Tài liệu Default and the Maturity Structure in Sovereign Bonds∗ docx

Default and the Maturity Structure in So vereign Bonds
∗
Cristin a Ar ella no
†
University of Minnesota and
Federal Reserve Bank of Minneapolis
Ananth Ramanaraya nan
‡
Federal Reserve Bank of Dallas
November 2008
Abstract
This paper s tu dies the matu rity composition and the term structure of i nterest rate spreads
of gove rnm ent debt in emerging mark ets. In the data, wh en interest rate sp reads rise, d eb t
matu rity shortens and the spread on short-term bonds is higher than on long-term bonds.
To account f o r this pa ttern, w e build a dyn am ic model of intern a tional borrow ing with
endogenous default and multiple maturities of debt. Short-term debt can deliver h igher
imm ed iate consum ptio n than long-term d eb t; la rge long-term lo an s are not available because
theborrowercannotcommittosaveinthenearfuturetowardsrepaymentinthefarfuture.
Howeve r, issuin g long -term debt can insure against the need to roll-over short-term debt
at high interest rate spreads. T h e trade-off between these two benefits is qua ntitatively
importan t for understanding the maturity composition in emerging ma rkets. W hen calibrated
to data from Brazil, the model matches the dynamics in the maturity of debt i ssu ances and
its como vement with the le vel of spreads across maturities.
∗
We thank V. V. Chari, Tim Kehoe, Patrick Kehoe, Naray ana Kocherlakota, Hanno Lustig, Enrique
Mendoza, Fabrizio Perri, and Victor Rios-Rull for many useful comments. The views expressed herein are
those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis, the Federal
Reserve Bank of Dallas, or the Federal Reserve System. All errors remain our own.
†

‡

anan
1 Introduction
Em erging markets f ace recurrent and costly financial c rises th at are characterize d by limited
access to credit and high interest rates on foreign debt. As crises approac h , not only is d eb t
limited but also the maturity of debt shortens, as docum ented by Broner, Lorenzoni, and
Schmukler (2007).
1
During these periods, however , the interest rate spread on short-term
bonds rises more than the spread on long-term bonds. Why do countries s horten their debt
maturity during crises ev en though spreads appear higher for shorter maturity debt? To
answer this question, this paper develops a dyna m ic model of the matu rity composition in
which debt prices reflect endogenous default risk and debt maturity responds to the prices
of short- and long-term debt contr acts. Our model can ration alize shorter debt m a turity
during crises as the result of a liquidity advantage in short-term debt contracts; although
these contracts carry higher spread s than longer term deb t, they can deliv er la rger resources
to the coun try in times of high default risk.
We first analyze the dynamics of the maturity composition of in ternational bonds and the
term structure of in terest rate spreads for four emerging m arket countries: Arg entina, Brazil,
Mexico, and Russia . We use data on prices and issuances of foreign-currency denom inated
bonds to estimate spread curves — interest rate spreads over U .S. Treasury bonds across
maturit y — as w ell as duration,ameasureoftheaveragetimetomaturityofpaymentson
coupon pa ying bonds. We find that governm ents issue short-term d eb t mor e heavily w hen
spreads are high and spread curves are down ward sloping, and they issue long-term debt
more heavily wh en sprea ds are lo w an d spread curves are upward slo ping. Across these four
countries, w ithin periods in which 2-y ea r spreads are belo w their 25th percentile, the average
duration of new debt is 7.1 y ears, and the average d ifference bet ween the 10-yea r spread and
the 2 -yea r spread is 2.3 percenta ge points. But w hen the 2-year s pread s are abo ve their 75th
per centile, the average du ra tion s horte ns to 5 .7 years, while th e ave rage differ en ce bet ween
the 1 0-year spread and the 2-year spread is −0.5 percentage points. From this evidence we
conclude that the maturity of deb t shortens in tim es of high spreads and do w nwa r d-slo ping

spread curves.
We then dev elop a dynamic model with defaultable bonds to study the c hoice of debt
matu rity and i ts co variation with t he term structure o f spreads. In ou r model, a risk averse
bo rrowe r faces persistent incom e sh oc ks an d ca n issue l on g a nd sh o rt d uration bonds. T h e
borrowercandefaultondebtatanypointintime,butfacescostsofdoingso. Default
1
Calvo and Mendoza (1996) document in detail how in Mexico during 1994, most of the public debt
was converted to 91-day Tesobonos. Bevilaqua and Garcia (2000) document a similar rise in short-term
government debt in Brazil during the 1999 crisis.
2
occ urs in equ ilibrium in low-income, high-debt times be cau se the cost of coupon paym ents
outweighs the co sts of default when con su m ption is low. Interest rate sp reads on lon g and
short bonds compensa te foreign len ders for the expected loss from future d efau lts. T hu s, t he
supply of credit is more string ent in time s of lo w income and high outstanding debt, becau se
the probability of default is high. In fact, cou nte rcyclica l default risk su bsta nt ially limits
the degree of risk sh aring, and the model can generate cap ital outflows in r ecession s, when
in terest rate spreads are at t heir highest.
The model generates the observ ed dynamics of spread curv es because the endogenous
probab ility of default is persistent, yet mean rever ting, as a result of the dynamics of deb t
and income. Wh e n debt is lo w and income is h ig h, default is unlikely in th e near future, s o
spreads are lo w . H owe ver, long-terms spreads are higher than short-term spreads because
default m ay become likely in the far future if t he borrower receives a sequence of bad shoc ks
and accum ulates debt. On the other hand, when income is low a nd debt i s high, default is
lik ely i n t he near future, so spreads are h igh . Lo ng-term spreads, ho wever, increa se by l ess
than short-term spread s becau se the borrowe r’s lik elihood of repa yin g may rise if it receives
a sequence of good shocks and red u ces its debt. Although cumu lative d efault p roba bilities
on long-ter m debt are a lways larger th an o n sh ort-term debt, the long spread can be lower
than the short spread because it reflects a lowe r a verage future default probability.
The m odel can rationalize the covariation observ ed in th e data between the maturit y
structure of debt issuances and the term structure of spreads as reflectingatrade-off bet ween

insurance benefits of long-term debt and liqu i d it y benefits of short-term debt, both due to
the presence of default. Long-term deb t provides insurance against the uncertaint y o f short-
term interest rate spreads. Since short-term spreads rise during periods of low income, w hen
default risk is high, issuing long-term d eb t allow s the borro we r to avo id rollin g over sho rt-
term debt at high spreads in states when con sum p tion is low . Moreove r, long-term d eb t
insures against futur e periods of limite d credit availabilit y ; in particular, the borrowe r can
a void capital outflows in recessions by i ssuin g long-term debt.
Ev en though long debt dom inates short debt in term s of insurance, it is not as effective in
delivering high i mmediate consumption; hence the l iquidity benefit of sh ort-term debt. Short-
termdebtallowstheborrowertopledgemoreofhisfutureincometowarddebtrepayment
becau se in eac h subseq uent period the threat of default punishm ent gives him incentives for
repayment before an y further short debt is issued. Long-term debt contr acts do not allo w
suc h large transfers because the borrowe r is unable to commit to saving in the near fu ture
to ward repa ym en t in the further future. Effectively, th e threat of default pu nish m ent is lowe r
with long-te rm debt given th at it w ill be relevant o nly in the fu tu re, when the lon g-te rm d eb t
3
is due. This greater efficacy of short-term debt in alleviating commitmen t problems for debt
repa yment is reflected in more lenien t price sc hedules and smaller drops in short-term prices
with in creases in the lev el of debt issues. In th is sen se, sho rt d ebt is a more liqu id a sset, and
consum ptio n can alw ays be margin ally increased b y more with short-term debt than with
long-term deb t.
The time-varying maturity structure responds to a time-varying valuation of the in surance
benefit of long-term debt and the liquidity benefitofshort-termdebt.Periodsoflowdefault
probabilities and upward spread curves correspond to states when the borrower is wealthy
and values insurance. Thus, the portfolio is shifted to wa rd long debt. Periods of high defa ult
probabilities and inverted sprea d cur ves correspond to states when the borrower is poor and
credit is limited. Th ese are times when liquidity is most va lua b le , an d thus the portfolio
is shifted to ward shorter-term debt. We can therefore rationalize higher short-term debt
po sitions in times of crises as an optimal response to the illiquidity of long-ter m d ebt, and
the tigh t er availability of its supply.

W hen calibra ted to Brazilia n dat a, the mod el quantitatively m a tches the dyna m ics of the
matu rity compo sition of new d ebt issuances an d its co variation w ith spreads observe d in the
data. In connecting our model to the data, a me thodological contribution of the p aper is to
develop a tractable fram ewo rk with bonds that have empirically releva nt duration. Bonds in
our model are perpetuity contra cts w ith non-state-contingent cou pon pa ym ents that decay
at differe nt rates. B ond s with paym ent s that decay quick ly have more of their va lu e paid
early, and so have short dura tion. This gives a recursive structure to debt a ccu mu lation that
allows the model to be cha ra cteriz ed in t e rm s of a sm a ll nu mber of state variables although
decisions at any date are contingent on a long sequence of future expected payments. Our
findings indicate that the insurance benefits of long-term debt and the liquidity benefits of
short-term debt are quantitatively important in understanding the dynamics of the m aturity
structure o b served i n Brazil. Importantly, the m a turity s tructur e in the model responds to
the underly in g dy na mics of default pr ob a bilities re flected in spread curves, wh ich ma tch the
data well.
Rela ted L itera tur e
This paper is related to the literature on the optimal maturity structure of government debt.
Angeletos (2002), B uera and Nicolini (2004) and Shin (2007) show that, when debt is not
state contingent, a rich m aturity structure of gov ernm en t bonds can be used to replicate
the allocations obtained w ith state-co nting ent deb t in econ om ies w ith distortiona ry taxes as
in Lucas a nd Stok ey (1983). In th ese closed econo my mod els, s hort- and long-ter m inter est
4
rate dynamics reflect the variation in the representative a gent’s margin al rate o f substitutio n,
which changes with the state of the economy. Thus, ha ving a ric h enough maturity structure
is equivalent to ha vin g assets with state-contin gent pa yo ffs.
2
Our paper shares with these
papers the message that man aging th e m atu rity com position of debt can provide benefits to
the governm ent because of uncertain ty over fu tu re in terest rates. The message is particularly
relevant for the case of emergin g m arket econo mies. As N eum eyer a nd Pe rri (2005) have
sho wn, fluctuations in c ountry specific i nterest rate sp rea ds play a major r ole in a ccountin g

for the large business cycle fluctuations in emerging markets. T h e lesson th at our paper
prov id es in this con text is that the volatility of the maturity composition of debt in these
countries is an optimal response to these int erest rate fluctuatio ns. Ho wever, in c ontra st
to these papers, the fluctuation s in interest rates in o ur model reflect time variation in the
endogenous country’s o wn prob ability of default.
3
The maturity of d ebt in e m erg ing coun tries is also of interest because of the general
view that coun tries could alleviate their vulnerability to v er y costly crises by c h oosing the
appropriate maturity structure. For example, Cole and Kehoe (1996) argue that the 1994
Mexican debt crisis could ha ve been a voided if the maturit y of go vernm ent debt had been
longer. Long er mat urity debt would allow count ries to better man age external shocks and
sudden stops. Broner, Lorenzoni, and Schmu kler (2007) formalize this idea in a model where
the govern ment can avoid a crisis in th e s hort term by is suing l ong -term debt. In their model,
with risk a ver se lenders w h o face liquidit y s h oc ks, lon g -term debt is more expensive, s o the
maturity com position is the result of a trade-off between safer long -term d eb t and cheaper
short-term debt. In line with their paper, w e a lso find that short-term debt provid es larger
liquidit y benefits. In c ontrast to Bron er, L orenzoni, and Schmu kler, in our m odel the time-
varying availabilit y of short- a nd long-term deb t is a n equilibriu m respons e to compensate fo r
the economy’s default r isk, rath er th an to compensate for foreign lenders’ shock s. Moreover,
our paper is the first to develop a dynamic framework with defaulta ble debt and m ultiple
matu rities with which these questions can be analyzed and assessed quantitative ly.
The larger liquidity benefits of short-term debt relativ e to lon g -term debt arise in our
model because short-term contracts are more effectiv e in solving the commitm ent prob lem of
the borrower in term s of future debt and default policies. In this regar d, o u r paper is related
to Jeanne’s (2004) model where short-term deb t gives m ore incentive s for the gover nm ent
2
Lustig, Sleet, and Yeltekin (2006) develop a general equilibrium model with uninsurable nominal frictions
to study the optimal maturity of government debt. They find that higher interest rates on long-term debt
relative to short-term debt reflect an insurance premium paid by the government, for the benefits long-term
debt provides in hedging against future shocks.

3
The idea that credit risk makes longer term debt attractive is also present in Diamond (1991) in a three
period model of corporate debt where firms have private information about their future credit rating.
5
to implement better policies. Wh en short-term debt needs to be rolled o ver, creditors can
discipline the go vern ment b y rolling over the debt only after desired polic ies are implement ed .
4
Moreo ver, when defaulted debt is renegotiated, Bi (2007) shows th at long-term debt is more
expe nsive also to co m pensate for debt dilution. Absent explicit sen iority clau ses, issuin g
short-term debt can dilute the reco very of long-term debt in case of default.
The theoretical model in this p aper builds o n t he work of A guiar and Gopinath (2006) a nd
Arellan o (2008), w ho mode l e q uilibriu m default with incomplete marke ts, a s in the seminal
paper on sovereign debt b y Eaton and G erso vitz (1981). This paper extends this framework
to incorporate lon g debt of multip le maturities. In recen t wo rk, Chatterjee and Eyigun gor
(2008) and Hatchondo and Martinez (2008) show that long-term defaultable debt allow s a
better fit of emerging market data in terms of the v olatility and mean of the coun try spread
as well as debt levels . All t hese models ge nerate a time - varyin g probability of d efault that is
linked to the dynamics of debt and income. The dynamics of the spread curv e in o ur model
reflect the time-varying default probab ility, in t he sam e way that Merton (1974) derived for
credit spread curves on defau ltable corporate bond s. In Merton’s model, w h en the exo genou s
default probability is low, the credit spread curve is up wa rd sloping, and w hen the default
probabilit y is hi gh, credit spread curves are downward sloping or hump shaped. The s pread
curve dyn am ics in t his p aper follow Merto n’s resu lts. Howe ve r, our framework d iffers from
Merton’s in that the probability of default and the level and maturity composition of debt
issuances are endogen ous variables.
The outline of the paper is as f o llows. Sect io n 2 documen ts the dynamics of the spread
curve and matu rity composition for four emerging markets: Argentina , Brazil, Mexico, a nd
Russia. Section 3 p resents the theor etical model. Section 4 presents so m e examp les t o
illustrate the mech anism for the optima l deb t portfolio. Section 5 p res ents all the quan titat ive
results, and Section 6 c onclu des.

2 Emerging M ark ets Bond Data
We examine data on sovereign bon ds issued in in terna tional financial mark ets b y four emerging-
market countries: A rgentina, Brazil, M exico, and Russia. We look at th e behavior of th e
in terest rate spreads over d efa ult-free bond s, across di fferen t maturities, and at the w a y the
matu rity of n ew debt issued covaries with spreads. We find that when spread s are low, g overn -
ments issue long-ter m bonds m ore heavily an d lon g-term spreads are higher than short-term
4
Commitment problems have been shown to reduce the level of sustainable debt in the literature of
optimal policy without commitment, as in Krusell, Martin, and Rios-Rull (2006).
6
spreads. When spreads rise, the maturity of bond issuances shortens a nd short-term spr eads
are higher than long-term spreads. Our findin gs also confirm the earlier results of Broner,
Lorenzon i, and Schmu kler (2007), who showed in a samp le o f eight emerging eco nom ies tha t
debt maturit y shortens when spreads a re v ery high.
5
2.1 Spread Curv es
We define the n-year sp read fo r an emerging market count ry as the difference between the
yield o n a d efaulta ble, zer o-coupon bond maturing in n years i ssu ed by the count ry and on
a zero-coupon bo nd of the same matu rity with negligible default risk (for example, a U.S.
Treasury note). The spread is the implicit interest rate premium required b y investors to
be willing to purchas e a defaultable bond of a given ma tur ity.
6
The spre ad curve depicts
spreads as a function of maturity.
We denote the ann ually com pounded yield at date t on a zero-coupon bond issued b y
country i,maturinginn years, as r
n
t,i
. The yield is related to the price p
n

t,i
of an n-y ea r
zero-coupon bond, wi th face value 1, through
p
n
t,i
=(1+r
n
t,i
)
−n
. (1)
We define country i’s n-year spread as the differ ence in zero-coupon yields betwe en a
bo nd issued b y country i relative to a default-free bond. The n-year spread for coun try i at
date t is given by: s
n
t,i
= r
n
t,i
− r
n
t,rf
,wherer
n
t,rf
is the yield of a n-year defau lt-free bon d .
7
Since governments do not issue zero-coupon bonds i n a wide range of maturities, we
estimate a country’s spread curve by using secondary market data on the prices at whic h

coupon-bearing bonds trade. The estimation procedure, described in the Appendix, follows
Svensson (1994) and Broner, Lorenzon i, and Schmukler ( 2007 ).
We comp ute spread s starting in M arch 19 96 at the earliest and e nding in M ay 2 004 at th e
latest, dependin g on th e availability of data for each co untry. Figure 1 displays the estim ated
spreads f or 2-year and 1 0-y ear bonds for A rgentina, Brazil, Mexico, and Russia.
5
Broner, Lorenzoni and S chmukler (2007) focus on the relationship betw een the term structure of risk
premia (compensation for risk aversion) and the average maturity of debt. In this section we construct
measures of the t erm structure of yield spreads and t he average duration of debt because these statistics
prov ide the basis for the quantitative assessment of our model.
6
Yield spreads on bonds issued by emerging markets could also arise due to risk premia or liquidity
differences. However, g iven the incidence of sovereign defaults in emerging markets, in o ur model we abstract
from these other factors and examine the extent t o which default risk can rationalize t hese spread dynamics.
7
Our data include bonds denominated in U.S. dollars and European currencies, so we take U.S. and
Euro-area government bond yields as default-free.
7
96 97 98 99 00 01 02 03 04
0
5
10
15
20
25
30
date
spread (%)
Argentina
96 97 98 99 00 01 02 03 04

0
5
10
15
20
25
30
date
spread (%)
Brazil
96 97 98 99 00 01 02 03 04
0
5
10
15
20
25
30
date
spread (%)
Mexico
96 97 98 99 00 01 02 03 04
0
5
10
15
20
25
30
date

spread (%)
Russia


2 year
10 year
Figure 1: Tim e series o f 2-year and 10-year spreads.
Spreads are v ery v olatile, and the difference between long-term and short-term spr eads
va ries substan tially over time. W hen spreads are low, long-term spreads are generally h igher
than short-term spreads. Howe ver, w h en the level o f spreads rises, the gap betwe en long and
short-term spreads tends to n arrow and sometimes reve rses; the spread curv e is flatter or
inverted. The tim e series in Figure 1 show sharp in crea ses in interest ra te spreads associated
with R u ssia ’s de fault in 1998, Argentin a’s default in 2001, and Braz il’s financial crisis in
2002.
8
The expectation that the countries w o uld default in these episodes is reflectedinthe
high spreads charged on de faultable bonds.
To em ph asize the pattern observed in the time series that sho rt-term spreads tend to rise
more than long-term spreads, in Figure 2 we display spread curv es a veraged across different
8
For Argentina and Russia, we do not report spreads after d efault on external debt, unless a restructu ring
agreement was largely completed at a later date. We use dates taken from Sturzenegger and Zettelmeyer
(2005). For Argentina, we report spreads until the last week of December 2001, when the country defaulted.
The restructuring agreement for external debt was not offered until 2005. For Russia, we report spreads un til
the second week of August 1998 and beginning again after August 2000 when 75% of external debt had been
restructured.
8
time periods for each country: the overall avera ge, the a vera ge w ithin periods with t he 2-yea r
spread belo w its 1 0th percentile, and the a ve rage within periods with t he 2 -year sp read above
its 9 0th percen tile. When spreads are lo w, the spread curve is up ward sloping: long-term

spreads are higher than short-term spreads. Wh en spreads are high, short-term spreads rise
more than l ong-term spreads. For A rgen tina, Brazil, a nd R ussia, the spread curve becomes
do wnward sloping in these tim es. For M exico, which had relatively smaller in creases in
spreads during this time period, the spread curv e flattens as short spreads rise more than
long spreads.
9
2.2 The Maturity Composition of Debt and Spreads
We no w examine the maturity of new debt issued by the four emerging mark et economies
during the s ample period, and relate t he changes in t he maturity of debt to changes in
spreads.
10
In each week in the sample, we measure the m atu rity of debt as a q u antity-we ighted
a verag e maturity of bonds issu ed that week. We measure the maturity of a bond using two
alternativ e statistics. T h e first is simply the number of y ears from the issue date until the
maturity date. The second is the bond’s duration,defined in M acaulay ( 1938) as a weighted
a verage o f the nu mber of years until each of the bond’s future payments. A bond issued at
date t by country i,payingannualcouponc at dates n
1
,n
2
, n
J
years into the future, and
face value of 1 has d uration d
t,i
(c) defined by
d
t,i
(c)=
1

p
t,i
(c)
Ã
J
X
j=1
n
j
c(1 + r
n
j
t,i
)
−n
j
+ n
J
(1 + r
n
J
t,i
)
−n
J
!
, (2)
where p
t,i
(c) is the coupon bond’s price, and r

n
t,i
is the zero-co upon yield curve. Th e time
until each future p ay m e nt is weig hted by the discounted value of that pay m ent rela tive to th e
price of the bond. A zero-coupon bond has duration equal to the number of years until its
maturity da te, but a coupon-paying bond m aturing on t he s ame d ate h as shorter dura tion.
We consid er duration as a measure of m atu rity because i t is m ore comparable a cross bonds
9
The findings are similar to empirical findings on spread curves in corporate debt markets. Sarig and
Warga ( 1989), for example, find that highly rated corporate bonds have low levels of spreads, and spread
curves that are flat or upward-sloping, while low-grade corporate bonds have high levels of spreads, and
average spread curves that are hump-shaped or downward-sloping.
10
In addition to external bond debt, emerging countries also have debt obligations with multilateral
institutions and foreign banks. However, marketable debt constitutes a large fraction of the external debt.
The average marketable debt from 1996 to 2004 is 56% of total external d ebt in Argentina, 59% in Brazil,
and 58% in Mexico (Cowan et al. 2006).
9
5 10 15 20
0
5
10
15
20
25
Argentina
years to maturity
spread (%)
5 10 15 20
0

5
10
15
20
25
Brazil
years to maturity
spread (%)
5 10 15 20
0
5
10
15
20
25
Mexico
years to maturity
spread (%)
5 10 15 20
0
5
10
15
20
25
years to maturity
spread (%)
Russia

average
high short spread
low short spread
Figure 2: Av erage s pread c urves: over all, and w ith in periods in th e h igh est an d lowest deciles
of the 2 -year spread.
with differen t coupon rates.
We calculate the a vera ge ma tur ity and avera ge dur ation of new bonds issued in each
week b y eac h country. Table 1 displays eac h country’s averages of th ese weekly maturit y
and duration series within periods of high (above median) and low (below med ia n) 2-y ea r
spreads.
First, the table shows that duration tends to be much shorter than maturity. Because the
yield on an e m erg ing market bond i s ty p ically high, the p r incipa l pay m e nt at the matu rity
date is severely d iscou nted, and much o f the bond’s value comes fr om coupon payments made
soon er in the future. This weig ht o n coupon pa ym ents shortens th e d uratio n measure relative
10
Table 1: Average Maturity and D uration o f New De bt
Maturity (y ears) Dura tion (y ears)
2-year spread: < median ≥ median
< median ≥ median
Arg ent in a 9.15 9.05 5.70 5.10
Brazil 14.02 6.60 6.59 4.47
Mexico 13.50 10.30 7.72 6.52
Russia 8.8 9 10.98 6.11 5.42
to the t im e-to -m a tur ity measure.
Second, the average duration of debt is s ho rter when sprea ds are high than when they are
lo w . M ex ico, for example, issues debt that averages about 1.2 years longer in duration w hen
the 2-year spread is below its median than when it is above its median. Fo r all countries
except Russia, this pattern also holds for the a v erage time-to-m aturity of bonds issued during
per iods of high spreads co m p ared to lo w spreads: Mexico issues bonds that m ature 3.2 years
soon er when spreads are high . Our unconditional point estimates for a shorter debt d ur ation

when spreads are high mirrors the findings in Broner, Lorenzoni, and Sc hmukler ( 2007). They
show th at a high sp rea d level is a statistically significan t determ in ant for a shorter m a turity
of debt issuances ev en after con tro lling for selection effects due the fact that the timing of
debt issuances is very irregular.
InTable2,weemphasizetherelationshipbetweenthespreadcurveslopes and a v erage
duration. The s lope o f the spread c urve, defined here as the difference bet ween the 10-year
(long-term) and 2-year (short-term ) spread, falls when the 2-year spread is high — the numbers
in colum n 4 of Table 2 are smaller than those in column 3. Dur ing these times, ho we ver,
the coun tries shift to ward s hort-term debt, ev en th ou gh the s p reads on long-term debt rise
less than for short-term debt. In Brazil, for example, while the spread curve chan ges from
depicting a 10-y ear spread that is 4 percentage poin ts above the 2-year spread to one that
is 1.33 percentage poin ts belo w t he 2 -year sp read, the avera ge d u ration of new ly issu ed d eb t
reduces by more than 2 years.
The message of th is section is that the sp rea d curv e and the maturity of bond issuances
in emerg ing markets are time-varying. In particular, the slope of th e sp read c u rve covaries
positively with the maturity of new debt, and negativ ely with the lev els of spreads: when
short-term spread s are lo w , the slope of the spread curv e is higher, and the maturit y of new
debt is longer, than when short-term spr eads are high. In what follows, we build a dyna m ic
11
Table 2: Slope of S pr ead Curve and Av erage Duration of Issuances
Duratio n (years) Spread curve slope ( % )
s
10
− s
2
short spread: < 25th pct ≥ 75th pct < 25th pct ≥ 75th pct
Argentina 6 .4 0 5.64 2.47 -1.16
Brazil 6.80 4.63 4.01 -1.33
Mexico 8.45 6.39 2.30 1.24
Russia 6.576.190.57-0.67

model that r ationalizes this pattern, in which spreads reflect the gove r nment’s like lih ood of
defaulting, and the a v erage maturity of ne w debt endogenously varies over time.
3 The Model
Cons ider a dyn am ic model of defaulta ble deb t that includes bonds of short and long dura tion.
A small open econom y receiv es a stochastic stream of outpu t, y, of a tradable good. Th e
output shock fo llows a Markov process with com p act s up port and transition function f(y
0
,y).
The economy trades t wo bonds of different duration with in terna tional lenders. F in ancial
contracts are unenforceab le: the economy can defau lt on its debt at a ny tim e. If the economy
defaults, it tem porarily loses access to international financial markets and also incurs direct
output costs.
The r epresent ative ag ent in the sm all open economy (henceforth, the “borrowe r”) r eceives
utility from consum pt ion c
t
and has preferences given b y
E
∞
X
t=0
β
t
u(c
t
), (3)
where 0 <β<1 is the tim e discou nt facto r and u(·) is increasing and concave .
The borrowe r issues debt in the form of two types of perpetuity contracts with coupon
payments that decay geometr ically. We let {δ
S
,δ

L
} ∈ [0, 1] denote the “decay factors” of the
pa yments for th e two bonds. A perpetuity with deca y factor δ
m
is a contra ct t ha t s pecifies
apriceq
m
t
and a loan face value 
m
t
suc h that the borro we r r eceive s q
m
t

m
t
unitsofgoodsin
period t and p romises to pa y, conditional on not defaulting, δ
n−1
m

m
t
unitsofgoodsinevery
12
futur e period t + n. T he decay of eac h perpetuity is related to its dura tion: a bond of
this t ype with rapidly declining pa yments has a l a rge r proportion of its value paid early on,
and therefore a shorter duration, than a bond with mor e slowly dec lining payments. We let
δ

S
<δ
L
,sothatδ
S
is the decay o f the perpetu ity w ith s hort d ur atio n an d δ
L
is the decay
of the perpetuity w ith long du ration. We w ill refer to the perpetuities with deca y factors δ
S
and δ
L
throughou t as short a n d long bonds, respective ly.
At ev ery time t the econo my has outstanding all past perpetuit y issuances. Define b
m
t
,
the stoc k o f perpetuities of duration m at time t, as the total payments due in period t on
all past i ssuan ces of type m, c ond itional on not defaulting:
b
m
t
=
t
X
j=1
δ
j−1
m


m
t−j
= 
m
t−1
+ δ
m

m
t−2
+ δ
2
m

m
t−3
+ + δ
t
m

m
0
+ b
m
0
,
where b
m
0
is given. Thus, the accu mu lation for t h e stocks of short and long perpetuities can

be wr itten recursively by the following laws of motion:
b
S
t+1
= δ
S
b
S
t
+ 
S
t
(4)
b
L
t+1
= δ
L
b
L
t
+ 
L
t
W ith these definitions, w e can compactly write the borrower’s budget constrain t condi-
tional on n ot d efaultin g. P urch ases of co nsu m ptio n a re c onstrain ed by t h e end ow m ent less
paym en ts on outstanding debt, b
S
t
+ b

L
t
, plus the issues of new perpetuities of short dur ation

S
t
at price q
S
t
and long duration 
L
t
at a price q
L
t
:
c
t
= y
t
− b
S
t
− b
L
t
+ q
S
t


S
t
+ q
L
t

L
t
(5)
The borrowe r chooses new issuances of perpetuities from a menu o f con tra cts wh ere prices
q
S
t
and q
L
t
for are quoted for eac h pair (b
S
t+1
,b
L
t+1
).
If the econo my defau lts, we assum e that all outstan ding debts and assets (b
S
t
+ b
L
t
) are

erased from the budget constraint, and the economy cannot borrow or sa ve, so that con-
sump tion equals output. In addition , the country incu rs output costs:
c
t
= y
def
t
,
where y
def
t
= h(y
t
) ≤ y
t
.
13
3.1 R ecursive Problem
We no w represen t the bo rrowe r’s infinite horizon decision problem as a recursiv e dyna m ic
program m in g problem . T he model h a s two endogenous s ta tes, which are the s tocks of each
ty pe of debt, b
S
t
and b
L
t
, an d o n e exogen ou s state, t he output of the economy, y
t
. T h e s ta te
of the economy at date t is then given b y (b

S
,b
L
,y) ≡ (b
S
t
,b
L
t
,y
t
).
At an y given state, the value of th e option to default is given by
v
o
(b
S
,b
L
,y)=max
c,d
©
v
c
(b
S
,b
L
,y),v
d

(y)
ª
, (6)
where v
c
(b
S
,b
L
,y) is the value associa ted with not defaulting and sta yin g in the contra ct and
v
d
(y) is the value associated with default.
Since we assume that default costs are incurred w henever the borro wer fails to repa y its
obligation s in full, the model will only generate complete default on all outstan d ing debt, both
short and long term. When the borro wer defa ults, output falls to y
def
, and the economy is
temporarily in financial autarky; θ is the p roba bility t h at it will rega in a ccess to international
credit markets each period. The value of de fault is then given by the follo w ing:
v
d
(y)=u(y
def
)+β
Z
y
0
£
θv

o
(0, 0,y
0
)+(1− θ)v
d
(y
0
)
¤
f(y
0
,y)dy
0
. (7)
We are taking a simple route t o mod el both costs of default that seem empirically r elevant:
exclusion from financial mark ets and direct costs in output. Moreover, w e assume that the
default value does not depend on the maturity composition of debt prior to default. This
captures the idea that the maturity compo sition of defaulted debt is not r elevant for the
restructuring procedures that allow the economy to reenter the credit market.
11
When the borro we r c hooses to remain in th e con tract, the value i s the f ollo wing:
v
c
=max
{
b
0
S
,b
0

L
,
S
,
L
,c
}
µ
u(c)+β
Z
y
0
v
o
(b
0
S
,b
0
L
,y
0
)f(y
0
,y)dy
0
¶
(8)
subject to th e budg et constraint:
c − q

S
(b
0
S
,b
0
L
,b
S
,b
L
,y) 
S
− q
L
(b
0
S
,b
0
L
,b
S
,b
L
,y) 
L
= y − b
S
− b

L
(9)
11
This is consistent with empirical evidence regarding actual restructuring processes, where the maturity
composition of the new debt obligations is part of the restructuring agreement (Sturzenegger and Zettelmeyer
2005).
14
and to the laws of motion for the stock of perpetuities of short and long duration:
b
0
S
= δ
S
b
S
+ 
S
b
0
L
= δ
L
b
L
+ 
L
.
The borrowe r decides on optimal d ebt levels b
0
S

and b
0
L
to maximize utilit y. The borrowe r
takes as g iven that each co ntract {b
0
S
,b
0
L
} ∈ B com es with specific prices {q
S
,q
L
} that
are conting ent o n t oda y’s sta tes (b
S
,b
L
,y). Th e d ecision of whether to remain in t h e credit
contract o r default i s a period -by-period decision, so t hat t h e expected value from next period
forw ard in ( 8) incorporates the option t o default in the future.
The default po licy can be c hara cterized by default sets and repayment sets. Let the
repa yment set, R(b
S
,b
L
), be the set of output levels for which repa yment is optimal when
short- and l ong-term deb t are (b
S

,b
L
):
R(b
S
,b
L
)=
©
y ∈ Y : v
c
(b
S
,b
L
,y) ≥ v
d
(y)
ª
,(10)
and let the complem ent, th e default set D(b
S
,b
L
), be the set of out put lev els for whic h def ault
is optimal for debt positions (b
S
,b
L
):

D(b
S
,b
L
)=
©
y ∈ Y : v
c
(b
S
,b
L
,y) <v
d
(y)
ª
. (11)
W hen the borrower does not default, optimal new debt tak es the form of two decision
rules mapping today’s state into tomo rrow ’s debt levels:
b
0
S
=
˜
b
S
(b
S
,b
L

,y) (12)
b
0
L
=
˜
b
L
(b
S
,b
L
,y)
Given this cha racterization of debt a nd default decisions, w e can n ow define the equilib-
rium bond prices at which lender s are willing to offer contracts.
3.2 Bond Prices, Spreads, and Duration
Lenders are risk neutral and h ave an o pportunit y cost of funds e qual to the risk-free rate r.
Lenders are therefor e willing t o purc ha se a defaultable bond at a price equal to the expected
discounted va lu e of pay m ent s receiv ed from the bond. Each new issue of debt 
S
t
> 0 or

L
t
> 0 bytheborrowerisapromisetopayacouponpaymenteveryperiodinthefuture,
conditional on not defaulting up to that period. The price of a new debt i ssue, then , i s th e
15
sum of the valu e of these coupon payments, eac h discoun ted by the risk-free rate and the
probability of repa yment up to the date of the pay m ent. If the borrower’s state is

¡
y
t
,b
S
t
,b
L
t
¢
,
the p rices q
S
t
and q
L
t
for loans 
S
t
and 
L
t
given future s equences of de bts
©
b
S
t+n
,b
L

t+n
ª
∞
n=1
are
given by
q
m
t
=
∞
X
n=1
δ
n−1
m
(1 + r)
n
Z
R(b
S
t+1
,b
L
t+1
)
···
Z
R(b
S

t+n
,b
L
t+n
)
f (y
t+n
,y
t+n−1
) ···f (y
t+1
,y
t
) dy
t+n
···dy
t+1
(13)
for m = {S, L}. In each element of the sum on the right-h and side, the term δ
n−1
m
corresponds
tothecouponratedueinperiodt + n; (1 + r)
−n
is the lender’s n-pe riod discount factor;
and the term u nd er th e integral calcula tes the prob ability tha t the borro we r receives ou tpu t
shocks tha t are in the repaym ent set each period up to t + n —thatis,theborrowerrepays
up to period t + n. If defa ult never occu rs, t hat i s
R
R(b

S
t+1
,b
L
t+1
)
f (y
t+1
,y
t
) dy
t+1
=1for a ll t,
then the price at date t is equal to t he risk-free p rice,
q
m
t
=
1
1+r − δ
m
.
Note that the p rice q
m
t
of new debt issuances depends on current out put, y
t
,asitinfluences
expectations of future out put rea lizations whi c h d etermine future de fault decisions. The price
also depends on the en tire future sequence of debts,

©
b
S
t+n
,b
L
t+n
ª
∞
n=0
, since the outsta n ding
debt in a ny period determines t h e decision to default, giv en the output shock. Ho we ve r, w e
can transform the infinite sum in (13) into a recursive expression for q
m
t
b y assum in g that
the lender for ecasts th e futu re d ebt leve ls usin g th e borro we r’s ow n decision rules for debt,
definedin(12),whicharefunctionsonlyofthedebt choice next period. T he sum in (13) can
then be written with recursiv e notation as
Z
R(b
0
S
,b
0
L
)
f(y
0
,y)

1+r
dy
0
+ δ
m
Z
R(b
0
S
,b
0
L
)
"
Z
R(
˜
b
S
(b
0
S
,b
0
L
,y),
˜
b
L
(b

0
S
,b
0
L
,y))
f(y
00
,y
0
)
(1 + r)
2
dy
00
#
f(y
0
,y)dy
0
+
Each future debt level is replaced i n sequence by the optim al decision r ules
˜
b
S
(b
0
S
,b
0

L
,y) and
˜
b
L
(b
0
S
,b
0
L
,y). Pr ices for debt then satisfy t he functional equations:
ˆq
S
(b
0
S
,b
0
L
,y)=
1
1+r
Z
R(b
0
S
,b
0
L

)
h
1+δ
S
ˆq
S
³
˜
b
S
(b
0
S
,b
0
L
,y
0
) ,
˜
b
L
(b
0
S
,b
0
L
,y
0

) ,y
0
´i
f (y
0
,y) dy
0
(14)
ˆq
L
(b
0
S
,b
0
L
,y)=
1
1+r
Z
R(b
0
S
,b
0
L
)
h
1+δ
L

ˆq
L
³
˜
b
S
(b
0
S
,b
0
L
,y
0
) ,
˜
b
L
(b
0
S
,b
0
L
,y
0
) ,y
0
´i
f (y

0
,y) dy
0
(15)
16
If at any state (y, b
S
,b
L
) the borrower chooses to sa ve, 
S
< 0 or 
L
< 0 , the contract
constitutes a promise from the lender to the borrower to pay thereafter the coupon paym en t.
We assum e th at savings rates for the borrowe r are risk-free, so th at th e e ffective pr ices th e
bo rrowe r faces in the budget constraint in (9) are
q
S
(b
0
S
,b
0
L
,b
S
,b
L
,y)=

(
ˆq
S
(b
0
S
,b
0
L
,y) if b
0
S
≥ δ
S
b
S
1
1+r−δ
S
if b
0
S
<δ
S
b
S
(16)
q
S
(b

0
S
,b
0
L
,b
S
,b
L
,y)=
(
ˆq
L
(b
0
S
,b
0
L
,y) if b
0
L
≥ δ
L
b
L
1
1+r−δ
L
if b

0
L
<δ
L
b
L
(17)
We are modeling saving s contra cts as r isk-free because they seem the m ost em p irically rel-
evant for em erging markets where saving s are generally done at the inter nation al inter est
rates (gen era lly with T -b ills), yet borrow ing contract s compensate investor s fo r d efa ult. Ad-
ditionally for computational convenience we are assuming that after default any savings that
the go vern m e nt has in intern ationa l financial m ark ets are d issipated.
12 ,13
We define the y ield -to-m atu rity on each bon d as in the d ata, as the im p licit constant
interest rate at which the discounted value of the bond’s coupons equal its price. That is,
given a price q
m
,theyieldr
m
is defined from
q
m
=
∞
X
n=1
δ
n−1
m
(1 + r

m
)
n
.
So,
r
S
=
1
q
S
+ δ
S
− 1 and r
L
=
1
q
L
+ δ
L
− 1.
We define s preads as the difference between th e yield on a d efau ltab le bo nd and the
default-free rate:
s
S
= r
S
− r and s
L

= r
L
− r.
As output and debt c ha ng e, the period-by -period proba bility of default va ries over time,
12
Ideally, one could have a model with four e ndogenous s t ate variables, two for s hort- a nd long- t erm debt
issuances and two for short- and long term savings. H owever this specification is computationally unfeasible.
Th us, under the assumption that after default any savings that the government has in international financial
markets are dissipated, we can maintain risk-free savings and defaultable short- and long-term debt with only
two endogenous states.
13
We could alternatively assume that savings contracts also carry the defaultable price, i.e. i nterest rates
on savings are higher than the risk-free rate. Results are similar with this alternative specification. However,
by having sav ings contracts being risk-free, we avoid having cases that seem empirically implausible w here
the government borrows large long-term loans just to increase its default probability and be able to save at
excessively high interest rates.
17
and therefore the prices of long-term and short-term debt d iffer, since they each put differ ent
weights on repaym ent probabilities in the futu re, as seen in (13). Spre ads on short-term
and long-term bon ds th erefore gen erally differ, and the relation ship bet ween the two spreads
c hanges over time, so that the spread c urve is tim e-varyi ng.
Finally, we define as in the data, the duration of debt issued at eac h date as the w eigh ted
a verag e of the time until eac h coupon p ayment, with th e we ights d eter m ined by t he fra ction
of the bond’s value on each paym ent date:
d
m
=
1
q
m

∞
X
n=1
n
δ
n−1
m
(1 + r
m
)
n
.
So,
d
S
=
1+r
S
(1 + r
S
− δ
S
)
and d
L
=
1+r
L
(1 + r
L

− δ
L
)
. (18)
For comparison, no te that if th e bonds were default-free, yields, a nd duration w ould be
r
rf
m
= r
d
rf
m
=
1+r
1+r − δ
m
.
We now define eq u ilibrium. A recursive e q uilib riu m for th is economy is (i) a set of policy
functions for consumption ˜c(b
S
,b
L
,y), n ew issua nces for sh ort-term debt
˜

S
(b
S
,b
L

,y) and
long-term d eb t
˜

L
(b
S
,b
L
,y), perpetuity stocks for short-term debt
˜
b
S
(b
S
,b
L
,y) and long-term
debt
˜
b
L
(b
S
,b
L
,y), repaymen t sets R(b
S
,b
L

), and default sets D(b
S
,b
L
), and (ii) price functions
for short debt q
S
(b
0
S
,b
0
L
,b
S
,b
L
,y) and long d ebt q
L
(b
0
S
,b
0
L
,b
S
,b
L
,y), suc h that:

1. Taking as given the bond price functions q
S
(b
0
S
,b
0
L
,b
S
,b
L
,y) and q
L
(b
0
S
,b
0
L
,b
S
,b
L
,y) ,
the po licy functio ns
˜
b
S
(b

S
,b
L
,y),
˜
b
L
(b
S
,b
L
,y),
˜

S
(b
S
,b
L
,y),
˜

L
(b
S
,b
L
,y) and ˜c(b
S
,b

L
,y),
repa ym ent sets R(b
S
,b
L
), and default sets D(b
S
,b
L
) satisfy the borro wer’s op tim ization
problem.
2. Th e bond price fu nction s q
S
(b
0
S
,b
0
L
,b
S
,b
L
,y) and q
L
(b
0
S
,b

0
L
,b
S
,b
L
,y) reflect the bor-
ro we r’s default probabilities and lenders break ev en in expected value: equations (14),
(15), ( 16), and ( 17) hold.
18
4 Default and O ptimal M aturity
In th is section we illu strate the mech anism s that determ ine the optima l matu rity composition
of debt in two simplified example econom ies. We view the borrowe r’s choice as a portfolio
alloca tion problem, in w h ich the benefits a nd costs o f short-term and lon g-term debt deter-
mine the relative amounts of eac h type issued. In the fir st example, we show that, in the
presence of l ack of commitm ent in future debt and default policies, short-term debt is m o re
effectiv e than long -term d ebt in transfer ring future resources to the present. If the borrower
would try to borro w a lot o f l ong-term debt, its p rice wo uld fall to zero faster than if instead
the l arg e lo an w ould be short-term; hence, short-term debt is beneficial for liquidity. In the
second example, we show that lon g-term debt allo ws the borro wer to avoid the risk of rolling
o ver short-term debt at prices that differ across future states due to differences in default
risk; hence, lon g-term debt provides insurance.
We constru ct the simplest possib le example s to illustrate the mec h an isms clearly. The
economy lasts for th ree periods. In period 0 , in com e equals zero, and in periods 1 and 2
income is stoc hastic (with details to be specified in each example). The borro we r can default
at any time, in which case consum ption f rom then on is equal t o y
def
.
In each examp le, we compare the allocation with only one maturity of d ebt — o ne- or
t wo-period bonds — against the allocation with both maturities of debt.

14
In each economy,
with both maturities available, in period 0 the borrower can issue one- and two-period bonds
b
1
0
and b
2
0
giv en price schedules q
1
0
(b
1
0
,b
2
0
) and q
1
0
(b
1
0
,b
2
0
), a nd consumption is
c
0

= q
1
0
(b
1
0
,b
2
0
)b
1
0
+ q
1
0
(b
1
0
,b
2
0
)b
2
0
.
In period 1, conditional on not defaulting, new short bonds b
1
1
are issued given price schedu le
q

1
1
(b
1
1
). Consumption is equal to income plus net debt:
c
1
= y
1
+ q
1
1
(b
1
1
)b
1
1
− b
1
0
.
In period 2, c onditional on n ot def aulting, the borrower pays off long- and short-term debt,
and consump tion equals income minus the repa ym ent:
c
2
= y
2
−

¡
b
1
1
+ b
2
0
¢
.
In the cases with only one type of d ebt ava ilable, the budget constraints are modified accord-
14
It is straightforward to extend these examples for the case where long bonds pay a coupon in period 1
in addition to the payment in period 2, as long as y
1
and y
2
are sufficiently different.
19
ingly.
The risk neutral lenders discoun t time at rate r and offer debt contracts that compensate
them for the r isk of default and give them zero expected profits.
4.1 Example 1: S h o r t- Term Deb t Provide s Liqu idity
For this examp le we consider the follow ing incom e process. Income in period 0 is equal to
0. Income in period 1 is equal to y. In co m e in period 2 can take 2 values, y
H
or y
L
with
y
H

>y
L
=0, and the proba bility of y
H
is equ al to g with 0 <g<1. Also, consumption
in de fault, y
def
, is equal to 0. To abstract from any insurance properties of debt, we a ssume
that pre ferences are linear in co nsumption and giv en b y
U = E[c
0
+ βc
1
+ β
2
c
2
].
We assume that th e borrower likes to front-lo ad con sum p tion , while len ders do n ot discou nt
the future: β<
1
1+r
=1, and we impose that consumption must be non-negativ e: c
t
≥ 0 for
t =0, 1, and 2.
4.1.1 Only Two-Period Bonds
First, consider the borrower’s problem w h en only two-period bonds are available in period
0, and one-period bonds are available in per iod 1. Under t he assump tion th at β<(gy
H

−
y
L
)/(y
H
− y
L
), t h e solution to the borrower’s problem is the following . In period 2 , the
borrower defaults w hen income equals y
L
. In period 0, the borrowe r borrows against all his
period 2 income, at price g, and in period 1 the borro we r consumes his period 1 incom e, so
consumption is
c
0
= gy
H
c
1
= y
c
2
¡
y
H
¢
=0,c
2
¡
y

L
¢
= y
def
=0.
Although the borro we r does n ot have preferences for smoothing c on sum ption over time,
and wou ld prefer to consume everyth ing up front, it is not possible to consume everythin g in
period 0, because none of the income in period 1 can be borrowed against using two-period
debt. This is because such a contract would requ ire a t wo-period loan with face value larger
than y
H
, so that the borrower would h a ve t o save p art of the period 1 income t o r epa y the
loan in period 2. Since the borrower cannot commit to this policy in period 0 , however, t he
20
optima l choice in period 1 wo uld be not to save, and then to defa ult in period 2 regardless o f
the lev el of income. That is, a debt con tr act that offered q
2
0
b
2
0
= a+gy
H
,foranya>0,isnot
possible, because the probability of default on the loan would be equal to one, and hence the
price q
2
0
would be zero. Effectively, the threat of punishment for default in period 2 when the
two -period loan is due does not induce the borrowe r to repay, because the borrower discounts

the future, so that reducing consumption in period 1 is worse than facing the punishment
for d efault in period 2. At the same time, the thr eat of punishm ent for default in period 1 is
irrelevant, because none of the debt is d ue in period 1, and the threat of punishment cannot
be used to induce savings.
4.1.2 One- and Two-Pe riod Bonds
Now , if the borrower were able to issue o n e-period debt in period 0, consumption wo uld be
c
0
= y + gy
H
c
1
=0
c
2
¡
y
H
¢
=0,c
2
¡
y
L
¢
= y
def
=0.
Mu ltiple possible portfolios allow this co nsu m p tion pattern. The borrower could use
short-term debt to borro w against all period 1 income and long-term debt to borro w against

all period 2 incom e (b
1
0
= y with q
1
0
=1,b
2
0
= y
H
with q
2
0
= g, b
1
1
=0); or, the borrowe r could
use only short-term d eb t, issuing bo nd s in pe riod 0 and period 1 (b
1
0
= y + gy
H
with q
1
0
=1,
b
1
1

= y
H
with q
1
1
= g). Since all consumption occurs in the first period, utilit y in this case
is higher than in the case with long-term debt only. With one-period bond s, the threat of
punishm ent for default is being used in both periods to i n duce repay m ent .
In t h is example, long-term debt is illiquid in th e sense th at a loa n that wo uld provide the
same level of consu m p tion in the firstperioddoesnotexist,becausethepriceoflong-term
debt falls to zero. T his example illustrates that in the presence of lack of comm itm ent in
debt policies an d default risk, short-term debt is more liquid du e to m ore lenient bond prices,
and th us it is a superior instrum ent to pro vide up-front resources.
15
15
It is easy to extend this example to an infinite horizon environment with deterministic and time varying
output. A one-period bond economy can deliver higher initial consumption than a longer-term bond — two-
period or perpetuity — economy. The main idea is again that the threat of punishment can be used more
effectively with one- period bonds because longer-term contracts might require savings in the future which
are impossible to induce with default punishments.
21
4.2 E xamp le 2: Long-Term D ebt Pro v ides I nsurance
For the second e xam ple, we focus on the motive for insura nce by assu m ing that the borrowe r’s
preferences are given b y
U = E[u(c
0
)+βu(c
1
)+β
2

u(c
2
)]
with u (·) strictly concave an d β =1. We also no w consider a differen t income process.
Income in period 0 is equal to 0,incomeinperiod1isequaltoy, andincomeinperiod2
can take two values: y
H
or y
L
with y
H
>y
L
. The pro b ability o f y
H
islearnedinperiod1
and can be either g or p with 0 <g<1 and 0 <p<1.
4.2.1 Only One-P eriod Bonds
First, consider the borrowe r’s c h oice under the assum ption th at only one-period bo nds are
a vailable. Under the assum ption that
y+
p+g
2
y
H
2+
p+g
2
>y
def

>y
L
−
2y
H
−y
2+
p+g
2
, the solution to the
borrowe r’s p roblem is the follow ing. The borrowe r defau lt s i n period 2 if income is y
L
and does not default in all other states. H ence, c
L
2
(p)=c
L
2
(g)=y
def
. Contingent on the
realization of the probabilit y p or g, c o nsum p tion is equalized between period 1 and the
high-income state in period 2:
c
1
(p)=c
H
2
(p)
c

1
(g)=c
H
2
(g)
Fina lly, consump tion i n period 0 is set to equalize ex pected ma rg in a l utility in period 1 to
margin al utility in period 0:
u
0
(c
0
)=
1
2
(u
0
(c
1
(p)) + u
0
(c
1
(g)))
Importantly, c
1
(p) 6= c
1
(g), so that consumption is not e qualized across states w ithin a
per iod. W ith only short-term deb t a vailable, the borrowe r borrows in period 0, then borrow s
again in period 1. Debt issues are b

1
0
= c
0
, b
1
1
(p)=
y
H
+c
0
−y
1+p
, and b
1
1
(g)=
y
H
+c
0
−y
1+g
. The price
of debt issu ed in period 1 depends on the state realized: q
1
1
(p)=p and q
1

1
(g)=g.Therefore,
as long as p 6= g, the price at w hic h debt is rolled over i n period 1 differs across states, a nd
consumption differsaswell.
22
4.2.2 One- and Two-Pe riod Bonds
Now , if the borrower has access to both one-and two -period bo nds, it i s po ssible to equalize
consum ptio n across all states in w hich the borrowe r does not default:
c
0
= c
1
(p)=c
1
(g)=c
H
2
(p)=c
H
2
(g)=
p+g
2
y
H
+ y
p+g
2
+2
The portfolio required invo lves using lon g-ter m and s ho rt-ter m debt i n period 0, while bor-

ro wing nothing in period 1:
b
2
0
=
2y
H
− y
¡
p+g
2
+2
¢
b
1
0
=
¡
1+
p+g
2
¢
y −
¡
p+g
2
¢
y
H
¡

p+g
2
+2
¢
b
1
1
=0
In th is example th e borrower faces r isk because of the varia tion in bond pr ices across
states in period 1 due to differences in default risk in period 2. U sing long-term debt in
per iod 0 allows the borrowe r to avoid the risk invo lved with r ollin g over short-term debt i n
period 1. The borrower benefits from this insurance with sm oother consumption an d high er
utility.
Note that in period 0 short debt has a higher price than long debt, q
1
0
=1>
p+g
2
= q
2
0
,yet
the borro we r issu es lon g -term debt. T h e lower discount p r ice o n lo ng debt is t he insurance
prem ium the borrowe r is willing to p ay for in su ran ce against the variation in bond prices
in period 1. This insurance mec hanism is the same as that emphasized in Kreps (1982),
Ang eletos (20 02) an d B u era a nd Nicolini (2004) in their m odels of the optima l ma tu rity
structure of debt with incomplete markets. The difference in our model is that the variation
in bond prices comes from the government’s inability to commit to r ep ay ing, r ath er than
from variation in t he lender ’s mar ginal rate of substitution.

4.3 Summary
In a standard incomplete markets model with fluctuating output a nd without default, a
borrowe r would find the portfolio of long and short debt indeterminate if the risk-free rate
were constan t across time; the two assets wo uld ha ve pa yoffsthatmakethemequivalent.
Howeve r, in our model, the risk of d efau lt makes the two assets distinct. The first exam ple
illustrated that long-term debt is more illiquid than short-term debt due t o the inability of
23
theborrowertocommittofuturedebtanddefaultpolicies. However,thesecondexample
illustrated that long -term is beneficial because it hedg es against variatio ns in sho rt rates a nd
pro vides insurance for default risk.
Insurance and liquidity shape the optimal maturit y structure of debt for a borro wing
go v ernment. The quantitative relevance of each of these forces depends on the specifics of
preferences and th e incom e p rocess. Thus, in the next section we quantify th ese two so urces
b y calibrating our general model to an actual emerging market economy.
5 Quantitativ e Analysis
5.1 C alibration
We solve the mod el nu m e rically to evaluate its q ua ntitative predictio ns regarding the dyn am ic
beha vior of the optim al maturity composition of debt and the spread curve in emerging
marke ts. We calibrate an ann ua l model t o the Brazilian economy.
The utilit y function of th e borrower i s u(c)=
c
1−σ
1 − σ
. The r isk aversion coefficient i s set
to 2, which is a comm on value used in real business cycle studies. Th e risk-free interest rate
is set t o 4.0% annually, which equa ls the a vera ge annua l yield of a two ye ar U.S. bond from
1996 to 2004. Th e stochastic process for outpu t is assumed to be a log-normal AR(1) process
log(y
t
)=ρ log(y

t−1
)+ ε with E[ε
2
]=η
2
y
. S hoc ks are discretized into a seven-state Ma rkov
c ha in using a quadrature-based procedure (Tauc h en and Hu ssey 1991). We use annual series
of G DP gro wth for 1960—2004 taken from the World Developm en t Indicators to calibrate the
v o latility of out put. Due to the short sam p le, rather than estimating the a utocorrelation
coe fficient w e c hoose an autocorrelation coefficient for the o utpu t proc ess of 0.9 , which is in
line with standard estim a tes fo r developed countr ies. T h e d ecay parameters of th e short and
long bonds, δ
S
and δ
L
, are set su ch that the default-free duratio ns equal 2 a nd 10 y ears.
Follo wing Arellano (2008) we assume that after defa ult, output before reentering financial
mark ets rem ains low and below some t hreshold, according t o the follo wing:
h(y)=
(
y if y ≤ (1 − λ)¯y
(1 − λ)¯y if y>(1 − λ)¯y
,
where ¯y is the mean level of o u tpu t.
The output cost after default λ, t he time preference parameter β, a nd the probability of
reen tering financial m arkets after d efault θ are calibrated jointly to m atch th ree moment s in
Brazil: the average 2-yea r spread of 6% , the volatility of the 2-year spread of 5.3 and the
24
a verag e dura tion of debt issuances in Brazil of 5.5 y ea rs. Table 3 sum m arizes the param e ter

values.
Table 3: Parameters
Value Target
Discount factor lender r=4% U.S. annual interest rate 4%
Risk aversion
σ =2 Standard value
Perpetuity decay factors
δ
S
=0.52 Default-free durations of 2 and 10 years
δ
L
=0.936
Stochastic structure ρ =0.9,η=0.022 Brazil output
Probabilit y of reentry θ =0.24 Mean 2-year spread of 6%
Output after default
λ =0.025 Volatility of 2-year spread of 5.3
Discount factor borrower
β =0.935 Average bond duration of 5.5 years
5.2 Resu lts
We simu late th e model, an d in the follo w ing su bsection s we report s tatistics on the dynam ic
beh avior of spread s and the maturity com position of debt from the limitin g distrib ution o f
debt holdings. The model contains a dynam ic portfolio problem where the borro we r chooses
holdings of tw o defaultable bonds of shorter and longer duration. B elow , we show h ow
move m e nts in the probability of d efa ult generate time-varying differences i n the prices, and
in the liquidity and insurance benefits o f these two assets, which rationalize the mo vements
in s pread curv es and maturity composition observed in t he data.
5.2.1 Prices and Spreads
In the model all decision rules are functions o f three state variables (b
S

,b
L
,y). Howev er, for
the purpose of illustration, we consider a sing le artificial state variable, the w ealth of the
economy: w = y − b
S
− b
L
. T his variable is informative because it is h ighly correlated
with the true state variables: the correla tions between wealth and income, short debt and
long debt equal 0.99, -0.56, 0.65 respectively. In what follows we analyze decision rules as
functions of w ealth, constructed as scatter plots from the model simulation.
We first analyze the default de cision and s preads, and their relationship to wealth. Default
happens w hen t he economy has a lo w lev el of wealth, as the left panel of Figure 3 i ndicates. In
the righ t panel of Figure 3, we see that, conditional on not defaultin g, spreads are higher for
relatively lower levels of wealth. H owever i n equilib rium, for very low wealth l eve ls th e spread
25