## 2.1    Debt Dynamics with Plain-Vanilla Bonds

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### Debt dynamics with plain-vanilla bonds
- Government nominal debt D_t satisfies: D_t = (1 + i_t)D_{t−1} − PB_t. (2)
- First-order approximation (small i_t and g_t): ∆d_t = (i_t − g_t)d_{t−1} − pb_t. (3)
- One-period-ahead conditional variance of the change in the debt ratio (conditioning on F_{t−1}):  
  Var_{t−1}(∆d_t) = Var_{t−1}(pb_t) + d^2_{t−1} Var_{t−1}(i_t − g_t) − 2 d_{t−1} Cov_{t−1}(pb_t, i_t − g_t). (4)
- Implication: higher Var(pb_t), higher Var(i_t − g_t), and more negative Cov(pb_t, i_t − g_t) raise uncertainty in the future public debt ratio.

### Debt dynamics with simple GIBs (growth-indexed bonds)
- If share X of outstanding debt is indexed to growth, indexed coupon per (1) with k_t = i_t − E_{t−1}[g_t] ∈ F_{t−1}. (5)
- Effective predetermined indexed rate within period t: \bar{i}_t = g_t + i_t − E_{t−1}[g_t] ≡ k_t. (5)
- Debt dynamics with partial indexation: ∆d_t = [X k_t + (1−X)(i_t − g_t)] d_{t−1} − pb_t. (6)
- Conditional variance under partial indexation:  
  Var_{t−1}(∆d_t) = Var_{t−1}(pb_t) + (1−X)^2 d^2_{t−1} Var_{t−1}(i_t − g_t) − 2(1−X)d_{t−1} Cov_{t−1}(pb_t, i_t − g_t). (7)
- Special case X = 1 (full indexation): Var_{t−1}(∆d_t) = Var_{t−1}(pb_t). (8)
  - Interpretation: full indexation fully hedges growth shocks; remaining variance is driven solely by primary balance volatility.

### Reducing the uncertainty around the debt ratio
- GIB issuance does not change the conditional mean of ∆d_t because k_t is defined to make risk-neutral investors indifferent between GIBs and conventional bonds.
- GIBs reduce conditional variance iff: X < 2 − 2 Cov_{t−1}(pb_t, i_t − g_t) / (d_{t−1} Var_{t−1}(i_t − g_t)). (9)
- Converting entire debt stock to GIBs reduces debt-ratio uncertainty whenever Cov_{t−1}(pb_t, i_t − g_t) < 0.
  - Empirical finding: using IMF April 2023 WEO database, this sufficient condition is satisfied for 25 of the 31 countries in the sample over 1996–2021.
- Variance-minimizing indexed share: X^⋆_{t−1} = 1 − Cov_{t−1}(pb_t, i_t − g_t) / (d_{t−1} Var_{t−1}(i_t − g_t)). (10)
  - When Cov_{t−1}(pb_t, i_t − g_t) < 0 (empirically dominant), X^⋆_{t−1} > 1: “over-indexation” further reduces debt variance by lowering coupon payments when primary balance is depressed.
- Limitation: the simple indexation formula does not insure against primary balance fluctuations. Section 3 develops richer formulas.

### Alternative indexation formulas — theoretical benchmark
- Fully contingent formula that would eliminate one-period conditional uncertainty: \bar{i}_t = pb_t / d_{t−1} + g_t + k_t. (11)
  - Not implementable due to moral hazard: sovereign incentives to manipulate the primary balance.
  - Serves as a theoretical benchmark with minimum achievable conditional debt variance equal to zero.

### Growth-indexed bonds with an optimal coefficient
- More general indexed coupon: \bar{i}_t = c g_t + k_t with c ≥ 0. (12)
- Debt dynamics: ∆d_t = [X k_t + (1−X) i_t − (1 − cX) g_t] d_{t−1} − pb_t. (13)
- Variance-minimizing coefficient for given X:  
  c^⋆_{t−1} = (1/X) [1 + Cov_{t−1}(g_t, pb_t) / (d_{t−1} Var_{t−1}(g_t)) − (1−X) Cov_{t−1}(i_t, g_t) / Var_{t−1}(g_t) ]. (14)
- If X = 1 (full indexation), simplifies to: c^⋆_{t−1} = 1 + Cov_{t−1}(g_t, pb_t) / (d_{t−1} Var_{t−1}(g_t)). (15)
  - Resulting conditional variance when X = 1: Var_{t−1}(∆d_t) = Var_{t−1}(pb_t) [1 − ρ^2_{g,pb,t−1}], where ρ_{g,pb} is the conditional correlation between growth and the primary balance. (16)
  - Interpretation: efficiency of growth indexation with optimal c depends on how well growth proxies the primary balance. If ρ_{g,pb} = 1, variance collapses to zero; if ρ_{g,pb} = 0, no gain beyond simple indexation (c^⋆ = 1).
- Note: optimization minimizes one-period-ahead conditional variance Var_{t−1}(∆d_t); optimal rules are state-dependent. Appendix A.1 derives conditions for longer-horizon variance minimization.

### Growth-gap indexed bonds
- For sub-sample of 15 advanced economies with output gap data, indexed coupon: \bar{i}_t = a g_t + b z_t + k_t, where z_t is the output gap. (17)
- Optimal coefficients (a^⋆, b^⋆) and additional variance reduction derived in Appendix A.2.
- Including output gap can further reduce variance when gap has independent explanatory power for the primary balance, but marginal gains often small and additional complexity may raise issuance premium.

### Simulations — methodology and data
- Objective: quantify gains as reductions in the upper tail of the debt-to-GDP ratio distribution using an extension of Blanchard et al. (2016) fan-chart methodology to a cross-country setting.
- Sample: 31 advanced and emerging economies; annual data from April 2023 IMF WEO database covering 1996–2021.
  - Variables: nominal GDP growth rate, effective nominal interest rate on debt (net interest payments / lagged debt), primary balance as share of GDP.
- Baseline expected values for interest rates, growth, and primary balance set equal to April 2023 IMF WEO projections through 2028, held constant at the 2028 level thereafter.
- Shocks drawn from multivariate normal distribution with covariance matrix estimated over 1996–2021. Benchmark assumes shocks i.i.d. across time.
- Simulations based on 10,000 Monte Carlo draws over 10-year and 20-year horizons.
- Implementation of k_t in simulations: at each forecast date t, k_t = \hat{i}_t − \hat{g}_t where \hat{i}_t and \hat{g}_t are IMF WEO baseline forecasts for nominal interest and nominal growth; for values beyond 2028 assume k_t remains constant at its 2027 value.

### Simple nominal growth-indexed bonds — simulation results
- Empirical confirmation: interest-growth differential and the primary balance are negatively correlated for most countries; simple GIBs would reduce debt-ratio variance for most of the sample.
- Two-step assessment used: fan charts (1st, 50th, 99th percentiles) and percentile-matching (Table 2).
- Heterogeneous results:
  - Average country: full indexation reduces the 99th-percentile debt-to-GDP ratio after 10 years to a level corresponding to the 91st percentile of the non-indexed distribution (equivalent to reducing a 9 percent tail probability to 1 percent).
  - Mexico and Japan: full indexation reduces a tail probability of more than 20 percent to 1 percent.
  - Cameroon: reduction negligible (99th to 98th percentile).
- Partial indexation limited: indexing 20 percent of the debt stock reduces the 99th-percentile tail by at most 2 percentile points for all 31 countries — essentially negligible from a policy standpoint.
  - Key conclusion: the share of debt indexed to growth is a first-order consideration; typical feasible issuance shares generate little insurance.

### Growth-indexed bonds with optimal coefficient — simulation results
- Re-estimate tail-risk reductions using formula (12) with c = c^⋆; simulations set k_t = \hat{i}_t − c \hat{g}_t to ensure similar expected returns and debt paths.
- Table 3 reports c^⋆ evaluated at 2021 debt levels:
  - Most countries have c^⋆ > 1 because primary balance is positively correlated with growth; values range from 0.8 for Egypt to 2.2 for the Netherlands (where ρ_{g,pb} = 0.77).
- Gains from using c^⋆ instead of c = 1 depend on how well growth proxies the primary balance:
  - Average additional variance reduction modest—around 4 percentile points.
  - Spain (R^2 = 0.73) and Costa Rica (R^2 = 0.75): additional reduction reaches 15 and 17 percentile points, respectively.
- Coefficients evaluated at 2021 debt levels used in simulations; coefficients remained fairly stable over time (Appendix, Table A.2).

### Appendix A.3 — stochastic time-varying k and related robustness
- Allowing a stochastic time-varying k_t (k_t = i_t − ˆg_t) does not change main results significantly; indexation benefits become on average higher across countries (Table A.7).
- The optimal growth indexation coefficient is on average reduced by 0.1 given the empirical positive correlations between i_t and g_t, yet the optimal coefficients remain above 1 for most countries (Table A.8).
- Use of the optimal indexation coefficient brings additional benefits in terms of tail-risk reductions—around 3 percentile points on average (Table A.9).

### Zero floor on indexed coupons (floored bond design)
- Baseline allows indexed coupon to be negative when realized growth is sufficiently low (fully symmetric insurance).
- Fairly priced floored bond sets kadjt each period so that E[max(g_t + kadjt, 0)] = ˆi_t.
- With i.i.d. normal growth shocks, kadjt solved numerically period-by-period using ˆg_t and historical moments.
- Results (Table A.10): a fairly priced floored bond provides materially less tail-risk protection than the fully symmetric design:
  - On average, the upper-tail drops from the 99th to 95th percentile (versus 91st in the baseline).
  - Japan and Greece suffer largest losses as the floor binds more tightly.
- Design implication: instrument design is important; practical fix suggestion: reduce principal repayment in years when the implied coupon would have been negative (International Capital Market Association (ICMA), 2017).

### Serial correlation (VAR(1) specification)
- Benchmark i.i.d. shocks reproduced using country-specific VAR(1) in (i_t, g_t, pb_t).
- VAR(1) results:
  - For most countries, tail-risk reductions qualitatively similar but lower—by around 3 percentile points on average (Table A.11).
  - For high-persistence countries (Mexico, Greece) VAR(1) generates different dynamics; main message remains.

### Introducing a risk premium
- Investors may require a net premium ρ to hold GIBs; indexed interest rate with premium: ̄i_t = g_t + k_t + ρ. (Equation (19))
- A positive ρ shifts baseline debt trajectory upward, partly or fully offsetting variance reduction.
- Break-even premium ̄ρ: ̄ρ = [(dnt / d0)^(1/t) − (dn,∗t / d0)^(1/t)]. (Equation (20))
  - Table 6 reports ̄ρ for 99th and 75th percentiles at 10-year and 20-year horizons under full indexation with c = 1.
  - Break-even premia are low for most countries—well below 100 basis points at 10-year horizon for the 75th percentile—though substantially higher for some high-variance countries.
  - Spain, Mexico, and Japan retain positive net benefits at premia above 100 basis points at the 99th-percentile threshold (10-year horizon).
  - For advanced economies with debt-to-GDP ratios near 100 percent, a sustained net annual premium of around 1 percent of GDP would be more than enough to eliminate the insurance value of GIBs.
- Two considerations:
  - Premia on growth-linked instruments likely countercyclical—higher during recessions when sovereigns most need insurance (Mouabbi et al., 2024).
  - A small share of indexed debt may face a liquidity discount that increases with market incompleteness, limiting gains from partial issuance (Moretti, 2020).

### Conclusion and key findings
- Headline findings:
  - Simple growth indexation (c = 1) does not provide complete insurance for debt stabilization because it does not directly hedge primary balance fluctuations. Median tail-risk reduction under full growth indexation corresponds to a shift from a 9 percent to a 1 percent tail probability—economically meaningful, but heterogeneous across countries.
  - Using an optimal coefficient (c = c⋆ > 1) improves outcomes more substantially for countries where growth is a strong proxy for fiscal conditions; issuing bonds with different indexation coefficients could help keep a portfolio indexation coefficient close to the time-varying optimal c∗t.
  - The share of indexed debt is critical: indexing 20 percent of the debt stock generates negligible tail-risk reduction at the 99th percentile for all countries in the sample; substantial gains require a coordinated initiative involving a sufficiently large share of sovereign debt.
- The size of the premium is crucial: a premium of 100 basis points, and even lower for some countries, may cancel potential benefits from indexation.
- Final assessment: there might still be a case for issuing growth-linked bonds in some countries, but benefits for debt ratio stabilization would remain elusive unless issuance is at substantially large scale and low premium.
- Suggested extensions and caveats:
  - Allow primary balance response to indexation-induced savings; model interaction between GIB issuance and monetary policy; analyze market liquidity and novelty premium determination.
  - Empirical covariance matrix assumed stable; sovereigns assumed able to issue arbitrary bond formulas; primary balance treated as exogenous—each assumption may affect quantitative outcomes.

*Source: wpiea2026080-source-pdf - 2.1 Debt Dynamics with Plain-Vanilla Bonds and Appendix A.3.*

### 2.1    Debt Dynamics with Plain-Vanilla Bonds

### 2.1    Debt Dynamics with Plain-Vanilla Bonds

### Debt dynamics with plain-vanilla bonds
- Government nominal debt D_t satisfies: D_t = (1 + i_t)D_{t−1} − PB_t. (2)
- Using first-order approximation for small i_t and g_t, the debt dynamics equation is: ∆d_t = (i_t − g_t)d_{t−1} − pb_t. (3)
- One-period-ahead conditional variance of the change in the debt ratio (conditioning on F_{t−1}):  
  Var_{t−1}(∆d_t) = Var_{t−1}(pb_t) + d^2_{t−1} Var_{t−1}(i_t − g_t) − 2 d_{t−1} Cov_{t−1}(pb_t, i_t − g_t). (4)
- Implication: higher variance of the primary balance and of the interest-growth differential, and more negative correlation between them, raise uncertainty in the future public debt ratio.

### Debt dynamics with simple GIBs (growth-indexed bonds)
- If share X of outstanding debt is indexed to growth, indexed coupon per (1) with k_t = i_t − E_{t−1}[g_t] ∈ F_{t−1}. (5)
- Within period t the effective predetermined indexed rate is: \bar{i}_t = g_t + i_t − E_{t−1}[g_t] ≡ k_t. (5)
- Debt dynamics with partial indexation: ∆d_t = [X k_t + (1−X)(i_t − g_t)] d_{t−1} − pb_t. (6)
- Conditional variance under partial indexation:  
  Var_{t−1}(∆d_t) = Var_{t−1}(pb_t) + (1−X)^2 d^2_{t−1} Var_{t−1}(i_t − g_t) − 2(1−X)d_{t−1} Cov_{t−1}(pb_t, i_t − g_t). (7)
- Special case X = 1 (full indexation): Var_{t−1}(∆d_t) = Var_{t−1}(pb_t). (8)
  - Interpretation: full indexation fully hedges growth shocks; remaining variance is driven solely by primary balance volatility.

### Reducing the uncertainty around the debt ratio
- GIB issuance does not change the conditional mean of ∆d_t because k_t is defined to make risk-neutral investors indifferent between GIBs and conventional bonds.
- GIBs reduce conditional variance iff: X < 2 − 2 Cov_{t−1}(pb_t, i_t − g_t) / (d_{t−1} Var_{t−1}(i_t − g_t)). (9)
- Converting entire debt stock to GIBs reduces debt-ratio uncertainty whenever Cov_{t−1}(pb_t, i_t − g_t) < 0.
  - Empirical finding: using IMF April 2023 WEO database, this sufficient condition is satisfied for 25 of the 31 countries in the sample over 1996–2021.
- Variance-minimizing indexed share: X^⋆_{t−1} = 1 − Cov_{t−1}(pb_t, i_t − g_t) / (d_{t−1} Var_{t−1}(i_t − g_t)). (10)
  - When Cov_{t−1}(pb_t, i_t − g_t) < 0 (empirically dominant), X^⋆_{t−1} > 1: “over-indexation” further reduces debt variance by lowering coupon payments when primary balance is depressed.
- Limitation: the simple indexation formula does not insure against primary balance fluctuations. Section 3 develops richer formulas.

### Alternative indexation formulas — theoretical benchmark
- Fully contingent formula that would eliminate one-period conditional uncertainty: \bar{i}_t = pb_t / d_{t−1} + g_t + k_t. (11)
  - Not implementable due to moral hazard: sovereign incentives to manipulate the primary balance.
  - Serves as a theoretical benchmark with minimum achievable conditional debt variance equal to zero.

### Growth-indexed bonds with an optimal coefficient
- More general indexed coupon: \bar{i}_t = c g_t + k_t with c ≥ 0. (12)
- Debt dynamics: ∆d_t = [X k_t + (1−X) i_t − (1 − cX) g_t] d_{t−1} − pb_t. (13)
- Variance-minimizing coefficient for given X:  
  c^⋆_{t−1} = (1/X) [1 + Cov_{t−1}(g_t, pb_t) / (d_{t−1} Var_{t−1}(g_t)) − (1−X) Cov_{t−1}(i_t, g_t) / Var_{t−1}(g_t) ]. (14)
- If X = 1 (full indexation), simplifies to: c^⋆_{t−1} = 1 + Cov_{t−1}(g_t, pb_t) / (d_{t−1} Var_{t−1}(g_t)). (15)
  - Resulting conditional variance when X = 1: Var_{t−1}(∆d_t) = Var_{t−1}(pb_t) [1 − ρ^2_{g,pb,t−1}], where ρ_{g,pb} is the conditional correlation between growth and the primary balance. (16)
  - Interpretation: efficiency of growth indexation with optimal c depends on how well growth proxies the primary balance. If ρ_{g,pb} = 1, variance collapses to zero (fully contingent); if ρ_{g,pb} = 0, no gain beyond simple indexation (c^⋆ = 1).
- Note: optimization minimizes one-period-ahead conditional variance Var_{t−1}(∆d_t); optimal rules are state-dependent. Appendix A.1 derives conditions for longer-horizon variance minimization.

### Growth-gap indexed bonds
- For sub-sample of 15 advanced economies with output gap data, indexed coupon: \bar{i}_t = a g_t + b z_t + k_t, where z_t is the output gap. (17)
- Optimal coefficients (a^⋆, b^⋆) and additional variance reduction derived in Appendix A.2.
- Including output gap can further reduce variance when gap has independent explanatory power for the primary balance, but marginal gains often small and additional complexity may raise issuance premium.

### Simulations — methodology and data
- Objective: quantify gains defined as reductions in the upper tail of the debt-to-GDP ratio distribution, extending Blanchard et al. (2016) fan-chart methodology to cross-country setting.
- Sample: 31 advanced and emerging economies; annual data from April 2023 IMF WEO database covering 1996–2021.
  - Variables: nominal GDP growth rate, effective nominal interest rate on debt (net interest payments / lagged debt), primary balance as share of GDP.
- Baseline expected values for interest rates, growth, and primary balance set equal to April 2023 IMF WEO projections through 2028, held constant at the 2028 level thereafter.
- Shocks drawn from multivariate normal distribution with covariance matrix estimated over 1996–2021. Benchmark assumes shocks i.i.d. across time.
- Simulations based on 10,000 Monte Carlo draws over 10-year and 20-year horizons.
- Implementation of k_t in simulations: at each forecast date t, k_t = \hat{i}_t − \hat{g}_t where \hat{i}_t and \hat{g}_t are IMF WEO baseline forecasts for nominal interest and nominal growth; for values beyond 2028 assume k_t remains constant at its 2027 value.
- Countries in sample listed for completeness in the source (31 countries).

### Simple nominal growth-indexed bonds — simulation results
- Empirical confirmation: interest-growth differential and the primary balance are negatively correlated for most countries; simple GIBs would reduce debt-ratio variance for most of the sample.
- Two-step assessment:
  - Fan charts for 1st, 50th, and 99th percentiles of debt/GDP under 100% and 20% indexation for selected countries (Figure 2).
  - Table 2 reports the percentile of the non-indexed distribution that coincides with the 99th or 75th percentile of the indexed distribution after 10 years.
- Heterogeneous results:
  - For average country in sample, full indexation reduces the 99th-percentile debt-to-GDP ratio after 10 years to a level corresponding to the 91st percentile of the non-indexed distribution (equivalent to reducing a 9 percent tail probability to 1 percent).
  - For Mexico and Japan, full indexation reduces a tail probability of more than 20 percent to 1 percent.
  - For Cameroon, reduction is negligible (99th to 98th percentile).
- Partial indexation limited: indexing 20 percent of the debt stock reduces the 99th-percentile tail by at most 2 percentile points for all 31 countries — essentially negligible from a policy standpoint.
  - Key conclusion: the share of debt indexed to growth is a first-order consideration; typical feasible issuance shares generate little insurance.

### Growth-indexed bonds with optimal coefficient — simulation results
- Re-estimate tail-risk reductions using formula (12) with c = c^⋆; simulations set k_t = \hat{i}_t − c \hat{g}_t to ensure similar expected returns and debt paths between indexed and non-indexed cases.
- Table 3 reports c^⋆ evaluated at 2021 debt levels:
  - Most countries have c^⋆ > 1 because primary balance is positively correlated with growth; values range from 0.8 for Egypt to 2.2 for the Netherlands (where ρ_{g,pb} = 0.77).
- Gains from using c^⋆ instead of c = 1 depend on how well growth proxies the primary balance:
  - Average additional variance reduction modest—around 4 percentile points.
  - For Spain (R^2 = 0.73) and Costa Rica (R^2 = 0.75), additional reduction reaches 15 and 17 percentile points, respectively.
  - Country-specific c^⋆ evaluated at 2021 debt levels used in simulations; coefficients remained fairly stable over time (Appendix, Table A.2).

### Takeaways and robustness
- Main conclusions:
  - Indexed debt reduces debt-ratio variance; gains vary significantly across countries and indexed shares and depend on co-movements between macro variables.
  - Simple c = 1 indexation offers partial insurance but does not directly hedge primary balance fluctuations.
  - Optimal indexation (c = c^⋆) improves outcomes modestly on average but substantially for countries with tight fiscal-cycle co-movement.
  - Growth-gap indexation provides further improvements for the 15-country subsample with output gap data, but marginal gains over c^⋆ are typically small and added complexity likely raises issuance premium.
- Important caveats to baseline simulations:
  - Empirical covariance matrix assumed stable; structural changes could alter results.
  - Sovereigns assumed able to issue arbitrary bond formulas; in practice standardization may limit feasible designs.
  - Primary balance treated as exogenous; lower interest payments under GIBs could allow more counter-cyclical fiscal policy, offsetting some direct hedging benefits on the upper-tail.
- Robustness checks summarized:
  - Endogenous interest rates: specification i^{end}_t = i^{base}_t + φ max(d_{t−1} − d^0, 0) with φ = 0.03 calibrated to Laubach (2009). Results remain equal or close to 0 difference relative to exogenous-rate baseline for all countries (Table A.6).
  - Stochastic k_t: main analysis treats k_t as time-varying but pre-determined; alternative treatment relaxing this assumption is considered (discussion continues beyond provided excerpt).

*Source: wpiea2026080-source-pdf - 2.1    Debt Dynamics with Plain-Vanilla Bonds*

### Appendix A.3 and analyze how a stochastic time-varying k

### Appendix A.3 and analyze how a stochastic time-varying k

### Stochastic time-varying k and indexation results
- Growth expectations remain time-varying but pre-determined; shocks to the interest rate translate one-for-one into the indexed coupon.
- Allowing a stochastic time-varying kt does not change main results in a significant way; indexation benefits become on average higher across countries (Table A.7).
- The optimal growth indexation coefficient is on average reduced by 0.1 given the empirical positive correlations between it and gt, yet the optimal coefficients remain above 1 for most countries (Table A.8).
- Use of the optimal indexation coefficient brings additional benefits in terms of tail-risk reductions—around 3 percentiles points on average (Table A.9).

### Zero floor on indexed coupons (floored bond design)
- Baseline specification allows indexed coupon to turn negative when realized growth is sufficiently low (fully symmetric insurance design).
- Investors may not accept negative coupons in practice; analysis considers a fairly priced floored bond where the contract term kadjt is set each period so that the expected coupon equals the expected non-indexed coupon, i.e. E[max(gt + kadjt, 0)] = ˆit.
- Given i.i.d. normal distribution of growth shocks, kadjt is solved numerically for each country, period by period using expected growth ˆgt and moments from the historical covariance matrix.
- Results (Table A.10) show a fairly priced floored bond provides materially less tail-risk protection than the fully symmetric design, especially for high-debt low-growth countries where negative coupons would have the highest insurance value:
  - On average, the upper-tail drops from the 99th to 95th percentile (versus 91st in the baseline).
  - Countries such as Japan or Greece suffer from the largest losses as the constraint is more binding for these countries.
- Design implication: the instrument design is important and has strong implication for the debt-stabilizing properties of indexed bonds.
- Practical fix suggestion: reduce the principal repayment in years when the implied coupon would have been negative (International Capital Market Association (ICMA), 2017).

### Serial correlation (VAR(1) specification)
- Benchmark assumes i.i.d. shocks over time.
- Reproducing main results using country-specific VAR(1) in (it, gt, pbt), allowing each variable to be serially correlated, yields:
  - For most countries, tail-risk reductions are qualitatively similar but lower in this specification—by around 3 percentiles points on average (Table A.11).
  - For a handful of high-persistence countries (Mexico, Greece), the VAR(1) generates somewhat different dynamics, but the main message remains unchanged.

### Introducing a risk premium
- Previous simulations assumed risk-neutral investors and costless contracts; in practice investors may require a net premium ρ to hold GIBs due to novelty, illiquidity, and state-contingent payoff risks.
- Indexed interest rate under a net premium ρ:  ̄it = gt + kt + ρ. (Equation (19))
- A positive ρ shifts the entire baseline debt trajectory upward, partly or fully offsetting the variance reduction.
- Fan charts under a 100-basis-point net premium show impacts for selected countries.
- Break-even premium ̄ρ—the largest ρ at which indexation still reduces the nth percentile of the debt distribution—is:
  - ̄ρ = [(dnt / d0)^(1/t) − (dn,∗t / d0)^(1/t)]. (Equation (20))
  - dnt and dn,∗t are the n−th percentiles of the non-indexed and indexed distributions at horizon t; d0 is the 2021 debt ratio.
- Table 6 reports ̄ρ for the 99th and 75th percentiles at 10-year and 20-year horizons under full indexation with c = 1:
  - Break-even premia are low for most countries—well below 100 basis points at the 10-year horizon for the 75th percentile—but substantially higher for a few high-variance countries.
  - Spain, Mexico, and Japan retain positive net benefits at premia above 100 basis points at the 99th-percentile threshold (10-year horizon), but this advantage erodes at the 75th-percentile or 20-year horizons.
  - For advanced economies with debt-to-GDP ratios near 100 percent, a sustained net annual premium of around 1 percent of GDP would be more than enough to eliminate the insurance value of GIBs—a threshold well below the premia observed in past GIB issuances.
- Two additional considerations:
  - Premia on growth-linked instruments are likely to be countercyclical, with investors demanding higher compensation during recessions—exactly when the sovereign most benefits from the insurance (Mouabbi et al., 2024).
  - A small share of indexed debt may face a liquidity discount that increases with the degree of market incompleteness, limiting gains from partial issuance (Moretti, 2020).

### Conclusion and key findings
- Three headline findings:
  - Simple growth indexation (c = 1) does not provide complete insurance for debt stabilization because it does not directly hedge primary balance fluctuations. Median tail-risk reduction under full growth indexation corresponds to a shift from a 9 percent to a 1 percent tail probability—economically meaningful, but heterogeneous across countries.
  - Using an optimal coefficient (c = c⋆ > 1) improves outcomes more substantially for countries where growth is a strong proxy for fiscal conditions; issuing bonds with different indexation coefficients could help keep a portfolio indexation coefficient close to the time-varying optimal c∗t.
  - The share of indexed debt is critical: indexing 20 percent of the debt stock generates negligible tail-risk reduction at the 99th percentile for all countries in the sample; substantial gains require a coordinated initiative involving a sufficiently large share of sovereign debt.
- The size of the premium is crucial: a premium of 100 basis points, and even lower for some countries, may cancel potential benefits from indexation.
- Under reasonable assumptions, benefits from growth-indexed bonds in terms of debt stabilization are much lower than traditionally assumed; difficulties issuing a large share of indexed bonds plus an implicit high premium help explain the non-contingency puzzle.
- Links to related literature and future research directions:
  - Findings consistent with Demertzis and Zenios (2019) on coordination failures and first-mover problems.
  - Consistent with Mouabbi et al. (2024) showing endogenous time-varying risk premia eliminate debt-stabilization benefits in a structural model.
  - Complementary reduced-form framework provides closed-form expressions for key design parameters and enables cross-country comparisons.
  - Suggested extensions: allow primary balance response to indexation-induced savings; model interaction between GIB issuance and monetary policy; analyze market liquidity and novelty premium determination (Moretti, 2020).
- Final assessment: there might still be a case for issuing growth-linked bonds in some countries, but benefits for debt ratio stabilization would remain elusive unless issuance is at substantially large scale and low premium.

*Source: Appendix A.3 and related sections of the provided content unit.*

### References

### wpiea2026080-source-pdf - References

### Literature and empirical context
- Cited works cover theory and practice of GDP-linked / growth-indexed debt instruments, pricing, hedging, and sovereign default implications (examples: Barr, D., O. Bush, and A. Pienkowski (2014); Benford, J., M. Joy, and M. Kruger (2016); Borensztein, E. and P. Mauro (2004); Shiller, R., M. Joy, J. Ostry, and J. Benford (2018)).
- Empirical and policy-relevant studies cited include IMF and European Commission reports: International Monetary Fund (2021) “Review of the Debt Sustainability Framework for Market Access Countries,” IMF Policy Paper 2021/003; European Commission (2024) “Fiscal Surveillance Under the Stability and Growth Pact: Debt Sustainability Analysis.”
- Topics in the references emphasize: pricing and no-arbitrage of GDP-linked bonds (Eguren Martin et al. 2021), state-contingent sovereign instruments as insurance (Demertzis and Zenios 2019), novelty premiums and liquidity premia (Chamon et al. 2008; Moretti 2020), and macro-finance properties of such securities (Mouabbi, Renne, and Sahuc 2024).

### Key empirical summaries and descriptive statistics (selected)
- Table 1: Descriptive statistics by country (1996–2021) reports d2021 (debt-to-GDP ratio in 2021) and moments including var(i−g), var(pb), corr(pb,i−g), corr(pb,g), corr(pb,i), corr(i,g). Examples:
  - United States: d2021 = 126.4; var(i−g) = 7.3; var(pb) = 12.8; corr(pb,i−g) = −0.31; corr(pb,g) = 0.41; corr(pb,i) = 0.37; corr(i,g) = 0.13.
  - Japan: d2021 = 255.4; var(i−g) = 4.5; var(pb) = 5.3; corr(pb,i−g) = −0.53; corr(pb,g) = 0.51; corr(pb,i) = −0.17; corr(i,g) = 0.01.
  - Greece: d2021 = 200.7; var(i−g) = 28.2; var(pb) = 14.2; corr(pb,i−g) = −0.24; corr(pb,g) = 0.27; corr(pb,i) = 0.16; corr(i,g) = 0.58.

### Simulation results on debt distribution and indexation
- Figure 2 and related text: Debt/GDP ratio forecasts for selected countries under baseline IMF projections, non-indexed debt, fully growth-indexed (100%), and partially growth-indexed (20%). Dashed lines show 1st and 99th percentiles of simulated debt/GDP distributions; blue solid line is baseline IMF projection.
- Table 2 (Simple GIBs, c = 1, 10-year horizon): Percentile of the non-indexed distribution corresponding to the 99th and 75th percentiles of indexed distribution under full (100%) and partial (20%) indexation. Example entries:
  - United States: 99th percentile → 92 (100%), 86 (20%); 75th percentile → 73 (100%), 73 (20%).
  - Japan: 99th percentile → 78 (100%), 75 (20%); 75th percentile → 71 (100%), 71 (20%).
  - Mexico: 99th percentile → 76 (100%), 75 (20%); 75th percentile → 70 (100%), 70 (20%).
- Interpretation note from table: A lower reported number indicates a larger tail-risk reduction (e.g., for the United States, full indexation reduces the 99th-percentile debt-to-GDP ratio after 10 years to a level corresponding to the 92nd percentile of the non-indexed distribution).

### Optimal indexation coefficients and their effects
- Table 3: Optimal indexation coefficient c⋆ to the nominal growth rate (evaluated at 2021 debt levels, assuming all debt is indexed, X = 1). Selected values:
  - United States: c⋆ = 1.4
  - United Kingdom: c⋆ = 1.8
  - Netherlands: c⋆ = 2.2
  - Argentina: c⋆ = 0.9
  - India: c⋆ = 1.2
- Figure 3: Debt/GDP forecasts comparing c = 1 vs. c = c⋆, showing percentile bands (1st and 99th) and baseline projections.
- Table 5 (Tail-Risk Reduction: c = 1 vs. c = c⋆, 10-year horizon): Percentiles of the non-indexed distribution corresponding to the 99th and 75th percentiles under c = 1 and c = c⋆, and the difference. Examples:
  - United States 99th percentile: c = 1 → 92; c = c⋆ → 90; Diff. = −2.
  - Spain 99th percentile: c = 1 → 86; c = c⋆ → 71; Diff. = −15.
  - Costa Rica 99th percentile: c = 1 → 86; c = c⋆ → 69; Diff. = −17.
- Note: Negative Diff. values indicate larger reductions under the optimal coefficient.

### Cost considerations: net premium and acceptable premia
- Figure 4 and Table 6 quantify effects of a net premium on growth-indexed bonds.
- Table 6: Maximum Acceptable Net Premium on Growth-indexed Bonds (percent per annum), c = 1, Full Indexation — thresholds such that the indicated percentile of the indexed debt distribution equals the same percentile of the non-indexed distribution. Selected entries (10-year / 20-year):
  - United States: 10-year 99th = 1.05; 75th = 0.29; 20-year 99th = 0.71; 75th = 0.21.
  - Mexico: 10-year 99th = 3.47; 75th = 1.08; 20-year 99th = 2.81; 75th = 0.95.
  - Costa Rica: 10-year 99th = 2.75; 75th = 0.89; 20-year 99th = 2.18; 75th = 0.81.
  - Cameroon: 10-year 99th = 0.06; 75th = 0.01; 20-year 99th = 0.04; 75th = 0.04.
- Interpretation example: For the United States, a sustained net premium of 105 basis points would equalize the 99th-percentile tails of the indexed and non-indexed debt distributions after 10 years (full indexation, c = 1).

### Robustness, extensions, and technical propositions (appendix)
- Proposition 1 (Horizon invariance): Under assumptions (local linearization, predetermined contract terms, conditional serial independence, negligible linearization errors), for any horizon h ≥ 1:
  - Var_{t−1}(d_{t+h−1}) = h V_{t−1}(X,c) + o(1)
  - arg min_{X,c} Var_{t−1}(d_{t+h−1}) = arg min_{X,c} V_{t−1}(X,c)
  - Implication: one-period variance-minimizing hedge ratios remain optimal at longer horizons under the stated conditions.
- Growth-Gap Indexed Bonds (A.2): For growth-gap coupon ̄i_t = a g_t + b z_t + k_t with X = 1, optimal coefficients:
  - a⋆ and b⋆ given by closed-form expressions (A.9) and (A.10).
  - If ρ_{g,z} = 0, minimum achievable conditional variance = Var_{t−1}(pb_t) [1 − ρ_{g,pb}^2 − ρ_{z,pb}^2] (equation A.11).
  - Empirical finding: corr(g,z) averages 0.62 in a 15-country subsample, implying output-gap terms add modest incremental benefit once growth is included.
  - Table A.3 reports a⋆ and b⋆ for the 15-country subsample (examples: United States a⋆ = 1.2, b⋆ = 0.3; United Kingdom a⋆ = 1.5, b⋆ = 0.7).
- Stochastic k_t and optimal indexation (A.3):
  - When k_t is stochastic (k_t = i_t − ˆg_t), optimal growth loading becomes c⋆,k = 1 + Cov(g_t, pb_t)/(d_{t−1} Var(g_t)) − Cov(i_t,g_t)/Var(g_t) (equation A.16).
  - Because Cov(i_t,g_t) > 0 for 24 of 31 countries, c⋆,k < c⋆ for most countries; average reduction of 0.1 (Table A.8).
  - Quantitative robustness: for most countries (21 of 31) tail-risk reduction is the same or larger under stochastic k_t versus predetermined k_t; average improvement about 2 percentile points (Table A.7).
- VAR(1) robustness check (A.4):
  - A 3-variable VAR(1) in (i_t, g_t, pb_t) is estimated country-by-country; intercepts aligned to IMF WEO forecasts for simulations.
  - Table A.11 compares i.i.d. vs. VAR(1) specifications for c = 1 and c = c⋆ at the 99th percentile, 10-year horizon. Selected comparisons (IID / VAR / Diff.):
    - United States, c = 1: IID = 92; VAR = 96; Diff. = +4.
    - France, c = c⋆: IID = 89; VAR = 95; Diff. = +6.
    - Mexico, c = c⋆: IID = 76; VAR = 90; Diff. = +14.
  - Implication: VAR(1) specification can change percentile-mapping outcomes materially for several countries; differences reported in Table A.11.

### Additional appendix tables and diagnostics (selected)
- Table A.1: Predetermined contract term k_t = ˆi_t − ˆg_t, 2022–2029 — annual values by country (examples: United States 2022 = −7.4; United Kingdom 2022 = −5.8; T̈urkiye 2022 = −100.9; Argentina 2022 = −74.1).
- Table A.2: Time-varying optimal coefficient c⋆_t, 2017–2021 — annual re-evaluations (examples: United States 2017–2021 all = 1.5 except 2021 = 1.4; Netherlands 2017 = 2.0, 2021 = 2.2).
- Tables A.5–A.10: Quantify marginal gains from growth-gap formula, endogenous interest-rate effects, stochastic k_t impacts, fairly priced zero-floor indexed coupon impacts, and more — with country-level percentile mappings and differences (see tables for full numeric detail).

*Source: wpiea2026080-source-pdf - References (Working Paper No. WP/2026/080).*

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