## _wp05216

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---

### I. Introduction — purpose and approach
- Focus: assess pricing of growth-indexed bonds as a way to reduce procyclicality and default risk associated with existing debt structures in emerging market countries.
- Motivations and observations:
  - Emerging markets often rely on short-term debt or foreign currency debt, making debt servicing more costly in downturns.
  - Fiscal policies more procyclical in emerging markets than in advanced countries (cited studies).
  - Growth-indexed bonds pay returns indexed to real GDP growth, increasing payments in strong-growth years and lowering them in weak-growth years, acting as automatic stabilizers.
- Scope and limits:
  - Paper focuses on parameter values relevant for emerging markets, where macroeconomic fluctuations and perceived default risk (bond spreads) are larger.
  - Abstracts from subjective premia (novelty, liquidity) and concentrates on estimating a model-based “theoretical price” for GDP-indexed bonds.

### II. Pricing methodology — core steps and assumptions
- Broad strategy:
  - Assume risk-neutral investors require, on average, the same return as on U.S. Treasury bonds of similar duration.
  - Extract combinations of the probability of default and recovery rates from observed spreads between emerging-market and U.S. bonds.
  - Estimate the joint distribution of real GDP growth, the primary balance, and the real exchange rate from past data.
- Simulation and calibration:
  - Simulate 250,000 paths for economic variables and the debt/GDP ratio over a 10-year horizon.
  - Extract a default trigger level for the debt/GDP ratio and a recovery rate that match expected repayments implied by observed spreads.
  - Compute payoffs for both growth-indexed bonds and plain-vanilla bonds across simulated paths and across different shares of growth-indexed debt in total debt.
- Instrument specification:
  - Two debt types considered:
    - Plain-vanilla debt paying fixed interest i_pv.
    - Growth-indexed debt paying interest i_ind where i_ind = max(0, i_pv + g_t − g), with g the contractual growth threshold; principal is not indexed.
- Key calibration choices for illustrative country "Emergingland":
  - Initial debt to GDP ratio: 60 percent.
  - Share  of dollar debt: ½.
  - Baseline interest rate i_pv: 6¾ percent.
  - Discount rate (10-year U.S. yield): 4.0 percent.
  - U.S. GDP deflator *: 2.0 percent.
  - Joint-normal distribution for {g_t, _t, pb_t} based on Brazil, Mexico, Turkey (1981-2004).
  - Expected values: g_t = 3.0% and pb_t = 2.1%.
  - Expected real depreciation: zero.
  - Standard deviations (in percentage points): σ_g = 3.8, σ_ε = 16.1, σ_pb = 3.3.
  - Correlations: ρ_{g, ε} = −0.63, ρ_{g,pb} = −0.34, ρ_{ε,pb} = 0.16.
  - Growth-indexed bond contract specifies g at 3 percent.

### III. Estimating default probabilities and recovery scenarios
- No-arbitrage constraint: equate expected return on Emergingland’s plain-vanilla bond to the return on a U.S. bond with similar duration (scenario where all debt is plain-vanilla).
- Two recovery scenarios and implied baseline default probabilities (within 10 years):
  - Recovery Value: 25 Percent → Baseline Default Frequency: 27.8 Percent (implicit default trigger: Debt/GDP > 73.2 percent).
  - Recovery Value: 50 Percent → Baseline Default Frequency: 35.9 Percent (implicit default trigger: Debt/GDP > 69.5 percent).
- Default trigger chosen so plain-vanilla bonds trade at par in the baseline (all plain-vanilla) scenario.
- Simulations: 250,000 joint paths over a 10-year horizon.

### IV. Simulation experiments and scenarios
- Three debt compositions simulated:
  - Share of Indexed Debt in Total Debt (in percent): .0001 (virtually all plain-vanilla), 50, 99.9999 (virtually all GDP-indexed).
- Instruments priced:
  - 10-year plain-vanilla bond with coupon equal to 6¾ percent.
  - Indexed bond with coupon equal to Max(6¾ % + growth − 3 %,0); principal not indexed.
- Selected results (Assumptions in Table 3 — 25 Percent Recovery Value and a 27.8 Percent Baseline Default Frequency):
  - Default Frequency (in percent) (within 10 years): 27.8, 23.0, 18.9
  - Price of Indexed Bond with Coupon=Max(6¾ % + growth − 3 %,0): 101.3, 105.2, 108.4
  - Price of Plain-Vanilla Bond with 6¾ % Coupon: 100.0, 104.1, 107.6
  - Coupon (in percentage points) Required for Plain-Vanilla Bond to Sell at 100: 6¾; 6¾ − 0.58; 6¾ − 1.03
- Selected results (Assumptions in Table 4 — 50 Percent Recovery Value and a 35.9 Percent Baseline Default Frequency):
  - Default Frequency (in percent) (within 10 years): 35.9, 31.4, 26.9
  - Price of Indexed Bond with Coupon=Max(6¾ % + growth − 3 %,0): 101.5, 104.3, 107.0
  - Price of Plain-Vanilla Bond with 6¾ % Coupon: 100.0, 103.0, 106.0
  - Coupon (in percentage points) Required for Plain-Vanilla Bond to Sell at 100: 6¾; 6¾ − 0.48; 6¾ − 0.90

### V. Sensitivity exercises: growth shocks and volatility shocks
- Table 5 — Default Rates and Expected Loss if Growth is on Average 1 Percent Lower than Initially Expected (under assumptions and prices of Table 3):
  - Default Frequency (in percent) (within 10 years): 36.27, 27.79, 19.98
  - Expected Loss on Indexed Bond with Coupon=Max(6¾ % + growth − 3 %,0): 12.42, 9.61, 7.09
  - Expected Loss on Plain-Vanilla Bond with 6¾ % Coupon: 6.74, 3.58, 0.81
- Table 6 — Default Rates and Expected Loss if Growth is 1½ Times as Volatile as Initially Expected (under assumptions and prices of Table 3):
  - Default Frequency (within 10 years) (in percent): 33.47, 27.97, 22.93
  - Expected Loss on Indexed Bond with Coupon=Max(6¾ % + growth − 3 %,0): 2.65, 1.84, 0.94
  - Expected Loss on Plain-Vanilla Bond with 6¾ % Coupon: 5.04, 4.04, 3.06

### VI. Numerical findings and interpretation
- Growth-indexed debt can lower the default frequency by 1/4 to 1/3 of its initial level in the simulations.
- Moving from no indexation to indexation of all debt reduces the default probability from 28 percent to 19 percent — a reduction of the default risk by one third (baseline with recovery = 25 percent and default = 27.8 percent).
- For the scenario with recovery rate = 50 percent and baseline default = 35.9 percent, the benefits of indexation are slightly smaller because higher recovery makes defaults less costly to investors.
- Indexed bond payoff distribution is smoother than plain-vanilla: bell-shaped in non-default cases; value depends on realized growth even in default cases.
- Indexed bond in the example is always worth more in expectation than its plain-vanilla counterpart because asymmetric indexation (coupon bounded below at zero) raises average coupon payments.
- Sensitivity to mean growth shock:
  - A reduction in the mean growth rate by one percentage point reduces bond prices more for growth-indexed bonds than for plain-vanilla bonds (direct effect on indexation).
  - However, higher shares of indexed debt attenuate the increase in default frequency, reducing losses on both plain-vanilla and indexed bonds.
- Reported loss examples:
  - Increasing the share of growth-indexed bonds from zero to 100 percent reduces the loss from 6.74 percent to 0.81 percent for plain-vanilla debt and from 12.42 percent to 7.09 percent for growth-indexed bonds (under the 1 percentage point lower growth scenario in Table 5).

### VII. Existing debt structures — stylized facts (selected statistics)
- Currency composition (central government debt, 2001, in percent of total):
  - Emerging market countries: Foreign-Currency Debt 48.0; Long-Term Domestic-Currency Debt 32.4; Total Debt as a Ratio to GDP 50.4.
  - Advanced Economies: Foreign-Currency Debt 5.6; Long-Term Domestic-Currency Debt 75.9; Total Debt as a Ratio to GDP 51.8.
  - Regional and country examples:
    - Latin America: Foreign-Currency Debt 67.9; Long-Term Domestic-Currency Debt 15.2; Total Debt/GDP 37.0.
    - Asia: Foreign-Currency Debt 28.5; Long-Term Domestic-Currency Debt 44.2; Total Debt/GDP 62.8.
    - China: Foreign-Currency Debt 17.7; Long-Term Domestic-Currency Debt 82.3; Total Debt/GDP 24.0.
    - India: Foreign-Currency Debt 14.5; Long-Term Domestic-Currency Debt 69.9; Total Debt/GDP 65.1.
    - Russia: Foreign-Currency Debt 90.3; Total Debt/GDP 50.0.
- Structure of domestically issued government bonds (end-2001, in percent of total) — emerging market aggregate examples:
  - Long term 28.24; Short term 41.51; Inflation 18.62; Domestic-Currency-Denominated Not indexed 26.47; Indexed to Domestic interest rate 7.2; Foreign-Currency-Denominated 6.3.
  - Country examples (selected):
    - Brazil: Long term 52.1; Short term 9.5; Inflation-indexed 0.0; Domestic-Currency-Denominated Not indexed 53.0; Indexed to Domestic interest rate 7.0; Foreign-Currency-Denominated 30.5.
    - Mexico: Long term 11.0; Short term 12.8; Inflation-indexed 23.6; Domestic-Currency-Denominated Not indexed 60.1; Indexed to Domestic interest rate 3.5; Foreign-Currency-Denominated 0.0.
    - India: Long term 27.0; Short term 8.1; Inflation-indexed 18.4; Domestic-Currency-Denominated Not indexed 0.0; Indexed to Domestic interest rate 0.0; Foreign-Currency-Denominated 0.0.
- Interpretation:
  - Emerging markets on average had almost one half of central government debt denominated in foreign currency in 2001, compared with approximately 5 percent in advanced countries.
  - Long-term domestic-currency debt represented three quarters of total debt in advanced countries, whereas it amounted to one third in emerging markets.
  - Considerable cross-country variation exists; Asian emerging markets typically have a higher share of long-term local currency debt than Latin American emerging markets.
- Institutional determinants:
  - Lack of credibility of monetary and fiscal policies, institutional quality, and size of domestic investor base influence ability to issue long-term domestic-currency bonds.
  - Some countries (Chile, Israel, Mexico, Poland) improved debt structures via inflation stabilization, central bank reforms, pension reforms, and issuance of inflation-indexed bonds.

### VIII. Advantages of growth-indexed bonds and obstacles
- Main advantages:
  - Lower likelihood of default and costly debt crises.
  - Less procyclical fiscal policy.
  - Greater international risk sharing: investors share in good and bad growth outcomes.
- Historical and empirical precedents:
  - Value Recovery Rights in restructurings for Bosnia, Bulgaria, and Costa Rica.
  - GDP warrants attached to bonds in Argentina’s early-2005 debt exchange.
- Principal obstacles:
  - Financial-innovation barriers: first-mover costs, coordination problems, concerns about secondary-market liquidity.
  - Political-economy concerns: governments with limited horizons may undervalue future benefits; potential incentive to misreport growth.
  - Investor concerns from surveys: verifiability of GDP data, secondary-market liquidity, analytical difficulty in pricing.
- Considerations addressing data-misreporting:
  - Political incentive: high growth, not low growth, aids reelection.
  - Domestic holders provide a lobby against underreporting growth.
  - Markets adapt to misreporting; private alternatives to official statistics have emerged historically.
  - Repeated cheating harms reputation and future market access.
  - Policy measures: greater independence for statistical agencies; outside monitoring of data quality in accordance with international standards.
- Authors’ view: obstacles are surmountable; development of a simple, market-acceptable pricing method could improve acceptance.

### IX. Policy-relevant conclusions
- Sound macroeconomic policies and institutions are key to improving debt structures; financial innovation alone is not a quick fix.
- Growth-indexed bonds can play a helpful role as part of a broader package to improve debt structures and reduce vulnerabilities and procyclicality in fiscal policies.
- Pricing growth-indexed bonds is feasible: a method using information extracted from observed plain-vanilla bond prices and a Monte Carlo exercise shows growth-indexed bonds are not much more difficult to price than plain-vanilla bonds.
- Issuance of growth-indexed bonds generates externalities: by lowering default probabilities they also lower required coupons on standard bonds.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2005/_wp05216.pdf*

### 1. Currency Composition of Central Government Debt in 2001 ..........................................6

### _wp05216 - 1. Currency Composition of Central Government Debt in 2001 ..........................................6

### 1. Currency Composition of Central Government Debt in 2001
- Section title: "Currency Composition of Central Government Debt in 2001"
- Page reference: 6
- Year referenced: 2001

### 2. Structure of Domestically Issued Government Bonds at End-2001
- Section title: "Structure of Domestically Issued Government Bonds at End-2001"
- Page reference: 8
- Date precision: end-2001

### 3. Default Rates and Expected Discounted Payoffs (25 Percent Recovery, 27.8 Percent Baseline Default Frequency)
- Section title: "Default Rates and Expected Discounted Payoffs Assuming a 25 Percent  
               Recovery Value and a 27.8 Percent Baseline Default Frequency"
- Page reference: 20
- Key numeric parameters preserved exactly:
  - Recovery Value: 25 Percent
  - Baseline Default Frequency: 27.8 Percent

### 4. Default Rates and Expected Discounted Payoffs (50 Percent Recovery, 35.9 Percent Baseline Default Frequency)
- Section title: "Default Rates and Expected Discounted Payoffs Assuming a 50 Percent 
               Recovery Value and a 35.9 Percent Baseline Default Frequency"
- Page reference: 20
- Key numeric parameters preserved exactly:
  - Recovery Value: 50 Percent
  - Baseline Default Frequency: 35.9 Percent

### 5. Default Rates and Expected Loss if Growth Is on Average 1 percent Lower than Initially Expected
- Section title: "Default Rates and Expected Loss if Growth is on Average 1 percent Lower than 
               Initially Expected, Under the Assumptions and Prices of Table 3"
- Page reference: 21
- Key numeric parameter preserved exactly:
  - Growth shortfall: 1 percent
- Reference to modeling inputs: "Under the Assumptions and Prices of Table 3"

### 6. Default Rates and Expected Loss if Growth Is 1½ Times as Volatile as Initially Expected
- Section title: "Default Rates and Expected Loss if Growth is 1½ Times as Volatile as 
               Initially Expected, Under the Assumptions and Prices of Table 3"
- Page reference: 21
- Key numeric parameter preserved exactly:
  - Volatility multiplier: 1½ Times
- Reference to modeling inputs: "Under the Assumptions and Prices of Table 3"

### Figure
- Figure title: "Figure 1. Distribution of Payoffs for Plain-Vanilla and Growth-Indexed Bonds"
- Page reference: 18

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2005/_wp05216.pdf*

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

### References

### I. Introduction — purpose and approach
- Focus: assess pricing of growth-indexed bonds as a way to reduce procyclicality and default risk associated with existing debt structures in emerging market countries.
- Key motivations and observations:
  - Emerging markets often rely on short-term debt or foreign currency debt, making debt servicing more costly in downturns.
  - Several studies find fiscal policies more procyclical in emerging markets than in advanced countries (Gavin and Perotti, 1997; International Monetary Fund, 2003, Ch. 3; Talvi and Vegh, forthcoming).
  - Growth-indexed bonds pay returns indexed to real GDP growth, increasing payments in strong-growth years and lowering them in weak-growth years, acting as automatic stabilizers.
- Scope and limits:
  - Paper focuses on parameter values relevant for emerging markets, where macroeconomic fluctuations and perceived default risk (bond spreads) are larger.
  - Abstracts from subjective premia (novelty, liquidity) and concentrates on estimating a model-based “theoretical price” for GDP-indexed bonds.

### II. Pricing methodology — core steps and assumptions
- Broad strategy:
  - Assume risk-neutral investors require, on average, the same return as on U.S. Treasury bonds of similar duration.
  - Extract combinations of the probability of default and recovery rates from observed spreads between emerging-market and U.S. bonds to minimize ad hoc assumptions.
  - Estimate the joint distribution of real GDP growth, the primary balance, and the real exchange rate from past data.
- Simulation and calibration:
  - Simulate 250,000 paths for economic variables and the debt/GDP ratio.
  - Extract a default trigger level for the debt/GDP ratio and a recovery rate that match expected repayments implied by observed spreads.
  - Defaults may occur from an initially moderate debt/GDP ratio if large adverse shocks (e.g., sharp real depreciation) materialize.
  - Compute payoffs for both growth-indexed bonds and plain-vanilla bonds across simulated paths and across different shares of growth-indexed debt in total debt.
- Key pricing contention:
  - The hardest part is pricing plain-vanilla bonds (estimating default likelihood and implied loss); using a consistent method for both plain-vanilla and growth-indexed bonds makes the additional complexity minimal.

### III. Numerical findings and implications
- Illustrative numerical outcomes reported:
  - Growth-indexed debt can lower the default frequency by 1/4 to 1/3 of its initial level in the simulations.
  - When the share of indexed debt rises, both growth-indexed bonds and plain-vanilla bonds become less sensitive to surprises in the growth rate and to growth volatility because indexation attenuates the effect of surprises on default probability.
- Observations on investor concerns:
  - Surveys of investors highlighted three frequent concerns: verifiability of GDP data, need for liquidity in secondary markets, and difficulty in pricing growth-indexed bonds.

### IV. Existing debt structures — stylized facts (selected statistics)
- Currency composition (Table 1, central government debt, 2001, in percent of total):
  - Emerging market countries: Foreign-Currency Debt 48.0; Long-Term Domestic-Currency Debt 32.4; Total Debt as a Ratio to GDP 50.4.
  - Advanced Economies: Foreign-Currency Debt 5.6; Long-Term Domestic-Currency Debt 75.9; Total Debt as a Ratio to GDP 51.8.
  - Regional and country examples (selected):
    - Latin America: Foreign-Currency Debt 67.9; Long-Term Domestic-Currency Debt 15.2; Total Debt/GDP 37.0.
    - Asia: Foreign-Currency Debt 28.5; Long-Term Domestic-Currency Debt 44.2; Total Debt/GDP 62.8.
    - China: Foreign-Currency Debt 17.7; Long-Term Domestic-Currency Debt 82.3; Total Debt/GDP 24.0.
    - India: Foreign-Currency Debt 14.5; Long-Term Domestic-Currency Debt 69.9; Total Debt/GDP 65.1.
    - Russia: Foreign-Currency Debt 90.3; Long-Term Domestic-Currency Debt ...; Total Debt/GDP 50.0.
- Structure of domestically issued government bonds (Table 2, end-2001, in percent of total):
  - Emerging market countries (aggregate): Long term 28.24; Short term 41.51; Inflation 18.62; Domestic-Currency-Denominated Not indexed 26.47; Indexed to Domestic interest rate 7.2; Foreign-Currency-Denominated 6.3 (table formatting retained as in source).
  - Country examples (selected):
    - Brazil: Long term 52.1; Short term 9.5; Inflation-indexed 0.0; Domestic-Currency-Denominated Not indexed 53.0; Indexed to Domestic interest rate 7.0; Foreign-Currency-Denominated 30.5.
    - Mexico: Long term 11.0; Short term 12.8; Inflation-indexed 23.6; Domestic-Currency-Denominated Not indexed 60.1; Indexed to Domestic interest rate 3.5; Foreign-Currency-Denominated 0.0.
    - India: Long term 27.0; Short term 8.1; Inflation-indexed 18.4; Domestic-Currency-Denominated Not indexed 0.0; Indexed to Domestic interest rate 0.0; Foreign-Currency-Denominated 0.0.
- Interpretation:
  - Emerging markets on average had almost one half of central government debt denominated in foreign currency in 2001, compared with approximately 5 percent in advanced countries.
  - Long-term domestic-currency debt represented three quarters of total debt in advanced countries, whereas it amounted to one third in emerging markets.
  - Considerable cross-country variation exists; Asian emerging markets typically have a higher share of long-term local currency debt than Latin American emerging markets.
- Institutional and structural determinants:
  - Lack of credibility of monetary and fiscal policies, institutional quality (political stability, rule of law), and size of domestic investor base influence ability to issue long-term domestic-currency bonds.
  - Some countries (Chile, Israel, Mexico, Poland) improved debt structures via inflation stabilization, central bank reforms, pension reforms, and issuance of inflation-indexed bonds.

### V. Advantages of growth-indexed bonds and obstacles
- Main advantages:
  - Lower likelihood of default and costly debt crises.
  - Less procyclical fiscal policy.
  - Greater international risk sharing: investors share in good and bad growth outcomes.
- Historical and empirical precedents:
  - Value Recovery Rights in restructurings for Bosnia, Bulgaria, and Costa Rica.
  - GDP warrants attached to bonds in Argentina’s early-2005 debt exchange.
  - Small markets for options on economic statistics (e.g., U.S. nonfarm payroll).
- Principal obstacles:
  - General financial-innovation barriers: first-mover costs, coordination problems, concerns about secondary-market liquidity.
  - Political-economy concerns: governments with limited horizons may undervalue future benefits; potential incentive to misreport growth.
  - Specific investor concerns identified in surveys: verifiability of GDP data, secondary-market liquidity, analytical difficulty in pricing.
- Considerations addressing data-misreporting concern:
  - Five considerations discussed:
    - Political incentive: high growth, not low growth, aids reelection.
    - Domestic holders of bonds provide a lobby against underreporting growth.
    - Markets adapt to misreporting; private alternatives to official statistics have emerged in the past.
    - Repeated cheating harms reputation and future market access, potentially costing more than short-term savings.
    - Policy measures can improve data reliability: greater independence for statistical agencies; outside monitoring of data quality in accordance with international standards.
- Conclusion on obstacles:
  - Authors view obstacles as surmountable; development of a simple, market-acceptable pricing method could improve acceptance.

### VI. Questions on model development and market adoption
- Raised issues:
  - Is a pricing model required for a market to operate?
  - Should academics or practitioners develop the model?
  - Should a model be developed before issuance, or will markets create it when instruments are issued?
- Practical stance taken:
  - Given investor concerns about pricing complexity, authors develop a simple pricing method grounded in standard market practice and consistent treatment of plain-vanilla and growth-indexed bonds.

*Source: IMF working paper content (References section and main text excerpts).*

### 1.     Is a pricing model needed for a financial market to operate? It seems clear that no formal

### Is a pricing model needed for a financial market to operate?

### Need for a pricing model
- No formal pricing model, and certainly not a generally accepted model, is a prerequisite for a financial market to operate. Historical example: Joseph de la Vega’s 1688 Confusion de Confusiones documents thriving trading in stocks, futures, and options well before formal stochastic pricing theories.
- A standard pricing model or methodology can nonetheless help start up a market.

### Who should develop a pricing model
- Financial market participants are best placed to price complicated instruments, but market participants do not always develop pricing models for new instruments.
- Academics can and have produced breakthrough pricing formulas (example: Black and Scholes) that expanded markets after the instruments were already in use.

### Timing and market response
- Market participants rapidly develop pricing methods when a new instrument becomes available (example: methods developed quickly for Argentine GDP warrants once inclusion in the debt restructuring became clear).
- Improvements in pricing techniques can further increase investor interest even after initial market pricing methods are developed.

### Distinctive features of the authors' pricing method
- Explicit modeling of conditions for default (endogenous changes in default probabilities) rather than incorporating default indirectly through observed spreads.
- Tracing implications of introducing growth–indexed bonds for pricing plain-vanilla bonds.

### Pricing growth-indexed bonds — approach overview
- Use Monte Carlo simulations of debt dynamics and resulting payoffs over a ten year horizon.
- Simulations recover a default-trigger rule and estimate effects of different debt compositions (shares of indexed versus non-indexed debt) on the probability distribution of defaults.
- Investors are assumed risk-neutral and value bonds at the expected net present value of payment streams.
- The illustrative example uses a hypothetical country "Emergingland" calibrated on averages for Brazil, Mexico and Turkey.

### Debt dynamics (model specification)
- The debt-to-GDP ratio evolves according to a standard recursive equation incorporating:
  - share of dollar debt (),
  - nominal exchange rate dynamics (e_t/e_{t−1}),
  - dollar and local interest rates (i_dollar and i_local),
  - real GDP growth (g_t),
  - change in GDP deflator (),
  - primary balance as percent of GDP (pb_t).
- Real exchange rate change denoted _t with 1+_t = (e_t/e_{t−1}) (1+* )/(1+), where * is U.S. GDP deflator change.
- For simplicity the real interest rate is assumed the same for dollar and local currency debt:
  (1+i_dollar)/(1+*) = (1+i_local)/(1+).
- Two debt types considered:
  - Plain-vanilla debt paying fixed interest i_pv.
  - Growth-indexed debt paying interest i_ind where i_ind = max(0, i_pv + g_t − g), with g the contractual growth threshold; principal is not indexed.

### Calibration for Emergingland
- Initial debt to GDP ratio set to 60 percent.
- Share  of dollar debt set to ½.
- Baseline interest rate i_pv set to 6¾ percent based on average EMBI spreads.
- Future payments discounted at the 10 year risk-free rate set to 4.0 percent (10-year U.S. yield).
- U.S. GDP deflator * set to 2.0 percent.
- Joint-normal distribution for {g_t, _t, pb_t} based on historical sample for Brazil, Mexico, Turkey in 1981-2004.
- Expected values set to historical sample averages: g_t = 3.0% and pb_t = 2.1%.
- Expected real depreciation set to zero.
- Standard deviations (in percentage points): σ_g = 3.8, σ_ε = 16.1 and σ_pb = 3.3.
- Correlations: ρ_{g, ε} = −0.63, ρ_{g,pb} = −0.34 and ρ_{ε,pb} = 0.16.
- Growth-indexed bond contracts specify g at 3 percent.

### Estimating default probabilities and recovery scenarios
- Parameters for expected probability of default and recovery extracted by imposing a no-arbitrage constraint equating expected return on Emergingland’s plain-vanilla bond to the return on a U.S. bond with similar duration (scenario where all debt is plain-vanilla).
- Two scenarios considered:
  - Recovery rate = 25 percent → resulting probability of default within 10 years = 27.8 percent.
  - Recovery rate = 50 percent → resulting probability of default within 10 years = 35.9 percent.
- Default occurs as soon as the simulated debt/GDP ratio crosses a trigger level chosen to make plain-vanilla bonds trade at par in the baseline (all plain-vanilla) scenario.
- Simulations: 250,000 joint paths over a 10-year horizon.

### Simulation and pricing experiments
- Three debt compositions simulated:
  (i) virtually all debt plain-vanilla;
  (ii) half plain-vanilla, half GDP-indexed;
  (iii) virtually all GDP-indexed.
- i_pv and the contracted i_ind formula are held constant across scenarios.
- Default trigger held constant at the calibration value.
- Instruments priced:
  - 10-year plain-vanilla bond with coupon equal to 6¾ percent.
  - Indexed bond with coupon equal to max(0, 6¾% + 3% − g_t) and principal not indexed.
- Results (selected findings):
  - In the baseline scenario (recovery rate = 25 percent, default probability 27.8 percent), the plain-vanilla bond pays full amount 72.2 percent of the time; in 27.8 percent of cases it pays only coupons up to default plus the recovery rate on principal, producing 10 discrete payoff spikes corresponding to defaults in each of the 10 years.
  - Growth-indexed bond payoff distribution is smoother: bell-shaped in non-default cases, value depends on realized growth even in default cases.
  - Moving from no indexation to indexation of all debt reduces the default probability from 28 percent to 19 percent — a reduction of the default risk by one third (baseline with recovery = 25 percent and default = 27.8 percent).
  - For the scenario with recovery rate = 50 percent and baseline default = 35.9 percent, the benefits of indexation are slightly smaller because higher recovery makes defaults less costly to investors.
  - Estimated savings on borrowing costs exceed 100 basis points when comparing coupon rates necessary to achieve baseline expected returns across debt compositions.
  - Indexed bond in the example is always worth more in expectation than its plain-vanilla counterpart because asymmetric indexation (coupon bounded below at zero) raises average coupon payments.
  - Sensitivity test: a reduction in the mean growth rate by one percentage point reduces bond prices more for growth-indexed bonds than for plain-vanilla bonds (direct effect on indexation). However, higher shares of indexed debt attenuate the increase in default frequency, reducing losses on both plain-vanilla and indexed bonds.
  - Reported loss examples: increasing the share of growth-indexed bonds from zero to 100 percent reduces the loss from 6.74 percent to 0.81 percent for plain-vanilla debt and from   12.42 percent to

*Italic: IMF working paper excerpt (pricing growth-indexed bonds; simulation calibration and results).*

### 7.09 percent for growth-indexed bonds. (In fact, the expected loss on a plain-vanilla bond when

### _wp05216 - 7.09 percent for growth-indexed bonds. (In fact, the expected loss on a plain-vanilla bond when

### Simulation results: default frequencies, bond prices, and required coupons (Tables 3 and 4)
- Assumptions in Table 3: 25 Percent Recovery Value and a 27.8 Percent Baseline Default Frequency (Implicit Default Trigger is Debt/GDP > 73.2 percent).
  - Share of Indexed Debt in Total Debt (in percent): .0001, 50, 99.9999
  - Default Frequency (in percent) (within 10 years): 27.8, 23.0, 18.9
  - Price of Indexed Bond with Coupon=Max(6¾ % + growth – 3 %,0): 101.3, 105.2, 108.4
  - Price of Plain-Vanilla Bond with 6 ¾ % Coupon: 100.0, 104.1, 107.6
  - Coupon (in percentage points) Required for Plain-Vanilla Bond to Sell at 100: 6 ¾; 6 ¾  - 0.58; 6 ¾  - 1.03

- Assumptions in Table 4: 50 Percent Recovery Value and a 35.9 Percent Baseline Default Frequency (Implicit Default Trigger is Debt/GDP > 69.5 percent).
  - Share of Indexed Debt in Total Debt (in percent): .0001, 50, 99.9999
  - Default Frequency (in percent) (within 10 years): 35.9, 31.4, 26.9
  - Price of Indexed Bond with Coupon=Max(6¾ % + growth – 3 %,0): 101.5, 104.3, 107.0
  - Price of Plain-Vanilla Bond with 6 ¾ % Coupon: 100.0, 103.0, 106.0
  - Coupon (in percentage points) Required for Plain-Vanilla Bond to Sell at 100: 6 ¾; 6 ¾ - 0.48; 6 ¾ - 0.90

### Sensitivity exercises: growth shocks and volatility shocks (Tables 5 and 6)
- Table 5 — Default Rates and Expected Loss if Growth is on Average 1 Percent Lower than Initially Expected (under assumptions and prices of Table 3):
  - Share of Indexed Debt in Total Debt (in percent): .0001, 50, 99.9999
  - Default Frequency (in percent) (within 10 years): 36.27, 27.79, 19.98
  - Expected Loss on Indexed Bond with Coupon=Max(6¾ % + growth – 3 %, 0): 12.42, 9.61, 7.09
  - Expected Loss on Plain-Vanilla Bond with 6 ¾ % Coupon: 6.74, 3.58, 0.81

- Table 6 — Default Rates and Expected Loss if Growth is 1½ Times as Volatile as Initially Expected (under assumptions and prices of Table 3):
  - Share of Indexed Debt in Total Debt (in percent): .0001, 50, 99.9999
  - Default Frequency (within 10 years) (in percent): 33.47, 27.97, 22.93
  - Expected Loss on Indexed Bond with Coupon=Max(6 ¾ % + growth – 3 %, 0): 2.65, 1.84, 0.94
  - Expected Loss on Plain-Vanilla Bond with 6 ¾ % Coupon: 5.04, 4.04, 3.06

### Key findings and interpretation (from Conclusions)
- Growth-indexation attenuates the effect of growth surprises on the probability of default.
- The larger the share of indexed debt in total debt, the lower the loss implied by growth shocks, for both plain-vanilla and growth-indexed bonds.
- When growth becomes more volatile (1½ times as expected), the shock can result in a lower expected loss for indexed bonds than for plain-vanilla bonds because coupon payments are bound at zero and average coupon can increase with volatility.
- Large issuance of growth-indexed bonds reduces the probability of default, thereby reducing the required coupon on standard bonds.
- Growth-indexed bonds reduce the sensitivity of bond payoffs to surprises in growth averages or volatilities by reducing the extent to which such surprises affect default probabilities.

### Policy-relevant conclusions
- Sound macroeconomic policies and institutions are key to improving debt structures; financial innovation alone is not a quick fix.
- Growth-indexed bonds can play a helpful role as part of a broader package to improve debt structures and reduce vulnerabilities and the degree of procyclicality in fiscal policies.
- Pricing growth-indexed bonds is feasible: a method using information extracted from observed plain-vanilla bond prices and a Monte Carlo exercise shows growth-indexed bonds are not much more difficult to price than plain-vanilla bonds.
- Issuance of growth-indexed bonds generates externalities: by lowering default probabilities they also lower required coupons on standard bonds.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2005/_wp05216.pdf*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2005/_wp05216.pdf_