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Working Paper Series
Debt Dilution and Sovereign Default Risk
WP 10-08R
Juan Carlos Hatchondo
Federal Reserve Bank of Richmond
Leonardo Martinez
International Monetary Fund
César Sosa Padilla
University of Maryland
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Debt dilution and sovereign default risk∗
Working Paper 10-08R
Juan Carlos Hatchondo Leonardo Martinez César Sosa Padilla
Richmond Fed
IMF
U. of Maryland
February 21, 2012
Abstract
We measure the effects of debt dilution on sovereign default risk and show how these
effects can be mitigated with debt contracts promising borrowing-contingent payments.
First, we calibrate a baseline model à la Eaton and Gersovitz (1981) to match features of
the data. In this model, bonds’ values can be diluted. Second, we present a model in which
sovereign bonds contain a covenant promising that after each time the government borrows
it pays to the holder of each bond issued in previous periods the difference between the
bond market price that would have been observed absent current-period borrowing and the
observed market price. This covenant eliminates debt dilution by making the value of each
bond independent from future borrowing decisions. We quantify the effects of dilution by
comparing the simulations of the model with and without borrowing-contingent payments.
We find that dilution accounts for 84% of the default risk in the baseline economy. Similar
default risk reductions can be obtained with borrowing-contingent payments that depend
only on the bond market price. Using borrowing-contingent payments is welfare enhancing
because it reduces the frequency of default episodes.
JEL classification: F34, F41.
Keywords: Sovereign Default, Debt Dilution, Debt Covenant, Long-term Debt, Endogenous Borrowing Constraints.
∗
For comments and suggestions, we thank Fernando Alvarez, Luis Brandao-Marques, Fernando Broner, Satyajit Chatterjee, Pablo D’Erasmo, Bora Durdu, Enrique Mendoza, Dirk Niepelt, Jorge Roldos, Diego Saravia, and
seminar participants at the U.S. Board of Governors of the Federal Reserve System, the University of Virginia, the
Federal Reserve Bank of Richmond, the Federal Reserve Bank of Atlanta, the Universidad Nacional de Tucumán,
the 2009 Wegmans conference, the Pontificia Universidad Católica de Chile, the Central Bank of Chile, the 2010
Royal Economic Society Meeting, the 2010 Eastern Economic Association Meeting, the 2010 SED conference,
the 2010 Berlin Conference on Sovereign Debt and Default, the 2010 and 2011 AEA meetings, the ECB, and the
2011 NBER International Finance and Macroeconomics Workshop. Remaining mistakes are our own. The views
expressed herein are those of the authors and should not be attributed to the IMF, its Executive Board, or its
management, the Federal Reserve Bank of Richmond, or the Federal Reserve System.
E-mails: hatchondo@gmail.com; leo14627@gmail.com; sosa@econ.umd.edu.
1
1
Introduction
This paper presents a measure of the effects of the debt dilution induced by governments’ borrowing decisions and shows how these effects can be mitigated with debt contracts that specify
borrowing-contingent payments. If governments could commit to not dilute the value of current
bond issuances with future borrowing, this would allow them to sell bonds at a higher price.
Sovereign debt dilution may become a problem when governments do not have the ability to
make such commitment.
Participants in various credit markets have made efforts to mitigate the dilution problem,
which is suggestive of the relevance assigned to this issue. Corporate debt contracts often include
covenants intended to limit debt dilution (Asquith et al. (2005), Smith and Warner (1979) and
Rodgers (1965) discuss corporate debt covenants). In some cases, debt claims differ in their
seniority—if existing debt is senior to new issuances, this may mitigate the dilution problem.
A seniority structure is common in corporate debt and collateralized loans to households. In
contrast, sovereign bonds typically do not present differences in legal seniority but include a pari
passu clause and a negative pledge clause that prohibits future issuances of collateralized debt.1
These clauses are intended to avoid making new debt senior to previously issued debt, but do
not make existing debt senior to debt that will be issued in the future.
The weaker protection against sovereign debt dilution may be due in part to the weak enforcement of sovereign debt claims.2 Overall, it seems clear that existing sovereign debt contracts
do not eliminate the risk of debt dilution.
The possibility of sovereign debt dilution has also received considerable attention in both
academic and policy discussions. Several studies describe the benefits of eliminating debt dilution.
1
Sturzenegger and Zettelmeyer (2006) explain how in Ecuador’s 2000 sovereign debt restructuring, the government exchanged defaulted bonds for new bonds that included a clause specifying that if a default occurred
within 10 years following the restructuring agreement, the government would extend new bonds to the holders of
the restructured debt. Sturzenegger and Zettelmeyer (2006) argue that the “effect of this was to offer a (limited)
protection of bond holders against the dilution of their claims by new debt holders in the event of default.”
However, the inclusion of such debt covenants is much more an exception than a rule. It has also been argued
that loans from institutions such as the International Monetary Fund or the World Bank receive de facto seniority
over loans from private agents (see, for example, Saravia (2010)).
2
This weak enforcement has lead to several proposals to induce more orderly sovereign debt restructurings
(see, for example, Bolton and Skeel (2005), Borensztein et al. (2004), G-10 (2002), IMF (2003), Krueger and
Hagan (2005), and Paulus (2002)).
2
For instance, Bizer and DeMarzo (1992) show how dilution may lead to equilibria with higher
debt levels and higher interest rates implied by higher default probabilities. It has also been
argued that dilution may lead to excessive issuance of short-term debt (Kletzer (1984)), or of
debt that is hard to restructure after a default (Bolton and Jeanne (2009)), which in turn could
increase the likelihood and/or severity of sovereign debt crises. Bolton and Skeel (2005) argue for
the importance of being able to grant seniority to debt issued while the country is negotiating with
holders of debt in default, as observed in corporate bankruptcy procedures. Borensztein et al.
(2004) suggest changes in national and international laws that may facilitate the introduction of
debt contracts that provide some protection against debt dilution.3 While these studies suggest
that debt dilution may be an important source of inefficiencies in debt markets, they do not
quantify the effects of dilution.
We do so using a default framework à la Eaton and Gersovitz (1981).4 Formally, we analyze a
small open economy that receives a stochastic endowment stream of a single tradable good. The
government’s objective is to maximize the expected utility of private agents. Each period, the
government makes two decisions. First, it decides whether to default on previously issued debt.
Second, it decides how much to borrow. The government can borrow by issuing non-contingent
long-duration bonds, as in Hatchondo and Martinez (2009). The cost of defaulting is represented
by an endowment loss that is incurred in the default period.
The most common modeling approach for the study of debt dilution is to focus on the
effect of seniority clauses.5 However, it is well known that seniority does not fully eliminate
debt dilution if new borrowing increases the default probability (see, for example, Bizer and
DeMarzo (1992)). Therefore, in general, one cannot measure accurately the effects of dilution
by comparing equilibria with and without seniority. Furthermore, seniority clauses may not be
3
Detragiache (1994), Eaton and Fernandez (1995), Niepelt (2008), Sachs and Cohen (1982), Tirole (2002), and
UN (2004) also discuss inefficiencies raised by debt dilution.
4
This framework has been used in many recent studies. See, for instance, Aguiar and Gopinath (2006), Alfaro
and Kanczuk (2009) Arellano (2008), Boz (2011), Cuadra et al. (2010), D’Erasmo (2008), Durdu et al. (2010),
Lizarazo (2005, 2006), and Yue (2010). These models share blueprints with the models used in studies of household
bankruptcy—see, for example, Athreya et al. (2007), Chatterjee et al. (2007), Li and Sarte (2006), Livshits et al.
(2008), and Sanchez (2010).
5
For instance, Bi (2006) analyzes a model with one-quarter and two-quarter bonds and studies the effects of
making earlier issuances senior to new issuances.
3
useful to eliminate sovereign debt dilution in reality. Weak enforcement of sovereign debt claims
may constitute an obstacle to implementing a meaningful seniority structure.
A second approach for the study of debt dilution is to compare equilibria obtained with longduration and with one-period bonds.6 However, one-period bonds do not only eliminate dilution
but also increase rollover risk. We show that, in general, one cannot measure accurately the effects
of dilution by comparing equilibria with one-period and with long-duration bonds: Lower default
probabilities with one-period bonds are not only the result of the elimination of debt dilution
but also the result of the lower debt levels chosen by the borrower to mitigate rollover risk.
Furthermore, while eliminating dilution with borrowing-contingent payments increases welfare,
replacing long-duration bonds by one-period bonds in the presence of rollover risk decreases
welfare.
We propose a new approach for the study of the effects of debt dilution. We modify the
baseline model by assuming that sovereign bonds include a covenant specifying that after each
time the government borrows it compensates existing bondholders by paying the difference between the bond market price that would have been observed absent current-period borrowing
and the observed market price. With this borrowing-contingent payment, bonds’ values become
independent from future borrowing and thus, there is no dilution caused by borrowing decisions.
We measure the effects of dilution by comparing simulations of the baseline model (with dilution)
with the ones of the modified model (without dilution). We impose discipline to our quantitative
exercise by calibrating the baseline model to match data from an economy facing default risk
(Argentina before its 2001 default).
We find that, if the sovereign eliminates debt dilution, the number of defaults per 100 years
decreases from 5.6 (with dilution) to 0.9 (without dilution). That is, dilution accounts for 84%
of the default risk in the simulations of the baseline model. Reducing the default frequency is
beneficial for welfare because defaulting is ex-ante inefficient. Thus, our exercise is indicative of
6
Intertemporal debt dilution only appears with long-duration bonds. With one-period bonds, when the government decides its current issuance level, the outstanding debt level is zero (either because the government honored
its debt obligations at the beginning of the period or because it defaulted on them). Thus, the government cannot
dilute the value of debt issued in previous periods. Chatterjee and Eyigungor (forthcoming) and Hatchondo and
Martinez (2009) show that in a sovereign default framework, equilibrium default risk is significantly higher with
long-duration bonds than with one-period bonds.
4
the quantitative importance of dilution and supports the view that dilution should be a central
issue in discussions of sovereign debt management and the international financial architecture
(e.g., Borensztein et al. (2004)).
Eliminating dilution allows the government to choose a low default risk. With dilution,
default risk is high even if the government chooses very low debt levels that almost certainly
would not trigger a default in the following period. This is the case because future governments
would increase the debt level. As long as a government cannot control the choices of future
governments, it cannot choose low default risk. Promising borrowing-contingent payments allow
the government to moderate the borrowing levels that future governments will choose.
The government’s ability to choose low default risk is reflected in its borrowing opportunities
(i.e., the set of combinations of levels of debt and interest rates the government can choose
from). Thus, eliminating dilution shifts the set of government’s borrowing opportunities. The
equilibrium combinations of debt and interest rate levels in the simulations without dilution are
not part of the government’s choice set with dilution. For no-dilution equilibrium debt levels, the
equilibrium interest rate would be about 400 basis points higher in the economy with dilution.
Borrowing-contingent payments weaken the government’s incentives to issue debt and thus imply
lower future issuance levels. For any debt level, the expectation of lower future issuance levels
implies a lower default probability. This in turn allows the government to pay a lower interest
rate.
The borrowing-contingent payments that eliminate dilution may be difficult to implement in
practice. This is because determining these payments requires knowledge of the price at which
bonds would have traded in the absence of current-period borrowing. While that price can be
easily computed in our simulations, it may be difficult to determine in practice.
However, we show that most gains from eliminating dilution can be obtained with two simple
borrowing-contingent payments that depend only on the sovereign bond market price. In one,
the sovereign promises to pay a predetermined share of current borrowing revenues to the holder
of each bond issued in previous periods. In the other, borrowing-contingent payments are a
decreasing function of the bond market price. The benchmark default frequency is reduced 69%
with the first scheme and 86% with the second scheme.
5
It should be emphasized that our findings are not based on the assumption that the government cannot default on borrowing-contingent payments in the same way it can default on other
payments. Sovereign debt contracts often contain an acceleration clause and a cross-default
clause (for example, see IMF (2002)). The first clause allows creditors to call the debt they hold
in case the government defaults on a payment. The cross-default clause states that a default in
any government obligation constitutes a default in the contract containing that clause. These
clauses imply that in practice, when the government chooses to default on a payment it chooses
to default on all its debt. The implementation of borrowing-contingent payments only requires
that defaulting on borrowing-contingent payments would trigger acceleration and cross-default
clauses and, therefore, a default on all government debt.
The borrowing-contingent payments studied in this paper resemble covenants commonly used
in corporate debt contracts to transfer resources from debtors to creditors when credit quality
deteriorates (for instance, because of an increase in indebtedness). For example, Asquith et al.
(2005) document such “interest-increasing performance pricing” and find lower interest rates for
contracts with this pricing.
Borrowing-contingent payments also resemble taxes used in previous studies for eliminating
overborrowing by private debtors (see Bianchi (forthcoming) and the references therein). In these
studies, borrowing by one agent increases other agents’ cost of borrowing and the probability of
a crisis. Taxing private borrowing reduces the frequency of crises. In this paper, the borrowing
by future governments increases the current government’s cost of borrowing and the default
probability. Borrowing-contingent payments “tax” borrowing by future governments and thus
reduce default risk.
The rest of the article proceeds as follows. Section 2 introduces the model. Section 3 discusses the calibration. Section 4 presents the results. Section 5 concludes and discusses possible
extensions of our analysis.
6
2
The model
We first discuss the baseline model with debt dilution and later introduce borrowing-contingent
payments that allows us to quantify the role of debt dilution.
2.1
The baseline environment
There is a single tradable good. The economy receives a stochastic endowment stream of this
good yt , where
log(yt ) = (1 − ρ) µ + ρ log(yt−1 ) + εt ,
with |ρ| < 1, and εt ∼ N (0, σǫ2 ).
The government’s objective is to maximize the present expected discounted value of future
utility flows of the representative agent in the economy, namely
E
"∞
X
#
β t u (ct ) ,
t=0
where E denotes the expectation operator, β denotes the subjective discount factor, and the
utility function is assumed to display a constant coefficient of relative risk aversion denoted by
γ. That is,
u (c) =
c(1−γ) − 1
.
1−γ
As in Hatchondo and Martinez (2009), we assume that a bond issued in period t promises an
infinite stream of coupons, which decreases at a constant rate δ. In particular, a bond issued in
period t promises to pay one unit of the good in period t + 1 and (1 − δ)s−1 units in period t + s,
with s ≥ 2.
Each period, the government makes two decisions. First, it decides whether to default.
Second, it chooses the number of bonds that it purchases or issues in the current period.
As in previous studies of sovereign default, the cost of defaulting is not a function of the
size of the default. Thus, as in Arellano and Ramanarayanan (2010), Chatterjee and Eyigungor
(forthcoming), and Hatchondo and Martinez (2009), when the government defaults, it does so
on all current and future debt obligations. This is consistent with the behavior of defaulting
7
governments in reality. As mentioned in the introduction, sovereign debt contracts often contain
acceleration and cross-default clauses. These clauses imply that after a default event, future debt
obligations become current.7 Following previous studies, we also assume that the recovery rate
for debt in default is zero.
Lenders are risk neutral and assign the value e−r to payoffs received in the next period. Bonds
are priced in a competitive market inhabited by a large number of identical lenders, which implies
that bond prices are pinned down by a zero expected profit condition.
When the government defaults, it faces an income loss of φ (y) in the default period. In
Section 4.3 we show that our findings are robust to assuming that the government is excluded
from capital markets after a default episode.8 Following Chatterjee and Eyigungor (forthcoming),
we assume a quadratic loss function φ (y) = d0 y + d1 y 2.
The government cannot commit to future default and borrowing decisions. Thus, one may
interpret this environment as a game in which the government making the default and borrowing
decisions in period t is a player who takes as given the default and borrowing strategies of other
players (governments) who will decide after t. We focus on Markov Perfect Equilibrium. That
is, we assume that in each period, the government’s equilibrium default and borrowing strategies
depend only on payoff-relevant state variables. As discussed by Krusell and Smith (2003), there
may be multiple Markov perfect equilibria in infinite-horizon economies. In order to avoid this
problem, we solve for the equilibrium of the finite-horizon version of our economy, and we increase
the number of periods of the finite-horizon economy until value functions and bond prices for
the first and second periods of this economy are sufficiently close. We then use the first-period
equilibrium functions as the infinite-horizon-economy equilibrium functions.
7
The type of acceleration clauses depend on the details of each bond contract and on the jurisdiction under
which the bond was issued (see IMF (2002)). For instance, in some cases it is necessary that creditors holding a
minimum percentage of the value of the bond issue request their debt to be accelerated for their future claims to
become due and payable. In other cases, no such qualified majority is needed.
8
Hatchondo et al. (2007) solve a baseline model of sovereign default with and without the exclusion punishment
and show that eliminating this punishment only affects significantly the debt level generated by the model.
8
2.2
Recursive formulation of the baseline framework
Let b denote the number of outstanding coupon claims at the beginning of the current period,
and b′ denote the number of outstanding coupon claims at the beginning of the next period. A
negative value of b implies that the government was a net issuer of bonds in the past. Let d
denote the current-period default decision. We assume that d = 1 if the government defaulted
in the current period and d = 0 if it did not. The number of bonds issued by the government is
given by − [b′ − (1 − d)(1 − δ)b]. Let V denote the government’s value function at the beginning
of a period, that is, before the default decision is made. Let Ṽ denote its value function after the
default decision has been made. Let F denote the conditional cumulative distribution function
of next-period endowment y ′. For any bond price function q the function V satisfies the following
functional equation:
V (b, y) = max {dṼ (1, b, y) + (1 − d)Ṽ (0, b, y)},
dǫ{0,1}
(1)
where
Ṽ (d, b, y) = max
u (c) + β
′
b ≤0
Z
V (b , y )F (dy | y) ,
′
′
′
(2)
and
c = y − dφ (y) + (1 − d)b − q(b′ , y) [b′ − (1 − d)(1 − δ)b] .
The bond price is given by the following functional equation:
Z
′
q(b , y) =
e−r [1 − h (b′ , y ′)] F (dy ′ | y)
Z
+ (1 − δ) e−r [1 − h (b′ , y ′)] q(g(h(b′ , y ′), b′ , y ′), y ′)F (dy ′ | y) ,
(3)
(4)
where h and g denote the future default and borrowing rules that lenders expect the government
to follow.9 If the government defaults (does not default), h = 1 (h = 0). The function g
9
The price of a bond issued at date t consists of the discounted sum of future payoffs. Formally,
q(bt+1 , yt ) =
∞
X
i=1
i−1 −ir
(1 − δ)
e
Z
...
Z
[1 − h(bt+i , yt+i )] F (dyt+i | yt+i−1 )...F (dyt+1 | yt ),
(5)
where bt+i denotes the government’s equilibrium asset position i periods ahead, which is a function of bt+1 and
9
determines the number of coupons that will mature next period. The first term in the righthand side of equation (4) equals the expected value of the next-period coupon payment promised
in a bond. The second term in the right-hand side of equation (4) equals the expected value of
all other future coupon payments, which is summarized by the expected price at which the bond
could be sold in the next period.
Equations (1)-(4) illustrate that the government finds its optimal current default and borrowing decisions taking as given its future default and borrowing decision rules h and g. In
equilibrium, the optimal default and borrowing rules that solve problems (1) and (2) must be
equal to h and g for all possible values of the state variables.
Definition 1 A Markov Perfect Equilibrium is characterized by
1. a set of value functions Ṽ and V ,
2. a default rule h and a borrowing rule g,
3. a bond price function q,
such that:
(a) given h and g, V and Ṽ satisfy equations (1) and (2) when the government can trade
bonds at q;
(b) given h and g, the bond price function q is given by equation (4); and
the sequence of income realizations yt+1 , ..., yt+i−1 (we omit these arguments to simplify the notation).
Equation (5) can be decomposed as
Z
−r
q(bt+1 , yt ) = e
[1 − h(bt+1 , yt+1 )] F (dyt+1 | yt ) +
(1 − δ) e
−r
Z "X
∞
i=1
= e−r
Z
i−1 −ir
(1 − δ)
e
Z
...
Z
#
[1 − h(bt+1+i , yt+1+i )] F (dyt+1+i | yt+i )...F (dyt+2 | yt+1 ) F (dyt+1 | yt )
[1 − h(bt+1 , yt+1 )] F (dyt+1 | yt ) + (1 − δ) e−r
10
Z
q(bt+2 , yt+1 )F (dyt+1 | yt ).
(c) the default rule h and borrowing rule g solve the dynamic programming problem defined
by equations (1) and (2) when the government can trade bonds at q.
2.3
A framework without debt dilution
In this section, we propose a modification to the model presented in Section 2.1 that will allow
us to study an economy without debt dilution and, in turn, to measure the effects of debt dilution. We eliminate debt dilution—caused by borrowing decisions—by introducing a borrowingcontingent debt covenant. The covenant specifies that if the sovereign borrows, it has to pay to
the holder of each previously-issued bond the difference between the counterfactual bond price
one would have observed absent new borrowing and the observed bond market price.10 This
covenant eliminates debt dilution by making the value of each bond independent from future
borrowing decisions.
We assume that a default on borrowing-contingent payments triggers acceleration and crossdefault clauses that make all the government’s debt obligations become current: if the government
selectively defaults on borrowing-contingent payments, it has to cancel all current and future debt
obligations, discounting future debt obligations at the risk-free rate. Consequently, selectively
defaulting on borrowing-contingent payments cannot be better than buying back all government
debt. Therefore, the next subsection presents the recursive formulation of the framework with
borrowing-contingent payments without giving the government the option to selectively default
on borrowing-contingent payments (but giving the government the option to buy back its debt).
2.4
Recursive formulation of the framework without debt dilution
As before, let q denote the price function of sovereign bonds. Let b̃ ≡ (1 − δ)b < 0 denote
the interim number of next-period coupon obligations. As in Section 2.1, when the government
10
To be precise, these payment obligations do not only depend on the borrowing decision (how many bonds are
issued in the current period). They also depend on the current income realization and the debt level, given that
the counterfactual bond price one would have observed absent new borrowing requires knowledge of the income
and debt level, and of the bond price function. While this price is easy to compute in our simulations, it would
be difficult to determine in reality. In Section 4.4 we show that most benefits from eliminating dilution can be
obtained with borrowing-contingent payments that do not depend on this counterfactual price.
11
wants to buy back its bonds, it does so at the secondary-market price. If the government issues
b̃ − b′ > 0 bonds, borrowing-contingent payments are given by −b̃[q(b̃, y) − q(b′ , y)]. Suppose the
bond price is higher when the debt level is lower because the default probability is increasing
with respect to the debt level (as is always the case for the parameterizations we study). The
equilibrium bond price is then given by
′
q(b , y) =
Z
e−r [1 − h (b′ , y ′)] F (dy ′ | y)
Z
n
o
−r
′
′
′
′
′
′
′
′
′
+ (1 − δ) e [1 − h (b , y )] max 0, q(b̃ , y ) − q(g(h(b , y ), b , y ), y ) F (dy ′ | y)
Z
+ (1 − δ) e−r [1 − h (b′ , y ′)] q(g(h(b′ , y ′), b′ , y ′), y ′)F (dy ′ | y) .
(6)
The first term of the right-hand side of equation (6) represents the expected value of the nextperiod coupon payment. The second term represents the expected value of borrowing-contingent
payments. The third term represents the expected value of a bond at the end of next period—
after the lender received the coupon and borrowing-contingent payments. Note that, because of
the borrowing-contingent payments, the future value of a lender’s investment may be affected by
the income shock, a debt buyback, and a default, but not by new borrowing. Thus, there is no
debt dilution in this framework.
The government’s budget constraint reads
c = y − dφ (y) + (1 − d)b + q(b′ , y)(b̃ − b′ ) + b̃ max{0, q(b̃, y) − q(b′ , y)}.
(7)
The last term of the right-hand side of equation (7) represents the government’s borrowingcontingent payment. After replacing equations (3) and (4) by equations (6) and (7) in the
dynamic programming problem described in Section 2.2, we obtain the problem without debt
dilution.
3
Calibration
Table 1 presents the calibration. We assume that the representative agent in the sovereign
economy has a coefficient of relative risk aversion of 2, which is within the range of accepted
12
Borrower’s risk aversion
γ
2
Interest rate
r
1%
Output autocorrelation coefficient
ρ
0.9
Standard deviation of innovations
σǫ
2.7%
Mean log output
µ
(-1/2)σǫ2
Duration
δ
0.0341
Discount factor
β
0.969
Default cost
d0
-0.69
Default cost
d1
1.01
Risk premium
α
4
Table 1: Parameter values.
values in studies of business cycles. A period in the model refers to a quarter. The risk-free
interest rate is set equal to 1%. The parameter values that govern the endowment process are
chosen so as to mimic the behavior of GDP in Argentina from the fourth quarter of 1993 to the
third quarter of 2001, as in Hatchondo et al. (2009). The parameterization of the output process
is similar to the parameterization used in other studies that consider a longer sample period (see,
for instance, Aguiar and Gopinath (2006)).
With δ = 3.41%, bonds have an average duration of 4.19 years in the simulations of the
baseline model.11 Cruces et al. (2002) report that the average duration of Argentinean bonds
included in the EMBI index was 4.13 years in 2000. This duration is not significantly different
from what is observed in other emerging economies. Using a sample of 27 emerging economies,
Cruces et al. (2002) find an average duration of 4.77 years, with a standard deviation of 1.52.
We calibrate the discount factor and the output cost (two parameter values) to target three
11
We use the Macaulay definition of duration, which with the coupon structure in this paper is given by
1 + r∗
,
δ + r∗
where r∗ denotes the constant per-period yield delivered by the bond.
D=
13
moments: a mean spread—i.e., the difference between the sovereign bond yield and the risk-free
interest rate—of 7.4, a standard deviation of the spread of 2.5, and a mean debt level of 28% of
the mean quarterly output in the pre-default samples of our simulations (the exact definition of
these samples is presented in Section 4.1).12 The targets for the spread distribution are taken
from the spread behavior in Argentina before its 2001 default (see Table 2). Regarding the debt
level, for the period we study, Chatterjee and Eyigungor (forthcoming) target a mean level of
unsecured sovereign debt of 70% of quarterly output. Since our model is a model of external debt
and Sturzenegger and Zettelmeyer (2006) estimate that 60% of the debt Argentina defaulted on
was held by residents, we choose to target a mean debt level that is roughly 40% of the value
targeted by Chatterjee and Eyigungor (forthcoming). In Section 4.3 we show that our findings
are robust to targeting a higher debt level.
4
Results
As in Hatchondo et al. (2010), we solve the models numerically using value function iteration
and interpolation.13 First, we show that debt dilution accounts for most of the default risk in
the benchmark economy. Second, we present the welfare gains from eliminating dilution. Third,
we discuss the robustness of our measurement of the effects of debt dilution. Fourth, we show
that most gains from eliminating dilution can be obtained with simpler borrowing-contingent
payment schemes that may be easier to implement. Fifth, we compare long-duration debt with
borrowing-contingent payments to one-period debt. Finally, we compare the allocation without
dilution with the allocation that the government could attain if it could trade a full range of
one-period state contingent bonds.
12
The discount factor value we obtain is relatively low but higher than the ones assumed in previous studies
(for instance, Aguiar and Gopinath (2006) assume β = 0.8). Low discount factors may be a result of political
polarization in emerging economies (see Amador (2003) and Cuadra and Sapriza (2008)).
13
We use linear interpolation for endowment levels and spline interpolation for asset positions. The algorithm
finds two value functions, Ṽ (1, ·, ·) and Ṽ (0, ·, ·). Convergence in the equilibrium price function q is also assured.
14
4.1
Dilution and default risk
This subsection measures the effects of debt dilution. In order to do so, it presents simulation
results from the models with and without debt dilution. Table 2 reports moments in the data and
in our simulations.14 As in previous studies, we report results computed for pre-default simulation
samples. The exception is the default frequency, which we compute using all simulation periods.
We simulate the model for a number of periods that allows us to extract 500 samples of 32
consecutive periods before a default. We focus on samples of 32 periods because we compare the
artificial data generated by the model with Argentine data from the fourth quarter of 1993 to
the third quarter of 2001.15 In order to facilitate the comparison of simulation results with the
data, we only consider simulation sample paths in which the last default was declared at least
two periods before the beginning of each sample.
The moments reported in Table 2 are chosen so as to illustrate the ability of the model to
replicate distinctive business cycle properties of emerging economies. These economies feature a
high, volatile, and countercyclical interest rate, and high consumption volatility. To compute the
quarterly interest rate spread we first calculate the yield that makes the present value of future
payments promised in a bond—which for the no-dilution case includes borrowing-contingent
payments—equal to the bond price. Then, we calculate the quarterly spread as the difference
between this yield and the risk-free rate. The annualized spread (Rs ) is four times the quarterly
14
The data for output and consumption were obtained from the Argentine Finance Ministry. The spread before
the first quarter of 1998 is taken from Neumeyer and Perri (2005), and from the EMBI Global after that.
15
The qualitative features of this data are also observed in other sample periods and in other emerging markets
(see, for example, Aguiar and Gopinath (2007), Alvarez et al. (2011), Boz et al. (2011), Neumeyer and Perri (2005),
and Uribe and Yue (2006)). The only exception is that in the data we consider, the volatility of consumption is
slightly lower than the volatility of income, while emerging market economies tend to display a higher volatility
of consumption relative to income.
15
Data Dilution
No dilution
Defaults per 100 years
5.59
0.91
Mean debt market value
0.21
0.20
Mean debt face value
0.28
0.30
0.21
E(Rs )
7.44
7.21
0.95
σ (Rs )
2.51
2.54
0.68
σ(ỹ)
3.17
2.97
3.28
σ(c̃)/σ(ỹ)
0.94
1.03
1.10
ρ (c̃, ỹ)
0.97
1.00
0.99
ρ (Rs , ỹ)
-0.65
-0.81
-0.63
Table 2:
Business cycle statistics. The second column is computed using data from Argentina from
1993 to 2001. Other columns report the mean of the value of each moment in 500 simulation samples.
Each sample consists of 32 periods before a default episode.
spread.16
In Table 2, the logarithm of income and consumption are denoted by ỹ and c̃, respectively. The
standard deviation of x is denoted by σ (x) and is reported in percentage terms. The coefficient
of correlation between x and z is denoted by ρ (x, z). Moments are computed using detrended
series. Trends are computed using the Hodrick-Prescott filter with a smoothing parameter of
1, 600. Table 2 also reports the mean debt market value (computed as the mean b divided by
δ + r ∗ , where r ∗ is the mean equilibrium interest rate) and the mean debt face value (computed
16
In the economy without dilution, let
C(b, y) = max {q(b(1 − δ)(1 − h(b, y)), y) − q(g(h(b, y), b, y), y), 0}
denote the per-bond borrowing-contingent payment when the initial state is given by the vector (b, y). Let
Z
q DF (b′ , y; i) = e−i
1 + C(b′ , y ′ ) + (1 − δ)q DF (g(h(b′ , y ′ ), b′ , y ′ ), y ′ ; i) F (dy ′ | y)
denote the price of a default-free bond that pays the coupon and C every period, where i denotes the constant
rate at which future payments are discounted. In state (b, y), the yield of a defaultable bond is defined as the
rate i∗ that satisfies
q DF (g(h(b, y), b, y), y, i∗) = q(g(h(b, y), b, y), y).
The annualized interest rate spread is therefore defined as Rs = 4(i∗ − r).
16
as the mean b divided by δ + r).
Table 2 shows that the baseline model with dilution matches the data reasonably well. As in
the data, in the simulations of the baseline model consumption and income are highly correlated,
and the consumption volatility is higher than the income volatility. The model also approximates
reasonably well the moments used as targets (the mean debt level, and the mean and standard
deviation of the spread). Estimating the default probability in the data is difficult. Using a
sample of 68 countries between 1970 and 2010, Cruces and Trebesch (2011) find a frequency of
6.6 defaults every 100 years. Arellano (2008) targets a frequency of 3 defaults per 100 years
because that is the number of defaults observed in Argentina during the last 100 years. The
default frequency in our benchmark simulation is between those numbers. In Section 4.3, we
show that, when the risk aversion of lenders is calibrated to generate a yearly default frequency
of three defaults per 100 years, the effects of debt dilution are similar to the ones reported using
our baseline calibration.
What are the quantitative effects of debt dilution? Table 2 shows that debt dilution accounts
for 84% of the default risk in the simulations of the baseline model. The number of defaults per
100 years decreases from 5.59 in the baseline to 0.91 in the model without debt dilution. Debt
dilution also accounts for 87% of the spread paid by the sovereign. The standard deviation of the
spread decreases from 2.54 with debt dilution to 0.68 without debt dilution. The mean face value
of outstanding bonds declines 32%. Most of this decline is explained by the lower interest rate
in the simulations of the model without debt dilution: The mean market value of outstanding
bonds decreases only by 9%.
In order to shed light on how eliminating debt dilution affects the government’s optimal
decisions it is illustrative to consider its first-order conditions. We use fj (x1 , ..., xn ) to denote the
first-order derivative of the function f with respect to the argument xj . The first-order condition
in the benchmark economy (with dilution) is given by
u1 (c)q
Dil
′
(b , y) = β
Z
V1Dil (b′ , y ′) F (dy ′ | y) − u1 (c)q1Dil (b′ , y)[b′ − (1 − d)b̃],
(8)
where the superindex Dil denotes variables in the economy with dilution. The left-hand side
17
of equation (8) represents the marginal benefit of borrowing. By issuing one extra bond today,
the government can increase current consumption by q Dil (b′ , y) units. The right-hand side of
equation (8) represents the marginal cost of borrowing. The first term in the right-hand side
represents the future cost of borrowing: by borrowing more, the government decreases expected
future consumption. The second term in the right-hand side represents the current cost of
borrowing: by borrowing more, the government decreases the issuance price of every bond it
issues in the current period, which in turn decreases current consumption.
Assuming that the zero-profit bond price is decreasing in the debt level (as we find it is the
case for the parameterizations we study) and the government chooses to borrow (b′ < b̃), the
first-order condition in the economy without dilution is given by
u1 (c)q
No dil
′
(b , y) = β
Z
V1No dil (b′ , y ′) F (dy ′ | y) − u1 (c)q1No dil (b′ , y)b′ ,
(9)
where the superindex No dil is used to denote variables in the economy with no dilution.
The comparison of equations (8) and (9) shows how our modification to the baseline model
affects the tradeoffs faced by the government when it issues debt. In equation (8), the current
cost of borrowing depends on the new issuances: The government only internalizes as a cost the
decline in the value of bonds issued in the current period. It does not internalize as a cost the
decrease in the value of the debt issued in previous periods. In contrast, equation (9) shows
that with our modification to the baseline model, the current cost of borrowing depends on the
entire debt stock at the end of the period. That is, the government chooses its issuance level
internalizing the dilution in the value of debt issued in previous periods.
Equation (9) illustrates the tradeoffs faced by the government for a given bond price. But the
change in the government’s tradeoffs also affect the bond price schedule the government faces
when issuing debt. We illustrate that in Figure 1. The figure presents the spread demanded
by lenders as a function of the face value of next-period debt. The figure also presents the
combination of spread levels and next-period debt chosen by the government when its initial
debt level is the average level in the simulations of each case considered in the graph.
Figure 1 shows that a shift in the government’s choice set plays an important role in accounting
18
Low y
High y
9
9
8
8
7
7
6
5
4
6
5
4
3
3
2
2
1
1
0
−0.3
−0.25
−0.2
−0.15
Debt
−0.1
−0.05
Low y
High y
10
Annual spread
Annual spread
10
0
0
−0.3
−0.25
−0.2
−0.15
Debt
−0.1
−0.05
0
′
b
Figure 1: Menu of combinations of spreads and next-period debt levels ( δ+r
) from which the government
can choose. The left panel corresponds to the baseline case. The right panel corresponds to the
case without debt dilution. In each of these two cases, solid dots illustrate the optimal decision of a
government that inherits a debt level equal to the average debt observed in our simulations for that case.
Vertical lines mark the government’s debt level before its issuance decision. The low (high) value of y
corresponds to an endowment realization that is one standard deviation below (above) the unconditional
mean.
for the reduction in spreads implied by the elimination of debt dilution: Even for the same
debt levels, spread levels are higher in the benchmark than in the no-dilution model. For the
equilibrium debt levels without dilution, equilibrium spread levels would be about 400 basis
points higher in the economy with dilution. In the model without dilution, borrowing-contingent
payments weaken the governments incentives to issue debt and thus imply lower future issuance
levels. For any debt level, the expectation of lower future issuance levels implies a lower default
probability. This in turn allows the government to pay a lower interest rate.
Figure 1 also helps us understand why consumption volatility is higher in the economy without
dilution. As illustrated in Figure 1, when income is low, issuance levels tend to be lower in the
economy without dilution than in the benchmark with dilution (in the figure, issuance levels
are represented by the horizontal distance between the dark dots and the vertical solid line).17
17
Notice that, for the same debt level, the spread curves in Figure 1 are steeper when income is lower. This
implies that in the economy without dilution, for the same issuance level, borrowing-contingent payments would
be larger when income is lower.
19
Thus, the government is more effective in mitigating the effects of low income realizations on
consumption in the benchmark with dilution.
4.2
Welfare gains from eliminating dilution
We next show that it is welfare enhancing to implement the borrowing-contingent payments that
eliminate debt dilution. Eliminating dilution reduces the frequency of defaults, and with that
it reduces the deadweight losses caused by defaults. We measure welfare gains as the constant
proportional change in consumption that would leave a consumer indifferent between continuing
living in the benchmark economy with dilution and moving to an economy without dilution. The
welfare gain of moving from the benchmark economy to the economy without dilution is given
by
1
V No dil (b, y) ( 1−γ )
− 1.
V Dil (b, y)
Figure 2 presents welfare gains from implementing the borrowing-contingent payments that
eliminate dilution. The figure considers two initial debt levels: zero, and the mean debt level
in the simulations of the economy with dilution. The figure shows that for both cases there are
positive welfare gains from eliminating dilution.
In order to eliminate dilution in an economy with positive debt levels, the government
promises borrowing-contingent payments to holders of existing debt. This is costly for the government and explains why in Figure 2 welfare gains from eliminating dilution are lower when
there is initial debt (except for lower income levels for which the government chooses to default).18
Figure 2 also presents welfare gains from the case in which the government captures existing bondholders’ capital gains from the introduction of borrowing-contingent payments that
eliminate debt dilution. We assume the government captures these gains through a debt exchange: The government makes a take-it-or-leave-it debt buyback offer with the promise that
18
Welfare gains are larger in a default period (when the government writes off all debt liabilities) than when the
government enters the period without debt. After a default, the government wants to smooth out the income cost
of defaulting. Thus, in a default period, the government has stronger incentives to borrow than when it enters
the period without debt. This explains why the improvement in borrowing terms implied by the elimination of
dilution is more valuable in default periods than when the government enters the period without debt.
20
0.8
Proportion of cons. (in %)
0.7
Initial debt = 0
Initial debt = avg. debt in baseline econ.
Bond exchange & initial debt = avg. debt in baseline economy
0.6
0.5
0.4
0.3
0.2
0.1
0
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
y
Figure 2: Consumption compensation (in percentage terms) that makes domestic agents indifferent
between living in an economy with or without dilution. The figure was constructed assuming that the
initial debt level is equal to zero or to the mean debt level in the simulations of the economy with
dilution. A positive number means that agents prefer the economy without dilution.
these borrowing-contingent payments will be implemented only if this offer is accepted. Thus,
the government offers bondholders to buy back previously issued bonds at the price that would
have been observed if borrowing-contingent payments were never implemented. That price is
lower than the no-dilution price at which the government would be able to issue debt after
implementing borrowing-contingent payments. By assuming that the government makes a takeit-or-leave-it offer, we focus on the extreme case in which it reaps all capital gains.19 The case in
which borrowing-contingent payments are introduced without a debt exchange constitutes the
other extreme case in which bondholders enjoy all these gains.
It should be mentioned that one may want to take our measure of the welfare gain with
a grain of salt. In particular, one could argue that our measure is too low. Since there is
no production in our setup, we cannot capture productivity gains from reducing the level and
volatility of interest rates.20 Several studies find evidence of a significant effect of interest rates
19
It is also assumed that there are no output costs triggered by that debt exchange.
The development of a sovereign default framework that accommodates effects of interest rates on factors
allocation is the subject of ongoing research (see, for example, Mendoza and Yue (forthcoming) and Sosa Padilla
20
21
on productivity (through the allocation of factors of production), and of a significant role of
interest rate fluctuations in the amplification of shocks (see, for example, Mendoza and Yue
(forthcoming), Neumeyer and Perri (2005), and Uribe and Yue (2006)). Furthermore, the welfare
gain is increasing in the level of debt, and we chose to calibrate our model to a low debt level
to resemble the level of sovereign defaultable debt held by foreigners. In the next subsection
we present a calibration with a higher debt level and show that eliminating dilution results in a
larger welfare gain.
4.3
Robustness
In this subsection, we show that our finding of a strong effect of debt dilution on sovereign default
risk is robust to changes in the benchmark economy we study. We focus on three changes to our
benchmark. First, we modify the cost of defaulting to allow for a higher debt stock. Second,
we assume that lenders are risk averse. Third, we introduce sudden-stop shocks into the model
(while we still assume risk-averse lenders). For all cases, we show that eliminating dilution leads
to a significant reduction of the default frequency and increases welfare.
In order to present a case with higher debt levels, we assume a larger cost of defaulting.
A defaulting economy is banned from international capital markets for a stochastic number of
periods and it loses a fraction of its output in every period it remains excluded. The probability of
reentry to capital markets is constant over time. This is the same cost of defaulting considered
in most previous studies of sovereign default (e.g., Aguiar and Gopinath (2006) and Arellano
(2008)). We assume that the probability of reentry equals 0.282, which is the value used by
Arellano (2008). The only parameters that change compared to the benchmark parameterization
are the ones that determine the output loss after a default. We set d0 = −0.7043 and d1 = 0.9236.
This calibration implies a debt level of 78 percent of quarterly output, a value slightly higher
than the one targeted by Chatterjee and Eyigungor (forthcoming).
We introduce risk premium as Arellano and Ramanarayanan (2010). We assume that the price
of sovereign bonds satisfies a no arbitrage condition with stochastic discount factor M(y ′ , y) =
(2010)).
22
′
2 σ2
ǫ
e−r−αε −0.5α
. Several studies document that the risk premium is an important component of
sovereign spreads and that a significant fraction of the spread volatility in the data is accounted
for by the volatility in the risk premium (see, for example, Borri and Verdelhan (2009), Broner
et al. (2007), Longstaff et al. (2011), and González-Rozada and Levy Yeyati (2008)).
The discount factor M(y ′ , y) is a special case of the discrete-time version of the Vasicek
one-factor model of the term structure (see Vasicek (1977) and Backus et al. (1998)). With our
formulation, the risk premium is determined by the income shock in the borrowing economy. The
advantage of our formulation is that it avoids introducing additional state variables to the model.
However, it may be more natural to assume that the lenders’ valuation of future payments is not
perfectly correlated with the sovereign’s income.
Our third robustness exercise shows that our measurement of the quantitative effect of dilution
on default risk is robust to assuming that there is a shock to the cost of borrowing that is
not perfectly correlated with the sovereign’s income. To this end, we introduce sudden-stop
shocks. These shocks have received considerable attention in the international macroeconomics
literature (see Durdu et al. (2009) and the references therein) but have been mostly ignored in the
quantitative sovereign default literature (Chatterjee and Eyigungor (forthcoming) is a notable
exception). We add a shock s such that when s = 1 (s = 0) the government can (cannot) issue
debt—the government can always buy back previously issued debt. We denote by π1 (π0 ) the
probability of st+1 = st conditional on st = 1 (st = 0). Thus, the value functions with sudden
stops are given by
V (b, y, s) = max {dṼ (1, b, y, s) + (1 − d)Ṽ (0, b, y, s)},
dǫ{0,1}
Z
′
′
′
′
′
u (c) + β [π1 V (b , y , 1) + (1 − π1 )V (b , y , 0)] F (dy | y) ,
Ṽ (d, b, y, 1) = max
′
b ≤0
and
Ṽ (d, b, y, 0) =
max
b(1−δ)≤b′ ≤0
u (c) + β
Z
[π0 V (b , y , 0) + (1 − π0 )V (b , y , 1)] F (dy | y) .
′
23
′
′
′
′
We calibrate π1 and π0 assuming that there is a 5.5% unconditional probability of observing
s = 0 and sudden-stop periods last for one year on average (using annual data for a sample of
developing countries from 1980 to 2003, Eichengreen et al. (2008) estimate a 5.5% probability of
observing a sudden stop).
Table 3 shows that there is a strong effect of debt dilution on sovereign default risk in the
three economies studied in this subsection. The effects of dilution on other variables are also very
similar across the different exercises. The exception is the debt level. When the government is
excluded from capital markets after a default, the market value of debt is higher in the economy
without dilution. The better borrowing terms that the government receives in the economy
without dilution increases the value of having access to debt markets and, therefore, it makes
defaulting more costly in that case. Eliminating dilution is also beneficial for welfare in all the
cases studies.
Higher debt
Risk-averse lenders
Sudden Stops
Dilution No dilution
Dilution No dilution
Dilution
No dilution
Defaults per 100 years
4.52
1.02
3.10
0.42
5.07
0.86
Mean debt market value
0.55
0.56
0.20
0.18
0.23
0.19
Mean debt face value
0.78
0.59
0.28
0.19
0.32
0.20
E(Rs )
7.12
1.11
7.04
0.99
7.64
0.52
σ (Rs )
2.72
0.75
2.19
0.71
3.05
0.65
σ(y)
2.98
3.22
2.03
3.36
3.07
3.25
σ(c)/σ(y)
1.08
1.26
1.04
1.08
1.03
1.11
ρ (c, y)
0.99
0.98
1.00
0.99
0.99
0.99
ρ (Rs , y)
-0.79
-0.65
-0.82
-0.63
-0.80
-0.62
Welfare gain (% of cons.)
0.33
0.13
0.12
Table 3: Robustness exercises. The welfare gain corresponds to the average gain for the case of zero
debt.
24
4.4
Alternative borrowing-contingent payments
Implementing the borrowing-contingent payments that eliminate dilution would require knowledge of the fundamentals that determine bond prices (income and debt in the model) and the
mapping from fundamentals onto bond prices. In this section, we study the effects of two simpler
debt covenants.
We first study the case in which borrowing-contingent payments are a predetermined fixed
share of borrowing revenues per unit of outstanding debt. We search for the optimal share of
borrowing revenues the government should promise to existing creditors. We find that this share
is such that on average the government pays to holders of debt issued in previous periods 36%
of its issuance revenues.21
Table 4 presents simulation results for the economy with the fixed-share borrowing-contingent
payments. The table shows these payments allow the government to achieve 83% of the decline
in the default frequency and 69% of the ex-ante welfare gain it achieves with the borrowingcontingent payments that eliminate dilution.
The main difference between fixed-share borrowing-contingent payments and the borrowingcontingent payments that eliminate dilution is that the former are an increasing function of the
post-issuance bond price q(b′ , y) while the latter are a decreasing function of q(b′ , y). Next, we
study the effects of introducing borrowing-contingent payments that are a decreasing function
of the post-issuance bond price (but do not depend on the bond price that would have been
observed in the absence of borrowing). In particular, we assume that the covenant specifies that
if the government issues debt in the current period, it has to pay A − q(b′ , y) per bond issued
in previous periods. We search for the optimal value of A and find that this value is 1.75%
higher than the price of a risk-free bond without borrowing-contingent payments. Table 4 shows
that this optimal borrowing-contingent function would allow the government to achieve the same
welfare gain as with the borrowing-contingent payments that eliminate dilution, with a slightly
lower default frequency. In summary, our findings indicate that simple borrowing-contingent
payments could also reduce default risk significantly and be welfare enhancing.
21
It should be mentioned that promising higher borrowing-contingent payments is not necessarily costly for the
government because it allows the government to sell bonds at a higher price.
25
Dilution No dilution
Fixed share
Linear
Defaults per 100 years
5.59
0.91
1.71
0.81
Mean debt market value
0.21
0.20
0.17
0.19
Mean debt face value
0.30
0.21
0.20
0.20
E(Rs )
7.21
0.95
2.01
0.79
σ (Rs )
2.54
0.68
1.31
0.66
σ(y)
2.97
3.27
3.05
3.28
σ(c)/σ(y)
1.03
1.10
0.96
1.10
ρ (c, y)
1.00
0.99
0.99
0.99
ρ (Rs , y)
-0.81
-0.63
-0.64
-0.61
0.13
0.09
0.13
Welfare gain (% of cons.)
Table 4: Simulation results for different borrowing-contingent payments. The welfare gain corresponds
to the average gain for the case of zero debt.
4.5
Borrowing-contingent payments vs. one-period bonds
In this subsection, we show that in general one cannot measure the effects of debt dilution by
comparing equilibria with long-duration and one-period bonds. With one-period bonds, when
the government issues debt, the outstanding debt level is zero (either because the government
honored its debt at the beginning of the period or because it defaulted on it) and, thus, the
government cannot dilute the value of debt issued in previous periods. But with one-period
bonds the government has to pay back its entire debt stock in every period, which increases its
exposure to rollover risk.
Table 5 presents simulation results for two models without dilution: Our model with the
borrowing-contingent payments that eliminate dilution, and a one-period bond model (i.e., a
model with δ = 1). The table presents results for both the baseline and the sudden-stop versions
of the model. Table 5 shows that simulation results obtained with one-period bonds differ from
those obtained with borrowing-contingent payments. Because one-period bonds may increase
the government’s exposure to rollover risk, differences in results are wider for the sudden-stop
26
model, for which rollover risk is more significant. With sudden stops, lower default probabilities
with one-period bonds are not only the result of the elimination of debt dilution but also the
result of the lower debt levels chosen by the borrower to mitigate rollover risk.
Baseline
Sudden stops
Borrowing-cont.
One-period
Borrowing-cont.
One-period
Defaults per 100 years
0.91
0.34
0.86
0.03
Mean debt (market value)
0.20
0.19
0.19
0.08
Mean debt (face value)
0.21
0.20
0.20
0.08
E(Rs )
0.95
0.44
0.52
0.15
σ (Rs )
0.68
0.45
0.65
0.27
σ(y)
3.28
3.10
3.25
3.22
σ(c)/σ(y)
1.10
1.23
1.11
1.28
ρ (c, y)
0.99
0.97
0.99
0.80
ρ (Rs , y)
-0.63
-0.73
-0.62
-0.41
Welfare gain (% of cons.)
0.13
0.13
0.12
-0.10
Table 5: Business cycle statistics without debt dilution. The welfare gain corresponds to the average
gain for the case of zero debt.
In terms of welfare, when the government is subject to sudden-stop shocks and rollover risk
is significant, replacing long-duration bonds by one-period bonds results in welfare losses. The
ex-ante welfare loss from issuing one-period bonds instead of long-duration bonds is equivalent to
a permanent consumption decline of 0.1%. In contrast, the ex-ante welfare gain from introducing
borrowing-contingent payments is equivalent to a permanent consumption increase of 0.12%.
4.6
Borrowing-contingent payments vs. one-period state-contingent
claims
Equation (7) makes clear that the borrowing-contingent payments needed to eliminate dilution
are state-contingent. In this subsection, we explore how those borrowing-contingent payments
27
perform against the best possible case with state-contingent one-period claims. It would be best
for the government to transfer resources across periods using contracts with payoffs that are
conditional on the past history of state realizations. For tractability reasons, in this subsection
we consider a market structure in which the government can issue one-period Arrow-Debreu securities that pay off conditional on the next-period domestic income realization. The government
is subject to the same limited liability constraint that is present in the benchmark economy.22
We assume that the government chooses how much it promises to pay next period for each
realization of next-period income y ′ (payments can be negative). Let b′ (y ′ ) denote such government’s promise. The government is subject to the same cost of defaulting that is present
in the benchmark economy. Without loss of generality, we assume that the government only
promises payments b′ (y ′) for which it would not choose to default. Lenders would not pay for a
government’s promise b′ (y ′) that the government would default on.
Let W̃ (d, b, y) denote the value function of a government that has chosen the default decision
d after starting the period with debt b and income y. For any b0 and b1 , W̃ (1, b0 , y) = W̃ (1, b1 , y).
Thus, since W̃ (0, b, y) is decreasing in b, for any income level y, there exists a debt level b(y ′)
¯
such that the government defaults if and only if its debt level is higher than −b(y ′ ). For any y,
¯
b(y) satisfies W̃ (0, b(y), y) = W̃ (1, b(y), y), where
¯
¯
¯
Z
′
′
′
W̃ (d, b, y) = max
u (c) + β W (b , y )F (dy | y)
b′ (y ′ )
Z
−r
s.t. c = y − dφ (y) + (1 − d)b − e
b′ (y ′)F (dy ′ | y)
b′ (y ′) ≥ b(y ′) for all y ′ ,
¯
and
W (b, y) = max {dW̃ (1, b, y) + (1 − d)W̃ (0, b, y)}.
dǫ{0,1}
Table 6 summarizes simulation results in the benchmark, and in the economies with the
borrowing-contingent payments that eliminate dilution, and with state-contingent claims. Since
22
Issuing long-term debt allows the government to bring resources forward from future periods, not only from
the subsequent period. That is not an option in the economy with one-period Arrow-Debreu securities. Clearly,
this limitation is not an issue in the absence of the limited liability constraint. But it can dampen the welfare
gain from issuing Arrow-Debreu securities in the presence of such constraint.
28
there are no defaults with state-contingent claims, we cannot use the same criterion for selecting
samples that we used for economies with defaults. In order to facilitate the comparison of
simulations across economies, for the economy with state-contingent claims we use the same
500 samples of 32 periods used to compute the simulations in the benchmark economy (income
shocks are the same, though the initial debt levels may differ).
Benchmark
No dilution
State-contingent claims
Defaults per 100 years
5.59
0.91
0
Mean debt market value
0.21
0.20
0.30
Mean debt face value
0.30
0.21
0.30
σ(y)
2.97
3.28
2.97
σ(c)/σ(y)
1.03
1.10
0.67
ρ (c, y)
1.00
0.99
0.87
0.13
0.53
Welfare gain (% of cons.)
Table 6: Business cycle statistics in the benchmark economy and in the economies with the borrowingcontingent payments that eliminate dilution and with one-period state-contingent bonds. The welfare
gain corresponds to the average gain for the case of zero debt.
Table 6 shows that the average ex-ante welfare gain from moving to an economy with statecontingent claims amounts to a permanent increase in consumption of 0.53%. There are three
sources of welfare gains from using state-contingent claims. First, issuing state-contingent claims
allows the government to avoid defaults. Second, when borrowing, the government is able to
obtain more resources with state-contingent claims (as reflected in the higher market value of
its debt). Third, the consumption process is more disentangled from the income process in the
economy with state-contingent claims (as reflected in the lower consumption volatility).
Welfare gains from moving to an economy with borrowing-contingent payments are 25%
of those from introducing state-contingent claims. In contrast with state-contingent claims,
borrowing-contingent payments do not eliminate defaults completely, reduce slightly the market
value of the government’s debt, and increase consumption volatility.
Our stylized framework is likely to overstate the advantages of introducing state-contingent
29
claims over the introduction of borrowing-contingent payments. First, identifying the state is
much more difficult in the real world than in our stylized model. In our model income shocks
are the only source of uncertainty. However, Tomz and Wright (2007) argue that other determinants of the sovereigns’ willingness to repay besides aggregate income play an important role
in accounting for sovereign defaults. Identifying the shocks that affect default risk, measuring
these shocks, and writing debt contracts with payments contingent on these shocks may be difficult. Second, writing debt claims contingent on income may suffer from verifiability and moral
hazard issues that are not present in our stylized model. A government could manipulate the
GDP calculation and final GDP data are available with a significant lag. It has also been argued
that debt claims contingent on GDP may introduce moral hazard problems by weakening the
government’s incentives to implement growth-promoting policies, which improves the likelihood
of repayment.23
5
Conclusions
We solved a baseline sovereign default framework à la Eaton and Gersovitz (1981) assuming
that sovereign bonds contain a debt covenant promising that after each time the government
borrows, it pays to the holder of each bond issued in previous periods the difference between
the bond market price that would have been observed absent current-period borrowing and the
observed market price. This covenant eliminates debt dilution—caused by borrowing decisions—
by making the value of each bond independent from future borrowing decisions. We measured the
effects of debt dilution by comparing the simulations of this model with the ones of the baseline
model without borrowing-contingent payments. We found that even without commitment to
future repayment policies and without optimally designed contingent claims, if the sovereign
eliminates debt dilution, the default probability decreases 84%. We also showed that most gains
from eliminating dilution can be obtained with borrowing-contingent payments that depend only
on the bond market price.
23
See, for instance, Krugman (1988). These issues could be addressed by indexing debt contracts to variables
that the government cannot control such as commodity prices or trading partners’ growth rates (see for instance
Caballero (2002)).
30
Our findings indicate that governments could benefit from committing to lower future borrowing levels, which governments could achieve through fiscal rules. Eliminating debt dilution
should be an important motivation for the implementation of fiscal rules that could reduce significantly the risk of debt crises and the mean and volatility of interest rates (see Hatchondo et al.
(2011)). Fiscal crises are occurring in countries with fiscal rules in part because of the weak
enforcement of these rules. The borrowing-contingent covenants studied in this paper could enhance the enforcement of fiscal rules by providing incentives for lower future borrowing levels.
Implementing these covenants would necessitate the same strict accounting norms necessary for
the successful implementation of fiscal rules.
As in Chatterjee and Eyigungor (forthcoming) and Hatchondo and Martinez (2009), we assume that the government cannot choose the duration of its debt. Relaxing this assumption
could enhance our understanding of the effects of debt dilution. It has been argued that the dilution problem may lead to excessive issuances of short-term debt (e.g., Kletzer (1984)). However,
allowing the government to choose the duration of its debt would increase the computation cost
significantly.24
24
If one allows the government to choose a different duration of sovereign bonds each period, one would have
to keep track of how many bonds the government has issued of each possible duration to determine government’s
liabilities (Arellano and Ramanarayanan (2010) study a model in which the government can choose to issue bonds
with two possible durations). The computation cost of including additional state variables may be significant (see
Hatchondo et al. (2010)).
31
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