Sustainability gap of public debt: importance of interest rates and a new decomposition with premia

Sustainability gaps (S2 indicators) are frequently used in national and international reports to assess the sustainability of public finances. For instance, in the European Commission’s Debt Sustainability Monitor (DSM) the indicators are analyzed in comparisons across (policy) scenarios, countries and time. The report’s findings play a crucial role in the context of the Stability and Growth Pact and the European Semester. As a result, sustainability gaps have a significant indirect influence on policy decisions. In this paper, we analyze two non-transparent properties of these indicators. First, the response of these indicators to changes in the interest rate-growth (r-g) differential is not readily predictable in terms of both strength and direction. Second, in our examples for low values of r-g (in a range of 0.5%), highly uncertain projections for distant periods after 2070 explain about 80% of the indicators’ values. To address these problems, we develop a new decomposition that takes into account the notion of premia (Reis 2021) and hence allows for a more transparent discussion of the sustainability of public debt.


Introduction
The sustainability gap (S2 indicator) is frequently used in national and international debt sustainability reports to assess the long-term sustainability of public finances. The S2 indicator measures the permanent adjustment in the primary deficit that is necessary to satisfy the government's intertemporal budget constraint (IBC) which is defined for an infinite period. The S2 indicator is analyzed in detail in the European Commission's Debt Sustainability Monitor (DSM) in comparisons across (policy) scenarios, countries and time. The report's findings play a crucial role in the context of the Stability and Growth Pact and the European Semester. As a result, the S2 indicator has a significant indirect influence on policy decisions.
Arguably, sustainability indicators used for policy decision making should be transparent and simple to analyze. In this paper we look at two properties of the S2 indicator that do not satisfy these requirements. First, the response of the indicator to changes in the interest rate-growth (r-g) differential is not readily predictable in terms of both strength and direction. The reason is that a change in the interest rate-growth (r-g) differential acts through two channels, sometimes with opposite effects. These channels can be illustrated by representing the S2 indicator as the sum of two annuities-one annuity translating the baseline debt (d) level, S2(d), and a second annuity translating all primary deficits (pd) expected in the future, S2(pd). The first annuity (S2(d)) clearly increases (decreases) as the discount rate (r-g) increases (decreases). For the second annuity (S2(pd)), the response depends on the trajectory of primary deficits since the discount rate affects the weighting of primary deficits over time. When the discount rate is increasing (decreasing), primary deficits that are close (distant) in time are weighted more heavily. For instance, this means that when the discount rate is low, the indicator gives strong weight to projected primary deficits in the distant future. The sum of the two annuities (S2(d) + S2(pd)) responds to a change in interest rates in a clearly determinable direction only when deficits are projected to decline. When primary deficits are projected to increase over time, the direction of change in the S2 indicator is not easily determinable, since the interest rate responsiveness of S2(d) and S2(pd) partly cancel out each other. Second, for low values of r-g (in a range of 0.5%), the influence of highly uncertain projections for distant periods on the S2 indicators is very large. For example, projection values for periods after 2070 explain up to 80% of the S2 indicator in some scenarios.
To capture the unexpected properties of the S2 indicator, we propose a new decomposition of the S2 indicator. The new decomposition exploits the observation that many sovereigns pay interest rates on their sovereign debt that are empirically lower than capital market interest rates on other asset classes, and hence, adopts the notion of premia (Reis 2021). Our decomposition allows for a more transparent discussion of the extent to which changes in the S2 indicators' values are due to direct fiscal policy actions or due to more indirect changes in the discount rate of the government's inter-temporal budget constraint. Additionally, the decomposition allows to disentangle parts of the indicator which are less dependent on very distant periods. Related Literature. This paper relates to a broader literature that seeks to understand the sustainability of public debt based on indicators (Blanchard 1990;Escolano 2010;Debrun et al. 2019;Furman and Summers 2020;Gründler et al. 2022). Many of these indicators seek to gauge the fiscal adjustment that will bring the public debt level to a targeted debt level (in % of GDP) within a given number of periods (e.g. S1 indicator). These indicators depend on a chosen debt target level and time horizon. Despite its practical appeal, the choice of target public debt levels and number of adjustment periods lacks theoretical rigor. Instead, the S2 indicator's definition of long-term fiscal sustainability directly follows from the intertemporal budget constraint of the government and can be related to Blanchard (1990).
Several papers focus on discussing important properties of the S2 indicator. In a recent paper Werding (2021) describes how to calculate the S2 indicator without imposing special restrictions. Andersen (2020) and Werding et al. (2020) provide an intuition for unexpected responses of the S2 indicator with respect to changes in the interest rates. We complement their studies by showing analytically that an indepth decomposition analysis is indeed required to make the mechanisms transparent. Additionally, this paper contributes to a literature investigating the impact of the discount factor on the inter-temporal government budget constraint. In this regard, several papers provide explanations for why sovereigns pay interest rates on their debt that are empirically lower than the capital market interest rates on other assets (Reis 2021; Bayer et al. 2021;Krishnamurthy and Vissing-Jorgensen 2012). Some papers (e.g. Jiang et al. 2019) argue that the discount factor used for the government budget constraint should be the same as for pricing other risky assets. We use the findings of this literature for a new decomposition of the S2 indicator.
The remainder of the paper is organized as follows: Sect. 2 outlines the properties of the S2 indicator and provides an in-depth decomposition of the response of the S2 indicator to changes in the r-g differential. Additionally, we explore the reliance of the indicator on distant periods. In Sect. 3, we introduce our new decomposition of the S2 indicator using the notion of premia. A concluding analysis is presented in Sect. 4.

Ssustainability indicator
The S2 indicator is used to assess the sustainability of government debt by relating it to future primary deficits and current debt levels. Let D t be the absolute level of government debt and d t = D t ∕GDP t be the relative debt as a percentage of GDP t at the end of period t . Let the "growth-corrected real interest rate" be represented as t ∶= where r t is the annual real interest rate on government debt and g t is the annual growth rate of real GDP. 1 The primary deficit as a percentage of GDP at time t is pd t . The debt level at the end of period t can then be described as According to the standard government inter-temporal budget constraint (IBC), a fiscal policy is sustainable in the long run if the present value of all future primary surpluses ( −pd t+i ∀i ≥ 1 ) is equal to the current debt level ( d t ) . Given actual projected primary deficits, pd t+i ∀i ≥ 1 , the IBC is not necessarily satisfied and a residual Z arises. If Z > 0 , then fiscal adjustment is required to comply with the IBC. 2 Sustainability indicators are a measure of the level of fiscal adjustment needs. In case of the S2 indicator, the fiscal adjustment requirement Z is translated into an annuity. 3 Each primary deficit p d t is thus adjusted in each period by an annuity amount k (constant in % of the respective GDP t ) so large that the IBC is exactly fulfilled (e.g. Sustainability Report of the German Federal Ministry of Finance (BMF) 2020; European Commission 2006Commission , 2018Commission , 2020. It follows: The S2 indicator at a time t corresponds exactly to the level of k . We next discuss important properties of this indicator.

Two sub-indicators
From Eq. (2) we directly get The S2 t indicator 4 at time t can be divided into two sub-indicators: one that maps the importance of already explicit debt (S2(d)) and one that maps the importance of future primary deficits (S2(pd)) for the S2 indicator. Note that S2(d) interacts directly with and S2(pd) with the weights w t+i ( ) = (1+ ) i , which depend on i and . The S2 indicator is the exact sum of the two sub-indicators. 5 2 Since for > 0 there is always a risk that the debt ratio will "explode", the Ponzi condition, which prevents an explosion qua assumption, is imposed (see, inter alia, Escolano 2010). The Ponzi condition is not necessary for < 0 , since the evolution of debt dynamics is stable in this case. For example, the debt equation converges to the value d ∞ = − pd for constant pd and < 0 . Note that the assumption t+i < 0 ∀i is very strong as it ignores the fact that t+i might increase above 0 for a large i = I , possibly in response to a change in pd t+i . 3 In some publications (e.g. Peters et al. 2019), the level Z is shown as an indicator. This total debt adjustment is rarely used as an fiscal policy indicator in a decision making context. 4 We only make use of subscripts when necessary. 5 Werding et al. (2020) describe the possibility of splitting the S2 indicator, as does the Debt Sustainability Monitor (DSM) of the European Union (European Commission, 2018. In the DSM, this is referred to as Initial Budgetary Positions (IBP) and Cost of Ageing (CoA), with primary deficits reflecting ageing costs.

Explicit debt, S2(d) sub-indicator
The sub-indicator is described by S2(d) = d t . The S2(d) sub-indicator shows which annuity payment k would have to be chosen to satisfy the IBC if total debt consisted of explicit debt only. To keep government debt constant at the level of year t , a S2(d) sub-indicator would always cover (growth-corrected) interest expenditures from current primary surpluses. With d t = 90% and � pd t+i = 0%∀i > 0 , and a growth-corrected real interest rate of = 2% , according to S2(d) the IBC would be satisfied with an annual primary surplus equal to S2(d) ∶= k = 0.02 * 0.9 = 1.8% of GDP. The following general statements regarding the S2(d) sub-indicator can be made: (a) The higher the initial debt level d t , the higher the value of S2(d).
(b) The higher the growth-corrected real interest rate γ , the higher the value of S2(d).

Future primary deficits, S2(pd) sub-indicator
To understand the S2(pd) sub-indicator, it is instructive to look first at the isolated effect of the weighting of time and then at its interaction with the projected trajectories of primary deficits.
Step I: Weighting of time Figure 1 shows periodic and cumulative weights for the case where a S2 indicator is calculated for year t = 2019 , i.e. S2 t=2019 . As in the usual present value calculation, Eq. (2) weights a single future period t + i by a factor of 1 (1+ ) i . Figure 1a plots this for alternative values of γ. The usual result holds: when the growth-corrected real interest rate is γ > 0, more distant time points are weighted less. This effect is stronger the larger γ is.
Alternatively, the "importance" of a period can be put in relation to all other considered . The weighting of a period in this case  Figure 1c further shows the cumulative weights from t = 2020 to t = 2080 (i.e., 1 ≤ i ≤ 61 ). When calculating the S2 indicator for the year t = 2019, all future periods ( ≥ 1, i. e. ≥ 2020) together always yield exactly a cumulative weight of 100%, i.e. ∑ ∞ i=1 w t+i (γ) = 1 . In this context, distant periods have a larger weight for small γ. For example, with = 0.5% the period for t > T = 2070 receives a weight of Step II: Weighting of time in interaction with trajectories of primary deficits. Now, we consider the entire S2(pd) sub-indicator. The S2(pd) sub-indicator shows which annuity payment would have to be chosen to meet the IBC if total debt consisted solely of future primary deficits. The sub-indicator is described by S2(pd) For purposes of illustration, Fig. 2 shows three examples for trajectories of primary deficits: (+) strongly monotonically increasing primary deficits, (o) comparatively moderately monotonically increasing primary deficits, and (-) monotonically decreasing primary deficits. By assumption, the values of the primary deficits for t ∈ [2071, ∞) remain at the value as in T = 2070 . The lower the value of γ, the more weight is put on distant time points.
The following general statements can be made regarding the S2(pd) sub-indicator: Ceteris paribus, (a) The higher the value of p d t+i for any i ≥ 1 , the higher the value of S2 t (pd). (b) In the case of monotonically rising primary deficits, cf. Fig. 2a,b: -the higher the growth-corrected real interest rate , the more weight is put on earlier periods and the lower the value of S2 t (pd). -In earlier periods, k < p d t+i ; in later periods, k > p d t+i .  (2020). 7 It can be seen that due to holds. Accordingly, the annuity payment is described by the level of k that ensures that the w t+i -weighted differences of annuity payments and primary deficits add up to zero.
Empirica (2022) 49:1009-1030 -The higher the growth-corrected real interest rate , the more weight is put on earlier periods and the higher the value of S2 t (pd).
(c) In case of monotonically declining primary deficits, cf. Fig. 2c: -the higher the growth-corrected real interest rate , the more weight is put on earlier periods and the lower the value of S2 t (pd).
(d) The more the primary deficits increase/decrease in level, i.e., the more extreme the difference between near-time and far-time primary deficits, the more S2 t (pd) responds to a change in .
(e) Long-term projections are more uncertain than short-term projections. Since with lower γ, more distant, and thus, more uncertain projections receive more weight, the S2(pd) indicator itself becomes more uncertain.

Two drawbacks of the S2 indicator
The S2 indicator is associated with two drawbacks. First, sensitivity to changes in the growth-corrected interest rate is not immediately transparent. Second, projection periods in the very distant future significantly drive the indicator's value.

Non-unique dependence on effective interest rate
The S2 indicator's sensitivity in response to a change in the growth-corrected real interest rate can be illustrated through numerical examples: Fig. 3 shows S2(d), S2(pd) and S2 for the primary deficit trajectories shown in Fig. 2, for an assumed initial debt level of d t=2019 = 90% , and for various values of (horizontal axis). The sub-indicator of explicit debt S2(d) is clearly positively dependent on γ . The indicator increases from 0.5% * 90% = 0.45% for γ = 0.5% to 2.5% * 90% = 2.25% for γ = 2.5% , i.e. the lower the effective interest rate, the lower the sustainability gap according to the S2(d) sub-indicator. This is independent of the trajectory of primary deficits and thus holds for all three cases described above. The dependence of the S2(pd) sub-indicator on γ , on the other hand, depends on the paths of primary deficits. In the case of increasing primary deficits, S2(pd) decreases when γ increases, cf. (a) and (b), i.e., the lower the effective interest rate, the higher the sustainability gap according to the S2(pd) sub-indicator. In the case of declining primary deficits, S2(pd) increases when γ increases, cf. (c), i.e., the lower the effective interest rate, the lower the sustainability gap according to the S2(pd) sub-indicator.
The overall interest rate sensitivity of the S2 indicator depends on how the interest rate sensitivities of the two sub-indicators relate to each other quantitatively. When primary deficits are projected to increase, the interest rate sensitivity of the S2 indicator is relatively small because the sensitivities of S2(d) and S2(pd) with respect γ run in opposite directions and partially offset each other. The direction of change of the S2 indicator is not even easily predictable a priori in this case. In example (a), the S2 indicator follows a U-shaped path. In example (b), the indicator increases continuously because the change in S2(d) outweighs the change in S2(pd) for all γ considered. Since in the case of falling primary deficits in example (c) both S2(d) and S2(pd) increase when γ increases, the response of the S2 indicator is clear and interest rate sensitivity is high. 8

Dominant importance of distant forecast periods
When the effective interest rate γ decreases, the influence of the S2(d) = d t γ sub-indicator decreases. At the same time, the S2(pd) sub-indicator becomes comparatively important, cf. Figure 3. Decomposing the S2(pd) sub-indicator into subperiods also provides an understanding of the contribution of the periods 2020 ≤ t ≤ 2070 = T and t > 2070 , cf. Fig. 4. To show this, we split S2(pd) for the year 2019 into two projection sub-periods, i.e.
. In particular for low values of γ the projection period after 2070 receives a lot of weight. As in Figs. 3 and 4 shows the plots of the S2 indicator and the S2(pd) sub-indicator for all three cases described above. In addition, the staggered bar graphs indicate how quantitatively important the sub-periods T > 2070 and T ≤ 2070 are for the level of the S2(pd) sub-indicator. For instance, for ≈ r − g = 0.5% , the S2 indicator in example (a) shows a value of 3.31%. The value is composed of S2(pd) = 2.86% and S2(d) = 0.45%. The period after T = 2070 is very important with S2(pd) >T = ∑ ∞ i=52 w t+i (γ) � pd t+i = 2.71%. Thus, in total, the forecast of primary deficits for the period after 2070 explains

S2 indicator: a new decomposition
It has been shown that the S2 indicator does not respond in an easily determinable way to a change in γ. Moreover, the indicator is highly dependent on assumptions regarding the very distant future. In order to interpret the S2 indicator in a meaningful way against the background of non-transparent interest rate reactions, we amend the model which allows for a new decomposition of the S2 indicator. The decomposition is presented below.

Debt decomposition with premia
The starting point is the observation that many sovereigns pay interest rates r on their sovereign debt that are empirically lower than capital market interest rates m on other asset classes, i.e. r < m . Investors are therefore willing to forego part of the yield achievable on the capital market in order to hold government bonds and thus pay a premium (convenience yield, seignorage). This premium has been justified, among other things, by government bonds having a liquidity and safety feature (Reis 2021; Bayer et al. 2021;Krishnamurthy and Vissing-Jorgensen 2012). In this regard investors value that government bonds carry minimal credit risk and are highly liquid. Other complementary explanations emphasize the role of market power. The argumentations follow the line that with an increase in mark ups, profits increase, and hence drive a wedge between the average return to capital and the interest rate on government bonds (Fahri and Gourio 2018;Eggertsson et al. 2021;Ball and Mankiw 2021).
If the government had to pay the standard capital market interest rate m, or in growthcorrected form ∶= (1+ m) (1+g) − 1 , for its government debt, the adjustment requirement to comply with the IBC would be different from the adjustment requirement Z shown in Eq. (1). The difference in the adjustment requirement due to the difference between δ and γ (premium) represents a premium revenue. Equation (1) can be rewritten in light of the above consideration into a part that captures the value of government debt as the sum of a fundamental value ( v t ), the premium revenue, and the residual adjustment requirement Z from above: The fundamental value v t does not depend on γ . In contrast, the premium revenue, (δ − γ) PV(d), depends on γ 9 : Imagine an initial situation with positive government debt, in which the government pays positive interest equal to the capital market rate of return ( m = r → = ) , there is a positive amount of debt throughout (d t+i > 0 ∀i) , and hence PV(d) > 0 ∀t also holds. If, ceteris paribus, the interest rate r is lowered from this starting point, this has an effect on the premium revenue via two channels: • On the one hand, the interest rate differential (m-r) increases and so does ( − ) . This is favorable for the government. For a given PV (d) > 0 this results in a positive premium revenue − the government additionally generates an implicit surplus. • On the other hand, the present value of all future government debt, PV(d), decreases because it grows more slowly due to the lower interest rate (d t+1 = d t (1 + ) + pd t+1 ) . In other words, the amount of government debt at which the interest rate differential can be exploited decreases.
The decomposition of government debt presented here can be used for the S2 indicator.

S2 indicator decomposition with premia
Based on Eq. (4) and in analogy to the derivation of the S2 indicator in Sect. 2, the S2 indicator can now be decomposed into three parts. (4) As long as the economy is dynamically efficient (g < m and therefore δ > 0 ), Eq. (4) is clearly defined. If the interest rate differential in a dynamically efficient economy is sufficiently large so that not only r < m holds but even r < g < m holds, then a deficit can be financed permanently due to the premium and debt converges to a constant level.
Here PV(d;S2) describes the present value of debt given an adjustment according to the S2 indicator. The new decomposition of the S2 indicator is similar to the original decomposition from Eq. (3). F2(d) and F2(pd) correspond to S2(d) and S2(pd) in Eq. (3), with the difference that is now used instead of , thus replacing the government bond yield r with the capital market yield m. 10 The sum of F2(pd) and F2(d) gives a variant of the S2 indicator when valued at the growth-corrected capital market rate and is independent of . By analogy with the interpretation introduced in Sect. 2, the sum of F2(d) and F2(pd) describes by how much the primary deficit would have to be permanently adjusted to satisfy the IBC if the government debt were to earn interest at the capital market rate m. This is not the case, however. In fact, the interest rate on government debt is not m , but r . The difference is reflected in F2(S). For example, if m − r > 0 (hence − > 0 ) and PV(d;S2) > 0 , then F2(S) > 0 , i.e. the S2 indicator decreases due to the implicit surplus being favorable to the government. Note that PV(d; S2) depends on the path of d t+i ∀i ≥ 1 which inter alia depends on S2. From Sect. 2 we know that the response of S2 with respect to changes in γ crucially depends on the trajectory of primary deficits. As a result, the response of PV(d;S2) with respect to changes in γ hinges on the trajectory of the primary deficits as well. In other words, F2(S) being a function of PV(d; S2) inherits the opaqueness of S2, and therefore is less straightforward to interpret than PV(d) = PV(d;S2 = 0). Furthermore, note that the S2 indicator can increase even with a favorable interest rate differential m − r > 0 if PV(d;S2) < 0. 11 In Fig

Applying the new decomposition
Overall, the new decomposition can be used to make the comparison of the S2 indicator across (policy) scenarios, time, and countries more transparent. Consider the example of a comparison between country A and B, S2 j=A and S2 j=B : In Fig. 6 below we assume the primary deficits for country A to follow the path of the ( + ) scenario and for country B to follow the path of the ( − ) scenario of the examples introduced in Sect. 2. In country A the growth adjusted real interest rate A is assumed to be 1.0% , and of country B it is assumed to be B = 0.5% . Like above, both countries are assumed to start with a debt level of 90%. Furthermore, we assume A = B = = 2.5%. On the left-hand table one can see that S2 A = S2 A (d) + S2 A (pd) > S2 B . On the righthand side table we observe that F2 A (d) + F2 A (pd) < F2 B (d) + F2 B (pd) . Discounting with instead of leads to F2 A (d) = F2 B (d) because both countries start with the same debt level. Additionally, the comparison of the sub-indicators reflecting the effect of the primary deficits is revealing. It holds that S2 A (pd) > S2 B (pd) but F2 A (pd) < F2 B (pd) . The reason is that distant periods are discounted more heavily given the assumption of > . This leads to F2(pd) < ( >) S2(pd) if primary deficits are projected to be increasing (decreasing). Hence, with > the decomposition allows to shift more weight of the analysis to the part of the sustainability gap that is less dependent on the distant future. However, as discussed in sub-Sect. 3.2 the part capturing PV(d;S2) still hinges fundamentally on the assumptions regarding the very distant future.
Moreover, the effect of the implicit return earned by the government due to the premia is very different comparing both countries, F2 A (S) < F2 B (S) . The reason is a combination of different ( − ) differentials and present values of debt PV(d;S2) . In other words, the sum F2 j (d) + F2 j (pd) allows to compare sustainability gaps as if there were no additional effects through premia in a ceteris paribus framework.

Discussion of the new decomposition
Comparisons across countries, time, and/or (policy) scenarios hinge on choosing appropriate values for . How should be chosen? We motivate the choice of by the fact that m > r (Reis 2021). As a result, for a given country j and time i it holds that j,i ≈ m j,i − g j,i > r j,i − g j,i ≈ j,i . However, in comparisons across time for a given country j (and similarly across countries given a point in time i) it might hold that j,i ≠ j,i+1 even if m j,i = m j ∀i, simply because g j,i ≠ g j,i+1 . 12 If one wants to separate the effect of different (or changing) growth-adjusted real interest rates j,i = r j,i − g j,i across countries (or over time), i,j should be chosen to be the same. In fact, our decomposition approach does not rely on a very specific value for other than being larger than . To put it differently, the observed relationship m > r rather motivates the assumption > than it does necessarily dictate a specific value for . Instead, in analyses the level of can be chosen taking additional considerations, such as uncertainty regarding the future or risk aversion, into account.
Differences in sustainability gaps across countries and time may be due to fiscal policy decisions, differences in projection assumptions, the economic environment and/or different discount rates. Our new decomposition facilitates explaining differences in sustainability gaps caused by different discount rates. In this type of analysis is imposed a ceteris paribus assumption as it abstracts from the fact that primary deficits and the level of (or ) may interact in a direct way. Relatedly, a question to ask is: What drives premia, and to what extent should these premia be considered as exogenously given or endogenous? More specifically, to what extent do government spending behavior and monetary policy decisions affect m and r ? Reis (2021) explains premia as arising from liquidity and safety, and studies how fiscal (redistributive) policies, inflation and financial repression have an influence on premia. Additionally, governments may face a trade-off in exploiting the premium by increasing public debt and sustaining the premium by not increasing public debt levels too much. Bayer et al. (2021) have made some progress in that direction of research. Future research should expand the analysis to shed light on the effect of fiscal and monetary policies on premia within a monetary union. For instance, under which circumstances does a centralized monetary authority affect premia of member states differently. Similarly, do fiscal policies of a country have spill-over effects on premia of other countries? Answering these questions remains for future research.
We have shown that the S2 indicator can be decomposed in various ways (e.g. projection intervals, premia) to enrich the analysis of the long-term sustainability of public debt. The broader result is that the decomposition suggested in our paper can be extended in many ways. For instance, one could decompose the present value of debt,

Conclusion
The S2 indicator (sustainability gap) is an established measure for assessing the long-term sustainability of public finances and shows the level of fiscal policy action required, taking into account trajectories of future primary deficits. The factors influencing the S2 indicator are diverse, and in some cases may be irritating for non-specialists.
First, a change in the discount rate acts through two channels, sometimes with opposite effects. These can be illustrated by representing the S2 indicator as the sum of two annuities-one annuity translating the baseline debt level (S2(d)) and a second annuity translating all primary deficits expected in the future (S2(pd)). Clearly, the first annuity (S2(d)) increases (decreases) as the discount rate increases (decreases). For the second annuity (S2(pd)) the response depends on the trajectory of primary deficits since the discount rate affects the weighting of primary deficits over time. When the discount rate is increasing (decreasing), primary deficits that are close (distant) in time are weighted more heavily. For instance, this means that when the discount rate is low, the indicator gives strong weight to projected primary deficits in the distant future. The sum of the two annuities (S2(d) + S2(pd)) responds to a change in interest rates in a clearly determinable direction only when deficits are projected to decline. When primary deficits are projected to increase over time, the direction of change in the S2 indicator is not easily determinable, since the interest rate responsiveness of S2(d) and S2(pd) partly cancel each other out.
Second, for low values of r-g, the influence of highly uncertain projections for distant periods on the S2 indicators is very large. This feature may be desirable as long-term analyses are meant to uncover long-term risks. Yet, long-term projections are more uncertain than short-term projections. Hence, the S2 indicator itself becomes more uncertain. For example, projection values for periods after 2070 explain up to 80% of the S2 indicator in some scenarios with r-g in a range of 0.5%. Importantly, since projected primary deficits are to a large extent driven by projected future policies, one important driver of uncertainty is the credibility of the implementation of these policies.
We propose a new decomposition of the S2 indicator to capture these two properties. The decomposition makes use of the observation that many sovereigns pay interest rates on their sovereign debt that are empirically lower than capital market interest rates on other asset classes. In other words, our decomposition explicitly takes into account the notion of premia (Reis 2021). These premia may differ across countries, (policy) scenarios, and over time. Therefore, explicitly introducing premia makes a comparison of sustainability gaps more transparent. Additionally, our new decomposition mitigates the reliance of highly uncertain projections for the distant future. By imposing the assumption that > , the decomposition allows to perform the sustainability analysis by attaching more weight to less distant periods, and thus, is less prone to projection uncertainty.
Sustainability gaps indicate a need for action but do not constitute a recommendation for a particular fiscal policy. The assumed constant adjustment level of the primary deficit S2 ∶= k is the technical result of the assumed annuity translation under assumptions and does not specify the adjustment path of the primary deficit that should be targeted with respect to welfare maximization. This is evident, among other things, from the property of non-uniqueness in changes to the indicator, which can show the same indicator value for different combinations of initial debt, trajectories of discount rates and primary deficits (see Sect. 2). For instance, in most cases, the policy recommendations for a particular value of the S2 indicator would be different if primary deficits were trending upwards than if primary deficits were trending downwards.
Overall, the S2 indicator remains an important measure for estimating the need for action to ensure fiscal sustainability. However, in contrast to purely present-oriented indicators (e.g. debt-to-GDP ratio), the notion that there can be only one indicator value for forward-looking indicators needs to be overcome. The S2 indicator (and its decompositions) is an indicator alongside debt-to-GDP ratios, primary deficits and refinancing costs. In the case of low discount rates, analysing S1 indicators that refrain from including infinite time periods, may be a complementary tool to S2 indicators. 13 However, the orientation on S1 indicators relinquishes the IBC for a target value for debt levels and adjustment periods which are rather arbitrary. Under no circumstances should S1/S2 indicators be the sole and binding benchmark for political action. Thus, even with primary surpluses, − � pd t+I > 0 , an "explosion" of the debt ratio can occur. For sustainable government finances, the primary surpluses necessary for debt sustainability depend on the level of the debt ratio at time t + I . In case (a), the debt explodes, i.e. a classic Ponzi game is played. In case (b), savings are even infinite in the long run, which cannot be optimal in welfare terms, since consumption opportunities would be foregone (transversality condition, TVC). In case (c) the debt ratio converges to a constant level and only in this case the Ponzi condition and the TVC is fulfilled.
for debt levels and adjustment periods which are rather arbitrary. Mechanically however, the S1 indicator shares many properties of the S2 indicator. Formally, the S1 indicator is described as where is defined as = and w t+i ( ) = 1 (1+ ) i . It should be noted that, unlike the S2 indicator, no special restrictions need to be placed on the interest rate. However, analogously to the S2 indicator the development of S1(pd) does not have an unambiguous response to changes in the interest rate-growth differential. The dependence on distant projection periods can be gauged by the distance of the target point t + I.
Amending the S1 indicator to allow for a decomposition analogous to the decomposition for the S2 indicator yields the following result: