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.
†
[email protected]
‡

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

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

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.

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

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

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

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

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

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

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


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

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

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

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

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

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

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
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
,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

©
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)

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

)
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

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

(
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
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

= 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

− 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

), 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:

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

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.

= 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

¡
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
)/(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

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

= 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

>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
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

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
¡

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

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