## 3.1   Probability Distortions

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---

### Parametrization and setup
- One period equals five years.
- First-period endowment y1 set so that E[y2(z)] = 1 = y1(1+g)^5, making g the expected growth rate.
- y2(z) = exp(z) where z ~ Normal(mean 0, std. dev. σz).
- Utility cost of default ξ is a mixture: with probability 3%, ξ = ξ̄ (a (negative) value so large that the government defaults regardless of fundamentals); with remaining probability, ξ ~ Uniform[0, 0.01].
- Threshold τ for repayment set at the mean of output in the second period.
- Parameter targets and values:
  - Borrower’s discount rate β: 6% ann.
  - Risk-free rate β: 3% ann.
  - Borrower’s risk aversion γ: 2
  - Output cost of default Δ: 20%
  - Weight of linear cost φ: 0
  - Expected growth rate g: 8% ann.
  - Threshold for repayment τ: 1
  - Std. deviation of log output σz: 0.15
- Calibration choice: parameters chosen to generate high debt and low default probability in the stylized one-period model.
- When the amount of debt is fixed at the optimal level for θ = 0 (rational-expectations case), the default probability does not vary with θ.

### Probability distortions under simple instruments
- Noncontingent debt:
  - Expected repayment is almost a step function of state z; default probability jumps to 1 when endowment is low enough.
  - Lenders’ stochastic discount factor is almost a step function of z.
  - As robustness parameter θ increases, lenders' worst-case distortions place more probability mass on the default set; expected return under distorted density falls and lenders require higher spreads.
- Linearly-indexed debt (α = 1 in illustration):
  - Lower repayments when output is lower shrink the ex-post default set.
  - Robust lenders distort probabilities downward for states with higher repayments, effectively acting as if the output process had a lower mean.
  - The amount of distortion increases with θ (shape remains similar).
  - Higher α shrinks the default set but enables stronger probability distortions — tradeoff depends on θ.
- Threshold bond (repay 1 unit if state is above its mean, 0 otherwise):
  - The jump in stipulated repayments creates a jump in probability distortions.
  - Large distortions lead to high spreads that can negate gains from contingency.

### Optimal debt design and robustness
- R★(z; θ) = repayment function that maximizes government equilibrium value V(θ, R) given lenders’ robustness θ, subject to R ≥ 0.
- Debt designed for rational-expectations lenders (θ → 0):
  - Optimal R(z) has a region with R = 0 followed by increasing repayments with output.
  - Design makes the default set empty (government only defaults with ξ = ξ̄) and exploits contingency, but is subject to large probability distortions when evaluated by robust lenders.
- Debt designed for robust lenders (illustration for θ = 4):
  - Optimal design varies much less with the state, reducing both ex-ante and ex-post contingency.
  - Probability distortions are much milder across all θ.
- As θ increases, the optimal debt structure:
  - Features less contingency, lower slopes, and avoids regions with zero or low stipulated repayments.
  - Balances minimizing contingency in stipulated repayments (to limit probability distortions) and minimizing ex-post default-contingency by promising as much as credibly possible in low states.

### Spreads: effects of ambiguity on pricing, issuance, and welfare
- Government welfare measured as equivalent increase in consumption relative to an equilibrium with same θ but noncontingent debt.
- Noncontingent debt:
  - More robust lenders charge higher spread for ambiguity of default.
  - Government issues less debt in response.
  - In parametrization, decrease in default probability roughly compensates increase in ambiguity-induced spreads.
- Linearly-indexed debt:
  - Under rational-expectations, reduces equilibrium default probability and yields welfare gains equivalent to about 0.9% of consumption compared to noncontingent debt benchmark.
  - As θ increases, ambiguity-of-contingency and ambiguity-of-default spreads open up, eroding issuance capacity and welfare gains.
  - At θ = 4, government still values moving from noncontingent to linearly-indexed debt at about 0.75% of consumption.
- Threshold debt:
  - Eliminates default risk related to y2(z), but faces spreads from event ξ = ξ̄.
  - Large probability distortions generate large ambiguity-of-contingency spreads that quickly turn welfare gains into welfare losses.
- Optimally-designed instruments:
  - Debt optimized for RE lenders: large ambiguity-of-contingency spreads arise as lenders become more robust, but because contingency is less severe than threshold debt, gains evaporate more slowly and remain positive across θ values.
  - Debt optimized for robust lenders: creates little ex-ante and ex-post contingency, keeping ambiguity spreads low and sustaining gains as θ increases.

---

### 4.2   Calibration

### Framework and objectives
- Embed state-contingent instruments in a sovereign default framework à la Eaton and Gersovitz (1981) with long-term debt and international lenders who fear model misspecification.
- Key mechanism: foreign lenders with multiplier (robustness) preferences distort probabilities toward low-utility states, boosting the perceived likelihood of low repayments and high default risk.
- Government: benevolent, issues long-term debt, cannot commit to repay; while not in default can issue debt purchased by competitive foreign lenders.
- Benchmark exercise: calibrate a model to Argentina’s experience (Argentina GDP warrants issued in 2005) with baseline noncontingent debt and then introduce a state-contingent “threshold” bond resembling the Argentina GDP warrant.

### State-contingent debt designs evaluated
- Four repayment specifications:
  - Noncontingent debt: R(z) = 1
  - Linear indexing: Rα(z) = 1 + α (y2(z) − E1[y2(z)])
  - Threshold debt: Rτ(z) = 1(y2(z) > τ)
  - Optimal design: R★(z; θ) chosen state-by-state (non-negative repayments)
- Argentina GDP-warrant features summarized:
  - Payments occur only if three conditions hold: (i) real GDP growth must exceed ‘baseline’ GDP growth in the reference year; (ii) the level of real GDP must be higher than the compounding of the baseline growth rates; (iii) cumulative past payments must be below another threshold (of 48 cents of the currency of denomination per unit of security).
  - Baseline GDP growth set to converge from an initial level of 4.3% in 2005 to a long-run level of 3% at maturity in 2034.

### Lenders’ preferences for robustness and pricing implications
- Lenders modeled with multiplier preferences (Hansen and Sargent, 2001) characterized by robustness parameter θ (reciprocal of marginal cost of relative entropy).
- An “evil agent” selects a probability distortion subject to an entropy penalty; lenders price debt using distorted expectations.
- Pricing equation (first-order condition) for debt:
  - u′(cL1) q(b; R) = β E [ exp(−θ u(cL2)) / E[exp(−θ u(cL2))] · u′(cL2) · (1 − d(b, z, ξ)) R(z) ]
- As θ → 0 the model converges to expected-utility rational-expectations pricing.

### Ambiguity premia and spread decomposition
- Debt price decomposition:
  - q(b, R) = q_RE + q_cont_θ + q_def_θ
    - q_RE: rational-expectations component
    - q_cont_θ: ambiguity component due to contingency in stipulated repayments
    - q_def_θ: ambiguity component due to default strategy
- Both ambiguity components are typically negative (lenders’ marginal utility decreasing in repayments), lowering bond prices and increasing spreads.
- Spread definitions:
  - r = E[R] / q is the implicit interest rate.
  - r − r★ is the spread, where r★ = 1/β − 1 is the international risk-free rate.
  - spr_RE = E[R]/q_RE − r★
  - spr_cont_θ = E[R]/(q_RE + q_cont_θ) − E[R]/q_RE
  - spr_def_θ = E[R]/(q_RE + q_cont_θ + q_def_θ) − E[R]/(q_RE + q_cont_θ)
- Distorted expectation used by lenders:
  - ̃E[X] = E [ exp(−θ u(cL2)) / E[exp(−θ u(cL2))] · X ]
- Distorted expectation magnifies likelihood of states with low lender utility.

---

### Quantitative findings and calibration results

### Preferences and endowment process (long-run calibration)
- Utility: u(c) = c^(1−γ) − 1 / (1 − γ) with γ denotes coefficient of relative risk aversion.
- log(y_t) = ρ log(y_{t−1}) + ε_t, with |ρ| < 1 and ε_t ∼ N(0, σ_ε^2).
- Default-income loss: φ(y) = d_0 y + d_1 y^2 (quadratic loss).

### Baseline parameter values (Table 3)
- Sovereign’s risk aversion γ: 2
- Interest rate r: 0.01
- Income autocorrelation coefficient ρ: 0.9485; 0.9484
- Standard deviation of innovations σ_ε: 0.027; 0.02
- Reentry probability ψ: 0.0385
- Duration of debt δ: 0.05
- Discount factor β: 0.9540; 0.9627
- Default cost linear d_0: -0.18819; -0.255
- Default cost quadratic d_1: 0.24558; 0.296
- Degree of robustness θ: 0; 1.62
- Linear coupon indexation α: 0
- Coupon repayment threshold τ: −∞
- Notes:
  - Period = quarter.
  - Average financial exclusion of 6.5 years (implied by ψ = 0.0385).
  - With δ = 0.05, noncontingent bonds have average maturity of 5 years.

### Calibration targets and identification
- Two parameters (β and income cost of default) calibrated to target three moments:
  - Mean spread: 8.15 percent
  - Std deviation of spread: 4.43 percent; 4.58 percent
  - Mean debt to (annual) GDP ratio: 70 percent; 46 percent
- Pouzo and Presno (2016) calibrate θ to target default frequency of three defaults per 100 years.

### Simulation design
- Extract 1000 samples of 35 consecutive periods before a default (to match Argentine sample Q1 1993–Q4 2001).
- Consider only sample paths where the last default was declared at least four periods before the beginning of each sample.
- Default frequencies computed on entire simulated sample.

### Key simulation statistics (Chatterjee and Eyigungor (2012) calibration)
- Rational Expectations, Noncontingent:
  - Spread: 8.5
  - Std Spread: 4.3
  - Debt: 69.9
  - Std(c)/Std(y): 1.24
  - Corr(y,c): 0.98
  - Corr(y,tb/y): -0.7
  - Corr(y,spread): -0.77
  - Default Prob: 5.8
  - Welfare Gains: -1.86
- Rational Expectations, Threshold (α = 1):
  - Spread: 0.6
  - Std Spread: 0.43
  - Debt: 159.6
  - Std(c)/Std(y): 0.83
  - Corr(y,c): 0.53
  - Corr(y,tb/y): 0.52
  - Corr(y,spread): -0.87
  - Default Prob: 0.56
  - Welfare Gains: 0.27
- Robust lenders θ = 1.6155, Noncontingent:
  - Spread: 8.4
  - Std Spread: 4.4
  - Debt: 62.6
  - Std(c)/Std(y): 1.25
  - Corr(y,c): 0.98
  - Corr(y,tb/y): -0.67
  - Corr(y,spread): -0.75
  - Default Prob: 2.3
  - Welfare Gains: 55
- Robust lenders θ = 1.6155, Threshold (α = 1):
  - Spread: 15.5
  - Std Spread: 2.3
  - Debt: 87.7
  - Std(c)/Std(y): 0.82
  - Corr(y,c): 0.94
  - Corr(y,tb/y): 0.58
  - Corr(y,spread): -0.61
  - Default Prob: 0.12
  - Welfare Gains: 1.8

### Key simulation statistics (Pouzo and Presno (2016) calibration)
- Rational Expectations, Noncontingent:
  - Spread: 8.1
  - Std Spread: 4.5
  - Debt: 48.7
  - Std(c)/Std(y): 1.24
  - Corr(y,c): 0.98
  - Corr(y,tb/y): -0.71
  - Corr(y,spread): -0.77
  - Default Prob: 5.5
  - Welfare Gains: -1.19
- Rational Expectations, Threshold (α = 1):
  - Spread: 0.3
  - Std Spread: 0.23
  - Debt: 116.5
  - Std(c)/Std(y): 0.82
  - Corr(y,c): 0.55
  - Corr(y,tb/y): 0.54
  - Corr(y,spread): -0.87
  - Default Prob: 0.3
  - Welfare Gains: 0.09
- Robustness θ = 1.6155, Noncontingent:
  - Spread: 8.15
  - Std Spread: 4.61
  - Debt: 44.06
  - Std(c)/Std(y): 1.25
  - Corr(y,c): 0.98
  - Corr(y,tb/y): -0.68
  - Corr(y,spread): -0.76
  - Default Prob: 3.0
  - Welfare Gains: 155
- Robustness θ = 1.6155, Threshold (α = 1):
  - Spread: 11.1
  - Std Spread: 1.58
  - Debt: 67.6
  - Std(c)/Std(y): 0.84
  - Corr(y,c): 0.93
  - Corr(y,tb/y): 0.52
  - Corr(y,spread): -0.63
  - Default Prob: 0.02
  - Welfare Gains: 0.37

### Optimal state-contingent bond design (selected results)
- Objective: maximize sovereign welfare by choosing τ and α.
- Chatterjee and Eyigungor (2012):
  - Rational Expectations: τ = 0.75, α = 4
    - Spread: 0.02
    - Std Spread: 0.02
    - Debt: 119.89
    - Std(c)/Std(y): 0.8
    - Corr(y,c): 0.99
    - Corr(y,tb/y): 0.98
    - Corr(y,spread): -0.42
    - Default Prob: 0.04
    - Welfare Gains: 3.2
  - Robustness: τ = 0.8, α = 3
    - Spread: 2.8
    - Std Spread: 0.11
    - Debt: 95.7
    - Std(c)/Std(y): 0.99
    - Corr(y,c): 0.98
    - Corr(y,tb/y): 0.13
    - Corr(y,spread): -0.17
    - Default Prob: 0.17
    - Welfare Gains: 1.4
- Pouzo and Presno (2016):
  - Rational Expectations: τ = 0.875, α = 7
    - Spread: 0.1
    - Std Spread: 0.04
    - Debt: 79.36
    - Std(c)/Std(y): 0.76
    - Corr(y,c): 0.99
    - Corr(y,tb/y): 0.98
    - Corr(y,spread): -0.91
    - Default Prob: 0.1
    - Welfare Gains: 1.79
  - Robustness: τ = 0.875, α = 5
    - Spread: 2.8
    - Std Spread: 0.13
    - Debt: 65.9
    - Std(c)/Std(y): 0.96
    - Corr(y,c): 0.98
    - Corr(y,tb/y): 0.25
    - Corr(y,spread): -0.67
    - Default Prob: 0.23
    - Welfare Gains: 0.79
- Welfare gains for a given state-contingent structure decrease in robustness parameter θ.

### Interpretations and mechanisms
- Robust lenders distort probabilities toward states with lower repayments, generating an ambiguity premium related to contingency even when default risk is negligible.
- Threshold-style GDP-linked bonds (pay only when income above mean) can produce large ambiguity premia under robust lending and thus higher spreads and welfare losses for the sovereign.
- Under rational-expectations lenders, threshold and optimally indexed bonds reduce spreads, default risk, and consumption volatility and yield welfare gains.
- The optimal bond design trades off ex-ante ambiguity premia (from stipulated payments) and ex-post ambiguity premia (from default). Optimal designs aim to eliminate ex-post default probability via ex-ante contingency while minimizing ambiguity premia.

---

### Policy-relevant implications
- Model rationalizes historical “novelty premia” on threshold-style GDP-linked instruments as ambiguity premia arising from lenders’ fears of model misspecification and the concentration of low repayments in particular states.
- Certain commonly issued state-contingent bond designs (threshold payments, regions with zero payments) can lead to equilibrium welfare losses relative to noncontingent debt once lenders’ robustness is considered.
- Optimal indexation depends critically on lenders’ degree of robustness; designing state-contingent debt to avoid sharp discontinuities or low-payment regions can reduce ambiguity spreads and improve welfare outcomes.
- Government trade-offs:
  - Reduce contingency to limit lenders’ probability distortions (which raises bond prices).
  - Use contingency to reduce ex-post default risk and smooth consumption.
  - Robust lenders limit feasible risk-sharing by penalizing structures that induce large probability distortions.

*Source: wpiea2021076-print-pdf — 3.1 Probability Distortions; 4.2 Calibration.*

### 3.1   Probability Distortions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

### wpiea2021076-print-pdf - 3.1   Probability Distortions. . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

### 3.1 Probability Distortions
- Section title: "3.1   Probability Distortions."
- Location in source PDF: page 15

### 3.2 Spreads
- Section title: "3.2   Spreads."
- Location in source PDF: page 21

### 4 The quantitative model
- Section title: "4  The quantitative model"
- Location in source PDF: page 24

### 4.1 Recursive formulation
- Section title: "4.1   Recursive formulation."
- Location in source PDF: page 26

*Source: wpiea2021076-print-pdf — https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021076-print-pdf.pdf*

### 4.2   Calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28

### 4.2   Calibration.

### Framework and objectives
- Objective: explain why state-contingent debt has been issued only modestly and at large discounts by embedding state-contingent instruments in a sovereign default framework à la Eaton and Gersovitz (1981) with long-term debt and international lenders who fear model misspecification.
- Key mechanism: foreign lenders with multiplier (robustness) preferences distort probabilities toward low-utility states, boosting the perceived likelihood of low repayments and high default risk.
- Government: benevolent, issues long-term debt, cannot commit to repay; while not in default can issue debt purchased by competitive foreign lenders.
- Benchmark exercise: calibrate a model to Argentina’s experience (Argentina GDP warrants issued in 2005) with baseline noncontingent debt and then introduce a state-contingent “threshold” bond resembling the Argentina GDP warrant.

### State-contingent debt designs evaluated
- Four repayment specifications (Table 1):
  - Noncontingent debt: R(z) = 1
  - Linear indexing: Rα(z) = 1 + α (y2(z) − E1[y2(z)])
  - Threshold debt: Rτ(z) = 1(y2(z) > τ)
  - Optimal design: R★(z; θ) chosen state-by-state (non-negative repayments)
- Argentina GDP-warrant features summarized:
  - Payments occur only if three conditions hold: (i) real GDP growth must exceed ‘baseline’ GDP growth in the reference year; (ii) the level of real GDP must be higher than the compounding of the baseline growth rates; (iii) cumulative past payments must be below another threshold (of 48 cents of the currency of denomination per unit of security).
  - Baseline GDP growth set to converge from an initial level of 4.3% in 2005 to a long-run level of 3% at maturity in 2034.

### Lenders’ preferences for robustness and pricing implications
- Lenders modeled with multiplier preferences (Hansen and Sargent, 2001) characterized by robustness parameter θ (reciprocal of marginal cost of relative entropy).
- Lenders choose actions to perform well under worst-case distributions; an “evil agent” selects a probability distortion subject to an entropy penalty.
- Pricing equation (first-order condition) for debt:
  - u′(cL1) q(b; R) = β E [ exp(−θ u(cL2)) / E[exp(−θ u(cL2))] · u′(cL2) · (1 − d(b, z, ξ)) R(z) ]
- As θ → 0 the model converges to expected-utility rational-expectations pricing.

### Ambiguity premia and spread decomposition
- Debt price decomposition (equation (1)):
  - q(b, R) = q_RE + q_cont_θ + q_def_θ
    - q_RE: rational-expectations component
    - q_cont_θ: ambiguity component due to contingency in stipulated repayments (covariance between stochastic discount factor and contingency)
    - q_def_θ: ambiguity component due to default strategy (covariance between stochastic discount factor and repayment strategy)
- Both ambiguity components are typically negative (lenders’ marginal utility decreasing in repayments), lowering bond prices and increasing spreads.
- Spread definitions:
  - r = E[R] / q is the implicit interest rate.
  - r − r★ is the spread, where r★ = 1/β − 1 is the international risk-free rate.
  - spr_RE = E[R]/q_RE − r★ (rational-expectations spread)
  - spr_cont_θ = E[R]/(q_RE + q_cont_θ) − E[R]/q_RE (premium from ambiguity of contingent repayment)
  - spr_def_θ = E[R]/(q_RE + q_cont_θ + q_def_θ) − E[R]/(q_RE + q_cont_θ) (premium from ambiguity of default)
- Distorted expectation used by lenders:
  - ̃E[X] = E [ exp(−θ u(cL2)) / E[exp(−θ u(cL2))] · X ]
  - Distorted expectation magnifies likelihood of states with low lender utility.

### Quantitative findings (overview from setup and motivation)
- In the rational-expectations case, the threshold bond resembling Argentina’s GDP warrants provides welfare gains relative to noncontingent debt and approximates the structure of optimal state-contingent debt facing rational lenders.
- With robust lenders (θ > 0), threshold-style instruments are heavily discounted because probability distortions magnify the likelihood of low-payment states, producing an “ambiguity spread.”
- The ambiguity spread can explain large observed premia on instruments like Argentina’s GDP warrants (empirical evidence: Costa, Chamon, and Ricci (2008) report spreads between 300 and 400 bps attributed to default risk and a residual “novelty” premium of 1200 bps declining to about 600 bps over time).
- Optimal state-contingent design under robustness:
  - Less contingency in stipulated repayments, lower slopes, and avoidance of regions with zero or low stipulated repayments relative to commonly used threshold bonds.
  - Gains from state-contingent debt decrease as robustness (θ) increases.
- Intuition: government trades off (i) reducing contingency to limit lenders’ probability distortions (which raises bond prices) and (ii) using contingency to reduce ex-post default risk and smooth consumption. Robust lenders limit feasible risk-sharing by penalizing structures that induce large probability distortions.

### Policy-relevant implications
- The model rationalizes historical “novelty premia” on threshold-style GDP-linked instruments as ambiguity premia arising from lenders’ fears of model misspecification and the concentration of low repayments in particular states.
- Consequence: certain commonly issued state-contingent bond designs (threshold payments, regions with zero payments) can lead to equilibrium welfare losses relative to noncontingent debt once lenders’ robustness is considered.
- Optimal indexation depends critically on lenders’ degree of robustness; designing state-contingent debt to avoid sharp discontinuities or low-payment regions can reduce ambiguity spreads and improve welfare outcomes.

*Source: wpiea2021076-print-pdf — 4.2   Calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28*

### 3.1  Probability Distortions

### 3.1  Probability Distortions

### Parametrization and setup
- One period equals five years.
- First-period endowment y1 set so that E[y2(z)] = 1 = y1(1+g)^5, making g the expected growth rate.
- Shock processes:
  - y2(z) = exp(z) where z ~ Normal(mean 0, std. dev. σz).
  - Utility cost of default ξ is a mixture: with probability 3%, ξ = ξ̄ (a (negative) value so large that the government defaults regardless of fundamentals); with remaining probability, ξ ~ Uniform[0, 0.01].
- Threshold τ for repayment set at the mean of output in the second period.
- Parameter targets and values (as summarized in Table 2):
  - Borrower’s discount rate β: 6% ann.
  - Risk-free rate β: 3% ann.
  - Borrower’s risk aversion γ: 2
  - Output cost of default Δ: 20%
  - Weight of linear cost φ: 0
  - Expected growth rate g: 8% ann.
  - Threshold for repayment τ: 1
  - Std. deviation of log output σz: 0.15
- Calibration choice: parameters chosen to generate high debt and low default probability in the stylized one-period model; default cost Δ and expected growth g set to high values to achieve this.
- When the amount of debt is fixed at the optimal level for θ = 0 (rational-expectations case), the default probability does not vary with θ.

### Probability distortions under simple instruments
- Noncontingent debt:
  - Expected repayment is almost a step function of state z; default probability jumps to 1 when endowment is low enough.
  - Lenders’ stochastic discount factor is almost a step function of z.
  - As robustness parameter θ increases, lenders' worst-case distortions place more probability mass on the default set; expected return under distorted density falls and lenders require higher spreads.
- Linearly-indexed debt (indexing coefficient α = 1 in illustration):
  - Lower repayments when output is lower shrink the ex-post default set.
  - Robust lenders distort probabilities downward for states with higher repayments, effectively acting as if the output process had a lower mean.
  - The amount of distortion increases with θ (shape remains similar).
  - Higher α shrinks the default set but enables stronger probability distortions — tradeoff depends on θ.
- Threshold bond (repay 1 unit if state is above its mean, 0 otherwise):
  - The jump in stipulated repayments creates a jump in probability distortions.
  - Large distortions lead to high spreads that can negate gains from contingency.

### Optimal debt design and robustness
- Define R★(z; θ) = repayment function that maximizes government equilibrium value V(θ, R) given lenders’ robustness θ, subject to R ≥ 0.
- Debt designed for rational-expectations lenders (θ → 0):
  - Optimal R(z) has a region with R = 0 followed by increasing repayments with output.
  - This design makes the default set empty (government only defaults with ξ = ξ̄) and exploits contingency, but is subject to large probability distortions when evaluated by robust lenders.
- Debt designed for robust lenders (illustration for θ = 4):
  - Optimal design varies much less with the state, reducing both ex-ante and ex-post contingency.
  - Resulting probability distortions are much milder across all θ.
- As θ increases, the optimal debt structure:
  - Features less contingency, lower slopes, and avoids regions with zero or low stipulated repayments.
  - Balances two goals: minimize contingency in stipulated repayments (to limit probability distortions) and minimize ex-post default-contingency by promising as much as credibly possible in low states.

### Spreads: effects of ambiguity on pricing, issuance, and welfare
- Measurement: government welfare measured as equivalent increase in consumption relative to an equilibrium with same θ but noncontingent debt.
- When issuing noncontingent debt:
  - More robust lenders charge higher spread for ambiguity of default.
  - Government issues less debt in response.
  - In parametrization, decrease in default probability roughly compensates increase in ambiguity-induced spreads.
- Linearly-indexed debt:
  - Under rational-expectations, reduces equilibrium default probability and yields welfare gains equivalent to about 0.9% of consumption compared to noncontingent debt benchmark.
  - As θ increases, ambiguity-of-contingency and ambiguity-of-default spreads open up, eroding issuance capacity and welfare gains.
  - At θ = 4, government still values moving from noncontingent to linearly-indexed debt at about 0.75% of consumption.
- Threshold debt:
  - Eliminates default risk related to y2(z), but faces spreads from event ξ = ξ̄.
  - Large probability distortions generate large ambiguity-of-contingency spreads that quickly turn welfare gains into welfare losses.
- Optimally-designed instruments:
  - Debt optimized for RE lenders: large ambiguity-of-contingency spreads arise as lenders become more robust, but because contingency is less severe than threshold debt, gains evaporate more slowly and remain positive across θ values.
  - Debt optimized for robust lenders: aims to create little ex-ante and ex-post contingency, keeping ambiguity spreads low and sustaining gains as θ increases.

### Key qualitative tradeoffs highlighted
- Introducing state-contingency in repayments can reduce ex-post default risk but increases lenders’ scope to distort probabilities (ambiguity), which raises spreads.
- Higher lender robustness parameter θ amplifies probability distortions, especially when repayment schedules feature jumps or strong state-dependence.
- Optimal debt design under robustness tends toward smoother, less state-contingent repayment profiles to limit ambiguity premia while still managing ex-post default incentives.

*Source: 3.1 Probability Distortions (wpiea2021076-print-pdf).*

### 4.2  Calibration

### 4.2 Calibration

### Preferences and endowment process
- Utility: u(c) = c^(1−γ) − 1 / (1 − γ) with constant coefficient of relative risk aversion denoted by γ.
- Persistent endowment y_t follows AR1 in logs: log(y_t) = ρ log(y_{t−1}) + ε_t, with |ρ| < 1 and ε_t ∼ N(0, σ_ε^2).
- Default-income loss function (during a default episode): φ(y) = d_0 y + d_1 y^2 (quadratic loss), following Chatterjee and Eyigungor (2012).

### Baseline parameter values (Table 3)
- Sovereign’s risk aversion γ: 2
- Interest rate r: 0.01
- Income autocorrelation coefficient ρ: 0.9485 (Chatterjee and Eyigungor (2012)); 0.9484 (Pouzo and Presno (2016))
- Standard deviation of innovations σ_ε: 0.027 (Chatterjee and Eyigungor (2012)); 0.02 (Pouzo and Presno (2016))
- Reentry probability ψ: 0.0385
- Duration of debt δ: 0.05
- Discount factor β: 0.9540 (Chatterjee and Eyigungor (2012)); 0.9627 (Pouzo and Presno (2016))
- Default cost: linear d_0: -0.18819 (Chatterjee and Eyigungor (2012)); -0.255 (Pouzo and Presno (2016))
- Default cost: quadratic d_1: 0.24558 (Chatterjee and Eyigungor (2012)); 0.296 (Pouzo and Presno (2016))
- Degree of robustness θ: 0 (Chatterjee and Eyigungor (2012)); 1.62 (Pouzo and Presno (2016))
- Linear coupon indexation α: 0
- Coupon repayment threshold τ: −∞

Notes:
- Period = quarter. Parametrizations chosen to mimic Argentina GDP behavior from Q1 1993 to Q4 2001.
- Noncontingent bonds implied by α = 0 and τ = −∞.
- Both calibrations assume average financial exclusion of 6.5 years (implied by ψ = 0.0385).
- With δ = 0.05, noncontingent bonds have average maturity of 5 years.

### Calibration targets and identification
- Two parameters (discount factor β and income cost of default) calibrated to target three moments:
  - Mean spread: 8.15 percent (both papers target 8.15 percent)
  - Std deviation of spread: 4.43 percent (Chatterjee and Eyigungor, 2012 target 4.43 percent); 4.58 percent (Pouzo and Presno, 2016 target 4.58 percent)
  - Mean debt to (annual) GDP ratio: 70 percent (Chatterjee and Eyigungor, 2012 target); 46 percent (Pouzo and Presno, 2016 target)
- Pouzo and Presno (2016) calibrate robustness θ to target default frequency of three defaults per 100 years.

### Simulation design and sample construction
- Simulations extract 1000 samples of 35 consecutive periods before a default (to match Argentine sample Q1 1993–Q4 2001).
- Consider only sample paths where the last default was declared at least four periods before the beginning of each sample.
- Default frequencies computed on entire simulated sample.

### Simulation results: Chatterjee and Eyigungor (2012) calibration (Table 4)
- Case: Rational Expectations (benchmark; Noncontingent and Threshold α = 1 reported)
  - Noncontingent:
    - Spread: 8.5
    - Std Spread: 4.3
    - Debt: 69.9
    - Std(c)/Std(y): 1.24
    - Corr(y,c): 0.98
    - Corr(y,tb/y): -0.7
    - Corr(y,spread): -0.77
    - Default Prob: 5.8
    - Welfare Gains: -1.86
  - Threshold (α = 1):
    - Spread: 0.6
    - Std Spread: 0.43
    - Debt: 159.6
    - Std(c)/Std(y): 0.83
    - Corr(y,c): 0.53
    - Corr(y,tb/y): 0.52
    - Corr(y,spread): -0.87
    - Default Prob: 0.56
    - Welfare Gains: 0.27
- Case: Robust lenders θ = 1.6155
  - Noncontingent:
    - Spread: 8.4
    - Std Spread: 4.4
    - Debt: 62.6
    - Std(c)/Std(y): 1.25
    - Corr(y,c): 0.98
    - Corr(y,tb/y): -0.67
    - Corr(y,spread): -0.75
    - Default Prob: 2.3
    - Welfare Gains: 55
  - Threshold (α = 1):
    - Spread: 15.5
    - Std Spread: 2.3
    - Debt: 87.7
    - Std(c)/Std(y): 0.82
    - Corr(y,c): 0.94
    - Corr(y,tb/y): 0.58
    - Corr(y,spread): -0.61
    - Default Prob: 0.12
    - Welfare Gains: 1.8
  - Additional entry: Threshold column also shows an extra row “--0.87 0.2” (note in source).

Note: Threshold debt pays if income is above the mean and payments are linearly indexed with alpha = 1.

Key qualitative points from this calibration:
- With rational-expectations lenders, income-indexed (threshold) bond almost eliminates spreads and default probability, increases indebtedness, and reduces consumption volatility.
- With robust lenders, the threshold bond reduces default risk but raises spreads substantially (e.g., from 8.5 to 15.5 percent), delivering welfare losses for the sovereign due to ambiguity premia related to the contingency.

### Simulation results: Pouzo and Presno (2016) calibration (Table 5)
- Case: Rational Expectations (Noncontingent and Threshold α = 1 reported)
  - Noncontingent:
    - Spread: 8.1
    - Std Spread: 4.5
    - Debt: 48.7
    - Std(c)/Std(y): 1.24
    - Corr(y,c): 0.98
    - Corr(y,tb/y): -0.71
    - Corr(y,spread): -0.77
    - Default Prob: 5.5
    - Welfare Gains: -1.19
  - Threshold (α = 1):
    - Spread: 0.3
    - Std Spread: 0.23
    - Debt: 116.5
    - Std(c)/Std(y): 0.82
    - Corr(y,c): 0.55
    - Corr(y,tb/y): 0.54
    - Corr(y,spread): -0.87
    - Default Prob: 0.3
    - Welfare Gains: 0.09
- Case: Robustness θ = 1.6155 (benchmark)
  - Noncontingent:
    - Spread: 8.15
    - Std Spread: 4.61
    - Debt: 44.06
    - Std(c)/Std(y): 1.25
    - Corr(y,c): 0.98
    - Corr(y,tb/y): -0.68
    - Corr(y,spread): -0.76
    - Default Prob: 3.0
    - Welfare Gains: 155 (source shows “3.00.02.6” and then “--0.37 0.07”; primary noncontingent row values listed above)
  - Threshold (α = 1):
    - Spread: 11.1
    - Std Spread: 1.58
    - Debt: 67.6
    - Std(c)/Std(y): 0.84
    - Corr(y,c): 0.93
    - Corr(y,tb/y): 0.52
    - Corr(y,spread): -0.63
    - Default Prob: 0.02
    - Welfare Gains: 0.37

Note: Threshold debt pays if income is above the mean and payments are linearly indexed with alpha = 1.

Key qualitative points from this calibration:
- As with Chatterjee and Eyigungor (2012) parametrization, income-indexed bonds dramatically reduce default risk and consumption volatility under rational-expectations lenders.
- With robust lenders, threshold bond increases spreads (e.g., from 8.15 to 11.1 percent) due to ambiguity premium associated with contingency, producing welfare losses.

### Optimal state-contingent bond design (Table 6)
- Objective: For each calibration maximize sovereign welfare by choosing τ and α.
- Main findings:
  - Optimal bond design depends on lenders’ preference for robustness; τ similar within each calibration, optimal α lower when lenders are robust.
  - Optimal state-contingent bonds substantially reduce default risk and consumption volatility and allow higher indebtedness.
  - When lenders have rational expectations, default risk decline implies similar sovereign spread decline.
  - When lenders are robust, sovereign spread remains around 3% even with negligible default risk due to ambiguity premium related to contingency.
  - Welfare gains from optimal design are larger with rational-expectations lenders and decrease in robustness parameter θ.

- Chatterjee and Eyigungor (2012) — Optimal designs and resulting statistics:
  - Rational Expectations: τ = 0.75, α = 4
    - Spread: 0.02
    - Std Spread: 0.02
    - Debt: 119.89
    - Std(c)/Std(y): 0.8
    - Corr(y,c): 0.99
    - Corr(y,tb/y): 0.98
    - Corr(y,spread): -0.42
    - Default Prob: 0.04
    - Welfare Gains: 3.2
  - Robustness: τ = 0.8, α = 3
    - Spread: 2.8
    - Std Spread: 0.11
    - Debt: 95.7
    - Std(c)/Std(y): 0.99
    - Corr(y,c): 0.98
    - Corr(y,tb/y): 0.13
    - Corr(y,spread): -0.17
    - Default Prob: 0.17
    - Welfare Gains: 1.4

- Pouzo and Presno (2016) — Optimal designs and resulting statistics:
  - Rational Expectations: τ = 0.875, α = 7
    - Spread: 0.1
    - Std Spread: 0.04
    - Debt: 79.36
    - Std(c)/Std(y): 0.76
    - Corr(y,c): 0.99
    - Corr(y,tb/y): 0.98
    - Corr(y,spread): -0.91
    - Default Prob: 0.1
    - Welfare Gains: 1.79
  - Robustness: τ = 0.875, α = 5
    - Spread: 2.8
    - Std Spread: 0.13
    - Debt: 65.9
    - Std(c)/Std(y): 0.96
    - Corr(y,c): 0.98
    - Corr(y,tb/y): 0.25
    - Corr(y,spread): -0.67
    - Default Prob: 0.23
    - Welfare Gains: 0.79

Additional note:
- Welfare gains for a given state-contingent structure decrease in robustness parameter θ.

### Interpretations and mechanisms
- Robust lenders distort probabilities toward states with lower repayments, generating an ambiguity premium related to contingency even when default risk is negligible.
- Threshold-style GDP-linked bonds (pay only when income above mean) can produce large ambiguity premia under robust lending and thus higher spreads and welfare losses for the sovereign.
- Under rational-expectations lenders, threshold and optimally indexed bonds reduce spreads, default risk, and consumption volatility and yield welfare gains.
- The optimal bond design trades off ex-ante ambiguity premia (from stipulated payments) and ex-post ambiguity premia (from default). Optimal designs aim to eliminate ex-post default probability via ex-ante contingency while minimizing ambiguity premia.

*Italic: Source — wpiea2021076-print-pdf (section 4.2 Calibration) — IMF working paper content provided.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021076-print-pdf.pdf_