## wpiea2021282-print-pdf

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---

### Model overview
- The model estimates a time-varying premium of a GDP-linked warrant by comparing a model-implied price (based on country fundamentals) with actual trading prices, using a Monte Carlo pricing exercise following Costa, Chamon, and Ricci (2008).
- Model-implied end-of-month price for currency j and month m:
  - ˆp_m(j, r_scdi_m(j)) = E_m[∑_{t>=m} c_t(j) / (1 + r_{t,m}(j) + r_scdi_m(j))] (equation (1))
  - c_t(j) denotes annual coupon payment; maturity year T.
- Coupon sizes depend on nominal GDP in constant prices Y_t, real GDP growth Y_t/Y_{t−1}, and exchange rate e_t; institutional details and formulas vary across countries.
- Repayment of principal is typically not required.

### Discount rate and premia decomposition
- Discount rate components:
  - Standard discount rate r_{t,m}(j): risk-free interest rate plus default premium inferred from CDS.
  - Residual discount rate r_scdi_m(j): time-varying premium that equates model-implied price to observed bid price:
    - p_bid_m(j) = ˆp_m(j, r_scdi_m(j)) (equation (2))
    - {r_scdi_m(j)}_m interpreted as additional premium for buy-and-hold investors until maturity.
  - Liquidity premium r_liq_m(j): spread inferred from bid-ask gap:
    - p_ask_m(j) = ˆp_m(j, r_scdi_m(j) + r_liq_m(j)) (equation (3))
    - r_liq_m(j) captures liquidity gap between bid and ask.

### Stochastic processes for simulation
- Real GDP growth (log changes): y_t ≡ log(Y_t/Y_{t−1}); expected growth projected at time m is ˆy_{m,t}. Evolution:
  - y_{t+1} − ˆy_{m,t+1} = θ_y (y_t − ˆy_{m,t}) + ε_{y,t} for m ≤ t ≤ T (equation (4))
- GDP deflator log-change:
  - d_t = ˆd_{m,t} + ε_{d,t} for m ≤ t ≤ T (equation (5))
  - (ε_{y,t}, ε_{d,t}) drawn from bivariate normal N(0, Σ_{y,d}); ε_{d,t} is modelled to follow a chi-square distribution (text note).
- Nominal exchange rate:
  - e_t = ˆe_{m,t} + ε_{e,t} for m ≤ t ≤ T (equation (6))
  - ε_{e,t} drawn from N(0, σ_e^2).
- Simplifying assumptions:
  - Correlations between ε_{e,t} and ε_{y,t}, and between ε_{e,t} and ε_{d,t} are muted due to limited data on ˆe_{m,t} (1-year and 2-year forwards).

### State-contingent coupon formulas (three country cases)
- Focus: GDP-linked warrants where coupons disbursed only if GDP growth (or nominal level) exceeds thresholds and coupon rate is proportional to growth.
- Countries analyzed: Argentina (2005), Greece (2012), Ukraine (2015).

- Argentina (annual coupons):
  - c_ARG_t(j) = γ(j) / 20 · e_t(j) · (Y_t − Y^c_t) · D_t × I{Y_t > Y^c_t} × I{Y_t/Y_{t−1} > Y^c_t/Y^c_{t−1}}
  - Features:
    - Coupon proportional to gap between Y_t (nominal GDP in 1993 peso price) and contractual cutoff Y^c_t.
    - Scaled by GDP deflator D_t, adjusted by exchange rate e_t(j), and currency-specific constant γ(j).
    - Paid only if nominal GDP level and real GDP growth both exceed cutoffs.
    - Argentina issued in US dollar, euro, Japanese yen, and Argentinian peso; analysis focuses on US dollar-denominated, US-law issuance.

- Greece (annual coupons):
  - c_GRC_t = min{1.5 * (Y_t/Y_{t−1} − Y^c_t/Y^c_{t−1}), 0.01} × I{Y_t > Y^c_t} × I{Y_t/Y_{t−1} > Y^c_t/Y^c_{t−1}}
  - Features:
    - Coupon proportional to gap between real GDP growth and cutoff.
    - Payment denominated in euros.
    - Coupon capped by 1 percent (0.01).

- Ukraine (annual coupons):
  - For year 2024 onwards:
    - c_UKR_t = 0.15 * E_t * max{(Y_t/Y_{t−1} − 1.03), 0.01} * Y_{t−1} (1 + D_t) × I{Y_t > Y^c_t} × I{Y_t/Y_{t−1} > 1.03}
      + 0.4 * E_t * (Y_t/Y_{t−1} − 1.04) * Y_{t−1} (1 + D_t) × I{Y_t > Y^c_t} × I{Y_t/Y_{t−1} > 1.04}
  - Prior to 2024, maximum coupon capped at one percent of GDP.
  - Features:
    - Two cutoffs: 3 percent and 4 percent real growth.
    - Payment denominated in Ukrainian hryvnia but converted to US dollar.

### Calibration approach
- Three stochastic processes calibrated: GDP growth (eq 4), GDP deflator (eq 5), exchange rate (eq 6).
- Growth process:
  - Use Consensus Economics market consensus for GDP growth forecasts: first two-year forecasts monthly; T+3 through T+10 quarterly or biannual, linear interpolation to monthly; years 11 to maturity assume real GDP growth converges to 3 percent.
  - Persistence parameter θ_y estimated from historical data per country; Table 1 sample horizons use milestones to truncate series (e.g., end of hyper-inflation, euro accession).
  - For older periods lacking forecast data, historical average GDP growth substitutes for ˆy_{m,t} to estimate θ_y.
- GDP deflator:
  - Extract 1-, 2-, ..., 5-year GDP deflator forecasts from World Economic Outlook; convert to monthly by linear interpolation and extrapolate 5-year forecasts forward.
  - Given θ_y, jointly estimate Σ from residuals via Maximum Likelihood Estimation.
- Exchange rate:
  - Projected exchange rates ˆe_{m,t} derived from 1-year and 2-year forward exchange rates; assume nominal exchange rate remains constant after year 3.
- Baseline discount rate r_{t,m}(j):
  - Calibrated as risk-free rate plus default premium inferred from CDS.
  - If rate missing (e.g., trading halted), values interpolated between nearest existing records.

### Parameter values (Table 1)
- Argentina:
  - θ_y = 0.7239
  - Σ_{y,d} = [0.003, -0.001; -0.001, 0.006]
  - Period: 1992-2005 (Hyperinflation ended in 1992)
- Greece:
  - θ_y = 0.2296
  - Σ_{y,d} = [0.001, 0.000; 0.000, 0.000]
  - Period: 2002-2018 (Joined the eurozone in 2001)
- Ukraine:
  - θ_y = 0.6553
  - Σ_{y,d} = [0.004, -0.003; -0.003, 0.01]
  - Period: 1995-2016 (Hyperinflation ended in 1995)
- Notes: θ_y is persistence in real GDP growth process; Σ is variance-covariance matrix of GDP and deflator errors.

### Monte Carlo algorithm (four steps)
1. Calibrate θ, Σ, and σ_e prior to simulations.
2. For each month m in year T, extract market forecasts {ˆy_{m,T+t}}_{t=0,1,...,30}, {ˆd_{m,T+t}}_{t=0,1,...,30}, and {ˆe_{m,T+t}}_{t=0,1,2}. Draw random numbers and compute (4), (5), and (6) until maturity year.
3. For each month m, compute discounted expected values of the GDP-linked warrant.
4. Find r_scdi and r_liq such that discounted values equal actual trading prices.

- Monte Carlo example (Table 2): dispersion (20th, 40th, 60th, 80th percentiles) across 500 simulation seeds for T+1 (example year T=2018) reported for GDP growth, GDP deflator percentage change, and coupon rates. Example simulated percentiles and actuals:
  - GDP Growth (T=2018) Argentina: 20th −5.278, 40th −2.444, 60th 0.795, 80th 4.269; Actual −2.088
  - GDP Deflator % Change (T=2018) Argentina: 20th 23.329, 40th 9.33, 60th 4.94, 80th 0.95; Actual 0.6
  - (Table notes: 500 simulation seeds; model fits next-year growth relatively well; GDP deflator forecasted less accurately.)

### Estimated SCDI premia and empirical results
- Framework applied to estimate time-varying risk premium (SCDI premium) and liquidity premium r_liq; averages taken per year and country; numbers expressed in percentage terms.
- Visualization components (Figures 1–3) on monthly frequency include:
  - Panel (a): model-implied price (CDS + risk-free discounting only) vs market price.
  - Panel (b): estimated SCDI premium with HP-filtered trend (black dotted line).
  - Panel (c): decomposition of SCDI, default, and liquidity premia.
  - Panel (d): comovement between detrended SCDI premium and CBOE Volatility Index (VIX, indicator for global risk aversion).
- Empirical focus: document three components of SCDI premia — secular trend, business cycle comovement, and liquidity during crises — and comovements with economic indicators.

### Summary statistics (Table 3 excerpts)
- Argentina (annual averages since issuance; percentages):
  - SCDI examples: 2005 16.21, 2006 9.61, 2007 8.89, 2008 16.43, 2009 6.96, 2010 24.36, 2011 9.10, 2012 8.60
  - Default examples: 2005 3.35, 2006 2.75, 2007 2.96, 2008 11.96, 200^9 23.24, 2010 8.94, 2011 7.14, 2012 11.88
  - Liquidity examples: 2005 8.64, 2006 1.88, 2007 0.49, 2008 0.72
  - Later years include SCDI: 2013 −7.73, 2014 −10.91, 2015 −6.13, 2016 5.03, 2017 9.85, 2018 10.39, 2019 −7.03, 2020 −29.79; Default: 2013 27.84, 2014 18.86, 2015 10.95, 2016 4.69, 2017 3.25, 2018 4.33, 2019 28.07, 2020 98.67; Liquidity average 1.73.
- Greece:
  - SCDI examples: 2012 −1.38, 2013 2.59, 2014 8.69, 2015 12.55, 2016 15.22, 2017 17.03, 2018 18.15, 2019 11.58, Avg. 10.55
  - Default examples: 2012 17.07*, 2013 17.07*, 2014 17.07, 2015 10.16, 2016 6.79, 2017 3.36, 2018 2.52, 2019 1.28, Avg. 9.41
  - Liquidity average 2.26
  - Note: numbers with asterisks are extrapolated values from earliest traded prices where trading halted due to restructuring.
- Ukraine:
  - SCDI examples: 2016 4.67, 2017 6.32, 2018 7.36, 2019 6.17, 2020 5.27, Avg. 5.96
  - Default examples: 2016 6.93*, 2017 6.93, 2018 4.51, 2019 5.66, 2020 5.60, Avg. 5.93
  - Liquidity average 0.18
- Additional note on 2020 pandemic effect:
  - Sharp drop in model-implied price in 2020 reflects built-in high persistence in GDP process; investors expected faster V-shaped recovery reducing observed risk premia relative to theoretical price. Robustness checks include excluding pandemic period and fitting different θ persistence for that period.

### Persistence of the SCDI Premium (Section 4.1)
- Key finding: The SCDI premium is "high, persistent and shows no sign of a downward secular trend for most of our sample period."
- Two-year averages of the SCDI premium reported:
  - 9.36 %p
  - -0.21%p
  - 7
  - 6.12 %p
- Averages over the next three years for specific countries:
  - Argentina: 15.78 %p
  - Greece: 8.70%
  - Ukraine: 7.18%
- Stylized Fact 1: SCDI Premium is high, persistent and shows no sign of a downward trend over the first five years since issuance.
- Comparison with prior work:
  - Costa, Chamon, and Ricci (2008) documented a substantial decline (~600 basis points) in Argentina during first two years; current analysis finds early decline reverses after year 2 and SCDI premium increases during first five years once cyclical components are removed.
  - For Greece and Ukraine, estimations do not show any sign of a downward trend.
- Implication: a large portion of the SCDI premium may be attributable to a permanent feature of a GDP-linked warrant rather than only to short-term novelty.

### Cyclicality (Section 4.2)
- Empirical approach:
  - Investigate comovement between SCDI premium and business-cycle indicators: industrial production (IP), unemployment rate (Unemp.), average earning yield in local stock market, and global indicators VIX and US monetary policy shock (US MP Shock).
  - Convention: premium is "counter-cyclical if the premium tends to rise in recession."
- Findings:
  - Stylized Fact 2: The SCDI premium is less counter-cyclical than the default premium in government bond returns.
  - Default premium (proxied by 5-year CDS spread) exhibits negative correlation with growth in industrial production over 12 month, 6 month and 1 month horizons (more counter-cyclical).
  - SCDI premium displays positive correlations with industrial production growth measures (more procyclical).
  - SCDI premium shows a positive correlation with the VIX, but this correlation is weaker than the default premium’s correlation with the VIX.
  - Illustrative episode: At the onset of the COVID-19 crisis, SCDI premia dropped substantially while the default premium rose.
- Possible confounders:
  - Assumptions on GDP shock persistence may bias measurements during short-lived shocks like COVID.
  - Decomposition artifacts: if total discount factor for SCDIs remains relatively constant, high default premium in recessions could mechanically force the SCDI premium to be low; nevertheless, total discount factor for SCDIs moves less counter-cyclically than the default premium, leaving a procyclical residual.

### Liquidity (Section 4.3)
- Stylized Fact 3: Liquidity premium in GDP-linked warrant markets is higher and fluctuates more widely than liquidity premium in plain-vanilla government bond markets.
- Mean values of liquidity premia reported:
  - SCDI liquidity premia mean:
    - Greece: 210.42 bps
    - Ukraine: 18.43 bps
  - Liquidity premia of plain 10-year government bonds mean:
    - Greece: 40.97 bps
    - Ukraine: 11.40 bps
- Observation: During sample periods, liquidity premium in GDP-linked warrant markets tends to be higher and fluctuate more widely than in plain-vanilla government bond markets.
- Due to data limitations, plots of liquidity premia are presented only for Greece and Ukraine.

### Model: Mechanism and Numerical Results (Section 5)
- Conceptual mechanism:
  - Global investors price securities issued by a small open economy government and have ambiguity-averse (robust) preferences following Hansen and Sargent (2001).
  - Government issues two securities:
    - Perpetual plain-vanilla bond (PV): pays fixed coupon r_PV perpetually; defaults on coupon if growth g_t falls below threshold g_PV.
    - GDP-linked warrant (SCDI): coupon proportional to gap between realized growth and threshold g_SCDI; pays r_SCDI = α(g_t − g_SCDI) when g_t ≥ g_SCDI, otherwise 0. Assumption g_SCDI > ̄g > g_PV; α chosen so p_SCDI( ̄g) = p_PV( ̄g).
  - Lender utility: multiplier/robust form with parameters β and γ; ω = −1/((1−β)(1−γ)).
  - Stochastic discount factor includes expectation-tilting term exp(−U_{t+1}/ω)/E_t[exp(−U_{t+1}/ω)] reflecting ambiguity aversion.
- Intuition:
  - Ambiguity-averse investors overweight probabilities of bad states; premium rises when small deviations can cause large payoff changes next period.
  - PV: payment volatility largest near recession due to default risk → default premium is counter-cyclical.
  - SCDI: payment volatility surges when current growth is near the payment threshold → SCDI premium moves procyclically relative to default premium.
  - Under standard expected utility, these probability distortions are absent and SCDI premia are small and less cyclical.
- Numerical calibration (Table 5):
  - ω Ambiguity Aversion: 0.25
  - β Time Discount Factor: 0.95
  - α SCDI Coefficient: 10
  - ̄r Plain-vanilla Bond Coupon Rate: 0.04
  - (g_PV, g_SCDI) Cutoffs: (-0.02, 0.01)
  - (θ_c, θ_g) AR(1) Coefficients: (0.5, 0.85)
  - (σ_c, σ_g) Noise Standard Deviations: (0.5, 0.5)
  - ρ Noise Correlation: 0.3
  - ( ̄c, ̄g) Means: (0, 0)
- Main numerical findings:
  - Robust preferences generate a sizable SCDI premium and greater variation with current growth rate than standard preferences.
  - The SCDI premium is highest when growth is close to the SCDI payment threshold, driven by ambiguity-aversion distortions.
  - The SCDI premium under robust preferences exhibits a more pronounced positive contemporaneous correlation with growth relative to the standard-preference model.
  - Expected returns: robust preferences generate heavier price discounts (higher yields) on PV when growth is near the default range; the gap between robust and standard preferences for SCDI yields is widest when the economy is in a good state.
- Sensitivity checks (Table 6) — selected entries for SCDI premia by current growth rate and model specification (values represent SCDI premia):
  - Growth Rate columns: -0.03 | -0.01 | 0 | 0.01 | 0.03
  - Baseline:
    - Robust: 0.016 | 0.047 | 0.071 | 0.019 | 0.020
    - Standard: 0.004 | 0.013 | 0.017 | 0.019 | 0.020
  - High theta (ω = 0.5):
    - Robust: 0.011 | 0.028 | 0.037 | 0.019 | 0.019
    - Standard: 0.005 | 0.014 | 0.017 | 0.019 | 0.019
  - Low theta (ω = 0.1):
    - Robust: 0.022 | 0.091 | 0.142 | 0.021 | 0.022
    - Standard: 0.007 | 0.015 | 0.019 | 0.021 | 0.022
  - High alpha (α = 11):
    - Robust: 0.019 | 0.122 | 0.213 | 0.020 | 0.020
    - Standard: 0.003 | 0.013 | 0.017 | 0.020 | 0.020
  - Low alpha (α = 9):
    - Robust: 0.025 | 0.089 | 0.135 | 0.019 | 0.019
    - Standard: 0.005 | 0.013 | 0.016 | 0.019 | 0.019
- Caveat: The model SCDI depends only on growth thresholds and omits level conditions on GDP, whereas empirical SCDIs also include level conditions; some empirical procyclicality may arise from dynamics involving GDP levels that the model abstracts from.

### Debt Restructuring (Section 6.1)
- Timing and pricing implications:
  - Issuance of GDP-linked warrants is most effective when the economy is at the trough of a business cycle.
  - Rationale:
    - Investors require a relatively higher risk premium on plain-vanilla bonds because ambiguity in a country’s debt repayment capacity is worsened in recession.
    - GDP-linked warrants face a relatively milder discount during recessionary periods.
    - Payment volatility of GDP-linked warrants is maximized when the economy turns to a good state.
  - Policy implication:
    - The cost of capital for issuing GDP-linked warrants can be lower if current output is low and debt repayment capacity is questioned than in expansionary periods.
  - Empirical observation:
    - Time-varying ambiguity premium provides one explanation why most GDP-linked warrants issued to date came out when issuing countries underwent a debt restructuring process.
- Cross-country characteristics of SCDIs:
  - Contract design effects:
    - GDP-linked warrants with more kinks on their payment structure tend to have a higher SCDI premium especially when kinks are located on a more probable region of the state.
    - Optimally designed GDP warrant takes the form of linear indexation with minimal use of payment cutoffs.
  - Institutional and credibility effects:
    - Moral hazard or ambiguity about moral hazard may increase the discount rate applied to SCDIs.
    - Discount may be larger for countries with less credible institutions or poor sovereign debt track records.
    - Cyclical properties of SCDI premium can vary among countries with different fiscal credibility or track records; a country could start with a countercyclical SCDI risk premium and graduate to a procyclical one as it gains credibility.
  - Investor base and financial development effects:
    - Issuers may market SCDIs to investor classes less averse to ambiguity to keep SCDI premium reasonable.
    - Available data suggest a diverse set of investors in most cases, with the top identified debtholders accounting for less than 5 percent of outstanding bonds (exception: the peso-denominated Argentine SCDI, with a third of outstanding bonds held by domestic nonbanks).
  - Research gaps:
    - Limited issuance cases preclude formal empirical corroboration; future issuances outside debt restructuring episodes could shed light on benefits and costs.
    - Investor base data could explain cross-country variations such as different levels and volatility of liquidity premia in Greece and Ukraine.
  - Noted recent example: Suriname (debt restructuring proposal dated June 2021).
- Inflation-indexed bonds as related SCDIs:
  - Comparison and substitutability:
    - Inflation-indexed bonds offer similar benefits to GDP-linked warrants because lower output growth is often accompanied with low inflation.
    - In a hypothetical world with a monotonous one-to-one mapping between inflation and GDP growth, a GDP-linked warrant can be designed to substitute inflation-indexed bonds.
    - Absence of payment cutoffs would help lower the cost of capital when global investors are averse to model ambiguity.
  - Research direction: comparison, substitutability, or complementarity among different types of state-contingent instruments is an interesting venue for future research.

### Key stylized facts and contribution (Conclusion)
- Three stylized facts highlighted:
  - First, the risk premia in state-contingent instruments are high and persistent.
  - Second, the premia exhibit a pro-cyclical pattern.
  - Third, the liquidity premium is higher and more volatile than that for plain-vanilla government bonds issued by the same sovereign.
- Contribution:
  - The paper develops a quantifiable model with robust preferences to study time-varying premia of state-contingent debt instruments across countries and to explain the cyclical properties of the risk premium.
  - The framework is intended as a stepping stone to study more general cases such as bonds indexed to commodity prices or equipped with escape clauses.

*Source: wpiea2021282-print-pdf*

### 3.1  General Framework

### 3.1  General Framework

### Model overview
- The model estimates a time-varying premium of a GDP-linked warrant by comparing a model-implied price (based on country fundamentals) with actual trading prices, using a Monte Carlo pricing exercise following Costa, Chamon, and Ricci (2008).
- Model-implied end-of-month price for currency j and month m is defined as:
  - ˆp_m(j, r_scdi_m(j)) = E_m[∑_{t>=m} c_t(j) / (1 + r_{t,m}(j) + r_scdi_m(j))] (equation (1))
  - c_t(j) denotes annual coupon payment; maturity year T.
- Coupon sizes depend on nominal GDP in constant prices Y_t, real GDP growth Y_t/Y_{t−1}, and exchange rate e_t; institutional details and formulas vary across countries.
- Repayment of principal is typically not required.

### Discount rate and premia decomposition
- Discount rate components:
  - Standard discount rate r_{t,m}(j): risk-free interest rate plus default premium inferred from CDS.
  - Residual discount rate r_scdi_m(j): time-varying premium that equates model-implied price to observed bid price:
    - p_bid_m(j) = ˆp_m(j, r_scdi_m(j)) (equation (2))
    - {r_scdi_m(j)}_m interpreted as additional premium for buy-and-hold investors until maturity.
  - Liquidity premium r_liq_m(j): spread inferred from bid-ask gap:
    - p_ask_m(j) = ˆp_m(j, r_scdi_m(j) + r_liq_m(j)) (equation (3))
    - r_liq_m(j) captures liquidity gap between bid and ask.

### Stochastic processes for simulation
- Real GDP growth: let y_t ≡ log(Y_t/Y_{t−1}), expected growth projected at time m is ˆy_{m,t}. Evolution:
  - y_{t+1} − ˆy_{m,t+1} = θ_y (y_t − ˆy_{m,t}) + ε_{y,t} for m ≤ t ≤ T (equation (4))
- GDP deflator log-change:
  - d_t = ˆd_{m,t} + ε_{d,t} for m ≤ t ≤ T (equation (5))
  - (ε_{y,t}, ε_{d,t}) drawn from bivariate normal N(0, Σ_{y,d}); ε_{d,t} is modelled to follow a chi-square distribution (text note).
- Nominal exchange rate:
  - e_t = ˆe_{m,t} + ε_{e,t} for m ≤ t ≤ T (equation (6))
  - ε_{e,t} drawn from N(0, σ_e^2).
- Simplifying assumptions:
  - Correlations between ε_{e,t} and ε_{y,t}, and between ε_{e,t} and ε_{d,t} are muted due to limited data on ˆe_{m,t} (1-year and 2-year forwards).

### State-contingent coupon formulas (three country cases)
- Focus: GDP-linked warrants where coupons disbursed only if GDP growth (or nominal level) exceeds thresholds and coupon rate is proportional to growth.
- Countries analyzed: Argentina (2005), Greece (2012), Ukraine (2015).

- Argentina (annual coupons):
  - c_ARG_t(j) = γ(j) / 20 · e_t(j) · (Y_t − Y^c_t) · D_t × I{Y_t > Y^c_t} × I{Y_t/Y_{t−1} > Y^c_t/Y^c_{t−1}}
  - Features:
    - Coupon proportional to gap between Y_t (nominal GDP in 1993 peso price) and contractual cutoff Y^c_t.
    - Scaled by GDP deflator D_t, adjusted by exchange rate e_t(j), and currency-specific constant γ(j).
    - Paid only if nominal GDP level and real GDP growth both exceed cutoffs.
    - Argentina issued in US dollar, euro, Japanese yen, and Argentinian peso; analysis focuses on US dollar-denominated, US-law issuance.

- Greece (annual coupons):
  - c_GRC_t = min{1.5 * (Y_t/Y_{t−1} − Y^c_t/Y^c_{t−1}), 0.01} × I{Y_t > Y^c_t} × I{Y_t/Y_{t−1} > Y^c_t/Y^c_{t−1}}
  - Features:
    - Coupon proportional to gap between real GDP growth and cutoff.
    - Payment denominated in euros.
    - Coupon capped by 1 percent (0.01).

- Ukraine (annual coupons):
  - For year 2024 onwards:
    - c_UKR_t = 0.15 * E_t * max{(Y_t/Y_{t−1} − 1.03), 0.01} * Y_{t−1} (1 + D_t) × I{Y_t > Y^c_t} × I{Y_t/Y_{t−1} > 1.03}
      + 0.4 * E_t * (Y_t/Y_{t−1} − 1.04) * Y_{t−1} (1 + D_t) × I{Y_t > Y^c_t} × I{Y_t/Y_{t−1} > 1.04}
  - Prior to 2024, maximum coupon capped at one percent of GDP.
  - Features:
    - Two cutoffs: 3 percent and 4 percent real growth.
    - Payment denominated in Ukrainian hryvnia but converted to US dollar.

### Calibration approach
- Three stochastic processes calibrated: GDP growth (eq 4), GDP deflator (eq 5), exchange rate (eq 6).
- Growth process:
  - Use Consensus Economics market consensus for GDP growth forecasts: first two-year forecasts monthly; T+3 through T+10 quarterly or biannual, linear interpolation to monthly; years 11 to maturity assume real GDP growth converges to 3 percent.
  - Persistence parameter θ_y estimated from historical data per country; Table 1 sample horizons use milestones to truncate series (e.g., end of hyper-inflation, euro accession).
  - For older periods lacking forecast data, historical average GDP growth substitutes for ˆy_{m,t} to estimate θ_y.
- GDP deflator:
  - Extract 1-, 2-, ..., 5-year GDP deflator forecasts from World Economic Outlook; convert to monthly by linear interpolation and extrapolate 5-year forecasts forward.
  - Given θ_y, jointly estimate Σ from residuals via Maximum Likelihood Estimation.
- Exchange rate:
  - Projected exchange rates ˆe_{m,t} derived from 1-year and 2-year forward exchange rates; assume nominal exchange rate remains constant after year 3.
- Baseline discount rate r_{t,m}(j):
  - Calibrated as risk-free rate plus default premium inferred from CDS.
  - If rate missing (e.g., trading halted), values interpolated between nearest existing records.

### Parameter values (Table 1)
- Argentina:
  - θ_y = 0.7239
  - Σ_{y,d} = [0.003, -0.001; -0.001, 0.006]
  - Period: 1992-2005 (Hyperinflation ended in 1992)
- Greece:
  - θ_y = 0.2296
  - Σ_{y,d} = [0.001, 0.000; 0.000, 0.000]
  - Period: 2002-2018 (Joined the eurozone in 2001)
- Ukraine:
  - θ_y = 0.6553
  - Σ_{y,d} = [0.004, -0.003; -0.003, 0.01]
  - Period: 1995-2016 (Hyperinflation ended in 1995)
- Notes: θ_y is persistence in real GDP growth process; Σ is variance-covariance matrix of GDP and deflator errors.

### Monte Carlo algorithm (four steps)
1. Calibrate θ, Σ, and σ_e prior to simulations.
2. For each month m in year T, extract market forecasts {ˆy_{m,T+t}}_{t=0,1,...,30}, {ˆd_{m,T+t}}_{t=0,1,...,30}, and {ˆe_{m,T+t}}_{t=0,1,2}. Draw random numbers and compute (4), (5), and (6) until maturity year.
3. For each month m, compute discounted expected values of the GDP-linked warrant.
4. Find r_scdi and r_liq such that discounted values equal actual trading prices.

- Monte Carlo example (Table 2): dispersion (20th, 40th, 60th, 80th percentiles) across 500 simulation seeds for T+1 (example year T=2018) reported for GDP growth, GDP deflator percentage change, and coupon rates. Example simulated percentiles and actuals:
  - GDP Growth (T=2018) Argentina: 20th −5.278, 40th −2.444, 60th 0.795, 80th 4.269; Actual −2.088
  - GDP Deflator % Change (T=2018) Argentina: 20th 23.329, 40th 9.33, 60th 4.94, 80th 0.95; Actual 0.6
  - (Table notes: 500 simulation seeds; model fits next-year growth relatively well; GDP deflator forecasted less accurately.)

### Estimated SCDI premia and empirical results
- Apply framework to estimate time-varying risk premium (SCDI premium) and liquidity premium r_liq; averages taken per year and country; numbers expressed in percentage terms.
- Visualization components (Figures 1–3) on monthly frequency include:
  - Panel (a): model-implied price (CDS + risk-free discounting only) vs market price.
  - Panel (b): estimated SCDI premium with HP-filtered trend (black dotted line).
  - Panel (c): decomposition of SCDI, default, and liquidity premia.
  - Panel (d): comovement between detrended SCDI premium and CBOE Volatility Index (VIX, indicator for global risk aversion).
- Empirical focus: document three components of SCDI premia — secular trend, business cycle comovement, and liquidity during crises — and comovements with economic indicators.

### Summary statistics (Table 3 excerpts)
- Argentina (annual averages since issuance; percentages):
  - SCDI examples: 2005 16.21, 2006 9.61, 2007 8.89, 2008 16.43, 2009 6.96, 2010 24.36, 2011 9.10, 2012 8.60
  - Default examples: 2005 3.35, 2006 2.75, 2007 2.96, 2008 11.96, 2009 23.24, 2010 8.94, 2011 7.14, 2012 11.88
  - Liquidity examples: 2005 8.64, 2006 1.88, 2007 0.49, 2008 0.72
  - Later years include SCDI: 2013 −7.73, 2014 −10.91, 2015 −6.13, 2016 5.03, 2017 9.85, 2018 10.39, 2019 −7.03, 2020 −29.79; Default: 2013 27.84, 2014 18.86, 2015 10.95, 2016 4.69, 2017 3.25, 2018 4.33, 2019 28.07, 2020 98.67; Liquidity average 1.73.
- Greece:
  - SCDI examples: 2012 −1.38, 2013 2.59, 2014 8.69, 2015 12.55, 2016 15.22, 2017 17.03, 2018 18.15, 2019 11.58, Avg. 10.55
  - Default examples: 2012 17.07*, 2013 17.07*, 2014 17.07, 2015 10.16, 2016 6.79, 2017 3.36, 2018 2.52, 2019 1.28, Avg. 9.41
  - Liquidity average 2.26
  - Note: numbers with asterisks are extrapolated values from earliest traded prices where trading halted due to restructuring.
- Ukraine:
  - SCDI examples: 2016 4.67, 2017 6.32, 2018 7.36, 2019 6.17, 2020 5.27, Avg. 5.96
  - Default examples: 2016 6.93*, 2017 6.93, 2018 4.51, 2019 5.66, 2020 5.60, Avg. 5.93
  - Liquidity average 0.18
- Additional note on 2020 pandemic effect:
  - Sharp drop in model-implied price in 2020 reflects built-in high persistence in GDP process; investors expected faster V-shaped recovery reducing observed risk premia relative to theoretical price. Robustness checks include excluding pandemic period and fitting different θ persistence for that period.

*Italic line indicating source attribution provided by pipeline.*

### 4.1  Persistence of the SCDI Premium

### 4.1  Persistence of the SCDI Premium

### Key findings on persistence
- The SCDI premium is described as "high, persistent and shows no sign of a downward secular trend for most of our sample period."
- Two-year averages of the SCDI premium reported in the text:
  - 9.36 %p
  - -0.21%p
  - 7
  - 6.12 %p
- Averages over the next three years for specific countries:
  - Argentina: 15.78 %p
  - Greece: 8.70%
  - Ukraine: 7.18%
- The authors note that the SCDI premium fluctuates widely over business cycles, but a sizable premium exists even after a reasonable timeframe since issuance.

### Interpretation and comparison with prior work
- Stylized Fact 1: SCDI Premium is high, persistent and shows no sign of a downward trend over the first five years since issuance.
- Contrast with Costa, Chamon, and Ricci (2008):
  - Earlier work documented a substantial decline (~600 basis points) in the SCDI premium during the first two years after issuance of Argentinian GDP-linked warrants, attributing decay to a "novelty premium."
  - The current analysis finds that the apparent early decline for Argentina (years 0–2) reverses after year 2 and shows a steady increase in the SCDI premium during the first five years once cyclical components are removed.
- For Greece and Ukraine, estimations do not show any sign of a downward trend.
- Implication: a large portion of the SCDI premium may be attributable to a permanent feature of a GDP-linked warrant rather than only to short-term novelty.

---

### 4.2  Cyclicality

### Empirical approach
- Investigate comovement between SCDI premium and business-cycle indicators: industrial production (IP), unemployment rate (Unemp.), average earning yield in local stock market, and global indicators VIX and US monetary policy shock (US MP Shock).
- A negative correlation with growth in industrial production (or positive with cyclical unemployment) indicates counter-cyclicality (premium rises in recessions). Convention: premium is "counter-cyclical if the premium tends to rise in recession."

### Findings on cyclicality
- Stylized Fact 2: The SCDI premium is less counter-cyclical than the default premium in government bond returns.
- Summary of contemporaneous correlation patterns (from Table 4):
  - Default premium (proxied by 5-year CDS spread) exhibits negative correlation with growth in industrial production over 12 month, 6 month and 1 month horizons (i.e., more counter-cyclical).
  - SCDI premium displays positive correlations with industrial production growth measures (i.e., more procyclical in these measures).
  - SCDI premium shows a positive correlation with the VIX, but this correlation is weaker than the default premium’s correlation with the VIX.
- Illustrative episode: At the onset of the COVID-19 crisis, SCDI premia dropped substantially while the default premium rose.
- Possible confounding factors discussed:
  - Assumptions on GDP shock persistence: the duration of the COVID shock may be shorter than typical recessions, which could lead to underestimation of the model-implied price of a GDP warrant and thus affect measured procyclicality.
  - Decomposition artifacts: if the total discount factor for SCDIs remains relatively constant, high default premium in recessions could mechanically force the SCDI premium to be low; nevertheless, the authors find the total discount factor for SCDIs moves less counter-cyclically than the default premium, leaving a procyclical residual.

---

### 4.3  Liquidity

### Empirical liquidity comparisons
- Stylized Fact 3: Liquidity premium in GDP-linked warrant markets is higher and fluctuates more widely than liquidity premium in plain-vanilla government bond markets.
- Mean values of liquidity premia reported for sample countries (Greece and Ukraine):
  - SCDI liquidity premia mean:
    - Greece: 210.42 bps
    - Ukraine: 18.43 bps
  - Liquidity premia of plain 10-year government bonds mean:
    - Greece: 40.97 bps
    - Ukraine: 11.40 bps
- Due to data limitations, plots of liquidity premia are presented only for Greece and Ukraine.
- Observation: During sample periods, liquidity premium in GDP-linked warrant markets tends to be higher and fluctuate more widely than in plain-vanilla government bond markets.

---

### 5  Model (Mechanism and Numerical Results)

### Conceptual mechanism
- Model setting:
  - Global investors price securities issued by a small open economy government and have ambiguity-averse (robust) preferences following Hansen and Sargent (2001).
  - Government issues two securities:
    - Perpetual plain-vanilla bond (PV): pays fixed coupon r_PV perpetually; defaults on coupon if growth g_t falls below threshold g_PV.
    - GDP-linked warrant (SCDI): coupon proportional to gap between realized growth and threshold g_SCDI; pays r_SCDI = α(g_t − g_SCDI) when g_t ≥ g_SCDI, otherwise 0. Assumption g_SCDI > ̄g > g_PV; α chosen so p_SCDI( ̄g) = p_PV( ̄g).
- Investor preferences and stochastic discount factor:
  - Lender utility specified as the multiplier/robust form with parameters β and γ; ω = −1/((1−β)(1−γ)).
  - Stochastic discount factor includes an expectation-tilting term exp(−U_{t+1}/ω)/E_t[exp(−U_{t+1}/ω)] reflecting ambiguity aversion.
- Intuition:
  - Ambiguity-averse investors overweight probabilities of bad states; the premium required rises when small deviations can cause large payoff changes next period.
  - For PV, payment volatility is largest near recession due to default risk → default premium is counter-cyclical.
  - For SCDI, payment volatility surges when current growth is near the payment threshold → SCDI premium moves procyclically relative to the default premium.
  - Under standard expected utility, these probability distortions are absent and SCDI premia are small and less cyclical.

### Numerical example and calibration
- Parameter values used (Table 5):
  - ω Ambiguity Aversion: 0.25
  - β Time Discount Factor: 0.95
  - α SCDI Coefficient: 10
  - ̄r Plain-vanilla Bond Coupon Rate: 0.04
  - (g_PV, g_SCDI) Cuto‚s: (-0.02, 0.01)
  - (θ_c, θ_g) AR(1) Coefficients: (0.5, 0.85)
  - (σ_c, σ_g) Noise Standard Deviations: (0.5, 0.5)
  - ρ Noise Correlation: 0.3
  - ( ̄c, ̄g) Means: (0, 0)
- Main numerical findings:
  - Robust preferences generate a sizable SCDI premium and greater variation with current growth rate than standard preferences.
  - The SCDI premium is highest when growth is close to the SCDI payment threshold, driven by ambiguity-aversion distortions.
  - The SCDI premium under robust preferences exhibits a more pronounced positive contemporaneous correlation with growth relative to the standard-preference model.
  - Expected returns:
    - Robust preferences generate heavier price discounts (higher yields) on PV when growth is near the default range.
    - The gap between robust and standard preferences for SCDI yields is widest when the economy is in a good state.
- Sensitivity checks (Table 6) — selected entries for SCDI premia by current growth rate and model specification (values represent SCDI premia):
  - Growth Rate columns: -0.03 | -0.01 | 0 | 0.01 | 0.03
  - Baseline:
    - Robust: 0.016 | 0.047 | 0.071 | 0.019 | 0.020
    - Standard: 0.004 | 0.013 | 0.017 | 0.019 | 0.020
  - High theta (ω = 0.5):
    - Robust: 0.011 | 0.028 | 0.037 | 0.019 | 0.019
    - Standard: 0.005 | 0.014 | 0.017 | 0.019 | 0.019
  - Low theta (ω = 0.1):
    - Robust: 0.022 | 0.091 | 0.142 | 0.021 | 0.022
    - Standard: 0.007 | 0.015 | 0.019 | 0.021 | 0.022
  - High alpha (α = 11):
    - Robust: 0.019 | 0.122 | 0.213 | 0.020 | 0.020
    - Standard: 0.003 | 0.013 | 0.017 | 0.020 | 0.020
  - Low alpha (α = 9):
    - Robust: 0.025 | 0.089 | 0.135 | 0.019 | 0.019
    - Standard: 0.005 | 0.013 | 0.016 | 0.019 | 0.019
- Model caveat noted by authors:
  - The model SCDI depends only on growth thresholds and omits level conditions on GDP, whereas empirical SCDIs also include level conditions. Thus some empirical procyclicality may arise from dynamics involving GDP levels that the model abstracts from.

*Source: wpiea2021282-print-pdf - 4.1  Persistence of the SCDI Premium*

### 6.1  Debt Restructuring

### 6.1  Debt Restructuring

### Timing and pricing implications of GDP-linked warrants
- Issuance of GDP-linked warrants is most effective when the economy is at the trough of a business cycle.
- Rationale:
  - Investors require a relatively higher risk premium on plain-vanilla bonds because the ambiguity in a country’s debt repayment capacity is worsened in recession.
  - GDP-linked warrants face a relatively milder discount during recessionary periods.
  - The payment volatility of GDP-linked warrants is maximized when the economy turns to a good state.
- Policy implication:
  - The cost of capital the government has to pay when issuing GDP-linked warrants can be lower if the current output is low and debt repayment capacity is questioned than that in expansionary periods.
- Empirical observation:
  - This time-varying ambiguity premium provides one explanation on why most GDP-linked warrants that have been issued to date came out when the issuing countries underwent a debt restructuring process (Table 7).

### Cross-country characteristics of SCDIs (state-contingent debt instruments)
- Contract design effects:
  - GDP-linked warrants with more kinks on their payment structure tend to have a higher SCDI premium especially when kinks are located on a more probable region of the state.
  - The optimally designed GDP warrant takes the form of linear indexation with minimal use of payment cutoffs (see Roch and Roldan (2021) referenced in text).
- Institutional and credibility effects:
  - Moral hazard on the government’s end (or ambiguity about moral hazard) may increase the discount rate applied to SCDIs.
  - Such a discount may be larger for countries with less credible institutions or a poor track record in sovereign debt markets.
  - The cyclical properties of SCDI premium can vary among countries with different fiscal credibility or a poor track record in international sovereign debt markets, or both.
  - A country could start with a countercyclical SCDI risk premium and graduate to a procyclical one as it gains institutional credibility and reputation over time.
- Investor base and financial development effects:
  - Countries wishing to keep SCDI premium at a reasonable level may want to market their SCDIs to particular classes of investors that are less averse to ambiguity in economic variables determining payouts.
  - Available data, though sparse and limited, suggests a diverse set of investors in most cases, with the top identified debtholders accounting for less than 5 percent of outstanding bonds (Table 8).
  - Exception: the peso-denominated Argentine SCDI, with a third of outstanding bonds held by domestic nonbanks.
- Research gaps and future data needs:
  - Due to the limited number of issuance cases, the authors do not attempt formal empirical corroboration and leave it as future research.
  - As more countries consider issuing GDP-linked warrants, more issuances outside debt restructuring episodes could shed light on benefits and costs of SCDIs.
  - Investor base data could potentially explain cross-country variations such as the very different levels and volatility of liquidity premia in Greece and Ukraine.
- Noted recent example:
  - Suriname is a recent example, see debt restructuring proposal dated June 2021.

### Inflation-indexed bonds as related SCDIs
- Comparison and substitutability:
  - Inflation-indexed bonds, issued mostly by advanced economies, offer similar benefits to GDP-linked warrants for issuing governments because lower output growth is often accompanied with low inflation (Phillips Curve linkage).
  - In a hypothetical world with a monotonous one-to-one mapping between inflation and GDP growth, a GDP-linked warrant can be designed to substitute inflation-indexed bonds.
  - Absence of payment cutoffs would help lower the cost of capital when global investors are averse to model ambiguity.
- Research direction:
  - Comparison, substitutability, or complementarity among different types of state-contingent instruments is an interesting venue for future research.

### Key stylized facts and model contribution (from Conclusion)
- Three stylized facts highlighted by the paper:
  - First, the risk premia in state-contingent instruments are high and persistent.
  - Second, the premia exhibit a pro-cyclical pattern.
  - Third, the liquidity premium is higher and more volatile than that for plain-vanilla government bonds issued by the same sovereign.
- Contribution:
  - The paper develops a quantifiable model with robust preferences to study time-varying premia of state-contingent debt instruments across countries and to explain the cyclical properties of the risk premium.
  - The framework is intended as a stepping stone to study more general cases such as bonds indexed to commodity prices or equipped with escape clauses.

*IMF Working Paper chapter: 6.1 Debt Restructuring (from provided PDF content)*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021282-print-pdf.pdf_