## _wp13174

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### Introduction
- Research question: optimal duration of sovereign debt using a model of sovereign defaults à la Eaton and Gersovitz (1981) augmented to allow for long-term debt.
- Trade-off:
  - Longer-term instruments reduce rollover risk but exacerbate the debt dilution problem.
  - Shorter-term debt mitigates debt dilution; one-period bonds eliminate dilution in the extreme.
- Calibration reference: Bianchi et al. (2012), using Mexico as a reference.
- Key simulation comparisons:
  - Average duration without sudden stops = 4.5 years; with sudden stops = 5.4 years.
  - Ex-ante optimal average durations:
    - Without sudden stops: 0.8 years.
    - With sudden stops: 3.7 years.
  - Sequential-choice versus ex-ante commitment:
    - In the economy with sudden stops, the duration chosen sequentially is almost 2 years longer than the ex-ante preferred duration.
    - In the economy without sudden stops, the duration chosen sequentially is on average almost 4 years longer than the ex-ante preferred duration.
- Policy interpretation:
  - Time inconsistency and debt dilution lead governments to choose longer durations sequentially than under ex-ante commitment.
  - Results question the convenience of policies that increase sovereign debt duration for economies facing significant sovereign risk.

### Model (overview and recursive formulation)
- Economy and government:
  - Dynamic small-open-economy with non-state-contingent defaultable debt of short and long maturity.
  - Endowment y ∈ Y ⊂ R++ follows a Markov process.
  - Benevolent government maximizes E_t Σ_{j=t}^∞ β^{j−t} u(c_j); u strictly increasing and concave.
- Sudden stops:
  - During a sudden stop, government cannot issue new debt and suffers income loss φ_s(y).
  - Sudden-stop shock starts with probability π ∈ [0,1] and ends with probability ψ_s ∈ [0,1]; government can buy back debt while in a sudden stop.
- Default:
  - Default triggers exclusion for a stochastic number of periods; income during exclusion: y − φ_d(y).
  - Regains access with probability ψ_d ∈ [0,1]; recovery rate in default assumed zero.
- Bonds and budget constraint (with access to credit):
  - One-period (short) bonds and long bonds promising infinite coupons that decrease at rate δ.
  - Long-bond dynamics: b_L,t+1 = (1−δ) b_L,t + i_L,t.
  - Budget: c_t = y_t − b_S,t − b_L,t + b_S,t+1 q_S,t + i_L,t q_L,t.
  - If s = 1 (sudden stop), then b_L′ − (1−δ) b_L ≤ 0 and b_S′ = 0.
- Default behavior and investor pricing:
  - When default occurs, it applies to all current and future obligations.
  - Foreign investors risk-neutral, discount at rate r; bond prices satisfy zero-expected-profit conditions.
- Recursive value functions:
  - V(b_S, b_L, y, s) = max{ V_R(b_S, b_L, y, s), V_D(y, s) } with specified V_R and V_D functional forms and constraints.
- Bond-price functional equations:
  - One-period: q_S(b_S′, b_L′, y, s) (1 + r) = E_{(y′, s′)|(y,s)} [ 1 − d̂(b_S′, b_L′, y′, s′) ].
  - Long-term: q_L(b_S′, b_L′, y, s) (1 + r) = E_{(y′, s′)|(y,s)} [ 1 − d̂(b_S′, b_L′, y′, s′) ] [ 1 + (1−δ) q(b_S′′, b_L′′, y′, s′) ].
- Equilibrium concept: Markov Perfect Equilibrium characterized by value functions V, V_R, V_D; rules d̂, b̂_S, b̂_L, ĉ; and bond-price functions q_S, q_L consistent with investor indifference.

### Calibration
- Functional forms:
  - u(c) = (c^{1−γ} −1)/(1−γ), γ̸= 1.
  - log(y_t) = (1−ρ)μ + ρ log(y_{t−1}) + ε_t, ε_t ∼ N(0,σ^2_ε).
  - Default cost: φ_d(y) = d_0 y + d_1 y^2.
  - Sudden-stop income loss: φ_s(y) = λ φ_d(y).
- Calibration choices (period = quarter):
  - Risk aversion γ 2
  - Risk-free rate r 1%
  - Probability default exclusion ends ψ_d 0.083 (average duration of sovereign default events = 3 years)
  - Probability of sudden-stop exclusion ends π 0.025 (one sudden stop every 10 years)
  - Probability of reentry after sudden stop ψ_s 0.25 (average duration of sudden stops = 1 year)
  - Debt duration δ 0.03
  - Income autocorrelation coefficient ρ 0.94
  - Standard deviation of innovations σ_ε 1.5%
  - Mean log income μ (-1/2) σ^2_ε
  - Discount factor β 0.9745
  - Income cost of defaulting d_0 -1.01683
  - Income cost of defaulting d_1 1.18961
  - Income cost of sudden stops λ 0.5
- Calibration targets:
  - Average debt-to-GDP ratio of 43 percent.
  - Mean spread of 3.4 percent.
  - Spread standard deviation of 1.5 percent.
  - Average accumulated income cost of a sudden stop of 14 percent of annual income.
- Sovereign spread and debt reporting:
  - Yield if held to maturity: q^M_t = 1/(1+i) + Σ_{j=1}^∞ (1−δ)^j/(1+i)^{j+1}, where δ = 1 for one-period bonds.
  - Sovereign spread = i − r.
  - Annualized spread r^s_t = ( (1+i)/(1+r) )^4 − 1.
  - Debt levels reported as present value discounted at risk-free rate: b^M_t (1+r)(δ+r)^{−1}, with δ = 1 for one-period bonds.
  - Debt reported as percentage of annualized income.
  - Macaulay duration used.

### Computation
- Numerical method:
  - Value function iteration.
  - Value and bond price functions correspond to first period of a finite-horizon economy with horizon chosen so max deviation between first- and second-period value and bond price functions < 10^{−6}.
  - Optimal portfolio: grid search over debt levels, then nonlinear optimizer initialized at best grid portfolio.
- Function approximation and grids:
  - V_D, V_R, q_S, q_L approximated using linear interpolation over y and cubic spline over debt levels.
  - Grid points: 20 for one-period and long-term debt; 25 for income realizations.
  - Expectations: 50 quadrature points for income shock.

### Results
- Benchmark properties:
  - Model with sudden stops replicates: high and countercyclical spread, consumption more volatile than income, countercyclical trade balance.
  - Sudden stops increase debt duration chosen by government.
- A. Effect of sudden stops on debt duration (simulation moments)
  - With sudden stops vs Without sudden stops:
    - Average duration (years): 5.4 vs 4.5
    - Mean Long-Debt-to-GDP: 39.7 vs 29.7
    - Mean Short-Debt-to-GDP: 0.0 vs 9.7
    - Mean Long-Debt r^s: 3.0 vs 1.6
    - Mean Short-Debt r^s: na vs 1.5
    - Long-Debt σ(r^s): 1.7 vs 1.0
    - Short-Debt σ(r^s): na vs 5.7
    - σ(c)/σ(y): 1.5 vs 1.4
    - σ(tb): 1.3 vs 1.9
    - ρ(tb,y): -0.9 vs -0.5
    - ρ(c,y): 0.99 vs 0.93
  - Mechanisms:
    - In low income, lenders anticipate lower future cost of defaulting → higher interest rates → government reduces borrowing → countercyclical trade balance.
    - Absence of sudden stops lowers average duration by almost one year and reduces average spread.
    - Change in duration does not significantly affect level of nominal debt obligations.
- B. Optimal ex-ante debt duration (commitment case, b_S = 0)
  - Method: solve model for different δ values; measure ex-ante welfare as continuation value from zero debt and income at unconditional mean.
  - Optimal δ:
    - With sudden stops: δ = 0.056 (implying duration of 3.7 years).
    - Without sudden stops: δ = 0.29 (implying duration of 0.8 years).
  - Committed-optimal simulation moments:
    - With sudden stops vs Without sudden stops:
      - Average duration (years): 3.6 vs 0.8
      - Mean Debt-to-GDP: 34.4 vs 34.8
      - Mean r^s: 1.7 vs 0.5
      - σ(r^s): 1.2 vs 0.7
      - σ(c)/σ(y): 1.8 vs 2.1
      - σ(tb): 2.2 vs 2.2
      - ρ(tb,y): -0.3 vs -0.7
      - ρ(c,y): 0.76 vs 0.91
  - Time inconsistency:
    - Optimal ex-ante duration is shorter than sequentially chosen duration in both economies.
    - Sequential decision-making fails to internalize the effect of current debt composition on future borrowing terms, leading to longer durations under sequential choices.

### Conclusions and policy implications
- Main results:
  - Sudden stops increase the preferred duration of sovereign debt.
  - There is a time inconsistency problem: governments would like to commit to a shorter debt duration than the one chosen under sequential decision-making.
- Policy implication:
  - Results cast doubt on the convenience of policies that increase sovereign debt duration for economies facing significant sovereign risk.

*Source: _wp13174 - 3. and bond price functionsq (PDF chapter).*

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

### _wp13174 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

### Introduction
- Research question: optimal duration of sovereign debt using a model of sovereign defaults à la Eaton and Gersovitz (1981) augmented to allow for long-term debt.
- Trade-off highlighted:
  - Longer-term instruments reduce rollover risk but exacerbate the debt dilution problem (see Hatchondo et al., 2010b).
  - Shorter-term debt mitigates debt dilution; in the extreme, one-period bonds eliminate dilution.
- Model setup and calibration:
  - Calibration presented by Bianchi et al. (2012), using Mexico as a reference.
  - Simulations replicate: a high and countercyclical spread, a high consumption volatility, and a countercyclical trade balance.
- Key quantitative findings from simulations:
  - Presence of sudden stops increases government debt duration: average duration without sudden stops = 4.5 years; with sudden stops = 5.4 years.
  - Ex-ante optimal average durations:
    - Without sudden stops: 0.8 years.
    - With sudden stops: 3.7 years.
  - Sequential-choice versus ex-ante commitment:
    - In the economy with sudden stops, the duration chosen sequentially is almost 2 years longer than the ex-ante preferred duration.
    - In the economy without sudden stops, the duration chosen sequentially is on average almost 4 years longer than the ex-ante preferred duration.
- Interpretation and policy implication:
  - Time inconsistency and debt dilution lead governments to choose longer durations sequentially than they would commit to ex-ante.
  - Results question the convenience of policies that increase sovereign debt duration for economies facing significant sovereign risk.
- Relation to literature:
  - Extends Arellano and Ramanarayanan (2012) and Hatchondo and Martinez (2009) by comparing optimal durations with and without sudden stops and contrasting sequentially chosen durations with ex-ante optimal ones.
  - Notes robustness exercises in Chatterjee and Eyigungor (2012) and Hatchondo et al. (2010b): under high probability of rollover crises or sudden stops, longer maturities can be preferred in some calibrations.

### Model (Overview)
- Economy:
  - Dynamic small-open-economy with non-state-contingent defaultable debt of short and long maturity.
  - Endowment y ∈ Y ⊂ R++ follows a Markov process.
- Government:
  - Benevolent government maximizes E_t sum_{j=t}^∞ β^{j−t} u(c_j).
  - Utility u is strictly increasing and concave.
  - Timing: income and sudden-stop shocks realized; government decides default and portfolio subject to constraints from sudden-stop shock and default decision.
- Sudden stops:
  - During a sudden stop, government cannot issue new debt and suffers income loss φ_s(y).
  - Sudden-stop shock follows a Markov process: starts with probability π ∈ [0,1] and ends with probability ψ_s ∈ [0,1].
  - Government can buy back debt while in a sudden stop.
- Default:
  - Default triggers exclusion from credit markets for a stochastic number of periods.
  - Income during exclusion: y − φ_d(y).
  - Regains access with constant probability ψ_d ∈ [0,1].
  - Recovery rate in default assumed zero.
- Bonds:
  - Government can issue one-period (short) bonds and long-duration bonds promising an infinite stream of coupons that decrease at rate δ.
  - Long-bond dynamics: b_L,t+1 = (1−δ) b_L,t + i_L,t, where b_L,t is number of long-bond coupons due at beginning of period t and i_L,t is number of long bonds issued in t.
  - Budget constraint (with access to credit):
    - c_t = y_t − b_S,t − b_L,t + b_S,t+1 q_S,t + i_L,t q_L,t,
    - where q_M,t is price of bond of maturity M ∈ {S,L}, depending on exogenous shocks and policy pair (b_S,t+1, b_L,t+1).
- Default behavior:
  - When government defaults, it does so on all current and future debt obligations.
  - Foreign investors are risk-neutral and discount at rate r.
  - Bonds priced competitively; price functions satisfy zero-expected-profit conditions.

### Recursive Formulation
- Value functions and choices:
  - V(b_S, b_L, y, s) = max{ V_R(b_S, b_L, y, s), V_D(y, s) }.
  - Repay value:
    - V_R(b_S, b_L, y, s) = max_{b_S′ ≥ 0, b_L′ ≥ 0, c ≥ 0} { u(c) + β E_{(y′, s′)|(y,s)} V(b_S′, b_L′, y′, s′) }
    - Subject to c = y − s φ_s(y) − b_S − b_L + q_S(b_S′, b_L′, y, s) b_S′ + q_L(b_S′, b_L′, y, s) [ b_L′ − (1−δ) b_L ].
    - If s = 1, then b_L′ − (1−δ) b_L ≤ 0 and b_S′ = 0.
  - Default value:
    - V_D(y, s) = u(y − φ_d(y)) + β E_{(y′, s′)|(y,s)} [ (1−ψ_d) V_D(y′, s′) + ψ_d V(0,0,y′, s′) ].
- Decision rules:
  - Solution yields rules for default d̂, one-period debt b̂_S, long-term debt b̂_L, and consumption ĉ.
  - d̂ = 1 if government defaults, 0 otherwise.

### Bond Pricing and Equilibrium Conditions
- Bond-price functional equations (investor indifference / zero expected profit):
  - One-period bond:
    - q_S(b_S′, b_L′, y, s) (1 + r) = E_{(y′, s′)|(y,s)} [ 1 − d̂(b_S′, b_L′, y′, s′) ].
  - Long-term bond:
    - q_L(b_S′, b_L′, y, s) (1 + r) = E_{(y′, s′)|(y,s)} [ 1 − d̂(b_S′, b_L′, y′, s′) ] [ 1 + (1−δ) q(b_S′′, b_L′′, y′, s′) ].
    - Where b_S′′ = b̂_S(b_S′, b_L′, y′, s′) and b_L′′ = b̂_L(b_S′, b_L′, y′, s′).
- Economic interpretation:
  - Price equations reflect investor indifference between selling today (investing risk-free) and keeping the bond (receiving coupon and selling next period).
- Equilibrium concept:
  - Focus on Markov Perfect Equilibrium where strategies depend only on payoff-relevant state variables.

### Recursive Equilibrium (characterization)
A Markov Perfect Equilibrium is characterized by:
- A set of value functions V, V_R, and V_D.
- Rules for default d̂, borrowing b̂_S and b̂_L, and consumption ĉ.
- Bond-price functions q_S and q_L consistent with investor zero-expected-profit conditions and the government's decision rules.

*Source: _wp13174 - References (source PDF content provided).*

### 3. and bond price functionsq

### 3. and bond price functionsq

### III. Calibration
- Utility: u(c) = (c^{1−γ} −1)/(1−γ), with γ̸= 1.
- Endowment process: log(y_t) = (1−ρ)μ + ρ log(y_{t−1}) + ε_t, with |ρ|<1, ε_t ∼ N(0,σ^2_ε).
- Default cost (asymmetric): φ_d(y) = d_0 y + d_1 y^2 (quadratic loss function).
- Sudden-stop income loss: φ_s(y) = λ φ_d(y).
- Calibration choices (period = quarter):
  - Risk aversion γ 2
  - Risk-free rate r 1%
  - Probability default exclusion ends ψ_d 0.083 (average duration of sovereign default events = 3 years)
  - Probability of sudden-stop exclusion ends π 0.025 (one sudden stop every 10 years)
  - Probability of reentry after sudden stop ψ_s 0.25 (average duration of sudden stops = 1 year)
  - Debt duration δ 0.03
  - Income autocorrelation coefficient ρ 0.94
  - Standard deviation of innovations σ_ε 1.5%
  - Mean log income μ (-1/2) σ^2_ε
  - Discount factor β 0.9745
  - Income cost of defaulting d_0 -1.01683
  - Income cost of defaulting d_1 1.18961
  - Income cost of sudden stops λ 0.5
- Calibration targets and rationale:
  - Parameters governing the endowment process, level of debt, and mean and standard deviation of the spread are taken from Bianchi et al. (2012) using Mexico as reference.
  - Targets include average debt-to-GDP ratio of 43 percent, mean spread of 3.4 percent, spread standard deviation of 1.5 percent, and average accumulated income cost of a sudden stop of 14 percent of annual income.
- Sovereign spread computation:
  - Yield if held to maturity: q^M_t = 1/(1+i) + Σ_{j=1}^∞ (1−δ)^j/(1+i)^{j+1}, where δ = 1 for one-period bonds.
  - Sovereign spread = i − r.
  - Annualized spread r^s_t = ( (1+i)/(1+r) )^4 − 1.
- Debt levels reported as present value of future payment obligations discounted at the risk-free rate: b^M_t (1+r)(δ+r)^{−1}, with δ = 1 for one-period bonds.
- Debt reported as percentage of annualized income.
- Note on duration: Macaulay definition used; with coupon structure D = [1+r]/[ε + r] (?) — original text notes one-period bond equivalent has duration = 1 period.

### IV. Computation
- Solution method:
  - Value function iteration.
  - Value and bond price functions correspond to first period of a finite-horizon economy with sufficiently many periods so max deviation between first- and second-period value and bond price functions < 10^{−6}.
  - Optimal portfolio allocation: search over grid of debt levels, then use best grid portfolio as initial guess in nonlinear optimizer.
- Function approximation and grids:
  - V_D and V_R and bond price functions q_S and q_L approximated using linear interpolation over y and cubic spline interpolation over debt levels.
  - Grid points: 20 for one-period and long-term debt; 25 for income realizations.
  - Expectations: 50 quadrature points for income shock.
- References to numerical method choices: Hatchondo et al. (2010a) discussion motivating interpolation and finite-horizon equilibrium.

### V. Results
- Overview:
  - Benchmark model with sudden stops matches features of emerging economies: high and countercyclical spread, consumption more volatile than income, countercyclical trade balance.
  - Sudden stops increase debt duration chosen by government.
  - Computation of optimal ex-ante debt duration; presence of sudden stops influences optimal duration.
  - Sequential debt-composition choice yields longer durations than ex-ante optimal duration (time inconsistency).
- A. Effect of sudden stops on debt duration (Simulation moments, Table 2)
  - With sudden stops vs Without sudden stops:
    - Average duration (years): 5.4 vs 4.5
    - Mean Long-Debt-to-GDP: 39.7 vs 29.7
    - Mean Short-Debt-to-GDP: 0.0 vs 9.7
    - Mean Long-Debt r^s: 3.0 vs 1.6
    - Mean Short-Debt r^s: na vs 1.5
    - Long-Debt σ(r^s): 1.7 vs 1.0
    - Short-Debt σ(r^s): na vs 5.7
    - σ(c)/σ(y): 1.5 vs 1.4
    - σ(tb): 1.3 vs 1.9
    - ρ(tb,y): -0.9 vs -0.5
    - ρ(c,y): 0.99 vs 0.93
  - Key mechanisms:
    - In low income, lenders anticipate lower future cost of defaulting → higher interest rates → government reduces borrowing → countercyclical trade balance.
    - Absence of sudden stops lowers average duration by almost one year and reduces average spread.
    - Change in duration does not significantly affect level of nominal debt obligations.
- B. Optimal ex-ante debt duration (commitment case, b_S = 0)
  - Method: solve model for different δ values; measure ex-ante welfare as continuation value from zero debt and income at unconditional mean.
  - Optimal δ:
    - With sudden stops: δ = 0.056 (implying duration of 3.7 years).
    - Without sudden stops: δ = 0.29 (implying duration of 0.8 years).
  - Effects of debt duration (Figure 2):
    - Shorter debt duration → larger debt obligations for given debt level → larger consumption fall when borrowing cost increases.
    - Shorter debt duration → mitigates debt dilution problem → reduces interest rate spread.
  - Simulation moments for committed-optimal δ (Table 3):
    - With sudden stops vs Without sudden stops:
      - Average duration (years): 3.6 vs 0.8
      - Mean Debt-to-GDP: 34.4 vs 34.8
      - Mean r^s: 1.7 vs 0.5
      - σ(r^s): 1.2 vs 0.7
      - σ(c)/σ(y): 1.8 vs 2.1
      - σ(tb): 2.2 vs 2.2
      - ρ(tb,y): -0.3 vs -0.7
      - ρ(c,y): 0.76 vs 0.91
  - Time inconsistency and debt dilution:
    - Optimal ex-ante duration is shorter than sequentially chosen duration in both economies with and without sudden stops.
    - Sequential decision-making fails to internalize effect of current debt composition on future borrowing terms, leading to longer chosen durations than the initial-state-committed optimum.

### VI. Conclusions
- Two main results:
  - Sudden stops increase the preferred duration of sovereign debt.
  - There is a time inconsistency problem: governments would like to commit to a shorter debt duration than the one chosen under sequential decision-making.
- Policy implication:
  - Results cast doubt on convenience of policies that increase sovereign debt duration for economies facing significant sovereign risk.

*Source: _wp13174 - 3. and bond price functionsq (PDF chapter).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13174.pdf_