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2012/SFOM10/015
Session: 2
On Debt Sustainability: Debt Projections and Fiscal
Risk
Purpose: Information
Submitted by: World Bank
10th Senior Finance Officials’ Meeting
St. Petersburg, Russia
28-29 June 2012
On Debt Sustainability: Debt Projections and Fiscal Risk.
Juan Pradelli
Economic Policy and Debt Department, World Bank
First Draft. June 21, 2012.
Abstract
We review standard notions and tools for the analysis of debt sustainability, with emphasis in the
probabilistic approach based on econometric forecasting. We develop a norm for assessing sustainability
requiring the net public debt-to-GDP ratio to stabilize or decrease in the long run. Our norm is more
stringent than the standard solvency condition and addresses the accumulation of financial assets by the
sovereign and the return differential between a government’s debts and assets. We illustrate a simple
methodology for assessing sustainability based on the probabilistic approach with a World Bank study on
Malaysia. Fiscal risks emanating from oil dependence are analyzed in a stochastic framework and using
fan charts. The methodology can be extended to other APEC middle-income countries.
Keywords: Debt – Sustainability – VAR
JEL Classification: F31 – F32 – F34
2
Introduction
The concept of public debt sustainability refers to the ability of the government to honor its current and
future financial obligations. Since fiscal and borrowing policies largely determine such obligations, fiscal
sustainability also refers to the ability of the government to maintain sound policies over time without
having to introduce major budgetary adjustments in the future. Conversely, fiscal and borrowing policies
are deemed unsustainable when they lead to excessive accumulation of public debt, which could
eventually cause the government to take action to address the unwanted consequences of a heavy debt
burden.1
Keeping fiscal and borrowing policies on a sustainable path represents a key condition for any
sovereign country to guarantee economic efficiency and inter-generational equity in the allocation of
resources. The accumulation of public debt, if temporary and not excessive, can contribute to fostering
long-term economic growth, achieving fairness across generations in terms of welfare, and stabilizing
business cycle fluctuations. When the accumulation of public debt becomes excessive, on the other hand,
it no longer contributes to achieving policy goals but instead brings about adverse effects on economic
growth and macroeconomic stability.
Preventing the public debt from embarking in an explosive unsustainable trajectory finds a
rationale in the well-documented facts that government debt does affect economic performance through
several real and financial channels, and, more importantly, that these effects tend to be negative
(positive) at high (low) levels of debt. In particular, at low debt levels, government borrowing is an
adequate means for financing growth-enhancing expenditure to build public infrastructure, provide health
and education, and develop a social safety net. As long as the government is able to tax the prospective
growth dividend of such expenditure (thus mobilizing resources from the private economy to the fiscal
budget), it is possible to afford the cost of servicing debt without experiencing destabilizing budgetary and
financial pressures in the future. Government borrowing also contributes to growth and macroeconomic
stability by allowing the use of active fiscal policies to cope with cyclical fluctuations and global shocks
(e.g. automatic stabilizers and ad hoc stimulus). On the other hand, at high debt levels, public
indebtedness tends to hamper economic growth through crowding out effects on private investment
(associated with higher interest rates, debt overhang problems, etc.) and the imposition of heavy tax
burdens that distort incentives to produce, save, and invest.
Several academic and policy-oriented studies report total public debt thresholds laying
somewhere in the range of 80-90 percent of GDP, which would draw the line between growth-enhancing
According to the IMF (2002, p.4-5), a debt is sustainable when “a borrower is expected to be able to continue
servicing its debts without an unrealistically large future correction to the balance of income and expenditure”,
which “does not rule out a situation in which a major correction is needed to adjust to a shock”.
1
3
effects of government debt and growth-hampering ones.2 These studies differ in terms of samples and
statistical techniques used, but nevertheless they often suggest that it makes sense to monitor debt
developments using thresholds (i.e., norms) that help indicating broadly which levels of debt are safe or
risky in terms of their effects on economic growth. In addition, the studies report little correlation (if any)
between growth and debt for low-debt countries, thus implying that, provided the thresholds are not
breached and adequate institutions are in place, debt financing can safely be used to expand investment
and boost economic growth.
A forward-looking fiscal sustainability analysis gravitates around a public debt indicator (e.g. a
country’s stock of government liabilities or flow of financing needs, scaled by a measure of repayment
capacity such as GDP or revenues) and basically compares a threshold or norm for that indicator with
projected paths over a medium- or long-term horizon. Whenever the projected paths violate the norm, the
public debt (and/or its underlying revenue and spending policies) is deemed unsustainable and the
sovereign is expected either to experience debt servicing difficulties in the future (which in turn might
trigger from an abrupt budgetary adjustment to an outright default) or to preemptively adopt policies to
correct financial imbalances. There are then two related notions of sustainable levels of public debt: a
maximum sustainable level of the public debt indicator beyond which a crisis episode is highly likely to
occur (i.e., the norm), and the long-run level at which the indicator converges (i.e., the projection)
provided that it does not rise above the maximum sustainable level (Ghosh et al. (2011)). Different
approaches to sustainability analysis derive the public debt norm from theoretical arguments or empirical
observations, often suggesting the norm is country-specific and depends on relevant financial,
macroeconomic, and institutional variables. Empirical approaches also formulate methodologies for
projecting public debt paths.
This paper reviews relevant concepts involved in the analysis of public debt (or fiscal)
sustainability. It also develops two refinements to standard tools for assessing fiscal sustainability. First,
while standard tools do not treat properly the accumulation of assets (e.g. international reserves, anticyclical funds) and the spread between the interest rates paid on government debts and the returns of
these assets, in the paper we introduce assets and return spread into the analysis of public debt
dynamics. A financial burden thus arises influencing both the intertemporal budget constraint and the ad
hoc restrictions on indebtedness ratios that are often used to assess sustainability. 3 Second, while the
2
Reinhart and Rogoff (2010) analyze 44 industrialized and developing economies over two centuries and find that
the GDP growth rate for countries whose debt exceeds 90 percent of GDP is lower than that for low-debt countries.
Kumar and Woo (2010) focus on 38 industrialized and emerging economies in a more recent period (1970-2007)
and also conclude the debt threshold is around 90 percent of GDP. Caner, Grennes, and Koehler-Geib (2010) find a
lower threshold at 77 percent of GDP based on the period 1980-2008.
3
We propose a long-term solvency condition that gives due consideration to the return spread and requires the net
public debt-to-GDP ratio to either stabilize or decline in the long run. In the traditional approach to sustainability
analysis, instead, studies assume a common rate of return on government liabilities and assets, and rely on a
solvency condition that allows the net public debt-to-GDP ratio to increase steadily in the long run provided that it
does not grow faster than the difference between the real interest rate and the growth rate of real GDP [e.g. IMF
(2002), IMF (2003), Burnside (2004)]. The traditional long-term solvency condition underlying the intertemporal
4
standard tools treat many macroeconomic and fiscal variables as exogenous, efforts have been made to
give due consideration to the effects on debt sustainability of the dynamic interactions between key
variables (as captured by correlations and feedbacks). The paper sketches a methodology based in
vector autoregression (VAR) models to estimate dynamic interactions and to construct consistent
scenarios for assessing public debt sustainability and fiscal risk. The methodology is illustrated with a
debt sustainability analysis for Malaysia. 4
The paper is divided into two sections. Section 1 discusses basic notions in the analysis of debt
sustainability and reviews studies using a probabilistic approach. Refinements to public debt sustainability
assessment are introduced in Section 2 and illustrated with a stochastic debt sustainability analysis for
Malaysia in Section 3. An Annex presents the European Union’s methodology for assessing long-term
fiscal sustainability, which emphasizes the effect of population aging on growth, public expenditure, and
debt.
1. Standard Tools to Assess Debt Sustainability
A debt sustainability analysis requires choosing a framework to operationalize the concept of sustainability. The
dynamics of the public debt is a framework widely used because of two practical advantages: firstly, debt is a
notion easy to understand and interpret in analyzing current and future policies; secondly, debt statistics are
constructed according to methodologies agreed at international level and reported on a regular basis, thus
facilitating cross-country comparisons and institutional transparency. The stock of public debt is the result of past
borrowings through which the government has financed budgetary deficits and conducted special fiscal
interventions [e.g., debt restructuring, financial bailouts, recapitalizations]. Future deficits and interventions would
then drive the public debt dynamics going forward. Since the public debt stock is the financial outcome of revenue
and spending policies, it is reflective of the two key elements involved in the concept of sustainability: financial
obligations and fiscal policies. For this reason, the public debt dynamics turns out to be a suitable framework for
assessing sustainability.
Debt dynamics
The analysis of public debt dynamics addresses the issue of how and why the debt stock evolves over
time. The debt stock changes during a given period of time as long as there is an imbalance between
budget constraint permits a country to run unrealistically-large primary deficits (relative to GDP) for a protracted
period of time before offsetting surpluses are achieved, and in addition, it sets no upper bound for the net public debt
ratio (Roubini, 2002). But such a permissive dynamics of sovereign indebtedness is at odds with the profuse
empirical evidence on abrupt reversals in the financing to sovereigns and default episodes triggered by loss of
market access and high levels of debt. Thus, the more stringent solvency condition we postulate, in which the net
public debt cannot grow forever, is worth exploring..
5
expenditure and revenues. Thus, if expenditure exceeds revenues, the government has to borrow to finance the
difference and thus the public debt stock increases; instead, if revenues exceed expenditure, the government has
resources that could be used to partly redeem the outstanding debt and thus the public debt stock decreases.
The so-called dynamic government budget constraint formalizes this accounting principle and states that the
change in the public debt stock in year t is
(1.1)
Dt  Dt 1  i Dt 1  Tt  Gt 
where Dt denotes the public debt stock at the end of year t, Tt is total revenues, Gt is the primary expenditure,
the interest bill is assumed to depend on the inherited debt stock Dt-1 and an average nominal interest rate i. The
right side of the equation is the overall deficit, i.e. the difference between total expenditure and total revenues,
with the former disaggregated into primary expenditure and the interest bill. The primary balance is Tt - Gt .5
The stock of public debt is not an important variable per se in analyzing fiscal sustainability because its
relevance has to be assessed in relation to the repayment capacity of the government, often captured by fiscal
revenues or the GDP as a summary of tax bases. The public debt-to-GDP ratio is widely used in practice as a
measure of debt burden. The change in public debt-to-GDP ratio in year t is
(1.2)
dt  dt 1  r dt 1  tt  gt 
where dt denotes the ratio between the public debt stock Dt at the end of year t and the nominal GDP Yt , and tt
and gt are the revenue-to-GDP and primary expenditure-to-GDP ratio, respectively. The rate r is defined as r = (i
- y)/(1+y), where the numerator is the difference between the nominal interest rate i and the growth rate of
nominal GDP y, which is often referred to as the interest rate-growth differential (or growth-adjusted interest
rate).6
From the point of view of long-term fiscal sustainability of public finances, it is also important to consider
the government’s financial assets. An overall surplus makes resources available that can be used either to repay
debt or to accumulate financial assets, and in both cases fiscal sustainability improves. An overall deficit, on the
4
A revised version of this paper will report results of ongoing work aimed at estimating a VAR model involving
return spread, public debt accumulation, output growth, and government revenues. This VAR can be applied to
APEC middle-income economies.
5
A meaningful way to account for the change in the public debt stock focus on the primary balance and the interest
bill: (i) the debt stock increases when the primary balance is either a deficit or a surplus smaller than the interest bill,
thus producing an overall deficit to be financed through new borrowings from the financial market; (ii) the debt
stock remains constant when the primary balance is a surplus equal to the interest bill, and hence no borrowing is
needed; and (iii) the debt stock decreases when the primary balance is a surplus larger than the interest bill and thus
budgetary resources can partly redeem the outstanding debt. In strict terms, the government in case (ii) does not
need to engage in net borrowing to finance an imbalance between revenues and expenditures; however, it is likely
that it borrows to rollover principal coming due.
6
Similarly to the previous footnote, a meaningful way to account for the change in the public debt-to-GDP ratio focus on the
primary balance-to-GDP ratio (tt - gt ) and the interest bill adjusted by the nominal GDP growth (r dt-1, to which we refer to as
the growth-adjusted interest bill). It should be noted that when the primary surplus as a percentage of GDP equals the growthadjusted interest bill, the government is actually borrowing, but since the public debt stock and the nominal GDP grow at the
same rate, the ratio between both remains constant. In contrast, when the primary surplus in nominal terms equals the interest
bill, the government does not have to borrow, the debt stock remains unchanged, and the debt-to-GDP ratio decreases due to
the nominal GDP growth.
6
other hand, can be financed with debt or though the depletion of financial assets. The notion of net debt then
allows take into account the effect on fiscal sustainability of investing or disinvesting in financial assets. For the
sake of simplicity, the analysis here disregards revenues resulting from money printing (seigniorage) and stockflow adjustments.
Sustainability indicators
The long-term sustainability of fiscal and borrowing policies means the accumulation of public debt is not
excessive, and thus is equivalent to saying that the public debt is not growing too fast or that the level of public
debt is not too high. Several definitions of sustainability therefore focus on either the rate of debt growth over a
given time horizon or the level of debt at the end of such horizon. These definitions are often complemented with
well-defined formal conditions that apply to the variables involved in the public debt-to-GDP ratio dynamics. When
a projected debt dynamics does not satisfy a formal sustainability condition, the fiscal adjustment needed to
restore sustainability is also a useful indicator.
The government’s intertemporal budget constraint (IBC), a widely used definition of solvency, asserts
that the public debt is sustainable as long as it does not grow explosively over an infinite time horizon.7 A fiscal
policy preventing an explosive growth of the debt-to-GDP ratio satisfies the IBC:8


(1.3)
s t
ts
1  r 

s 1t

s t
gs
1  r 
s 1t
 dt 1
In equation (1.3), the present discounted value (PDV) of the revenue-to-GDP ratio (left-hand side) must be
equal to the PDV of the primary expenditure-to-GDP ratio plus the inherited public debt stock (right-hand side).
The PDV is computed over an infinite time horizon with using the rate r discussed above. It is therefore necessary
that current and future revenues cover current and future primary expenditure plus the inherited public debt stock.
If such an intertemporal balance is achieved, the debt-to-GDP ratio does not grow too fast.
Equation (1.3) can be computed using projections for revenues (ts), the primary expenditure (gs), the
interest rate (i), and the growth rate of nominal GDP (y) in order to determine whether the IBC is satisfied or not.
Failure to comply with the IBC implies an intertemporal imbalance where the future borrowings needed to finance
projected budget deficits fuel an excessively-rapid growth of the public debt over time. The intertemporal budget
gap (IBG) measures such imbalance as the difference between the debt stock and the PDV of projected primary
balances,
7
One example of an unsustainable debt dynamics is the case in which a government systematically has a balanced
primary budget, and therefore must always borrow from the financial markets in order to pay for the growthadjusted interest bill. A sustainable debt dynamics, instead, refers to the case in which a government runs budgetary
imbalances but primary surpluses tend to prevail over time and consequently resources are made available to pay for
the growth-adjusted interest bill (partially or entirely) and perhaps even to reduce the stock of outstanding debt.
8
In the remaining part of the chapter, it is assumed that the nominal interest rate i, the nominal GDP growth rate y,
and the rate r remain constant over time. In addition, it is assumed that i exceeds y.
7

IBG  dt 1  
s t
(1.4)
pbs
1  r 
s 1t
where pbs denotes the primary balance-to-GDP ratio in year s and by definition pbs=ts - gs . Fiscal policy aimed at
closing the gap must increase revenues ts or reduce the primary expenditure gs (i.e. budget consolidation), or
reduce the inherited public debt stock dt-1 (i.e. debt renegotiation), or a combined actions. Over the long term,
however, it is not only fiscal policies that help in achieving sustainability but also other policies that can raise the
growth rate of GDP, e.g. structural reforms.9
An alternative definition of sustainability is based on a value of the debt-to-GDP ratio that is deemed
acceptable; such a value is referred to as a debt target at a certain future date. If the target is achieved at the end
of a finite time horizon, public debt is sustainable. Given the initial debt-to-GDP ratio dt-1 and a sequence of
primary balances between t and t+h, the debt-to-GDP ratio dt+h is
dt  h  1  r 
h 1
(1.5)
t h
dt 1   1  r 
s t
t hs
 ts  g s 
Consider a government that wants to attain a debt-to-GDP ratio dt+h in year t+h equal to the debt-toGDP ratio dt-1 inherited in year t. The fiscal policies implemented between t and t+h must ensure that the debt-toGDP ratio remains unchanged over the time horizon. The government could run some primary deficits between t
and t+h but eventually primary surpluses must prevail to counteract the snowball effect induced by interests and
the primary deficits (if any). As a special case, if the government wants to attain a debt-to-GDP ratio dt in year t
equal to the inherited debt-to-GDP ratio dt-1, it needs to run a primary surplus equal to the growth-adjusted
interest bill, i.e. pb* = r dt-1 ; this is the so-called debt-stabilizing primary balance pb*, usually computed by the
IMF and the World Bank.
Equation (1.5) can be computed using projections for ts, gs, i, and y in order to determine whether the debt-toGDP ratio hits the target value or not. If the debt target is missed, there is an imbalance within the finite projection
horizon since the future borrowings needed to finance projected budget deficits lead to an excessive level of
public debt in t+h. If instead the target is hit, the debt-to-GDP ratio does not reach a too high level.
9
In assessing fiscal sustainability on the basis of the IBC, three issues should be taken into account. First, it is
necessary to make assumptions regarding future values of the variables beyond the projection horizon. Even though
the constraint and the IBG refer to an infinite time horizon, the long-term projections of revenues, primary spending,
interest rates, and growth rates of nominal GDP typically cover a finite number of years. Second, the IBC imposes
only a mild restriction on public finances because a supposedly sound fiscal policy is allowed to fuel a rapid
indebtedness process as long as large primary surpluses are expected in the distant future. For this reason, the IBG is
a rough measure of the fiscal adjustment needed to restore sustainability the IBG provides no guideline on how and
when the adjustment should be made because the same value of IBG is compatible with different combinations of
changes in revenues and spending and the adjustment could be made in the short term or in the distant future. Third,
satisfying the IBC is compatible with the worsening of economic conditions and places excessive emphasis on the
public debt dynamics over an infinite time horizon, without considering the negative effects on the economy of
persistently high levels of public debt. For instance, in a case where the debt stock is too high, even a low growth
rate of debt implies that the government is borrowing a large amount of resources from the financial markets,
potentially crowding out private investment, and using resources to pay the interest bill rather than for financing
expenditures or tax cuts.
8
The fiscal gap (FG) measures the permanent adjustment in the primary balance-to-GDP ratio between t
and t+h that must be undertaken for dt+h to attain the target value,
FG  rdt 1 
(1.6)
r  dt 1  d
1  r 
h 1
*
t h

1
t h
r  1  r 
s t
1  r 
t hs
h 1
pbs
1
where d*t+h denotes the debt target value to be attained in t+h. Fiscal policy aimed at closing the gap between
projected debt and target must increase revenues, reduce primary expenditure, reduce the inherited public debt
stock, or undertake a combined action.10
Norms to assess sustainability
Different approaches to fiscal sustainability analysis derive a norm for a public debt indicator from
theoretical arguments. A main example is precisely the long-term solvency condition (or transversality condition)
whereby the debt cannot grow explosively in the long run. The IBC, obtained precisely by combining this norm
with the dynamic government budget constraint, allows calculate an upper-bound limit for the initial public debt
indicator, conditional upon projected primary balances, interest rates, and GDP growth rates. Such a limit is a
simple norm and can be compared to the actual indicator to assess fiscal sustainability (Burnside, 2004). The IBC
is indeed fulfilled when two conditions are met: (i) the sovereign is solvent, with its expected cash flows balanced
over an infinite time horizon [alternatively, the face value of government liabilities does not exceed the present
discounted value of the expected primary surpluses]; and (ii) the sovereign is liquid, with its period-by-period
temporary cash-flow mismatches expected to be financed by local and foreign creditors, under contractual
conditions compatible with solvency [alternatively, the creditors are willing to finance primary deficits and debt
servicing if and when it is necessary]. Whereas liquidity drives the public debt dynamics because the stock of
government liabilities increases when a sovereign borrows, solvency limits indebtedness because the
intertemporal balance of cash flows cannot be achieved if the public debt dynamics is explosive.
A rather different type of norms are derived from empirical observations and reflect the actual level of
indebtedness exhibited by sovereigns that did undergo debt servicing problems in historical data (e.g., rollover
crisis, forced debt restructuring, debt default) compared to a control country group who did not. Early-warning
models identify debt distress episodes and use the signal approach to derive a benchmark for a public debt
10
In assessing fiscal sustainability on the basis of the debt target, three issues should be taken into account. First, as the
debt target is to be reached at the end of a finite time horizon, the long-term projections of revenues, primary spending, interest
rates, and growth rates of nominal GDP can be used to determine the future evolution of the debt-to-GDP ratio and the FG,
without having to make assumptions about the behaviour of these variables beyond the projection horizon. Second, attaining a
debt target at a certain future date imposes a strong restriction on the government in terms of primary balances. Thus, the debt
target reduces the possibility that running large primary surpluses in the distant future will be enough to make the current fiscal
policies sustainable, regardless of an indebtedness process already in motion. On the other hand, the debt target entails another
problem: since it does not impose any constraint on the debt dynamics beyond the projection horizon, the debt-to-GDP ratio
might explode after year t+h or the debt-to-GDP ratio might stabilise around a level above the target.
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indicator by minimizing the noise-to-signal ratio (Hemming et al., 2003). Binary response models investigate the
determinants of debt distress episodes and estimate their probability of occurrence conditional upon debt
indicators and other explanatory variables [e.g., IMF-IDA (2004), IMF-IDA (2012)]. In addition, given any arbitrary
probability of experiencing debt problems, the models permit calibrate public debt thresholds that can be
compared to the actual indicators to assess sustainability. Barring structural changes, the debt thresholds
estimated using historical data can be used in forward-looking fiscal sustainability analysis as well [as in IMF-IDA
(2004) and IMF-IDA (2012)].
In the intersection of theoretical and empirical approaches for determining public debt norms and
assessing sustainability lie econometric tests that confront the model-based long-term solvency condition with a
bubble-generating process for the stock of public debt [Hamilton and Flavin (1986), Wilcox (1989)]. The tests
introduce simplifying assumptions with regard to real interest rates and examine the stationarity of primary
balance and public debt series, individually or in a joint model. Provided that no unit roots are found in the series,
or that the income-expenditure patterns underlying the primary balance do react to the accumulation of
government liabilities, one concludes that fiscal sustainability was achieved in the historical data. Bohn (1998)
shows that if a policy reaction function adequately adjusts the primary balance to changes in the public debt-toGDP ratio, the government’s long-run solvency condition would be met. Thus, the author proposes to test
whether such a reaction is supported by empirical evidence in historical series as a means for assessing public
debt sustainability. This approach is subject to standard criticism directed towards unit-root testing and cannot be
used to assess public debt sustainability going forward.
Another model-based approach aimed at empirical implementation is the natural public debt limit
proposed by Mendoza and Oviedo (2007). The authors calculate the maximum level of debt that the government
would be able to service with absolute certainty given its capacity to generate fiscal savings and the historical
volatility of revenue and expenditure. The natural debt limit is equal to the annuity value of the primary balance in
the event of a fiscal crisis, which in turn constitutes a persistent sequence of negative shocks to revenue where
public expenditure adjusts to a tolerable minimum. The government would always find any debt stock below the
natural debt limit to be affordable, even in the worst-case scenario concerning its fiscal position, and thus the
probability of default is zero. The natural debt limit can be compared with the current debt level to assess debt
sustainability, and if the later exceeds the former, the government faces a non-zero probability of default on
sovereign debt.
Projections of public debt in sustainability analysis
The dynamic government budget constraint describing the evolution of the public debt is the
common denominator underlying all methodologies for projecting paths of government liabilities.
Traditional studies rely on steady state or long-term deterministic paths for the forcing variables driving
public indebtedness [i.e., primary balance, real interest rate, and real GDP growth], often treating them as
exogenous [e.g., Simonsen et al. (1985), IMF (2002), Goldstein (2003)]. These paths are plugged into the
dynamic government budget constraint and thus a baseline projection of public debt is obtained. The
10
baseline outlook represents the most likely outcome [loosely defined because there is no probability
attached to it] and is complemented with alternative deterministic scenarios constructed to address the
uncertainty surrounding the central projections.
More recent studies develop a probabilistic approach that acknowledges the joint endogeneity
among the variables involved in the public debt dynamics and resorts to stochastic simulations to better
treat projection uncertainty and model unforeseen macroeconomic shocks hitting the economy. Several
studies produce medium-term forecasts of the forcing variables driving public indebtedness by estimating
VAR models that uncover the pattern of correlations and feedbacks between these variables in the
historical data [e.g. IMF (2003), Hevia (2012)]. Data availability often constrains researchers to investigate
emerging market, e.g., IMF (2003) analyzes 41 emerging markets and estimates country-specific VAR
models using annual data in 1980-2000. Hevia (2012) is a notable exception as the author examines 76
low-income countries and specifies panel VAR models to deal with data scarcity and pool parameter
estimates across countries, using annual observations in 1971-2007. In these papers, the central forecast
feeds into the baseline projection of public debt in a straightforward manner.
Noting that VAR-based forecasts are driven by past macroeconomic co-movements and might fail
to capture structural breaks going forward, some studies propose to average out forecasts coming from
different models and introduce an analyst’s expert judgment on future events not observed in the past
[e.g., policy adjustments, capital account liberalization, structural reforms] [e.g., Clemen and Winkler
(1999), Osterholm (2006)]. In the context of public debt sustainability, for instance, Ariala et al. (2008)
combine VAR-based forecasts with the IMF’s World Economic Outlook projections to improve predictive
performance.
Stochastic simulations permit to characterize the probability distribution of the public debt
indicator and to analyze risks to fiscal sustainability with confidence intervals and fan charts. IMF (2003)
and Hevia (2012) utilize stochastic simulations to ascertain the likelihood of occurrence of debt paths
associated with deterministic scenarios constructed with ad hoc calibration of shocks’ magnitude and
persistence. Along with the central forecast, the authors generate thousands of alternative debt paths
through Monte Carlo simulations, in which shocks are either drawn from a Gaussian distribution with the
estimated VAR residuals’ covariance matrix [IMF (2003)] or bootstrapped from the estimated VAR
residuals themselves [Hevia (2012)]. While drawing shocks from a standard probability distribution is a
rather simple procedure and ensures well-known stochastic properties, bootstrapping shocks better
captures the fat tails in residuals associated with low-probability large-size disturbances that are
commonly observed in developing countries [e.g., financial, currency, and debt crises].
The probabilistic approach has been largely experimented in analyses of public debt
sustainability. For instance, IMF (2011) estimates country-specific unconstrained VAR models with annual
observations in 1995-2010, and run stochastic simulations to evaluate the likelihood of occurrence of
public debt-to-GDP paths associated with partial-equilibrium shocks to macroeconomic variables. Garcia
and Rigobon (2004) specify a VAR model for Brazil with monthly data in 1995-2002, and use stochastic
11
simulations to estimate probabilities of public debt-to-GDP exceeding a 75 percent threshold over a fixed
ten-year horizon. Interestingly, these probabilities, which are seen as model-based risk measures, move
in sync with the Brazil EMBI+ sovereign bond spread, which is a proxy of financial markets’ assessment
of default risk. Celasun et al. (2007) consider country-specific VAR models for Argentina, Brazil, Mexico,
South Africa, and Turkey, using quarterly data, and enrich the stochastic simulations by introducing
estimated fiscal policy rules in which the primary balance reacts to cyclical conditions and debt dynamics,
proxy by output gap and public debt-to-GDP ratio, respectively [Bohn (1998) emphasizes that a policy
reaction function plays a major role in ensuring public debt sustainability and intertemporal solvency]. 11
Tanner and Samake (2006), Penalver and Thwaites (2006), and Hostland and Karam (2005, 2006)
pursue a similar avenue. Frank and Ley (2009) extend the work of Celasun et al. (2007) by introducing
structural breaks in the VAR models (identified with Markov-switching methods) and fat tails in the
residuals, using quarterly data for Argentina, Brazil, and South Africa. Running stochastic simulations
over crisis and non-crisis periods, the authors find that confidence intervals are significantly wider for the
VAR model estimated over crisis periods. In addition, as the estimated residuals are sometimes skewed,
the authors argue that sampling from a Gaussian distribution underestimates the probability of large
shocks and thus narrows the estimated confidence intervals around a central forecast. A number of
studies impose more economic structure on the empirical models, e.g., Mendoza and Oviedo (2006)
apply stochastic simulation methods to a stylized dynamic general equilibrium model and Hostland and
Karam (2005, 2006) use a structural model calibrated to a generic emerging-market economy. These
models provide a sound rationale for feedback mechanisms such as fiscal policy rules and fiscal
multipliers, which ultimately constrain the accumulation of public debt in response to adverse shocks and
thus narrow the estimated confidence intervals around a central forecast.
2. Refining Tools to Assess Debt Sustainability
The public debt sustainability models reviewed in Section 1 focus on the sovereign’s financing
needs associated with the budget deficit and maturing debts, but often miss the accumulation of financial
assets (or its opposite movement: the depletion of assets) and the return spread between government
debts and assets.12 By leaving the government’s financial assets out of the picture, a public debt
sustainability model underestimates the cash flow mismatch that the gross borrowing should cover as well
as the protection provided by asset holdings against rollover risk. Even when the accumulation of assets
is fully debt-financed and thus the net debt does not increase, the return spread implies higher net
11
The IMF Fiscal Monitor (2010) reports fan charts for public debt of four advanced economies (Germany, Greece,
UK, and US) based on the methodology developed by Celasun et al. (2007).
12
Developing countries’ interest rates on government’s foreign liabilities are higher than returns on external assets (e.g.,
reserves and sovereign funds) and the spreads are explained by global and domestic factors [Damill and Kampel (1999),
González Rozada and Levy Yeyati (2005)].
12
interest payments in the future and a reduction in the sovereign’s wealth in present-value terms.13 In the
remainder of this section, we explore the implications on the IBC of incorporating the government’s
financial assets and the return spread into the net public debt dynamics. We also formulate a norm for the
net public debt-to-GDP ratio that can be used to assess fiscal sustainability.
Net public debt and the intertemporal budget constraint, revisited
Our analysis starts identifying main sources and uses of funds for the sovereign. We assume the
Dt pays an interest rate rt , the stock of financial assets At yields a rate of return it ,
public debt stock
and the return spread is
t
income received from assets
=
rt - it .14 In period t, the sources of funds are government revenues Tt ,
it At , principal amortizations of debts owed to the sovereign AmtA , and new
D
(gross) issuances of government liabilities Ft . The uses of funds, on the other hand, are primary
expenditure Gt , interest payments on public debt
rt Dt , principal amortizations of public debt AmtD , and
A
new (gross) purchases of financial assets Ft . An accounting identity states that sources and uses of
funds must be identical ex post:
(2.1)
where magnitudes are nominal and expressed in local currency.
The sovereign borrows funds Ft
D
to cover financing needs associated with the budget
D
, the maturing liabilities Amt , and the new (net) purchases of financial
deficit
assets Ft  Amt . For simplicity, we consider continuous time and denote derivatives with an upper dot.
A
The dynamics of
A
Dt and At reflect the new (net) issuances of government liabilities and the new (net)
purchases of financial assets, respectively:
(2.2)
and
,
13
Consider a sovereign whose fiscal accounts are balanced so the net public debt remains unchanged. Under these
circumstances, cash inflows and outflows offset each other. Because of the return spread, however, the PDV of
future debt services paid to investors in government debt is higher than that received from the government’s asset
holdings. Therefore, the public net wealth decreases and the fiscal sustainability deteriorates. The discount rate used
to compute the present value of public-sector net wealth should be the interest rate charged on public debts, which is
the opportunity cost of allocating resources to purchase financial assets rather than buyback government debt.
14
International reserves and sovereign wealth funds would be included in At . For simplicity, the current rates rt and
it apply to existing stocks and thus average and marginal return rates are equal. We exclude money printing
(seigniorage). Magnitudes are nominal and expressed in local currency.
13
whereas the dynamics of the net public debt
Dt  At reflect the budget deficit:
(2.3)
.
Given an initial value
 D0  A0  , the path of the net public debt is:
(2.4)
.
The long-run solvency (or transversality) condition requires the present value of net public debt to
be non-positive:
(2.5)
.
The intertemporal budget constraint (IBC) obtained from (2.4) and (2.5) is
(2.6)
.
According to (2.6), a sovereign is solvent if the face value of the net public debt outstanding does
not exceed the present value of the primary surpluses adjusted downwards by the present value of terms
 t At . In the right-hand side of (2.6), the present
value of terms
 t At
represents the adverse effect on
the sovereign’s wealth of assets accumulation-cum-return spread. It depends on the stock of assets and
thus is potentially large if the sovereign invests heavily. Admittedly the stock of assets in the left-hand
side of (2.6) improves fiscal sustainability because a higher level of government debt would be
sustainable when the sovereign also holds assets. But on the other hand, the return spread between
government debts and assets deteriorates fiscal sustainability because a lower level of government debt
would be sustainable when a return spread exists.
A norm for the net public debt-to-GDP ratio
We now consider a norm for the net public debt-to-GDP ratio that requires the ratio to stabilize or
decrease in the long run.15 This is indeed a more stringent requirement compared to the solvency
condition (2.5) (adapted to the public debt ratio) because solvency allows the ratio to increase in the long
run provided that its growth rate is below the interest rate-growth differential.
15
Restrictions in the same vein are elaborated by Simonsen et al. (1985) and Frenkel (2004).
14
Rt   Dt  At  / Yt , evolves according to
The net public debt-to-GDP ratio, denoted
(2.7)
.
For the sake of analytical tractability, we assume a steady state path for the forcing variables in
(2.7), with constant values for return rates (denoted r and i), spread (σ), and logarithmic growth rates of
GDP (y), revenues (rev), primary expenditure (pe), and public debt (d).16 Thus, given an initial value
R0 ,
the path of the net public debt-to-GDP ratio is:17
(2.8)
The net public debt-to-GDP ratio stabilizes or decreases in the long run if and only if there is a
maximum value of
periods t >
Rt at a certain period T * and stable or declining trajectory of Rt for all subsequent
T * . Formally, the period T* when the ratio reaches a maximum exists provided that three
conditions are satisfied:
dRt / dt t T  0
*
2
d Rt / dt
,
2
t T
*
0
, and
dRt / dt
t T
*
0
. Taking derivatives in
(2.8), these three conditions are:
(2.9.a)
(2.9.b)
(2.9.c)
The existence and magnitude of
T * depend on the return rates, spread, growth rates, and initial
conditions found in the expression (2.9). Three cases can arise. First, if
T * exists and is a positive value,
the debt ratio is on an increasing path but will cease to grow at some future period and hence fiscal
sustainability will be achieved sooner or later [the value of
stabilize or start decreasing]. Second, if
T * indicates how long it will take for the ratio to
T * exists and is a negative value, the debt ratio is on a
decreasing path and hence fiscal sustainability has been already achieved. Third, if
T * does not exist,
then the debt ratio is on an increasing path and will never cease to grow, so fiscal sustainability cannot be

16
For a variable Z t , the time path is given by Z t  Z 0 e and z  Z t / Z t denotes the rate of change.
17
Expression (2.8) holds provided that i differs from y, rev, and pe.
zt
15
attained. Following the standard practice, we can compute the difference between the current primary
balance and the balance that would ensure achieving fiscal sustainability immediately. Thus, if we find a
case where
T * does not exist, we set T * = 0 in (2.9.a), solve for the primary surplus, and check that
(2.9.b) and (2.9.c) hold. It then turns out that the primary surplus required to immediately stabilize the net
public debt-to-GDP ratio is
(2.10)
Equations (2.9) and (2.10) can be used to assess public debt sustainability. One has to determine
the existence and magnitude of
T * based on initial conditions and long-term projections of return rates,
spread, and growth rates. VAR models provide estimates of long-run steady state values of some of
these variables, as discussed below.
VAR models and the norm for the net public debt-to-GDP ratio18
Following the probability approach reviewed in Section 1, we explore using VAR models to
identify relationships between variables determining the net public debt dynamics and to produce a
central forecast that can be confronted with the conditions (2.9) to assess sustainability. The VAR
specification assumes the return spread (σt) to be exogenous and the growth rate of government debt
(dt), real output (ρt), and real revenues (μt) to be endogenous. A possible specification is 19
(11a)
3
2
l 1
l 0
Z t  C   Vl Z t l   Bl t l   t
where
Zt   dt
t
Bl   b1,l
b2,l
t 
C   c1 c2
T
b3,l 
T
Vl  vnj ,l 
c3 
T
 t  1t
2t 3t 
T
j , n 1,2,3
*
The VAR estimated coefficients allow to compute long-run steady state values d ,
for alternative values of
 * , and  *
 * : 20
1
(11b)
3

 
 2
 
Z   I 3   Vl OLS   C OLS    BlOLS   * 
l 1

 
 l 0
 
*
18
A revised version of this paper will report results of ongoing work aimed at estimating a VAR model involving
return spread, public debt accumulation, output growth, and government revenues. VAR estimates would help
computing T* and thus assessing fiscal sustainability. This VAR can be applied to APEC middle-income economies.
19
 t is a vector of random errors with standard properties.
20
OLS denotes a matrix of estimated coefficients, and a star denotes a long-run value.
16
We expect to find a long-run negative relationship between the return spread and the growth rate
of government debt, real output, and real revenues. If so, depending on whether the effect is stronger in
the numerator or in the denominator, a higher steady-state return spread could accelerate or decelerate
the growth of the net public debt-to-GDP ratio (ceteris paribus the exogenous projection of real primary
expenditure, the return on government assets, and the relevant deflators). Possible trajectories of the
Rt with t between 0 and 15 and  * set equal to
debt ratio are illustrated below, with three dynamics for
4%, 7%, and 10%.
5
4
3
2
Rt
1
0
-1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-2
-3
-4
-5
spread = 4%
spread = 7%
spread = 10%
With the estimated long-run steady state values we can also compute
T * for different
magnitudes of the exogenous variable and thus evaluate the sensitivity of the public debt sustainability
assessment to deviations from the central forecast.
3. Debt Sustainability Analysis (DSA): the case of Malaysia
As discussed in Section 1, the stock of public debt (measured as a share of GDP) depends on
four key variables: the public debt stock inherited from the past (which results from past borrowing
choices), the primary balance (that reflects the current fiscal policies and institutions concerning taxation
and spending patterns), the cost of borrowing (represented by the average interest rate charged on the
government liabilities) and the GDP growth rate (as a proxy of repayment capacity). A basic DSA model
postulates a debt dynamics equation to determine the public debt-to-GDP ratio:
(3.1)
17
where Dt denotes the public debt-to-GDP ratio in year t, PDt is the primary deficit as a share of GDP, it is
the average interest rate paid on public debt (determining the interest costs of carrying debt), and Ŷt is
the growth rate of nominal GDP.
The macroeconomic and fiscal variables driving the dynamics of public debt are neither isolated
nor determined independently. On the contrary, they depend on each other through several interactions
and feedback effects reflected in the observed co-movements among time series. For instance, in
Malaysia, output growth depends on oil prices and real interest rates, and government revenues depend
on the level of output and oil prices. A more realistic DSA model would enrich the debt dynamics equation
(3.1) by introducing a set of behavioral hypotheses (expressed as functional forms) that do capture
interactions and feedbacks as well as the specificities of the Malaysian economy. We then postulate the
following debt dynamic equation:
(3.2)
where relevant improvements are introduced: (i) the public debt-to-GDP ratio Dt is disaggregated into
domestic debt Dd,t and foreign debt Df,t (converted into local currency using the nominal exchange rate
Et); (ii) the primary deficit-to-GDP ratio is broken down into oil and non-oil components; (iii) the oil-related
revenue ORRt and oil-related primary expenditure ORPEt explicitly depend on the Tapis oil price POt;
(iv) the non-oil-related revenue NORRt and non-oil-related primary expenditure NORPEt explicitly depend
on the nominal GDP Yt as a proxy for the relevant tax bases and the scale of expenditure programs; and
(v) the interest expenditure reflects the cost of carrying both domestic and foreign debt, with nominal
interest rates denoted id,t and if,t; (vi) the valuation effect of currency depreciation Êt on the foreign debt
converted into local currency is explicitly added to the cost of carrying foreign debt
Four simple decompositions can be added to treat real and nominal variables:
(3.3.a)
where the growth rate of nominal GDP Ŷt is broken down into
the growth rate of real GDP ŷt and the inflation rate measured by GDP deflator pt.
(3.3.b)
where the growth rate of real GDP ŷt depends on a weighted
average of growth rates in oil and non-oil output, denoted ŷo,t and ŷno.t , and the weight αoil is fixed for
simplicity.
(3.3.c)
where the domestic nominal interest rate id,t on domestic
debt is broken down into the real interest rate rd,t and the inflation rate measured by GDP deflator pt.
18
(3.3.d)
where the currency depreciation Êt (i.e. the relative
change in the nominal exchange rate) is decomposed into the relative change in the real exchange rate
ȇt, the domestic inflation rate, and the foreign inflation rate.
In the DSA model for Malaysia, eight macroeconomic variables influence the fiscal outcomes: the
oil price POt, the growth rates of real oil GDP and non-oil GDP (ŷo,t and ŷno.t), the relative variation in
the real exchange rate ȇ, the domestic real interest rate rd,t, the foreign nominal interest rate if,t, and the
domestic and foreign inflation rates (pt and pf,t). Given projections for these eight macroeconomic
variables, the parameters and functions linking macroeconomic and fiscal variables (e.g. the functions in
ORRt, ORPEt, POt, and NORRt), and a few basic accounting identities, the DSA model easily computes
projections for the budgetary and debt variables of interest, e.g. revenues, expenditures, budget
balances, net borrowing, and debt-to-GDP ratio.
With additional assumptions on the financing terms (i.e. amortizations, maturity, grace period)
corresponding to existing public debts and prospective borrowings, the DSA model can also compute
additional variables such as gross borrowing to cover financing needs. Gross financing needs arise out of
the overall budget deficit and the debt amortizations (i.e., payments of principal falling due). We assume
these needs are covered with special receipts and use of assets (e.g. privatization, drawdown of
government deposits) as well as with borrowing (i.e., new debt flows). A debt strategy is the choice of
instruments of a certain type and financial terms (e.g., currency of denomination, tenor, interest rate)
issued to meet the borrowing needs in a given year. The debt strategy specifies the shares of borrowing
needs that is met with each instrument. In the DSA model, we consider only domestic debt and foreign
debt, which can be seen as two highly-aggregated instruments. The corresponding formal expressions
are:
(3.4.a)
(3.4.b)
(3.4.c)
with
(3.4.d)
(3.4.e)
19
where the gross financing needs, the special receipts and use of assets, the borrowing needs, and the
amortizations of domestic and foreign debt are scaled by nominal GDP; and the debt strategy specifies
shares of domestic and foreign debt instruments to be issued, wd,t ad wf,t, respectively.
VAR model
We estimate a VAR model involving five endogenous variables: the Tapis oil price POt, the
growth rates of real oil GDP and non-oil GDP (ŷo,t and ŷno.t), the relative variation in the real exchange
rate ȇt, and the domestic real interest rate rd,t. We use annual data in 1992-2009, obtained from World
Bank’s World Development Indicators and IMF’s World Economic Outlook. As times series are not long,
we consider only one lag of the endogenous variables, i.e., a VAR(1) model. The estimated coefficients
and statistics are reported in Table 3.1, whereas the estimated residuals’ covariance matrix (used to
perform stochastic simulations below) is reported in Table 3.2. The VAR is an empirical macroeconomic
dynamic model often used by governments to assess the impact of fiscal and monetary policies, as well
as to analyze the effect of shocks on the economy. It is important to emphasize that we do not deem the
VAR model as a perfect forecasting device. We instead see it as a useful model to represent interactions
and feedbacks among variables involved in the public debt dynamics and to simulate alternative
macroeconomic scenarios that can be compared with the baseline scenario.
20
Table 3.1. VAR estimated coefficients and statistics.
Domestic real interest
rate (%) - Lagged
Relative variation in the
RER (%) - Lagged
Real non-oil GDP growth
rate (%) - Lagged
Real oil GDP growth rate
(%) - Lagged
Tapis oil price
(USD/barrel) - Lagged
Intercept
Dummy Crisis 1998
R-squared
Adj. R-squared
Sum sq. resids
S.E. equation
F-statistic
Log likelihood
Akaike AIC
Schwarz SC
Mean dependent
S.D. dependent
Domestic
real
interest
rate (%)
Relative
variation in
the real
exchange
rate (%)
Real nonoil GDP
growth rate
(%)
Real oil
GDP
growth rate
(%)
Tapis oil
price
(USD/barrel)
-0.45086
[-0.92733]
-0.20815
[-0.30648]
0.50108
[ 0.59000]
0.57409
[ 1.76356]
-1.14269
[-0.64457]
0.36247
[ 1.12742]
0.43901
[ 0.97751]
0.60240
[ 1.07264]
0.13713
[ 0.63703]
-2.40493
[-2.05147]
0.00025
[ 0.00122]
0.04646
[ 0.16344]
0.15287
[ 0.43004]
0.20396
[ 1.49687]
-0.59883
[-0.80700]
0.45947
[ 0.92648]
0.91400
[ 1.31937]
0.79472
[ 0.91739]
0.28032
[ 0.84424]
-3.52196
[-1.94769]
-0.01606
[-0.19362]
0.03689
[ 0.31830]
-0.01720
[-0.11867]
0.00235
[ 0.04230]
0.50666
[ 1.67502]
2.05349
[ 0.29151]
-7.32545
[-0.74444]
-3.47138
[-0.28211]
2.78172
[ 0.58978]
48.19123
[ 1.87619]
-8.31008
[-1.57433]
25.86224
[ 3.50748]
-10.69319
[-1.15973]
-17.39788
[-4.92279]
14.36219
[ 0.74621]
0.45
0.13
142.54
3.78
1.39
-42.20
5.79
6.13
2.95
4.04
0.78
0.65
278.14
5.27
6.06
-47.88
6.46
6.80
1.25
8.97
0.34
-0.06
434.94
6.59
0.86
-51.68
6.90
7.25
2.48
6.42
0.83
0.73
63.90
2.53
8.35
-35.38
4.99
5.33
5.85
4.90
0.83
0.73
1895.15
13.77
8.28
-64.19
8.38
8.72
38.65
26.59
Determinant resid covariance (dof adj.)
Determinant resid covariance
Log likelihood
Akaike information criterion
Schwarz criterion
t-statistics in [ ]
394704.50
27798.89
-207.5882
28.53978
30.25522
Table 3.2. VAR estimated residuals’ covariance matrix.
Relative
Domestic real
variation in the
interest rate
real exchange
(%)
rate (%)
Domestic real
interest rate (%)
Relative variation in
the real exchange
(%)GDP
Real rate
non-oil
growth rate (%)
Real oil GDP
growth rate (%)
Tapis oil price
(USD/barrel)
Real non-oil
GDP growth
rate (%)
Real oil GDP
growth rate
(%)
Tapis oil price
(USD/barrel)
14.25
5.56
-6.19
-5.42
-49.77
5.56
27.81
-19.49
-6.26
-16.57
-6.19
-19.49
43.49
-0.52
16.59
-5.42
-6.26
-0.52
6.39
17.51
-49.77
-16.57
16.59
17.51
189.51
Stochastic DSA and fiscal risks
A baseline scenario underlying a DSA relies on projections of macroeconomic and fiscal variables
that capture key interactions and feedbacks among them. As no one knows with certainty how these
variables will behave in the future, the baseline projections are meant to reflect their expected values as
perceived by the analyst. The notion of fiscal risk refers to the possibility of experiencing unexpected
shocks that induce deviations of fiscal outcomes from those expected in the baseline scenario. Thus, to
introduce fiscal risk into the DSA model, it is necessary to produce alternative projections by shocking the
21
macroeconomic and fiscal variables and then calculate the corresponding paths of the budgetary and
debt variables of interest. Assessing the fiscal risk facing the government’s budget and debt boils down to
quantifying and analyzing the discrepancies between the (expected) baseline projections and the
(unexpected, yet possible) alternative projections generated with shocks hitting the economy.
A standard method for generating alternative projections with shocks employs stochastic
simulations. Shocks are drawn from a joint probability distribution that captures the pattern of
contemporaneous correlations between relevant variables. The shocks are added to the corresponding
variables to generate new projections and then the DSA model computes the corresponding budgetary
and debt paths. The procedure is repeated several times to construct the (simulated) probability
distribution (density) of the variables of interest, e.g., the debt-to-GDP ratio. A fan chart is a widely-used
devise to report the results.
In the stochastic DSA for Malaysia, we consider alternative projections for five variables: the
Tapis oil price POt, the growth rates of real oil GDP and non-oil GDP (ŷo,t and ŷno.t), the relative
variation in the real exchange rate ȇt, and the domestic real interest rate rd,t. We construct an alternative
projection for any of these five variables by adding a shock to the value under the baseline projection.
Thus, for instance, the Tapis oil price in an alternative projection (indexed by i) is POi,t = PO*t +
ShockPOt, where a star ‘*’ denotes the value in the baseline scenario. We assume that shocks to the five
variables follow a multivariate Gaussian distribution, with a mean vector of zeros and the covariance
matrix discussed below, thus capturing contemporaneous correlations between them. We draw shocks
5000 times and run the DSA model accordingly with the resulting alternative projections. 21
For illustrative purposes, Figures 3.3, 3.4, 3.5, and 3.6 show the probability distribution of the
Tapis oil price, the growth rate of real GDP (computed as the average of oil GDP and non-oil GDP, with
weight αoil fixed to 10 percent), the Federal Government (FG) revenues, and the FG debt-to-GDP ratio,
obtained after 5000 simulations. These fan charts depict only a few selected percentiles that are
summary indicators of the probability distribution (density), e.g., if the random variable X’s percentile 25 is
h, it means that X takes values smaller (larger) than h with 25 (75) percent probability.
The stochastic DSA is used to measure and understand the risks around the projected path of
debt under a baseline scenario. Given Malaysia’s dependence on oil revenues, quantifying the potential
deviations around the projected fiscal path (in the form, for example, of fan charts) can be very helpful for
policy making.22 We formulate a stochastic DSA model to address the uncertainty regarding future oil
21
In all the stochastic simulations, the foreign nominal interest rate if,t, and the domestic and foreign inflation rates,
pt and pf,t, are unchanged with respect to the baseline scenario. The oil-related revenue in an alternative projection i
is computed as ORRi,t = ORR*t POi,t / PO*t, while the oil-related primary expenditure ORPEi,t, the non-oil-related
revenue NORRi,t, and the non-oil-related primary expenditure NORPEi,t are generically calculated as Xi,t = X*t Yi,t /
Y*t .
22
How to quantify the fiscal risk stemming from Tapis oil price volatility? Three factors are relevant: (i) the
volatility of Tapis oil price, i.e. intuitively, how much Tapis oil price fluctuates; (ii) the sensitivity of oil-related
revenues and expenditures (e.g. subsidies) to changes in Tapis oil price, i.e. an elasticity measuring how much
individual budgetary items react to percentage fluctuations in Tapis oil price; (iii) the shares of oil-related revenues
and expenditures, respectively, in total revenues and expenditures, i.e. how much budgetary aggregates react to
22
prices and economic growth by introducing shocks to the main macroeconomic variables. 23 Shocks affect
the economy through the pattern of co-movements among these variables that has characterized the
Malaysian economy over the last two decades. In the stochastic DSA, therefore, fiscal outcomes are
captured more comprehensively through modelling explicitly macro-financial interactions that largely
ignored in deterministic debt sustainability assessments. Our results suggest that, under very adverse
circumstances with regard to oil prices and economic growth, in 2015 the FG debt could reach 55 percent
of GDP (Figure 3.6). Also, in 2015 the FG debt would exceed 50 percent of GDP with a probability of 25
percent. The fan chart is centered around the debt-to-GDP path under the baseline scenario.
As an extension of the basic analysis, in which Tapis oil price is assumed to be an endogenous
variable that affects only the revenue side of the FG budget, we also consider the Tapis oil price to be an
exogenous variable. In fact, the Tapis oil price is likely to be determined by developments in the global
economy, not within Malaysia. Furthermore, we acknowledge that the Tapis oil price also affects the
spending side of the FG budget through subsidies to fuel consumption. To capture these two features, we
explore another version of the stochastic DSA model in which Tapis oil price is an exogenous variable
and determines both oil-related revenues and expenditures through econometric regressions. The FG
debt projections, reported in Figure 3.7, are more concentrated around the median path than those
presented in Figure 3.6. The fiscal risk facing the FG budget therefore appears smaller on two accounts:
first, the impact on FG budget of variations in oil-related revenues is attenuated by the simultaneous
variations in oil-related expenditure; second, the Tapis oil projection used in this stochastic DSA is taken
from the baseline scenario, applied to all stochastic simulations, and not subject to shocks.
changes in their individual components. While factor (i) refers to the variations in the Tapis oil price, factors (ii) and
(iii) determine the exposure of the FG budget to those variations. It is difficult to measure the volatility of Tapis oil
price -especially at annual frequency- because the available series is short and shows an accelerated growth
concomitant with significant fluctuations in recent years. In 2000-2009, in fact, the annual growth rate of Tapis oil
price averaged 12 percent, with a standard deviation of 26 percent. More important, it is also difficult to assess how
much of the large variations in the Tapis oil price had been expected (ex ante facto) and how much unexpected (thus
known ex post facto). Such a distinction is analytically relevant: while expected variations should be taken into
account when building the baseline scenario's projections of fiscal variables, the unexpected variations induce
surprises in fiscal outcomes and hence concern the assessment of fiscal risk.
23
The stochastic DSA model runs on an Excel-based tool developed by World Bank staff. More advanced results
are computed using a stochastic DSA programmed by World Bank staff with the software Analytica.
23
Figure 3.3. Tapis oil price (USD/barrel).
160.0
150.0
95% - 99%
140.0
90% - 95%
130.0
75% - 90%
120.0
67% - 75%
110.0
50% - 67%
33% - 50%
100.0
25% - 33%
90.0
10% - 25%
80.0
5% - 10%
70.0
1% - 5%
2015
2014
2013
2012
2011
2010
2009
2008
2007
60.0
Scenario S1
Source: World Bank’s calculations.
Figure 3.4. Real GDP growth rate (%).
12.0%
95% - 99%
10.0%
90% - 95%
8.0%
75% - 90%
67% - 75%
6.0%
50% - 67%
4.0%
33% - 50%
25% - 33%
2.0%
10% - 25%
5% - 10%
0.0%
1% - 5%
2015
2014
2013
2012
2011
2010
2009
2008
2007
-2.0%
Scenario S1
Source: World Bank’s calculations.
Figure 3.5: FG total revenues (% of GDP).
24.0%
95% - 99%
23.0%
90% - 95%
75% - 90%
22.0%
67% - 75%
50% - 67%
21.0%
33% - 50%
25% - 33%
20.0%
10% - 25%
19.0%
5% - 10%
1% - 5%
2015
2014
2013
2012
2011
2010
2009
2008
2007
18.0%
Source: World Bank’s calculations.
24
Scenario S1
Figure 3.6: FG debt (% of GDP).
65.0%
95% - 99%
60.0%
90% - 95%
75% - 90%
67% - 75%
55.0%
50% - 67%
33% - 50%
50.0%
25% - 33%
10% - 25%
45.0%
5% - 10%
1% - 5%
40.0%
2015
2014
2013
2012
2011
2010
2009
2008
2007
S1 scenario
Source: World Bank’s calculations.
Figure 3.7. FG Debt (% of GDP).
65.0%
95% - 99%
60.0%
90% - 95%
75% - 90%
67% - 75%
55.0%
50% - 67%
33% - 50%
50.0%
25% - 33%
10% - 25%
45.0%
5% - 10%
1% - 5%
40.0%
2015
2014
2013
2012
2011
2010
2009
2008
2007
Central proj.
Source: World Bank’s calculations.
25
Annex: The European Union methodology for assessing long-term fiscal sustainability
The Stability and Convergence Program, an annual report on macroeconomic and fiscal developments and policies
that the EU Member States are required to submit each year to the European Commission and the ECOFIN
Council, includes a chapter analyzing long-term sustainability of public finances. The objective of analyzing fiscal
sustainability over a quite long projection horizon (up to 2060) is to provide the qualitative and quantitative
information necessary to identify sources of risk associated to budgetary policies and the population ageing.
The pivotal variable for the sustainability analysis is gross public debt and its evolution is obtained by means
of projections of demographic, macroeconomic, and public-finance variables prepared by the Ageing Working Group
(AWG, a working group attached to the European Commission). Using an approach agreed at European level
makes it possible to compare the analyses conducted by all the Member States. The definition of gross public debt
is contained in the Maastricht Treaty’s protocol governing the Excessive Deficit Procedure (EDP). Debt is therefore
relatively easy to report and to compare across Member States. However, from the standpoint of fiscal sustainability,
it is also important consider the financial assets held by the government. In this regard, there are criteria for defining
the concept of adjusted gross public debt, which is calculated by subtracting the consolidated liquid assets
accumulated in pension funds and earmarked for covering pension expenditures from the gross public debt. The
assets include cash, transferrable deposits, government securities and publicly traded equities, and the Member
States are required to indicate the assets’ current market value and returns earned. The analysis of the debt
dynamics thus looks at the adjusted gross public debt relative to GDP.
The EU methodology for assessing long-term fiscal sustainability includes three distinct phases: (i)
computing projections of the public debt stock in relation to GDP, using demographic, macroeconomic, and publicfinance projections over a time horizon that currently extends out to 2060 and starts after the medium-term horizon
covered by the Stability and Convergence Program of each Member State; (ii) calculating sustainability indicators to
measure the fiscal adjustment that would be necessary to get the public accounts back on a sustainable path; and
(iii) conducting a sensitivity analysis in order to evaluate the robustness of baseline results, taking into consideration
alternative scenarios characterized by different demographic, macroeconomic, and public-finance assumptions.
Some of the analytical issues involved in the first two phases are discussed below.
(i) Evolution of the public debt
The analysis of the public debt dynamics is based on projections of revenues, primary expenditure, nominal
interest rates, and growth rates of nominal GDP. The AWG has developed a comprehensive methodology for
projecting country-specific growth rates of real GDP and age-related expenditures (European Commission, 2009). In
the case of revenues and primary expenditure not related to the population ageing, the AWG assumes that such
variables remain constant in relation to GDP over the entire projection period, at the level achieved after the
26
medium-term horizon covered by the Stability and Convergence Program of each Member State.24 In addition, both
revenues and non-age-related expenditures are calculated in structural terms, i.e. net of cyclical effects and one-off
measures, thereby making it possible to obtain a structural primary balance. For all Member States, the real interest
rate is held constant at 3 per cent per annum and the inflation rate measured with the GDP deflator is held constant
at 2 per cent per annum; therefore, the nominal interest rate is equal to 5 per cent.25
The analysis of debt dynamics is carried out considering two scenarios. The program scenario is the
baseline case and assumes that the Member State achieves the public-finance objectives set out in its Stability and
Convergence Program. Thus, the initial conditions for structural primary balance and public debt stock are given by
Program. The second scenario is an alternative case and assumes that the Member State is not able to achieve the
objectives of the Stability and Convergence Program. Thus, the country’s primary balance-to-GDP ratio remains
constant at the level achieved in the recent past. The initial conditions for projecting public debt are therefore
different between the two scenarios.
(ii) Sustainability indicators
In the EU, fiscal and borrowing policies are deemed sustainable if they meet the government’s intertemporal
budget constraint (IBC). In addition, given the budgetary requirements within the EU’s fiscal framework, there is
another definition used: policies are considered sustainable if they steer the public debt stock to a level below the
Maastricht criterion of 60 per cent of GDP (which can be seen as an indicative debt target).
In order to make a quantitative sustainability assessment, the EU Member States compute two indicators,
known as sustainability gaps S1 and S2, which measure the budgetary adjustment that would be necessary to
satisfy the two sustainability conditions. The indicators can be broken down in order to identify two sources of risk to
fiscal sustainability: (i) the initial budgetary position, associated with the structural primary balance and the debt stock
carried forward; and (ii) the cost of ageing, associated with the projected deterioration in the primary balance due to
rising age-related expenditure.
The S1 indicator measures the permanent adjustment in the structural primary balance (as a percentage of
GDP) necessary for the debt-to-GDP ratio to reach the value of 60 per cent by 2060. Hence, S1 is a particular case
of the fiscal gap presented in Section 1. Given macroeconomic and fiscal projections starting in year t0+1 and ending
in 2060, S1 is computed by:26
24
The public spending other than age-related expenditure consists of public investments, other social expenditures,
intermediate consumption (purchase of goods and services) not related to age-related expenditures, and income from
full-time employment other than the income earned by workers in the healthcare and education sectors.
25
There are no stock-flow adjustments contemplated unless specifically indicated in the Stability and Convergence
Program of each Member State.
26
It is assumed that the nominal interest rate i and the growth rate of nominal GDP y remain constant over time, and
consequently the rate r also remains constant. The derivation of S1 and S2 in the case where the nominal GDP
growth rate varies over time is presented at the end of the Annex.
27
2050
S1  r dt0  pbt0 
A

r dt0  60
1  r 
2050  t0
B

1

1
 1  r  pb
1
 1  r 
s  t0
s t0 1
s
2050
s  t0
s  t0 1
C
(A1)
where dt0 denotes the debt-to-GDP ratio at the end of year t0 , and Δpbs is the difference between the projected
primary balance-to-GDP ratio in year s and the ratio observed in year t0 , i.e. Δpbs = pbs - pbt0 . For the years between
t0+1 and 2060, the primary balance-to-GDP ratio pbs can be seen as the sum of two elements: (i) the value of the
initial primary balance pbt0 that would be observed if the future budgetary conditions in s were identical to those in t0 ;
and (ii) the value Δpbs that captures the expected change in the budgetary conditions associated with the effects of
ageing on the age-related spending items.27 In equation (A1), S1 is broken down into three parts. The first
component A (denoted ‘initial budgetary position’) indicates the adjustment in the primary balance-to-GDP ratio
between t0+1 and 2060 that is necessary to keep the debt-to-GDP ratio constant over the projection horizon at the
inherited value dt0 if no budgetary changes were expected in the future, i.e. if Δpbs were zero. Intuitively, the primary
balance should be adjusted permanently for the purpose of fully offsetting the snowball effect, even before taking
into account the effects of ageing. The second component B (denoted ‘debt requirement in 2060’) indicates the
additional adjustment in the primary balance-to-GDP ratio between t0+1 and 2060 that is necessary to take the debt
from the inherited value dt0 to the target value of 60 per cent of GDP in 2060, still assuming no future budgetary
changes. Intuitively, given an initial level of debt exceeding 60 per cent of GDP, the fiscal adjustment that
counteracts the snowball effect (measured by A) must be complemented with an additional fiscal effort in order to
reduce the debt-to-GDP ratio and so hit the 60 per cent target by 2060 (measured by B). Finally, the third
component C (denoted ‘long term changes in the primary balance’) indicates the adjustment in the primary balanceto-GDP ratio between t0+1 and 2060 that is necessary to offset fully the expected changes in the budgetary
conditions due to ageing. The values of Δpbs are often negative because the population ageing leads to increasing
age-related spending and therefore the primary balance deteriorates over time. In other words, the primary balance
should be adjusted permanently for the purpose of fully offsetting the effects of ageing on the age-related spending
items over the projection period. 28
An implicit shortcoming in the definition of S1 motivates the development of the S2 indicator. Since the
population ageing is likely to exert stronger pressure on the public finances by the end of the projection horizon, the
fiscal adjustment indicated by S1 probably does not ensure meeting the IBC. This shortcoming is the basis for
introducing the S2 indicator in the sustainability analysis.
27
According to the AWG’s criteria for projecting the components of the primary balance, the expected changes in
the primary balance-to-GDP ratio over the projection period depend exclusively on changes in the age-related
expenditure-to-GDP and property income-to-GDP ratios.
28
This adjustment is the weighted average of the projected changes in the primary balance-to-GDP ratio Δpbs from
t0+1 to 2060, with weights given by the discounting factors. When the primary balances are adjusted by the
component C, the effects of ageing are implicitly absorbed by other budgetary items (e.g. higher taxes or lower nonage-related expenditure).
28
The S2 indicator measures the permanent adjustment in the structural primary balance (as a percentage of
GDP) necessary for the IBC to be satisfied. Hence, S2 is closely related to the intertemporal budget gap presented
in Section 1.29 The S2 indicator is computed by:

S 2  r dt0  pbt0 
D
1
 1  r  pb
1
 1  r 
s  t0
s t0 1
s

s  t0
s  t0 1
E
(A2)
where it is assumed that Δpbs from 2061 onwards (i.e. in years outside the projection horizon) are equal to the value
projected for 2060. In other words, outside the projection period, the budgetary effects of ageing are assumed to be
identical to those in 2060. In equation (A2), S2 is broken down into two parts. The first component D (denoted ‘initial
budgetary position’, which is similar to the A component of S1) measures the adjustment in the primary balance-toGDP ratio that is necessary to keep the debt-to-GDP ratio constant over an infinite horizon at the inherited value dt0
(offsetting the snowball effect induced by interests) if no budgetary changes were expected in the future. The second
component E (denoted ‘long term changes in the primary balance’, which is similar to the C component of S1)
measures the adjustment in the primary balance-to-GDP ratio that is necessary to offset fully the future budgetary
effects of ageing in an infinite time horizon (thus overcoming the shortcoming of S1 discussed above).
Breaking down S2 is useful for identifying two sources of risk to the long-term sustainability of public
finances: (i) the initial budgetary position, indicated by the component D and associated with the structural primary
balance and the inherited debt stock; and (ii) the cost of ageing, indicated by the component E and associated with
the expected rise in the age-related expenditure. Furthermore, the magnitude of the S2 components quantifies the
level of risk stemming from both sources. For instance, a positive and large value for S2 could stem from a weak
fiscal position at present (a large D), together with a negligible increase in the age-related expenditure expected for
the future. But it could also stem from a balanced initial fiscal position (a small D), coupled with large effects of
ageing on future fiscal budgets (a large E). On the other hand, a negative value for S2 typically is the result of a very
strong initial fiscal position (a negative D) that, if persistent, can offset a potentially large projected increase in agerelated expenditure (a positive E). Therefore, the S2 indicator gives the order of magnitude of the permanent
budgetary adjustment needed for the public finances to reach a sustainable position and to satisfy the IBC.
A third sustainability indicator used in the EU methodology is the required primary balance (RPB). The RBP
indicates the level of the structural primary surplus (as a percentage of GDP) that should be achieved in the
medium-term according to the adjustment implied by S2. The RPB is the average of the primary balance-to-GDP
ratios expected over the first five years of the projection period plus the value of S2. By comparing the RPB with the
current primary balance-to-GDP ratio, or with the planned ratio over the medium-term, it is possible to gauge
whether there is consistency between a sustainable budgetary policy and the current budgetary conditions. The
RPB is computed by:
29
More precisely, S2 is a flow indicator that measures the adjustment in the primary balance in each year, while the
intertemporal budget gap is a stock indicator that can be computed as the PDV of an infinite series of terms equal to S2.
29
RPB 
(A3)
t0  5

s t0 1
pbs
 S2
5
Critical aspects of the European Union methodology for assessing sustainability
It is widely acknowledged that the European Union methodology for assessing long-term fiscal sustainability
has limitations and drawbacks that have prompted further development and refinement of the methodology itself.
There are two kinds of shortcomings. The first is related to the sources of data and the use of projections, while the
second results from adopting a partial equilibrium approach in analyzing debt dynamics.
Regarding the first issue, there are data availability problems in several EU countries and the AWG
projection methodologies leave room for some idiosyncrasies since national models are used to estimate spending
on public pensions, the single most important age-related expenditure item. Furthermore, since there is no
econometric model underpinning most AWG projections, it is not possible to characterize the error margins of the
projections as if they were forecast errors. For the same reason, it is equally unfeasible to attach probabilities of
occurrence to the alternative scenarios explored in the sensitivity analysis. Finally, projections over several decades
are very sensitive to the estimation of potential GDP at the beginning of the projection period and this estimation is
subject to a high degree of uncertainty.
Other limitations of the European Union methodology flow from modeling public debt dynamics by using a
partial equilibrium framework in which only a limited number of the relationships between demographic,
macroeconomic, and public-finance variables are taken into account. Four important relationships not considered in
the European approach are the following. First, the impact of demographic trends on primary spending items is fairly
well addressed by the AWG methodology. But since the impact on tax revenues is not yet considered, the
projections offer only a partial picture of the fiscal effects of ageing. Second, as far as economic growth is concerned,
ageing affects the evolution of long-run potential real GDP solely through changes in the employment level, e.g. a
declining working-age population and a rising employment rate jointly determine changes in the number of
employees, which, in turn, affects the growth rate of potential real GDP. But the effect of ageing on productivity
growth has not been explored yet. Third, despite the fact that interaction between macroeconomic and fiscal
variables heavily influences public debt dynamics, the AWG framework neglects an important consideration, i.e. how
public expenditure on education affects economic growth. In the European methodology, public expenditure on
education has no influence on labor productivity, while such a link has been emphasized by the literature on human
capital and growth. This has implications on long-term fiscal sustainability: education financed by government
borrowing raises the public debt stock, but since it contributes to human capital accumulation and potential GDP
growth, it can improve long-term fiscal sustainability. Fourth, the framework misses the feedback effects between
public debt and budgetary imbalances, on the one hand, and, on the other, interest rates and growth. Interest rates
are assumed to be independent of fiscal developments in the AWG methodology. If persistent budget deficits result
in an increasing debt-to-GDP ratio, investors may require higher interest rates to compensate for additional risks,
30
e.g. default risk, and then the public debt dynamics would accelerate because of a stronger snowball effect. In
addition, if higher interest rates crowd out private investment and thus reduce potential GDP growth, the debt-toGDP ratio dynamics accelerates further. Therefore, a long-term fiscal-sustainability assessment that relies on a
partial equilibrium analysis is likely to underestimate the explosive effects of persistent budget deficits. By the same
token, such an assessment also underestimates the positive effects of a fiscal consolidation program that improves
budgetary positions, reduces interest rates, and decelerates the growth of public debt.
In the EU’s sustainability assessment, the no-policy-change assumption concerning revenues and non-agerelated expenditures, the limited number of relationships considered, and the exogenous nature of many relevant
variables often give rise to projections indicating a strong accentuation of the debt profile. In practice, however, it is
likely that a government will react to an explosive accumulation of either public debt or assets.
Taking these limitations into account, the EU’s sustainability assessments always emphasize that the
purpose of analyzing debt dynamics is to signal possible budgetary imbalances on the basis of current policies and
projected changes in age-related expenditure. Consequently, the EU uses the sustainability indicators as a tool to
facilitate policy debate and to identify the timing and magnitude of budgetary pressures that could arise in the future
in light of ageing and the no-policy-change assumption. Moreover, the EU does not take the S1 and S2 indicators at
face value, nor does it recommend undertaking a fiscal adjustment of the size implied by the indicators. Instead, the
indicators’ values are used to classify the EU countries by their levels of risk to public finance sustainability in quite
broad categories: low-, medium-, or high-level risk. In this regard, the quantitative indicators are complemented by
qualitative information about factors that have a bearing on the long-term fiscal sustainability, such as the level of
debt-to-GDP ratio and the budget position at the beginning of the projection period, the current level of tax-to-GDP
ratio, the effects of structural reforms on public pension and healthcare systems, and the reliability of projections on
age-related expenditure. These factors facilitate interpretation of the quantitative results and discussion of the
budgetary risks facing EU countries.
Derivation and breakdown of S1 and S2 - Debt dynamics and the IBC when the growth rate of nominal GDP
varies over time
In the AWG projections, the nominal interest rate i is assumed to be constant at 5 percent per annum.
Instead, the growth rate of nominal GDP yt varies over time as a result of growth in employment and labor
productivity in any given country. Therefore, equations need to be slightly modified in order to describe the
dynamics of the debt-to-GDP ratio. The rate rt in year t is defined by the formula rt = (i - yt)/(1+yt), so that
now rt is a variable that changes over time following the growth rate of nominal GDP yt. Consequently, for a
government that inherits a debt-to-GDP ratio dt-1 and shows a series of primary balance-to-GDP ratios pbs
between t and t+h, the debt-to-GDP ratio dt+h at the end of year t+h is given by:
t h
dt  h  Rt ,t  h dt 1   Rs 1,t  h pbs
(A4)
s t
31
where the rate Ri,j is defined by Ri,j = (1+ri)(1+ri+1)…(1+rj) if i≤j and Ri,j = 1 if i>j. Dividing both sides of (A4) by
Rt,t+h and imposing the condition that the PDV of the debt-to-GDP ratio converge to zero as the time horizon
lengthens (i.e. the no-Ponzi-game condition), the IBC becomes:


(A5)
s t

ts
gs

 dt 1
Rt , s s t Rt ,s
The S1 indicator ensures that equation (A4) is satisfied with a target value of 60 percent in 2060 for
the debt-to-GDP ratio. Using macroeconomic and fiscal projections starting in t0+1 and ending in 2060,
imposing d2060=60, substituting pbs=pbt0+Δpbs+S1, and substituting Δpbs= (pis - pit0) – (ares - aret0) where pis
is the property income-to-GDP ratio and ares is the age-related spending-to-GDP ratio, the formula of S1
becomes:

2050
S1 
Rt0 1,2050  1
2050

s t0 1
dt0  pbt0 

Rs 1,2050 pis  pit0
s t0 1

2050

Rs 1,2050
s t0 1
Rs 1,2050
A
2050
dt0  60

2050

s  t0 1
(A6)


s t0 1

Rs 1,2050 ares  aret0

2050

Rs 1,2050
s t0 1
B
Rs 1,2050
C
The S2 indicator ensures that the equation (A5) is satisfied. Using macroeconomic and fiscal
projections starting in t0+1, substituting ts - gs=pbt0+Δpbs+S2, and substituting Δpbs= (pis - pit0) – (ares - aret0),
the formula of S2 becomes:

S2 
1


s t0 1
(A7)
1
dt0  pbt0 

s t0 1
Rt0 1, s
1
Rt0 1, s


s t0 1
 pi
s
 pit0

 

1
Rt0 1, s
D
s t0 1

1
ares  aret0
Rt0 1, s


s t0 1

1
Rt0 1, s
E
No-Ponzi-game condition, debt growth, and sustainability
The no-Ponzi-game condition has a simple interpretation in terms of the rate of growth of the debtto-GDP ratio. A useful concept is the ‘average’ growth rate of the debt-to-GDP ratio between years t and t+h,
denoted zt,t+h , which is the rate that delivers the debt-to-GDP ratio dt+h in year t+h if it is applied in a
compounded fashion to the debt-to-GDP ratio dt-1 inherited in year t. Thus, zt,t+h provides a summary
description of the dynamics of the debt-to-GDP ratio since it depends on the initial and final value of the
debt-to-GDP ratio and on the length of the time horizon. Formally, zt,t+h is given by:
32
(A8)
zt ,t  h  h1 dt  h / dt 1  1
Another similar concept is the ‘average’ rate r between years t and t+h, denoted rt,t+h :
(A9)
rt ,t h  h1 1  rt 1  rt 1  ... 1  rt h   1
Using the two concepts defined above, the no-Ponzi-game condition is:
(A10)
d
lim t  h  0
h  R
t ,t  h

1  z 
lim
1  r 
h 1
t ,t  h
h 
h 1
dt 1  0
t ,t  h
Equation (A10) imposes the condition that the average growth rate of the debt-to-GDP ratio zt,t+h
must be lower than the average rate rt,t+h over an infinite time horizon. Two special cases illustrate this
condition. First, for a government running balanced primary budgets systematically, the public finances are
not sustainable because the systematic rollover of the growth-adjusted interest bill leads to a debt-to-GDP
ratio that grows too fast. In this case, zt,t+h is equal to rt,t+h and so the left side of (A10) is equal to dt-1 (a
positive value for an indebted government) for any time horizon, thus violating the sustainability condition.
Second, for a government running primary imbalances but with primary surpluses prevailing over time, the
public finances are sustainable because the primary surpluses offset the snowball effect associated to the
growth-adjusted interest bill. In this case, zt,t+h is lower than rt,t+h and so the left side of (A10) converges to
zero as the time horizon gets longer, thus fulfilling the sustainability condition. The analysis above is
consistent with the idea that the IBC entails a restriction on the average speed at which the debt-to-GDP
ratio grows over a infinite time horizon; such average speed must be lower than the average rate rt,t+h. This
restriction refers strictly to the growth of the debt-to-GDP ratio, so it may be fulfilled by a debt-to-GDP ratio
exhibiting a high (or low) level. The analysis also suggests that the sustainability condition based on a debt
target implicitly imposes a restriction on the average growth rate of the debt-to-GDP ratio within the finite
time horizon considered. Given the target d*t+h
and the inherited debt-to-GDP ratio dt-1, equation (A8)
indicates the average growth rate allowed by the debt target condition.
33
References
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