## _wp1054

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---

### I. Introduction — context and research question
- Over the last decade many emerging and developing countries improved their net foreign asset position (Lane and Milesi-Ferretti, 2007).
- Two hypotheses for reserve accumulation:
  - Export-led growth strategies (Dooley, Folkerts-Landau and Garber, 2003).
  - Self-insurance / precautionary demand against balance of payments crises (Jeanne and Ranciere, 2006; Kim, 2008; Durdu, Mendoza, and Terrones, 2009; Aisenman and Lee, 2007; Alfaro and Kanczuk, 2009; Caballero and Panageas, 2008; others).
- Objective: derive estimates of optimal levels of net foreign assets (NFA) for Bolivia using a calibrated precautionary savings model with borrowing constraints, focusing on current account shocks.
- Model extension: variant where agents also choose investment to capture saving for shielding consumption and financing investment.
- Key high-level finding:
  - Model simulations focusing on current account shocks suggest an optimal level of net foreign assets for Bolivia between 29 to 37 percent of GDP.
  - Bolivia became a net external creditor in 2008.
  - Even under fairly conservative calibrations, Bolivia would be around the optimal NFA range estimated by the model.
- Caveat: model yields NFA for the whole economy; empirical comparison uses financial system foreign reserves (including the central bank) as the observable proxy.

### II. Bolivia: external sector developments (calibration context)
- Reserve-accumulation drivers:
  - Major terms of trade improvement from sharp increases in key export commodity prices during 2004–2008.
  - Important expansion in natural gas export volumes.
  - Migrant remittances increased fivefold between 2003 and 2008 to about 6 percent of GDP.
  - Debt relief contributed substantially to improvement in Bolivia’s net foreign asset position.
- Policy/institutional context:
  - Crawling peg exchange rate regime.
  - Absence of an institutional arrangement to manage hydrocarbons wealth; public sector deposits of hydrocarbon receipts were held at the central bank.
- Empirical implication: terms of trade behavior for Bolivia resembles that of energy exporters as natural gas exports gained importance.
- Period emphasized for export-price-driven accumulation: 2004–2008.

### III. Model — structure and assumptions
- Baseline: standard precautionary savings model in partial equilibrium (variant of Carroll, 2004).
- Preferences: maximize E[Σ β^t u(C_t)] with CRRA utility u(.).
- Consumers consume only tradable goods; income process exogenous.
- Sequence of events:
  - Start of period: consumer holds net foreign assets X_t and chooses consumption C_t.
  - Middle of period: remainder invested in a risk-free security yielding R.
  - End of period: income realized determining assets for t+1.
- Law of motion (normalized by permanent income):
  - x_{t+1} = R (x_t - c_t) + a + τ_{t+1}, where τ = transitory shocks (mean-one, i.i.d.), and a denotes time-invariant A/Y.
- Borrowing constraint: c_t ≤ x_t (consumption cannot exceed current net foreign assets).
- Shocks decomposition: τ = ϛ γ, with ϛ representing terms of trade shocks and γ representing export volume shocks; ϛ and γ assumed i.i.d. and independent.
- Euler equation (normalized): u'(c_t) = β R E[u'(c_{t+1})].
- Modeling choices and implications:
  - Non-tradables excluded — likely yields conservative (lower) optimal NFA estimates.
  - Borrowing constraints capture inability to tap international capital markets (sudden stops).
  - Impatience condition: 1/β > R (i.e., Rβ < 1) required for a well-defined solution.

### IV. Analytical illustrative case — closed-form insights
- Simplifying assumptions for analytical insight:
  - No terms of trade shocks: ϛ_{t+1} = 1.
  - Volume shock γ takes values 1 or 0 with probabilities 1-Θ and Θ respectively (Θ is sudden depletion probability).
  - Relative risk aversion ρ = 1 for compact expressions.
- Perfect-foresight solution when resources exhausted: c_t = (1 - (Rβ)^{1/ρ}) x_t.
- Comparative statics:
  - An increase in Θ raises the optimal level of reserves; marginal impact largest at low Θ.
- Policy implication:
  - If central bank objectives align with private smoothing motives, central bank-held reserves provide insurance to private agents.

### V. Numerical solution approach and calibration overview
- Full model (with both terms of trade and export volume shocks) solved numerically via dynamic programming using Carroll (2006)’s endogenous gridpoints method.
- Baseline calibration choices (Appendix B):
  - Utility function: CRRA u(c_t) = c_t^(1-ρ)/(1-ρ) with ρ = 2.
  - Real return on net foreign assets: 2 percent a year.
  - Discount factor β = 0.94 (chosen to guarantee impatience condition).
  - Period interpretation: roughly 1 year.
  - Terms of trade and export volume shocks: i.i.d., log-normally distributed.
    - Baseline parameters match volatility of terms of trade shocks for energy exporters (WEO, period 1970−2010).
    - Alternative calibration matches Bolivia’s terms of trade volatility over 1970−2010 (less volatile).
  - Export volume shocks: volatility computed from World import volumes growth over 1980−2010; regression of Bolivian export volume growth on world import growth had slope not statistically different from one → assume one-for-one transfer.
  - With investment model: α = 0.2; depreciation δ = 10 percent a year.
  - Net current receipts of non-export components of the current account: average over 1998–2008 yields 11 percent of exports.

### VI. Optimal reserves mapping and baseline quantitative results
- Optimal reserves expressed as ratio to long-run consumption (optimal reserves/imports ratio), then converted to percent of GDP:
  - Optimal reserves/imports ratio:
    - 1.02 for the lower volatility (Bolivia-like) case.
    - 1.20 for the baseline (energy-exporter volatility) case.
  - Procedure: HP-filter with λ=1600 over 1980–2008 on observed imports-to-GDP to approximate long-run imports, then back out reserves as percentage points of GDP.
  - Corresponding optimal levels (as of 2009):
    - 25 percent of GDP for the lower volatility case (as of 2009).
    - 29 percent of GDP for the baseline case (as of 2009).
- Model extension with investment:
  - Precautionary savings both self-insure and finance capital.
  - Optimal reserves/imports ratio increases to 1.52.
  - Backed out level in terms of GDP: 37 percent of GDP (with investment).
- Consolidated model result:
  - Both model versions yield optimal levels between 29 and 37 percent of GDP; larger role for entrepreneurs → closer to 37 percent.

### VII. Sensitivity analysis (discount factor β and risk aversion ρ)
- Table 1: Optimal Net Foreign Assets, in percent of GDP

  Without Investment (columns labeled Β = 0.92, 0.94*, 0.96)
  - ρ = 1: 25.4, 26.6, 32.2
  - ρ = 2*: 27.9, 29.4, 37.9
  - ρ = 3: 29.8, 31.8, 42.8

  With Investment (columns labeled Β = 0.92, 0.94*, 0.96)
  - ρ = 1: 25.7, 28.4, 35.7
  - ρ = 2*: 32.5, 37.4, 50.6
  - ρ = 3: 39.0, 46.1, 63.9

- Main sensitivity findings:
  - Substantial variation in optimal NFA as β and ρ vary.
  - As risk aversion increases and impatience decreases, the excess of observed reserves relative to optimal reserves virtually disappears when compared to the current actual level of slightly below 60 percent.
  - However, such calibrations imply opportunity cost of holding reserves would fall below 4 percent, which is argued to be unrealistic given average interest rates on existing foreign debt or open market instruments.
  - Interpreting β = 0.94 as an upper bound for impatience and allowing a higher ρ (say 3) implies Bolivia would be at the optimum when stripping gold and NFPS dollar-deposits; even under fairly conservative calibrations, the model yields Bolivia around the optimum.

### VIII. Mapping to observed reserves and alternative definitions
- Observed reserves measure: consolidated financial system (central bank and financial institutions).
- Alternative reserve definition excludes:
  - Gold holdings (assumed minimum level; historically not used in severe crisis of 1982–1985).
  - Dollar-denominated public sector deposits in the Central Bank (likely not part of reserves if government had a dedicated hydrocarbons-wealth arrangement).
- Comparison results:
  - Actual level exceeds model-suggested optimal under baseline scenario by about 25 percentage points of GDP.
  - Under alternative definition (excluding gold and NFPS dollar deposits), the excess is about 15 percent of GDP.
- Note: model-based optimal levels are not sensitive to the business cycle; observed time-variation reflects method to measure long-run level of imports. State-dependent modification could introduce business-cycle dependence.

### IX. Benchmarks: rules of thumb
- Months-of-imports rule: conventional wisdom is at least three months of import coverage. Bolivia far exceeds this threshold.
- Guidotti-Greenspan rule: reserves should cover 100 percent of short-term foreign debt. Bolivia’s short-term foreign debt is very low (even on remaining maturity basis) given limited international capital market integration and mostly long-term public debt, resulting in a very large indicator of reserve adequacy under this rule.

### X. Conclusions and policy implications
- Bolivia became a net external creditor in 2008 following a surge in commodity prices and export volumes.
- Calibrated precautionary savings model with borrowing constraints and an extension including investment produced optimal net foreign asset estimates in the range 29–37 percent of GDP.
- Standard rule-of-thumb measures suggest excess reserves substantially above model-based calculations.
- Results sensitive to parameter calibration and to the definition of net foreign assets; even under conservative calibrations, Bolivia would be close to the model-implied optimum.
- Comparing economy-wide optimal NFA with financial-system reserves is conservative because it omits the non-financial private sector.
- Policy implication: need to properly account for country-specific balance-of-payments risks when deriving reserve adequacy indicators.

### XI. Appendix highlights (derivations and calibration)
- Appendix A: closed-form derivations solve for consumption when resources are exhausted and for the target level of reserves x* using budget constraints and Euler equations; intermediate algebra sets ρ = 1 for compactness while full derivations maintain general ρ.
- Appendix B: calibration summary (see Section V above for parameter list).

*Source: _wp1054 — Appendix and extracted sections on parameter choices, model structure, results, and conclusions.*

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

### _wp1054 - References

### I. Introduction — context and research question
- Over the last decade many emerging and developing countries improved their net foreign asset position (Lane and Milesi-Ferretti, 2007).
- Two key hypotheses for reserve accumulation:
  - Export-led growth strategies (Dooley, Folkerts-Landau and Garber, 2003).
  - Self-insurance against balance of payments crises / precautionary demand (Jeanne and Ranciere, 2006; Kim, 2008; Durdu, Mendoza, and Terrones, 2009; Aisenman and Lee, 2007; Alfaro and Kanczuk, 2009; Caballero and Panageas, 2008; others).
- Paper objective: derive estimates of optimal levels of net foreign assets (NFA) for Bolivia using a calibrated precautionary savings model with borrowing constraints, focusing on current account shocks.
- Model extension: a version where agents also choose investment levels to capture saving for shielding consumption and financing investment.

Key findings summarized:
- Model simulations focusing on current account shocks suggest an optimal level of net foreign assets for Bolivia between 29 to 37 percent of GDP.
- Bolivia became a net external creditor in 2008.
- Even under fairly conservative calibrations, Bolivia would be around the optimal NFA range estimated by the model.
- Comparison caveat: model yields NFA for the whole economy; empirical comparison uses financial system foreign reserves (including the central bank) as the readily available measure.
- Additional roles of reserves (e.g., signaling) are acknowledged but not modeled.

### II. Bolivia: external sector developments
- Recent sharp increase in foreign reserves in Latin America was particularly accentuated for Bolivia, Brazil, and Peru.
- Drivers of Bolivia’s reserve accumulation:
  - Major terms of trade improvement from sharp increases in key export commodity prices during 2004–2008.
  - Important expansion in natural gas export volumes.
  - Migrant remittances increased fivefold between 2003 and 2008 to about 6 percent of GDP.
  - Debt relief also contributed substantially to improvement in Bolivia’s net foreign asset position.
- Policy and institutional context:
  - Bolivia operated a crawling peg exchange rate regime.
  - Absence of an institutional arrangement to manage hydrocarbons wealth; public sector deposits of hydrocarbon receipts were held at the central bank.
- Empirical implication for calibration:
  - Terms of trade behavior for Bolivia resembles that of energy exporters as natural gas exports gained importance.

Key statistics and timeline:
- Remittances: increased fivefold between 2003 and 2008 to about 6 percent of GDP.
- Bolivia became a net external creditor in 2008.
- Period emphasized for export-price-driven accumulation: 2004–2008.

### III. The model — structure and assumptions
- Baseline: standard precautionary savings model in partial equilibrium (variant of Carroll, 2004).
- Preferences: maximize E[Σ β^t u(C_t)] with constant relative risk aversion utility u(.).
- Consumers consume only tradable goods; income process exogenous.
- Sequence of events:
  - Beginning of period: consumer holds net foreign assets X_t and chooses consumption C_t.
  - Middle of period: remainder invested in a risk-free security yielding R.
  - End of period: income realized determining assets for t+1.
- Law of motion for net foreign assets (normalized by permanent income):
  - x_{t+1} = R (x_t - c_t) + a + τ_{t+1}, where τ = transitory shocks (mean-one, i.i.d.), and a denotes time-invariant A/Y.
- Borrowing constraint introduced: c_t ≤ x_t (consumption cannot exceed current net foreign assets).
- Shocks decomposition: τ = ϛ γ, with ϛ representing terms of trade shocks and γ representing export volume shocks; ϛ and γ assumed i.i.d. and independent.
- Euler equation (normalized problem): u'(c_t) = β R E[u'(c_{t+1})].

Modeling choices and implications:
- Non-tradables excluded — likely yields conservative (lower) estimates of optimal NFA because inclusion of non-tradables would typically raise optimal NFA (but sensitive to elasticity of substitution).
- Borrowing constraints capture inability to tap international capital markets (sudden stops).
- Impatience condition needed for well-defined solution: 1/β > R (i.e., Rβ < 1).

### IV. Analytical illustrative case — optimal NFA and risk
- Simplifying assumptions for analytical insight:
  - No terms of trade shocks: ϛ_{t+1} = 1.
  - Volume shock γ takes values 1 or 0 with probabilities 1-Θ and Θ respectively (interpreted as sudden depletion probability Θ).
  - Coefficient of relative risk aversion set to 1 (ρ = 1) for closed-form expression.
- Perfect-foresight solution for an economy that has exhausted resources: c_t = (1 - (Rβ)^{1/ρ}) x_t.
- Analytical formula for optimal normalized net foreign assets under these assumptions:
  - x* = [1/(1 - Rβ)] * { (1 + (1 - Θ) * [ ... ] ) }  (presented in the paper as equation (11); the expression shows x* increasing in Θ).
- Comparative statics:
  - An increase in Θ (risk of resource depletion) raises the optimal level of reserves.
  - Marginal impact largest at initially low levels of Θ.
- Policy implication:
  - If central bank objectives align with private consumers’ smoothing motives, central bank-held reserves provide insurance to private agents.

### V. Numerical solution approach (preview)
- Full model (with both terms of trade and export volume shocks) solved numerically via dynamic programming using Carroll (2006)’s endogenous gridpoints method.
- Calibration details and quantitative experiments follow in subsequent sections (not fully included in the extracted content).

Final methodological notes:
- The paper emphasizes properly accounting for country-specific balance-of-payments risks when deriving reserve adequacy measures.
- Model produces NFA estimates for the economy as a whole; empirical comparison uses financial-system foreign reserves (central bank included) as observable proxy.

*Source: _wp1054 - References (extracted content).*

### section in the appendix discusses the chosen values for the parameters of the model. The

### Appendix: Chosen Parameter Values and Model Results

### Consumption rule and baseline calibration
- The model solution is a time-invariant consumption rule c(x) as a function of net foreign assets that determines consumers’ decisions; the baseline optimal consumption rule is depicted in Figure 4.
- Baseline calibration: parameters of the distribution of terms of trade shocks calibrated to match the volatility of the terms of trade series for energy exporters (thick solid line). Alternative calibration: match Bolivia’s historical terms of trade volatility (thin solid line), which is less volatile.
- In both calibrations, the variance of export volume shocks is set equal to the baseline calibration.
- The 45 degree line in Figure 4 shows when the borrowing constraint binds: consumption of tradable goods can at most equal the level of net foreign assets.
- The vertical line denotes the baseline-optimal level of foreign assets, which lies to the right of the point at which the liquidity constraint begins to bind.
- The buffer stock of net foreign assets minimizes the likelihood of consumers falling into the constrained region.

### Mapping optimal reserves to data
- The optimal level of reserves is expressed as a ratio to the long-run consumption level (optimal reserves/imports ratio) and then taken to the data.
- Resulting ratios:
  - 1.02 for the lower volatility (Bolivia-like) case.
  - 1.20 for the baseline (energy-exporter volatility) case.
- Procedure to express as percent of GDP: apply an HP-filter with λ=1600 over the period 1980–2008 to the observed imports-to-GDP ratio to approximate long-run imports, then back out reserves as percentage points of GDP.
- Corresponding optimal levels (calibrations shown in Figure 4) yield:
  - 25 percent of GDP for the lower volatility case (as of 2009).
  - 29 percent of GDP for the baseline case (as of 2009).

### Comparison of suggested optimal reserves with observed reserves
- Measure of observed reserves: consolidated financial system (central bank and financial institutions).
- Additional alternative measure: excluding gold holdings and dollar-deposits from the public sector in the central bank because:
  - Gold is assumed to be a minimum level of reserves (gold not used during severe crisis of 1982–1985).
  - Dollar-denominated public sector deposits in the Central Bank are likely not part of reserves if the government had a dedicated arrangement to manage hydrocarbons wealth; local-currency component assumed destined for non-tradable goods and not deducted.
- Comparison results:
  - Actual level exceeds the model-suggested optimal under the baseline scenario by about 25 percentage points of GDP.
  - Under the alternative definition of reserves (excluding gold and NFPS dollar deposits), the excess is about 15 percent of GDP.
- Note: model-based optimal levels are not sensitive to the business cycle; observed time-variation reflects the approach to measure long-run level of imports. A state-dependent modification could introduce business-cycle dependence.

### Model extension: Adding investment
- Extended model includes investment (capital goods used in production of export goods) following a structure similar to Sandri (2009). Income is endogenous; entrepreneurs own projects producing export-related goods.
- Key model features:
  - Investment decision variable k; income from exports τ t+1 k_t^α with decreasing marginal returns; depreciation fraction δ.
  - Capital enters the budget constraint contemporaneously to simplify tracking of the capital stock.
  - First-order conditions yield Euler equations (Equations (20) and (21)); Equation (21) relates marginal utility of current consumption to expected marginal utility of investing in the risky production project with rate of return 1 + [ (1-α) τ_t+1 k_t^α ] / (something derived in text).
- Solution results:
  - Consumption function with investment c_e(x) lies below the one without investment: consumer cuts consumption to finance capital.
  - Precautionary savings serve both to self-insure and to finance capital stock; ratio of optimal reserves to long-run consumption increases to 1.52 (up from 1.20 without investment).
  - Backed out level in terms of GDP: 37 percent of GDP (with investment).
  - Both model versions yield optimal levels between 29 and 37 percent of GDP; larger importance of entrepreneurs implies closer to 37 percent.

### Sensitivity analysis (discount factor β and risk aversion ρ)
- Table 1: Optimal Net Foreign Assets, in percent of GDP

  Without Investment (columns labeled Β = 0.92, 0.94*, 0.96)
  - ρ = 1: 25.4, 26.6, 32.2
  - ρ = 2*: 27.9, 29.4, 37.9
  - ρ = 3: 29.8, 31.8, 42.8

  With Investment (columns labeled Β = 0.92, 0.94*, 0.96)
  - ρ = 1: 25.7, 28.4, 35.7
  - ρ = 2*: 32.5, 37.4, 50.6
  - ρ = 3: 39.0, 46.1, 63.9

  - * Denotes baseline calibrations.
- Main findings from sensitivity experiments:
  - Substantial variation in optimal net foreign assets as β and ρ vary.
  - As risk aversion increases and impatience decreases, the excess amount of reserves relative to actual holdings virtually disappears when compared to the current actual level of slightly below 60 percent.
  - However, such calibrations imply opportunity cost of holding reserves would fall below 4 percent, which is unrealistic given average interest rates on existing foreign debt or open market instruments.
  - Interpreting β=0.94 as an upper bound for impatience and allowing a higher ρ (say 3) implies Bolivia would be at the optimum when stripping gold and NFPS dollar-deposits; even under fairly conservative calibrations, the model yields Bolivia around the optimum.

### Benchmarks: Rules of thumb
- Months-of-imports rule: conventional wisdom is at least three months of import coverage. Bolivia far exceeds this threshold (Figure 8).
- Guidotti-Greenspan rule: reserves should cover 100 percent of short-term foreign debt. Bolivia’s short-term foreign debt is very low (even on remaining maturity basis) due to limited international capital market integration and mostly long-term public debt, resulting in a very large indicator of reserve adequacy under this rule.

### Conclusions
- Bolivia became a net external creditor in 2008 following a surge in commodity prices and export volumes; similar trends observed elsewhere.
- Calibrated precautionary savings model with borrowing constraints and an extension including investment produced optimal net foreign asset estimates in the range 29–37 percent of GDP.
- Standard rule-of-thumb measures suggest excess reserves substantially above model-based calculations.
- Results are sensitive to parameter calibration and to the definition of net foreign assets; even under conservative calibrations, Bolivia would be close to the model-implied optimum.
- Comparing economy-wide optimal net foreign assets with financial-system reserves is conservative because it omits the non-financial private sector.
- Policy implication: need to properly account for country-specific balance of payment risks when deriving reserve adequacy indicators.

### Appendix A: Derivation highlights
- Closed-form derivations in the appendix solve for consumption when resources are exhausted and for the target level of reserves x* using the budget constraints and Euler equations.
- A simplifying analytical assumption is setting relative risk aversion ρ = 1 in intermediate algebra to obtain a more compact expression; full derivations maintain general ρ.

### Appendix B: Calibration (parameters used)
- Utility function: CRRA u(c_t) = c_t^(1-ρ)/(1-ρ) with ρ = 2.
- Real return on net foreign assets: 2 percent a year.
- Discount factor β = 0.94 (chosen to guarantee impatience condition).
- Period interpretation: roughly 1 year.
- Terms of trade and export volume shocks: i.i.d., log-normally distributed.
  - Parameters chosen to match volatility of terms of trade shocks for energy exporting countries (WEO, period 1970−2010).
  - Alternative: parameters matching Bolivia’s terms of trade volatility over 1970−2010.
- Export volume shocks: volatility computed from World import volumes growth over 1980−2010; regression of Bolivian export volume growth on world import growth had slope not statistically different from one → assume one-for-one transfer.
- With investment model:
  - Share of capital in production function α = 0.2.
  - Depreciation rate δ = 10 percent a year.
- Net current receipts of non-export components of the current account: average over 1998–2008 yields 11 percent of exports.

*Source: Appendix section on parameter choices and model calculations from the provided IMF working paper content.*

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