## wp1835

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### I. Introduction — objective and headline quantitative findings
- Objective: assess benefits and costs of using market insurance (put options) to hedge commodity price risk and enhance macroeconomic resilience by augmenting a standard sovereign default model with access to put options calibrated to Mexican data.
- Key quantitative findings from full model simulations:
  - Welfare gains equivalent to a permanent increase in consumption of 0.44 percent.
  - Approximately 90 percent of the welfare gains stem from the borrowing-costs channel; remaining from income smoothing.
  - Risk spreads on debt are 19 basis points lower in the hedging economy relative to no-hedging.
  - Welfare gains decline if option cost includes a premium above the actuarially fair price; only a sizable premium would reduce gains to zero.
  - Welfare gains increase with strike price of put options, hedged volume of oil, volatility of oil prices, and with risk aversion of foreign investors.
  - Selling oil forward can generate larger welfare gains than buying put options because forwards avoid the upfront cost of insurance, but forwards entail forgoing revenue windfalls if oil prices rise and carry political economy costs.

### II. Mexico’s oil hedging program — empirical context and mechanics
- Program history and scope:
  - Program known since 2001; Mexico used market hedging instruments as early as 1990.
  - Mexico is the 12th largest oil producer (source: U.S. International Energy Administration).
  - Oil sector controlled by state-owned Petroleos Mexicanos (PEMEX); oil-related risks directly affect public finances.
- Fiscal shares and exposures (averages and 2016 values):
  - Over 2000-2016, oil-related revenues represented 32 percent of total fiscal revenues, of which 47 percent corresponded to oil exports and the remainder to net domestic sales of petroleum products.
  - Over same period, oil exports averaged 11 percent of total exports.
  - By 2016, oil revenues represented about 16 percent of total fiscal revenues and close to 5 percent of total exports.
  - Average annual production over 2000-2016: 1 billion barrels; Mexico imported about 178 million barrels of petroleum products annually over the same period.
  - After offsetting domestic sales and imports, the Mexican treasury hedged on average 29 percent of total production over the past 10 years.
- Hedging mechanics and costs:
  - Treasury computes assumed export price for subsequent fiscal year as weighted average between 10 years of historical prices and up to 5 years of future prices.
  - Treasury purchases Asian put options with strike equal or close to budgeted oil price; Asian options lock in a minimum average price for whole fiscal year.
  - Contracts executed with foreign banks; most contracts use Maya oil as underlying asset (about 80 percent of Mexico’s oil export volumes), some use Brent.
  - Since 2001, the cost of the options has averaged 0.1 percent of GDP per year.
  - Options exercised in 2009, 2015, and 2016 with payoffs reaching 0.5, 0.6, and 0.3 percent of GDP respectively.
- Correlations and institutional context:
  - Correlation coefficient between risk spreads on external sovereign debt and oil prices: −0.59 over the past twenty years.
  - 2013 constitutional reform opened oil sector to private investment; domestic fuel price liberalization began in 2016 and completed by end-2017.
  - Political economy risk exemplified by Ecuador’s experience in early 1993 where hedging losses triggered criticism and investigations.

### III. Two-period model — analytical illustration and propositions
- Model setup (t ∈ {0,1}):
  - Consumers choose bonds d at price q to maximize U0 = max_d log c0 + β E0 log c1, with β < 1 and u(c) = log c.
  - Period-1 income yH or yL < yH with probabilities p and 1−p; default yields income ydef > 0.
  - Risk-free rate r* assumed zero.
- Bond-pricing by risk-neutral foreign investors:
  - q = 1 if yL − d ≥ ydef (no default ever);
  - q = p if yL − d < ydef ≤ yH − d (default only in bad state);
  - q = 0 if yH − d < ydef (default always).
- Hedging (insurance) introduced in period 0 guaranteeing at least ȳ in period 1 at cost ξ where:
  - ξ = p(ȳ − yH) + (1−p)(ȳ − yL), if ȳ ≥ yH;
  - ξ = (1−p)(ȳ − yL), if yL < ȳ < yH;
  - ξ = 0, if ȳ ≤ yL.
- Key propositions and scenarios:
  - Proposition 1 (Default Incentives and Hedging): Introducing hedging increases the lower default threshold (ˆydef) and reduces the upper default threshold (ˆˆydef); hedging can either improve or worsen incentives to default depending on default cost.
  - Proposition 2 (No Default in Equilibrium): When default is too costly so economy does not default in equilibrium, introducing hedging increases social welfare and the country borrows more.
  - Proposition 3 (Default Only When Income is Low): When economy defaults only when y = yL, whether hedging increases or decreases welfare depends on its impact on default incentives.
- Three scenario outcomes summarized:
  - Case 1: if hedging reduces default incentives, hedging increases welfare, but borrowing might increase or decrease.
  - Case 2: if hedging does not change default incentives, it reduces welfare and increases borrowing.
  - Case 3: if hedging increases default incentives, both social welfare and borrowing decline.
- Intuition:
  - If default is very costly (never optimal), hedging raises welfare via income smoothing and higher borrowing capacity.
  - If default costs are moderate, hedging can lower borrowing costs but upfront insurance cost and induced borrowing changes can in some parameter regions worsen default incentives and reduce welfare.
  - Net welfare effect depends on default incentives and balance between income-smoothing benefits versus upfront insurance cost and its effect on borrowing.

### IV. Benchmark sovereign-default model with put options — structure and calibration
- Benchmark features:
  - Preferences: E0[ Σ_{t=0}^∞ β^t C_t^{1−γ}/(1−γ) ] with CRRA coefficient γ and discount factor β.
  - Income: Y_t = F_t + X_t ≡ F_t + p_t Q_t; normalize by F_t and set y_t ≡ Y_t/F_t = 1 + p_t Q with Q constant.
  - Financial instruments: zero-coupon one-period bonds b_{t+1} priced q_t; put options at unit price ξ( p̄_t ) hedging fraction αQ at strike p̄_t.
  - Budget: c_t + q_t G b_{t+1} + α Q G ξ( p̄_t ) = w_t where w_t = y_t + b_t.
  - Put payoff next period: α Q max{ p̄_t − p_{t+1}, 0 }.
  - Default: exclusion from markets, inability to borrow or buy puts, income loss h(y_t), redemption probability λ ∈ (0,1) each period.
  - No-arbitrage pricing:
    - Bond price: q(b_{t+1},p_t) = E_t[ 1 − D(y_{t+1} + α Q max{ p̄_t − p_{t+1},0 } + b_{t+1}, p_{t+1}) ] / (1 + r*).
    - Put option price: ξ( p̄_t ) = E_t[ max{ p̄_t − p_{t+1}, 0 } ] / (1 + r*).
- Calibration to Mexican data (1996-2016); selected parameter values:
  - Risk-free rate r∗ = 0.64 percent
  - Risk aversion γ = 2
  - Probability of redemption λ = 0.11
  - Growth rate G = 1.0313
  - Unconditional mean p = 54.60
  - Persistence ρ = 0.71
  - Volatility σ = 0.25
  - Oil to non-oil GDP ratio pQ = 6 percent
  - Hedging share α = 0.29
  - Strike price ̄p_t = μ E_t[p_{t+1}|p_t] with μ = 0.77 in baseline (data-based μ average = 0.85)
  - Discount rate β = 0.76
  - Output loss parameter y∗ = 0.98 E[y] = 1.03 (chosen to match spreads)
  - Oil production over 1996-2016 averages 1.03 billion barrels per year.
  - Deterministic annual real growth rate of non-oil income computed as 3.13 percent (G = 1.0313).
- Numerical solution details:
  - Value function iteration; oil-price grid via Rouwenhorst method; 21 grids for oil price and 500 grids for bond holdings.

### V. Main quantitative results, mechanisms, and event dynamics
- Welfare measurement and simulation:
  - Welfare gains Δ(w_t,p_t) = 100 * [ (V(w_t,p_t) / ̃V(w_t,p_t))^{1/(1−γ)} − 1 ].
  - Conditional welfare gains Δ(w_t,p_t) vary from 0 to 0.45 percent (with p_t at unconditional mean).
  - Unconditional welfare gains E[Δ(w_t,p_t)] = 0.44 percent (permanent increase in annual consumption).
  - Simulation: 100 Monte Carlo runs of 2,000 periods each for benchmark and no-hedging; first 500 periods discarded.
- Stochastic steady-state averages (Table 3):
  - Hedging: debt ratio 11.97 percent, sovereign spreads 1.40 percent, default probability 1.27 percent.
  - No-Hedging: debt ratio 10.50 percent, sovereign spreads 1.59 percent, default probability 1.41 percent.
- Decomposition of welfare gains:
  - Replacing no-hedging bond prices with hedging bond prices implies welfare gains equal to a permanent consumption increase of 0.40 percent — i.e., 90 percent of total gains stem from borrowing-cost channel; income-smoothing accounts for remaining 0.04 percent.
- Event dynamics around default (11,000-period simulation, 20-period windows):
  - Sharp decline in oil price at time 0 triggers option payoffs in hedging economy, compensating income fall.
  - No-hedging economy defaults and debt stock reduces to zero.
  - Default probability spikes in event window, peaking at 89 percent even under hedging.
  - Hedging economy sustains higher consumption despite option cost due to lower cost of debt.

### VI. Robustness checks and extensions
- Option price premium sensitivity:
  - Baseline assumes actuarially fair price.
  - Introducing additional cost x per barrel (fraction of actuarially fair price) reduces welfare gains as x increases.
  - To reduce welfare gains to zero requires premium on the order of 2.3 times the actuarially fair price.
  - Equivalent dollar value for zero welfare gains: US$2.1 per barrel (in 2009 constant dollars).
  - Historical average cost paid by Mexico for put options (2006-2016): US$3.5 per barrel.
- Strike-price sensitivity:
  - Baseline μ = 0.77; alternative μ choices show welfare gains increase with strike price.
  - With μ = 1.03 welfare gains rise to 0.75 percent in comparisons.
- Oil-price process parameters:
  - Increasing persistence ρ reduces welfare gains (one-year options compensate smaller fraction of cumulative loss when prices are more persistent).
  - Increasing volatility σ increases welfare gains (higher default risk strengthens borrowing-cost channel).
- Hedged volume and risk aversion:
  - Welfare gains increase when a larger volume α is hedged.
  - Welfare gains increase with higher γ (greater risk aversion).
- Other sensitivities:
  - Welfare gains increase with higher G (higher non-oil income growth).
  - Welfare gains decline when r∗ increases.
  - Welfare gains decline when output-loss-from-default parameter y∗ increases (lower cost of default reduces hedging benefits).
  - Welfare gains decline with higher λ (higher probability of redemption reduces cost of default and thus hedging benefits).
  - Welfare gains decline with higher β (more patient consumers borrow less; borrowing-cost channel weakens).
- Forwards versus put options:
  - Forwards at conditional mean involve no upfront cost but forego upside.
  - Quantitative comparison (μ = 1):
    - Forwards: welfare gains 0.89 percent; debt 14.19 percent; default spreads 0.96 percent; default prob 0.92 percent.
    - Put options: welfare gains 0.75 percent; debt 13.32 percent; default spreads 1.14 percent; default prob 1.06 percent.
  - Conclusion: forwards yield larger welfare gains in these calibrations, lower default probability and spreads, and allow more borrowing; political economy tradeoffs (foregone upside) remain salient.
- Risk-averse international investors (time-varying pricing kernel with parameter ν):
  - Pricing equations adjusted by m_{t+1} = e^{−r∗} e^{−ν ε_{t+1}}.
  - Selected results (Table 6):
    - ν = 0: Welfare gains 0.44 percent; debt 11.97 percent; spreads 1.40 percent; default prob 1.27 percent; cost premium 2.52
    - ν = 0.25: Welfare gains 0.44 percent; debt 11.77 percent; spreads 1.24 percent; default prob 0.86 percent; cost premium 2.10
    - ν = 0.5: Welfare gains 0.59 percent; debt 13.03 percent; spreads 1.45 percent; default prob 0.50 percent; cost premium 2.06
    - ν = 0.75: Welfare gains 0.98 percent; debt 17.82 percent; spreads 2.20 percent; default prob 0.32 percent; cost premium 0.46
  - Interpretation: higher foreign-investor risk aversion (higher ν) increases both debt and option risk premia; welfare gains from hedging increase with ν in these simulations.

### VII. Policy-relevant conclusions and implications
- Hedging via market insurance (put options or forwards) can provide measurable welfare gains in presence of defaultable debt; baseline unconditional gain ≈ 0.44 percent of permanent consumption.
- Majority (about 90 percent) of welfare benefit accrues through reduction in borrowing costs rather than pure income smoothing.
- Hedging increases average borrowing capacity in calibrated model (higher debt ratio under hedging) because lower borrowing costs and additional borrowing to pay for options both operate.
- Welfare gains are robust across many parameter variations but sensitive to:
  - Option price premia above actuarially fair pricing (large premia needed to eliminate gains).
  - Strike choice μ (higher strikes raise welfare gains).
  - Oil-price persistence ρ and volatility σ (high ρ reduces gains; high σ increases gains).
  - Hedged volume α and risk aversion γ.
- Selling oil forward (locking price at conditional mean) yields larger simulated welfare gains than put options in baseline comparisons, but political economy considerations (foregone upside) may make options preferable in practice.
- When international investors are risk averse and price both bonds and options with time-varying kernels, hedging can be even more valuable (welfare gains increase with investor risk aversion in simulations).

*Source: wp1835 - References (IMF working paper; content from provided PDF excerpt).*

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

### References

### I. Introduction
- Sharp decline in oil prices started in late 2014; Mexico purchased put options in the fall of 2014 to hedge 228 million barrels of oil, about 28 percent of production, at a strike price of US$ 76.4 per barrel—US$ 31.1 above the actual average oil price in 2015.
- Objective: assess benefits and costs of using market insurance (put options) to hedge commodity price risk and enhance macroeconomic resilience, by augmenting a standard sovereign default model with access to put options calibrated to Mexican data.
- Benchmark economy features:
  - Exposure to commodity export price risk.
  - Borrowing through one-period defaultable debt acquired by risk-neutral foreign investors.
  - Default follows a willingness-to-pay framework a` la Eaton and Gersovitz (1981).
  - Access to put options from risk-neutral foreign investors to lock in a minimum price for commodity exports in the subsequent period.
- Key quantitative findings from full model simulations:
  - Using put options yields welfare gains equivalent to a permanent increase in consumption of 0.44 percent.
  - Approximately 90 percent of the welfare gains stem from the borrowing-costs channel; the remaining from income smoothing.
  - Risk spreads on debt are 19 basis points lower in the hedging economy relative to no-hedging.
  - Welfare gains decline if option cost includes a premium above the actuarially fair price; only a sizable premium would reduce gains to zero.
  - Welfare gains increase with strike price of put options, hedged volume of oil, volatility of oil prices, and with risk aversion of foreign investors.
  - Selling oil forward can generate larger welfare gains than buying put options because forwards avoid the upfront cost of insurance, but forwards entail forgoing revenue windfalls if oil prices rise and carry political economy costs.

### II. Mexico’s Oil Hedging Program (empirical context and mechanics)
- Program history and scope:
  - Known program set up in 2001; Mexico used market hedging instruments as early as 1990.
  - Mexico is the 12th largest oil producer (source: U.S. International Energy Administration).
  - Oil sector controlled by state-owned Petroleos Mexicanos (PEMEX); oil-related risks directly affect public finances.
- Fiscal shares and exposures (averages and 2016 values):
  - Over 2000-2016, oil-related revenues represented 32 percent of total fiscal revenues, of which 47 percent corresponded to oil exports and the remainder to net domestic sales of petroleum products.
  - Over the same period, oil exports averaged 11 percent of total exports.
  - By 2016, oil revenues represented about 16 percent of total fiscal revenues and close to 5 percent of total exports.
  - Average annual production over 2000-2016: 1 billion barrels; Mexico imported about 178 million barrels of petroleum products annually over the same period.
  - After offsetting domestic sales and imports, the Mexican treasury hedged on average 29 percent of total production over the past 10 years.
- Hedging mechanics:
  - The treasury computes assumed export price for the subsequent fiscal year as a weighted average between 10 years of historical prices and up to 5 years of future prices.
  - Mexican treasury purchases Asian put options with strike price equal or close to the budgeted oil price; Asian options lock in a minimum average price for the whole fiscal year.
  - Program executed through several contracts with foreign banks; most contracts use Maya oil as underlying asset (about 80 percent of Mexico’s oil export volumes), some use Brent.
- Costs and payoffs:
  - Since 2001, the cost of the options has averaged 0.1 percent of GDP per year.
  - Options exercised in 2009, 2015, and 2016 with payoffs reaching 0.5, 0.6, and 0.3 percent of GDP respectively.
- Correlations and institutional context:
  - High negative correlation between risk spreads on external sovereign debt and oil prices: correlation coefficient of -0.59 over the past twenty years.
  - 2013 constitutional reform opened the oil sector to private investment; private sector remained in infancy as of end-2017 but expected to gain importance as investment picks up.
  - A process of liberalization of domestic fuel prices began in 2016 and was completed by end-2017.
  - Political economy risk illustrated by Ecuador’s experience in early 1993 where hedging losses triggered heavy criticism and investigations.

### III. Benefits and Costs of Hedging in a Two-Period Model (analytical illustration)
- Model setup (two periods t ∈ {0,1}):
  - Consumers choose bonds d at price q to maximize U0 = max_d log c0 + β E0 log c1, with discount factor β < 1 and u(c) = log c.
  - Income: y in period 0; period-1 income takes values yH or yL < yH with probabilities p and 1−p respectively.
  - Default yields income ydef > 0 in period 1.
  - Risk-free rate r* assumed zero.
- Bond-pricing by risk-neutral foreign investors:
  - q = 1, if yL − d ≥ ydef (no default ever);
  - q = p, if yL − d < ydef ≤ yH − d (default only in bad state);
  - q = 0, if yH − d < ydef (default always).
- Introduction of hedging (insurance bought in period 0 guaranteeing at least ȳ in period 1 at cost ξ):
  - ξ defined piecewise:
    - ξ = p(ȳ − yH) + (1−p)(ȳ − yL), if ȳ ≥ yH;
    - ξ = (1−p)(ȳ − yL), if yL < ȳ < yH;
    - ξ = 0, if ȳ ≤ yL.
  - With hedging, period-1 consumption c_i1 = max{ max{ȳ, y_i} − d, ydef }, i ∈ {H, L}.
- Roles and channels of hedging:
  - Income smoothing: choosing ȳ = p yH + (1−p) yL (the unconditional mean) yields period-1 incomes ȳ or yH, reducing income volatility.
  - Borrowing/default incentives: hedging can alter default thresholds and thus affect bond prices and borrowing costs.
- Formal propositions (summarized):
  - Proposition 1 (Default Incentives and Hedging): Introducing hedging increases the lower default threshold (ˆydef) and reduces the upper default threshold (ˆˆydef); hedging can either improve or worsen incentives to default depending on default cost.
  - Proposition 2 (No Default in Equilibrium): When default is too costly so that the economy does not default in equilibrium, introducing hedging increases social welfare and the country borrows more.
  - Proposition 3 (Default Only When Income is Low): When the economy defaults only when y = yL, whether hedging increases or decreases welfare depends on its impact on default incentives.
- Intuition and implications:
  - If default is very costly (never optimal), hedging increases welfare via smoother income and higher borrowing capacity.
  - If default costs are moderate, hedging can reduce downside risk and lower borrowing costs, but the upfront cost of insurance may require increased borrowing, which can in some parameter regions worsen default incentives.
  - The net welfare effect depends critically on default incentives and on the relative magnitudes of the income-smoothing benefits versus the upfront insurance cost and its effect on borrowing.

*Source: wp1835 - References (IMF working paper; content from provided PDF excerpt).*

### 1.  if hedging reduces default incentives, hedging increases welfare, but borrowing might

### 1.  if hedging reduces default incentives, hedging increases welfare, but borrowing might

### Key propositions and scenarios
- Proposition outcomes:
  - Case 1: if hedging reduces default incentives, hedging increases welfare, but borrowing might increase or decrease.
  - Case 2: if hedging does not change default incentives, it reduces welfare and increases borrowing.
  - Case 3: if hedging increases default incentives, both social welfare and borrowing decline.
- Intuition:
  - Case 1: both the income-smoothing and borrowing cost channels imply a welfare gain despite the upfront cost of insurance; impact on borrowing is ambiguous because (i) borrowing to purchase insurance raises borrowing, (ii) higher borrowing raises default likelihood and debt costs, reducing incentives to borrow.
  - Case 2: hedging raises income in the low state only if there is no default; if the economy defaults at y = yL, hedging does not change default incentives nor income (default yields same income ydef as without hedging). Consumers borrow more in period 0 to purchase insurance, lowering current disposable income and reducing welfare.
  - Case 3: hedging increases default incentives and moves the economy from defaulting only in the bad state to always defaulting, hence reducing welfare.

### Interpretation of Figure 3 (two-period model regions)
- Left regions: areas where the cost of default is high; hedging is always desirable:
  - Either both borrowing-cost and income-smoothing channels operate (when hedging reduces default risk).
  - Or only income-smoothing operates (when there is no default in equilibrium).
- Right regions: where hedging reduces welfare because costs of default are small.
- Empirical note: defaults are rare events — Mexico has defaulted only 8 times since 1821 — and empirical literature documents significant output losses following sovereign defaults, implying left regions in Figure 3 are likely more empirically relevant.

### Benchmark model structure (main features)
- Preferences and objective:
  - Representative agent maximizes E0[ sum_{t=0}^∞ β^t C_t^{1−γ}/(1−γ) ] with CRRA coefficient γ and discount factor β.
- Income and normalization:
  - Total income Y_t = F_t + X_t ≡ F_t + p_t Q_t.
  - Normalize by F_t and define y_t ≡ Y_t/F_t = 1 + p_t Q, with Q ≡ Q_t/F_t and assume Q constant.
  - Normalized objective E0[ sum_{t=0}^∞ (β G^{1−γ})^t c_t^{1−γ}/(1−γ) ] where G is deterministic growth of F_t.
- Financial instruments and budget constraint:
  - Wealth w_t = y_t + b_t.
  - Choices: consumption c_t; zero-coupon one-period bonds b_{t+1} priced q_t; put options at unit price ξ( p̄_t ) hedging fraction αQ at strike p̄_t.
  - Period budget: c_t + q_t G b_{t+1} + α Q G ξ( p̄_t ) = w_t.
  - Next-period wealth under put exercise: w_{t+1} = y_{t+1} + α Q max{ p̄_t − p_{t+1}, 0 } + b_{t+1}.
- Continuation (no-default) value function:
  - V^c(w_t,p_t) = max_{c_t,b_{t+1}} [ c_t^{1−γ}/(1−γ) + β G^{1−γ} E_t[V(w_{t+1},p_{t+1})] ]
  - Subject to the budget and w_{t+1} laws above; V(w_t,p_t) = max(V^c(w_t,p_t), V^d(p_t)).
- Default decision and value:
  - Default implies exclusion from international credit markets, inability to borrow or buy put options, and income loss h(y_t) each period.
  - Default is not permanent: redemption probability λ ∈ (0,1) each period returns economy to markets with zero net assets.
  - Value under default:
    - V^d(p_t) = [c_t^{1−γ}/(1−γ)] + β G^{1−γ}[ λ E_t V(w_{t+1},p_{t+1}) + (1−λ) E_t V^d(p_{t+1}) ]
    - Subject to c_t = y_t − h(y_t) and w_{t+1} = y_{t+1}.
  - Default occurs iff V^d(p_t) > V^c(w_t,p_t).

### Pricing and external investors
- Risk-neutral foreign investors and no-recovery assumption:
  - Foreign investors are risk-neutral; if default occurs they recover zero from bonds and reneged put options (any recoveries assumed consumed in transaction costs/legal fees).
  - This implies bond pricing may understate welfare gains from hedging if recoveries are actually positive.
- No-arbitrage pricing:
  - Bond price: q(b_{t+1},p_t) = E_t[ 1 − D(y_{t+1} + α Q max{ p̄_t − p_{t+1},0 } + b_{t+1}, p_{t+1}) ] / (1 + r^*).
  - Put option price: ξ( p̄_t ) = E_t[ max{ p̄_t − p_{t+1}, 0 } ] / (1 + r^*).
  - Default indicator D(·) enters bond pricing because hedging income affects default probabilities and risk spreads.

### Counterfactual: economy without put options
- No-hedging economy variables denoted with tilde (˜).
- Continuation and default value functions ˜V^c, ˜V^d defined analogously but without option terms in budget constraints.
- Bond pricing in no-hedging economy:
  - ˜q(b_{t+1},p_t) = E_t[ 1 − ˜D(y_{t+1} + b_{t+1}, p_{t+1}) ] / (1 + r^*).

### Recursive equilibrium definition (Markov Perfect Equilibrium)
- Equilibrium characterized by value functions {V, V^c, V^d}, default function D, consumption c_t, bond choice b_{t+1}, and price q_t such that:
  - Given states {w_t,p_t}, option cost ξ( p̄_t ), and strike p̄_t, choices solve optimization problems (continuation and default).
  - V = max(V^c, V^d) and q_t satisfies the no-arbitrage bond pricing equation.

*Source: Authors’ construction.*

### 2.  The Markov Perfect Equilibrium of the economy without put options is characterized by a

### 2.  The Markov Perfect Equilibrium of the economy without put options is characterized by a

### Calibration
- Model calibrated to Mexican data over 1996-2016. Benchmark model has 13 parameters divided in three groups.
- Key parameter values (as stated in the source):
  - Risk-free rate r∗ = 0.64 percent
  - Risk aversion γ = 2
  - Probability of redemption λ = 0.11
  - Growth rate G = 1.0313
  - Unconditional mean p = 54.60
  - Persistence ρ = 0.71
  - Volatility σ = 0.25
  - Oil to non-oil GDP ratio pQ = 6 percent
  - Hedging share α = 0.29
  - Strike price ̄p_t = μ E_t[p_{t+1}|p_t] with μ = 0.77 in baseline (data-based μ average = 0.85)
  - Discount rate β = 0.76
  - Output loss parameter y∗ = 0.98 E[y] = 1.03 (chosen to match spreads)
- Oil production over 1996-2016 averages 1.03 billion barrels per year.
- Deterministic annual real growth rate of non-oil income computed as 3.13 percent (reported as G = 1.0313 in Table 2).
- Strike-price calibration:
  - μ chosen so long-run simulated probability of exercising options = 18.75 percent (Mexico exercised options 3 times between 2001 and 2016).
  - μ = 0.77 in baseline; alternative data-based μ average = 0.85.
- Numerical solution details:
  - Solve by value function iteration; oil-price grid via Rouwenhorst method.
  - Use 21 grids for oil price and 500 grids for bond holdings.

### Welfare gains from hedging (main quantitative findings)
- Welfare measurement:
  - Welfare gains expressed as a permanent increase in annual consumption using equation (14): Δ(w_t,p_t) = 100 * [ (V(w_t,p_t) / ̃V(w_t,p_t))^{1/(1−γ)} − 1 ].
  - Conditional welfare gains Δ(w_t,p_t) vary from 0 to a 0.45 percent permanent increase in consumption (with p_t set to its unconditional mean).
  - Unconditional welfare gains E[Δ(w_t,p_t)] = 0.44 percent (permanent increase in annual consumption).
- Simulation procedure:
  - Run 100 Monte Carlo simulations of 2,000 periods each for benchmark and no-hedging economies.
  - Initial 500 periods discarded.
  - Welfare approximated via present discounted value of simulated consumption paths.
- Key dynamics and mechanisms:
  - Two channels of welfare gains: income smoothing and reduction in default incentives (borrowing costs channel).
  - Stochastic steady-state comparisons (Table 3):
    - Hedging: debt ratio 11.97 %, sovereign spreads 1.40 %, default probability 1.27 %
    - No-Hedging: debt ratio 10.50 %, sovereign spreads 1.59 %, default probability 1.41 %
  - Hedging leads to systematically higher bond prices (lower borrowing costs) except where default risk is zero.
  - Decomposition: replacing no-hedging bond prices with hedging bond prices implies welfare gains equivalent to a permanent consumption increase of 0.40 percent — i.e., 90 percent of total gains stem from the borrowing-cost channel (income-smoothing accounts for remainder).

### Event dynamics around default
- 11,000-period simulation used to construct 20-period event windows around defaults (10 years before and after default).
- A sharp decline in oil price at time 0:
  - Options payoff compensates income fall in hedging economy.
  - No-hedging economy defaults and stock of debt reduces to zero.
  - Default probability spikes even under hedging, peaking at 89 percent in the event window.
  - Hedging economy sustains higher consumption levels despite option cost, due to lower cost of debt.

### Robustness checks
- Cost premium on put options:
  - Baseline assumes actuarially fair price.
  - Introducing an additional cost x per barrel (fraction of actuarially fair price) reduces welfare gains as x increases.
  - To reduce welfare gains to zero would require a premium on the order of 2.3 times the actuarially fair price.
  - Equivalent dollar value for zero welfare gains: US$2.1 per barrel (in 2009 constant dollars).
  - Historical context: Mexico paid on average US$3.5 per barrel during 2006-2016 to purchase put options.
- Strike-price sensitivity:
  - Benchmarks with μ = 0.74 and μ = 1.03 show welfare gains increase with strike price.
  - With μ = 1.03 welfare gains rise to 0.75 percent (see comparison to forwards below).
- Oil-price process parameters:
  - Increasing persistence ρ reduces welfare gains (one-year options compensate smaller fraction of cumulative loss when prices are more persistent).
  - Increasing volatility σ increases welfare gains (higher default risk strengthens borrowing-cost channel).
- Other parameter sensitivities (Table 4 highlights many cases):
  - Welfare gains increase when a larger volume α is hedged.
  - Welfare gains increase with higher γ (greater risk aversion).
  - Welfare gains increase with higher G (higher non-oil income growth).
  - Welfare gains decline when r∗ increases.
  - Welfare gains decline when output-loss-from-default parameter y∗ increases (lower cost of default reduces hedging benefits).
  - Welfare gains decline with higher λ (higher probability of redemption reduces cost of default and thus hedging benefits).
  - Welfare gains decline with higher β (more patient consumers borrow less; borrowing-cost channel weakens).

### Extensions: Selling oil forward vs put options
- Alternative hedging vehicle: selling oil forward at conditional mean (no upfront cost, but forego upside windfall).
- Two-period model implication (Proposition 4): introducing forwards at conditional-mean price can increase the no-default income threshold ˆy_def (i.e., reduce default incentives).
- Quantitative comparison (Table 5):
  - Forwards (μ = 1): welfare gains 0.89 percent; debt 14.19 %; default spreads 0.96 %; default prob 0.92 %
  - Put options (μ = 1): welfare gains 0.75 percent; debt 13.32 %; default spreads 1.14 %; default prob 1.06 %
  - Conclusion: hedging via forwards yields larger welfare gains than put options in these calibrations, and forwards lower default probability and spreads while allowing more borrowing. Political economy tradeoffs (forgoing upside) remain relevant.

### Extension: Risk-averse international investors
- Introduced time-varying pricing kernel m_t with m_{t+1} = e^{−r∗} e^{−ν ε_{t+1}}; log m_t = −r∗ − ν ε_t; var(log m_t) = ν^2 σ^2.
- Pricing equations:
  - q_t(b_{t+1},p_t) = E_t[m_{t+1} (1−D(y_{t+1}+b_{t+1},p_{t+1}))]
  - ξ_t(p_t) = E_t[m_{t+1} max(̄p − p_{t+1},0)]
- Table 6 results (selected):
  - ν = 0: Welfare gains 0.44 %; debt 11.97 %; spreads 1.40 %; default prob 1.27 %; cost premium 2.52
  - ν = 0.25: Welfare gains 0.44 %; debt 11.77 %; spreads 1.24 %; default prob 0.86 %; cost premium 2.10
  - ν = 0.5: Welfare gains 0.59 %; debt 13.03 %; spreads 1.45 %; default prob 0.50 %; cost premium 2.06
  - ν = 0.75: Welfare gains 0.98 %; debt 17.82 %; spreads 2.20 %; default prob 0.32 %; cost premium 0.46
- Interpretation: as foreign investors become more risk averse (higher ν), both debt and option prices embed larger risk premia; nonetheless, welfare gains from hedging increase with ν in the simulations.

### Quantitative summary tables and key statistics (selected)
- Baseline unconditional welfare gain: 0.44 percent (permanent increase in annual consumption).
- Conditional welfare gains up to 0.45 percent.
- Stochastic steady-state (averages across simulations):
  - Hedging: debt 11.97 %, spreads 1.40 %, default prob 1.27 %
  - No-Hedging: debt 10.50 %, spreads 1.59 %, default prob 1.41 %
- Decomposition: ~90 percent of welfare gains (0.40 percent out of 0.44 percent) arises from reduced borrowing costs.
- Forwards vs options (μ = 1):
  - Forwards welfare gains 0.89 percent; Put options welfare gains 0.75 percent.
- Cost premium required to eliminate welfare gains: ~2.3 times actuarially fair price (equivalent to US$2.1 per barrel in 2009 dollars).
- Historical average cost paid by Mexico for put options (2006-2016): US$3.5 per barrel.

### Main policy-relevant conclusions
- Hedging via market insurance (put options or forwards) can provide measurable welfare gains in the presence of defaultable debt; baseline unconditional gain ≈ 0.44 percent of permanent consumption.
- The majority (about 90 percent) of the welfare benefit accrues through a reduction in borrowing costs rather than pure income smoothing.
- Hedging increases average borrowing capacity in the calibrated model (higher debt ratio under hedging) because lower borrowing costs and additional borrowing to pay for options both operate.
- Welfare gains are robust across many parameter variations but are sensitive to:
  - Option price premia above actuarially fair pricing (large premia needed to eliminate gains).
  - Strike choice μ (higher strikes raise welfare gains).
  - Oil-price persistence and volatility (high persistence reduces gains; high volatility increases gains).
  - Degree of hedged volume α and risk aversion γ.
- Selling oil forward (locking price at conditional mean) yields larger simulated welfare gains than put options in baseline comparisons, but political economy considerations (foregone upside) may make options preferable in practice.
- When international investors are risk averse and price both bonds and options with time-varying kernels, hedging can be even more valuable (welfare gains increase with investor risk aversion in the simulations).

*Source: IMF working paper — authors’ calculations and tables as provided in the content unit.*

### REFERENCES

### REFERENCES

### References
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- Arellano, Cristina, 2008, “Default risk and income fluctuations in emerging economies,” The American Economic Review, pp. 690–712.
- Baffes, John, M. Ayhan Kose, Franziska Ohnsorge, and Marc Stocker, 2015, “The Great Plunge in Oil Prices: Causes, Consequences, and Policy Responses,” World Bank Group Policy Research Note No. PRN/15/01.
- Blass, Javier, 2017, “Uncovering the Secret History of Wall Streets Largest Oil Trade,” https://www.bloomberg.com/news/features/2017-04-04/uncovering-the-secret-history-of-wall-street-s-largest-oil-trade.
- Borensztein, Eduardo, Eduardo Cavallo, and Olivier Jeanne, 2017, “The welfare gains from macro-insurance against natural disasters,” Journal of Development Economics, Vol. 124, pp. 142–156.
- , Olivier Jeanne, and Damiano Sandri, 2013, “Macro-hedging for commodity exporters,” Journal of Development Economics, Vol. 101, pp. 105–116.
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- Daniel, James, 2001, “Hedging Government Oil Price Risk,” IMF working papers.
- Duclaud, Javier, and Gerardo Garcia, “Mexicos Oil Price Hedging Program,” 2012, in Rabah Arezki, Catherine Pattillo, Marc Quintyn, and Min Zhu, eds., Commodity Price Volatility and Inclusive Growth in Low-Income Countries, (Washington D.C.: International Monetary Fund).
- Eaton, Jonathan, and Mark Gersovitz, 1981, “Debt with potential repudiation: Theoretical and empirical analysis,” The Review of Economic Studies, Vol. 48, No. 2, pp. 289–309.
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- Lucas, Robert E, 1987, Models of business cycles, Vol. 26 (Basil Blackwell Oxford).
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### Appendix I — Normalized Economy
- Original preferences: E0[Σt=0∞ βt C1−γt /(1−γ)].
- Total income: Yt = Ft + pt Qt.
- Budget constraint under no default:
  - Ct + qt Bt+1 + αQt+1 ξ( p̄t ) = Yt + Bt  (agents hedge Qt+1 production of oil at period t).
- Budget constraint under default:
  - Ct = Yt − H(Yt), where H(Yt) = h(yt) Ft.
- Normalization given Ft grows at constant rate G every period:
  - Normalized preference derivation leads to preferences expressed in terms of ct and G: E0[Σt=0∞ βt (Gt ct)1−γ /(1−γ)] and an overall factor F0.
- Normalized total income:
  - yt = Yt / Ft = 1 + pt Q / Ft = 1 + pt Q̄ (written as 1 + pt Q).
- Normalized budget constraint under no default:
  - yt + bt = ct + qt G bt+1 + α Q G ξ( p̄t ).
- Normalized budget constraint under default:
  - ct = yt − h(yt).

### Appendix II — Proofs (Propositions 1–3)
- Proposition 1 (default ordering and hedging effects):
  - If economy defaults in H state, it must default in L state since yH > yL.
  - Conditional on no default, optimal borrowing d* satisfies:
    - 1/(y + d*) = β ( p/(yH − d*) + (1−p)/(yL − d*) ).
  - Define ŷdef = yL − d* and ŷ̂def = yH − d**; establish default thresholds.
  - With hedging, optimal borrowing d*,hedge satisfies:
    - 1/(y + d*,hedge) − ξ = β ( p/(yH − d*,hedge) + (1−p)/( ̄y − d*,hedge) ).
  - d*,hedge > d* and d**,hedge > d** imply ̄ŷdef,hedge < ŷ̂def, but ̄ydef,hedge > ŷdef due to cL,hedge1 > cL1.
- Proposition 2 (welfare with hedging when no default):
  - Without default, optimal allocation satisfies:
    - 1/(y + d*) = β ( p/(yH − d*) + (1−p)/(yL − d*) ).
  - With hedging:
    - 1/(y + d*,hedge) − ξ = β ( p/(yH − d*,hedge) + (1−p)/( ̄y − d*,hedge) ).
  - d*,hedge > d*.
  - Welfare comparison: show Uhedge0(d* + ξ( ̄y)) − U0(d*) > 0 via:
    - Uhedge0(d* + ξ( ̄y)) − U0(d*) = β [ p log(yH − ξ( ̄y) − d*) + (1−p) log( ̄y − ξ( ̄y) − d*) ] − β [ p log(yH − d*) + (1−p) log(yL − d*) ] > 0.
  - Uses function f(x) = β [ p log(yH − (1−p)x − d*) + (1−p) log(yL + p x − d*) ] increasing in x ∈ [0, yH − yL].
- Proposition 3 (hedging when default only in low state):
  - First-order conditions for debt when default only in L state:
    - p/(y + p d**) = β p/(yH − d**).
  - With hedging:
    - 1/(y + d*,hedge) = β ( p/(yH − d*,hedge) + (1−p)/( ̄y − d*,hedge) ).
  - Sign of d** versus d*,hedge is ambiguous; marginal benefit and cost both change.
  - If hedging does not change default incentives, social welfare is lower because economy borrows more and consumption streams are unambiguously lower. If hedging increases default incentives, welfare is further reduced; in that case there is no borrowing.

### Appendix III — Two-Period Model with Forwards; Algorithm; Estimation of Oil Price Process
- Proposition 4 (forwards in two-period model):
  - Two-period setup with forwards:
    - Uforwards0 = maxd log c0 + β log c1 s.t. c0 = y + d, c1 = max{ ̄y − d, ydef } where ̄y = p yH + (1−p) yL.
  - Optimality when no default:
    - 1/(y + d*,forwards) = β 1/( ̄y − d*,forwards ).
  - d* < d*,forwards since marginal cost of borrowing decreases with forwards.
  - Inequality chain:
    - β 1/( ̄y − d*,forwards ) = 1/(y + d*,forwards ) < 1/(y + d*) = β ( p/(yH − d*) + (1−p)/(yL − d*) ) < β 1/(yL − d*),
    - implies ̄y − d*,forwards > yL − d* and thus ŷdef,forwards > ŷdef.
  - Welfare comparisons depend on default patterns: forwards can increase or reduce welfare depending on which states default under each regime.
- Algorithm for solving model (value function iteration):
  1. Start with initial guess {Vi(b,p), Vdi(p), Vci(b,p)} for each b ∈ B and p ∈ P for iteration i = 0.
  2. Update Vd i+1(p) using equation (8).
  3. Update Vc i+1(b,p) according to equation (7).
  4. Update Vi+1(b,p) using Vi+1(w,p) = max{ Vc i+1(b,p), Vd i+1(p) }.
  5. Calculate implied bond price:
     - qi+1(b′, p) = Ep′|p [ 1{ Vc i+1(b′, p′) ≥ Vd i+1(p′) } ] / (1 + r*).
  6. Iterate until qj(b′, p), Vj(b,p), Vc j(b′, p′) and Vd j(p) are sufficiently close for j = i and j = i + 1.
- Estimation of oil price process (AR(1) in logs):
  - Unconditional oil price estimate: p̂ = (1/T) ΣT t=1 pt.
  - Model:
    - log pt = (1 − ρ) [ log( p̂ ) − 1/2 σ2 /(1 − ρ2) ] + ρ log pt−1 ︸︷︷︸ μt−1 + εt.
  - Conditional density:
    - f(log pt | pt−1, p, ρ, σ2) = 1/√(2πσ2) e−(log pt − μt−1)2 /(2σ2).
  - Likelihood:
    - L = ΠT t=2 1/√(2πσ2) e−(log pt − μt−1)2 /(2σ2).
    - log L = −(T − 1)/2 log(2πσ2) − 1/(2σ2) ΣT t=2 (log pt − μt−1)2.
  - First-order conditions:
    - ∂log L / ∂ρ = −1/(2σ2) ΣT t=2 2 (log pt − μt−1)(−log pt−1) = 0.
    - ∂log L / ∂σ2 = −(T − 1)/(2σ2) + 1/(2(σ2)2) ΣT t=2 (log pt − μt−1)2 = 0.

### Appendix V — Option Pricing
- Put option payoff: max{ ̄p − pt+1, 0 } for strike price ̄p and current price pt at time t.
- Risk-neutral pricing formula:
  - ξ(pt) = Et[ max{ ̄p − pt+1, 0 } ] / (1 + r*).
- With log pt following AR(1):
  - log pt+1 ∼ N( μt, σ2 ), where
    - μt = (1 − ρ) [ log(p) − 1/2 σ2 /(1 − ρ2) ] + ρ log pt.
- Closed form for ξ(pt):
  - ξ(pt) = ̄p/(1 + r*) Φ( (log ̄p − μt)/σ ) − 1/(1 + r*) eμt+σ2/2 Φ( (log ̄p − μt − σ2)/σ ).

*Italic: Source: wp1835 - REFERENCES (PDF chapter/section).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp1835.pdf_