## 2.1 Recursive Formulation

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### Recursive formulation of the government problem
- Exogenous state vector: (y; p).
- Value function for government not in default:
  - V(b; b_C; s) = max { V_R(b; b_C; s); V_D(b; b_C; s) }.
- Value of repaying:
  - V_R(b; b_C; s) = max_{i ≥ 0, i_C ≥ 0, c ≥ 0} { u(c) + β E_{s′|s} V(b′; b′_C; s′) }
  - Subject to:
    - c = y − g − δ b − [1 − I(p)] δ_C b_C + q(b′; b′_C; s) i + q_C(b′; b′_C; s) i_C
    - i = b′ − b(1 − δ)
    - i_C = b′_C − [1 − I(p)] b_C (1 − δ_C) − I(p) b_C e^{r_C}
    - q(b′; b′_C; s) ≥ q̲ for b′ > b(1 − δ)
    - q_C(b′; b′_C; s) ≥ q̲ for b′_C > [1 − I(p)] b_C (1 − δ_C) + I(p) b_C e^{r_C}
  - I(p) is an indicator equal to 1 if risk premium shock p takes the high value, and 0 otherwise.
- Value of defaulting:
  - V_D(b; b_C; s) = u(y − g) − U_D(y) + β E_{s′|s} [ V( b̄; b̄_C; s′ ) ].
- Price equations:
  - Non-contingent bond price:
    - q(b′; b′_C; s) = E_{s′|s} [ M(s′; p) [ d′ e^{−} q(b̄′; b̄′_C; s′) + (1 − d′) [ δ + (1 − δ) q(b′′; b′′_C; s′) ] ] ].
  - Cocos price:
    - q_C(b′; b′_C; s) = E_{s′|s} [ M(s′; p) [ d′ e^{−} q_C(b̄′; b̄′_C; s′) + (1 − d′) [ [1 − I(p′; g′)] [ δ_C + (1 − δ_C) q_C(b′′; b′′_C; s′) ] + I(p′; g′) e^{r_C} q_C(b′′; b′′_C; s′) ] ] ].
  - d′ = d̂(b′; b′_C; s′): next-period equilibrium default decision.
  - b′′ = b̂(b′; b′_C; s′): next-period equilibrium non-contingent debt decision.
  - b′′_C = b̂_C(b′; b′_C; s′): next-period equilibrium cocos decision.

### Recursive equilibrium
- Markov Perfect Equilibrium characterized by:
  1. Rules for default d̂, non-contingent borrowing b̂, and cocos borrowing b̂_C.
  2. Bond price functions q and q_C.
- Equilibrium conditions:
  - Given q and q_C, policy functions d̂, b̂, and b̂_C solve the Bellman equations.
  - Given policy rules { d̂; b̂; b̂_C }, bond price functions q and q_C satisfy the pricing conditions.

### Calibration for the economy without cocos
- Preferences and default cost:
  - u(c) = c^{1−γ} − 1 / (1 − γ) with γ ≠ 1.
  - Utility cost of default: U_d(y) = max{0; ω_0 + ω_1 y^g }.
- Time and key parameter values:
  - Period length: a period is a year.
  - Risk-free rate r = 4%
  - Discount factor β = 0.92
  - Income autocorrelation coefficient ρ = 0.66
  - Minimum bond price q̲ = 0.45
  - Standard deviation of innovations σ_ε = 3.4%
  - Mean log income μ = (−1/2) σ_ε^2
  - Government consumption g = 0.12
  - Recovery rate ϕ = 63%
  - Debt duration δ = 0.286
  - Probability of exiting high risk premium θ_HL = 0.8
  - Cost of defaulting ω_0 = 0.5305
  - Cost of defaulting ω_1 = 4.64
  - Borrower’s risk aversion γ = 2.19
  - Probability of entering high risk premium θ_LH0 = 0.38
  - Probability of entering high risk premium θ_LH1 = 38
  - Risk-premium shock p_H = 3.8
- Targets and identification:
  - Use Mexico (1993–2014) data for endowment process, level/duration of debt, mean spread.
  - g set to 12 percent of average income.
  - δ = 0.2845 gives average sovereign debt duration of 3 years in simulations.
  - Assume three high risk-premium episodes every twenty years; average episode duration 1.25 years.
  - θ_HL = 0.8; p_L = 0; calibrate p_H to match average spread increase in high-risk episodes.
  - Calibration targets (six moments): mean spread of 2.4 percent; mean public debt/GDP of 43.5 percent; volatility of consumption equal to volatility of income; three high-risk episodes every twenty years; average income 4 percent lower during episodes; spread increase during high-risk episodes of 2 percentage points.
- Numerical solution details:
  - Value function iteration with search over grid of debt levels and nonlinear optimization.
  - V_D and V_R and q, q_C approximated with linear interpolation over y and cubic spline over debt.
  - 20 grid points for debt levels; 25 grid points for income realizations.
  - Expectations: 50 quadrature points for income shocks.
  - Finite-horizon approximation with horizon large enough that maximum deviation between first and second-period value and price functions ≤ 10^{−6}.

### Calibration moments and baseline simulation
- Key reported moments (Data | Benchmark | With cocos):
  - Mean debt/y (%) = 43.0 | 43.1 | 53.9
  - Mean cocos debt/y (%) = n:a: | n:a: | 49.0
  - Mean r_s (%) = 2.4 | 2.4 | 2.4
  - Mean cocos r_s (%) = n:a: | n:a: | 2.8
  - Defaults per 100 years = n:a: | 6.2 | 6.8
  - Duration = 3.0 | 3.0 | 3.0
  - Duration cocos = n:a: | n:a: | 3.6
  - σ(c) = σ(y) = 1.0 | 0.99 | 0.97
  - σ(r_s) = 0.9 | 1.4 | 1.6
  - σ(r_s) cocos = n:a: | n:a: | 1.8
  - ρ(c; y) = 0.8 | 0.97 | 0.90
  - Probability high-risk-premium starts (%) = 15.0 | 15.0 | 15.0
  - Lower income during high-risk-premium (%) = 4.0 | 4.1 | 4.4
  - ∆r_s with high-risk-premium shock = 2.0 | 2.1 | 3.1
  - ∆r_s cocos with high-risk-premium shock = n:a: | n:a: | 2.7
  - Fraction of defaults triggered by liquidity (%) = 3.2 | 0.0
- Computation notes:
  - Moments computed using detrended series (Hodrick-Prescott filter smoothing parameter 100).
  - Simulation moments are means over 250 samples, each with 120 periods (30 years) without a default episode. Samples start at least five years after a default; default episodes excluded.
  - For cocos, yield, duration, and debt stock computed using expected payments incorporating payment time uncertainty.

### Numerical and interpretive remarks from calibration
- q̲ = 0.45 chosen to eliminate consumption booms before defaults and is never binding in simulations.
- Borrower’s risk aversion γ = 2.2 results from calibration.
- Domestic risk aversion is a key parameter for willingness to tolerate liquidity shocks; targeting σ(c) = σ(y) brings model closer to non-durable consumption data.
- Finite-horizon approximation approach similar to Hatchondo et al. (2010).

### The effects of cocos
- Experimental setup:
  - Compare benchmark model without cocos (i_C = 0) to model where government can issue both non-contingent bonds and cocos.
  - Assume suspended payments earn the risk-free rate (r_C = r), so nominal haircut from triggering coco contingency equals zero.
- Main simulation findings:
  - Introducing cocos increases default frequency and raises spreads because overall debt level increases with cocos (government borrows more as rollover risk is mitigated).
  - Cocos exacerbate the spread increase triggered by the risk-premium shock, even though reprofiling of cocos eliminates defaults triggered by risk-premium shocks, because:
    - Lenders dislike postponement of payments when risk premium is high.
    - Suspension of payments triggered by financing shocks increases debt levels while government faces these shocks.
  - Introducing cocos increases welfare.
    - Welfare metric: constant proportional change in consumption that makes a consumer indifferent between economies.
    - Initial consumption increase due to government sustaining higher indebtedness with cocos accounts for bulk of welfare gains.
    - Cocos improve consumption smoothing.
    - Higher lenders’ risk aversion lowers consumption in benchmark economy but increases consumption in economy with cocos.
    - Welfare gains from improved consumption smoothing outweigh welfare losses from higher default frequency.
- Robustness:
  - Simulating cocos economy without allowing government buybacks yields almost identical results: buybacks do not play a significant role.
  - Economy with only cocos (no non-contingent debt) yields similar results with slightly higher debt level (54.6), spread (2.9), and default probability (7.2).
  - Duration differences have relatively mild effects.

### How much debt relief?
- Research question: Is coco design that suspends all debt payments with no nominal haircut (r_C = r) optimal?
- Main findings:
  - Sovereign does not want less debt relief: welfare gains would be lower if cocos suspend only a fraction of debt payments.
  - Government would benefit from cocos that provide additional debt relief (haircuts): optimal rate of growth of suspended cocos payments r_C is negative, indicating optimality of haircuts after adverse shocks.
  - With haircuts, cocos lead to:
    - Significantly lower default probability.
    - Lower spreads.
    - Lower spread increase triggered by risk-premium shocks.
    - Improved consumption smoothing and lower consumption volatility.
  - Cocos that only suspend payments increase debt (due to automatic rollover of suspended payments), increasing default probability and spreads, impairing borrowing for consumption smoothing.
  - With haircuts, cocos lower debt, default probability, and spreads, improving borrowing for consumption smoothing.
  - Welfare gains from increasing haircuts level off after sufficient debt relief is provided because government can compensate by issuing fewer cocos and more non-contingent bonds.
- Table 3 excerpts for different r_C (Benchmark r_C = r | r_C = −0.25 | r_C = −0.45):
  - Mean debt/y (%) = 43.09 | 93.90 | 103.61 | 115.24
  - Mean cocos debt/y (%) = n:a: | 49.03 | 45.28 | 39.95
  - Mean r_s (%) = 2.40 | 2.42 | 1.69 | 1.57
  - Mean cocos r_s (%) = n:a: | 2.76 | 1.76 | 1.48
  - Defaults per 100 years = 6.20 | 6.79 | 4.56 | 4.11
  - σ(c) = σ(y) = 0.99 | 0.97 | 0.92 | 0.93
  - ∆r_s with shock = 2.10 | 3.09 | 1.98 | 1.71
  - ∆r_s cocos with shock = n:a: | 2.82 | 1.92 | 1.65
- Fraction of coupons paid during suspension κ and next-period cocos stock:
  - b′_C = [1 − I(p)] b_C (1 − δ_C) + I(p) b_C [ κ (1 − δ) + (1 − κ) e^{r_C} ] + i_C
  - κ = 1 → debt non-contingent; κ = 0 → all cocos payments suspended.
- Additional remark:
  - If cocos provide too much debt relief (haircuts too high), government can issue fewer cocos and more non-contingent bonds to compensate; commitment differences and duration effects limit perfect compensation, but welfare effects of effective duration are not large in simulations.

### Conclusions
- Cocos that suspend debt payments in periods of low global liquidity:
  - Reduce frequency of sovereign defaults triggered by liquidity shocks.
  - Increase consumption in low-liquidity periods.
- Cocos increase overall default frequency because they increase indebtedness.
- Full suspension without nominal haircuts (r_C = r) is not sufficient debt relief.
- Allowing cocos to trigger haircuts is welfare enhancing; cocos with haircuts reduce default frequency.
- Results hold when triggers other than global liquidity shocks are modeled, including shocks to government financing needs.

*Source: wpiea2022078-print-pdf - 2.1 Recursive Formulation*

### 2.1 Recursive Formulation

### 2.1 Recursive Formulation

### Recursive formulation of the government problem
- Let (y; p) denote the vector of exogenous states. Let V denote the value function of a government that is not currently in default. The function V satisfies:
  - V(b; b_C; s) = max { V_R(b; b_C; s); V_D(b; b_C; s) }.
- The government’s value of repaying:
  - V_R(b; b_C; s) = max_{i ≥ 0, i_C ≥ 0, c ≥ 0} { u(c) + β E_{s′|s} V(b′; b′_C; s′) }
  - subject to:
    - c = y − g − δ b − [1 − I(p)] δ_C b_C + q(b′; b′_C; s) i + q_C(b′; b′_C; s) i_C
    - i = b′ − b(1 − δ)
    - i_C = b′_C − [1 − I(p)] b_C (1 − δ_C) − I(p) b_C e^{r_C}
    - q(b′; b′_C; s) ≥ q̲ for b′ > b(1 − δ)
    - q_C(b′; b′_C; s) ≥ q̲ for b′_C > [1 − I(p)] b_C (1 − δ_C) + I(p) b_C e^{r_C}
  - where I(p) is an indicator equal to 1 if the risk premium shock p takes the high value, and 0 otherwise.
- The value of defaulting:
  - V_D(b; b_C; s) = u(y − g) − U_D(y) + β E_{s′|s} [ V( b̄; b̄_C; s′ ) ].
- Price of non-contingent bonds:
  - q(b′; b′_C; s) = E_{s′|s} [ M(s′; p) [ d′ e^{−} q(b̄′; b̄′_C; s′) + (1 − d′) [ δ + (1 − δ) q(b′′; b′′_C; s′) ] ] ].
- Price of cocos:
  - q_C(b′; b′_C; s) = E_{s′|s} [ M(s′; p) [ d′ e^{−} q_C(b̄′; b̄′_C; s′) + (1 − d′) [ [1 − I(p′; g′)] [ δ_C + (1 − δ_C) q_C(b′′; b′′_C; s′) ] + I(p′; g′) e^{r_C} q_C(b′′; b′′_C; s′) ] ] ].
  - d′ = d̂(b′; b′_C; s′) denotes next-period equilibrium default decision.
  - b′′ = b̂(b′; b′_C; s′) denotes next-period equilibrium non-contingent debt decision.
  - b′′_C = b̂_C(b′; b′_C; s′) denotes next-period equilibrium cocos decision.

### Recursive equilibrium
- A Markov Perfect Equilibrium is characterized by:
  1. Rules for default d̂, non-contingent borrowing b̂, and cocos borrowing b̂_C.
  2. Bond price functions q and q_C.
- Equilibrium conditions:
  - Given q and q_C, policy functions d̂, b̂, and b̂_C solve the Bellman equations (1), (2), and (3).
  - Given policy rules { d̂; b̂; b̂_C }, bond price functions q and q_C satisfy conditions (4) and (5), respectively.

### Calibration for the economy without cocos
- Utility and default cost:
  - u(c) = c^{1−γ} − 1 / (1 − γ) with γ ≠ 1.
  - Utility cost of default: U_d(y) = max{0; ω_0 + ω_1 y^g }.
- Period definition: a period is a year.
- Parameters explicitly reported:
  - Risk-free rate r = 4%
  - Discount factor β = 0.92
  - Income autocorrelation coefficient ρ = 0.66
  - Minimum bond price q̲ = 0.45
  - Standard deviation of innovations σ_ε = 3.4%
  - Mean log income μ = (−1/2) σ_ε^2
  - Government consumption g = 0.12
  - Recovery rate ϕ = 63%
  - Debt duration δ = 0.286
  - Probability of exiting high risk premium θ_HL = 0.8
  - Cost of defaulting ω_0 = 0.5305
  - Cost of defaulting ω_1 = 4.64
  - Borrower’s risk aversion γ = 2.19
  - Probability of entering high risk premium θ_LH0 = 0.38
  - Probability of entering high risk premium θ_LH1 = 38
  - Risk-premium shock p_H = 3.8
- Targets and identification:
  - Use Mexico (1993–2014) data for endowment process, level/duration of debt, mean spread.
  - Government consumption g set to 12 percent of average income.
  - δ = 0.2845 gives average sovereign debt duration of 3 years in simulations.
  - Assume three high risk-premium episodes every twenty years; average episode duration 1.25 years.
  - θ_HL = 0.8; p_L = 0; calibrate p_H to match average spread increase in high-risk episodes.
  - Calibration targets (six moments): mean spread of 2.4 percent; mean public debt/GDP of 43.5 percent; volatility of consumption equal to volatility of income; three high-risk episodes every twenty years; average income 4 percent lower during episodes; spread increase during high-risk episodes of 2 percentage points.
- Numerical solution details:
  - Value function iteration.
  - Search over grid of debt levels; then nonlinear optimization.
  - V_D and V_R and q, q_C approximated with linear interpolation over y and cubic spline over debt.
  - 20 grid points for debt levels; 25 grid points for income realizations.
  - Expectations: 50 quadrature points for income shocks.
  - Solve finite-horizon version with horizon large enough that maximum deviation between first and second-period value and price functions ≤ 10^{−6}.

### Calibration moments and baseline simulation
- Table 2 reports data and simulation moments (benchmark without cocos vs. with cocos). Key reported moments (Data | Benchmark | With cocos):
  - Mean debt/y (%) = 43.0 | 43.1 | 53.9
  - Mean cocos debt/y (%) = n:a: | n:a: | 49.0
  - Mean r_s (%) = 2.4 | 2.4 | 2.4
  - Mean cocos r_s (%) = n:a: | n:a: | 2.8
  - Defaults per 100 years = n:a: | 6.2 | 6.8
  - Duration = 3.0 | 3.0 | 3.0
  - Duration cocos = n:a: | n:a: | 3.6
  - σ(c) = σ(y) = 1.0 | 0.99 | 0.97
  - σ(r_s) = 0.9 | 1.4 | 1.6
  - σ(r_s) cocos = n:a: | n:a: | 1.8
  - ρ(c; y) = 0.8 | 0.97 | 0.90
  - Probability high-risk-premium starts (%) = 15.0 | 15.0 | 15.0
  - Lower income during high-risk-premium (%) = 4.0 | 4.1 | 4.4
  - ∆r_s with high-risk-premium shock = 2.0 | 2.1 | 3.1
  - ∆r_s cocos with high-risk-premium shock = n:a: | n:a: | 2.7
  - Fraction of defaults triggered by liquidity (%) = 3.2 | 0.0
- Notes:
  - Moments computed using detrended series (Hodrick-Prescott filter smoothing parameter 100).
  - Simulation moments are means over 250 samples, each with 120 periods (30 years) without a default episode. Samples start at least five years after a default; default episodes excluded.
  - For cocos, yield, duration, and debt stock computed using expected payments incorporating payment time uncertainty.

### Numerical and interpretive remarks from calibration
- q̲ = 0.45 chosen to eliminate consumption booms before defaults and is never binding in simulations.
- Borrower’s risk aversion γ = 2.2 results from calibration.
- Domestic risk aversion is a key parameter for willingness to tolerate liquidity shocks; targeting σ(c) = σ(y) brings model closer to non-durable consumption data.
- Finite-horizon approximation approach similar to Hatchondo et al. (2010).

### 4 The effects of cocos
- Experimental setup:
  - Compare benchmark model without cocos (i_C = 0) to model where government can issue both non-contingent bonds and cocos.
  - Assume suspended payments earn the risk-free rate (r_C = r), so nominal haircut from triggering coco contingency equals zero.
- Main simulation findings:
  - Introducing cocos increases default frequency and is reflected in higher spreads because overall debt level increases with cocos (government borrows more as rollover risk is mitigated).
  - Cocos exacerbate the spread increase triggered by the risk-premium shock, even though reprofiling of cocos eliminates defaults triggered by risk-premium shocks, because:
    - Lenders dislike postponement of payments when risk premium is high.
    - Suspension of payments triggered by financing shocks increases debt levels while government faces these shocks.
  - Introducing cocos increases welfare.
    - Measured welfare gains: constant proportional change in consumption that makes a consumer indifferent between economies.
    - Initial consumption increase due to government sustaining higher indebtedness with cocos accounts for bulk of welfare gains.
    - Cocos improve consumption smoothing.
    - Higher lenders’ risk aversion lowers consumption in benchmark economy but increases consumption in economy with cocos.
    - Welfare gains from improved consumption smoothing outweigh welfare losses from higher default frequency.
- Robustness notes:
  - Simulating cocos economy without allowing government buybacks yields almost identical results: buybacks do not play a significant role.
  - Economy with only cocos (no non-contingent debt) shows similar results with slightly higher debt level (54.6), spread (2.9), and default probability (7.2).
  - Duration differences have relatively mild effects.

### 5 How much debt relief?
- Question: Is the coco design that suspends all debt payments with no nominal haircut (r_C = r) optimal?
- Findings from Figure 2 and Table 3:
  - The sovereign does not want less debt relief: welfare gains would be lower if cocos suspend only a fraction of debt payments.
  - The government would benefit from cocos that provide additional debt relief (haircuts): the optimal rate of growth of suspended cocos payments r_C is negative, indicating optimality of haircuts after adverse shocks.
  - With haircuts, cocos lead to:
    - Significantly lower default probability.
    - Lower spreads.
    - Lower spread increase triggered by risk-premium shocks.
    - Improved consumption smoothing and lower consumption volatility.
  - Cocos that only suspend payments increase debt (due to automatic rollover of suspended payments), increasing default probability and spreads, impairing borrowing for consumption smoothing.
  - With haircuts, cocos lower debt, default probability, and spreads, improving borrowing for consumption smoothing.
  - Welfare gains from increasing haircuts level off after sufficient debt relief is provided because the government can compensate by issuing fewer cocos and more non-contingent bonds.
- Table 3: Cocos paying different interest rates r_C during suspensions (Benchmark r_C = r | r_C = −0.25 | r_C = −0.45)
  - Mean debt/y (%) = 43.09 | 93.90 | 103.61 | 115.24
  - Mean cocos debt/y (%) = n:a: | 49.03 | 45.28 | 39.95
  - Mean r_s (%) = 2.40 | 2.42 | 1.69 | 1.57
  - Mean cocos r_s (%) = n:a: | 2.76 | 1.76 | 1.48
  - Defaults per 100 years = 6.20 | 6.79 | 4.56 | 4.11
  - σ(c) = σ(y) = 0.99 | 0.97 | 0.92 | 0.93
  - ∆r_s with shock = 2.10 | 3.09 | 1.98 | 1.71
  - ∆r_s cocos with shock = n:a: | 2.82 | 1.92 | 1.65
- Notes on fraction of coupons paid during suspension:
  - Let κ denote fraction of coupons paid during suspension. Next-period cocos stock:
    - b′_C = [1 − I(p)] b_C (1 − δ_C) + I(p) b_C [ κ (1 − δ) + (1 − κ) e^{r_C} ] + i_C
  - κ = 1 → debt non-contingent; κ = 0 → all cocos payments suspended.
- Additional remark:
  - If cocos provide too much debt relief (haircuts too high), government can compensate by issuing fewer cocos and more non-contingent bonds. Commitment differences and duration effects limit perfect compensation, but welfare effects of effective duration are not large in simulations.

### 6 Conclusions
- Main conclusions:
  - Cocos that suspend debt payments in periods of low global liquidity reduce frequency of sovereign defaults triggered by liquidity shocks and increase consumption in low-liquidity periods.
  - Cocos increase overall default frequency because they increase indebtedness.
  - Full suspension of payments without nominal haircuts (r_C = r) is not sufficient debt relief.
  - Allowing cocos to trigger haircuts is welfare enhancing; cocos with haircuts reduce default frequency.
  - Results hold when triggers other than global liquidity shocks are modeled, including shocks to government financing needs.

*Source: wpiea2022078-print-pdf - 2.1 Recursive Formulation*

### References

### References

### Core topics covered in the cited literature
- Sovereign default models and quantitative properties:
  - Aguiar, M., Chatterjee, S., Cole, H., and Stangebye, Z. (2016). ‘Quantitative Models of Sovereign Debt Crises’.
  - Arellano, C. (2008). ‘Default Risk and Income Fluctuations in Emerging Economies’. American Economic Review, volume 98(3), 690–712.
  - Eaton, J. and Gersovitz, M. (1981). ‘Debt with potential repudiation: theoretical and empirical analysis’. Review of Economic Studies, volume 48, 289–309.
  - Hatchondo, J. C., Martinez, L., and Sapriza, H. (2010). ‘Quantitative properties of sovereign default models: solution methods matter’. Review of Economic Dynamics, volume 13, no. 4, 919–933.
  - Hatchondo, J. C., Martinez, L., and Sosa-Padilla, C. (2014). ‘Voluntary Sovereign Debt Exchanges’. Journal of Monetary Economics, volume 61, 32–50.
  - Hatchondo, J. C., Martinez, L., and Sosa Padilla, C. (2016). ‘Debt dilution and sovereign default risk’. Journal of Political Economy, volume 124(5), 1383–1422.
  - Hatchondo, J. C., Martinez, L., and Sosa-Padilla, C. (2020b). ‘Sovereign Debt Standstills’. NBER Working Paper 28292.
  - Dvorkin, M., Sanchez, J. M., Sapriza, H., and Yurdagul, E. (2020). ‘Sovereign Debt Restructurings’. AEJ: Macroeconomics, volume forthcoming.

- Debt maturity, term structure, and rollover risk:
  - Aguiar, M., Amador, M., Hopenhayn, H., and Werning, I. (2019). ‘Take the Short Route: Equilibrium Default and Debt Maturity’. Econometrica, volume 87, no. 2, 423–462.
  - Arellano, C. and Ramanarayanan, A. (2012). ‘Default and the Maturity Structure in Sovereign Bonds’. Journal of Political Economy, volume 120, no. 2, 187–232.
  - Hatchondo, J. C. and Martinez, L. (2009). ‘Long-duration bonds and sovereign defaults’. Journal of International Economics, volume 79, 117 – 125.
  - Vasicek, O. (1977). ‘An equilibrium characterization of the term structure’. Journal of Financial Economics, volume 5, 177–188.
  - Bianchi, J., Hatchondo, J. C., and Martinez, L. (2018). ‘International Reserves and Rollover Risk’. American Economics Review, volume 108(9), 2629–2670.
  - Mihalache, G. (2020). ‘Sovereign default resolution through maturity extension’. Journal of International Economics, volume 125C.

- State-contingent instruments, GDP-indexing, and contingent debt:
  - Brooke, M., Mendes, R., Pienkowski, A., and Santor, E. (2011). ‘Sovereign Default and State-Contingent Debt’. Joint study of the Bank of Canada and the Bank of England.
  - Hatchondo, J. C. and Martinez, L. (2012). ‘On the Benefits of GDP-Indexed Government Debt: Lessons from a Model of Sovereign Defaults’. Economic Quqrterly, volume 98 (2), 139–157.
  - IMF (2017a). ‘State-Contingent Debt Instruments for Sovereigns’. IMF Policy Paper. International Monetary Fund.
  - IMF (2017b). ‘State-Contingent Debt Instruments for Sovereigns—Annexes’. IMF Policy Paper. International Monetary Fund.
  - Önder, Y. K. (2021). ‘Optimal GDP-indexed Bonds’. Ghent University Working Paper.
  - Consiglio, A. and Zenios, S. A. (2015). ‘The case for contingent convertible debt for sovereigns’. The Wharton Financial Institutions Center, Working Paper 15-13.

- Macroeconomic vulnerabilities, sudden stops, and emerging market cycles:
  - Aguiar, M. and Gopinath, G. (2007). ‘Emerging markets business cycles: the cycle is the trend’. Journal of Political Economy, volume 115, no. 1, 69–102.
  - Calvo, G., Izquierdo, A., and Talvi, E. (2006). ‘Sudden Stops and Phoenix Miracles in Emerging Markets’. The American Economic Review, volume 96 (2), 405–410.
  - Calvo, G. A., Izquierdo, A., and Mejia, L.-F. (2004). ‘On the Empirics of Sudden Stops: The Relevance of Balance-Sheet Effects’. NBER Working Paper 10520.
  - Neumeyer, P. and Perri, F. (2005). ‘Business cycles in emerging economies: the role of interest rates’. Journal of Monetary Economics, volume 52, 345–380.
  - Uribe, M. and Yue, V. (2006). ‘Country spreads and emerging countries: Who drives whom?’ Journal of International Economics, volume 69, 6–36.

### Risk management, insurance, and commodity/mineral revenue issues
- Commodity price risk management and macro-hedging:
  - Borensztein, E., Jeanne, O., and Sandri, D. (2013). ‘Macro-hedging for commodity exporters’. Journal of Development Economics, volume 101, 105–116.
  - Lopez-Martin, B., Leal, J., and Martinez Fritscher, A. (2019). ‘Commodity price risk management and fiscal policy in a sovereign default model’. Journal of International Money and Finance, volume 96, 304–323. ISSN 0261-5606.
  - Ma, C. and Valencia, F. (2017). ‘Welfare Gains from Market Insurance: The Case of Mexican Oil Price Risk’. Manuscript.

- Natural disasters, climate risk, and sovereign risk:
  - Borensztein, E., Cavallo, E., and Jeanne, O. (2017). ‘The welfare gains from macro-insurance against natural disasters’. Journal of Development Economics, volume 124, 142–156.
  - Mallucci, E. (2020). ‘Natural Disasters, Climate Change, and Sovereign Risk’. International Finance Discussion Papers 1291. Washington: Board of Governors of the Federal Reserve System, https://doi.org/10.17016/IFDP.2020.1291.
  - Phan, T. and Schwartzman, F. (2021). ‘Climate Defaults and Financial Adaptation’. Unpublish manuscript.

### Policy, institutional frameworks, and proposals
- IMF and multilateral policy work:
  - IMF (2013). ‘Staff Guidance Note for Public Debt Sustainability Analysis in Market-Access Countries’. International Monetary Fund.
  - IMF (2014). ‘The Fund’s Lending Framework and Sovereign Debt - Preliminary Considerations’. Technical report.
  - IMF (2017a). ‘State-Contingent Debt Instruments for Sovereigns’. IMF Policy Paper. International Monetary Fund.
  - IMF (2017b). ‘State-Contingent Debt Instruments for Sovereigns—Annexes’. IMF Policy Paper. International Monetary Fund.

- Proposals for crisis management and institutional design:
  - Buiter, W. H. and Sibert, A. C. (1999). ‘UDROP: A Contribution to the New International Financial Architecture’. International Finance, volume 2, 227–247.
  - Barkbu, B., Eichengreen, B., and Mody, A. (2012). ‘Financial crises and the multilateral response: What the historical record shows’. Journal of International Economics, volume 88, 422–435.
  - Weber, A. A., Ulbrich, J., and Wendorff, K. (2011). ‘Safeguarding financial market stability, strengthening investor responsibility, protecting taxpayers A proposal to reinforce the European Stability Mechanism through supplementary bond issuance terms’. Research Paper, Deutsche Bundesbank.
  - FT (2013). ‘Creditors likely losers from IMF rethink, by Robin Wigglesworth’. Financial Times, June 4.
  - FT (2014). ‘Finance: In search of a better bailout’. Financial Times, January 26.

### Other relevant methodological and thematic contributions
- Hedging, precautionary contractions, and financial frictions:
  - Caballero, R. and Panageas, S. (2008). ‘Hedging Sudden Stops and Precautionary Contractions’. Journal of Development Economics, volume 85, 28–57.
  - Alvarez, F., Brandao, L., and Toledo, M. P. (2013). ‘Durable goods, financial frictions, and business cycles in emerging economies’. Journal of Monetary Economics, volume 60, 720–736.

- Welfare, calibration, and theoretical contributions:
  - Jeanne, O. and Ranciere, R. (2011). ‘The Optimal Level of Reserves for Emerging Market Countries: a New Formula and Some Applications’. Economic Journal, volume 121(555), 905–930.
  - Hatchondo, J. C. and Martinez, L. (2017). ‘Calibrating the income cost in models of sovereign defaults’. In Rethinking Fiscal Policy after the Crisis, Cambridge University Press.
  - Bocola, L. and Dovis, A. (2015). ‘Self-Fulfilling Debt Crises: A Quantitative Analysis’. Manuscript.
  - Fernández, R. and Martin, A. (2015). ‘The Long and the Short of It: Sovereign Debt Crises and Debt Maturity’. Manuscript.
  - Önder, Y. K., Alejandra, R. S. M., and Villamizar-Villegas, M. (2022). ‘Debt Moratorium: Theory and Evidence’. Unpublished Manuscript.
  - Mallucci, E. (2020). ‘Natural Disasters, Climate Change, and Sovereign Risk’. International Finance Discussion Papers 1291.

*Sovereign Cocos — Working Paper No. WP/2022/078*

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_Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022078-print-pdf.pdf_