## Boosting Fiscal Space: The Roles of GDP-Linked Debt and Longer Maturities

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### Executive summary: background, proposition, and mechanisms
- Background and motivation:
  - Fiscal policy's role has risen amid fears of secular stagnation and constrained monetary easing.
  - High sovereign debt ratios have made authorities cautious about using fiscal policy despite record low interest rates.
  - Fiscal space is defined as the distance between the current public debt ratio and the debt limit where the sovereign is shut out of the debt market.
- Core proposition:
  - Debt management policies can expand fiscal space for a given path of primary fiscal balances by reducing sovereign default risk in bad states and lowering real borrowing costs.
  - Two policies explored: issuance of GDP-linked debt (state-contingent debt) and issuance of longer-maturity debt.
- Mechanisms:
  - GDP-linked debt: aligns debt service with ability to pay; reduces variability of debt ratios and default probabilities, allowing lower average interest rates while preserving lenders’ expected profit.
  - Longer-maturity debt: provides equity-like risk sharing via expected future prices; favorable future states lift current prices while default censors downside, stabilizing bond prices and debt ratios.

### Quantitative findings (simulations) and headline magnitudes
- Simulated gains in fiscal space from debt management:
  - Gains on the order of 5–40 percent of GDP are plausible taking account of a reasonable allowance for investor risk aversion.
  - Gains of 10–60 percent of GDP are plausible under the assumption of investor risk neutrality.
- Specific simulation highlights (representative advanced-economy calibration):
  - GDP-linked bonds:
    - With growth uncertainty = Low: Nominal Bond (A) debt limit = 179.7 (percent of GDP); GDP-linked Bond (B) debt limit = 199.4; B – A = 19.7.
    - With growth uncertainty = High: Nominal Bond (A) debt limit = 159.1; GDP-linked Bond (B) debt limit = 199.4; B – A = 40.3.
    - Interpretation: GDP-linked bond debt limit is unaffected by growth uncertainty in the model; gains are ~20 percent of GDP (low uncertainty) and ~40 percent of GDP (high uncertainty).
  - Long-duration nominal bonds (Table 2):
    - Growth Uncertainty: Low
      - 1 year (A) = 87.6; 10 years (B) = 103.5; 20 years (C) = 135.0; Debt Limit of GLB (D) = 204.8.
      - B – A = 15.9; C – A = 47.4; D – A = 117.2.
    - Growth Uncertainty: High
      - 1 year (A) = 65.3; 10 years (B) = 104.6; 20 years (C) = 137.6; Debt Limit of GLB (D) = 204.8.
      - B – A = 39.4; C – A = 72.3; D – A = 139.5.
    - Interpretation:
      - Lengthening duration from 1 year to 10 years may raise fiscal space by 16–39 percent of GDP depending on uncertainty.
      - Increasing duration to 20 years yields larger gains—on the order of 50–70 percent of GDP.
      - Under simulated assumptions, GDP indexation (GLBs) yields larger gains than maturity extension.

### Model intuition: debt limits, fiscal fatigue, and uncertainty
- Debt dynamics (simplified budget constraint):
  - d_{t+1} − d_t = [ ( r_t − g_{t+1} ) / (1 + g_{t+1}) ] d_t − s^*
  - d is debt-to-GDP ratio; r is interest rate on debt; g is GDP growth rate; s^* is primary balance as a share of GDP.
- Debt limit concept:
  - Debt limit is the highest debt level that can be serviced at a finite interest rate; fiscal space = current debt subtracted from this limit.
  - Under growth uncertainty, expected debt service schedule is convex and lies above the certainty-case schedule; bond yields rise faster once default risk emerges.
  - Result: debt limit under uncertainty (d̄_1) lies to the left of the certainty-case limit (d^*); growth uncertainty reduces fiscal space.
- GLB mechanics in the model:
  - GLB payout: ρ_{t+1} = (1 + g_{t+1}) / (1 + g^*).
  - GLBs adjust payout proportionally to realized growth; expected payout equals nominal bond payout under risk neutrality.
  - In model setup, GLBs restore debt service schedule to the no-uncertainty line and restore fiscal space to the certainty level d^*.
- Long-duration bond mechanics:
  - Coupons decay geometrically at rate δ; q_t depends on expected no-default future price q_{t+1}^E and default probability p_{t+1}:
    - q_t = (1 − p_{t+1}) [ 1 + (1 − δ) q_{t+1}^E ] / (1 + r^*)
  - Expected-future-price term raises current prices by capturing value of favorable future states while censoring unfavorable states that induce default, allowing higher debt limits relative to one-period debt.

### Calibration, simulation design, and parameter values
- Fiscal reaction function:
  - Primary balance ratio is stochastic and follows a cubic fiscal reaction per Ostry et al. (2010): s_{t+1} = f(d_t) + e_{t+1}.
  - Intercept term of f(d) set to 1.15 percent of GDP (cross-country median).
  - Primary balance shock e_{t+1} ~ triangular distribution over [–2.16, 2.16] (percent of GDP).
- Growth-rate distribution calibration:
  - Annual data for 23 advanced economies over 1980–2020 (including IMF WEO forecasts).
  - Growth discretized over n = 25 values.
  - Panel summary statistics:
    - Cross-country average growth rate mean = 2.4 percent; standard deviation = 1.4 percent (low-uncertainty case).
    - Pooled individual-country growth rates mean = 2.4 percent; standard deviation = 2.5 percent (high-uncertainty case).
- Risk-free rate and recovery:
  - Risk-free interest rate r^* set equal to 1.1 percent in the representative simulations (Introduction calibration).
  - Alternative statement in Appendix: risk-free interest rate is set to 4.4 percent; recovery value set to 90 percent.
  - Recovery on defaulted debt assumed to be 90 percent in cross-country averages; reducing recovery from 90 percent to 70 percent nearly doubles simulated gains in fiscal space in the low-uncertainty case (from about 20 percent of GDP to 36 percent of GDP).
- Simulation technicals for long-duration bond:
  - Duration (no default) = (1 + r*) / (δ + r*); risk-free price q* = 1 / (δ + r*).
  - Equilibrium selection uses q_{t+1}^E = α q̄ + (1 − α) q* with 0 < α < 1 and α > 0.5 in simulations; additional restriction q_{t+2}^E = q_{t+1}^E at the debt limit provides a conservative lower-bound debt limit.

### Investor preferences, pricing, and sensitivity to risk aversion
- Baseline assumption: investors are risk neutral (expected receipts equalize across instruments), justifying GLB issuance without extra insurer compensation in model.
- Considerations if investors are risk averse:
  - Risk-averse investors demand premiums for GLB return volatility or for longer maturities, reducing issuer gains.
  - Table 3 simulated maximum risk premiums and Sharpe ratios for GLBs (risk-neutral baseline):
    - Low uncertainty: Risk Premium = 15 basis points; Sharpe Ratio = 31.09.
    - High uncertainty: Risk Premium = 260 basis points; Sharpe Ratio = 1.03.
  - Willingness-to-pay calculations suggest issuing countries could pay premiums on the order of 150–260 basis points to transfer debt-service risk under greater uncertainty.
  - Literature evidence:
    - Barr et al. (2014): GLBs yield benefits unless coefficient of relative risk aversion ≥ 24.
    - Kamstra and Shiller (2009): U.S. risk premium would be at most 150 basis points.
    - Cabrillac et al. (2016): GLBs can provide better risk diversification for international investors than equities.
- Conclusion on investor aversion:
  - Unless investors are exceptionally risk averse or severely constrained in diversification, desirable risk transfer via GLBs appears feasible; risk aversion reduces but does not necessarily eliminate fiscal-space gains.

### Fiscal cyclicality, debt composition, and marginal effects
- Fiscal cyclicality (extended fiscal reaction function): s_{t+1} = f(d_t) + β(g_{t+1} – g*) + e_{t+1} with β = 0.4 (countercyclical) or β = −0.4 (procyclical).
- Table 4 excerpt (debt limits in percent of GDP) — neutral vs. countercyclical vs. procyclical outcomes:
  - Low growth uncertainty, Acyclical:
    - Nominal Bond (A) = 179.7; GLB-RN (B) = 199.4; GLB-RA (C) = 191.1; B – A = 19.7; C – A = 11.4.
  - High growth uncertainty, Countercyclical:
    - Nominal Bond (A) = 139.2; GLB-RN (B) = 195.7; GLB-RA (C) = 179.0; B – A = 56.5; C – A = 39.8.
  - Interpretation:
    - Countercyclical fiscal policy tends to amplify stochastic variation in the debt ratio and lowers the nominal debt limit; gains from GLBs are largest under countercyclical policy and smallest under procyclical policy.
    - Under high uncertainty, B – A = 56.5 (countercyclical) vs. B – A = 20.6 (procyclical) in reported entries.
- Debt composition and marginal gains (Table 5):
  - ω denotes share of GLBs in total debt.
  - For ω increasing from 0 to 1, marginal gains in fiscal space are positive but diminishing.
  - Reported marginal gains (percent of GDP) under Low uncertainty: ω = 1/4: 6.4; ω = 1/2: 5.4; ω = 3/4: 4.5; ω = 1: 3.4.
  - Under High uncertainty: ω = 1/4: 13.6; ω = 1/2: 11.1; ω = 3/4: 9.7; ω = 1: 5.9.

### Policy implications, benefits, and implementation challenges
- Policy recommendations:
  - Debt managers should consider issuing GDP-linked instruments and lengthening maturities to expand fiscal space without immediate fiscal contraction or binding forward fiscal promises.
  - These tools can make fiscal consolidation more growth-friendly by producing larger gains in fiscal space for a given path of primary fiscal balances.
- Benefits:
  - GLBs stabilize debt ratios by insulating debt service from growth shocks; long-duration debt stabilizes through asymmetric price responses and expectation effects.
  - Significant gains in fiscal space are feasible when GLBs account for only a negligible share of total debt.
  - GLBs can be attractive to defined-benefit pension funds due to correlation with GDP and natural hedging properties.
- Challenges and caveats:
  - Investor appetite and risk preferences matter: if investors demand premiums for GLB return volatility or term premia for longer maturities, issuer gains decline.
  - Practical obstacles to GLB issuance include potential misreporting of GDP data, ex post GDP revisions, methodological changes, limited liquidity, and novelty premiums.
  - Political commitment to countercyclical fiscal policy affects realized gains because debt service is procyclical.
  - Financial-stability implications: issuance of state-contingent instruments migrates risk from sovereign to private investors and may concentrate risk in private balance sheets.

*Source: Excerpted material from "Boosting Fiscal Space: The Roles of GDP-Linked Debt and Longer Maturities."*

### Executive Summary ......................................................................................................

### Executive Summary

### Background and motivation
- Fears about secular stagnation and diminishing returns from monetary easing have increased the role of fiscal policy to support growth.
- High sovereign debt ratios in many advanced economies and some emerging markets have led to caution in using fiscal policy, even with record low interest rates.
- The paper asks whether fiscal space can be expanded without paying down debt today or promising to do so in the future (markets might take such promises "cum grano salis").
- Fiscal space is defined here as the distance between the current public debt ratio and the debt limit where the sovereign is shut out of the debt market.

### Core proposition: debt management to expand fiscal space
- Debt management policies can expand fiscal space for a given path of primary fiscal balances by reducing the risk of sovereign default in bad states and lowering real borrowing costs.
- The paper explores two debt management policies:
  - Issuance of GDP-linked debt (state-contingent debt).
  - Issuance of longer-maturity debt (as opposed to short-term debt).

### Mechanism: GDP-linked debt
- Sovereign debt/GDP ratios are volatile because GDP is subject to random shocks; in bad states (recessions, crises), debt ratios rise and default risk increases.
- GDP-linked debt aligns debt service with the sovereign’s ability to pay by giving the sovereign a break on debt service in bad states in exchange for higher service in booms.
- By reducing the variability of debt ratios and default probabilities, GDP-linked contracts can be struck at a lower average interest rate while preserving lenders’ expected profit, thereby boosting fiscal space.

### Mechanism: longer-maturity debt
- Longer maturities provide a risk-sharing benefit to the issuer similar to equity, reducing the default premium.
- The price of long-duration debt reflects the expected value of future prices; favorable future states lift current prices, while very bad states where default occurs censor downside impacts.
- This censoring stabilizes bond prices and debt ratios relative to short-term debt, reducing default risk, lowering average borrowing costs, and enhancing fiscal space.

### Quantitative findings (simulations)
- Simulation results suggest gains in fiscal space from managing debt along the two dimensions:
  - Gains on the order of 5–40 percent of GDP are plausible taking account of a reasonable allowance for investor risk aversion.
  - Gains of 10–60 percent of GDP are plausible under the assumption of investor risk neutrality.
- Gains are larger when GDP-linked bonds account for only a negligible share of debt as at present.
- Mutual gains for issuers and investors are feasible unless investors are exceptionally risk-averse or constrained in risk diversification opportunities.

### Additional insights and considerations
- The average interest rate on nominal debt is increasing in the extent of growth uncertainty due to convexity: as debt rises toward the debt limit, investors demand higher interest rates, which further raise default risk and rates.
- GDP-linked debt reduces variability in the debt ratio and, through lower default risk, lowers average borrowing costs.
- Longer-dated debt benefits from the asymmetric impact of shocks on bond prices (favorable shocks support prices while unfavorable shocks are censored when default occurs), stabilizing debt dynamics.

### Policy implications / recommendations
- Debt managers should consider issuance of GDP-linked instruments and lengthening maturities as tools to expand fiscal space without immediate fiscal contraction or forward fiscal promises.
- These tools can make fiscal consolidation more growth-friendly by producing larger gains in fiscal space for a given path of primary fiscal balances.
- Feasibility depends on investor appetite and risk preferences; terms should be structured so expected investor profit is preserved while lowering issuer default risk.

*Source: Executive Summary of "Boosting Fiscal Space: The Roles of GDP-Linked Debt and Longer Maturities."*

### Introduction

### Introduction

### Rationale and relevant literature
- GDP-linked bonds (GLBs) link payments on sovereign debt to the issuing country’s GDP and can help stabilize debt ratios and reduce vulnerability to external shocks and financial crises.
- Ad hoc simulation studies show debt-stabilizing effects of GLBs: Borensztein and Mauro (2004); Blanchard et al. (2016).
- Barr et al. (2014) and Bank of England (2016) find GLBs can significantly increase debt limits and reduce default risk.
- Council of Economic Advisers (2004) argues GLBs facilitate countercyclical fiscal policy, especially for emerging market economies.
- Welfare analyses in optimizing frameworks indicate GDP-linked debt can raise national welfare: Durdu (2009); Hatchondo and Martinez (2012); Onder (2016).
- Early and related proposals: Shiller (1993); Obstfeld and Peri (1998); Drèze (2000).

### Obstacles, investor concerns, and pricing issues
- Empirical rarity: GLBs have rarely been issued by emerging market economies outside restructuring contexts (Griffith-Jones and Sharma 2005).
- Practical obstacles:
  - Potential misreporting of GDP data and difficulties in ex post verification.
  - Complications from ex post GDP revisions and methodological changes.
  - Investor perceptions of risk and limited liquidity.
- Demand considerations:
  - Defined-benefit pension funds may have substantial demand because GLB yields closely correlate with GDP, resembling pension liabilities (Borensztein 2016).
- Issuer concerns:
  - Possible extra premiums to compensate investors for illiquidity, novelty, and return volatility.
- Pricing literature limitations:
  - Chamon and Mauro (2006) propose a pricing method based on a simple debt sustainability framework but assume a constant default probability equal across nominal and GDP-linked bonds.
  - Kruse et al. (2005) and Miyajima (2006) use Black-Scholes or present-value models but abstract from sovereign default risk.
- Literature gap: Lack of a coherent framework incorporating sovereign default risk to determine theoretical GLB prices.

### Debt duration, maturity trade-offs, and gaps in literature
- Earlier work links optimal maturity to tax smoothing and debtor moral hazard (Barro 1995; Angeletos 2002; Missale and Blanchard 1994; Jeanne 2009).
- Recent work emphasizes trade-offs between rollover risks and borrowing costs:
  - Broner et al. (2013): safer longer-term vs. cheaper short-term; emerging markets pay a positive term premium and shift to shorter maturities during crises.
  - Greenwood et al. (2015): comparative advantages of maturities in rollover risk and liquidity (abstracting from default risk).
  - Arellano and Ramanarayanan (2012): dynamic optimizing model where longer maturities hedge spread volatility while short maturities reduce borrower disincentives.
- Gap: Impact of debt maturities on fiscal space under frameworks incorporating sovereign default risk remains underexplored.
- Paper objective: develop an analytical framework to determine debt limits and theoretical prices of GDP-linked and long-maturity debt under uncertainty and default risk, and quantify gains in fiscal space via simulations grounded in the model.

### Theory: debt limits, fiscal fatigue, and the role of uncertainty
- Fiscal space definition (paper): difference between current debt and the debt limit (the level beyond which fiscal solvency is in doubt).
- Behavioral assumptions:
  - Sovereign summarized by a reaction function linking the primary balance to debt and other factors.
  - Atomistic creditors set lending rates reflecting default risk.
  - Governments behave responsibly up to a point by increasing the primary balance in response to rising debt (empirically supported by Bohn (2008) and Mendoza and Ostry (2008)).
  - Economic and political limits exist on how high primary balances can rise; fiscal fatigue (Ostry et al. 2010) describes the petersing out of fiscal effort at high debt ratios.
- Simple budget constraint (debt-to-GDP evolution, as in equation (1) of the source):
  - d_{t+1} − d_t = [ ( r_t − g_{t+1} ) / (1 + g_{t+1}) ] d_t − s^*
  - where d is debt-to-GDP ratio, r is interest rate on debt, g is GDP growth rate, and s^* is primary balance as a share of GDP.
- Debt limit intuition:
  - If the primary balance cannot offset debt service (increment to primary balance ratio smaller than interest-rate-growth differential), debt dynamics become explosive.
  - Rising default risk and interest rates are mutually reinforcing; the fixed-point yields a finite interest rate consistent with default risk.
  - Debt limit is the highest debt level that can be serviced at a finite interest rate; fiscal space is the distance between current debt and this limit.
- Uncertainty case:
  - Growth is random with average equal to g^*; default occurs when realized d_{t+1} exceeds the debt limit.
  - Default probability p_{t+1} = Pr[ d_{t+1} > d̄ ] defines a fixed-point problem because default probability depends on bond price/yield and vice versa.
  - Under growth uncertainty, expected debt service (EDS) schedule is convex and lies above the no-uncertainty debt service schedule; bond yields rise faster once default risk emerges.
  - Consequence: debt limit under uncertainty (d̄_1) lies to the left of the certainty-case limit (d^*); growth uncertainty lowers the debt limit and reduces fiscal space.
  - Mechanism: lender gets no upside from good shocks but absorbs downside from bad shocks; average borrowing cost increases with uncertainty.

### GDP-linked bonds: mechanics and effects on fiscal space
- GLB payout specification used in the framework (equation (3) in the source):
  - ρ_{t+1} = (1 + g_{t+1}) / (1 + g^*)
- Risk-sharing property:
  - GLBs adjust payout in proportion to realized growth, insulating issuer from growth uncertainty by reducing obligations in bad times and increasing them in good times.
  - Expected payout of GLB equals payout of nominal bond, ensuring same expected return to a risk-neutral lender.
  - In the model, introduction of GLBs restores the debt service schedule to the no-uncertainty (red) line and restores fiscal space to the level under certainty.
- Lender willingness:
  - Risk-neutral lenders are indifferent because expected receipts are independent of uncertainty.
  - Risk-averse lenders would require compensation for additional risk; debtors who value fiscal flexibility and face costly defaults may be willing to pay a premium for GLBs.
- Dynamic implications:
  - Higher debt limit with GLBs implies the convex segment of the debt service schedule emerges at a higher debt ratio, reducing dispersion in the debt ratio across shocks.
  - Simulation literature confirms reduced debt-ratio volatility with GLBs (Blanchard et al. 2016; Benford et al. 2016).

### Simulations: setup and calibration details
- Objective: quantify plausible gains in fiscal space from introducing GLBs using simulations for a representative advanced economy.
- Fiscal reaction:
  - Primary balance ratio is stochastic and follows a cubic fiscal reaction as in Ostry et al. (2010) to capture fiscal fatigue.
  - Specification: s_{t+1} = f(d_t) + e_{t+1}, where f(d) is a cubic function.
  - In the simulation, the intercept term of f(d) is set to 1.15 percent of GDP (cross-country median of country-specific intercepts).
- Primary balance shock:
  - e_{t+1} is independently and identically distributed and follows a triangular distribution over finite support [–2.16, 2.16] (percent of GDP).
  - The selected support is intended to capture observed deteriorations in primary balances in recessions; endpoints represent the median of country-specific averages of the worst five residuals from estimating f(d).
- Growth-rate distribution calibration:
  - Calibrated using sample histograms with annual data for 23 advanced economies over 1980–2020 (including IMF World Economic Outlook forecasts).
  - Growth rate distribution is discretized over n = 25 values (25 bins).
  - Panel summary statistics from calibration:
    - Cross-country average growth rate mean = 2.4 percent; standard deviation = 1.4 percent (used for low-uncertainty case).
    - Pooled individual-country growth rates mean = 2.4 percent; standard deviation = 2.5 percent (used for high-uncertainty case).
- Risk-free interest rate in simulations:
  - r^* is set to equal 1.1 percent (average of US Treasury and German Bund yields for 1, 5, and 10-year maturities).

*Italic source attribution: Introduction, "BOOSTING FISCAL SPACE: THE ROLES OF GDP-LINKED DEBT AND LONGER MATURITIES", source PDF chapter 45132-resdp-boosting-fiscal-space - Introduction*

### 1. Time Series Data (Cross-country averages)

### 1. Time Series Data (Cross-country averages)

### Calibrating growth uncertainty and recovery assumptions
- Recovery value on defaulted debt assumed to be 90 percent—consistent with estimates reported by Benjamin and Wright (2009) for upper-middle-income countries.
- Simulated gains in fiscal space are sensitive to recovery rates:
  - With recovery rate reduced from 90 percent to 70 percent, simulated gains in fiscal space almost double in the case of low uncertainty—from about 20 percent of GDP to 36 percent of GDP.

### Expanding fiscal space with GDP-linked debt (summarized findings)
- Table 1 simulated debt limits (in percent of GDP):
  - Growth Uncertainty: Low
    - Nominal Bond (A): 179.7
    - GDP-linked Bond (B): 199.4
    - B – A: 19.7
  - Growth Uncertainty: High
    - Nominal Bond (A): 159.1
    - GDP-linked Bond (B): 199.4
    - B – A: 40.3
- Interpretation:
  - For the nominal bond, the debt limit is about 180 percent of GDP in the case of low growth uncertainty, and falls to 159 percent for the high-uncertainty case.
  - For the GDP-linked bond, the debt limit is unaffected by the extent of growth uncertainty.
  - Gains in fiscal space from GDP-linked debt are some 20 percent of GDP when uncertainty is low and 40 percent in the case of high uncertainty (as reported in the text).

### Expanding fiscal space with long-maturity (long-duration) debt — theory
- Long-duration bond features:
  - Coupon payments decay geometrically at the rate δ over time.
  - Debt dynamics equation (as presented):
    - x_{t+1} = ( (x_t / q_t) / (1 + g_{t+1}) ) + ( (1 − δ) b_t / (1 + g_{t+1}) ) − s^*
    - where q is the bond price, b and x = b − s^* denote aggregate coupon payments and gross financing need (GFN) as a share of GDP.
- Equilibrium bond price depends on default risks throughout the life of the bond:
  - q_t = (1 − p_{t+1}) [ 1 + (1 − δ) q_{t+1}^E ] / (1 + r^*)
  - q_{t+1}^E = E_t[ q_{t+1} | q_{t+1} > 0 ] is the expected bond price in the next period in a no-default state.
- Mechanism:
  - Expected future price incorporates value from potential favorable future shocks but not unfavorable shocks that induce default (coupon payments zero in default).
  - This expectational term lowers the threshold growth rate for default relative to one-period debt, allowing a higher debt limit and larger fiscal space.
  - Long-duration debt thus exhibits an equity-like, risk-sharing character via price resilience.

### Simulations: magnitude of maturity effects (Table 2 results and interpretation)
- Table 2 debt limits (in percent of GDP) — columns correspond to:
  - (A) 1 year (actual one-period debt limit)
  - (B) 10 years (lower bound of actual debt limit for long-duration bond)
  - (C) 20 years (lower bound of actual debt limit for long-duration bond)
  - (D) Debt Limit of GLB (actual debt limit)
  - B – A, C – A, D – A are reported differences
- Reported values:
  - Growth Uncertainty: Low
    - 1 year (A): 87.6
    - 10 years (B): 103.5
    - 20 years (C): 135.0
    - Debt Limit of GLB (D): 204.8
    - B – A: 15.9
    - C – A: 47.4
    - D – A: 117.2
  - Growth Uncertainty: High
    - 1 year (A): 65.3
    - 10 years (B): 104.6
    - 20 years (C): 137.6
    - Debt Limit of GLB (D): 204.8
    - B – A: 39.4
    - C – A: 72.3
    - D – A: 139.5
- Interpretation and headline simulation results:
  - Lengthening debt duration from 1 year to 10 years may increase fiscal space by some 16–39 percent of GDP, with the range governed by the extent of growth uncertainty (higher end corresponding to larger uncertainty).
  - If duration increases to 20 years, gains in fiscal space are larger—on the order of 50–70 percent of GDP.
  - Comparing maturity extension with GDP indexation: the table shows larger gains in fiscal space for GDP indexation than for maturity extension under the simulation assumptions.

### Sensitivity to underlying assumptions
- Investor risk neutrality assumption:
  - The model assumes risk-neutral investors who accept additional risk from GLBs at zero cost because expected returns on state-contingent and non-state-contingent instruments are the same.
  - Justifications:
    - Large institutional investors can pool GLBs across countries and diversify away return volatility.
    - Idiosyncratic and diversifiable GDP variation appears far larger than systematic global GDP variability.
- Considering investor risk aversion:
  - Risk-averse investors would demand a premium for GLB-induced return volatility.
  - Literature findings referenced:
    - Barr et al. (2014): GLBs yield benefits in increased fiscal space unless coefficient of relative risk aversion ≥ 24.
    - Kamstra and Shiller (2009): the risk premium for the United States would be 150 basis points at most.
    - Cabrillac et al. (2016): simulations show GLBs can provide better risk diversification for international investors than equities.
  - Issuer willingness-to-pay analysis (model-based):
    - Maximum premiums that an issuing country would be willing to pay to transfer debt service risk to investors are on the order of 150–260 basis points, larger the greater the underlying growth uncertainty.
    - Reported Sharpe ratios for GLBs in the simulations are greater than one, suggesting GLBs could be attractive to investors under simulated conditions.
  - Conclusion:
    - Unless investors are exceptionally risk averse or severely constrained in diversification opportunities, desirable risk transfer via GLBs would appear feasible.
    - The scope of desirable risk transfer is reduced if investors are risk averse—the impact on fiscal space gains is addressed in the chapter.

### Contextual points on issuance and multiple shocks
- Empirical issuance context:
  - GLBs have rarely been issued outside sovereign debt restructuring contexts.
  - Greater incidence of issuance during restructuring may reflect larger scope for desirable risk sharing when default risk is initially high.
- Interactions among multiple shocks:
  - Stochastic variation in the debt ratio arises from multiple shocks to debt dynamics.
  - The impact of GLBs on debt paths and fiscal space depends on the configuration and correlation of shocks (including primary balance shocks).
  - Insulating against one source of uncertainty could be counterproductive depending on shock correlations; e.g., if shocks to the primary balance are negatively correlated with growth (procyclical fiscal policy), fiscal action will tend to reduce stochastic variation in the debt ratio.

*Source: 1. Time Series Data (Cross-country averages), BOOSTING FISCAL SPACE: THE ROLES OF GDP-LINKED DEBT AND LONGER MATURITIES (excerpt).*

### Chapter 3, which assume risk neutrality. In practice, investor risk aversion will result in a higher default pre-

### Chapter 3

### Risk Premiums and Sharpe Ratios for GDP-linked Bond
- Table 3 reports maximum risk premiums and Sharpe ratios for GDP-linked bonds under two growth uncertainty scenarios:
  - Low: Risk Premium = 15 basis points; Sharpe Ratio = 31.09
  - High: Risk Premium = 260 basis points; Sharpe Ratio = 1.03
- Note: Sharpe ratio is the ratio of the risk premium to the standard deviation of the bond yield. Risk premiums are in basis points.
- The chapter assumes investor risk neutrality; in practice, investor risk aversion will result in a higher default premium and a correspondingly lower debt limit for both nominal bonds and GLBs, but more so for nominal bonds. The authors note their willingness-to-pay calculations are conservative on this account.

### Sensitivity to Fiscal Policy Cyclicality
- The model is extended to allow cyclical responsiveness of fiscal policy via the fiscal reaction function s_{t+1} = f(d_t) + β(g_{t+1} – g*) + e_{t+1}, with β set to 0.4 (countercyclical) or −0.4 (procyclical).
- Simulations assume growth uncertainty is exogenous to fiscal policy.
- For risk-averse investors, premiums of 70 and 126 basis points are used for the low- and high-growth-uncertainty cases, respectively (constructed by applying a Sharpe ratio of 0.5 to the simulated maximum risk premiums in Table 3).

- Table 4: Effects of Cyclical Fiscal Policy on Fiscal Space (Debt limits in percent of GDP)
  - Low growth uncertainty, Acyclical:
    - Nominal Bond (A) = 179.7
    - GLB-RN (B) = 199.4
    - GLB-RA (C) = 191.1
    - B – A = 19.7
    - C – A = 11.4
  - Low growth uncertainty, Countercyclical:
    - Nominal Bond (A) = 174.1
    - GLB-RN (B) = 197.1
    - GLB-RA (C) = 188.7
    - B – A = 23.0
    - C – A = 14.6
  - Low growth uncertainty, Procyclical:
    - Nominal Bond (A) = 184.7
    - GLB-RN (B) = 196.8
    - GLB-RA (C) = 188.3
    - B – A = 12.1
    - C – A = 3.6
  - High growth uncertainty, Acyclical:
    - Nominal Bond (A) = 159.1
    - GLB-RN (B) = 199.4
    - GLB-RA (C) = 184.1
    - B – A = 40.3
    - C – A = 25.0
  - High growth uncertainty, Countercyclical:
    - Nominal Bond (A) = 139.2
    - GLB-RN (B) = 195.7
    - GLB-RA (C) = 179.0
    - B – A = 56.5
    - C – A = 39.8
  - High growth uncertainty, Procyclical:
    - Nominal Bond (A) = 169.8
    - GLB-RN (B) = 190.4
    - GLB-RA (C) = 171.0
    - B – A = 20.6
    - C – A = 1.2

- Key interpretation:
  - Countercyclical fiscal policy (β = 0.4) tends to amplify stochastic variation in the debt ratio caused by growth uncertainty and leads to a lower debt limit for nominal debt relative to acyclical policy; procyclical policy (β = −0.4) tends to mitigate stochastic variation and increases the nominal debt limit.
  - Gains in fiscal space from GDP-linked debt are largest under countercyclical fiscal policy and smallest under procyclical policy.
  - Under high uncertainty, the gain in fiscal space from GLBs is 56 percent of GDP under countercyclical policy but falls to 21 percent under procyclical policy (Table 4 entries correspond to B – A).
  - Gains in fiscal space are smaller if investors are risk averse, falling to 40 percent of GDP in the high-uncertainty, countercyclical case (C – A = 39.8), and becoming negligible in the case of procyclical policy (C – A = 1.2).

### Debt Composition and Marginal Gains in Fiscal Space
- The chapter analyzes mixed portfolios with both nominal bonds and GLBs, focusing on neutral fiscal policy (β = 0).
- ω denotes the share of GDP-linked debt out of total debt.

- Table 5: Marginal Gains in Fiscal Space and the Share of GLBs (Debt limits and gains in percent of GDP)
  - For ω = 0 (nominal debt), reported debt limits:
    - Low: 179.7
    - High: 159.1
  - Marginal gains in fiscal space when increasing ω:
    - Low growth uncertainty:
      - ω = 1/4: 6.4
      - ω = 1/2: 5.4
      - ω = 3/4: 4.5
      - ω = 1: 3.4
    - High growth uncertainty:
      - ω = 1/4: 13.6
      - ω = 1/2: 11.1
      - ω = 3/4: 9.7
      - ω = 1: 5.9

- Key interpretation:
  - Simulated debt limits increase with the share of GLBs; marginal gains in fiscal space are positive but diminishing as ω increases.
  - Example: Under high growth uncertainty, fiscal space increases by 14 percent of GDP when ω increases from 0 to 25 percent (reported as 13.6), then to 11 percent when going from 25 to 50 percent (reported as 11.1), and to 10 percent when going from 50 to 75 percent (reported as 9.7).

### Policy Findings, Benefits, and Challenges
- Main findings:
  - Issuing state-contingent debt such as GDP-linked bonds can help stabilize debt ratios by inoculating debt dynamics against growth uncertainty.
  - Increasing debt duration can lead to less volatile debt ratios and larger fiscal space.
  - Any debt management policy that stabilizes debt ratios under uncertainty could help increase fiscal space.
  - The reduction in stochastic variation in the debt ratio has a larger impact on fiscal space the greater the extent of growth uncertainty and the higher the actual debt ratio.
  - There is scope for significant gains in fiscal space when GLBs account for only a negligible share of total debt.

- Caveats and investor considerations:
  - Expect reduced scope for risk sharing between sovereign and investors if investors demand a premium for return volatility in GLBs or a term premium for longer maturities.
  - Simulations suggest the risk-sharing benefit of GLBs is plausible unless investors are exceptionally risk averse or opportunity for risk diversification is significantly constrained.
  - GLBs would be especially attractive to defined-benefit pension funds because these instruments provide a natural hedge against GDP-induced variation in pension liabilities.

- Policy challenges:
  - Political commitment to countercyclical fiscal policy would be necessary to reap the benefit from GLBs, given that debt service is procyclical.
  - Countries should be mindful of the financial-stability implications of risk migration from the sovereign to private investors that issuance of state-contingent instruments may entail.

*Source: Authors’ calculations (Chapter 3).*

### Conclusion

### Conclusion

### Risk-sharing mechanism and fiscal space
- GDP-linked debt increases fiscal space primarily through risk sharing between the sovereign and private investors.
- Risk migration from the sovereign to private investors is more pertinent in a closed-economy setting where growth uncertainty is a nondiversifiable risk.
- Conventional wisdom: the government is a better risk absorber than the private sector because of taxation powers.
- Counterpoint in the analysis: the government's shock-absorbing capacity depends on the sovereign retaining fiscal space to absorb risk; issuance of GLBs and other debt-management policies to enhance fiscal space is most relevant where fiscal space is limited.
- Risk migration is not unique to state-contingent instruments; long-term (nominal) debt can also migrate risk to private investors.
- Empirical clue cited: Berndt et al. (2012) find that, in the post-war period, about 9 percent of the US government’s unanticipated defense spending needs were financed by a reduction in the market value of debt.
- Conclusion: concerns about risk migration should not be overstated as an impediment to issuance of state-contingent public debt instruments at the current juncture, in either advanced or emerging market economies.

### Model setup and key definitions
- GDP growth rate (g) is distributed according to cumulative distribution G(g) over finite support [gL, gH] with mean g*.
- Primary balance as a share of GDP is constant at s*.
- Investors are risk neutral and recover nothing upon default.
- Three bond types considered: one-period nominal bond, one-period GDP-linked bond (GLB), long-duration nominal bond.
- One-period nominal bond total payout at maturity = one dollar.
  - Debt dynamics: d_{t+1} = (d_t / q_t) / (1 + g_{t+1}) − s*  (A.1)
  - Bond price: q_t = (1 − p_{t+1}) / (1 + r*)  (A.2)
  - Default probability solves fixed-point: p_{t+1} = G(H(p_{t+1}; d_t, d̄)) with H(...) = ((d_t / d̄ + s*) (1 + r*)/(1 − p_{t+1})) − 1  (A.3)
  - Debt limit d̄_1 is largest d̄ for which p_{t+1} < 1 interior solution exists; at debt limit d_t = d̄_1, default probability jumps to unity otherwise.
- Certainty case (growth constant at g*):
  - d_{t+1} − d_t = [((r* − g*) / (1 + g*))] d_t − s*  (A.4)
  - Debt limit d* = [((1 + g*) / (r* − g*))] s*.

### Properties of GLB
- GLB total payout at maturity ρ_{t+1} = (1 + g_{t+1}) / (1 + g*); expected value = one dollar absent default.
- GLB debt dynamics simplify to: d_{t+1} = (d_t / q_t) / (1 + g*) − s*  (A.5)
- GLB bond price reduces to q_t = (1 − p_{t+1}) / (1 + r*)  (A.7) with p_{t+1} = 0 and q_t = 1 / (1 + r*) if primary balance constant and conditional default probabilities independent of growth.
- Implication: debt dynamics under GLB do not depend on growth uncertainty; debt limit for GLB equals d*, the certainty-case debt limit.
- Proof sketch: under symmetry of G(g) and condition g_H − 2g* < 1, fixed-point for nominal bond at d_t = d* has only corner solution p_{t+1} = 1, showing debt limit is higher for GLB than for nominal bond under uncertainty.

### Effects of uncertainty and debt maturity
- Debt limit for the nominal bond is lower the larger the extent of uncertainty: a mean-preserving spread G̃(g) raises default probability and lowers the debt limit if H < g* in equilibrium.
- Long-duration nominal bond specification:
  - Coupons c_{t+k} = (1 − δ)^{k−1} for k ≥ 1, where 0 < δ ≤ 1.
  - Duration (no default) = (1 + r*) / (δ + r*); risk-free price q* = 1 / (δ + r*). One-period bond corresponds to δ = 1.
  - Gross financing need (GFN) x = b − s*; law of motion:
    x_{t+1} = (x_t / q_t) / (1 + g_{t+1}) + (1 − δ) b_t / (1 + g_{t+1}) − s*  (A.9)
  - Debt ratio d_t = [1 + (1 − δ) q_t] b_t − s* = [1 + (1 − δ) q_t] x_t + (1 − δ) q_t s*  (A.10)
  - Bond price satisfies: q_t = (1 − p_{t+1}) [1 + (1 − δ) q_{t+1}^E] / (1 + r*)  (A.11) where q_{t+1}^E = E_t[q_{t+1} | q_{t+1} > 0].
- Key observations:
  - For long-duration bond (δ < 1), future bond prices affect current price via q_{t+1}^E; absent for one-period bond.
  - There exists a finite GFN limit x̄ beyond which dynamics are unsustainable; default probability p_{t+1} = Pr[x_{t+1} > x̄] is a fixed-point.
  - Debt limit for long-duration bond expressed as d̄ = [1 + (1 − δ) q̄] x̄ + (1 − δ) q̄ s* where q̄ is bond price at GFN limit.
  - Q_t = [1 + (1 − δ) q̄] / [1 + (1 − δ) q_{t+1}^E] > 0 enters fixed-point threshold Z(...). For one-period bond Q_t = 1; for long-duration bond Q_t < 1 when default risk present, implying interior solution at a debt ratio higher than one-period bond — hence long-duration bond yields higher debt limit.
  - If no growth uncertainty, p_{t+1} = A_{t+1} = 0 and debt limit equals d* (same as one-period bond).

### Simulation approach and equilibrium selection for long-duration bond
- Multiple equilibria arise due to feedback from future to current prices (A.11); a sensible equilibrium selection rule is assumed where equilibrium price of long-duration bond is lower the higher the debt ratio.
- At debt limit, q_{t+1}^E expressed as weighted average: q_{t+1}^E = α q̄ + (1 − α) q* , 0 < α < 1  (A.16).
  - α restricted to be greater than 0.5 in simulations to rule out excessively optimistic expectations.
  - As α → 1 (q_{t+1}^E → q̄), debt limit converges to one-period debt case.
- Additional technical restriction in simulations: q_{t+2}^E = q_{t+1}^E at the debt limit, which yields a lower simulated q̄ and hence a lower simulated GFN limit x' (a conservative lower bound).
- Fixed-point for simulation rewritten in terms of W(...) involving x̄; because W decreasing in q_t, the simulated maximum interior solution is at a lower GFN x' than theoretical, producing a lower-bound debt limit via (A.10).
- Simulation assumptions stated:
  - Constant primary balance = 4 percent of GDP.
  - Truncated normal distribution for growth rate over [g* − 1.96σ, g* + 1.96σ], where g* = 2.4 percent, and σ = 1.4 percent in the low-uncertainty variant.

*Source: Conclusion, CHAPTER 6.*

### 2.5 percent in the high-uncertainty variant. The risk-free interest rate and the

### BOOSTING FISCAL SPACE: THE ROLES OF GDP-LINKED DEBT AND LONGER MATURITIES

### Model calibration and key parameters
- "2.5 percent in the high-uncertainty variant."
- The risk-free interest rate is set to 4.4 percent.
- The recovery value is set to 90 percent.

### Document location
- Appendix 37

*Source: BOOSTING FISCAL SPACE: THE ROLES OF GDP-LINKED DEBT AND LONGER MATURITIES*

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_Source: https://www.imf.org/-/media/files/publications/dp/2018/45132-resdp-boosting-fiscal-space.pdf_