## _wp11249

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### I. Introduction and paper objectives
- Develops a tractable analytical framework to quantify reserve levels rationalized as insurance against large external shocks faced by low-income countries.
- Empirically estimates the benefits of reserves in smoothing domestic absorption in response to external shocks.
- Calibrates optimal reserves for country groups using regression estimates, country-specific fundamentals, and simplified assumptions about the cost of holding reserves and risk-neutrality.
- Sample and definition notes:
  - “Low-income countries” refers to all countries shown on the IMF’s list of countries eligible for the Poverty Reduction and Growth Trust (PRGT) at end-December 2010.
  - External shock episode analysis uses data for the period 1990-2008.

### II. Reserve accumulation patterns and empirical model of precautionary demand
- Reserve accumulation patterns (descriptive)
  - Median reserve coverage in 2009: 4.7 months of imports, and 55 percent of broad money.
  - Since 2003, ratios of reserves to imports increased most sharply for oil exporters.
  - Sub Saharan African (SSA) countries have had persistently higher reserve coverage than the median low-income country, although this difference has narrowed in recent years.
  - Countries with GDP per capita less than US$ 500 have been well-below the low-income country median throughout the 2000s.
  - Reserve accumulation has outpaced the conventional rule of thumb of three months of import cover for both fixed and floating exchange rate regimes; countries with fixed regimes generally tend to have higher reserve coverage.
  - As of 2009, over a quarter of all countries had reserve levels lower than suggested by the import cover metric.
  - Drivers of accelerating reserve build-up: low initial reserve holdings, increasing openness of economies, a favorable global environment, and policy choice to build precautionary reserves against balance of payment risks.
  - Note: The increase in reserves in 2009 is largely attributable to the SDR allocation in response to the global financial crisis.

- Empirical model of precautionary demand (estimation)
  - Purpose: assess whether recent accumulation in low-income countries aligns with fundamentals via a cross-country empirical model of precautionary demand for reserves.
  - Regression sample and estimation:
    - Panel data for 62 low-income countries (excluding economies with a population less than 1 million) from 1992 to 2001 used for estimation.
    - Remaining years used to compare out-of-sample forecasts with actual reserve buildups (out-of-sample comparison period: 2002-2008; excluding 2009 SDR allocation).
  - Main regression findings (full sample, Column 1):
    - Reserve holdings positively and significantly related to indicators of current account vulnerability (import ratio and export volatility).
    - Reserve holdings positively and significantly related to indicators of capital account vulnerability, such as broad money.
    - Exchange rate volatility and fixed exchange rate regimes significantly associated with higher reserve holdings.
    - The proxy for cost of holding reserves (interest rate differential between government treasury bill and corresponding U.S. asset) has the expected sign but lacks statistical significance.
    - The empirical model for the full sample accounts for over 60 percent of the variation in reserves (excluding country fixed effects).
  - Subsample differences:
    - Exchange rate volatility is more important for non-commodity exporters.
    - Export volatility is highly significant in explaining reserve demand in commodity exporters.

### III. Analytical framework: cost-benefit approach to reserve adequacy
- Motivation and distinctions
  - Traditional metrics offer rough guidance and lack empirical/theoretical foundations; propose a cost-benefit framework to determine optimal reserve levels for low-income countries.
  - Low-income countries face different shocks than EMs: sharp swings in foreign aid, remittances, FDI, and natural disasters; higher frequency/incidence of terms-of-trade shocks.
  - Limited market access implies external drains primarily on the current account; current-account based measures (e.g., months of imports) remain useful indicators.

- Insurance perspective and cost of holding reserves
  - Reserves can reduce likelihood and magnitude of abrupt drops in consumption/absorption from external shocks.
  - Empirical evidence: cumulative consumption losses over five years were about 6-17 percentage points for countries with reserve coverage of less than three months of imports, whereas impact was limited among those with higher coverage.
  - Cost of holding reserves:
    - Typically defined as difference between return on short-term foreign currency assets and return on alternative investments.
    - For many low-income countries with limited market access, net financial cost may be negative because concessional borrowing rates are typically lower than reserve returns.
    - Opportunity cost proxy: domestic-foreign real interest rate differential; calibrations use unit cost r in reference values between 2 and 6 percent (adjusted for real financial return on reserves of about 1 percent a year).

- Objective function (precautionary motive)
  - Framework maximizes trade-off between expected crisis utility cost and unit cost of holding reserves:
    - Key parameters: conditional probability of a crisis P(R,Z), real absorption loss C(R,Z), unconditional probability of a large shock q, and unit cost of holding reserves r.
  - Limitations:
    - Utility cost measured by loss in real absorption (percent of GDP) assuming risk-neutral utility; framework simulates a lower bound of optimal reserves.
    - Static objective function—alternative insurance from multilateral/bilateral sources may justify simplification.

### IV. Empirical analysis: shocks, crisis definition, and probability estimation
- Shock identification and sample
  - Sample: 1990-2008 for 71 low-income countries.
  - Large negative external shock episodes identified when annual percentage change of relevant variable falls below the 10th percentile in country-specific left-tail distribution.
  - Six shock types: (i) external demand; (ii) terms-of-trade; (iii) FDI; (iv) aid; (v) remittances; (vi) climatic shocks (large natural disasters).
  - For natural disasters, top 25th percentile in number affected and economic damage (EM-DAT/CRED).
  - Total shock observations: 645 shock observations (out of 1,349 observations).
  - Correlation between different shocks ranges from -0.05 to 0.05.

- Crisis definition and descriptive findings
  - A crisis occurs when:
    - (i) The post-shock two-year average (t and t+1) level of real absorption per capita falls below the pre-shock three-year trend; and
    - (ii) Growth of real absorption per capita is negative at time t.
  - Drop in terms-of-trade: over 30 percent for bottom 25th percentile of the shock sample, versus less than 5 percent for non-shock sample.
  - Median growth of real absorption per capita is positive for the entire shock sample—implying not all shocks lead to absorption drops.
  - Unconditional probability of a crisis within the shock sample: less than 30 percent.
  - There is a statistically significant difference of more than 8 percentage points in real absorption growth per capita between crisis and non-crisis cases.

- Estimating the conditional probability of a crisis
  - Panel probit model for conditional probability of a crisis estimated for 49 low-income countries.
  - Final explanatory variables: reserves (in months of imports), ratio of government balance to GDP, World Bank’s CPIA index, dummy for flexible exchange rate regime, and dummy for Fund-supported programs. All lags by one year except contemporaneous Fund program dummy.
  - Results (Table 3 baseline):
    - Reserves, months of imports (t-1): -0.0896*** (0.0339)
    - Flexible exchange rate regime (t-1): -0.3801*** (0.1366)
    - Government balance, % of GDP (t-1): -0.0323*** (0.0125)
    - CPIA index (t-1): -0.3090*** (0.1056)
    - IMF program (t): -0.3021** (0.1409)
    - No. of observations: 445; Pseudo R2: 0.1099
  - Marginal effects (Table 4 example):
    - Sample mean of reserves: 3.2 months of imports.
    - Increasing reserves from 3.2 months to 5 months of imports reduces probability of an absorption drop by around 3.6 percentage points (assuming Fund-supported program and evaluating other variables at sample means under a flexible exchange rate regime).
    - Estimated change in crisis probability tends to be higher (over 5 percentage points) under fixed exchange rate regimes.

### V. Crisis probability and mitigation quantitative findings
- Crisis probability: selected marginal effects (preserve reported exact values)
  - A reserves increase from 3.3 to 3.7 (months of imports) reduces the crisis probability by 3 percentage points under a flexible regime and by 4.2 percentage points under a fixed regime.
  - An improvement in the government balance from 4.5 to 3 percent of GDP reduces the estimated annual probability of a crisis by over 1 percentage points.
  - Shifting from a fixed to a flexible exchange rate regime reduces the likelihood of an absorption drop by 11 percentage points.
  - Having an IMF Fund-supported program in the event of a shock reduces the crisis probability by:
    - approximately 8.8 percentage points under a flexible regime, and
    - approximately 10.9 percentage points under a fixed exchange rate regime.
  - Increasing reserve coverage from three to four months of imports reduces the probability of a crisis by about 3.5 percentage points under a fixed exchange rate regime; the marginal effect is smaller under a flexible regime.

- Crisis mitigation (severity) findings
  - Absorption loss regression (Table 5 baseline):
    - Log of reserves, months of imports (t-1): -2.2403*** (0.6677)
    - Flexible exchange rate regime (t-1): -8.6983*** (2.1689)
    - External demand growth (t): -0.9320** (0.4356)
    - Terms of trade growth (t): -0.0841* (0.0484)
    - N: 418; Adjusted R2: 0.34
  - Interpretation:
    - Higher reserve holdings associated with lower absorption losses; marginal effect diminishing in level of reserves (log specification).
    - A flexible exchange rate regime reduces real absorption loss by around 9 percent of GDP relative to the fixed regime, after controlling for shocks and country fixed effects.
    - Fixed-effects OLS absorption loss regression accounts for 35 percent of the variation in absorption loss across countries.

- Robustness
  - Results broadly robust across sample splits and specification checks.
  - Reserves variable remains highly significant and of expected sign except when substantially reducing sample by dropping African countries.
  - Similar results when using real consumption per capita drops instead of absorption drops.
  - Alternative reserve specifications yield very similar optimal reserve levels to the log specification.

### VI. Calibration of optimal precautionary reserves: setup, results, and sensitivity
- Calibration setup and parameters
  - Model parameters: conditional probability P(R,Z), real absorption loss C(R,Z), unconditional probability q, unit cost r.
  - Sample estimate of unconditional probability q = 0.5 (sample average).
  - Unit cost of holding reserves r considered at reference values between 2 and 6 percent.
  - Economic fundamentals (fiscal balance, CPIA index) set to five-year average over 2003-07 for each country group.
  - Shock values taken from sample median for country groups; estimated real absorption loss augmented by one standard deviation of residuals (roughly the upper 85th percentile) to capture risk aversion.
  - Calibration assumes availability of contingent IMF support in the event of large external shocks (this affects P(R,Z)).

- Calibration results and sensitivity (selected exact reported values)
  - Calibrated optimal reserves vary across country groups and assumptions; reported ranges in Table 6.
  - Fixed Regime, Availability of Fund Support = Yes (months of imports)
    - Cost = 2.00: ALL 9.9; AFR 9.4; COM 10.2; Non-COM 9.7; FRG 12.6
    - Cost = 3.00: ALL 7.3; AFR 7.0; COM 7.7; Non-COM 7.0; FRG 9.7
    - Cost = 4.00: ALL 5.5; AFR 5.3; COM 5.9; Non-COM 5.2; FRG 7.6
    - Cost = 5.00: ALL 4.2; AFR 4.1; COM 4.7; Non-COM 4.0; FRG 5.9
    - Cost = 6.00: ALL 3.3; AFR 3.3; COM 3.8; Non-COM 3.1; FRG 4.7
  - Flexible Regime, Availability of Fund Support = Yes (months of imports)
    - Cost = 2.00: ALL 3.9; AFR 4.7; COM 5.4; Non-COM 3.2; FRG 5.3
    - Cost = 3.00: ALL 2.7; AFR 3.2; COM 3.8; Non-COM 2.3; FRG 3.8
    - Cost = 4.00: ALL 2.1; AFR 2.4; COM 2.9; Non-COM 1.8; FRG 2.9
    - Cost = 5.00: ALL 1.6; AFR 1.8; COM 2.3; Non-COM 1.4; FRG 2.3
    - Cost = 6.00: ALL 1.4; AFR 1.5; COM 1.8; Non-COM 1.2; FRG 1.9
  - Calibration sensitivities:
    - Sensitivity to shock assumption: using 25th percentile shocks raises optimal reserves (example: for commodity exporters, r = 4 percent and shocks at the 25th percentile, optimal reserves = 3.4 months of imports under the flexible regime).
    - Sensitivity to unit cost r: higher r implies lower optimal reserves; reported for r between 2 and 6 percent.
    - For representative low-income country (r = 4 percent):
      - “Insurance” value of a flexible exchange rate regime is about 0.6 percent of GDP per year (or around three months of imports on average) in annual savings in the cost of holding optimal reserves.
      - Availability of contingent Fund support can result in annual savings in optimal reserves of about 0.3 percent of GDP per year (around two months of imports).
  - Cross-country comparison (end-2008 data):
    - Low-income countries with fixed exchange rate regimes, particularly commodity exporters and fragile states, were on average below the computed adequacy range.
    - Countries with flexible regimes were well above the computed adequacy range (with substantial cross-country heterogeneity).
  - Caveat: end-2009 SDR allocation likely distorted reserve holdings.

### VII. Policy implications, recommendations, and extensions
- Policy implications and recommendations
  - Reserve adequacy assessments should be country-specific and explicitly account for:
    - country characteristics (e.g., fragility, commodity dependence, exchange rate regime);
    - access to Fund financing in the event of a shock; and
    - the marginal cost of holding reserves.
  - Reliance on simple cross-country benchmarks (such as three months of imports) is inadequate; policymakers should use frameworks that internalize interactions between fundamentals and external support access.
  - Given calibration under risk-neutrality, calibrated levels are a lower bound for reserves if policymakers are risk-averse or if reserves serve additional objectives (e.g., exchange rate policy, intergenerational savings).
  - Reserves provide a temporary, partial solution; durable risk reduction requires:
    - strengthening policy frameworks,
    - increasing exchange rate flexibility, and
    - diversifying economies.

- Suggested extensions for future work
  - Improved estimates of the cost of holding reserves in low-income countries (consistent marginal product of capital estimates).
  - Analysis of optimal reserve levels in currency unions, accounting for institutional features and requirements of the union arrangement.

### VIII. Key statistics and empirical magnitudes (preserved exactly)
- Median reserve coverage in 2009: 4.7 months of imports, and 55 percent of broad money.
- Conventional rule of thumb: three months of import cover.
- Estimation sample for reserve demand regression: 62 low-income countries (excluding population < 1 million), 1992 to 2001.
- Out-of-sample comparison period: 2002-2008 (excluding 2009 SDR allocation).
- Shock sample period and size: 1990-2008 for 71 low-income countries; 645 shock observations out of 1,349 observations.
- Crisis unconditional probability within shock sample: less than 30 percent.
- Difference in real absorption growth per capita between crisis and non-crisis cases: more than 8 percentage points.
- Sample mean of reserves in probit marginal-effects example: 3.2 months of imports.
- Effect of raising reserves from 3.2 to 5 months: reduces crisis probability by around 3.6 percentage points (around over 5 percentage points under fixed regimes).
- Model explanatory power: empirical model for full sample accounts for over 60 percent of variation in reserves (excluding country fixed effects).
- Terms-of-trade shock magnitude: drop over 30 percent for bottom 25th percentile of shock sample versus less than 5 percent for non-shock sample.
- Panel probit baseline coefficient: Reserves, months of imports (t-1): -0.0896*** (0.0339).
- Absorption loss regression baseline coefficient: Log of reserves, months of imports (t-1): -2.2403*** (0.6677).
- Table 6 representative calibrated values reported above (Fixed and Flexible regimes, costs 2.00 to 6.00 percent).

*Source: _wp11249 - References*

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

##_wp11249 - References

### I. Introduction and paper objectives
- Develops a tractable analytical framework to quantify reserve levels rationalized as insurance against large external shocks faced by low-income countries.
- Empirically estimates the benefits of reserves in smoothing domestic absorption in response to external shocks.
- Calibrates optimal reserves for country groups using regression estimates, country-specific fundamentals, and simplified assumptions about the cost of holding reserves and risk-neutrality.
- Sample and definition notes:
  - “Low-income countries” refers to all countries shown on the IMF’s list of countries eligible for the Poverty Reduction and Growth Trust (PRGT) at end-December 2010.
  - External shock episode analysis uses data for the period 1990-2008.

### II. Key empirical findings
- Reserves have an insurance role:
  - The ratio of reserves to imports is highly statistically significant and has the expected sign in regressions estimating crisis prevention and mitigation benefits, even after controlling for other fundamentals.
  - Results are robust across country samples and time periods.
- Crisis definition and empirical strategy:
  - A crisis is defined as a sharp drop in absorption.
  - The paper estimates both crisis prevention and mitigation benefits of holding reserves against a range of shocks faced by low-income countries, using past severe shock episodes (1990-2008) and controlling for policy fundamentals and country characteristics.
- Interaction with country characteristics:
  - The shock-mitigation effect of reserves is particularly pronounced in highly indebted economies, small islands, commodity exporters, and countries with fixed exchange rate regimes (as noted in referenced event-study evidence).
  - The reserve shock-mitigation effect depends crucially on country characteristics and vulnerabilities.

### III. Calibration results and quantitative conclusions
- Standard rule-of-thumb limitations:
  - The standard metric of reserves equivalent to three months of imports provides an imprecise benchmark.
- Calibrated optimal reserves:
  - Calibrated optimal reserves vary from about 1 to 12 months of imports.
  - Higher estimated optimal reserves are found for fragile states and commodity exporters relative to other countries.
  - Optimal reserves are generally higher for:
    - fixed exchange rate regimes;
    - the absence of Fund support in the event of a shock; and
    - countries facing a lower cost of reserves.
  - Stronger policy fundamentals are associated with lower optimal reserve holdings.
- Cost assumptions:
  - For low-income countries with limited access to international markets, the net financial cost of reserves is implied by foregone investment opportunities or the differential between domestic and foreign real interest rates.
  - Calibrations combine regression estimates of benefits with country-specific data on fundamentals and assumptions about the marginal cost of holding reserves and risk-neutrality.

### IV. Policy implications and recommendations
- Reserve adequacy assessments should be country-specific and explicitly account for:
  - country characteristics (e.g., fragility, commodity dependence, exchange rate regime);
  - access to Fund financing in the event of a shock; and
  - the marginal cost of holding reserves.
- Reliance on simple cross-country benchmarks (such as three months of imports) is inadequate; policymakers should use frameworks that internalize interactions between fundamentals and external support access.
- Given the calibration under risk-neutrality, policymakers should recognize calibrated levels are a lower bound for reserves if policymakers are risk-averse or if reserves serve additional objectives (e.g., exchange rate policy, intergenerational savings).

### V. Caveats, scope, and contributions
- The calibrated reserve levels constitute a lower bound because:
  - the framework assumes risk-neutrality (greater risk aversion would imply higher optimal reserves);
  - countries may accumulate reserves for objectives beyond self-insurance (e.g., exchange rate policy, intergenerational savings), which are not captured by the calibration.
- Contribution to literature:
  - Provides empirically grounded estimates of both crisis prevention and mitigation benefits of reserves for low-income countries.
  - Presents a more tractable framework that facilitates calibration using country-specific data.

*italic: Source: _wp11249 - References*

### 4.7 months of imports, and 55 percent of broad money in 2009.

### _wp11249 - 4.7 months of imports, and 55 percent of broad money in 2009.

### Reserve accumulation patterns in low-income countries
- Median reserve coverage: 4.7 months of imports, and 55 percent of broad money in 2009.
- Since 2003, ratios of reserves to imports increased most sharply for oil exporters.
- Sub Saharan African (SSA) countries have had persistently higher reserve coverage than the median low-income country, although this difference has narrowed in recent years.
- Countries with GDP per capita less than US$ 500 have been well-below the low-income country median throughout the 2000s.
- Reserve accumulation has outpaced the conventional rule of thumb of three months of import cover for both fixed and floating exchange rate regimes; countries with fixed regimes generally tend to have higher reserve coverage.
- As of 2009, over a quarter of all countries had reserve levels lower than suggested by the import cover metric.
- Drivers of accelerating reserve build-up: low initial reserve holdings, increasing openness of economies, a favorable global environment, and policy choice to build precautionary reserves against balance of payment risks.
- Note: The increase in reserves in 2009 is largely attributable to the SDR allocation in response to the global financial crisis.

### Empirical model of precautionary demand for reserves
- Purpose: assess whether recent accumulation in low-income countries aligns with fundamentals via a cross-country empirical model of precautionary demand for reserves.
- Theoretical idea: countries choose reserves to balance macroeconomic adjustment costs if reserves are exhausted (precautionary motive) against opportunity cost of holding reserves.
- Regression sample and estimation:
  - Panel data for 62 low-income countries (excluding economies with a population less than 1 million) from 1992 to 2001 used for estimation.
  - Remaining years used to compare out-of-sample forecasts with actual reserve buildups.
- Main regression findings (full sample, Column 1):
  - Reserve holdings positively and significantly related to indicators of current account vulnerability (import ratio and export volatility).
  - Reserve holdings positively and significantly related to indicators of capital account vulnerability, such as broad money.
  - Exchange rate volatility and fixed exchange rate regimes significantly associated with higher reserve holdings.
  - The proxy for cost of holding reserves (interest rate differential between government treasury bill and corresponding U.S. asset) has the expected sign but lacks statistical significance.
  - The empirical model for the full sample accounts for over 60 percent of the variation in reserves (excluding country fixed effects).
- Subsample differences:
  - Breakdown into commodity exporters and non-commodity exporters reveals:
    - Exchange rate volatility is more important for non-commodity exporters.
    - Export volatility is highly significant in explaining reserve demand in commodity exporters.
- Out-of-sample comparison:
  - Growth in low-income country reserve holdings between 2002 and 2008 broadly in line with evolving fundamentals (Figure 3 comparison of model forecasts with actual buildups, excluding 2009 SDR allocation).

### Analytical framework: cost-benefit approach to reserve adequacy
- Motivation: Traditional metrics offer rough guidance and lack empirical/theoretical foundations; propose a cost-benefit framework to determine optimal reserve levels for low-income countries.
- Key distinctions for low-income countries:
  - Face different shocks than EMs: sharp swings in foreign aid, remittances, FDI, and natural disasters; higher frequency/incidence of terms-of-trade shocks.
  - Limited market access implies external drains primarily on the current account; current-account based measures (e.g., months of imports) remain useful indicators.
- Insurance perspective:
  - Reserves can reduce likelihood and magnitude of abrupt drops in consumption/absorption from external shocks.
  - Empirical evidence (Crispolti and Tsibouris, 2011): cumulative consumption losses over five years were about 6-17 percentage points for countries with reserve coverage of less than three months of imports, whereas impact was limited among those with higher coverage.
- Cost of holding reserves:
  - Typically defined as difference between return on short-term foreign currency assets and return on alternative investments.
  - For many low-income countries with limited market access, net financial cost may be negative because concessional borrowing rates are typically lower than reserve returns.
  - Given infrastructure needs, opportunity cost may be the foregone return on risky/high-yield domestic investments; domestic-foreign real interest rate differential commonly used as proxy.
- Objective function presented in text (precautionary motive):
  - Max(   ,   )(   ,   ) R NBRq  P  R  Z    C  R  Zr   R
  - Where P and C are conditional probability of a crisis given a large shock and the utility cost of a crisis (both depend on reserves R and controls Z); q is unconditional probability of a large shock; r is unit cost of holding reserves.
- Limitations noted:
  - Utility cost measured by loss in real absorption (percent of GDP) assuming risk-neutral utility; framework aims to simulate a lower bound of optimal reserves.
  - Static objective function—recognizing alternative insurance from multilateral/bilateral sources may justify this simplification.

### Empirical analysis: identifying shocks and estimating crisis probability
- Sample and shock identification:
  - Sample: 1990-2008 for 71 low-income countries.
  - Large negative external shock episodes identified when annual percentage change of relevant variable falls below the 10th percentile in country-specific left-tail distribution.
  - Six shock types: (i) external demand; (ii) terms-of-trade; (iii) FDI; (iv) aid; (v) remittances; (vi) climatic shocks (large natural disasters).
  - For natural disasters, top 25th percentile in number affected and economic damage (EM-DAT/CRED).
  - Total shock observations: 645 shock observations (out of 1,349 observations).
  - Correlation between different shocks ranges from -0.05 to 0.05.
- Crisis definition:
  - A crisis occurs when:
    - (i) The post-shock two-year average (t and t+1) level of real absorption per capita falls below the pre-shock three-year trend; and
    - (ii) Growth of real absorption per capita is negative at time t.
  - This definition focuses on extreme events combining level drops and growth declines.
- Descriptive findings:
  - Drop in terms-of-trade: over 30 percent for bottom 25th percentile of the shock sample, versus less than 5 percent for non-shock sample.
  - Median growth of real absorption per capita is positive for the entire shock sample—implying not all shocks lead to absorption drops.
  - Unconditional probability of a crisis within the shock sample is less than 30 percent.
  - There is a statistically significant difference of more than 8 percentage points in real absorption growth per capita between crisis and non-crisis cases.
- Estimating the probability of a crisis:
  - Panel probit model for conditional probability of a crisis (probability of a crisis given a shock) estimated for 49 low-income countries.
  - Final explanatory variables: reserves (in months of imports), ratio of government balance to GDP, World Bank’s CPIA index, dummy for flexible exchange rate regime, and dummy for Fund-supported programs. All lags by one year except contemporaneous Fund program dummy.
  - Results (Table 3):
    - Probability of a crisis decreases with better institutional quality (CPIA) and higher government balance to GDP.
    - Coefficient on reserves is of expected sign, statistically significant, and broadly similar across specifications.
    - Exchange rate regime and Fund support are important determinants: greater exchange rate flexibility and Fund-supported programs associated with lower likelihood of crisis given a shock.
  - Marginal effects (Table 4 top panel):
    - Sample mean of reserves: 3.2 months of imports.
    - Increasing reserves from 3.2 months to 5 months of imports reduces probability of an absorption drop by around 3.6 percentage points (assuming Fund-supported program and evaluating other variables at sample means under a flexible exchange rate regime).
    - Estimated change in crisis probability tends to be higher (over 5 percentage points) under fixed exchange rate regimes.

### Key statistics and empirical magnitudes (preserved exactly)
- Median reserve coverage in 2009: 4.7 months of imports, and 55 percent of broad money.
- Conventional rule of thumb: three months of import cover.
- Estimation sample for reserve demand regression: 62 low-income countries (excluding population < 1 million), 1992 to 2001.
- Out-of-sample comparison period: 2002-2008 (excluding 2009 SDR allocation).
- Shock sample period and size: 1990-2008 for 71 low-income countries; 645 shock observations out of 1,349 observations.
- Crisis unconditional probability within shock sample: less than 30 percent.
- Difference in real absorption growth per capita between crisis and non-crisis cases: more than 8 percentage points.
- Sample mean of reserves in probit marginal-effects example: 3.2 months of imports.
- Effect of raising reserves from 3.2 to 5 months: reduces crisis probability by around 3.6 percentage points (around over 5 percentage points under fixed regimes).
- Model explanatory power: empirical model for full sample accounts for over 60 percent of variation in reserves (excluding country fixed effects).
- Terms-of-trade shock magnitude: drop over 30 percent for bottom 25th percentile of shock sample versus less than 5 percent for non-shock sample.

### Conclusions and methodological implications highlighted in the text
- Median reserve accumulation has outpaced standard rules of thumb in recent years.
- Reserve demand regressions suggest recent growth in reserve holdings is largely in line with fundamentals, but do not provide optimal reserve levels in light of shocks faced by low-income countries.
- Traditional metrics are simple but lack empirical/theoretical foundations; a cost-benefit framework is proposed to determine optimal reserve levels for low-income countries.
- The proposed framework focuses on absorption smoothing benefits, opportunity costs of holding reserves, and empirically estimating both the conditional probability of a crisis and its utility cost.

*Source: IMF working paper content as provided in the supplied text.*

### 3.3 to 3.7 reduces the crisis probability by 3 percentage points under a flexible regime and by

### _wp11249 - 3.3 to 3.7 reduces the crisis probability by 3 percentage points under a flexible regime and by

### Crisis probability: key quantitative findings
- A reserves increase from 3.3 to 3.7 (months of imports) reduces the crisis probability by 3 percentage points under a flexible regime and by 4.2 percentage points under a fixed regime.
- An improvement in the government balance from 4.5 to 3 percent of GDP reduces the estimated annual probability of a crisis by over 1 percentage points.
- Shifting from a fixed to a flexible exchange rate regime reduces the likelihood of an absorption drop by 11 percentage points.
- Having an IMF Fund-supported program in the event of a shock reduces the crisis probability by:
  - approximately 8.8 percentage points under a flexible regime, and
  - approximately 10.9 percentage points under a fixed exchange rate regime.
- Increasing reserve coverage from three to four months of imports reduces the probability of a crisis by about 3.5 percentage points under a fixed exchange rate regime; the marginal effect is smaller under a flexible regime.

### Crisis mitigation (severity) findings
- Regressions estimate real absorption loss (normalized in percent of GDP) conditional on reserves (log of reserves in months of imports), exchange rate regime, and shock size, including country fixed effects.
- Higher reserve holdings are associated with lower absorption losses; the marginal effect is diminishing in the level of reserves (log specification).
- Positive external demand and terms of trade shocks are associated with lower absorption losses (foreign aid shock generally insignificant).
- A flexible exchange rate regime reduces real absorption loss by around 9 percent of GDP relative to the fixed regime, after controlling for shocks and country fixed effects.
- The fixed-effects OLS absorption loss regression accounts for 35 percent of the variation in absorption loss across countries.

### Robustness checks (regression robustness)
- Probit and fixed-effects OLS regression results are broadly robust across sample splits and specification checks.
- The reserves variable remains highly significant and of the expected sign in most sub-samples, except when the sample is substantially reduced by dropping African countries.
- Results are similar when using real consumption per capita drops instead of absorption drops.
- Alternative reserve specifications (e.g., R* = R/(1+R)) yield very similar optimal reserve levels to the log specification.
- Remittance shocks were excluded because their inclusion substantially reduces observations; foreign aid shock is frequently insignificant.

### Optimal precautionary reserves: calibration setup and parameters
- Model determined by four parameters: conditional probability of a crisis P(R,Z), real absorption loss C(R,Z), unconditional probability of a large shock event q, and unit cost of holding reserves r.
- Sample estimate of the unconditional probability q = 0.5 (sample average).
- Unit cost of holding reserves r considered in reference values ranging between 2 and 6 percent. These values derive from estimates of the marginal product of capital and differential between domestic and foreign real interest rates (adjusted for real financial return on reserves of about 1 percent a year).
- Economic fundamentals (fiscal balance, CPIA index) set to five-year average over 2003-07 for each country group.
- Shock values in calibration taken from the sample median for different country groups; estimated real absorption loss augmented by one standard deviation of residuals (roughly the upper 85th percentile) to capture risk aversion.
- Calibration assumes availability of contingent IMF support in the event of large external shocks; this affects the conditional probability P(R,Z).

### Calibration results and sensitivity
- Calibrated optimal reserves vary across country groups (ALL, AFR, COM, FRG) and across assumptions about r and shock severity; reported calibrated values range from less than 2 to over 12 months of imports depending on characteristics and parameter choices.
- Optimal reserves are generally higher for:
  - countries with fixed exchange rate regimes,
  - fragile states, and
  - commodity exporters.
- Sensitivity to shock assumption: using the 25th percentile shock rather than the median raises optimal reserves. Example: for commodity exporters, with r = 4 percent and shocks at the 25th percentile, optimal reserves = 3.4 months of imports under the flexible regime.
- Sensitivity to unit cost r: higher r implies lower optimal reserves; results reported for r between 2 and 6 percent.
- The calibration yields that the traditional rule of thumb of 3 months of imports is more appropriate for countries with flexible exchange rate regimes—particularly if Fund support is readily available—but is likely inadequate for countries with fixed exchange rate regimes.
- For the representative low-income country (r = 4 percent):
  - The “insurance” value of a flexible exchange rate regime is about 0.6 percent of GDP per year (or around three months of imports on average) in annual savings in the cost of holding optimal reserves.
  - Availability of contingent Fund support can result in annual savings in optimal reserves of about 0.3 percent of GDP per year (around two months of imports).
- Calibration acknowledges substantial unexplained variation in absorption loss regressions; sensitivity analyses include taking more extreme shock values (bottom 5th, 10th, 25th percentiles).

### Cross-country and policy implications
- Stronger fiscal positions and higher CPIA scores (better policy and institutional frameworks) are associated with lower optimal reserves.
- Sensitivity of optimal reserves to fundamentals differs across exchange rate regimes; fixed regimes show higher sensitivity.
- Comparison of actual vs. computed optimal reserves (end-2008 data):
  - Low-income countries with fixed exchange rate regimes, particularly commodity exporters and fragile states, were on average below the computed adequacy range.
  - Countries with flexible regimes were well above the computed adequacy range (with substantial cross-country heterogeneity).
- Cautions: end-2009 SDR allocation likely distorted reserve holdings; flexible-regime countries are more open and integrated with international financial markets, so capital flight risks and non-precautionary motives for holding reserves (monetary policy, exchange rate management) may matter.

### Policy recommendations and conclusions
- The role of reserves in reducing both the probability and severity of sudden stops is robustly established.
- Optimal reserve levels are sensitive to country fundamentals and exchange rate regimes; country-specific calibration is necessary.
- Rules of thumb (e.g., 3 months of imports) provide imprecise benchmarks and can be inadequate—especially for fixed exchange rate regimes.
- Reserves provide a temporary, partial solution; durable risk reduction requires:
  - strengthening policy frameworks,
  - increasing exchange rate flexibility, and
  - diversifying economies.
- Two suggested extensions for future work:
  - Improved estimates of the cost of holding reserves in low-income countries (consistent marginal product of capital estimates).
  - Analysis of optimal reserve levels in currency unions, accounting for institutional features and requirements of the union arrangement.

*Source: IMF working paper content provided in the supplied PDF excerpt.*

### References

### _wp11249 - References

### Key bibliographic sources
- Aizenman, Joshua, and Nancy Marion, 2003, "The high demand for international reserves in the Far East: What is going on?" Journal of the Japanese and International Economies, Vol. 17(3), pp. 370-400.
- Aizenman, Joshua, and Jaewoo Lee, 2008, "Financial versus Monetary Mercantilism: Long-run View of Large International Reserves Hoarding," The World Economy, Vol. 31(5), pp. 593-611.
- Barnichon, Regis, 2009, “The Optimal Level of Reserves for Low-Income Countries: Self-Insurance against External Shocks,” IMF Staff Papers, Vol.56 (4), pp. 852-875.
- Becker, Törbjörn I., Olivier Jeanne, Paolo Mauro, Jonathan David Ostry, and Romain Ranciere, 2007, “Country Insurance: The Role of Domestic Policies,” Occasional Paper No. 254 (Washington: International Monetary Fund).
- Caselli, Francesco, and James Feyrer, 2007, “The Marginal Product of Capital,” Quarterly Journal of Economics, 122, pp. 535-568.
- Dooley, Michael P., David Folkerts-Landau, and Peter Garber, 2004, "The Revised Bretton Woods System," International Journal of Finance and Economics, Vol. 9, pp.307-313.
- Dreher, Axel, 2006, “IMF and Economic Growth: The Effects of Programs, Loans, and Compliance with Conditionality,” World Development, Vol. 34, pp.769-788.
- Durdu, Ceyhun Bora, Enrique Mendoza, and Marco Terrones, 2009, "Precautionary Demand for Foreign Assets in Sudden Stop Economies: An Assessment of the New Mercantilism," Journal of Development Economics, Vol. 89, pp. 194-209.
- Hauner, David, 2006, "A Fiscal Price Tag for International Reserves," International Finance, Vol. 9(2), pp. 169-195.
- International Monetary Fund, 2003, World Economic Outlook.
- International Monetary Fund, 2011, “Managing Volatility: A Vulnerability Exercise for Low-Income Countries.”
- Loayza, Norman V., Romain Rancière, Luis Servén, and Jaume Ventura, 2007, “Macroeconomic Volatility and Welfare in Developing Countries: An Introduction,” World Bank Economic Review, Oxford University Press, Vol. 23(3), pp.343-357.
- Perry, Guillermo, 2009, “Beyond Lending: How Multilateral Banks Can Help Developing Countries Manage Volatility” (Washington: Center for Global Development).
- Rancière, Romain, and Oliver Jeanne, 2006 and 2008, on optimal international reserves (IMF Working Paper 06/229; CEPR Discussion Papers No. 6723).
- Valencia, Fabian, 2010, “Precautionary Reserves: An Application to Bolivia,” IMF Working Papers 10/54.

### Figures — central reported measures (as presented)
- Figure 1: Recent Trends in Reserve Accumulation in Low-Income Countries (LICs) and Emerging Market Economies (EMs)
  - Series shown: Reserves in Months of Imports (3 Months of Imports benchmark), Reserves to Broad Money (20 Percent of Broad Money benchmark).
  - Time span on charts: 1991 to 2009 (annual markers shown: 1991 1993 1995 1997 1999 2001 2003 2005 2007 2009).
- Figure 2: Median Reserve Coverage in Low-Income Countries (LICs) (in months of current year's imports)
  - Subpanels: Oil exporters, Oil importers, SSA, GDP per capita less than US$500, de facto fixed exchange regime in 2007, de facto floating exchange regime in 2007.
  - Time span on charts: 1990 to 2008 (annual markers shown: 1990 1993 1996 1999 2002 2005 2008).
- Figure 3: Actual and Predicted Reserves, 2002-2008 (mean, in percent of GDP)
  - Series plotted: Actual and Predicted reserves for 2002–2008.
- Figure 4: External Shock Episodes — distributions shown for Aid (change in the ratio to GDP), FDI (change in the ratio to GDP), Terms of trade (percentage change), External demand (percentage change).
  - Boxplot percentiles indicated: 25th percentile, 50th percentile, 75th percentile; separate plotting for Shock episode and Non-shock episode.
- Figure 5: Marginal Effects on the Probability of a Crisis
  - Note: Government balance and CPIA assumed at their respective mean values. Panel A assumes no Fund-supported program; Panel B assumes the fixed exchange rate regime.
- Figure 6: Sensitivity of Optimal Reserves to Country Fundamentals
  - Panels: A. Reserves and Exchange rate Regimes (Fixed regime vs Flexible regime); B. Reserves and IMF Program (No IMF program vs IMF program); Additional panels show Optimal Reserves vs Fiscal Balance and Optimal Reserves vs CPIA with Fixed and Flexible regimes.
- Figure 7: Actual vs. Computed Reserves, 2008-09
  - Notes: Calculation assumes access to Fund-support following a shock. Country groups: SSA = Africa, COM = commodity exporters, FRG = fragile states. Optimal reserves assume the cost of holding reserves is 4 percent; range shown for cost varying from 3 to 5 percent.

### Key table findings and exact reported estimates (selected)
- Table 1. Estimating Reserve Demand in LICs (Dependent variable: Reserves/GDP)
  - Column (1) ALL:
    - Income: -0.0045*** (0.0006)
    - Log(Population): -2.2280*** (0.3743)
    - Imports/GDP: 0.2611*** (0.0198)
    - Exchange rate volatility: -0.0351** (0.0147)
    - Export volatility (3y st. dev.): 0.0482** (0.0235)
    - Broad Money/GDP: 0.3374*** (0.0326)
    - Peg Dummy: 1.2851* (0.7792)
    - Interest rate differential with US: -0.2178 (0.5248)
    - Observations: 414
    - R-squared: 0.6390
  - Note: Standard errors in parentheses; *, **, *** indicate significance at 10 percent, 5 percent, and 1 percent, respectively. Sample period: 1992-2001.

- Table 3. Probability of Absorption Drops (Panel probit regression, 1990-2007)
  - Baseline (column 1):
    - Reserves, months of imports (t-1): -0.0896*** (0.0339)
    - Flexible exchange rate regime (t-1): -0.3801*** (0.1366)
    - Government balance, % of GDP (t-1): -0.0323*** (0.0125)
    - CPIA index (t-1): -0.3090*** (0.1056)
    - IMF program (t): -0.3021** (0.1409)
    - Constant: 0.8648** (0.3614)
    - No. of observations: 445
    - Pseudo R2: 0.1099
  - Robustness columns (various exclusions) report similar signs and significance for these main variables.

- Table 5. Absorption Loss Regression (Fixed Effects regression with the probit sample)
  - Selected coefficients (Baseline, column 1):
    - Log of reserves, months of imports (t-1): -2.2403*** (0.6677)
    - Flexible exchange rate regime (t-1): -8.6983*** (2.1689)
    - External demand growth (t): -0.9320** (0.4356)
    - Terms of trade growth (t): -0.0841* (0.0484)
    - Change in FDI to GDP (t): -0.0159 (0.3391)
    - Change in aid to GDP (t): 0.0527 (0.0839)
    - N: 418
    - Adjusted R2: 0.34
  - Country fixed effects included.

- Table 4. Fundamentals and Marginal Effects on Crisis Probability (evaluated at mean value, percent)
  - Sample means and parameter changes (selected):
    - Reserves, months of imports: 3.2; change 3.2 --> 5 leads to probability changes: Fixed regime -3.6; Flexible regime -5.1
    - Government balance, % of GDP: -4.5; change -4.5 --> -3 leads to probability changes: Fixed regime -1.1; Flexible regime -1.6
    - CPIA index: 3.3; change 3.3 --> 3.7 leads to probability changes: Fixed regime -3.0; Flexible regime -4.2
    - IMF program dummy: no program --> program leads to probability changes: -8.8 (Fixed), -10.9 (Flexible)
    - Exchange rate regime dummy: fixed --> flexible leads to probability change: -11.1
    - Probability of crisis (no IMF program): 25.2 and 38.5 (two values shown)
    - Probability of crisis (with IMF program): 16.5 and 27.6 (two values shown)

- Table 6. Calibrated Optimal Reserves for Low-Income Countries (In months of imports)
  - Fixed Regime, Availability of Fund Support = Yes:
    - Cost of Reserves (%) = 2.00: ALL 9.9; AFR 9.4; COM 10.2; Non-COM 9.7; FRG 12.6
    - Cost = 3.00: ALL 7.3; AFR 7.0; COM 7.7; Non-COM 7.0; FRG 9.7
    - Cost = 4.00: ALL 5.5; AFR 5.3; COM 5.9; Non-COM 5.2; FRG 7.6
    - Cost = 5.00: ALL 4.2; AFR 4.1; COM 4.7; Non-COM 4.0; FRG 5.9
    - Cost = 6.00: ALL 3.3; AFR 3.3; COM 3.8; Non-COM 3.1; FRG 4.7
  - Flexible Regime, Availability of Fund Support = Yes:
    - Cost = 2.00: ALL 3.9; AFR 4.7; COM 5.4; Non-COM 3.2; FRG 5.3
    - Cost = 3.00: ALL 2.7; AFR 3.2; COM 3.8; Non-COM 2.3; FRG 3.8
    - Cost = 4.00: ALL 2.1; AFR 2.4; COM 2.9; Non-COM 1.8; FRG 2.9
    - Cost = 5.00: ALL 1.6; AFR 1.8; COM 2.3; Non-COM 1.4; FRG 2.3
    - Cost = 6.00: ALL 1.4; AFR 1.5; COM 1.8; Non-COM 1.2; FRG 1.9
  - Note: Country groups: AFR = Africa, COM = commodity exporters, Non-COM = non commodity exporters, FRG = fragile states.

### Appendix — variables, robustness, sample composition
- Table A.I: Variables used in regression analysis (sources indicated)
  - Key variables and sources: Lag of real GDP per capita growth (WEO), Lag of government expenditure to GDP (WEO), Lag of inflation rate (WEO), Lag of current account deficit to GDP (WEO), Trade openness (WEO), De facto exchange rate regime dummies (Reinhart & Rogoff (2004)), CPIA index (World Bank), Reserves as months of imports (WEO), Lag of FDI to GDP (WEO), Lag of aid to GDP (OECD), IMF program dummy (Dreher (2006) and IMF).
- Robustness checks (selected excerpts)
  - Table A.II Absorption Drop Probit Regression — longer sample (1980-2009) and regional exclusions report consistent negative coefficients on Reserves, months of imports (t-1) and Flexible exchange rate regime (t-1).
  - Table A.III Probability of Consumption Drops — baseline and many robustness splits show Reserves, months of imports (t-1) negative and often significant (e.g., baseline -0.0866*** (0.0329)).
  - Table A.IV Absorption Loss Regression — across regional exclusions, Log of reserves, months of imports (t-1) mostly negative and significant in several columns (e.g., column 2 -2.2679*** (0.6556)).
  - Table A.V Consumption Loss Regression — Log of reserves, months of imports (t-1) negative and significant in multiple specifications (e.g., baseline -2.0199*** (0.5041)); Flexible exchange rate regime (t-1) negative and significant across specifications.
  - Table A.VI Sensitivity Analysis of Calibrated Optimal Reserves — shows optimal reserve months across alternative data types (Bot 5, Bot 10, Bot 25), availability of Fund support, cost of reserves, and exchange rate regimes; specific matrix values reported in the table.
- Table A.VII List of Countries in the Sample
  - 71 LICs listed.
  - 49 countries used in the probit analysis (explicit list provided).

*Source: _wp11249 - References (PDF content supplied).*

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