## _wp1211

## Source details

**Canonical URL:** [_wp1211](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1211.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1211.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1211.pdf.json)

---

### I. Introduction — purpose and scope
- Presents a range of indicators and analytical tools for assessing fiscal vulnerabilities and risks for advanced economies.
- Tools organized mainly by time horizon to highlight distinct elements of fiscal risks and vulnerabilities:
  - Short-term indicators: size of gross financing needs; high-frequency market-based measures of sovereign default risk (CDS, RAS); measure of potential spillovers in distress dependence among advanced economies.
  - Medium- to long-run indicators: required fiscal effort to stabilize debt; impact of adverse growth and interest rate shocks on baseline debt trajectory; probabilistic measure of the debt outlook via stochastic simulations.
- Objectives guiding tool choice:
  - Ensure wide coverage of advanced economies.
  - Minimize time lag of data availability.
  - Allow relative assessment (ranking) of countries to feed the joint IMF-FSB Early Warning Exercise (EWE).
- Notes relation to prior work: complements Baldacci, McHugh, and Petrova (2011); builds on Cottarelli (2011) framework and relates to early warning system approaches reviewed by Baldacci et al. (2011).

### II. Short-term pressures: indicators and interpretation
- Gross funding needs
  - Two complementary fiscal indicators:
    - Measure of gross financing needs in the current and following year:
      - Practical assumptions: (1) current year fiscal deficit financed linearly over the year (e.g., when indicator calculated at end-June, assume half of projected deficit for the current year still needs financing); (2) short-term debt projected to mature in current year assumed refinanced by new short-term debt that then falls due in next year.
    - Current stock of general government debt divided by the average debt maturity: indicates average amount of debt to be refinanced or repaid in future years.
  - Mitigating factors: government liquid assets, privatization proceeds, availability of non-market funding.

- Markets’ perceptions of sovereign default risk
  - Sovereign CDS spreads:
    - Measure cost of insurance against sovereign default; quoted as a percentage of the notional amount insured.
    - Five-year sovereign CDS contracts are generally the most liquid markets.
    - Sovereign CDS contracts are quoted in U.S. dollars, except for the United States for which they are quoted in euros.
  - Relative Asset Swap (RAS) spreads:
    - RAS spread = Ri – RSWi where Ri = yield on 10-year government bonds issued by country i; RSWi = 10-year fixed rate on interest rate swaps in the currency of country i.
    - Abstracts from exchange rate risk; negative RAS spreads indicate government paper assessed as less risky than swap flow.
  - Cautions:
    - Market indicators depend on factors beyond fiscal vulnerabilities (short-term interest rates, large-scale central bank purchases, perceived financial sector strength, expected global growth, VIX).
    - Market indicators tend to lag deterioration of the fiscal outlook.
    - Government bond yields are weaker measures of default risk than CDS and RAS.

- Distress dependence among sovereigns (spillovers)
  - Spillover coefficient: probability of sovereign distress in one country given default in another.
  - Methodology (Caceres, Guzzo, and Segoviano, 2010):
    - (1) Extract marginal probabilities of default from CDS spread series.
    - (2) Compute joint and conditional probabilities using a non-parametric technique.
    - (3) Compute spillover coefficient as weighted sum of conditional probabilities.
  - Caveats: rankings influenced by market perceptions; country coverage currently restricted to fourteen countries at a time.

### III. Medium- and long-run pressures: indicators I1 and I2
- Indicator I1 (medium- to long-term adjustment needed)
  - I1: cyclically-adjusted primary balance needed to reach a specified debt target by a reference date.
  - Illustrative setup:
    - Primary balance to be realized by 2020 and maintained until 2030 to achieve a given debt ratio by 2030.
    - Gross debt target set at 60 percent of GDP by 2030 (pre-crisis median) or the end-2012 level if end-2012 gross debt-to-GDP ratio is less than 60 percent.
    - For countries with large financial assets, net debt targets are assumed.
  - I1 is a function of:
    - initial and target level of debt;
    - path of the cyclical component of the primary balance;
    - projected increase in age-related spending;
    - initial cyclically-adjusted primary balance.
  - Decomposition components (applies to both indicators):
    - Effect of initial debt (B0 and BT).
    - Size of age-related costs during 2013-30.
    - Initial cyclically-adjusted primary balance.
    - Projected path of the cyclical component of the primary balance.
  - Assumptions used in illustrations:
    - Growth-adjusted interest rate r: r = 0 up to 2015; r = 1 percent thereafter.
    - Primary balance projections: current year and next five years from the latest WEO; thereafter add age-related expenditure flow until 2050; from 2050 onward primary balance-to-GDP ratio constant at 2050 level.

- Indicator I2 (very long-run stability)
  - I2: implied primary balance in 2016 consistent with stabilizing the debt level in the very long run and satisfying the intertemporal budget constraint.
  - Calculation relies on S2 sustainability indicator formula (European Commission): S2 uses tPB including cyclically-adjusted primary balance, increase in aging related costs after 2016, and cyclical component.
  - I2 = baseline primary balance in 2016 + required adjustment (S2).
  - Rationale: I2 focuses on the level of required primary balance that must be maintained, avoiding S2’s potential understatement when baseline includes significant fiscal adjustment.

### IV. Vulnerabilities to shocks: scenario design and practical assumptions
- Growth shock scenario
  - Practical assumption: lower real GDP growth by one percentage point compared to baseline (in many illustrations specified as 1 percentage point in the current year and the next five years).
  - Additional assumptions:
    - Potential GDP not affected by adverse developments.
    - Governments do not take discretionary corrective actions.
  - Transmission to primary balance:
    - pb_t = pb_WEO + η_R * (change in output gap) - η_G * (change in output gap)
    - Where η_R and η_G are semi-elasticities of revenues and expenditures to changes in the output gap.
  - Debt dynamics equation used:
    - d_t = (1 + r_t) / (1 + g_t) * d_{t-1} + pb_t.
  - Representative numeric pointers:
    - Primary balance year reference: 2016.
    - Practical shock magnitude: one percentage point (lower real GDP growth).
    - Historical context date: summer months of 2011.
    - Example debt-target figures:
      - Japan: net debt target is 80 percent, which corresponds to a gross debt target of 200 percent.
      - Australia, New Zealand and Canada: net debt target is 40 percent.
    - Empirical finding: for countries with debt ratios above 90 percent of GDP, a 10 percentage point increase in the initial debt-to-GDP ratio is associated with a slowdown in annual real per capita GDP growth of around 0.2 percentage points per year.

- Interest rate shock scenario
  - Simpler option: apply a common shock (example given: 100 basis points over the medium term) to all countries.
  - Impact computation:
    - c̃_it = t_nf × ĩ_t
    - Where c̃_it is difference in net financing costs relative to baseline, t_nf is accumulated amount of debt in percent of GDP financed since beginning of shock, and ĩ_t is deviation in interest rate relative to baseline.
  - Computation considerations:
    - Impact depends on amount of new borrowing subject to higher rates, gross financing needs, and debt maturity structure.
    - When future maturity structure unavailable, constant maturity structure assumption may be used.
    - Interest income on financial assets assumed to change by same amount as interest rate for new borrowing.
    - Average maturity of financial assets assumed to be five years for all countries where data sparse.

### V. Stochastic simulations: methodology and risk indicators
- Methodology overview
  - Stochastic model calibrated on past macroeconomic and financial shocks (growth, interest rates, exchange rate) and average policy response.
  - Shocks are random draws from joint normal distribution with zero mean and variance-covariance matrix from an unrestricted VAR.
  - Unrestricted VAR generates macro projections consistent with shocks and dynamics.
  - Key output: probability distributions of public debt depicted in “fan charts” showing likelihood of deviations from planned debt trajectory.
- Two risk indicators from simulations
  - “Good” outcome indicator: probability that the five-year-ahead model-based projected debt ratio will not be greater than the maximum median debt level observed during the seven years prior.
  - “Bad” outcome (“tail event”) indicator: the model-based projected five-year-ahead debt ratio corresponding to the 90th percentile.

- Fiscal reaction function and calibration
  - Fiscal reaction function estimated on panel of 33 advanced economies, annual data, up to 21 years (1990-2010); fixed effects specification chosen.
  - Key behavioral estimates:
    - Persistence of primary policies: coefficient of lagged primary balance is 0.74.
    - Cyclical sensitivity: countercyclical with 0.58 percent of GDP increase in primary deficit for each 1 percentage point widening of the output gap during recessions (asymmetric; no tightening during booms).
    - Fiscal response to rising debt: coefficient 0.04 percent of GDP on average.
  - Calibration uses unrestricted quarterly VAR for 1990-2010 with variables: real output growth, ln(real effective exchange rate), real long-term domestic bond yields, real long-term foreign bond yields.
  - Inputs to probability distributions:
    - Fiscal reaction coefficients, normally distributed fiscal policy shock with country-specific variance, most recent WEO projections, estimated VAR variance-covariance matrix.
  - Adjustments to projected growth-adjusted effective interest rates:
    - Observed divergence between historical and projected rates averaged -81 basis points and ranged up to 337 basis points.
    - Adjustment approach uses three sizes only:
      - 130 basis points reduction (applies to 21 economies).
      - 200 basis points increase (applies to three economies).
      - 300 basis points increase (applies to two economies).

### VI. Illustrative empirical findings (selected advanced economies)
- Gross funding needs (short and medium term)
  - Japan: financing needs in remainder of 2011 and 2012 exceeded 60 percent of GDP by the data cut-off.
  - Greece and the United States: about 30 percent of GDP.
- Market perceptions and spillovers (end-August 2011 / July 2011 context)
  - Sovereign CDS and RAS spreads surged for Greece; stabilized in Portugal; came down sharply in Ireland (since July 2011).
  - Spillover coefficient: Greece, Portugal, and Ireland identified as largest sources of contagion; their spillover coefficient almost doubled since March 2011.
  - Germany, Japan, and the United States appear most resilient to sovereign risk contagion by the spillover coefficient.
- Medium- and long-run primary balance requirements
  - Greece, Japan, and the United States would need to reach and maintain a cyclically-adjusted primary surplus of more than 7 percent of GDP to achieve a debt-to-GDP ratio of 60 percent of GDP by 2030 (indicator I1).
  - United Kingdom requires a higher primary surplus in the very long run (indicator I2) than Japan because of projected age-related spending increases:
    - United Kingdom: more than 9½ percent of GDP over the next four decades.
    - Japan: about 2 percent of GDP.
- Impact of shocks (selected results)
  - Impact of a permanent increase in interest rates on average financing costs during 2011-16 is largest in Japan, Greece, and the United States, ranging between 1 and ¾ percent of GDP for these three countries.
  - Impact of a one percentage point reduction in real GDP growth over 2011-16: largest impact for Japan and Greece; United States’ impact mitigated by stronger trend growth and smaller automatic stabilizers.
- Stochastic simulations (medium-term risks to public debt trajectories)
  - Trajectories most unfavorable in Japan, Greece, and the United States: all three have a less than fifty percent probability to stabilize their debt ratio by end-2016.
  - For the United States, the debt level not to be exceeded with 90 percent probability is substantially lower than for Japan and Greece.
  - Germany and the United Kingdom: simulations predict a greater than 50 percent probability to stabilize debt ratios over the next five years based on random shocks and past fiscal behavior.

- Selected model median vs WEO median debt projections (model median in 2016 vs WEO)
  - Australia: 24.8 20.2 / 24.1 20.6
  - Austria: 72.1 72.2 / 69.5 69.8
  - Belgium: 95.9 93.0 / 99.4 98.8
  - Czech Republic: 45.9 46.3 / 47.9 47.6
  - Denmark: 55.8 49.1 / 47.6 40.2
  - Finland: 51.7 52.3 / 59.4 61.1
  - France: 86.2 87.7 / 87.3 86.7
  - Germany: 80.1 76.6 / 78.3 71.9
  - Greece: 225.8 166.3 / 209.9 145.5
  - Ireland: 131.7 114.3 / 141.2 121.5
  - Italy: 103.4 112.9 / 118.0 118.0
  - Japan: 261.3 253.3 / 264.9 250.5
  - Korea: 14.1 22.2 / 10.6 19.8
  - Netherlands: 63.7 64.4 / 64.9 64.4
  - Portugal: 100.0 110.5 / 87.4 106.5
  - Spain: 77.0 77.4 / 74.1 75.9
  - Sweden: 35.0 19.3 / 39.6 22.8
  - Switzerland: 45.9 46.3 / 47.2 45.7
  - United Kingdom: 85.3 80.0 / 88.0 81.3
  - United States: 106.0 115.4 / 108.8 111.9
  - Note: Model-based values refer to median of 1000 replications; divergences with WEO can reflect financing operations (e.g., EU/IMF programs) included in WEO but not in the model.

### VII. Policy implications and recommended considerations
- Consider structural differences across countries when designing policy responses: trend growth, initial debt stock, automatic stabilizers, maturity profile.
- Trade-offs to manage:
  - Short-term financing needs versus medium-term maturity management (e.g., lengthening average maturity to reduce rollover vulnerability).
  - Potential feedback loops from higher debt/gross financing needs to higher interest rates via risk premia.
- Country-specific mitigation examples from illustrative sample:
  - United States: ensure strong trend growth to support debt dynamics.
  - Japan: contain age-related entitlement spending pressure to manage medium- and long-term pressure.
  - United Kingdom: maintain long average debt maturity to keep funding needs in check.
- Recommended approach: use a set of indicators and cross-country comparisons to identify and communicate key areas of concern and combine these with forceful fiscal measures addressing country-specific weaknesses.

*Content derived from the PDF chapter titled "References" in the source document _wp1211 - References.*

### References .............................................................................................................

### _wp1211 - References .............................................................................................................

### I. INTRODUCTION
- Recent developments in international financial markets reaffirm that concerns over fiscal sustainability can precipitate a crisis in advanced as well as emerging economies.
- Persistent fiscal imbalances can result in high levels of general government debt that raise concerns about sovereign debt rollover and, in the extreme, solvency, threatening macroeconomic stability and weighing on economic growth.
- The paper presents a range of indicators and analytical tools for assessing fiscal vulnerabilities and risks for advanced economies.
- Six tools are organized mainly by time horizon to highlight related but conceptually distinct elements of fiscal risks and vulnerabilities.
- Short-term indicators: size of gross financing needs; high-frequency market-based measures of sovereign default risk; measure of potential spillovers in distress dependence among advanced economies.
- Medium- to long-run indicators: required fiscal effort to stabilize debt; impact of adverse growth and interest rate shocks on the baseline debt trajectory; probabilistic measure of the debt outlook reflecting risks associated with baseline debt projections.
- Additional aspects to be covered going forward include assessing the investor base and level of contingent liabilities.
- Three objectives guided tool choice and development: ensure wide coverage of advanced economies; minimize time lag of data availability; allow relative assessment (ranking) of countries to provide input for the joint IMF-FSB Early Warning Exercise (EWE).
- Relation to other work:
  - Complements Baldacci, McHugh, and Petrova (2011) by assessing shocks to baseline projections, capturing market-based risk indicators, and analyzing spillover risks with shorter data lags.
  - Builds on Cottarelli (2011) framework used in April and September 2011 Fiscal Monitors; data limits of that approach restrict country-level application.
  - Early warning system approaches reviewed by Baldacci et al. (2011); caveats include typical limitations of early warning models and data constraints.
- Paper structure: Section II overview of six tools; Section III illustrative examples using recent advanced economy data; Section IV conclusions, caveats, and suggestions. Appendix provides technical details.

*Authors acknowledge Julio Escolano and Manmohan S. Kumar for initiating and guiding early steps of the analysis; Phil Gerson and Martine Guerguil for comments. One author (G. Callegari) is currently with the European Central Bank; paper prepared during IMF tenure and does not reflect views of the ECB.*

### II. FISCAL VULNERABILITIES AND RISKS: DIMENSIONS, INDICATORS AND ANALYTICAL TOOLS

#### A. Short-Term Pressures: Gross Funding Needs, Market Perceptions, and Susceptibility to Spillovers

- Rationale:
  - Short-term pressures can cause financing gaps and impose additional costs if borrowing is available only at sharply higher interest rates.
  - Tools use relatively high-frequency fiscal and financial market data.
  - Distinction across time horizons is blurred in practice (e.g., liquidity strains can lead to insolvency; medium-term credible consolidation can alleviate short-term pressures).

- Gross Funding Needs
  - Funding needs determined by budget deficit size and maturing debt to be rolled over; high past debt with significant near-term maturities can create large new borrowing needs even if current deficits are small.
  - Mitigating factors: government liquid assets (e.g., deposits with central bank or banking system), privatization proceeds, availability of non-market funding.
  - Two complementary fiscal indicators:
    - Measure of gross financing needs in the current and following year: captures simultaneous need for large new issuance to finance a fiscal deficit together with refinancing large maturing debt.
      - Practical assumptions used: (1) current year fiscal deficit financed linearly over the year (e.g., when indicator calculated at end-June, assume half of projected deficit for the current year still needs financing); (2) short-term debt projected to mature in current year assumed refinanced by new short-term debt that then falls due in next year.
    - Current stock of general government debt divided by the average debt maturity: indicates average amount of debt to be refinanced or repaid in future years; countries with high stock and low average maturity are more exposed.

- Markets’ Perceptions of Sovereign Default Risk
  - High-frequency financial market indicators used as relative measures of sovereign default risk: sovereign Credit Default Swap (CDS) spreads and Relative Asset Swap (RAS) spreads.
  - CDS spreads:
    - Measure direct cost of insurance against sovereign default; quoted as a percentage of the notional amount insured.
    - For sovereign issuers, often considered the default risk premium associated with insurance against a specific type of government bond.
    - In deep competitive markets, CDS spreads should reflect expected default-related loss = probability of default × loss given default.
    - Sovereign CDS contracts are quoted in U.S. dollars, except for the United States for which they are quoted in euros. Five-year sovereign CDS contracts are generally the most liquid markets. Sovereign CDS contracts do not trade on their local currency due to potential of the country issuing money to prevent a credit event. Protection bought in local currency tends to be cheaper than in the standard currency because of the absence of protection against this risk.
  - RAS spreads:
    - RAS spread = Ri – RSWi where Ri = yield on 10-year government bonds issued by country i; RSWi = 10-year fixed rate on interest rate swaps in the currency of country i.
    - RAS spread corresponds to spread between sovereign bond yields and fixed interest rate arm in interest rate swap contracts.
    - Swap rate and bond yield are applied to principals denominated in same currency, so RAS abstracts from exchange rate risk or other currency-specific factors.
    - Negative RAS spreads indicate government paper assessed as less risky than flow of funds exchanged in swaps.
    - Interest rates on swaps are effectively free from the risk of default of sovereign issuers, although they entail some residual counterparty risks.
  - Cautions:
    - Financial market indicators are relative assessments and depend on factors beyond fiscal vulnerabilities, including short-term interest rates, large-scale sovereign bond purchases by major central banks, perceived strength of financial sectors (e.g., relative stock price movements), expected global growth, and volatility of equity prices as measured by the VIX index.
    - Market indicators tend to lag deterioration of the fiscal outlook.
    - Government bond yields are an even weaker measure of sovereign default risk relative to CDS and RAS because yields depend on business cycle position, inflation expectations, and exchange rate risks.

- Distress Dependence Among Sovereigns
  - Sovereign CDS spreads have shown high volatility and synchronicity; co-movement may reflect linkages across sovereigns (trade, capital flows, simultaneous bank support measures, cross-border financial institution activities) or common global factors (e.g., global risk aversion).
  - Tool to quantify distress dependence: computes a spillover coefficient = probability of sovereign distress in one country given default in another.
  - Methodology (Caceres, Guzzo, and Segoviano, 2010):
    - (1) Extract marginal probabilities of default from each individual CDS spread series at each point in time.
    - (2) Compute joint and conditional probabilities of default using a non-parametric technique.
    - (3) Compute spillover coefficient as weighted sum of probability of distress of each country given distress in other countries in the sample.
  - Interpretation: spillover coefficient measures exposure of each country to distress dependence from others.
  - Caveats:
    - Uses market CDS spreads as input, so market perceptions significantly influence rankings.
    - Country coverage currently restricted to fourteen countries at a time due to computational/program limitations.

#### B. Medium- and Long-Run Pressures and Susceptibility to Shocks

- Medium- and Long-Term Adjustment Needs to Ensure Fiscal Sustainability
  - Large primary deficits and high debt levels can become unsustainable if not corrected.
  - Scale of fiscal adjustment required to achieve debt targets serves as an indicator of vulnerability; accounts for projected increases in entitlement spending (pensions and health care).
  - Two indicators to gauge medium- and long-term adjustment needs:
    - First indicator (I1): cyclically-adjusted primary balance needed to reach a certain debt target by a reference date.
      - Illustrative calculations cover next two decades: primary balance to be realized by 2020 and maintained until 2030 to achieve a given debt ratio by 2030.
      - Gross debt target set at 60 percent of GDP by 2030 (pre-crisis median of advanced economies) or the end-2012 level if gross debt-to-GDP ratio that year is less than 60 percent.
      - For some countries with large financial assets, net debt targets are assumed.
      - I1 is a function of:
        - (i) initial and target level of debt;
        - (ii) path of the cyclical component of the primary balance;
        - (iii) projected increase in age-related spending;
        - (iv) initial level of the cyclically adjusted primary balance.
    - Second indicator: specifies the required primary balance to stabilize the path of the debt-to-GDP ratio (methodology similar to I1; see Appendix C for details).
  - Methodological note: approach similar to European Commission sustainability indicators used in its Sustainability Report for EU member states; practical applications calculated as described above.

*Italic source attribution: Content derived from the PDF chapter titled "References" in the source document _wp1211 - References.*

### Section III illustrates how these sources of the medium-term spending pressures differ

### _wp1211 - Section III illustrates how these sources of the medium-term spending pressures differ

### Indicator I2: long-run primary balance required to stabilize debt
- The second indicator (I2) shows the primary balance in 2016 that would be consistent with stabilizing the debt level in the very long run, in order to satisfy the intertemporal budget constraint.
- Because this indicator includes an infinite-horizon projection, it gives much more weight to long-run rather than short- or medium-term projections of the primary balance.
- Consequence: countries with larger increases in age-related spending will tend to have a higher required primary balance.

### Vulnerabilities to adverse growth and interest rate shocks
- Debt dynamics are sensitive to the economic outlook; the interest rate-growth differential is key for the future debt path.
- Countries can be analyzed according to their susceptibility to changes in both interest rates and growth.
- Context: uncertainty regarding the likely pace of economic recovery and the impact on countries’ public finances increased during the summer months of 2011.
- With higher debt levels, most advanced economies have become more sensitive to interest rate shocks even though yields remain low for many.

### Growth shocks: scenario design and practical assumptions
- The impact of a “low growth” scenario is gauged by adjusting the real GDP assumptions that enter the debt dynamics equations.
- Approaches to construct shocks:
  - Based on historical growth patterns.
  - By accounting for the level of debt.
  - By applying the same relative or the same absolute shock to all economies (the latter allows comparison of susceptibility to the same size shock).
- Practical assumption used:
  - Lower real GDP growth by one percentage point compared to the baseline projections.
  - Assume potential GDP is not affected by the adverse growth developments.
  - Assume governments do not take any corrective discretionary actions to smooth the impact.
- Under these assumptions, the shock affects the deficit and debt-to-GDP ratios.

### Representative numeric and empirical pointers preserved from the source
- Primary balance year reference: 2016.
- Historical context date: summer months of 2011.
- Practical shock magnitude: one percentage point (lower real GDP growth).
- Example debt-target figures cited in related calculations:
  - Japan: net debt target is 80 percent, which corresponds to a gross debt target of 200 percent.
  - Australia, New Zealand and Canada: net debt target is 40 percent.
- Empirical finding cited: for countries with debt ratios above 90 percent of GDP, a 10 percentage point increase in the initial debt-to-GDP ratio is associated with a slowdown in annual real per capita GDP growth of around 0.2 percentage points per year.

*Source: _wp1211 - Section III illustrates how these sources of the medium-term spending pressures differ*

### Section II. F describes a methodology that allows assessing the debt dynamics under stochastic shocks.

### _wp1211 - Section II. F describes a methodology that allows assessing the debt dynamics under stochastic shocks.

### Methodology overview: stochastic simulations for debt dynamics
- Uses a stochastic model of debt sustainability calibrated on past constellations of macroeconomic and financial shocks affecting debt dynamics (growth, interest rates, and the exchange rate) and on the average policy response to these shocks.
- Shocks to key macroeconomic variables are random draws from the joint normal distribution with zero-mean and the variance-covariance matrix of the unrestricted VAR estimates.
- The unrestricted VAR generates macroeconomic projections consistent with each series of shocks and the underlying economic dynamics.
- Key output: series of probability distributions of public debt depicted in “fan charts” indicating, for each year, the likelihood of deviations from the planned debt trajectory.
- Two risk indicators derived from the simulations:
  - “Good” outcome indicator: probability that the five-year-ahead model-based projected debt ratio will not be greater than the maximum median debt level observed during the seven years prior.
  - “Bad” outcome (“tail event”) indicator: the model-based projected five-year-ahead debt ratio corresponding to the 90th percentile value.

### Interest rate shocks: calibration and indicators
- Simpler option: apply a common shock (example given: 100 basis points over the medium term) to all countries rather than country-specific shocks to avoid extra assumptions on determinants of interest rates and risk premia.
- Two useful indicators to assess vulnerability:
  - Impact of a permanent increase in interest rates on average financing costs over the next five years (in percent of GDP) compared to the baseline.
  - Total average level of financing costs during the five-year period (captures resources needed for debt servicing).
- Computation considerations:
  - Impact over time depends on amount of new borrowing subject to higher rates, related to gross financing needs and debt maturity structure.
  - Calculating total impact requires accounting for debt maturing each year and being refinanced at higher rates (future maturity structure); when unavailable, a constant maturity structure assumption may be used.
  - For transparency, assume interest rates for financial assets and new liabilities increase by the same amount when accounting for interest income on assets.

### Growth shocks: transmission factors and calibration
- Impact of lower-than-expected real GDP growth varies with:
  - Trend growth (especially important for low-projected-growth countries like Italy and Japan).
  - Size of the pre-shock stock of public debt (relevant for Japan, Italy, Greece, Ireland, Portugal, the United Kingdom, and the United States).
  - Size of automatic stabilizers (more important for many European countries).
- Example calibration: a one percentage point reduction in real GDP growth over the next six years is used to compare impacts across countries.
- Automatic stabilizers and change in the GDP base are explicitly accounted for (see Appendix D referenced).

### Illustrative application: selected advanced economies (findings)
- Countries illustrated: Germany, Greece, Japan, the United Kingdom, and the United States (data draw on September 2011 Fiscal Monitor and the World Economic Outlook).
- Gross funding needs (short and medium term):
  - Japan: financing needs in the remainder of 2011 and 2012 exceeded 60 percent of GDP by the data cut-off.
  - Greece and the United States: about 30 percent of GDP.
  - Japan’s position reflects large fiscal deficit, high debt stock, and relatively short maturity profile of debt.
- Market perceptions and spillovers:
  - Sovereign CDS and RAS spreads surged for Greece; stabilized in Portugal; come down sharply in Ireland (since July 2011).
  - Spillover coefficient identifies Greece, Portugal, and Ireland as largest sources of contagion in the sample (based on CDS data as of end-August 2011); their spillover coefficient almost doubled since March 2011.
  - Germany, Japan, and the United States appear most resilient to sovereign risk contagion by the spillover coefficient.
- Medium- and long-run primary balance requirements:
  - Greece, Japan, and the United States would need to reach and maintain a cyclically-adjusted primary surplus of more than 7 percent of GDP to achieve a debt-to-GDP ratio of 60 percent of GDP by 2030 (indicator I1).
  - The United Kingdom requires a higher primary surplus in the very long run (indicator I2) than Japan because of projected age-related spending increases (United Kingdom: more than 9½ percent of GDP over the next four decades; Japan: about 2 percent of GDP).
- Impact of shocks (selected results):
  - Impact of a permanent increase in interest rates on average financing costs during 2011-16 is largest in Japan, Greece, and the United States, ranging between 1 and ¾ percent of GDP for these three countries (reflects high gross financing needs).
  - Comparing the impact of a one percentage point reduction in real GDP growth over 2011-16: largest impact for Japan and Greece; the United States’ impact is mitigated by stronger trend growth and smaller automatic stabilizers.
- Stochastic simulations (medium-term risks to public debt trajectories):
  - Trajectories most unfavorable in Japan, Greece, and the United States: all three have a less than fifty percent probability to stabilize their debt ratio by end-2016.
  - For the United States, the debt level not to be exceeded with 90 percent probability is substantially lower than for Japan and Greece.
  - Germany and the United Kingdom: simulations predict a greater than 50 percent probability to stabilize debt ratios over the next five years based on random shocks and past fiscal behavior.

### Policy implications and recommended considerations
- Policy responses and debt management strategies should consider:
  - Structural differences across countries (trend growth, initial debt stock, automatic stabilizers, maturity profile).
  - The trade-offs between short-term financing needs and medium-term maturity management (e.g., lengthening average maturity to reduce vulnerability).
  - The potential for feedback loops from higher debt/gross financing needs to higher interest rates via risk premia.
- Country-specific mitigation examples in the illustrative sample:
  - United States: ensure strong trend growth to support debt dynamics.
  - Japan: contain age-related entitlement spending pressure to manage medium- and long-term pressure.
  - United Kingdom: maintain long average debt maturity to keep funding needs in check.
- Use a set of indicators and cross-country comparisons to identify and communicate key areas of concern and policy requirements; combine these with forceful fiscal measures addressing weaknesses in specific areas.

### Technical highlights from appendices (methodological details)
- Spillover Coefficient (SC):
  - Uses sovereign CDS spreads to extract marginal probabilities of default P(A) and P(B).
  - Joint probability P(A,B) obtained via the CIMDO non-parametric methodology (Segoviano, 2006) based on Kullback cross-entropy, constrained to match observed individual probabilities.
  - Conditional probability P(A/B) = P(A,B) / P(B) via Bayes’ law.
  - SC(Ai) = ∑ P(Ai/Aj) · P(Aj) for all j ≠ i; contribution of each country to changes in SC analyzed over time to identify largest sources of contagion.
- Medium- and long-term adjustment indicator I1 (calculation outline):
  - Two time horizons: 1T (10 years) during which the primary balance is adjusted gradually by factor α, and 2T (10 years) during which the primary balance is kept constant at the level projected at end of 1T.
  - r is the growth-adjusted interest rate where 1rλ = + (notation as in source); TB is the target debt/GDP ratio; B0 is starting public debt/GDP ratio.
  - Required primary balance in 2020 can be decomposed into four key components reflecting financing needs, cyclical components, and age-related spending (full derivation and decomposition given in Appendix C).
- Data sources noted:
  - Historical and projection data from World Economic Outlook and September 2011 Fiscal Monitor; financial and market data from Bloomberg L.P., Datastream, CMA (Markit), IMF International Financial Statistics, and IMF Information Notice System.
  - Projections of increases in health care and pension spending from September 2011 Fiscal Monitor and IMF Policy Paper (2010b).
  - Elasticities of expenditure and revenues to the output gap from Girouard and André (2005) and European Commission (2005).

*Source: IMF staff estimates and methodological exposition in Section II.F and related sections of the supplied document.*

### 1. The size of debt at the beginning and at the end of the adjustment period (B

### _wp1211 - 1. The size of debt at the beginning and at the end of the adjustment period (B

### Indicators and their decomposition
- Indicator I1
  - I1 is defined as the level of the cyclically-adjusted primary balance between 2010 and 2030 needed to reach the debt target.
  - I1 depends on:
    - The size of debt at the beginning and at the end of the adjustment period (B0 and BT) (the first term in the equation).
    - The size of the age-related costs during 2013-30 (the second term).
    - The level of the cyclically-adjusted primary balance at the beginning of the adjustment period (the third term).
    - The projected path of the cyclical component of the primary balance (the last term).
  - The adjustment path and in particular the value of T1 affects the required primary balance: a higher value suggests a slower adjustment and hence a higher required primary balance in 2020.
  - The coefficient on the initial primary balance is positive: a higher initial balance implies a lower required primary balance in 2020 because the initial balance affects the adjustment path.

- Indicator I2
  - I2 shows the implied primary balance in 2016 that is consistent with stabilizing the debt level in the very long run and satisfying the inter-temporal budget constraint.
  - I2 is calculated using the S2 sustainability indicator formula developed by the European Commission:
    - S2 uses tPB (primary balance-to-GDP ratio) that includes: cyclically-adjusted primary balance, increase in aging related costs after 2016, and the cyclical component of the primary balance.
    - S2 shows the required adjustment applied to the baseline primary balance every year in the future to satisfy the inter-temporal budget constraint.
  - I2 = baseline primary balance in 2016 + required adjustment (S2).
  - Rationale for using I2 instead of S2: S2 can understate fiscal risks for countries that already have a significant fiscal adjustment in the baseline since that reduces the degree of further needed adjustment. I2 focuses on the level of required primary balance that must be maintained for debt sustainability.

- Common decomposition (applies to both indicators)
  - Effect of the level of initial debt (first term).
  - Projected path of the increase in aging related costs.
  - Projected path of cyclically-adjusted primary balance using baseline assumptions.
  - Projected path of the cyclical component of the primary balance.

### Assumptions and baseline projection mechanics
- Growth-adjusted interest rate r assumptions used in illustrations in Section III:
  - r = 0 up to 2015.
  - r = 1 percent thereafter.
  - These are broadly in line with WEO projections for advanced economies and the standard illustrative scenario in the Fiscal Monitor.
- Primary balance projection mechanics:
  - Primary balance projections for the current year and the next five years are those of the latest WEO.
  - Thereafter until 2050 the primary balance is calculated by adding the flow of age-related expenditure (in percent of GDP) to the last year of the WEO projections for the primary balance.
  - From 2050 onwards, the primary balance-to-GDP ratio is assumed to remain constant at the 2050 level.
- Comparability note:
  - Results for I1 and I2 are not fully comparable: I1 includes adjustment required until 2020 to achieve a cyclically-adjusted balance consistent with a particular debt target; I2 is based on longer-term expenditure pressures applied on top of the baseline primary balance path.

### Vulnerabilities to adverse growth and interest rate shocks
- Low growth scenario
  - Two debt paths are compared: baseline (current WEO projection) and a low growth scenario.
  - Low growth scenario specification:
    - Real GDP growth is 1 percentage point lower in the current year and the next five years than in the baseline.
    - Potential GDP is assumed not impacted by the shock.
    - Governments do not take corrective discretionary action to smooth its impact.
  - Debt dynamics used:
    - Standard debt dynamic equation: dt = (1 + rt) / (1 + gt) * dt-1 + pb_t   (presented in text as 1: d_t = (1 + r_t)/(1 + g_t) d_{t-1} + pb_t) where pb_t is the primary balance and r_t is the growth adjusted interest rate.
  - Primary balance under shock:
    - pb_t = pb_WEO + η_R * (change in output gap) - η_G * (change in output gap)
    - Where pb_WEO is baseline primary balance-to-GDP ratio; η_R and η_G are semi-elasticities of revenues and expenditures to changes in the output gap; Δog_t is the change in the output gap between baseline and low growth scenario.
  - Interest rate derivation:
    - The interest rate is derived by dividing interest payments by the stock of debt observed at the end of the preceding year.
  - Footnote definition:
    - The growth adjusted interest rate is defined as r = (1+i)/(1+γ) - 1, where i is nominal interest rate and γ is growth rate of nominal GDP.

- Interest rate shock scenario
  - Impact computation:
    - Δc_t = t_nf × t_ĩ   (text expresses impact as: c̃_it = t_nf × ĩ_t)
    - Where c̃_it is the difference in net financing costs relative to the baseline, t_nf is the accumulated amount of debt in percent of GDP financed since the beginning of the shock, and ĩ_t is the deviation in interest rate relative to the baseline (interest rate shock).
  - Mechanics and assumptions:
    - As debt is rolled over and new deficits financed, more debt becomes subject to higher interest rates; total impact accumulates over time by tracking debt subjected to higher rates using gross financing needs and issuance structure.
    - Gross financing needs = rollover of existing debt + budget deficit.
    - Rollover needs estimated using maturity structure of outstanding marketable debt at end-2010 applied to outstanding total gross debt at end-2010 to calculate rollover needs for current year (t) and next five years (t+5).
    - Rollover needs are affected by structure of gross issuances during this time (some newly financed debt may be rolled over during t+1 to t+5).
    - Maturity structure of gross issuance is based on maturity structure of debt at end-2010.
    - Baseline budget balance path is adjusted for higher interest payments.
    - Remuneration of financial assets held by general government assumed to change by same amount as interest rate for new borrowing.
    - Average maturity of financial assets assumed to be five years for all countries (due to sparse country-specific data).

### Stochastic simulations: medium-term risks to public debt trajectories
- Risk-based approach steps
  - Fiscal reaction function estimation
    - Panel of 33 advanced economies, annual data, maximum of 21 years (1990-2010).
    - Fixed effects regression specification chosen.
    - Estimated fiscal reaction function (as presented):
      - b̂_{t,c} = ρ b_{t-1,c} + 0.74 b_{t-1,c} + ... (presentation in source shows coefficients and terms; key outcomes summarized below).
    - Key estimated behavioral coefficients and findings:
      - Persistence of primary policies: coefficient of lagged primary balance is 0.74.
      - Cyclical sensitivity: countercyclical with 0.58 percent of GDP increase in primary deficit for each 1 percentage point of output gap widening during recessions (asymmetry: no tightening during booms).
      - Policymakers attempt corrective fiscal action to rising debt: coefficient 0.04 percent of GDP (on average in advanced economies).
    - Estimation notes:
      - Except for coefficient on positive output gap, all coefficients significant at 5 percent using robust standard errors.
      - Alternative estimators (LIML and GMM with instruments for output gap) were considered; fixed effects chosen based on diagnostics.

  - Calibration exercise
    - Unrestricted quarterly VAR for 1990-2010 using: real output growth, ln(real effective exchange rate), real long-term domestic bond yields, real long-term foreign bond yields.
    - Shocks to macro variables are random draws from joint normal distribution with zero mean and variance-covariance matrix from the unrestricted VAR.

  - Probability distribution calculation
    - Inputs:
      - Coefficients of estimated fiscal reaction function (common to all panel countries).
      - Fiscal policy shock distributed normally with mean zero and country-specific variance of fiscal reaction function residuals.
      - Most recent WEO projections of debt, primary balance, output gap and nominal output for each advanced economy.
      - Estimated variance-covariance matrix of VAR residuals for each country.
    - Outcome:
      - Construct explicit probability distributions for evolution of debt stock over t to t+5.
      - Country-specific fixed effects adjusted to anchor primary balance projections to WEO for t to t+5; adjustment may not fully reconcile model median debt ratio with WEO forecasts.

  - Adjustments to projected growth-adjusted effective interest rates
    - Observed divergence between historical and projected growth-adjusted effective interest rates for several advanced economies: averaging -81 basis points and ranging up to 337 basis points.
    - Adjustment approach (not country-specific, three sizes only):
      - (i) 130 basis points reduction (applies to 21 economies).
      - (ii) 200 basis points increase (applies to three economies).
      - (iii) 300 basis points increase (applies to two economies).

- Model median vs WEO median debt projections (Table A.1, model-based median in 2016 vs WEO baseline projection)
  - Model 1 / WEO (Sept. 2011 WEO / Apr. 2011 WEO) selected entries:
    - Australia: 24.8 20.2 / 24.1 20.6
    - Austria: 72.1 72.2 / 69.5 69.8
    - Belgium: 95.9 93.0 / 99.4 98.8
    - Czech Republic: 45.9 46.3 / 47.9 47.6
    - Denmark: 55.8 49.1 / 47.6 40.2
    - Finland: 51.7 52.3 / 59.4 61.1
    - France: 86.2 87.7 / 87.3 86.7
    - Germany: 80.1 76.6 / 78.3 71.9
    - Greece: 225.8 166.3 / 209.9 145.5
    - Ireland: 131.7 114.3 / 141.2 121.5
    - Italy: 103.4 112.9 / 118.0 118.0
    - Japan: 261.3 253.3 / 264.9 250.5
    - Korea: 14.1 22.2 / 10.6 19.8
    - Netherlands: 63.7 64.4 / 64.9 64.4
    - Portugal: 100.0 110.5 / 87.4 106.5
    - Spain: 77.0 77.4 / 74.1 75.9
    - Sweden: 35.0 19.3 / 39.6 22.8
    - Switzerland: 45.9 46.3 / 47.2 45.7
    - United Kingdom: 85.3 80.0 / 88.0 81.3
    - United States: 106.0 115.4 / 108.8 111.9
  - Note: Model-based values refer to the median value of 1000 replications. Exceptions where model and WEO diverge include countries supported by EU/IMF programs (e.g., Greece and Ireland) where financing operations may be included in WEO projections but not accounted for in the model.

*Source: IMF staff calculations and WEO projections as presented in the source PDF.*

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1211.pdf_