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CESifo / LBI Conference on
Sustainability of Public Debt
22 - 23 October 2004
Evangelische Akademie Tutzing
Managing Debt Stability
Emanuele Bacchiocchi and Alessandro Missale
CESifo
Poschingerstr. 5, 81679 Munich, Germany
Phone: +49 (89) 9224-1410 - Fax: +49 (89) 9224-1409
E-mail: [email protected]
Internet: http://www.cesifo.de
Managing Debt Stability
Emanuele Bacchiocchi and Alessandro MissaleW
October 19, 2004
PRELIMINARY AND INCOMPLETE
Abstract
This paper presents a simple model in which debt management stabilizes the debt-toGDP ratio and thus helps to ensure debt sustainability. The optimal debt composition is
derived by looking at the relative impact of the risk and cost of alternative debt instruments
on the probability of stabilizing the debt ratio. The optimal debt structure is a function of
the expected return dierentials between debt instruments, of the conditional variance of debt
returns and of their covariances with output growth, inflation, and the short-term interest rate.
We then explore how the relevant covariances and thus the optimal choice of debt instruments
depend on the monetary regime and on the Central Bank preferences for output and price
stability and interest-rate smoothing. Finally, we estimate the debt structure that would have
stabilized the debt ratio in OECD countries over the last two decades. The empirical evidence
suggests that the public debt should have a long maturity and a large share of it should be
indexed to the price level.
JEL Classification: E63, H63.
Corresponding author: Alessandro Missale; Dip. di Economia Politica e Aziendale; Via Conservatorio 7; 20122 Milano; Italy. Tel. +39-02-50321512 Fax +39-02-50321505. E-mail: [email protected]
W
We thank Carlo Favero for suggestions. The authors are associated with Università di Milano.
1. Introduction
No mention is made of debt management in the debate on debt sustainability, but a careful
choice of debt instruments is needed to control interest payments and budget deficits. Interestcost minimization is important especially in countries where interest payments absorb a large
share of the budget. In the same countries avoiding the risk that interest-rate shocks lead to large
payments on short-term debt and floating-rate debt is equally important.
We want to examine whether a concern for debt sustainability justifies the lengthening of debt
maturity that has occurred in EMU member states. We are also interested in assessing the scope
for inflation-indexed bonds that have been issued only in France and Italy in the EMU while
have been discontinued in the US. To address these issues we rely on a simple model in which
debt management stabilizes the debt-to-GDP ratio and thus helps to ensure debt sustainability.
Reducing the uncertainty of the debt ratio, for any expected cost of debt service, is valuable in
that it lowers the probability that the fiscal adjustment may fail because of a bad shock to the
budget.
The optimal debt composition is derived by looking at the relative impact of the risk and cost
of alternative debt instruments on the probability of missing the stabilization target. This allows
to price risk against the expected cost of debt service and thus to find the optimal combination
along the trade o between cost and risk minimization.
The optimal debt structure is a function of the expected return dierentials between debt
instruments, of the conditional variance of debt returns and of their covariances with output
growth, inflation and the short-term interest rate.
We show that the debt-to-GDP ratio can be stabilized by issuing debt instruments that provide
a hedge against variations in the debt ratio due to lower-than-expected inflation and output
growth. For instance, if interest rates and output were negatively correlated, a long maturity
debt would insulate the budget from interest rate shocks, thus avoiding higher than expected
interest payments at times of cyclical downturns. Inflation-indexed bonds would also provide a
hedge against an increase in the debt ratio due to lower-than-expected inflation.
We find that a stronger fiscal reaction to the debt ratio reduces the importance of expected
return dierentials for the choice of the debt instruments. In fact, if debt sustainability is on
average ensured by a restrictive fiscal stance, minimizing the expected cost of debt service becomes
less important than minimizing the risk that debt stabilization may fail because of large shocks
to the budget.
We then explore how the relevant covariances between the short-term interest rate, inflation
and output growth and thus the optimal choice of debt instruments depends on the monetary
regime and on Central Bank preferences for output relative to inflation stabilization and interestrate smoothing. In particular, we compare the implications for debt management of an inflationtargeting regime with those derived from a fixed-exchange regime.
Finally, we estimate the relevant conditional covariances for OECD countries by approximating
the unanticipated components of output growth, inflation and the short-term interest rate using
the residuals of forecasting regressions run on yearly data for the period 1976 to 2003. The
empirical evidence suggests that the public debt should have a long maturity and a large share of
2
it should be indexed to the price level.
The paper is organized as follows. Section 2 introduces a simple model of debt stabilization
which trades o cost and risk minimization. Then, the optimal debt composition is derived in
Section 3 as a function of the risk premia on government bonds and the stochastic relations
between output growth, inflation and the interest rate. Section 4 examines the implications for
debt management of alternative monetary regimes. Section 5 presents estimates of the stabilizing
debt structure for OECD countries. Section 5 concludes.
2. The government problem
In this section we present a simple model where debt management stabilizes the debt ratio
and thus helps to ensure debt sustainability. Debt stabilization calls for funding at low cost but
also for minimizing the risk of large payments due to unexpected changes in interest rates and
inflation. Hence, the choice of debt instruments trades o the risk and the expected cost of debt
service.
Risk minimization is accomplished by choosing debt instruments which both ensure a low
return variability and provide a hedge against an unexpected economic slowdown (see e.g. Bohn
1990). Reducing the uncertainty of the debt ratio, for any expected cost of debt service, is valuable
in that it lowers the probability that debt stabilization may fail because of a lower than expected
output growth and/or an unanticipated shock to the budget.
To provide insurance against variations in the debt ratio due to lower economic growth, public
bonds should be indexed to nominal GDP. However, this would be a costly innovation. Indeed,
a high premium would have to be paid: i) for insurance; ii) for the illiquidity of the market and;
iii) for the delay in the release of GDP data and their revisions. Therefore, we focus on three
main funding instruments currently available to EU governments: short-term bills (or floating
rate notes), fixed-rate long-term bonds and inflation-indexed bonds. We do not consider debt
denominated in foreign currency as these instruments are no longer issued since the start of the
EMU (see Favero et al. 2000).
In order to ensure a sustainable debt, the government can follow a simple rule; it plans the
primary surplus as an increasing function of the debt ratio:
P
= Bt
St+1
(1)
P is the planned (or expected) primary surplus between period t and t+1 and B , denotes
where St+1
t
the debt-to-GDP ratio at the beginning of period t. As shown by Bohn (1995), under ”normal
conditions”, a suciently high can ensure debt sustainability —i.e. that the intertemporal budget
constraint holds. In his words ”the estimated policy rules suggest that US fiscal policy is sucient
to keep the debt-to-GDP stationary in the future unless interest rates and growth rates move very
unfavorably”.
We assume that, in order to avoid such circumstances, the government also aims at stabilizing
the debt ratio. Hence it minimizes the probability that the debt ratio increases:1
Min P rob[Bt+1 > Bt ]
(2)
There are at least three reasons while this is a sensible objective. First, even a rigorous
fiscal rule may not prevent that an unexpected economic slowdown leads the debt ratio onto an
1
The analysis can be extended to the case the debt ratio must not exceed a given threshold.
3
unsustainable path. Second, an increase in the debt ratio, according to the fiscal rule (1), requires
a revision in the government budget; i.e. a higher primary surplus has to be planned for the
following year. Such revisions are costly either because of distortionary taxation or because of
political reasons. Third, unlike debt sustainability, debt stabilization is a visible, well defined and
politically relevant goal. Therefore, there are economic, political and practical reasons as to why
the government eorts are in general directed at preventing a rise in the debt ratio rather than
at ensuring long-run sustainability. The Stability and Growth Pact is a point in case; it has been
designed to prevent or limit debt growth more than to ensure debt sustainability.
The debt ratio may rise either because of high real return on debt instruments, or because
of low output growth or because of shocks to the government budget. In particular, the primary
P , because
surplus can turn out to be lower than what originally planned by the government, St+1
of, say, unanticipated government spending, Gt+1 . Denoting the realized primary surplus with
P G
St+1
t+1 , the debt-to-GDP ratio is equal to
d
P
yt+1 )Bt St+1
+ Gt+1
Bt+1 = (1 + It+1 t+1
d
Bt+1 = (1 + It+1 t+1
yt+1 )Bt + Gt+1
(3)
d
is the rate of inflation measured
where It+1 denotes the nominal rate of return on public debt, t+1
by the GDP deflator, and yt+1 is the growth rate of output. Gt+1 is a zero mean shock to
government spending; i.e. the uncertain component of the primary surplus. Alternatively, Gt+1
can be viewed as a shock that occurs after the budget law has been implemented.
The nominal interest payments (per unit of debt), It+1 , depend on the composition of public
debt chosen at the end of period t 1. The government can choose between short-term bills (or
floating-rate notes), inflation-indexed bonds and fixed-rate long-term bonds. We take the time
period as corresponding to one year and assume that short-term bills have a one-year maturity
while bonds have a two-year maturity. Focusing on a two-year horizon is obviously a rough
approximation given that both inflation-indexed bonds and fixed-rate bonds are issued with much
longer maturities. A partial justification for this assumption is provided by the monetary policy
model presented in the following section, in which the eects of economic shocks last only two
periods.2
The nominal rate of return on public debt is equal to
I
It+1 = it s + (Rt31
+ t+1 )h + Rt31 (1 s h)
(4)
where s is the share of short-term debt, h is the share of inflation-indexed debt, t+1 is CPI
inflation and it denotes the short-term interest rate between period t and t + 1, which determines
the nominal rate of return on one-year bills. The interest rate it is not known at time t 1 when
the composition of the debt is chosen. Rt31 is the long-term interest rate at which fixed-rate (par)
bonds are issued and is equal to their nominal rate of return. The nominal return on fixed-rate
bonds is thus known at the time of issuance. Finally, the nominal rate of return on inflationI , known at the time of
indexed (par) bonds is equal to the sum of the real interest rate, Rt31
issuance, and the rate of CPI inflation, t+1 , to which the bonds are indexed.
3. The choice of debt maturity and indexation
2
We shall study the relative impact on the returns of the various debt instruments arising from the monetary
policy reaction to output and inflation shocks.
4
The Treasury chooses the composition of the debt, and thus s and h, with the objective of
minimizing the probability that debt stabilization fails because of government spending shocks
or other shocks that occur after the budget law has been implemented. Defining the real rate of
d
yt+1 , the government problem is
return on debt net of output growth as Xt+1 = It+1 t+1
equal to
Z
Min Et31 P rob[Gt+1 > ( Xt+1 )Bt ] = Min Et31
"
(3Xt+1 )Bt
!(Gt+1 )dGt+1
(5)
subject to 4.
where !(Gt+1 ) denotes the probability density function of Gt+1 .
Deriving (5) with respect to s and h yields
Et31 !(( Xt+1 )Bt )(it Rt31 ) = 0
I
+ t+1 Rt31 ) = 0
Et31 !(( Xt+1 )Bt )(Rt31
(6)
(7)
where ( Xt+1 )Bt is the reduction in the debt-to-GDP ratio in the absence of spending shocks,
and !(( Xt+1 )Bt ) is a function of s and h.
The first order conditions (6)-(7) have a simple interpretation: they show that the debt structure is optimal only if the increase in the probability that stabilization fails, that is associated
with the interest cost of additional funding in a particular type of debt, is equalized across debt
instruments. If this were not the case, the government could reduce the probability of failure by
modifying the debt structure; e.g. it could substitute fixed-rate bonds for short-term bills or vice
versa.3
To gain further intuition we observe that the dierence between the interest cost of short-term
bills and long-term fixed-rate bonds depends on the dierence between the short-term interest
rate between time t and t + 1 and its value as expected at time t 1, minus the term premium
on fixed-rate bonds. More precisely, the cost dierential is equal to
it Rt31 = it (1/2)(it31 + Et31 it ) T pt31 = it Et31 it + (1/2)(Et31 it it31 ) T pt31 (8)
where it31 is the short-term interest rate between time t 1 and t, Et31 it is the expected shortterm interest rate between time t and t + 1 and where T pt31 is the term premium on fixed-rate
bonds.
Equation (8) shows that the return dierential between time t and t + 1 depends on the
unanticipated component of the short-term interest rate, on the term premium and on the slope
of the yield curve Et31 it it31 . In fact, a shorter maturity debt allows to postpone interest
payments in the presence of a positive sloped yield curve. Noting that Et31 it it31 is known at
time t 1, it is convenient to write equation (8) as
it Rt31 = it Et31 it T Pt31
(9)
where T Pt31 = T pt31 (1/2)(Et31 it it31 ) denotes the term premium net of the yield slope. In
what follows we implicitly assume that the latter term is equal to zero.
3
The argument assumes that there are non-negative constraints to the choice of debt instruments.
5
Equation (9) shows that the expected cost of funding with short-term bills is lower than fixedrate bonds because of the term premium but, ex-post, the cost may be greater if the short-term
rate turns out to be higher than expected.4
The dierence between the nominal return on price-indexed bonds and fixed-rate long-term
bonds is equal to
RIt31 + t+1 Rt31 = t+1 Et31 t+1 + (1/2)(Et31 t+1 Et31 t ) Ipt31
(10)
where Ipt31 denotes the inflation risk premium.
Equation (10) shows that the return dierential depends on unexpected inflation, on the
inflation risk-premium and on the expected increase in inflation (1/2)(Et31 t+1 Et31 t ). This
is because fixed-rate bonds allow to smooth interest payments in the presence of a positive sloped
yield curve due to a rising expected inflation. Noting that Et31 t+1 Et31 t is known at time
t 1, we write equation (10) for notational convenience as
I
+ t+1 Rt31 = t+1 Et31 t+1 IPt31
Rt31
(11)
where IPt31 = Ipt31 (1/2)(Et31 t+1 Et31 t ) denotes the inflation-risk premium net of the
expected change in inflation. In what follows we implicitly assume that the latter term is equal
to zero.5
Substituting the return dierentials (9) and (11) in the first order conditions (6)-(7) yields
Et31 !(( Xt+1 )Bt )(it Et31 it ) = T Pt31 Et31 !(( Xt+1 )Bt )
Et31 !(( Xt+1 )Bt )(t+1 Et31 t+1 ) = IPt31 Et31 !(( Xt+1 )Bt )
(12)
(13)
Equations (12)-(13) show the trade o between the risk and expected cost of debt service that
characterizes the choice of debt instruments.
At the margin, the impact on the probability of debt stabilization of assuming more risk must
be equal to the impact of reducing the expected cost of debt servicing. Hence, the marginal
increase in probability can be used to price risk against the expected cost of debt service and
thus find the optimal combination along the trade o between cost and risk minimization. For
example, equation (12) shows that issuing short-term bills is optimal until the uncertainty of the
short-term interest rate raises the probability that the debt ratio will increase as much as paying
the term premium on fixed-rate long-term bonds.
Therefore, the objective of debt stabilization oers a solution to the identification of the
optimal debt structure which is independent of the government’s preferences towards risk. This
is because both the risk and the expected cost of debt service aect the probability of debt
stabilization.
To derive an explicit solution for the optimal shares of the various types of debt we must
specify the probability density function, !(Gt+1 ). Since this function cannot be estimated, we
4
It is also worth noting that equation (9) implicitly assumes that investors’ expectations coincide with the
expectations of the government. If this were not the case, the expected cost digerential relevant for the government,
I
I
I
T Pt31 , would include an informational spread: T Pt31 = T Pt31
+ (Et31
it 3 Et31 it ) where Et31
denotes investors’
I
is the true term premium.
expectations and T Pt31
5
It is worth noting that IPt31 is the inflation risk premium which is relevant to the government and may include,
in addition to the true premium, a spread reflecting the lack of credibility of the announced inflation target; i.e.
I
I
IPt31 = IPt31
+ EtI t+1 3 Et t+1 where IPt31
is the true inflation-risk premium.
6
take a linear approximation of !(Gt+1 ) over the range of bad realizations, Gt+1 > 0, of government
spending.6 This implies a triangular probability density function equal to
!(Gt+1 ) =
Ḡ Gt+1
Ḡ2
(14)
where Gt+1 > 0 and Ḡ is the worst possible realization of government spending.
In fact, the triangular density is the linear approximation of any density function decreasing
with Gt+1 (for Gt+1 > 0); it implies that bad realizations of the fiscal adjustment are less likely
to occur the greater is their size.
Substituting equations (14) and (4) in the first order conditions (12) and (13) yields the
optimal shares of short-term debt, sW , and inflation-indexed debt, hW :
sW =
hW =
d i )
Covt31 (t+1
Covt31 (yt+1 it )
t
+
V art31 (it )
V art31 (it )
Covt31 (t+1 it )
+
V art31 (it )
T Pt31
Ḡ
+
( Et31 Xt+1 )]
[
V art31 (it ) Bt31
hW
d Covt31 (t+1
Covt31 (yt+1 t+1 )
t+1 )
+
V art31 (t+1 )
V art31 (t+1 )
(15)
Covt31 (t+1 it )
+
V art31 (t+1 )
IPt31
Ḡ
+
( Et31 Xt+1 )](16)
[
V art31 (t+1 ) Bt31
sW
where V art31 (.) and Covt31 (.) denote variances and covariances conditional on the information
available at time t 1.
The optimal debt shares depend on both risk and cost considerations. Risk is minimized
if a debt instrument provides insurance against variations in the debt ratio due to output and
GDP-inflation uncertainty, and if the conditional variance of its returns is relatively low. This is
captured by the first two terms in equations (15) and (16).
Equation (15) shows that short-term debt is optimal for risk minimization when the short-term
interest rate and thus the interest payments are positively correlated with unanticipated output
and inflation. To pay less interests when output and inflation are unexpectedly low is valuable
because a lower nominal growth rate of output tends to increase the debt ratio. Instruments with
returns correlated to nominal output growth help to stabilize the debt ratio, thus reducing the
risk that it will grow above target. On the other hand, the case for short-term debt weakens
as the conditional variance of the short-term interest rate increases, thus producing unnecessary
fluctuations in interest payments.
Equation (16) shows that the optimal share of inflation-indexed debt increases with the covariance between output and inflation. If this covariance is positive, lower interest payments on
inflation-indexed debt provide an insurance against unexpected slowdowns in economic activity
that raise the debt-to-GDP ratio. However, inflation-indexed debt would be optimal even if the
covariance between output and inflation were zero. The reason is that bonds indexed to CPI
6
We assume that the planned surplus, Bt is expected to stabilize the debt ratio, so that > Et [It+1 3t+1 3yt+1 ].
7
inflation are expected to provide a good hedge against an increase in the debt ratio due to lower
than expected nominal output growth.
Risk minimization also depends on the conditional covariances between the returns on the
various debt instruments. For instance, a positive covariance between the returns of two types of
debt makes the two instruments substitutes in the government portfolio. This is captured by the
third term in equations (15) and (16).
Leaving aside cost considerations, the government should choose the debt composition which
oers the best insurance against the risk of deflation and low growth. But insurance is costly;
higher expected returns are generally required on hedging instruments, and this leads on average
to greater debt accumulation. Debt stabilization thus implies a trade o between cost and risk
minimization. The eect of expected return dierentials on the optimal debt composition is
captured by the last term on the right-hand-side of equations (15) and (16). This term increases
with the expected return dierentials (as perceived by the government), T Pt31 and IPt31 , of fixedrate bonds relative to the instrument considered. The impact of the expected cost dierential on
the optimal share of a given debt instrument decreases with the variance of its return. In fact, a
greater variance of returns reduces the impact of interest-cost dierentials as much as it reduces
the relevance of insurance considerations.
More importantly, the impact of the expected cost dierential depends on the dierence between the maximum shock to government spending and the planned (or expected) reduction of the
debt ratio. A stronger fiscal reaction to the debt ratio, as captured by a higher , clearly reduces
the importance of expected return dierentials for the choice of the debt instruments that stabilize the debt ratio. Intuitively, as debt sustainability is on average ensured by a restrictive fiscal
stance, cost considerations become less important than insurance considerations for the choice of
debt instruments. In other words, if debt stabilization may fail only for unusual large realizations
of spending shocks, then debt management should be mostly concerned with providing insurance
against such events. This result possibly explains why in countries where the dynamics of the
debt is out of control interest-cost minimization is the main goal of debt management.
While the result that the optimal debt composition relates to the stochastic structure of the
economy is not new in the literature, in the next section, building on the intuition in Bohn (1988),
we investigate the role of monetary policy in determining the relations between debt returns and
the other variables that aect the dynamics of the debt ratio.
4. Monetary policy and debt management
The stochastic relations between output, inflation and the short-term interest rate depend on
the reaction of the monetary authority to macroeconomic shocks aecting the economy. Therefore,
we expect the monetary regime to play an important role in the choice of debt maturity and
indexation.
In this section we use a simple model to examine the implications of alternative monetary
regimes for the choice of debt instruments. In particular, we compare an inflation targeting
regime with a fixed-exchange rate and with a monetary union.
The central bank controls aggregate demand and thus output and inflation with a lag through
the choice of the nominal interest rate, it set at the beginning of period t.
The aggregate demand is equal to
yt+1 = yt a(it Et t+1 r̄) + vt+1
8
(17)
where yt+1 is the output gap and (it Et t+1 ) is the real interest rate between period t and
t + 1. The impact of the interest rate depends on the parameter a, while measures output
auto-correlation. Finally, vt+1 is an i.i.d. demand shock with mean zero and variance equal to v2 .
The supply side of each economy is modeled as a backward looking Phillips curve:
t+1 = t + cyt+1 + ut+1
(18)
where c measures the impact of the output gap on inflation. Inflation is aected by an adverse
supply shock, ut , with mean zero and variance equal to u2 .
Equation (18) implies important nominal rigidities and backward looking behavior in that the
current inflation rate entirely depends on lagged inflation as opposed to expected inflation. The
backward looking Phillips curve has been introduced by Fuhrer and Moore (1995) and Obstfeld
(1995) and its empirical performance is quite satisfactorily (see e.g. Fuhrer 1997).
4.1 Inflation targeting
In an inflation-targeting regime the Central Bank aims at maintaining expected inflation close
to the target and, possibly, at stabilizing output. Then, assuming a concern for interest-rate
volatility, the loss function of the Central Bank is equal to
T 2
2
2
LIT
t = Et (t+i ) + Et yt+1 + (it it31 )
(19)
where T is the inflation target, and and are the (publicly known) weights given by the
Central Bank to output stabilization and interest-rate smoothing, respectively, relative to the
inflation target.
The Central Bank chooses the short-term interest rate, it , to minimize the loss function (19)
subject to equations (17) and (18). The interest rate rule in an inflation targeting regime is, thus,
equal to
(20)
it = µit31 + (1 µ)[T + r̄ + (t T ) + yt ]
where
=
c + a
;
ac2 + a
=
a
µ=
(1 ac)2
a2 c2 + a2 + (1 ac)2
and where it is assumed that 1 ac > 0 to ensure that an increase in the interest rate reduces
the inflation rate. Hence, the reaction to current inflation is greater than one, > 1, and, as
expected, it decreases with the weight, , assigned to output stabilization. Finally, the degree of
interest-rate smoothing is captured by µ which is increasing in .
The interest rate rule (19) can be combined with the aggregate demand (17) and the aggregate
supply (18) to derive the conditional covariances between output, inflation and the interest rate.
In what follows we examine how such covariances are aected by the preferences of the Central
Bank regarding output stabilization and interest rate smoothing.
4.1.1 Inflation targeting and debt management
The implications of the monetary regime for debt management depend on how monetary
policy aects the conditional covariances between output, inflation and the interest rate. Such
covariances are conditional on the information available at time t1, when the government chooses
9
the composition of the debt. The (two-period ahead) unanticipated components of inflation,
output and the interest rate are equal to
(t Et31 t ) + µzc[(t Et31 t ) + (yt Et31 yt )] + ut+1 + cv(21)
t+1
+
c
= 2
(t Et31 t ) + µz[(t Et31 t ) + (yt Et31 yt )] + vt+1 (22)
c +
= (1 µ)[(t Et31 t ) + (yt Et31 yt )]
(23)
t+1 Et31 t+1 =
yt+1 Et31 yt+1
it Et31 it
c2
where z = a/(1 ac) > 0.
Recalling that V art31 (t ) = c2 v2 + u2 , V art31 (yt ) = v2 and Covt31 (yt t ) = cv2 , we derive
the following correlation coecients of equations (15) and (16):
c[c( + c)v2 + u2 ]
µz
Covt31 (yt+1 it )
= 2
+
2
2
2
2
V art31 (it )
(c + )(1 µ)[( + c) v + u ] 1 µ
(24)
Covt31 (t+1 it )
[c( + c)v2 + u2 ]
µzc
= 2
+
2
2
2
2
V art31 (it )
(c + )(1 µ)[( + c) v + u ] 1 µ
(25)
2 2
2
cv2 (c2c
Covt31 (t+1 yt+1 )
+)2 (c v + u ) + µzH
=
2
V art31 (t+1 )
c2 v2 + u2 + (c2+)2 (c2 v2 + u2 ) + µzQ
(26)
where
2
)[c( + c)v2 + u2 ]
+
2c
[c( + c)v2 + u2 ]
Q = µc2 z[( + c)2 v2 + 2 u2 ] + 2
c +
Equations (24), (25) and (26) show the correlation coecients that are relevant for the choice of
debt maturity. To interpret these correlation it is useful to abstract from interest-rate smoothing
and assume, for the moment, that µ = 0. In this case the covariance between output and
the interest rate in equation (24) is always negative but decreases in absolute value with .
The conditional covariance between the output gap and the interest rate is negative because the
interest rate lowers inflation through a contraction of aggregate demand. This negative correlation
weakens as the inflation targeting becomes more flexible (as increases) because of the weaker
reaction of the interest rate to the inflationary pressure. It follows that short-term debt may
have a role in stabilizing the debt only if the conditional covariance between the interest rate and
inflation is positive. However, equation (25) shows that the coecient Covt31 (t+1 it )/V art31 (it )
is positive (and increases with ) only for > 0; i.e. only if the monetary authority cares about
output stabilization. The reason is that, in a strict inflation targeting –i.e. for = 0– the
interest rate is set so as to fully stabilize inflation. As shown in equation (21), if = 0 (and
no interest-rate smoothing), inflation may dier from its expectation two periods earlier only
because of contemporaneous shocks. Intuitively, if monetary policy has only one objective, at
time t 1, then inflation is expected to be uncorrelated with the policy instrument, it , otherwise
the monetary authority would intervene. On the other hand, if the authorities also care about
output stabilization (or interest-rate smoothing), the interest-rate reaction is not sucient to
eliminate inflationary pressures and a positive covariance between inflation and the interest rate
H = µcz[( + c)2 v2 + 2 u2 ] (1 10
c2
emerges. Therefore, in the absence of interest-rate smoothing, a role for short-term debt emerges
in an inflation-targeting regime only if > c, that is, only if the monetary authority cares
suciently about output stabilization.
Focusing on the choice of inflation-indexed debt, we observe that the correlation coecient in
equation (26) is positive for = 0 and increases with the variance of demand shocks relative to
the variance of supply shocks. Moreover, this coecient decreases with . The intuition for these
results is as follows. The covariance between output and inflation at time t + 1 conditional on
the information at time t 1 is uncertain because it depends on both demand and supply shocks
occurring at time t + 1 and by the correlation induced by monetary policy. Contemporaneous
shocks lead to a positive covariance which increases with the magnitude of demand relative to
supply shocks. On the other hand, the eect of monetary policy depends on the weight assigned
to output stabilization. In a strict inflation targeting, with = 0, unanticipated inflation only
depends on contemporaneous shocks and the policy eect vanishes. In a flexible inflation targeting,
once we abstract from contemporaneous shocks, the reaction of the monetary authority induce a
negative covariance between output and inflation which reduces the overall covariance. The eect
is stronger the greater the weight, , assigned to output stabilization.
To conclude, in the absence of interest-rate smoothing, there is little or no role for shortterm conventional debt or floating-rate debt. In particular, if the monetary authority only cares
about inflation, the debt should either have a long maturity or be indexed to the inflation rate. As
implied by tax-smoothing, the optimal share of indexed debt increases with the variance of demand
relative to supply shocks (see Missale 1997). On the other hand, if the monetary authority also
aims at output stabilization, then a lower share of inflation-indexed bonds is needed to stabilize
the debt ratio and a role for short-term debt may emerge.
4.1.2 Debt management and Central Bank preferences
The analysis of the previous section suggests that the debt structure that stabilizes the debt
ratio is strongly related to the preferences of the monetary authority, in particular, to the weight,
, that the Central Bank assigns to output relative to price stabilization.
Interestingly, the ’more conservative’ the Central Bank in its anti-inflationary policy, the
greater is the role of long-term debt and inflation-indexed debt in stabilizing the debt ratio. On
the contrary, a concern for output stabilization unambiguously favors short-term and floating-rate
debt since it induces a positive covariance between inflation and the interest rate and increases the
covariance between output and the interest rate (though the latter covariance remains negative).
Interestingly, a greater concern for output stabilization also reduces the opportunity to issue
inflation-indexed bonds. This happens for two reasons. First, the variance of inflation rises.
Second, the covariance between output and inflation falls, since the authorities give up inflation
stability in exchange for output stabilization.
The analysis thus provides insights in how the optimal debt structure for deficit stabilization
may have changed along with the the monetary authority, as policy making moved from national
central banks to the ECB. Clearly, if the ECB were less concerned with output fluctuations
than the national authorities, then, everything else being equal, the optimal policy would call for
lengthening the maturity of the debt and issuing inflation indexed debt.
4.1.3 Interest rate smoothing
11
We have, so far, assumed that the monetary authority does not smooth the interest rate,
that is, that µ = 0. It is important to see how previous conclusions change when we relax this
unrealistic assumption.
Equations (25) clearly shows that the correlation coecient between the short-term interest
rate and next period inflation increases with the degree of interest-rate smoothing µ. Intuitively,
a concern for interest rate volatility weakens the short-run reaction of the monetary authority
to inflationary pressure and leads to a lower variance of the interest rate. More importantly, to
the extent that next period inflation is not completely eradicated, current inflationary shocks will
tend to persist, and this generates a positive covariance between next period inflation and the
current interest rate.
The correlation coecient between output growth and the interest rate also increases with
interest-rate smoothing as can be shown by deriving equations (24) with respect to µ. The reason
is that the impact on output is lower the weaker is the interest-rate reaction and this reduces the
negative covariance between output and the interest rate.
These results hold independently of Central Bank preferences, that is, independently of .
It follows that interest-rate smoothing favors short-term debt relative to the benchmark case of
µ = 0. If the positive correlation of the interest rate with inflation prevails over the negative
correlation between the interest rate and output, which may happen for a suciently high , then
a positive amount of short-term debt may possibly be justified for debt stabilization.
The eect of interest-rate smoothing on the correlation coecient between output and inflation
is ambiguous and so are its implications for the choice of inflation-indexed debt. The reason is that
a weaker interest-rate reaction increases the covariance between output and inflation (because it
leads to more inflation persistence) while it reduces the negative impact on output. However,
interest-rate smoothing also implies higher inflation volatility and this makes the overall eect
on the correlation coecient uncertain. In particular, the total eect will depend on the relative
variance of supply and demand shocks. It can be shown that interest-rate smoothing reduces the
correlation between output and inflation, and thus the need for inflation indexed debt, when the
variance of demand shock is high relative to the variance of supply shock and thus when a high
share of inflation-indexed debt would otherwise be optimal.
Finally, it is worth noting that interest-rate smoothing does not alter the eect of the Central
Bank’s preference for output relative to inflation stability on the correlation coecients and thus
its implication for the choice of debt instruments. In particular, it can be shown that for any given
µ a greater concern for output stabilization, i.e. a higher , still reduces the need for inflation
indexation and may provide a role for short-term debt in stabilizing the debt ratio.
To conclude, interest-rate smoothing sets a case for short-term debt. In particular, a positive
amount of short-term debt can be optimal for debt stabilization even in a strict inflation-targeting
regime, that is, even if the Central Bank does not aim at stabilizing output.
4.2 Fixed exchange rate
In a fixed exchange-rate regime the Central Bank maintains the fixed parity by pegging the
interest rate to the interest rate of the leader country augmented by an eventual currency premium.
To the extent that the monetary authority has some flexibility in pursuing domestic stabilization
objectives, its loss function is equal to
2
+ %(it iWt pet )2
LFt E = Et (t+i T )2 + Et yt+1
12
(27)
where pet denotes the currency premium, that is, the sum of expected depreciation, the country
risk premium and the foreign-exchange premium. Finally, % is the weight given by the Central
Bank to maintaining a fixed exchange rate relative to price stability.
We assume that the structure of the economy is still represented by equations (17) and (18)
which implies to assume that the domestic economy is not aected by the exchange rate and
by foreign output. Obviously, this is an unrealistic assumption that is made for simplifying the
analysis. It is however worth noting that unanticipated changes in domestic output and inflation
would not be aected by the exchange rate if the the latter were maintained fixed.
The interest rate rule in a fixed exchange regime is derived by minimizing the loss function
(27) subject to (17) and (18) and is equal to
it = (iWt + pet ) + (1 )[T + r̄ + (t T ) + yt ]
(28)
where is an increasing function of % and captures the extent of interest-rate pegging.
Assuming that the monetary authority of the leader country sets the interest rate to stabilize
domestic inflation and output; i.e. iWt = T W + r̄W + W (tW T W ) W ytW , the domestic interest rate
rule is equal to
it = [pet + T W + r̄W + W (tW T W ) W ytW ] + (1 )[T + r̄ + (t T ) + yt ]
(29)
The interest rate rule (29) can be combined with the aggregate demand (17) and the aggregate
supply (18) to derive the conditional covariances between output, inflation and the interest rate.
4.2.1 Fixed exchange rate and debt management
The implications of interest rate pegging for debt management depend on how shocks to the
currency premium and foreign monetary policy aect the conditional covariances between output,
inflation and the interest rate. To understand the role of foreign policy it is useful to look at the
unanticipated component of the domestic interest rate that is equal to
it Et31 it = et + (1 )[(t Et31 t ) + (yt Et31 yt )] +
+ [ W (tW Et31 tW ) + W (ytW Et31 ytW )]
(30)
where t defines the unanticipated component of the currency premium, i.e. t = pet Et31 pet ,
that captures unanticipated variations in expected depreciation and/or in the country risk premium. We assume that et , has mean zero, variance equal to e2 and is uncorrelated to supply and
demand shocks ut and vt .
Equation (30) shows that the domestic interest rate diers from that prevailing under inflation
targeting either because of shocks to the currency premium or because of ’imported’ monetary
policy. The latter may dier from the optimal policy under inflation targeting for two reasons.
First, the timing and magnitude of foreign demand and supply shocks are, in general, dierent from
those of domestic shocks. Second, the economic structure and, thus, the transmission mechanism
of monetary policy vary across country, as captured by the possibly dierent reaction parameters,
W and W .
In what follows we focus on each aspect at a time starting from the eects of shocks to the
currency premium.
13
4.2.2 Currency premium shocks and debt management
To focus on the eect of changes in expected depreciation and/or in the country risk premium
that aect interest rate spreads, we assume that the domestic and foreign monetary authorities
face the same supply and demand shocks and that the structures of the two economies are identical.
The (two-period ahead) unanticipated components of inflation, output and the interest rate
are equal to
(t Et31 t ) + ut+1 + cvt+1 czet
c2 + c
(t Et31 t ) + vt+1 zet
= 2
c +
= et + [(t Et31 t ) + (yt Et31 yt )]
t+1 Et31 t+1 =
(31)
yt+1 Et31 yt+1
(32)
it Et31 it
(33)
where z = a/(1 ac) > 0.
Equations (31), (32) and (33) are the same as in an inflation targeting regime except for the
presence of the currency premium shock. The latter has the same impact on the economy as
an interest rate shock, that is, as a change in the interest rate that is unrelated to supply and
demand shocks.
The correlation coecients of equations (15) and (16) can be derived using (31), (32) and (33)
and be expressed in terms of their counterparts (24), (25) and (26) under inflation targeting:
F E (y
IT (y
2
2
Covt31
Covt31
t+1 it )
t+1 it ) ze
=
F E (i )
IT (i ) + 2 2
V art31
V art31
t
t
e
(34)
F E (
IT (
2
2
Covt31
Covt31
t+1 it )
t+1 it ) cze
=
F E (i )
IT (i ) + 2 2
V art31
V art31
t
t
e
(35)
F E (
IT (y
2 2 2
Covt31
Covt31
t+1 yt+1 )
t+1 t+1 ) + cz e
=
F E (
IT (
2 2 2 2
V art31
V art31
t+1 )
t+1 ) + c z e
(36)
Equations (33) and (34) show that the correlation coecients of output and inflation with
the interest rate are lower than the corresponding coecients under inflation targeting. This is
because shocks to the currency premium lead to changes in the interest rate which are unrelated
to demand and supply shocks. Since output and inflation fall as the interest rate rises and vice
versa, this generates a negative covariance of the interest rate with both output and inflation.
Therefore the consideration of changes in the country risk premium or in expected depreciation
in a fixed exchange regime sets a strong case against short-term debt. In particular, the share of
short-term debt should be lower than in an inflation targeting regime.
By contrast inflation-indexed debt may provide a partial insurance against changes in the
interest rate induced by shocks of the currency premium. Indeed, from equation (35) we can
verify that the correlation coecient between output and inflation is always higher than the
corresponding coecient under inflation targeting.7 The reason is that changes in the interest
rate make output and inflation move in the same direction thus reinforcing the positive covariance
induced by demand shocks. It is, however, worth noting that this result hinges on the ability of
the Central Bank to maintain the exchange rate fixed. If an increase in the country risk premium
7
IT
IT
The condition Covt31
(yt+1 t+1 )/V art31
(t+1 ) < 1/c is verified for = 0 and thus for any 14
triggered a depreciation, then the resulting inflation would lead to a negative covariance with
output and thus to the opposite result.
To conclude countries which experience substantial variations in the interest rate due to variation in the currency premium should abstain from short-term funding of public debt and rely on
long-term bonds. If the event of a currency crisis cannot be ruled out, fixed-rate bonds provide
the best insurance against variations of the debt ratio.
4.2.3 Imported monetary policy and debt management
The implications of a fixed exchange rate for debt management also depend on how foreign
monetary policy aects the conditional covariances between output, inflation and the interest
rate. To understand the eect of importing the monetary policy of the leader country it is useful
to focus on the case where the domestic and the foreign economy have the same structure but are
hit by dierent shocks. A simple formalization of this hypothesis is to assume that:
uWt = kut
vtW = kvt
By varying the parameter k we can examine the case of: i) greater foreign shocks, k > 1; ii)
smaller foreign shocks, 0 < k < 1, and; iii) asymmetric shocks, k < 0.
In what follows, we abstract from shocks to the currency premium and assume that the
structures of the domestic and foreign economies are identical.
Noting that tW Et31 tW = k(t Et31 t ) and ytW Et31 ytW = k(yt Et31 yt ), the unanticipated
components of inflation, output and the interest rate are equal to
(37)
t+1 Et31 t+1 = [1 cz( 1)](t Et31 t ) + cz(1 )(yt Et31 yt ) + ut+1 + cvt+1
(38)
yt+1 Et31 yt+1 = z( 1)(t Et31 t ) + +z(1 )(yt Et31 yt ) + vt+1
it Et31 it = [(t Et31 t ) + (yt Et31 yt )]
(39)
where = 1 + k
Equations (37), (38) and (39) allow us to derive the following correlation coecients:
Covt31 (yt+1 it )
z( 1)[c( + c)v2 + u2 ]
z(1 )( + c)v2
=
+
V art31 (it )
[( + c)2 v2 + 2 u2 ]
[( + c)2 v2 + 2 u2 ]
(40)
Covt31 (t+1 it )
[1 cz( 1)][c( + c)v2 + u2 ]
cz(1 )( + c)v2
=
+
V art31 (it )
[( + c)2 v2 + 2 u2 ]
[( + c)2 v2 + 2 u2 ]
(41)
Covt31 (t+1 yt+1 )
c2 z( 1)[1 cz( 1)](c2 v2 + u2 ) + [1 z(1 )]z(1 )cv2
= 2 2v 2
V art31 (t+1 )
(c v + u ){1 + [1 cz( 1)]2 } + z(1 )[z(1 ) 2 2cz( 1)]c2 v2
(42)
By comparing equations (40) and (41) to equations (24) and (25) we can show that whether
the correlation coecients of output and inflation with the interest rate are greater or lower than
the corresponding coecients under inflation targeting depends on = 1 + k. If 0 < < 1
both coecients are greater than those under inflation targeting and so is the share of shortterm debt needed for debt stabilization. In this case the shocks hitting the foreign economy are
smaller than domestic shocks (and possibly negatively correlated), (1 )/ < k < 1, and thus
15
imply a reaction by the foreign Central Bank that is weaker than what would be optimal for the
domestic country. On the contrary, if > 1 or < 0, both coecients are smaller than those
under inflation targeting, suggesting little or no role for short-term debt in stabilizing the debt
ratio. Interestingly, the case against short-term debt arises either because of greater foreign than
domestic shocks, k > 1, or because of negatively correlated shocks, k < (1)/). In the former
case the imported monetary policy leads to a too strong interest rate reaction, while in the latter
it implies a pro-cyclical interest rate reaction.
The role of inflation-indexed debt depends on the correlation between output and inflation that
is shown in equations (42). Whether this coecient is greater or smaller than the corresponding
coecient under inflation targeting does not only depend on but also on the values of the
parameters and the variance of demand relative to supply shocks. For instance, in the case of
= 0 we can show that the correlation between output and inflation in a fixed exchange regime is
greater than the coecient under inflation targeting if > 1 and if < 1 W + acW , where W is
greater than one and increasing with the variance of demand relative to supply shocks.8 It follows
that a greater share of inflation-indexed debt than under inflation targeting would be optimal for
large shocks to the foreign economy leading to a too strong interest rate reaction and in the case of
small or asymmetric foreign shocks implying a pro-cyclical imported monetary policy. However,
the exact range of shocks for which inflation-indexed debt has a stronger stabilizing role than
under inflation targeting would depend on the structure of the economy.
Therefore, dierences in the magnitude and correlation of domestic and foreign shocks in a
fixed exchange regime may favor inflation-indexed bonds, and possibly long-term conventional
bonds, as stabilizing instruments when the foreign policy reaction is too strong or asymmetric for
domestic stabilization purposes. By contrast, a passive imported policy would favor short-term
debt.
4.2.4 Dierent transmission mechanisms and debt management
The results of the previous section are derived for the unrealistic case that the structures of the
domestic and the foreign economy are the same. In what follows we show how the implications of
dierent economic structures for the choice of debt instruments can be derived using the framework
developed so far.
A first case to consider is when the aggregate demand of the foreign economy is more sensitive
to the interest rate than the demand of the domestic economic, that is, when aW > a. If we restrict
the analysis to the case where both monetary authorities do not care about output stabilization,
then W = (a/aW ) and W = (a/aW ). Then, assuming that the two countries experience the same
shocks, the unanticipated component of the interest rate is equal to
it Et31 it = ![(t Et31 t ) + (yt Et31 yt )]
(43)
where ! = (a/aW ) + (1 ) < 1
Equation (43) and ! < 1 show that the implications for debt management of a greater sensitivity of the aggregate demand of the leader country to the interest rate are the same of those
derived earlier under the assumption of a lower variance of foreign shocks. As the stronger foreign
transmission mechanism of monetary policy implies a weaker reaction by the foreign Central Bank
8
W = (1 3 )/[1 3 C 3 C + (1 3 )C 2 ] where C = c2 v2 + u2 .
16
to the same shocks, then a higher share of short-term debt would be optimal for stabilizing the
debt ratio.
Finally, consider the case that foreign inflation is less sensitive to variation in the output gap
than domestic inflation, that is, cW < c. If we restrict the analysis to the case where both monetary
authorities do not care about output stabilization, then W = (c/cW ) and W = . Then, assuming
that the two countries experience the same shocks, the unanticipated component of the interest
rate is equal to
¯ t Et31 t ) + (yt Et31 yt )]
(44)
it Et31 it = [(
where ¯ is the weighted average of the foreign and domestic interest-rate reactions to inflation
shocks, and ¯ = [(c/cW ) + (1 )] > .
Equation (44) and ¯ > show that the implications for debt management of a lower sensitivity
of foreign inflation to the output gap are equivalent to those derived earlier for a stronger preference
of the Central Bank for inflation control relative to output stabilization. In particular, the stronger
reaction to inflationary pressure would reduce the optimal share of short-term debt while it would
increase the share of inflation-indexed debt.
5. Estimating the optimal debt structure
The optimal debt composition for debt stabilization can be obtained by estimating the conditional covariances of output growth, inflation and the short-term interest rate. The ratios of the
conditional covariances relative to the conditional variances can be obtained in two steps. First,
the unanticipated components of output growth, CPI inflation, GDP inflation and the short-term
interest rate can be estimated as the residuals of forecasting equations in their second lags (in the
first lag for the interest rate). Secondly, the correlation coecient of equations (15) and (16) can
be obtained as the coecients of the regressions of the residuals of output growth and inflation on
the residuals of the interest rate and as the coecients of the regression of the residuals of output
growth on the residuals of inflation. The first stage regressions are run recursively using only the
information available up to the time when the forecast is made and using a constant window of
fifteen years. This implies that the series of residuals starts in 1976 or 1980, depending on data
availability, despite the fact that the sample period runs from 1960 to 2004.
Table 1 shows the estimated coecients when a distinction is made between CPI and GDP
inflation. In the case of Denmark, Italy, Spain and Sweden we use the long-term interest rate
because of the unavailability of suciently long series for the short-term interest rate.
The estimated coecients are remarkably consistent with the stochastic structure derived
from the monetary policy model developed in section 4. As expected, the coecients of output
growth on the interest rate are negative and significant for all countries considered except for
Spain and Sweden. The coecients of inflation on the interest rate, though positive, display a
dierent pattern across countries. In Japan, the Netherlands, Spain and the UK, the hypothesis
of no conditional correlation between inflation and the interest rate cannot be rejected at the 5%
significant level, while in Italy at the 10% level. The absence of a relation is consistent with strict
inflation targeting regimes where the authorities place no weight on output stabilization. (For
the UK, however, this result depends on the use of the GDP deflator instead of the CPI index for
measuring inflation as shown in Table 2).
A second group of countries presents instead a significant positive correlation between inflation
and the interest rate that should emerge in monetary regimes with a concern for output stabi17
lization and/or interest-rate smoothing. This evidence is also consistent with a fixed exchange
regime when the interest-rate reaction of the leader countries is weaker than that needed to fully
stabilize domestic inflation. Coecients of inflation on the interest rate in the 0.4-0.6 range are
reported for Belgium, Denmark, Germany and the US. The last column of Table 1 shows the
coecient of output on inflation which is negative but not significant for all countries considered
except for Denmark, France, the UK and the US.
The estimated coecients in Table 1 allow us to derive the debt compositions that would have
supported debt stabilization over the past two decades under the hypothesis of zero expected
return dierentials. The composition of public debt that would have provided insurance against
variations in the debt ratio is shown in Table 3.
In the case Denmark, France, Italy, Japan, the Netherlands, Spain, the UK and the US shortterm debt should not have been issued for stabilizing the debt ratio except possibly for its lower
expected return. In fact, these countries exhibit a significant negative correlation between output
growth and the interest rate that is not oset by the positive correlation between inflation and
the interest rate. The absence of a significant correlation between output growth and inflation
for Belgium, Germany, Ireland, Italy, Japan, the Netherlands, Spain and Sweden implies that
inflation-indexed debt would have played an important stabilizing role in these countries. However,
this role would be limited to the insurance that inflation indexed debt provides against unexpected
deflation. Finally, the optimal share of fixed-rate long-term debt is equal or greater than 40% in
Denmark, France, the Netherlands, the UK and the US where short-term debt plays no stabilizing
role while the insurance provided by inflation indexation is partly oset by the negative correlation
between inflation and output growth.
The only countries where short-term debt would have stabilized the debt ratio are Ireland,
Sweden and, to a lesser extent, Belgium where the correlation between inflation and the interest
rate has been high and significant.
5. Conclusions
TO BE WRITTEN
18
References
Bohn, H. (1988). “Why Do We Have Nominal Government Debt?,” Journal of Monetary Economics, 21: 127-140.
Bohn, H. (1990). “Tax Smoothing with Financial Instruments,” American Economic Review,
80(5): 1217-1230.
Favero, C., A. Missale and G. Piga (2000). “EMU and Public Debt Management: One Money,
One Debt?,” CEPR Policy Paper No. 3.
Fuhrer, J. C. (1997). “The (Un)Importance of Forward Looking Behavior in Price Setting,”
Journal of Money Credit and Banking, 29: 338-350.
Fuhrer, J. C. and G. R. Moore (1995). “Inflation Persistence,” Quarterly Journal of Economics,
110(1): 127-159.
Missale, A. (1997). “Managing the Public Debt: The Optimal Taxation Approach,” Journal of
Economic Surveys, 11(3): 235-265.
Missale, A. (2001), “Optimal Debt Management with a Stability and Growth Pact,” Public Finance and Management, 1(1): 58-91.
Obstfeld, M. (1995). “International Currency Experience: New Lessons and Lessons Relearned,”
Brookings Papers on Economic Activity, 1:1995: 119-196.
19
Table 1: Estimation results: residuals calculated from recursive regressions.
w-statistics in parentheses. signi• cant at 10% level. signi• cant at 5% level.
d long-term interest rate have been used in the analysis.
Years in parentheses indicate the recursive estimation sample.
Fry+CDE
Fry+CDE
Fry+|M.4 >lM ,
Fry+|M.4 >M.4 ,
M.4 >lM ,
M.4 > M.4 ,
Y du+lM ,
Y du+lM ,
Y du+M.4 ,
Y du+M.4 ,
+53==69
5;,
3=7:
3=::
=3<
+33=98,
Denmarkd
+63==:9
77,
3=97
3=93
+63==88
9<,
France
+63==6:
4<,
3=56
3=83
55
+43==:3,
Germany
+53==75
49,
3=77
3=88
+6=;8,
+3=;8,
Ireland
+53==89
8<,
3=;5
3=95
=4:
+43=5:,
Italyd
+53==:4
:6,
3=:4
3=<:
+4<=4,
+3=5:,
=44
+33=<<,
Belgium
+1976-2004,
+1976-2004,
+1980-2004,
+1976-2004,
+1985-2004,
+1980-2004,
+6=69,
+44=6,
+6=45,
+8=;7,
+5=3:,
+8=94,
+6=<4,
+6=:7,
+9=:4,
+4=::,
3=55
3=36
68
+43==;5,
+3=77,
3=48
3=;;
Netherlands
3=64
3=34
3=7<
+6=:7,
+3=:5,
Spaind
3=5<
3=4:
3=<7
3=3;
3=64
4=48
3=9<
3=34
3=55
3=5<
Japan
+1980-2004,
+1976-2004,
+1976-2004,
Swedend
+1976-2004,
+4=<:,
+3=;<,
+3=<9,
+43=7,
+3=3:,
+47=;,
+3=78,
+5=9:,
+8=:6,
UK
+6=48,
+3=6<,
3=48
3=;5
US
3=94
3=6<
3=86
+1980-2004,
+1976-2004,
3=7;
+6=8;,
+;=<4,
+6=84,
+;=<;,
1
3=45
+3=8:,
+3=94,
+6=33,
+4=<9,
Wdeoh 5= Hvwlpdwlrq uhvxowv= uhvlgxdov fdofxodwhg iurp uhfxuvlyh uhjuhvvlrqv1
vljqlfdqw dw 43( ohyho1 vljqlfdqw dw
M0vwdwlvwlfv lq sduhqwkhvhv1
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8( ohyho1 orqj0whup lqwhuhvw udwh kdyh ehhq xvhg lq wkh dqdo|vlv1
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+4<:905337 ,
Ghqpdun
+4<:905337 ,
d
Iudqfh
+4<;305337 ,
Jhupdq|
+4<:905337 ,
Luhodqg
+4<;805337 ,
Lwdo|
Fry+|M.4 >lM ,
Y du+lM ,
Fry+CDE
M.4 >lM ,
Y du+lM ,
Fry+CDE
M.4 > M.4 ,
Y du+M.4 ,
Fry+|M.4 >M.4 ,
Y du+M.4 ,
+53 73
95,
3=7<
3=:;
3 47
+3=8;,
+3=57,
+4=95,
+4=56,
3 ;5
3=97
3=93
3 85
=
=
3=86
=
+6=8<,
Qhwkhuodqgv
+4<:905337 ,
5=3;
+73 75
4:,
3=54
3=86
3=84
3=97
+8=43,
3=6:
+53 66
7:,
+5=64,
+4=37,
+53 7<
8<,
3=76
3=88
+4=5:,
=
=
+3=<;,
=
=
Vsdlq
+4<:805337 ,
Vzhghq
d
+4<:905337 ,
XN
+4<;305337 ,
XV
+4<:905337 ,
+4=:4,
+5=5;,
+6=86,
+5=46,
+5=45,
=
=
3=67
3=47
+6=98,
+3=86,
4=94
5=5<
+3=<4,
+3=<4,
+4=7<,
3 :7
3=:;
3=95
3 54
3=8:
4=38
+63 ;4
48,
+4=8;,
+3=95,
+4=3:,
73
+43 <8,
+3=65,
=
+7=78,
+6=33,
3=7:
+9=75,
3=96
4=:;
=
+4=;3,
3=97
+6=;4,
+4=78,
+3=:8,
3=9<
3=<:
3=<9
4=36
+3353,35
4=6:
3=45
3=;<
3 48
5=39
4=94
3=95
3=9<
+63 78
59,
+3367,39
=
3=7:
+4=67,
3=;<
+3=86,
+4=;;,
3=8;
+6=6;,
+4=<6,
+3=:9,
3 67
3 54
3=<8
3 43
3=75
3=73
+4=:7,
+3=9:,
+4=46,
+33<6,64
+3356,67
4=47
3=9<
3 43
4=5:
3=9<
+4=:4,
+4=4<,
3=3<
3=;5
3 56
+3=96,
3=;5
+5=:6,
=
=
4=75
=
d
=
+6=9:,
4=5<
=
Mdsdq
+8=76,
3=;9
3=93
=
+4<;305337 ,
+5=;<,
3=78
=
+3=<<,
+4=3:,
3=:7
=
+4<;305337 ,
3=43
+44=5,
+3=<4,
=
d
+6=56,
+3=<:,
+3=67,
=
=
=
=
3 87
=
+6=76,
=
=
+3=7:,
+5=6<,
+4=:8,
+4:=;,
+8=;3,
+<=86,
+6=;4,
+47=3,
3=55
+8=65,
=
=
+9=46,
=
+4=5:,
+6=7:,
3=3;
3=;9
=
+3=97,
4=86
=
+3=;7,
3=;7
=
+6=65,
+3=55,
+3=7<,
+4=39,
3=97
3=;9
+7=;:,
+3=79,
+73 :6
58,
3=73
3=87
+5=3:,
3=73
+53 67
59,
+3=78,
3=64
+;=66,
+6=68,
+3=<4,
3=55
=
=
3=57
+6=4;,
4
+:=<:,
+6=36,
3=46
=
=
3=85
Table 3 – Optimal Debt Composition for Debt Stabilization
Short Debt
Unconstrained
Constrained
Inflation Fixed-Rate Short Debt Inflation Fixed-Rate
Indexed Long Debt
Indexed Long Debt
Belgium
11
68
21
11
68
21
-12
5
107
0
5
95
-14
28
86
0
28
72
2
77
21
2
77
21
26
45
29
26
45
29
0
100
0
0
100
0
-20
77
43
0
77
23
-32
61
71
0
61
39
-46
86
60
0
86
14
84
68
-52
55
45
0
-33
60
73
0
60
40
-22
24
98
0
24
76
Denmark*
France
Germany
Ireland
Italy*
Japan
Netherlands
Spain*
Sweden*
UK
US
Notes: The debt composition is derived from equation (15) and (16) using the coefficients in
Table 1. The constrained debt composition is computed assuming that debt shares cannot be
negative. A star indicates that the long-term interest rate has been used to estimate the
coefficients in Table 2.