## _wp1666

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### Overview and central premise
- Emerging countries (EM) that have defaulted on external private debt in the past are more likely to default again than non-defaulters with the same external debt-to-GDP ratio.
- The paper extends a dynamic stochastic general equilibrium framework to model renegotiations between a defaulting country and its creditors, allowing bargaining over both recovery rates and the rate of return on newly-issued debt.
- Quantitative equilibrium implication highlighted: the equilibrium probability of default for a given debt-to-GDP level is weakly increasing with the number of past defaults.
- Empirical-theoretical linkage: lower recovery rates at renegotiation (i.e., higher haircuts) are associated with larger increases in yield spreads on new debt; countries that accept less-than-100 percent recovery face higher subsequent borrowing costs.

### Data, sample, and definitions
- EM sample defined by IMF World Economic Outlook (WEO): 83 countries.
- Cruces and Trebesch (2013) dataset covers 179 sovereign defaults and restructurings over 1978–2010.
- Sovereign defaulters: 36 (since 1985) and 34 (since 1990) EMs.
- Non-defaulters: 47 (since 1985) and 49 (since 1990) EMs.
- Two sample windows used: from 1985 (over 25 years) and from 1990 (over 20 years), reflecting episode coverage and finite investors’ memory.

### New stylized empirical facts (summarized)
- Past defaulters suffer higher borrowing costs in terms of interest spreads and yields on newly issued bonds.
- Past defaulters are more likely to default again than non-defaulters.
- EM serial defaulters have repeated 3.7 defaults and restructurings in 1978–2010.
- Serial defaulters reach subsequent defaults or restructurings more quickly than previous ones.
- Lower recovery rates (higher haircuts) at renegotiation are associated with larger increases in yield spreads.

### Visual and regression evidence (selected quantitative findings)
- Figure evidence (average 2005–2011):
  - EMBIG spreads vs PPG external debt/GDP: past defaulters have higher EMBIG spreads than non-defaulters at given PPG external debt/GDP.
  - S&P credit rating vs PPG external debt/GDP: past defaulters have lower credit ratings (average of Moody’s and Standard and Poor) than non-defaulters at given PPG external debt/GDP.
- Baseline cross-section regression on EMBIG spreads (benchmark specification):
  - Past defaulters suffer higher borrowing costs by 2.4 percentage points.
  - PPG external debt-to-GDP ratio and exchange rate depreciation are associated with higher EMBIG spreads.
  - Robust to including CPI inflation and to using past-defaults window from 1990.
- Appendix II (yields on newly issued bonds):
  - Baseline regression: past defaulters suffer higher yields on newly issued bonds after defaults/restructurings by 2.2 percentage points.
  - Table A1 selected coefficients:
    - Past Default Dummy (since 1985): 2.24*** (0.72)
    - Past Default Dummy (since 1990): 2.19*** (0.74)
    - External Debt/GDP ave, 2004-2010 (%): 0.028** (0.012) and 0.029** (0.012)
    - Exchange rate depreciation, ave. 2005-11: 0.43*** (0.12) and 0.44*** (0.12)
    - Sample countries: 32; Adj-R^2: 0.77 and 0.76; Root MSE: 2.44 and 2.47
  - Stylized fact A1: Defaulters suffer higher yields on newly issued bonds after defaults and restructurings than non-defaulters (by 2.2 percent).

### Regression table highlights (Table 1 selected entries)
- Past Default Dummy (since 1985) on EMBIG Spreads: 2.39*** (0.81).
- Past Default Dummy (since 1985) on EMBIG Spreads (alternative): 2.06** (0.89).
- Past defaulters estimated by credit ratings: -1.50** (0.68) (lower in credit ratings).
- Past Default Dummy (since 1990) on CDS spreads: 1.03* (0.52).
- CDS evidence: past defaulters suffer higher CDS spreads than non-defaulters by 1 percentage point (6th column).
- Other significant coefficients (selected):
  - PPG External Debt/GDP ave, 2004-10 (%): 0.070*** (0.022), 0.072*** (0.022), 0.070*** (0.022) across EMBIG specs; -0.094*** (0.019) and -0.093*** (0.019) in credit-rating specs; 0.057*** (0.018) and 0.055*** (0.018) in CDS specs.
  - CPI inflation rate, ave. 2005-11: 0.35*** (0.063) and 0.35*** (0.064) in CDS regressions; -0.19** (0.071) in credit-rating regressions.
  - Reserves / GDP ave. 2005-11 (%): 0.063*** (0.023) (credit-rating regressions); small negative coefficients in EMBIG regressions (-0.029 (0.031), -0.039 (0.033)).
- Sample sizes and fit (selected):
  - EMBIG regressions sample countries: 27; Adj-R2: 0.80; Root MSE: 2.12.
  - Credit rating regressions sample countries: 46; Adj-R2: 0.51; Root MSE: 2.08.
  - CDS regressions sample countries: 27; Adj-R2: 0.66; Root MSE: 1.30.

### Serial-default statistics and timing (stylized facts)
- Panel (A), Table 2: serial defaulters account for 61 percent of sovereigns that experienced at least one default or restructuring in 1978–2010.
  - Serial Defaulters: 41 countries; Num. of defaults/restructurings in 1978–2010: 3.7 (average).
  - Single-default countries: 26 countries; Num. defaults/restructurings: 1.0.
- EM focus (Panel B, Table 2):
  - EM serial defaulters account for 58 percent of total serial defaulters.
  - EM serial defaulters: 24 countries; Number of restructurings average: 4.1; Period between restructurings – Average (years): 3.2.
  - Representative entries:
    - Argentina 4 6.6
    - Brazil 6 2.2
    - Mexico 6 1.3
    - Pakistan 2 0.4
    - Poland 8 1.8
    - Russia 4 0.9
    - Ukraine 4 0.5
    - Uruguay 5 5.0
- Timing between restructurings (Panel C, Table 2):
  - From 1st to 2nd: 13.2 quarters (Number of countries: 41).
  - From 2nd to 3rd: 12.0 quarters (Number of countries: 28).
  - From 3rd to 4th: 11.8 quarters (Number of countries: 19).
- Stylized facts:
  - Stylized fact 3: Serial defaulters, accounting 61 percent of past defaulters, have repeated 3.7 defaults or restructurings in 1978–2010.
  - Stylized fact 4: Serial defaulters repeat next defaults or restructurings more quickly than the previous ones.

### Recovery rates and spread increases (66 restructuring episodes, 1986–2010)
- “Increases in spreads” defined as difference in spreads between completion of restructuring and one year before completion.
- Regression results on recovery rates (Table 3, baseline and variants):
  - Baseline (1):
    - Constant: 79.11*** (5.16)
    - Increase in spreads (%): -1.02*** (0.32)
    - External Debt/GDP ratio (%): -0.17** (0.067)
    - Sample: 60; Adj-R2: 0.20; Root MSE: 18.33
  - Political risks (2):
    - Constant: 113.77*** (19.21)
    - Increase in spreads (%): -1.03*** (0.32)
    - External Debt/GDP ratio (%): -0.21*** (0.068)
    - Political risk: -0.57* (0.31)
    - Sample: 59; Adj-R2: 0.24; Root MSE: 18.01
  - Increase in spreads below 10% (3):
    - Constant: 76.89*** (5.20)
    - Increase in spreads (%): -2.96*** (0.79)
    - External Debt/GDP ratio (%): -0.14* (0.068)
    - Sample: 56; Adj-R2: 0.30; Root MSE: 17.10
  - Global factors (4):
    - Constant: 91.94*** (27.30)
    - Increase in spreads (%): -0.98*** (0.32)
    - External Debt/GDP ratio (%): -0.20*** (0.069)
    - LIBOR (%): 2.14 (1.82)
    - Sample: 59; Adj-R2: 0.23; Root MSE: 18.11
- Stylized fact 5: Lower NPV recovery rates (higher haircuts) at renegotiation are associated with larger increases in yield spreads.

### Theoretical mechanism: renegotiation and endogenous spread premia
- Renegotiation modeled as Nash bargaining over both recovery rate α(b_t,h_t,y_t) and additional spread premia φ(b_t,h_t+1,y_t).
- Trade-offs:
  - Debtor chooses between repaying defaulted debt fully now at a high short-run cost to avoid long-term deterioration in credit conditions, or accepting a partial repayment (haircut) now and facing higher future borrowing costs.
  - Creditors choose between accepting partial short-term recovery or demanding higher rates of return on future debt issuances to recoup losses.
- Equilibrium implication: lower recovery rates at renegotiation lead to larger agreed increases in rates of return on new debt, raising future default probability and generating serial-default patterns.
- Distinctive model feature relative to Yue (2010): endogenous determination of increases in rate of return on new debt after renegotiation (φ), not externally imposed.

### Formal environment and key equations (preserved expressions)
- Sovereign expected utility: E0 Σ_{t=0}^{∞} β^{t} u(c_{t}) with 0 < β < 1.
- Credit history h_{t} ∈ H = [0,1,2, ..., h_{max}] tracks past renegotiations.
- Income shock y_{t} drawn from Y; transition μ(y_{t+1}|y_{t}).
- Bond price q(b_{t+1}, h_{t}, y_{t}) determined in equilibrium; one-period zero-coupon bonds.
- Default set: D(b_t, h_t) = { y_t ∈ Y : V_R(b_t,h_t,y_t) < V_D[b_t,h_t,y_t; α(·), φ(·)] }.
- Discounted expected payment to investors: P(b_t,h_t,y_t) = (1/(1+r)) ∫ I(b_{t+1}(b_t,h_t,y_t), h_t, y_{t+1}) b_{t+1}(b_t,h_t,y_t) dμ(y_{t+1}, y_t).
- Nash bargaining: (α(b_t,h_t,y_t), φ(b_t,h_t+1,y_t)) = argmax_{a_t, sp_t} [ Δ_B(a_t,sp_t)^{θ} Δ_L(a_t,sp_t)^{1−θ} ] subject to Δ_B(·) ≥ 0 and Δ_L(·) ≥ 0.
- Bond pricing (zero expected profit):
  - For h_t=0: q(b_{t+1},0,y_t) = 1/(1+r) if b_{t+1} ≥ 0; otherwise q = [1−p + p γ] /(1+r).
  - For h_t≥1: q(b_{t+1},h_t,y_t) = 1/(1+r) if b_{t+1} ≥ 0; otherwise q = [1−p + p γ] /(1+r+φ).
- Sovereign spread: s(b_{t+1},h_t,y_t) = 1/q(b_{t+1},h_t,y_t) − (1+r).

### Calibration and baseline parameter values (preserved)
- Time unit: quarter.
- Utility: CRRA u(c_t) = (c_t^{1−σ} − 1)/(1−σ) with σ = 2.
- Output process approximated via 21-grid Tauchen with:
  - μ_g = 0.0042
  - σ_g^2 = 0.0253
  - ρ_g = 0.41
- Baseline output loss parameter: λ_d = 0.02 (2 percent).
- Discount factor and bargaining power set to target observed stats:
  - β = 0.75
  - θ = 0.72
  - Targets achieved: Average default frequency 2.65 percent annually (0.66 percent quarterly); Recovery rate 31.3 percent.
- Additional parameters:
  - r = 0.017
  - φ_max = 0.01
  - h_max = 3
  - χ = 0.025
- Table 4 selected entries: σ = 2; r = 0.017; λ_d = 0.02; μ_g = 0.0042; σ_g^2 = 0.0253; ρ_g = 0.41; β = 0.75; θ = 0.72; φ_max = 0.01; h_max = 3; χ = 0.025.

### Quantitative simulation design and selected results
- Simulation design:
  - 1000 rounds, 2000 periods per round.
  - Extract 80 observations before and 25 observations after each default event in stationary distribution for statistics.
- Data sources for Argentina: J.P. Morgan EMBIG; MECON; IMF WEO; external debt series.
- Business cycle statistics (selected, Table 5):
  - Consumption Std./Output Std. before default: Data 1.03 / Model 1.24
  - Trade Balance/Output Std Dev.(%) before default: Data 1.23 / Model 3.71
  - Corr. (Trade Balance/GDP, Output) before default: Data -0.83 / Model -0.005
- Non-business cycle statistics (selected, Table 6):
  - Target Default Probability: Data 2.7 / Model 2.65
  - Average Recovery Rate (%): Data 33 / Model 31.3
  - Before Default: Average Debt/GDP ratio: Data 12.7 / Model 10.2
  - Before Default: Average Bond Spreads (%): Data 7.4 / Model 3.1
  - After Default: Average Debt/GDP ratio: Data 43.0 / Model 13.2 — Model 12.3 reported elsewhere
  - After Default: Average Bond Spreads (%): Data 6.7 / Model 3.9
- Model-generated averages:
  - Pre-default average bond spreads: 3.1 percent; Std Dev. 1.9 percent.
  - Post-default average bond spreads: 3.9 percent; Std Dev. 2.0 percent.
- Average spans between defaults (Table 7, quarters) vary with h_max; examples:
  - h_max = 3: 1st def. 57, 2nd def. 19, 3rd def. 8; Model default prob. 2.65
  - h_max = 4: 1st def. 57, 2nd def. 30, 3rd def. 8, 4th def. 8; Model def. prob. 2.99
  - h_max = 5: 59, 27, 16, 14, 8; Model def. prob. 3.67
  - h_max = 6: 59, 27, 22, 16, 14, 8; Model def. prob. 4.47

### Decomposition of interest spreads and contract-curve determinants
- Spread decomposition for h_t ≥ 1 and b_{t+1} < 0:
  - s = (1+r+φ(b_t, h_t, y_t)) / (1−p + p γ) − (1+r)
- Two spread components:
  - (A) Component based on “pure” default probability (future defaults) — credit-history irrelevant (Yue (2010) measure).
  - (B) Component arising from additional spread premia φ (past defaults) — raises spreads directly and indirectly via higher default probability.
- Empirical implication: total spreads deviate from “pure” default-probability spreads when debt-to-GDP exceeds threshold 0.175 in mean income state; deviation interpreted as spread component associated with past default history.
- Determinants of contract curve slope (Table 8 insights, selected preserved values):
  - Data slope: -0.01
  - Slope reacts to parameters:
    - Discount factor decreases → slope steeper (larger absolute value).
    - Reducing φ_max can steepen slope (creditors substitute spread premia with higher recovery demands).
    - Increasing λ_d increases slope absolute value.
    - Higher r increases slope absolute value.
    - Probability of upgrading χ does not materially affect slope.
- Duration and size of additional spread premia (Table 9 insights):
  - Increasing χ shortens deterioration in long-run credit and lowers average debt-to-GDP post-default.
  - Increasing φ_max raises average debt-to-GDP pre- and post-default, raises average bond spreads post-default, and increases default probability.
  - Table 9 selected entries (examples):
    - χ: 0 / 0.025 / 0.075 → Default Probability: 2.67 / 2.65 / 2.65; Average Recovery Rate (%): 31.9 / 31.3 / 31.9; Before Default Average Debt/GDP ratio: 10.4 / 9.5 / 10.4; After Default Average Debt/GDP ratio: 10.9 / 12.3 / 10.9; After Default Average Bond Spreads (%): 3.9 / 3.9 / 3.9
    - φ_max: 0.005 / 0.01 / 0.025 → Default Probability: 2.55 / 2.65 / 3.04; Average Recovery Rate (%): 32.2 / 31.3 / 31.5; Before Default Average Debt/GDP ratio: 10.4 / 9.5 / 11.8; After Default Average Debt/GDP ratio: 10.9 / 12.3 / 12.4; After Default Average Bond Spreads (%): 3.6 / 3.9 / 4.6

### Computation algorithm (Appendix I, stepwise)
- Grids and initial guesses:
  - Credit history grid H = [0,1,2,3] with h_max = 3.
  - Asset grid example B = [−0.3, .... . ,0].
  - Income Markov chain: 21 equally spaced grids.
  - Initial baseline bond price q_1 = q_f = (1 + r)^{-1}.
  - Baseline recovery rate α_0 = 0.5.
  - Baseline additional spread premia φ_0 = 0.01.
- Iterative steps (summary):
  1. Solve sovereign optimization (value functions V, V_D, V_R) given q_0, α_0, φ_0.
  2. Derive default set and default probabilities using transitions.
  3. Update bond price q using investors’ zero-profit condition.
  4. Solve bargaining problem to obtain α′ and φ′; iterate to fixed points for q^*, α^*, φ^*.

### EMBIG-spread extrapolation method (Appendix III)
- For each country, linear regression over 1999M1–2012M12:
  - Yields_{t,i} = c_i + β_1 LIBOR_t + β_2 Rating_{t,i} + ε_{t,i}
- Use estimated coefficients to impute EMBIG yields for 1986M1–1998M12.
- Convert to EMBIG spreads by differencing imputed yields and US Treasury 10-year yields.
- Selected country coefficients (Table A2, examples preserved):
  - Argentina: Constant 193.14***, LIBOR 1.94***, ICRG Ratings -2.60***, Adj-R^2 0.64
  - Chile: Constant 18.51***, LIBOR 0.50***, ICRG Ratings -0.19***, Adj-R^2 0.73
  - Mexico: Constant 36.98***, LIBOR 0.78***, ICRG Ratings -0.46***, Adj-R^2 0.82
  - Russia: Constant 89.22***, LIBOR 1.48***, ICRG Ratings -1.20***, Adj-R^2 0.80

### Key quantitative and policy-relevant conclusions
- Incorporating additional spread premia φ agreed at restructuring allows the model to reproduce the stylized fact that lower recovery rates (larger haircuts) are associated with larger increases in yield spreads.
- Default probability is weakly increasing with credit history (number of past renegotiations) for a given debt-to-GDP ratio.
- Additional spread premia create a durable component of higher borrowing costs after default, contributing to serial-default patterns by:
  - Directly increasing spreads via φ.
  - Indirectly increasing spreads via higher future default probability.
- Model accounts for multiple features of pre- and post-default periods, including higher post-default average debt-to-GDP and higher post-default average spreads (model: pre-default average spreads 3.1 percent; post-default 3.9 percent).
- Mechanism: creditors offset larger haircuts by demanding higher rates on newly issued bonds; this raises long-term borrowing costs and increases the likelihood of repeat defaults.

### Suggested extensions (from authors)
- Compare renegotiation outcomes under asymmetric information (e.g., unrevealed government type or output cost).
- Explore uninformed creditor coordination/composition effects during renegotiation.

*Source: _wp1666 (extracted content from the supplied IMF PDF chapter/section).*

### 1. External Debt/GDP, Bond Spreads, and Credit Ratings, Average 2005–10 .........................8

### 1. External Debt/GDP, Bond Spreads, and Credit Ratings, Average 2005–10 .........................8

### Overview
- Paper premise: Emerging countries (EM) that have defaulted on external private debt in the past are more likely to default again than non-defaulters with the same external debt-to-GDP ratio.
- Theoretical extension: dynamic stochastic general equilibrium framework that explicitly models renegotiations between a defaulting country and its creditors, allowing bargaining over both recovery rates and the rate of return on newly-issued debt.
- Quantitative result highlighted: the equilibrium probability of default for a given debt-to-GDP level is weakly increasing with the number of past defaults.
- Empirical and theoretical linkage: lower recovery rates at renegotiation (i.e., higher haircuts) are associated with larger increases in yield spreads on new debt; countries that accept less-than-100 percent recovery face higher subsequent borrowing costs.

### Data and sample
- EM sample defined by IMF World Economic Outlook (WEO): 83 countries.
- Cruces and Trebesch (2013) private external debt defaults and restructurings dataset covers 179 sovereign defaults and debt restructurings over 1978–2010.
- Definitions and counts:
  - Sovereign debt defaulters: sovereigns which have experienced at least one default or restructuring since 1985 (1990) – 36 (34) EMs.
  - Non-defaulters: sovereigns which have experienced neither a default nor a restructuring since 1985 (1990) – 47 (49) EMs.
- Two sample windows used: (i) from 1985 (over 25 years), (ii) from 1990 (over 20 years), motivated by episode coverage and finite investors’ memory.

### New stylized empirical facts (summary)
- From cross-sectional analysis and Cruces and Trebesch (2013) data, the paper confirms:
  - (i) Past defaulters suffer higher borrowing costs both in terms of interest spreads and yield on newly issued bonds.
  - (ii) Past defaulters are more likely to default again than non-defaulters.
  - (iii) EM serial defaulters have repeated 3.7 defaults and restructurings in 1978–2010.
  - (iv) Serial defaulters reach the next default or restructuring more quickly than previous ones.
  - (v) Lower recovery rates (higher haircuts) at renegotiation are associated with larger increases in yield spreads.

### Key empirical patterns and visual evidence
- Figure 1 (average 2005–2011) shows:
  - Panel A: EMBIG Spreads versus PPG external debt/GDP indicates past defaulters have higher EMBIG spreads than non-defaulters for a given PPG external debt/GDP.
  - Panel B: S&P credit rating versus PPG external debt/GDP indicates past defaulters have lower credit ratings (average of Moody’s and Standard and Poor ratings) than non-defaulters for a given PPG external debt/GDP.
- Sources cited for figures: Bloomberg, Cruces and Trebesch (2013), IMF WEO, Moody’s, Standard and Poor, WB International Debt Statistics.

### Regression evidence (selected result)
- Baseline cross-section regression on EMBIG spreads (benchmark specification following Eichengreen and Mody (1998) and Ardagna and others (2007)):
  - Past defaulters suffer higher borrowing costs by 2.4 percentage points.
  - Additional controls: PPG external debt-to-GDP ratio and exchange rate depreciation are associated with higher EMBIG spreads.
  - Robustness: result holds when including CPI inflation and when using the shorter interval for past defaults starting from 1990.

### Theoretical mechanism (concise)
- Renegotiation modeled as Nash bargaining over both recovery rate and increase in rate of return on new debt.
- Trade-offs:
  - Debtor: repay defaulted debt in full now at a high short-run cost to avoid long-term deterioration in credit conditions, or accept a partial repayment (haircut) now and face higher future borrowing costs.
  - Creditors: accept partial short-term recovery or demand higher rates of return on future debt issuances to recoup losses.
- Equilibrium implication: lower recovery rates at renegotiation lead to larger agreed increases in rates of return on new debt, raising future default probability and generating serial-default patterns.

### Relation to literature
- Builds on Eaton and Gersovitz (1981) sovereign-default models and on quantitative sovereign debt work (Aguiar and Gopinath (2006); Arellano (2008); Tomz and Wright (2007)).
- Closest to Yue (2010) but differs by endogenizing increases in rate of return on new debt in renegotiation; special case of full recovery yields results similar to Yue (2010).
- Empirical linkages to prior findings: Ozler (1992, 1993), Cantor and Packer (1996), Lindert and Morton (1989), Catao and others (2009), Cruces and Trebesch (2013), Benczur and Ilut (2016).

*Source: _wp1666 - 1. External Debt/GDP, Bond Spreads, and Credit Ratings, Average 2005–10, from the provided IMF PDF content*

### Appendix II demonstrates that defaulters’ yields on newly issued bonds in general are higher

### _wp1666 - Appendix II demonstrates that defaulters’ yields on newly issued bonds in general are higher

### Borrowing costs and default probability (regression evidence)
- Benchmark regressions follow Kohlscheen (2009) and Dreher and Walter (2010).
- Credit ratings (average of Moody’s and Standard and Poor) and Credit Default Swap (CDS) spreads are used as proxies for default probability.
- Key estimated impacts (Table 1 highlights):
  - Past Default Dummy (since 1985) on EMBIG Spreads: 2.39*** (0.81).
  - Past Default Dummy (since 1985) on EMBIG Spreads (alternative column): 2.06** (0.89).
  - Past defaulters estimated by credit ratings: -1.50** (0.68) (lower in credit ratings).
  - Past Default Dummy (since 1990) on CDS spreads: 1.03* (0.52).
  - Using CDS spreads: past defaulters suffer higher CDS spreads than non-defaulters by 1 percentage point (6th column).
- Other significant explanatory coefficients (from Table 1):
  - PPG External Debt/GDP ave, 2004-10 (%): 0.070*** (0.022), 0.072*** (0.022), 0.070*** (0.022) across EMBIG specifications; -0.094*** (0.019) and -0.093*** (0.019) in credit-rating specifications; 0.057*** (0.018) and 0.055*** (0.018) in CDS specifications.
  - CPI inflation rate, ave. 2005-11: 0.35*** (0.063) and 0.35*** (0.064) in CDS regressions; -0.19** (0.071) in credit-rating regressions.
  - Reserves / GDP ave. 2005-11 (%): 0.063*** (0.023) (credit-rating regressions); other specifications show small negative coefficients like -0.029 (0.031) and -0.039 (0.033) in EMBIG regressions.
- Sample sizes and fit (selected):
  - EMBIG regressions sample countries: 27; Adj-R2: 0.80; Root MSE: 2.12.
  - Credit rating regressions sample countries: 46; Adj-R2: 0.51; Root MSE: 2.08.
  - CDS regressions sample countries: 27; Adj-R2: 0.66; Root MSE: 1.30.

### Stylized facts from empirical results
- Stylized fact 1: Past defaulters suffer higher borrowing costs than non-defaulters.
  - EMBIG spreads for past defaulters are higher than those for non-defaulters by 2-2.4 percentage points.
- Stylized fact 2: Past defaulters are more likely to default again.
  - Default probability for past defaulters is higher than for non-defaulters, measured by lower credit ratings and higher CDS spreads.
- Note on CDS evidence: Despite limited sample of EMs with liquid markets, CDS spreads of past defaulters are much higher than those of non-defaulters, implicitly measuring sovereigns’ future default risk.

### Serial sovereign defaults and restructurings (stylized facts and statistics)
- Panel (A) (Table 2): Serial defaulters account 61 percent of sovereigns that have experienced at least one default or restructuring in 1978–2010.
  - Serial Defaulters: 41 countries; Num. of defaults/restructurings in 1978–2010: 3.7 (average).
  - Countries with a single default/restructuring: 26 countries; Num. of defaults/restructurings: 1.0.
- EM focus (Panel B, Table 2):
  - EM serial defaulters account 58 percent of total serial defaulters.
  - EM serial defaulters: 24 countries; Number of restructurings average: 4.1; Period between restructurings – Average (years): 3.2.
  - Representative entries (Country – Number of restructurings – Period between restructurings – Average (years)):
    - Argentina 4 6.6
    - Brazil 6 2.2
    - Mexico 6 1.3
    - Pakistan 2 0.4
    - Poland 8 1.8
    - Russia 4 0.9
    - Ukraine 4 0.5
    - Uruguay 5 5.0
- Timing between restructurings (Panel C, Table 2):
  - From 1st to 2nd: 13.2 quarters (Number of countries: 41).
  - From 2nd to 3rd: 12.0 quarters (Number of countries: 28).
  - From 3rd to 4th: 11.8 quarters (Number of countries: 19).
- Stylized fact 3: Serial defaulters, accounting 61 percent of past defaulters, have repeated 3.7 defaults or restructurings in 1978–2010.
- Stylized fact 4: Serial defaulters repeat next defaults or restructurings more quickly than the previous ones.

### Recovery rates and increases in spreads for restructuring episodes
- Figure 2 summarizes NPV recovery rates and increases in spreads for 66 sovereign debt restructuring episodes during 1986–2010.
- Definition: “Increases in spreads” = difference in spreads between the time of completion of the restructurings and one year before the completion.
- Data notes:
  - Most EMBIG spreads available from 1999; spread series extrapolated using LIBOR and ICRG rating as explained in Appendix III.
  - Fitted line obtained by regressing recovery rates on increases in spreads controlling for GDP deviation from the trend and external debt-to-GDP ratio.
- Regression results on recovery rates (Table 3):
  - Baseline (1):
    - Constant: 79.11*** (5.16).
    - Increase in spreads (%): -1.02*** (0.32).
    - External Debt/GDP ratio (%): -0.17** (0.067).
    - Sample: 60; Adj-R2: 0.20; Root MSE: 18.33.
  - Political risks (2):
    - Constant: 113.77*** (19.21).
    - Increase in spreads (%): -1.03*** (0.32).
    - External Debt/GDP ratio (%): -0.21*** (0.068).
    - Political risk: -0.57* (0.31).
    - Sample: 59; Adj-R2: 0.24; Root MSE: 18.01.
  - Increase in spreads below 10% (3):
    - Constant: 76.89*** (5.20).
    - Increase in spreads (%): -2.96*** (0.79).
    - External Debt/GDP ratio (%): -0.14* (0.068).
    - Sample: 56; Adj-R2: 0.30; Root MSE: 17.10.
  - Global factors (4):
    - Constant: 91.94*** (27.30).
    - Increase in spreads (%): -0.98*** (0.32).
    - External Debt/GDP ratio (%): -0.20*** (0.069).
    - LIBOR (%): 2.14 (1.82).
    - Sample: 59; Adj-R2: 0.23; Root MSE: 18.11.
- Stylized fact 5: Lower NPV recovery rates (higher haircuts) at renegotiation are associated with larger increases in yield spreads.
- Interpretation: Trade-off for defaulting countries — larger recovery at renegotiation is associated with smaller long-term borrowing costs; creditors offset larger haircuts by demanding higher rates on newly issued bonds.

### Model environment (overview of the sovereign-default model)
- Model builds on Eaton and Gersovitz (1981) and follows quantitative sovereign-default frameworks; closest reference: Yue (2010).
- Distinctive feature: introduction of effects of increases in rate of return on new debt after re-entry to the market; both recovery rates and increases in rate of return on new debt are determined endogenously.
- Country preferences and environment:
  - Expected utility: E0 Σ_{t=0}^{∞} β^{t} u(c_{t}) with 0 < β < 1.
  - Credit history h_{t} ∈ H = [0,1,2, ..., h_{max}] records number of past debt renegotiations.
  - Credit history can revert with exogenous probability χ conditional on the country choosing to pay spread returns after defaults.
  - Income shock y_{t} drawn from Y = [y_{min}, ..., y_{max}] ⊂ ℝ_{+}; μ(y_{t+1}|y_{t}) denotes transition probabilities.
- Investors and bonds:
  - Infinite risk-neutral, competitive investors who keep track of credit history and agreed additional spread premia.
  - Investors can borrow or lend at constant risk-free interest rate (r).
  - One-period zero-coupon bonds only; bond position b_{t+1} with set B = [b_{min}, ..., b_{max}] ⊂ ℝ.
  - Bond price q(b_{t+1}, h_{t}, y_{t}) determined in equilibrium.
- Default and renegotiation:
  - Country can default and then faces exclusion and direct output costs.
  - Upon default, country and investors negotiate reductions of unpaid debt via Nash bargaining; both recovery rates and additional spread premia on newly issued bonds are agreed.
  - Country regains market access after short exclusion periods; credit history records renegotiation.
- Model timing and implications:
  - The model allows multi-state credit history (not binary) to analyze how past renegotiations affect future default probability.
  - The bargaining environment and endogenous determination of spreads after renegotiation imply that outcomes at renegotiation affect future borrowing costs and default probabilities.

*Source: _wp1666 - Appendix II demonstrates that defaulters’ yields on newly issued bonds in general are higher*

### 1. The sovereign starts current period with initial assets/debt b

### _wp1666 - 1. The sovereign starts current period with initial assets/debt b

### Timing and state transitions
- Period t starts with initial assets/debt b_t and credit history h_t. We are in node (A).
- An income shock y_t realizes; the sovereign chooses between payment (node (B)) or default (node (C)).
- (B) Payment node:
  - Sovereign chooses consumption c_t and next-period assets/debt b_{t+1}.
  - Creditors choose b_{t+1}; bond price q(b_{t+1}, h_t, y_t) is determined in the market.
  - With exogenous probability χ, next period credit history upgrades: h_{t+1}=h_t−1; otherwise h_{t+1}=h_t.
- (C) Default node:
  - Debt renegotiation yields recovery rate α(b_t, h_t, y_t) and additional spread premia φ(b_t, h_t+1, y_t).
  - Sovereign pays recovered debt α(b_t, h_t, y_t) b_t and suffers output cost λ_d y_t.
  - Sovereign cannot raise funds this period: b_{t+1}=0, regains market access next period.
  - Credit history records renegotiation: h_{t+1}=h_t+1.
- Next period income shock y_{t+1} realizes.

### Sovereign country’s recursive problem
- Objective: maximize expected lifetime utility; value function V(b_t, h_t, y_t).
- Given bond price q(b_{t+1}, h_t, y_t), recovery rates α(·) and spread premia φ(·), country chooses c_t and b_{t+1}.

Key formulations (preserve original expressions):
- For h_t=0 and b_t≥0:
  - V(b_t, 0, y_t) = max_{c_t,b_{t+1}} u(c_t) + β ∫ V(b_{t+1}, 0, y_{t+1}) dμ(y_{t+1}, y_t)
  - subject to c_t + q(b_{t+1}, 0, y_t) b_{t+1} = y_t + b_t.  (Equation (1))
- For h_t=0 and b_t<0, option to default:
  - V(b_t,0,y_t) = max{ V_R(b_t,0,y_t), V_D[b_t,0,y_t; α(b_t,0,y_t), φ(b_t,1,y_t)] }.  (Equation (2))
  - V_R(b_t,0,y_t) = max_{c_t,b_{t+1}} u(c_t) + β ∫ V(b_{t+1},0,y_{t+1}) dμ(y_{t+1}, y_t)
    subject to c_t + q(b_{t+1},0,y_t) b_{t+1} = y_t + b_t.  (Equation (3))
  - V_D[·] = u((1−λ_d) y_t + α(b_t,0,y_t) b_t) + β ∫ V(0,1,y_{t+1}) dμ(y_{t+1}, y_t).  (Equation (4))
    - Here −α(b_t,0,y_t) b_t is amount repaid; λ_d y_t denotes output cost.
- For h_t≥1:
  - If b_t≥0:
    - V(b_t, h_t, y_t) = max_{c_t,b_{t+1}} u(c_t) + β ∫ V(b_{t+1}, h_t, y_{t+1}) dμ(y_{t+1}, y_t)
      subject to c_t + q(b_{t+1}, h_t, y_t) b_{t+1} = y_t + b_t.  (Equation (5))
    - Credit history remains: h_{t+1}=h_t.
  - If b_t<0:
    - V(b_t,h_t,y_t) = max{ V_R(b_t,h_t,y_t), V_D[b_t,h_t,y_t; α(b_t,h_t,y_t), φ(b_t,h_t+1,y_t)] }.  (Equation (6))
    - V_R(b_t,h_t,y_t) = max_{c_t,b_{t+1}} u(c_t) + β [ (1−χ) ∫ V(b_{t+1}, h_t, y_{t+1}) dμ + χ ∫ V(b_{t+1}, h_t−1, y_{t+1}) dμ ]
      subject to c_t + q(b_{t+1}, h_t, y_t) b_{t+1} = y_t + b_t.  (Equation (7))
      - With exogenous probability χ, h_{t+1}=h_t−1; otherwise h_{t+1}=h_t.
    - V_D[·] = u((1−λ_d) y_t + α(b_t,h_t,y_t) b_t) + β ∫ V(0,h_t+1,y_{t+1}) dμ(y_{t+1}, y_t).  (Equation (8))

- Default set and indicator:
  - Default set D(b_t, h_t) = { y_t ∈ Y : V_R(b_t,h_t,y_t) < V_D[b_t,h_t,y_t; α(·), φ(·)] }.  (Equation (9))
  - I(b_t, h_t, y_t) = 1 if y_t ∈ D(b_t, h_t); 0 otherwise.

- Discounted expected payment to investors next period:
  - P(b_t,h_t,y_t) = (1/(1+r)) ∫ I(b_{t+1}(b_t,h_t,y_t), h_t, y_{t+1}) b_{t+1}(b_t,h_t,y_t) dμ(y_{t+1}, y_t).  (Equation (10))
  - Discount factor used is 1/(1+r) (foreign investors’ discounting), not β.

### Debt renegotiation (generalized Nash bargaining)
- Debt renegotiation determines recovery rate α(b_t, h_t, y_t) and additional spread premia φ(b_t, h_t+1, y_t).
- After renegotiation, country pays fraction α(b_t,h_t,y_t) of defaulted debt; country value after renegotiation equals V_D[·] (see above).
- Present value to investors from future payments after re-entry:
  - R(b_t, h_t, y_t) = P(b_t, h_t, y_t) + (1/(1+r)) ∫ R(b_{t+1}, h_t, y_{t+1}) dμ(y_{t+1}, y_t)
    subject to b_{t+1} = b_{t+1}^*(b_t, h_t, y_t).  (Equation (11))
  - P(·) defined in (10); b_{t+1}^*(·) is sovereign policy if not defaulting.

- Autarky threat point (country stays in financial autarky permanently, investors get nothing):
  - V_AUT(y_t) = u((1−λ_d) y_t) + β ∫ V_AUT(y_{t+1}) dμ(y_{t+1}, y_t).  (Equation (12))

- Bargaining surpluses:
  - Country surplus: Δ_B(a_t, sp_t; b_t, h_t, y_t) = V_D[b_t,h_t,y_t; α(·), φ(·)] − V_AUT(y_t).  (Equation (13))
  - Investors surplus: Δ_L(a_t, sp_t; b_t, h_t, y_t) = − a_t b_t − R(b_t, h_t, y_t).  (Equation (14))

- Nash bargaining with country bargaining power θ (θ ∈ Θ ⊂ [0,1]):
  - (α(b_t,h_t,y_t), φ(b_t,h_t+1,y_t)) = argmax_{a_t, sp_t} [ Δ_B(a_t,sp_t)^{θ} Δ_L(a_t,sp_t)^{1−θ} ]
    subject to Δ_B(·) ≥ 0 and Δ_L(·) ≥ 0.  (Equation (15))
  - φ(b_t,h_t+1,y_t) specifies state-variant contracts depending on future streams of b_t and h_t.
- Negotiation assumed one-round per default event.

### Foreign investors’ problem and bond pricing
- With h_t=0, foreign investors’ expected profit π(b_{t+1},0,y_t):
  - π(b_{t+1},0,y_t) =
    - q(b_{t+1},0,y_t) b_{t+1} − (1/(1+r)) b_{t+1} if b_{t+1} ≥ 0,
    - (1−p(b_{t+1},0,y_t) + p(b_{t+1},0,y_t) γ(b_{t+1},0,y_t)) /(1+r) (−b_{t+1}) − q(b_{t+1},0,y_t) (−b_{t+1}) otherwise.  (Equation (16))
  - p(·) is expected default probability, γ(·) expected recovery rate for debt positions.
- Zero expected profit in competitive market implies bond price:
  - q(b_{t+1},0,y_t) = 1/(1+r) if b_{t+1} ≥ 0;
    otherwise q(b_{t+1},0,y_t) = [1−p(b_{t+1},0,y_t) + p(b_{t+1},0,y_t) γ(b_{t+1},0,y_t)] /(1+r).  (Equation (17))
  - q(b_{t+1},0,y_t) ∈ (0, 1/(1+r)].

- With h_t≥1, borrowing costs include additional spread premia φ(b_{t+1}, h_t, y_t):
  - π(b_{t+1},h_t,y_t) defined analogously with 1+r+φ(·) in denominator for debt positions.  (Equation (18))
  - Zero-profit bond price:
    - q(b_{t+1},h_t,y_t) = 1/(1+r) if b_{t+1} ≥ 0;
      otherwise q(b_{t+1},h_t,y_t) = [1−p(b_{t+1},h_t,y_t) + p(b_{t+1},h_t,y_t) γ(b_{t+1},h_t,y_t)] /(1+r+φ(b_{t+1},h_t,y_t)).  (Equation (19))
  - q(b_{t+1},h_t,y_t) ∈ (0, 1/(1+r+φ(b_{t+1},h_t,y_t))].

- Sovereign bond interest rate and spread:
  - r_S(b_{t+1},h_t,y_t) = 1/q(b_{t+1},h_t,y_t) − 1.
  - Country total spread: s(b_{t+1},h_t,y_t) = 1/q(b_{t+1},h_t,y_t) − (1+r).  (Equation (20))

### Recursive equilibrium definition and equilibrium objects
- A stationary recursive equilibrium is a set of functions:
  - (a) V^*(b_t,h_t,y_t), V_R^*(·), V_D^*(·; α^*(·), φ^*(·)), b_{t+1}^*(·), c_t^*(·), default set D^*(b_t,h_t), discounted expected payment P^*(·).
  - (b) Recovery rates α^*(b_t,h_t,y_t) and additional spread premia φ^*(b_t,h_t+1,y_t).
  - (c) Bond price function q^*(b_{t+1},h_t,y_t) and total spreads s^*(·).
- Equilibrium conditions:
  1. Given q^*, α^*, φ^*, sovereign optimization (1)-(10) is satisfied.
  2. Given q^*, sovereign value and P^*, α^*, φ^* solve bargaining problem (15).
  3. Given α^*, φ^*, q^* and s^* satisfy foreign investors’ optimality (17) and (19).

- Equilibrium default probability and expected recovery rate:
  - p^*(b_{t+1},h_t,y_t) = ∫_{D^*(b_t,h_t)} dμ(y_{t+1}, y_t).  (Equation (21))
  - γ^*(b_{t+1},h_t,y_t) = ∫ α^*(b_t,h_t,y_t) dμ(y_{t+1}, y_t)_{D^*(b_t,h_t)} / p^*(b_{t+1},h_t,y_t)
    = [∫ α^*(b_t,h_t,y_t) dμ(y_{t+1}, y_t)_{D^*(b_t,h_t)}] / p^*(b_{t+1},h_t,y_t).  (Equation (22))

### Quantitative analysis setup (overview from text)
- Three new model elements relative to Yue (2010):
  1. Maximum level of additional spread premia.
  2. Maximum level of credit history.
  3. Probability of upgrading in credit history.
- Rationale: upper limits ensure stationarity; probability of upgrading reflects finite record of defaults.
- Time unit: each period is a quarter.
- Utility: CRRA u(c_t) = (c_t^{1−σ} − 1)/(1−σ).  (Equation (23))
  - Parameter σ set to 2.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp1666.pdf*

### 1.7 percent. The baseline output loss parameter λ

### _wp1666 - 1.7 percent. The baseline output loss parameter λ

### Model setup: stochastic output process and detrending
- Output growth modeled as AR(1) for g_t = y_t / y_{t−1} with equation:
  - log(g_t) = (1−ρ_g) log(1+μ_g) + ρ_g log(g_{t−1}) + ε_t^g
  - ε_t^g ∼ i.i.d N(0, σ_g^2)
- Parameter values used to approximate the process via a 21-grid Tauchen (1986) Markov chain:
  - μ_g = 0.0042
  - σ_g^2 = 0.0253
  - ρ_g = 0.41
- Detrending:
  - Any variable x_t is detrended as x'_t = x_t / y_{t−1}
  - Equilibrium value function, bond price function, recovery rate and interest spreads are evaluated on detrended variables
- Baseline output loss parameter:
  - λ_d = 0.02 (2 percent) based on Sturzenegger (2004) estimate

### Calibration: time preference, bargaining power, spreads, and credit history
- Discount factor and bargaining power set to target observed default and recovery statistics:
  - β = 0.75
  - θ = 0.72
  - Targets achieved with these values:
    - Average default frequency 2.65 percent annually (0.66 percent quarterly)
    - Recovery rate 31.3 percent
- Target references:
  - Target default probability 2.7 percent annually and average recovery rate 33 percent from 2005 international debt restructuring (Sturzenegger and Zettelmeyer)
- Additional spread premia and credit history parameters:
  - φ_max = 0.01 (maximum level of additional spread premia, corresponding to increase in spreads less than 0.01)
  - h_max = 3 (maximum level of credit history, reflecting 3 defaults of Argentina in 1901–2002)
  - χ = 0.025 (probability of upgrading in credit history; investors' memory lasts for 10 years)
- Table 4 parameter summary (selected entries preserved):
  - σ = 2
  - r = 0.017
  - λ_d = 0.02
  - μ_g = 0.0042
  - σ_g^2 = 0.0253
  - ρ_g = 0.41
  - β = 0.75
  - θ = 0.72
  - φ_max = 0.01
  - h_max = 3
  - χ = 0.025

### Equilibrium properties: spreads, recovery rates, and credit history effects
- Relationship between increase in interest spreads and recovery rates:
  - Negative relationship: higher increase in spreads → lower recovery rate (larger haircut)
  - Additional spread premia agreed at restructurings drive differences from Yue (2010), where the contract curve slope is vertical
- Default probability under baseline case (mean income level):
  - Default probability is weakly increasing with credit history h_t
  - At h_t = 3, additional increase in spreads from prior renegotiation raises borrowing costs and default likelihood at given debt-to-GDP
- Bond price schedule:
  - Bond price is weakly decreasing with credit history due to past-agreed additional spread premia reducing bond price both directly and indirectly via higher default probability

### Simulation design and data sources
- Simulation details:
  - 1000 rounds, 2000 periods per round
  - Extract 80 observations before and 25 observations after each default event in stationary distribution to compute statistics
- Data sources for Argentina:
  - Bond spreads: J.P. Morgan EMBIG for 1997Q1–2001Q4 and 2005Q3–2011Q3
  - Output, consumption, trade balance: MECON (seasonally adjusted) for various sample windows (Output: 1980Q1–2001Q4 and 2005Q3–2011Q3; consumption and trade balance: 1993Q1–2001Q4 and 2005Q3–2011Q3)
  - External debt: IMF WEO for 1980–2001 and 2005–11
- Sovereign indebtedness measures computed:
  - Average external debt/GDP ratio
  - Debt service (including short-term debt) to GDP ratio

### Simulation results: business cycle and non-business cycle statistics
- Business cycle statistics (Table 5, before vs. after default):
  - Before Default (Data / Model / Yue (2010) / A. and G. (2006)):
    - Consumption Std./Output Std.: 1.03 / 1.24 / 1.04 / 1.05
    - Trade Balance/Output Std Dev.(%): 1.23 / 3.71 / 2.81 / 0.95
    - Corr. (Trade Balance/GDP, Output): -0.83 / -0.005 / -0.16 / -0.19
  - After Default (Data / Model):
    - Consumption Std./Output Std.: 1.00 / 1.31
    - Trade Balance/Output Std Dev.(%): 1.03 / 4.20
    - Corr. (Trade Balance/GDP, Output): -0.74 / -0.02
- Non-business cycle statistics (Table 6, selected entries preserved):
  - Target statistics:
    - Default Probability: Data 2.7 / Model 2.65 / Yue (2010) 2.67 / A. and G. (2006) 0.92
    - Average Recovery Rate (%): Data 33 / Model 31.3 / Yue (2010) 27.31 / A. and G. (2006) 0
  - Before Default (Data / Model / Yue (2010) / A. and G. (2006)):
    - Average Debt/GDP ratio: 12.7 / 10.2  — Model 9.5 / Yue 10.1 / A. and G. 5.95
    - Corr. (Spreads, Output): -0.86 / -0.19 / -0.11 / -0.29
    - Average Bond Spreads (%): 7.4 / 3.1 / 1.86 / 3.58
    - Bond Spreads Std Dev. (%): 2.9 / 1.9 / 1.58 / 6.36
    - Corr. (Debt/GDP, Spreads): Data 0.43 / Model 0.72
  - Debt Renegotiation:
    - Corr. (Default Prob., Recovery Rates): Model -0.31
    - Corr. (Defaulted Debt, Recovery Rates): Data 0.33 / Model 0.31 / Yue 0.31
    - Average Exclusion (years): Data 3.5 / Model 0.25 / Yue 0.25 / A. and G. 2.5
  - After Default:
    - Average Debt/GDP ratio: Data 43.0 / 13.2 — Model 12.3
    - Corr. (Spreads, Output): Data -0.43 / Model -0.32
    - Average Bond Spreads (%): Data 6.7 / Model 3.9
    - Bond Spreads Std Dev. (%): Data 4.1 / Model 2.0
    - Corr. (Debt/GDP, Spreads): Data 0.72 / Model 0.90
    - Span Between Defaults (Years): Data 14.25 (Model entry blank)
- Key qualitative matches and differences:
  - Model matches volatile consumption and trade balance volatility pre-default and negative correlation between trade balance and output
  - Model better matches post-default statistics (consumption volatility and trade balance-output correlation)
  - Model generates average debt-to-GDP higher in post-default period (12.3 percent) than pre-default (9.5 percent), consistent with data feature that post-default debt can be higher due to increased borrowing costs
  - Average bond spreads in simulations:
    - Pre-default: 3.1 percent
    - Post-default: 3.9 percent
    - Volatility: pre-default 1.9 percent; post-default 2.0 percent

### Average spans between defaults
- Computed from 2000 rounds, extracting initial 200 periods per round: average spans between defaults are weakly decreasing with number of past renegotiations; results robust to h_max changes
- Table 7 (quarters) — Data: group average (EM countries) in 1824-2001, 64 Def. Prob. 2.7:
  - h_max = 3: 1st def. 57, 2nd def. 19, 3rd def. 8, ... ; Model def. prob. 2.65
  - h_max = 4: 1st def. 57, 2nd def. 30, 3rd def. 8, 4th def. 8 ; Model def. prob. 2.99
  - h_max = 5: 59, 27, 16, 14, 8 ; Model def. prob. 3.67
  - h_max = 6: 59, 27, 22, 16, 14, 8 ; Model def. prob. 4.47

### Decomposition of interest spreads
- Interest spread s(b_{t+1}, h_t, y_t) for h_t ≥ 1:
  - If b_{t+1} ≥ 0 → s = 0
  - Otherwise:
    - s = (1+r+φ(b_t, h_t, y_t)) / (1−p(b_{t+1}, h_t, y_t) + p(b_{t+1}, h_t, y_t) γ(b_{t+1}, h_t, y_t)) − (1+r)
- Total spreads decomposed into two components:
  - (A) Spread components based on “pure” default probability (future defaults) — credit-history irrelevant; measure used in Yue (2010)
  - (B) Spread components based on impact of additional spread premia (past defaults) — via φ(b_t, h_t, y_t); increases spreads directly and indirectly through higher default probability
- Empirical implication from Figure 7:
  - Total spreads deviate from spreads based on “pure” default probability when debt-to-GDP exceeds threshold 0.175 in mean income state — difference interpreted as spread component associated with past default history

### Determinants of the contract curve slope and spread premia dynamics
- Determinants of slope (Table 8, selected comparative values):
  - Data slope: -0.01
  - Discount factor effects:
    - β = 0.81 → slope -0.03
    - β = 0.77... (?) → slope -0.07 (table entries preserved as presented)
  - φ_max effects:
    - φ_max = 0.025 → slope -0.03
    - φ_max = 0.005 → slope -0.12
  - Output cost (λ_d) effects:
    - λ_d = 0.025 → slope -0.10
    - λ_d = 0.0225 → slope -0.08
  - Risk-free rate r effects:
    - r = 0.03 → slope -0.08
    - r = 0.017 → slope -0.07
    - r = 0.01 → slope -0.05
  - Probability of upgrading in credit history χ:
    - χ = 0 → slope -0.07
    - χ = 0.075 → slope -0.07
  - Insights:
    - Slope gets steeper (larger absolute value) as discount factor decreases
    - Reducing φ_max can steepen the slope because creditors substitute spread premia with higher recovery rate demands
    - Increasing output cost (λ_d) increases slope absolute value: country prefers higher immediate recovery to avoid costly future spread payments
    - Higher risk-free rate increases slope absolute value (future spread returns relatively less valuable), leading creditors to demand higher current recovery rates
    - Probability of upgrading χ does not affect slope value materially

- Duration and size of additional spread premia (Table 9 insights):
  - Increasing χ (probability of upgrading) shortens deterioration in long-run credit, lowering average debt-to-GDP in post-default
  - Increasing φ_max (size of additional spread premia) raises average debt-to-GDP in pre- and post-default, raises average bond spreads in post-default, and increases default probability
  - Table 9 selected preserved values:
    - Prob. of Upgrading (χ): 0 / 0.025 / 0.075
    - Max. level—spread premia (φ_max): 0.005 / 0.01 / 0.025
    - Default Probability (corresponding entries): 2.67 / 2.65 / 2.65 ; (for φ_max columns) 2.55 / 2.65 / 3.04
    - Average Recovery Rate (%): 31.9 / 31.3 / 31.9 ; (φ_max) 32.2 / 31.3 / 31.5
    - Before Default: Average Debt/GDP ratio (examples) 10.4 / 9.5 / 10.4 ; (φ_max) 10.4 / 9.5 / 11.8
    - After Default: Average Debt/GDP ratio (examples) 10.9 / 12.3 / 10.9 ; (φ_max) 10.9 / 12.3 / 12.4
    - After Default: Average Bond Spreads (%) examples: 3.9 / 3.9 / 3.9 ; (φ_max) 3.6 / 3.9 / 4.6
  - Note: Changes in χ and φ_max do not affect business cycle statistics significantly

### Quantitative findings and policy-relevant mechanisms
- Main quantitative findings summarized:
  - Incorporating additional spread premia allows the model to match observed pattern: lower recovery rates (larger haircuts) associated with larger increases in yield spreads
  - Default probability is weakly increasing with credit history for a given debt-to-GDP ratio
  - The model accounts for business cycle and non-business cycle regularities in post-default periods, including higher average debt-to-GDP after default
  - Average spans between defaults are weakly decreasing as the country experiences more defaults
  - Interest spreads decompose into two parts: spread components of future defaults and spread components of past default history (additional premia)
- Mechanism driving serial defaults:
  - Additional spread premia agreed at past restructurings increase long-term borrowing costs, inducing higher debt accumulation and raising default likelihood in subsequent periods

### Conclusion and avenues for further research
- Theoretical and empirical exploration shows equilibrium default probability increases weakly with past defaults and aligns with stylized facts linking lower recovery rates to larger increases in yield spreads
- Model assumes symmetric information at renegotiation; suggested extensions:
  - Compare renegotiation outcomes under asymmetric information (e.g., unrevealed government type or output cost) or uninformed creditor coordination/composition as potential future research topics

*Source: _wp1666 - 1.7 percent. The baseline output loss parameter λ (extracted content from the supplied PDF chapter/section).*

### Appendix I. Computation Algorithm

### Appendix I. Computation Algorithm

### Computation procedure (stepwise)
- (1) Set discrete grids on the space of credit history as H = [0,1,2,3] corresponding to h_max = 3.
- (2) Set finite grids on the space of endowment and asset holdings as B = [−0.3, .... . ,0].
  - Limits of asset space are set to ensure that the limits do not bind in equilibrium.
  - Limits of endowment space are big enough to include large deviations from the average value of shocks.
  - Approximate the sovereign’s stochastic income process using a discrete Markov chain of 21 equally spaced grids.
  - Calculate the transition matrix based on the probability distribution μ(y_{t+1}, y_t).
- (3) Set finite grids on the space of recovery rate and additional spread premia.
  - Limits of both recovery rates and additional spread premia are set to ensure that they do not bind in equilibrium.
- (4) Set initial values for equilibrium bond price, recovery rate, and interest spreads.
  - Baseline equilibrium bond price: risk-free bond price q_1 = q_f = (1 + r)^{-1}.
  - Baseline recovery rate: α_0 = 0.5.
  - Baseline additional spread premia: φ_0 = 0.01.
- (5) Given baseline q_0 = q_f, α_0 = 0.5, and φ_0 = 0.01, solve the country's optimization problem for each credit history (h_t = 0,1,2 ...).
  - Guess the value functions (V_0, V_{D,0}, V_{R,0}) and iterate them using the Bellman equation to find fixed values (V^*, V_{D,*}, V_{R,*}), given the baseline bond price, recovery rate, and spreads.
  - By iterating the Bellman function, derive the optimal asset policy function for every value (a′, a′^D, a′^R).
  - For each credit history, obtain choices of default by comparing values of defaulting and non-defaulting to calculate the default set.
  - Based on the default set, evaluate the default probability using the transition matrix.
- (6) Using the default set in step (5) and the zero profit condition for foreign investors, compute the new price of discounted bond (q_1). Iterate step (5) to obtain the fixed value of equilibrium bond price.
- (7) Given the value functions (V^*, V_{D,*}, V_{R,*}), value of autarky (V_A), the payment of bonds (R^*) derived from the iterations above and the price of discounted bond (q^*), solve the bargaining problem and compute the new debt recovery schedule (α′) and additional spread premia (φ′) for every (b, h, y).
  - Iterate steps (5) and (6) to obtain the fixed optimal debt recovery rate (α^*) and the optimal additional spread premia (φ^*).

### Numerical and modeling specifics
- Credit history grid: H = [0,1,2,3], h_max = 3.
- Asset grid example: B = [−0.3, .... . ,0].
- Income Markov chain: 21 equally spaced grids.
- Baseline bond price: q_1 = q_f = (1 + r)^{-1}.
- Baseline recovery rate: α_0 = 0.5.
- Baseline additional spread premia: φ_0 = 0.01.
- Value function iteration yields (V^*, V_{D,*}, V_{R,*}) and policy functions (a′, a′^D, a′^R).

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### Appendix II. Yields on Newly Issued Bonds for Defaulters and Non-Defaulters

### Main empirical findings
- Baseline regression (1st column) result: past defaulters suffer higher yields on newly issued bonds after defaults or restructurings by 2.2 percentage points.
- The result is robust when using a shorter interval for past defaults starting from 1990 (2nd column).
- Sovereigns with higher external debt-to-GDP ratio and larger exchange rate depreciation have higher yields on newly issued bonds.
- Due to limited sample of non-defaulters with available PPG external debt data, external debt-to-GDP ratio is used.

### Table A1 (selected coefficient estimates and statistics)
- Past Default Dummy (since 1985): 2.24*** (0.72)
- Past Default Dummy (since 1990): 2.19*** (0.74)
- External Debt/GDP ave, 2004-2010 (%): 0.028** (0.012) and 0.029** (0.012)
- GDP growth rate, ave. 2005-11 (%): 0.23 (0.14) and 0.23 (0.14)
- Exchange rate depreciation, ave. 2005-11: 0.43*** (0.12) and 0.44*** (0.12)
- Capital account openness, ave. 2005-11: -0.22 (0.35) and -0.15 (0.36)
- Sample countries: 32 32
- Adj-R^2: 0.77 and 0.76
- Root MSE: 2.44 and 2.47
- Note: Standard errors in parentheses. ***, **, * show significance at 1, 5, and 10 percent levels respectively.

### Data and measurement notes
- Yields of newly issued external bonds denominated in foreign currency. Weighted average (%) using issuance amounts in the US dollar. If bonds are denominated in currencies other than the US dollar, issuance amounts converted by exchange rates at the end of months.
- Exchange rate depreciation: Change in end of period annual exchange rate from the previous level.
- Capital account openness index from Chinn and Ito (2006) ranges from -1.86 and 2.44 corresponding to the lowest and highest degree of capital openness respectively.

### Stylized fact
- Stylized fact A1: Defaulters suffer higher yields on newly issued bonds after defaults and restructurings than non-defaulters (by 2.2 percent).

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### Appendix III. Extrapolation Method of EMBIG Spreads

### Method summary
- Goal: obtain longer series of EMBIG spreads covering 1986–2010 by ICRG ratings and LIBOR.
- For each country, run a linear time-series regression of EMBIG yields on LIBOR and ICRG ratings for the period 1999M1–2012M12:
  - Yields_{t,i} = c_i + β_1 LIBOR_t + β_2 Rating_{t,i} + ε_{t,i} for i = 1 ...... N
- LIBOR captures influence of global liquidity on sovereign yields; ICRG ratings reflect country-specific variations of investor risk assessment.
- A constant c_i captures time-invariant components of yields for each country.
- Since most coefficients are significant at 1 percent level, use estimated coefficients for 1999M1–2012M12 to obtain imputed EMBIG yields for 1986M1–1998M12 for each country.
- Take a difference between imputed EMBIG yields and yields of the US Treasury 10-year bonds for each country to obtain estimates of EMBIG spreads.

### Selected country regression coefficients (Table A2)
- Argentina: Constant 193.14***, LIBOR 1.94***, ICRG Ratings -2.60***, Adj-R^2 0.64
- Chile: Constant 18.51***, LIBOR 0.50***, ICRG Ratings -0.19***, Adj-R^2 0.73
- Cote D’Ivore: Constant 105.44***, LIBOR 2.84***, ICRG Ratings -1.61***, Adj-R^2 0.62
- Egypt: Constant 18.88***, LIBOR 0.36***, ICRG Ratings -0.21***, Adj-R^2 0.15
- Jamaica: Constant 54.38***, LIBOR 0.65***, ICRG Ratings -0.71***, Adj-R^2 0.57
- Mexico: Constant 36.98***, LIBOR 0.78***, ICRG Ratings -0.46***, Adj-R^2 0.82
- Peru: Constant 51.14***, LIBOR 0.54***, ICRG Ratings -0.65***, Adj-R^2 0.58
- Russia: Constant 89.22***, LIBOR 1.48***, ICRG Ratings -1.20***, Adj-R^2 0.80
- South Africa: Constant 37.40***, LIBOR 0.74***, ICRG Ratings -0.48***, Adj-R^2 0.66
- Turkey: Constant 28.40***, LIBOR 0.30***, ICRG Ratings -0.34***, Adj-R^2 0.62

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### Appendix IV. Figures at Steady State Distributions

### Key graphical insights (descriptions)
- Figure A1: Shows relationship between the increase in interest spreads and recovery rates conditional on income realization.
  - There is a negative relationship between recovery rates and increase in interest spreads in the lowest, mean and highest mean income states.
  - The slope of the contract curve in the lowest income state is steeper than ones in both mean and the highest income states.
- Figure A2: Presents that the slope of the contract curve is vertical in the case of Yue (2010).
  - Since Yue (2010) does not consider any additional spread premia agreed at the debt renegotiation, there is no increase in spreads.

*Source: Appendix I–IV, _wp1666 - Appendix I. Computation Algorithm*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp1666.pdf_