Having estimated conditional probability of a crisis (P) and the cost of a crisis (C), this chapter elaborates on the calibration results. We first construct a benchmark calibration for various country groups and present some sensitivity analysis. We then present estimates of optimal reserves across different groups and illustrate the relationship with country fundamentals and IMF support.
Benchmark Calibration and Sensitivity Analysis
As described in Chapter 3, the behavior of the model economy is determined by four parameters: the conditional probability of a crisis in the event of external shocks, P(R, Z); real absorption loss, C(R, Z); the unconditional probability of a large shock event, q; and the unit cost of holding reserves, r. This chapter reports the calibration results for the optimal levels of reserves for various country groups, including all LICs (ALL), sub-Saharan African countries (AFR), commodity exporters (COM), non-commodity exporters (non-COM), and fragile states (FRG). Further disaggregation of country groups, albeit desirable in light of significant heterogeneity across LICs, was not considered because the number of countries is uneven across country groups, often with too few countries in a certain group to yield statistically meaningful results.
Optimal reserves are calibrated using the estimated conditional probability, P(R, Z), and real absorption loss, C(R, Z), regression equations reported in the previous chapter. Other parameter values were taken directly from the data. The unconditional probability of a large shock event (q) is estimated from the data to be 0.5 (the sample average). For the unit cost of holding reserves (r), several reference values were considered, ranging between 2 and 6 percent. These values are based on various existing estimates of the marginal product of capital and the differential between domestic and foreign real interest rates (adjusted for real financial return on reserves of about 1 percent a year).1 Economic fundamentals, such as fiscal balance and the CPIA index, are set to their respective five-year average during the period 2003–07 for each country group.
Shock values in the calibration are taken from the sample median for each country group.2 The estimated real absorption loss (for chosen values of shocks and country fundamentals) is augmented by one standard deviation of the residuals from the fixed effects ordinary least squares (OLS) absorption loss regression. Assuming normality, the augmented value corresponds roughly to the upper-85th percentile of the distribution of absorption losses. Given that there remains large unexplained variation in the fixed effects OLS absorption loss regression (the regression accounts for 35 percent of the variation in absorption loss across countries), this adjustment is intended as an attempt to capture possible risk aversion.3
The calibration assumes the availability of access to (contingent) IMF support in the event of large external shocks, which affects the conditional probability of a crisis.4 Calibrated optimal reserves are reported in Table 6.1 for different country groups. As can be seen from the table, these vary from less than 2 to more than 12 months of imports depending on country characteristics, fundamentals, and the cost of holding reserves. In all instances, optimal reserves are generally higher for countries with fixed exchange rate regimes, fragile states, and commodity exporters, reflecting their greater vulnerability to external shocks.
Calibrated Optimal Reserves for LICs
(Months of imports)
Calibrated Optimal Reserves for LICs
(Months of imports)
| Exchange rate regime | Availability of Fund support |
Cost of reserves (percent) |
Country Group | ||||
|---|---|---|---|---|---|---|---|
| ALL | AFR | COM | Non-COM | FRG | |||
| 2.00 | 9.9 | 9.4 | 10.2 | 9.7 | 12.6 | ||
| 3.00 | 7.3 | 7.0 | 7.7 | 7.0 | 9.7 | ||
| Fixed regime | Yes | 4.00 | 5.5 | 5.3 | 5.9 | 5.2 | 7.6 |
| 5.00 | 4.2 | 4.1 | 4.7 | 4.0 | 5.9 | ||
| 6.00 | 3.3 | 3.3 | 3.8 | 3.1 | 4.7 | ||
| 2.00 | 3.9 | 4.7 | 5.4 | 3.2 | 5.3 | ||
| 3.00 | 2.7 | 3.2 | 3.8 | 2.3 | 3.8 | ||
| Flexible regime | Yes | 4.00 | 2.1 | 2.4 | 2.9 | 1.8 | 2.9 |
| 5.00 | 1.6 | 1.8 | 2.3 | 1.4 | 2.3 | ||
| 6.00 | 1.4 | 1.5 | 1.8 | 1.2 | 1.9 | ||
Calibrated Optimal Reserves for LICs
(Months of imports)
| Exchange rate regime | Availability of Fund support |
Cost of reserves (percent) |
Country Group | ||||
|---|---|---|---|---|---|---|---|
| ALL | AFR | COM | Non-COM | FRG | |||
| 2.00 | 9.9 | 9.4 | 10.2 | 9.7 | 12.6 | ||
| 3.00 | 7.3 | 7.0 | 7.7 | 7.0 | 9.7 | ||
| Fixed regime | Yes | 4.00 | 5.5 | 5.3 | 5.9 | 5.2 | 7.6 |
| 5.00 | 4.2 | 4.1 | 4.7 | 4.0 | 5.9 | ||
| 6.00 | 3.3 | 3.3 | 3.8 | 3.1 | 4.7 | ||
| 2.00 | 3.9 | 4.7 | 5.4 | 3.2 | 5.3 | ||
| 3.00 | 2.7 | 3.2 | 3.8 | 2.3 | 3.8 | ||
| Flexible regime | Yes | 4.00 | 2.1 | 2.4 | 2.9 | 1.8 | 2.9 |
| 5.00 | 1.6 | 1.8 | 2.3 | 1.4 | 2.3 | ||
| 6.00 | 1.4 | 1.5 | 1.8 | 1.2 | 1.9 | ||
Sensitivity analysis undertaken for the calibration results suggests that optimal reserves are higher if more extreme shock values are considered (taking the bottom 5th, 10th, or 25th percentile of the group-specific distribution instead of the median; see Table A2.4). For example, assuming that the unit cost of holding reserves is 4 percent, optimal reserves for commodity exporters are 3.4 months of imports under the flexible regime if shock values were set to the 25th percentile instead of the median. Similarly, as can be seen in Table 6.1, optimal reserve levels are sensitive to assumptions about the unit cost of holding reserves.
Results across Country Groups
As discussed above, calibrated optimal reserves vary depending on country characteristics and the cost of holding reserves. Although the range of calibrated optimal reserves encompasses the traditional rule of thumb of three months of imports, the results suggest that this benchmark is more appropriate for countries with flexible exchange rate regimes, particularly if IMF support is readily available. It should be emphasized, however, that the calibration assumes risk-neutral utility and thus tends to yield a lower bound of optimal reserves. As such, the traditional rule of thumb is likely to be inadequate for countries with fixed exchange rate regimes. For the representative low-income country, assuming the unit cost of holding reserves is set at 4 percent, the “insurance” value of a flexible exchange rate regime—measured in terms of annual savings in the cost of holding optimal reserves—is about 0.6 percent of GDP per year (or around three months of imports on average). A similar calculation suggests that the availability of (contingent) IMF support can result in annual savings in optimal reserves of about 0.3 percent of GDP per year (about two months of imports), and could possibly be higher.
The overall policy framework plays an important role in the determination of optimal reserve levels. Figure 6.1 illustrates the sensitivity of calibrated optimal reserves to various country fundamentals for LICs in sub-Saharan Africa when the unit cost of holding reserves is set at 4 percent. It is evident that a stronger fiscal position is associated with lower optimal reserves. Similarly, better policy and institutional frameworks, as measured by a higher CPIA score, are also associated with lower optimal reserves. The results also suggest that the sensitivity of optimal reserves to varying fundamentals differs across exchange rate regimes, with a higher sensitivity for fixed regimes. This result underscores the importance of country-specific fundamentals in determining optimal reserves and suggests that applying a uniform metric for reserve adequacy across all LICs would be inappropriate.
Sensitivity of Optimal Reserves to Country Fundamentals
Source: IMF staff calculations.Note: CPIA: Country Policy and Institutional Assessment.Sensitivity of Optimal Reserves to Country Fundamentals
Source: IMF staff calculations.Note: CPIA: Country Policy and Institutional Assessment.Sensitivity of Optimal Reserves to Country Fundamentals
Source: IMF staff calculations.Note: CPIA: Country Policy and Institutional Assessment.An assessment of actual reserve holdings against the derived optimal reserves suggests that, on average, low-income country reserve holdings are broadly adequate. Figure 6.2 shows a comparison of actual reserve holdings in LICs against computed optimal reserve levels. Based on end-2008 data, LICs 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 range, although this masks significant differences across individual countries. A comparison of optimal reserves with end-of-2009 data shows a slightly different picture as the 2009 SDR allocation likely distorted reserve holdings for many LICs. A number of caveats should be borne in mind while drawing inferences from this comparison: Countries with flexible regimes are relatively more open and integrated with international financial markets as compared to other LICs, suggesting that capital flight risks may be playing a role; other non-precautionary motives for holding reserves, including monetary policy and exchange rate decisions by the central bank, could also be pertinent for managed float regimes.
Actual versus Computed Reserves, 2008–2009
Sources: IMF, World Economic Outlook; and IMF staff calculations.Notes: The calculation assumes access to Fund-support following a shock. Country groups are as follows: SSA: Africa; COM: commodity exporters; FRG: fragile states. Optimal reserves is calculated assuming the cost of holding reserves is 4 percent. A range of optimal reserves is also shown assuming the cost of reserves varies from 3 to 5 percent.Actual versus Computed Reserves, 2008–2009
Sources: IMF, World Economic Outlook; and IMF staff calculations.Notes: The calculation assumes access to Fund-support following a shock. Country groups are as follows: SSA: Africa; COM: commodity exporters; FRG: fragile states. Optimal reserves is calculated assuming the cost of holding reserves is 4 percent. A range of optimal reserves is also shown assuming the cost of reserves varies from 3 to 5 percent.Actual versus Computed Reserves, 2008–2009
Sources: IMF, World Economic Outlook; and IMF staff calculations.Notes: The calculation assumes access to Fund-support following a shock. Country groups are as follows: SSA: Africa; COM: commodity exporters; FRG: fragile states. Optimal reserves is calculated assuming the cost of holding reserves is 4 percent. A range of optimal reserves is also shown assuming the cost of reserves varies from 3 to 5 percent.The marginal product of capital is an important measure for LICs given their large investment needs. Caselli and Feyrer (2007) calculate a range of 3 to 8 percent for the marginal product of capital in LICs.
Alternatively, shock values could be simulated by assuming a multivariate normal distribution for shocks, with the variance-covariance matrix estimated from the sample. Optimal reserves could be calibrated for each set of simulated shock values and then averaged to yield final results. Although computationally demanding, this option allows for explicitly accounting for the correlation among shocks.
In view of the large uncertainty surrounding estimates of risk-aversion parameters, experimenting with more extreme shock values or larger adjustments, while assuming risk-neutral utility, could be a practical approach to address differences in the risk attitude across countries.
Because a large shock event is defined as a union of six types of shock events (defined at or below the 10th percentile of the country-specific sample distribution), the unconditional probability (q) should be close to 0.6 if individual shocks are uncorrelated. The sample estimate of 0.5 thus suggests that individual shocks are positively (albeit weakly) correlated in the sample.
