## _wp0685

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---

### I. Introduction and Purpose
- Develops a pricing model for GDP-linked warrants (GLWs) and evaluates numerically the risk profile of a specific GLW design from investor and sovereign issuer perspectives.
- Focus on cash-flow considerations: payments on GLWs are designed to be proportional to the tax revenues available in each period so that payments are always fully financed.
- Paper structure (as presented): history and characteristics of GLWs; analytical evaluation of a specific GLW design (price estimation and comparison of debt payments with resources); conclusion.

### II. GLW Design Principles and Practical Issues
- Objectives of GLW design proposed to avoid prior problems:
  - Maturity set to 20 years (t = 0,1,...,21 with a one-year payment lag).
  - Triggers based on real GDP in domestic currency and require (a) actual real GDP levels exceed potential levels determined upon issuance, and (b) actual real GDP growth rates are positive.
  - Reference date: the year preceding the calculation date (one-year lag reflecting GDP calculation lag).
  - Payments capped (Cap C) to prevent extreme growth surprises producing unserviceable cash flows; cap choice example: C = 3 percent of trend GDP in prior year.
  - Payment amount equals difference between actual and potential real GDP growth rates multiplied by nominal GDP level in the prior period.
- Practical challenges and historical experience:
  - GDP measurement complexity, reporting incentives, and data ambiguities have undermined past GLW issuances (examples: Bulgaria, Bosnia and Herzegovina).
  - Successful nontradable implementations include Singapore NSS and ERS (dividends: 3 percent plus real GDP growth with guarantee of at least 3 percent; dividend paid every March 1; NSS: 2002–2007; ERS: 2004–2008; 2005 dividend rate 11.4 percent).
  - Argentina issued detachable 30-year GLWs in June 2005 with annual payment rule: 5 percent of excess cash flows when (a) actual GDP (constant peso terms) exceeds threshold GDP and (b) annual growth rates exceed 3 percent; first payment scheduled for December 15, 2006; value calculated on November 1; warrants detachable and traded separately since end-November 2005.

### III. Formal Model, Equations, and Variable Definitions
- Key equations (symbol usage preserved):
  - Excess cash flows: ECF_{t,i} = (g_{i,t} - g_{i,t}^N) * v_{i,t-1} * (1 + π_{i,t}) ∏_{j=0}^{t-1} (1 + ε_{i,j})
  - Warrant payments: W_{t,i} = min(C * Q_{t-1,i}, ECF_{t,i})
  - Present value of warrant payments: WPV_t = Σ_{i=0}^{t-1} W_{i,t} / ∏_{i=0}^{t-1} (1 + R_{i,t})
  - Forward exchange rates (interest rate parity): E_{t} = E_0 ∏_{i=0}^{t-1} (1 + h_{i,t})
- Variable notation (selected, preserved):
  - v_{i,t}: realized nominal GDP; Q_{t,i}: nominal trend GDP; v_0: initial value of real GDP; g_i: average growth rates; σ: volatility of growth rates; ε_{i,t}: random shocks to growth; π_{i,t}: realized inflation rates; W_{i,t}: warrant payments; C: cap rates; R_t: issuer domestic real interest rate; E_0: initial exchange rate; h_{i,t}: realized rates of change in exchange rates.

### IV. Baseline Parameter Values (as used in simulations)
- g = 3.1 percent
- σ = 2.2 percent
- κ = 20 percent
- v_0 = 100
- E_0 = 1
- υ = 0
- μ = 0
- s = 0
- ψ = 0
- δ = 0
- r (t_r = r) = 3 percent
- R (t_R = R) = 8 percent
- χ (t_χ = χ) = 14 percent
- λ (t_λ = λ) = 2 percent
- γ (t_γ = γ) fall from 6.5 to 4 percent over periods
- Cap C example: 3 percent of trend GDP in the year before (Q_{t-1})

### V. Key Quantitative Results — Prices and WI Ratios
- Baseline present-value estimates:
  - Domestic-currency price (based on equation (3)): 6.36 (initial GDP value = 100).
  - Foreign-currency price (using equations (3) and (4)): 3.06 (initial value = 100).
  - If forward exchange rates assumed constant at initial exchange rate of 1, estimated foreign-currency price = 5.58.
- Table 1 (Growth surprises scenario with g = 5 percent) — Price and WI Ratio at maturity by cap:
  - Cap ∞: Price 29.2; WI Ratio 1.23
  - Cap 5 percent: Price 26.3; WI Ratio 1.00
  - Cap 4 percent: Price 24.0; WI Ratio 0.89
  - Cap 3 percent: Price 20.4; WI Ratio 0.74
  - Cap 2 percent: Price 15.2; WI Ratio 0.54
  - Cap 1 percent: Price 8.3; WI Ratio 0.29
- Baseline scenario WI Ratio (Table 2) — selected period values (WI Ratio preserved exactly):
  - Period 2: 0.42
  - Period 3: 0.37
  - Period 4: 0.35
  - Period 5: 0.33
  - Period 6: 0.36
  - Period 7: 0.37
  - Period 8: 0.36
  - Period 9: 0.35
  - Period 10: 0.40
  - Period 11: 0.39
  - Period 12: 0.36
  - Period 13: 0.33
  - Period 14: 0.37
  - Period 15: 0.31
  - Period 16: 0.35
  - Period 17: 0.37
  - Period 18: 0.35
  - Period 19: 0.36
  - Period 20: 0.35
  - Period 21: 0.33
- Probability that WI Ratio exceeds one (prob( WI > 1 ), Table 2 row representing Eq. (5); values preserved):
  - Period 2: 20.00
  - Period 3: 17.80
  - Period 4: 16.80
  - Period 5: 15.10
  - Period 6: 18.40
  - Period 7: 19.00
  - Period 8: 16.80
  - Period 9: 15.90
  - Period 10: 18.20
  - Period 11: 18.20
  - Period 12: 16.20
  - Period 13: 15.50
  - Period 14: 17.50
  - Period 15: 15.30
  - Period 16: 15.00
  - Period 17: 16.80
  - Period 18: 15.80
  - Period 19: 16.50
  - Period 20: 15.90
  - Period 21: 15.10
- Expected average cash shortfalls given WI Ratio > 1 (Eq. (6); current terms, billions of domestic currency units; Table 2 row representing Eq. (6) — values preserved):
  - Period 2: -0.87
  - Period 3: -1.12
  - Period 4: -1.30
  - Period 5: -1.41
  - Period 6: -1.39
  - Period 7: -1.64
  - Period 8: -1.91
  - Period 9: -1.95
  - Period 10: -2.30
  - Period 11: -2.31
  - Period 12: -2.55
  - Period 13: -2.46
  - Period 14: -2.86
  - Period 15: -2.88
  - Period 16: -3.67
  - Period 17: -3.63
  - Period 18: -3.89
  - Period 19: -4.01
  - Period 20: -4.44
  - Period 21: -4.21
- Interpretation of baseline results:
  - WI Ratio remains within 0.31 and 0.42 in baseline; initial higher value 0.42 falls to 0.33 toward maturity.
  - Probability that WI Ratio exceeds one remains moderate at around 15 to 20 percent across periods.
  - Expected cash shortfalls conditional on WI Ratio > 1 increase from 0.87 in period 2 to 4.21 in period 21 (current-value billions), indicating substantial fiscal pressure in tail events.

### VI. Sensitivity Analyses — Price and WI Ratio Responses
- Case I — changing g and σ (Table 3 domestic price matrix; g columns: 1.5 percent | 3.1 percent | 5.0 percent; σ rows):
  - σ = 1.0 percent: 0.03 | 3.29 | 23.24
  - σ = 2.2 percent: 0.59 | 6.36 | 20.35
  - σ = 10.0 percent: 6.05 | 9.26 | 14.03
- Observations (Case I):
  - Holding volatility constant, price increases significantly when growth rates outpace threshold growth rates of 3.1 percent.
  - When growth rates held below (above) threshold of 3.1 percent, price increases (decreases) with volatility.
  - Baseline price 6.36 rises by 1.5 times when volatility changes from 2.2 percent to 10 percent.
  - When g increases to 5 percent from 3.1 percent with volatility held at baseline, price increases by a factor of three.
- Case II — changing s and υ (Table 4 price matrix; s columns = -10 percent | -5 percent | 0 percent | 5 percent | 10 percent; υ rows = 0 percent | 10 percent | 20 percent):
  - υ = 0 percent: 7.03 | 6.68 | 6.36 | 6.07 | 5.81
  - υ = 10 percent: 7.13 | 6.76 | 6.43 | 6.13 | 5.86
  - υ = 20 percent: 7.43 | 7.01 | 6.65 | 6.32 | 6.02
- Observations (Case II):
  - Parallel increases in inflation reduce the real value of payments modestly (effect of observation-payment lag).
  - Random shocks increase price as volatility of random shocks increases, but impact is limited.
- Case III — changing δ (pass-through of growth shocks to inflation; Table 5 prices for δ = -3, -2, -1, 0, 1, 2, 3):
  - Prices: 6.4 | 6.4 | 6.3 | 6.3 | 6.3 | 6.3 | 6.38
- Observation (Case III):
  - Changes in δ have only a limited impact on the price of GLWs; baseline δ = 0 yields one of the lowest estimated prices.
- Case IV — changing ψ (parallel shifts in rates of change in exchange rates; Table 6 foreign prices):
  - ψ = -10 percent | 0 percent | 10 percent yields prices: 6.79 | 3.06 | 1.79
- Observation (Case IV):
  - If forward exchange rates depreciate by 10 percent relative to interest rate parity, price would be halved; if they appreciate by 10 percent, price would more than triple.
- Case V — changing μ and υ (Table 7 foreign price matrix; μ columns = 0 percent | 10 percent | 20 percent; υ rows = 0 percent | 10 percent | 20 percent):
  - υ = 0 percent: 3.06 | 3.28 | 4.1
  - υ = 10 percent: 3.09 | 3.33 | 4.18
  - υ = 20 percent: 3.2 | 3.47 | 4.46
- Observations (Case V):
  - Shocks to inflation rates have limited impact: raising υ to 20 percent while μ = 0 percent changes price from 3.06 to 3.20.
  - Shocks to exchange-rate change rates have larger effects: raising μ from 0 percent to 20 percent (with υ = 0) increases price from 3.06 to 4.10.
  - Foreign investors benefit from random shocks through both inflation rates and exchange-rate changes.

### VII. Indexation Premium and Investor Compensation
- CAPM/CCAPM approach for foreign investors:
  - Estimated β (CAPM) for foreign investors ranges from 0 to 0.06.
  - CCAPM estimations confirm low β findings.
- Conclusion on indexation premium:
  - Insurance/indexation premium on GLWs would likely be low; investors would require little additional value from the insurance premium, so stand-alone price is a reasonable approximation of market price.

### VIII. Historical Path Illustration and Comparative PVs
- Three hypothetical real GDP level paths compared (initial GDP value = 100):
  - Case (a) 1974–94 historical path: present-value warrant payments = 7.93
  - Baseline model prediction (trend g = 3.1 percent): 6.36
  - Case (b) 1984–2004 historical path: 4.54
- Interpretation:
  - Baseline prediction (6.36) lies between historical extremes 7.93 and 4.54, illustrating model responsiveness to historical growth trajectories.

### IX. Main Policy and Design Implications
- Caps are effective: Cap C (example 3 percent of trend GDP) contains extreme payments and limits WI Ratio under high-growth outcomes.
- Trade-offs:
  - GLWs share growth risk between issuer and creditors but may introduce moral hazard concerns; GDP indexing mitigates some moral hazard because GDP aggregates many agents.
  - Accurate, unambiguous GDP measurement and transparent reference definitions are critical for credible GLWs.
  - Callable features reduce insurance value for investors (as evidenced by Bulgaria case).
- Operational recommendations embedded in design:
  - Use long maturities (example: 20 years) to smooth payments over cycles.
  - Define triggers in real domestic-currency GDP and require recovery above long-term trend plus positive growth before payments.
  - Include caps and reference lags to align payment feasibility with fiscal capacity.
  - Consider foreign-currency pricing sensitivity to forward exchange-rate assumptions; deviations from interest-rate parity can materially alter foreign-currency prices.

### X. Model Extensions and Enrichment Suggestions
- Allow tax revenue-to-GDP ratio κ to be state-dependent and model tax revenue storability.
- Treat deterministic variables such as interest rates as stochastic to reflect state-dependent default risk.
- Assess issuer repayment capacity with alternative criteria beyond WI Ratio thresholds.

*Source: _wp0685 (PDF chapter/section).*

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

### _wp0685 - References..............................................................................................................

### Figures
- 1. Growth Rates of Real GDP, Average of Main EMBI Countries, 1974–2004 .................... 11
- 2. Hypothetical Historical Paths to Real GDP Levels ............................................................ 19

### Tables
- 1. Prices of GLWs and WI Ratios at Maturity Under Growth Surprises (g=5 percent) ......... 12
- 2. WI Ratios and Other Variables—Baseline Scenario .......................................................... 13
- 3. Price Sensitivities—Case I (changing g and σ)................................................................... 16
- 4. Price Sensitivities—Case II (changing s and υ).................................................................. 16
- 5. Price Sensitivities—Case III (changing δ).......................................................................... 16
- 6. Price Sensitivities—Case IV (changing ψ) ......................................................................... 17
- 7. Price Sensitivities—Case V (changing μ and υ) ................................................................. 18

### Appendix
- Appendix (no items listed on this page)

*Source: _wp0685 - References (PDF chapter/section).*

### 1.      Theoretical      Model .........................................................................................

### 1.      Theoretical      Model

### I. INTRODUCTION
- Recent research studies mechanisms to improve sovereign debt sustainability during downturns by sharing growth risk between issuer and creditors via GDP-linked instruments (GLWs).
- This paper:
  - Develops a pricing model for GDP-linked warrants (GLWs).
  - Evaluates numerically the risk profile of a specific GLW design from both investor and sovereign issuer perspectives.
  - Focuses on cash-flow considerations: payments on GLWs are designed to be proportional to the tax revenues available in each period so that payments are always fully financed.
- Paper structure:
  - Section II: history and characteristics of GLWs.
  - Section III: analytical evaluation of a specific GLW design (price estimation and comparison of debt payments with resources).
  - Section IV: conclusion.

### II. HISTORY AND CHARACTERISTICS
- Origin and motives:
  - GLWs proposed to share growth risk; link debt payments to GDP to cushion negative growth shocks (Froot, Scharfstein, and Stein (1989); Shiller (1993); Obstfeld and Peri (1989); Borensztein and Mauro (2002)).
  - GLWs expected to reduce incidence of defaults by restricting variation of coupon payments to revenues; potential moral hazard concerns noted but mitigated because GDP aggregates many agents.
  - Early proposals suggested indexing to exogenous variables to avoid moral hazard (Bailey (1983); Krugman (1988)); counterarguments support GDP indexing under asymmetric information.
- Practical challenges:
  - GDP measurement is complex and error-prone; ambiguity in reference definitions can undermine triggers.
  - Issuers may have incentives to misreport GDP; inclusion of population statistics increases uncertainty.
  - Callable features can be inconsistent with GDP-linked objectives.
- Prior pricing literature:
  - Kruse, Meitner, and Schröder (2005): Black-Scholes approach; differences from vanilla bonds may arise from liquidity or uncertainty premia.
  - Chamon and Mauro (2005): increasing share of GDP-linked bonds raises price of vanilla bonds and reduces default frequency.
  - This paper adds: detailed simulations of price sensitivity and explicit incorporation of issuer repayment capacity in GLW design.
- Recent experiences:
  - GLWs were issued in the Brady restructurings by Bulgaria, Bosnia and Herzegovina, and Costa Rica (only sovereign funding GLWs); limited market development partly due to design/data problems.
  - Singapore issued nontradable, GDP-linked shares for low-income citizens (New Singapore Shares (NSS) and Economic Restructuring Shares (ERS)); dividends paid as bonus shares at rate of 3 percent plus real GDP growth with a guarantee of at least 3 percent; dividends paid every March 1 (NSS: 2002–2007; ERS: 2004–2008); in 2005 the dividend rate was 11.4 percent.
  - Argentina included 30-year GLWs in its June 2005 restructuring: GLWs annually pay 5 percent of excess cash flows (difference between actual GDP and threshold GDP in nominal terms) when (a) actual GDP (constant peso terms as of the reference date) exceeds threshold GDP and (b) annual growth rates of actual GDP (constant peso terms) exceed 3 percent. First payment scheduled for December 15, 2006; value calculated on November 1; warrants detachable and traded separately since end-November 2005.
- Case studies of design/data issues:
  - Bulgaria: trigger required GDP at 125 percent of 1993 level plus year-to-year increase; ambiguity in data source and measurement units led to payments never triggering because government chose constant-value local currency units.
  - Bulgaria’s GLWs callable at par; issuer call option reduced insurance against growth slowdowns and investor upside.
  - Bosnia and Herzegovina: trigger required GDP target and two-year persistence and GDP per capita above US$2,800 (1997 units, adjusted for German CPI); high informal economy (as much as 37 percent) and unreliable population statistics; limited trading activity; maturity December 2017.
  - Singapore: NSS and ERS are nontradable, distributed to eligible citizens, exchangeable only for cash with the government.

### III. THE MODEL — Specific Design of GLWs
- Design characteristics proposed to avoid prior problems:
  - Terms sufficiently long (example: 20 years) to smooth payments over cycles.
  - Trigger conditions based on real GDP in domestic currency so inflation and exchange rates do not dictate triggers.
  - Repayment trigger: (a) actual real GDP levels exceed potential levels determined upon issuance, and (b) actual real GDP growth rates are positive — ensuring recovery above potential before payments.
  - Reference date: the year preceding the calculation date (one-year lag reflecting GDP calculation lag).
  - Payments are capped to prevent extreme growth surprises producing unserviceable cash flows.
  - Payment amount equals difference between actual and potential real GDP growth rates multiplied by nominal GDP level in the prior period.

### III. THE MODEL — Formal Model and Equations
- Time periods: t = 0,1,...,21 (20-year maturity with a one-year payment lag so period 21 modeled).
- Key modeling assumptions:
  - GDP in domestic currency follows a stochastic process.
  - Trigger conditions evaluated via Monte Carlo simulation.
  - Payable cash flows computed using a coupon formula incorporating the above GLW characteristics.
  - Matching of period-by-period payable cash flows with revenues available for servicing (incremental tax revenues).
  - Price estimated as present value of payable cash flows, computed in domestic currency and converted to foreign currency using implied forward exchange rates derived from interest rate parity.
  - Risks: random shocks and parallel shifts applied to domestic inflation rates and exchange-rate change rates in every period.
- Four main equations (symbols preserved as in source):
  - Excess cash flows:
    - ECF_{t,i} = (g_{i,t} - g_{i,t}^N) * v_{i,t-1} * (1 + π_{i,t}) ∏_{j=0}^{t-1} (1 + ε_{i,j})  [presented in source as equation (1) with exact symbol usage]
  - Warrant payments:
    - W_{t,i} = min(C * Q_{t-1,i}, ECF_{t,i})  (equation (2))
  - Present value of warrant payments:
    - WPV_t = Σ_{i=0}^{t-1} W_{i,t} / ∏_{i=0}^{t-1} (1 + R_{i,t})  (equation (3) representation)
  - Forward exchange rates (interest rate parity):
    - E_{t} = E_0 ∏_{i=0}^{t-1} (1 + h_{i,t})  (equation (4) representation)
- Variable definitions (symbols preserved):
  - v_{i,t}: realized nominal GDP
  - Q_{t,i}: nominal trend GDP
  - v_0: initial value of real GDP
  - g_i: average growth rates of real GDP
  - σ: volatility of growth rates of real GDP
  - ε_{i,t}: random shocks to growth rates of real GDP
  - π_{i,t}: realized inflation rates
  - W_{i,t}: warrant payments
  - C: cap rates
  - R_t: issuer country’s domestic real interest rate
  - E_0: initial value of exchange rates (domestic currency per U.S. dollar)
  - h_{i,t}: realized rates of change in exchange rates

### III. THE MODEL — Parameter Values (Baseline and Choices)
- Data sourcing and calibration:
  - Growth rates and volatility for baseline taken from historical values for main EMBI countries (simple average over 1974–2004 for Argentina, Brazil, Colombia, Ecuador, Mexico, Morocco, Nigeria, Panama, Peru, Philippines, and Venezuela). Bulgaria, Egypt, Poland, and Turkey excluded due to data limitations.
  - Figure 1 summary statistics: Average growth rates = 3.1 percent, standard deviation = 2.2 percent (1974–2004).
- Baseline parameter values:
  - g (average growth rates) = 3.1 percent
  - σ (volatility) = 2.2 percent
  - κ (ratio of total tax revenues to GDP) = 20 percent
  - v_0 (initial value of GDP) = 100
  - E_0 (spot exchange rate domestic per foreign) = 1
  - υ (volatility of shocks to inflation rates) = 0
  - μ (volatility of shocks to rates of change in exchange rates) = 0
  - s (parallel-shift parameter for inflation) = 0
  - ψ (parallel-shift parameter for exchange-rate change) = 0
  - δ (pass-through of growth shocks to inflation) = 0
  - r (foreign country real interest rate, t_r = r) = 3 percent
  - R (issuer domestic real interest rate, t_R = R) = 8 percent
  - χ (foreign discount rate for issuer sovereign bonds, t_χ = χ) = 14 percent
  - λ (foreign inflation rates, t_λ = λ) = 2 percent
  - γ (issuer domestic inflation rates, t_γ = γ) fall from 6.5 to 4 percent over periods
- Cap choice:
  - Cap C set to 3 percent of trend GDP in the year before (Q_{t-1}) to limit extreme payments. Source notes cap of 3 percent helps contain warrant payments and has minimal impact on price (Table 1).

### Key Results — Prices and WI Ratios (Selected Findings)
- Table 1 snapshot (Growth surprises scenario with g = 5 percent):
  - Cap ∞: Price 29.2
  - Cap 5 percent: Price 26.3
  - Cap 4 percent: Price 24.0
  - Cap 3 percent: Price 20.4
  - Cap 2 percent: Price 15.2
  - Cap 1 percent: Price 8.3
  - Corresponding WI Ratio at maturity under growth surprises (g = 5 percent):
    - Cap ∞: WI Ratio 1.23
    - Cap 5 percent: WI Ratio 1.00
    - Cap 4 percent: WI Ratio 0.89
    - Cap 3 percent: WI Ratio 0.74
    - Cap 2 percent: WI Ratio 0.54
    - Cap 1 percent: WI Ratio 0.29
- Baseline scenario WI ratio statistics (Table 2 — selected periods; WI Ratio values preserved exactly):
  - Periods 2–11 WI Ratio:
    - Period 2: 0.42
    - Period 3: 0.37
    - Period 4: 0.35
    - Period 5: 0.33
    - Period 6: 0.36
    - Period 7: 0.37
    - Period 8: 0.36
    - Period 9: 0.35
    - Period 10: 0.40
    - Period 11: 0.39
  - Periods 12–21 WI Ratio:
    - Period 12: 0.36
    - Period 13: 0.33
    - Period 14: 0.37
    - Period 15: 0.31
    - Period 16: 0.35
    - Period 17: 0.37
    - Period 18: 0.35
    - Period 19: 0.36
    - Period 20: 0.35
    - Period 21: 0.33
- Probability that WI Ratio exceeds one (prob( WI > 1 ), Table 2 row representing Eq. (5); values preserved):
  - Period 2: 20.00
  - Period 3: 17.80
  - Period 4: 16.80
  - Period 5: 15.10
  - Period 6: 18.40
  - Period 7: 19.00
  - Period 8: 16.80
  - Period 9: 15.90
  - Period 10: 18.20
  - Period 11: 18.20
  - Period 12: 16.20
  - Period 13: 15.50
  - Period 14: 17.50
  - Period 15: 15.30
  - Period 16: 15.00
  - Period 17: 16.80
  - Period 18: 15.80
  - Period 19: 16.50
  - Period 20: 15.90
  - Period 21: 15.10
- Expected average cash shortfalls given WI Ratio > 1 (Eq. (6); amounts in current terms, billions of domestic currency units; Table 2 row representing Eq. (6) — values preserved):
  - Period 2: -0.87
  - Period 3: -1.12
  - Period 4: -1.30
  - Period 5: -1.41
  - Period 6: -1.39
  - Period 7: -1.64
  - Period 8: -1.91
  - Period 9: -1.95
  - Period 10: -2.30
  - Period 11: -2.31
  - Period 12: -2.55
  - Period 13: -2.46
  - Period 14: -2.86
  - Period 15: -2.88
  - Period 16: -3.67
  - Period 17: -3.63
  - Period 18: -3.89
  - Period 19: -4.01
  - Period 20: -4.44
  - Period 21: -4.21
- Interpretation of baseline results:
  - WI Ratio remains within 0.31 and 0.42 in baseline (with 0.42 in initial period, falling to 0.33 toward maturity).
  - Probability that WI Ratio exceeds one remains moderate at around 15 to 20 percent across periods.
  - Expected cash shortfalls conditional on WI Ratio > 1 rise sharply from 0.9 in period 2 to 4.2 in period 21 (current-value billions of domestic currency units), implying significant fiscal pressure in tail events.

### III. THE MODEL — Sensitivity Analysis (Scenarios and Implications)
- Summary of parameter variations and effects on WI Ratio:
  - Average growth rates varied between -0.9 and 7.1 percent:
    - WI Ratio rises steeply at shorter maturities when growth rates > 4 percent.
    - WI Ratio tends toward zero under low-growth scenarios.
    - The 3 percent cap binds in high-growth cases, keeping WI Ratio below 0.9 even in worst-case scenarios.
  - Volatility of real GDP growth varied between 0 and 20 percent (mean growth = baseline 3.1 percent):
    - WI Ratio and frequency of trigger events increase with volatility up to a point; beyond that cap binds and limits WI Ratio.
    - WI Ratio generally kept below 0.5, except at short maturities where it approaches 0.6.
  - High average growth scenario (g = 5 percent > baseline 3.1 percent) with varying volatility:
    - Frequency of trigger events falls as volatility increases, reducing WI Ratio.
    - With zero volatility, WI Ratio reaches 1 across all maturities; with positive volatility WI Ratio declines dramatically.
  - Low average growth scenario (g = 1.5 percent) with varying volatility:
    - Surface shape similar to baseline volatility scenarios; high volatility can offset low growth, raising WI Ratio up to 0.62 at short ends, much lower at long ends.
  - Pass-through of growth shocks to inflation varied between -2 and 2 (baseline 0):
    - Inflation assumed affected by innovations in real GDP growth; parameter controls effect on realized inflation (further details in Appendix figures referenced).
- Key implications:
  - Caps (e.g., 3 percent of trend GDP) are effective at limiting WI Ratio under extreme growth outcomes.
  - Volatility plays a nuanced role: low volatility with high mean growth can raise WI Ratio substantially; moderate-to-high volatility often reduces the frequency of sustained trigger events due to cap interaction.
  - Tail events where WI Ratio > 1 are moderate in probability but can generate sizable cash shortfalls that compete with earmarked government expenditures.

*Source: _wp0685 - 1.      Theoretical      Model (excerpt).*

### Appendix illustrates how changes in the value of the pass-through of growth shocks

### _wp0685 - Appendix illustrates how changes in the value of the pass-through of growth shocks to inflation have a limited impact across all maturities, with the ratio remaining within the range of 0.31 and 0.42.

### Sensitivity to pass-through and random inflation shocks (overview)
- Random shocks υ between 5 and 15 percent were applied to inflation rates (compared with a baseline value of zero). Such shocks are applied to the inflation rates in each period, and Figure A6 in the Appendix illustrates how the impact of such random shocks on the value of the WI ratio is limited.
- Simulations with alternative average growth rate assumptions do not alter these conclusions.

### E. Price of GLWs in Domestic Currencies
- Present-value estimate of GLWs (based on equation (3)) in domestic currency: 6.36, compared with an initial GDP value of 100 in the domestic currency.
- Parameters changed for sensitivity analysis: real growth rates of GDP (g), volatility (σ), random shocks (υ), parallel shifts to inflation rates (s), and pass-through of growth shocks to inflation (δ).

Findings (Case I — changing g and σ):
- Average growth rates assumed: 1.5, 3.1 (baseline), and 5 percent.
- Volatility assumed: 1, 2.2 (baseline), and 10 percent.
- Table 3 price matrix (g columns: 1.5 percent | 3.1 percent | 5.0 percent; σ rows):
  - σ = 1.0 percent: 0.03 | 3.29 | 23.24
  - σ = 2.2 percent: 0.59 | 6.36 | 20.35
  - σ = 10.0 percent: 6.05 | 9.26 | 14.03
- Observations:
  - When volatility held constant, price of GLWs increased significantly when growth rates outpaced threshold growth rates of 3.1 percent.
  - When growth rates held below (above) threshold of 3.1 percent, price increased (decreased) with volatility.
  - Baseline price 6.36 rises by 1.5 times when volatility changes from 2.2 percent to 10 percent.
  - When g increases to 5 percent from 3.1 percent with volatility held at baseline, price increases by a factor of three.

Findings (Case II — changing s and υ):
- Parallel shifts to inflation s = -10 percent, -5 percent, 0 percent, 5 percent, 10 percent (baseline s = 0).
- Random shocks υ = 0 percent (baseline), 10 percent, 20 percent.
- Table 4 price matrix (s columns in order):
  - υ = 0 percent: 7.03 | 6.68 | 6.36 | 6.07 | 5.81
  - υ = 10 percent: 7.13 | 6.76 | 6.43 | 6.13 | 5.86
  - υ = 20 percent: 7.43 | 7.01 | 6.65 | 6.32 | 6.02
- Observations:
  - Parallel increases in inflation reduce the real value of payments modestly (effect of observation-payment lag).
  - Random shocks increase price as volatility of random shocks increases, but impact is limited.

Findings (Case III — changing δ, pass-through of growth shocks to inflation):
- δ values tested: -3, -2, -1, 0 (baseline), 1, 2, 3.
- Table 5 prices (in order of δ): 6.4 | 6.4 | 6.3 | 6.3 | 6.3 | 6.3 | 6.38
- Observations:
  - Changes in δ have only a limited impact on the price of GLWs.
  - Baseline δ = 0 yields one of the lowest estimated prices, implying baseline is somewhat aggressive from issuer’s point of view.

### F. Price of GLWs in Foreign Currencies
- Baseline price in foreign currency terms (using equations (3) and (4)): 3.06, compared with an initial value of 100.
- Note: If forward exchange rates are assumed constant at initial exchange rate of 1, estimated price is 5.58 in foreign currency terms (assumption of constant forward rates overvalues present value in favor of issuer).

Findings (Case IV — changing ψ, parallel shifts in rates of change in exchange rates):
- ψ = -10 percent | 0 percent | 10 percent yields prices: 6.79 | 3.06 | 1.79
- Observation:
  - If forward exchange rates depreciate by 10 percent relative to interest rate parity, price would be halved; if they appreciate by 10 percent, price would more than triple.

Findings (Case V — changing μ and υ, random shocks to inflation and exchange rates):
- μ (random shocks impact on exchange rates) values: 0 percent, 10 percent, 20 percent (columns).
- υ (random shocks impact on inflation) values: 0 percent, 10 percent, 20 percent (rows).
- Table 7 price matrix:
  - υ = 0 percent: 3.06 | 3.28 | 4.1
  - υ = 10 percent: 3.09 | 3.33 | 4.18
  - υ = 20 percent: 3.2 | 3.47 | 4.46
- Observations:
  - Shocks to inflation rates have limited impact: raising υ to 20 percent while μ = 0 percent changes price from 3.06 to 3.20.
  - Shocks to exchange-rate change rates have larger effects: raising μ from 0 percent to 20 percent (with υ = 0) increases price from 3.06 to 4.10.
  - Foreign investors benefit from random shocks through both inflation rates and exchange-rate changes.

### G. Indexation Premium
- Purpose: estimate premium over risk-free rate required by investors holding GLWs in portfolios.
- Method: CAPM and CCAPM estimated for foreign (U.S. dollar-based) investors whose existing portfolio is highly correlated with GDP growth of the U.S. or return to S&P stock index.
- Findings:
  - Estimated β (CAPM) for foreign investors ranges from 0 to 0.06.
  - CCAPM estimations confirm low β findings.
  - Conclusion: insurance/indexation premium on GLWs would likely be low; investors would require little additional value from the insurance premium, so stand-alone price is a reasonable approximation of market price.

### Key numeric summaries
- Baseline domestic-currency price: 6.36 (initial GDP = 100).
- Baseline foreign-currency price: 3.06 (initial value = 100).
- WI ratio remains within the range of 0.31 and 0.42 under pass-through changes.
- Sensitivity highlights:
  - Domestic: price matrix entries (Table 3) include values such as 0.03, 3.29, 23.24, 0.59, 6.36, 20.35, 6.05, 9.26, 14.03.
  - Domestic parallel shifts/random shocks (Table 4) include 7.03, 6.68, 6.36, 6.07, 5.81, 7.13, 6.76, 6.43, 6.13, 5.86, 7.43, 7.01, 6.65, 6.32, 6.02.
  - Domestic pass-through δ (Table 5) prices: 6.4, 6.4, 6.3, 6.3, 6.3, 6.3, 6.38.
  - Foreign exchange-shift (Table 6) prices: 6.79, 3.06, 1.79.
  - Foreign shocks matrix (Table 7) prices: 3.06, 3.28, 4.1, 3.09, 3.33, 4.18, 3.2, 3.47, 4.46.
- Indexation premium: implied β ranges from 0 to 0.06 (CAPM); CCAPM supports low premium conclusion.

*Source: _wp0685 - Appendix*

### 3.1 percent based on historical data, the issuer country’s actual historical path of GDP may

### _wp0685 - 3.1 percent based on historical data, the issuer country’s actual historical path of GDP may

### Illustration: Historical vs. Trend GDP paths and implications for GLW payouts
- Three hypothetical real GDP level paths (initial GDP value = 100) are compared:
  - (a) historical path during 1974–94,
  - (b) historical path during 1984–2004,
  - (c) trend real GDP growing at a constant rate of 3.1 percent.
- Present-value warrant payments (domestic currency) under each path:
  - Case (a) (1974–94 historical path): 7.93
  - Baseline model prediction (trend g = 3.1 percent): 6.36
  - Case (b) (1984–2004 historical path): 4.54
- Key interpretation:
  - Strong early-period GDP growth followed by slower growth (case a) reduced trigger events later but yielded a higher present value (7.93) than the baseline (6.36).
  - Less favorable overall growth patterns (case b) reduced trigger events and payments, yielding 4.54.
  - The baseline prediction of 6.36 lies between historical extremes, suggesting reasonable predictive capacity of the model for these exercises.

### Concluding remarks: benefits, risks, and design principles for Growth-Linked Warrants (GLWs)
- Objectives of the paper:
  - Numerically examine quantitative properties of GLWs to evaluate issuer government’s capacity to service debts and to estimate GLW prices from investors’ perspectives.
- Literature synthesis:
  - GLWs provide opportunities to share growth risks between issuer and holder.
  - Potential moral hazard issues exist (debt overhang may reduce issuer incentives to promote growth).
  - Under asymmetric information, GDP-linked payments can facilitate debt-service relief.
  - Poor design and low quality of statistics have historically undermined GLW experience; successful issuance (e.g., growth-linked shares in Singapore) shows proper design can manage GLWs successfully.
- Proposed GLW design features in the paper:
  - Maturity set to 20 years to smooth payments over business cycles.
  - Trigger conditions defined as function of GDP in real value and domestic currency terms.
  - Design ensures a country growing above potential in recovery does not pay until actual GDP exceeds long-term potential trend.
  - Reference dates account for time required to calculate GDP value.
  - All payments on GLWs are capped to ensure extreme growth surprises do not cause cash-flow payments to exceed the country’s capacity to service debt.

### Model framework: stochastic process, triggers, payments, and WI ratio
- Real GDP dynamics:
  - Real GDP in period t in state i, i_t v, follows geometric Brownian motion with drift g and volatility σ (equation (A1)); initial real GDP value = v_0.
  - Realized growth rates: i_t g = dv/v (equation (A3)).
- Threshold (trend) real GDP:
  - Threshold q_t starts at v_0 and grows at annual rate θ (equation (A4)).
- Nominal adjustments:
  - Realized and threshold nominal GDP (i_t V and Q_t) adjust for deterministic inflation rates t γ (equations (A5) and (A6)).
- Excess cash flows and warrant payments:
  - Excess cash flows in period t paid in period t+1 in state i are defined as difference between realized and trend GDP (real terms) multiplied by realized nominal GDP (equation (A7)).
  - Warrant payment W_{t+1,i} = min(C percent of trend nominal GDP in period t, excess cash flow) when trigger conditions hold (equation (A8)); otherwise zero.
  - Trigger conditions: 0 < i_t v, i_t g, and q_t (i_t v > q_t and i_t g > 0).
  - Cap parameter: payments capped at C percent of trend GDP to smooth warrant payments.
- WI ratio (warrant payment / incremental tax revenues):
  - Total tax revenues i_t T = κ × nominal GDP (equation (A9)); tax revenues include risk factor i_t ε.
  - Incremental tax revenues ΔT_{t} = T_{t-1} − T_{t} = κ (V_{t-1} − V_{t}) (equation (A10)).
  - WI ratio i_t WI = W_{t,i} / ΔT_{t} (equation (A11)); used to evaluate issuer’s capacity to service GLWs.

### Price estimation methodology (domestic and foreign currency)
- Domestic currency price:
  - Each coupon payment (nominal) is deflated by cumulative inflation and discounted by domestic real interest rate R_t (equation (A12)).
  - Domestic currency price = E[∑_t PV(W_{t,i})] (equation (A13)).
- Foreign currency price:
  - Forward exchange rates E_t estimated from domestic and foreign nominal interest rate term structures via interest rate parity (equation (A14)); E_t can be written with an annual rate of change η (equation (A15)).
  - Warrant payments converted into U.S. dollars using E_t and discounted by issuer’s nominal risky foreign rate χ_t (equation (A16)).
  - Foreign currency price = E[∑_t PV_{USD}(W_{t,i})] (equation (A17)).
- Sensitivities:
  - Estimated GLW prices in both domestic and U.S. dollar terms change significantly under different growth scenarios.
  - Prices are sensitive to changes in average growth rates g and volatility σ; other parameters have limited effects.
  - Price in foreign currency terms is sensitive to exchange rates; deviations of forward exchange rates from interest-rate-parity-implied paths alter foreign-currency prices and such exchange-rate risk is borne by foreign investors.
  - CAPM-based indexation premium was calculated and found to be even lower than results reported in the literature.

### Disturbances, extensions, and model enrichment
- Disturbances:
  - Price levels P_{t,i} and exchange rate levels E_{t,i} follow geometric Brownian motions with drifts γ_t and η_t and volatilities υ and μ (equations (A18) and (A19)).
  - Realized inflation π_{t,i} includes deterministic component γ_t, stochastic shock υ ζ_{t,i}, pass-through from growth shocks δ σε_{t,i}, and parallel shift s (equation (A21)).
  - Realized rates of change in exchange rates h_{t,i} include ψ parallel shift and stochastic component μ ω_{t,i} (equation (A22)).
- Suggested model enrichments:
  - Model tax revenue-to-GDP ratio κ as state-dependent and allow tax revenues to be intertemporally storable.
  - Model deterministic variables such as interest rates as stochastic and reflecting state-dependent default risk.
  - Assess issuer repayment capacity based on alternative criteria.

### Parameters and notation (as summarized in the appendix)
- Growth and volatility:
  - g: average growth rates of real GDP
  - i_t g: realized growth rates of real GDP
  - σ: volatility of growth rates of real GDP
  - v_0: initial value of real GDP
- Thresholds and triggers:
  - θ: growth rates of real threshold GDP
  - q_t: real threshold GDP
  - Q_t: nominal threshold GDP
  - ρ: trigger growth rate
- Fiscal and exchange parameters:
  - κ: ratio of total tax revenues to GDP
  - i_t T: total tax revenues
  - 0E: initial value of exchange rates (domestic currency per foreign currency)
  - m: fraction of excess cash flows payable
- Shocks and volatilities:
  - i_t ε: random shocks to growth rates of real GDP
  - i_t ζ: random shocks to domestic inflation rates
  - i_t ω: random shocks to rates of change in exchange rates
  - υ: volatility of random shocks to domestic inflation rates
  - μ: volatility of random shocks to rates of change in exchange rates
  - s: parallel shifts, domestic inflation rates
  - ψ: parallel shifts, rates of change in exchange rates
  - δ: pass-through of growth shocks to inflation
- Interest rates and inflation:
  - t r: foreign real interest rates
  - t R: domestic real interest rates
  - t χ: issuer’s risky foreign nominal interest rates
  - t λ: foreign inflation rates
  - t γ: domestic inflation rates

### Indexation premium and CAPM/CCAPM findings
- Insurance value and CAPM context:
  - GLWs provide insurance if their returns are negatively correlated with the investor’s portfolio returns.
  - Two approaches used for theoretical risk premium: mean-variance CAPM and consumption CAPM (CCAPM).
- Mean-variance CAPM:
  - Expected return relationship: E[r_i] = r_f + β_i (E[r_m] − r_f).
  - β estimated only for the foreign investor (no well-defined domestic asset prices for hypothetical issuer).
  - Approximation: r_i ~ real return on GLWs; r_m approximated by U.S. real GDP growth and S&P 500 real total return.
  - Table A1 reported estimated β values for two periods (1974-94 and 1984-2004) suggesting i_r ≤ f_r and indexation premium small positive or negative. (Table entries in source show estimated βs: for 1974-94: -0.06 (U.S. Real GDP Growth) and -0.03 (S&P Real Return); for 1984-2004: -0.02 and 0.)
- CCAPM:
  - Euler equation and consumption-based pricing used as robustness check.
  - Utility assumed u(c) = (c^{1−θ} − 1)/(1−θ); standard parameter values used: (θ, ρ) = (5.0, 0.95) notation in the source.
  - Consumption growth approximated by GDP growth; covariance term estimated to be very low for foreign investor.
- Overall conclusion:
  - CAPM/CCAPM implications are suggestive but limited in emerging-market context; estimated indexation premium is small and often lower than literature findings.

### Sensitivity analysis highlights (WI ratio)
- Baseline scenario: g = 3.1 percent.
- Sensitivity charts (figures in appendix) illustrate WI ratio responses to changes in:
  - Average growth rates of real GDP (Figure A1)
  - Volatility of GDP growth (Figures A2, A3, A4 for baseline, high-growth g = 5 percent, low-growth g = 1.5 percent)
  - Pass-through of growth shocks to inflation δ (Figure A5; baseline g = 3.1 percent)
  - Random shocks to inflation (Figure A6; baseline g = 3.1 percent)
- General finding:
  - WI Ratio is sensitive to average growth rates and volatility; pass-through and inflation shocks also affect WI ratio, with graphical reporting across scenarios in the appendix.

*Source: Excerpts from the PDF chapter provided in the content unit.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp0685.pdf_