , 𝑟 𝑡 )
𝑏𝑡

(19)

where 𝑏 𝑅 is the renegotiation outcome defined in (15). This is the haircut that would occur if arenegotiation were to happen in period 𝑡 with the shock realizations (𝑦 𝑡 , 𝑟 𝑡 ) and the defaulted debt
had been 𝑏 𝑡 . We simulate 10,000 default episodes and compute the average paths in a 20-period
window around each.
Figure 6: Paths around default events
Income Shock

Risk-free rate

1.00
NR
R
default

0.94
0.92 10

1.6

26

NR
R
default

1.5

percent

0.96

haircut
default

1.7
percentage points

y

0.98

Implied haircut to face value

28

22

1.4
5

0
t

5

10

24

10

5

0
t

5

10

10

5

0
t

5

While the pattern of income shocks is pretty similar in both models, interest-rate hikes are substantially more associated with default in the benchmark model. Also, note that the hypotheticalhaircut substantially increases in the periods leading up to default event, which stresses the rolethat better expected terms for the government play in triggering the default decision. Given the persistence of the income and risk-free interest rates, the anticipation of more favorable restructuringterms makes the default choice more attractive and borrowing more expensive. This mechanism isnonexistent in the two counterfactual models with exogenous fixed debt relief.
20

10Figure 7 shows the distributions of realized haircuts conditional on each interest rate level.
The distribution under low interest rates has a lower variance and haircuts are more concentrated
around the targeted average. In contrast, the distribution under high interest rates is much morevolatile and slightly skewed to the left. The higher mode and higher average of realized haircuts
capture the government’s improved bargaining conditions when interest rates are high.
Figure 7: Distribution of haircuts

The longer left tail under the high regime captures some other interesting dynamics of themodel. Governments gamble on receiving generous haircuts: their realized haircut is high in casethe persistent risk-free interest rate remains high, but their realized haircut is low if there is a regime
switch. And since the default set is larger when interest rates are high, high regimes feature morerenegotiation episodes involving governments with relatively low debt and high income. These are
the unlucky cases in the left tail of the distribution under the high regime.
Renegotiation took longer than the expected 1𝜃 ≈ 3.5 years. It was only in 1989/1990 thatthe Brady plan resolved the 1982 Mexican default episode. A potential explanation is that US
regulators did not allow banks to write down their default debt. From Lewis William Seidman,
former head of the U.S. Federal Deposit Insurance Corporation, in Seidman (2000):“Had these institutions been required to mark their sometimes substantial holdings of underwater debt to market or to increase loan-loss reserves to levels close to the expected losses on this21

debt (as measured by secondary market prices), then institutions such as Manufacturers Hanover,
Bank of America, and perhaps Citicorp would have been insolvent.”
It was not only US monetary policy but also banking regulatory decisions that have a bearing
on the 1980s debt crisis. Loans to a single borrower could not exceed 10% of bank’s capital; nevertheless, different government agencies in foreign countries were considered different borrowers.
During the 1980s, banks were exempted from building reserves for delinquent least developed
countries’ loans. Seidman (2000) claims there were non-profit maximizing incentives for lending
during the 1980s which we do not model in this paper:“The entire Ford administration, including me, told the large banks that the process of recycling
petrodollars to the less developed countries was beneficial, and perhaps a patriotic duty.”

4

Conclusion

We provide a framework in which external shocks may have a relevant role in triggering sovereign
debt crisis in emerging economies. For this, we provide a new mechanism through which risk-freeinterest rates affect potential future debt restructuring and, ultimately, default incentives.
This feature has not been present in many quantitative sovereign default models that studies
exactly the role of such shocks. We show that our model is able to reconcile sovereign default
models to the widespread narrative about the 1980’s.

22References
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A

ProofsLemma 1. If 𝛼 ∈ (0, 1), then there is a unique 𝜌 ∗ ∈ (0, 1) that solves the bargaining problem in (4).
Proof : The first-order-condition of the bargaining problem implies(1 − 𝛼) 𝑟 𝑢0 (1 − 𝜌 ∗ ) ∗𝑢 (1 − 𝜌 ∗ ) − 𝑢 (𝜆)𝛼=𝜌
1 − 𝛽 (1 − 𝜃)
𝜃 +𝑟
1− 𝛽


where 𝑢 is strictly increasing and concave. Note that when 𝜌 ∗ → 0 the left-hand-side is positive,
while the right-hand-side is zero. Similarly, when 𝜌 ∗ → 1 the left-hand-side is negative, while theright-hand-side is positive. Since 𝑢 and 𝑢0 are continuous, then the Intermediate Value Theorem
implies the existence of a solution 𝜌 ∗ . Moreover, since both sides of the equation are monotonic
then the solution is unique.

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