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# Appendix III. A Model of Optimal Reserves

Paolo Mauro, Torbjorn Becker, Jonathan Ostry, Romain Ranciere, and Olivier Jeanne
Published Date:
April 2007
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This appendix presents background information on the Jeanne and Rancière (2006) model of a growing open economy where reserves are accumulated, at a cost, in order to reduce the frequency of sudden stops in financial flows and smooth their impact on domestic consumption. The simple version of the model presented in this study considers only the smoothing role of reserves. At the time of a sudden stop, the country’s authorities lend reserves to domestic agents whose lines of credit have been cut off by private lenders, thereby mitigating the drop in domestic consumption. The cost of holding reserves (the difference between the return on reserves and the long-term interest rate) translates into lower domestic consumption during noncrisis times. The optimal level of reserves equates the marginal costs and benefits of holding reserves. The parameters of the benchmark calibration (Table A3.1) are based on information drawn from a panel of 33 middle-income countries during 1980–2003. Defining a sudden stop as a worsening in the financial account balance by more than 5 percent of GDP, 77 sudden stops are observed during the sample period. This yields an unconditional probability of a sudden stop of 10 percent per year. The magnitude of a sudden stop is the mean size of sudden stops in the sample (11 percentage points of GDP). The output cost of a sudden stop is set equal to 6 percent of GDP (the difference in average real GDP growth between the year of the sudden stop, on the one hand, the following year and, on the other hand, the remaining years for all countries).

Table A3.1.Calibration Parameters
ParametersBaselineRange of Variation
Size of sudden stopλ = 0.11[0, 0.3]
Probability of sudden stopπ = 0.10[0, 0.25]
Output lossγ = 0.06[0, 0.2]
Potential output growthg = 0.033
Risk-free rater* = 0.05
Risk aversionσ = 2[1, 10]
Sources: IMF staff calculations. The parameters were chosen based on data from the U.S. Board of Governors of the Federal Reserve System and the IMF’s International Financial Statistics database.
Sources: IMF staff calculations. The parameters were chosen based on data from the U.S. Board of Governors of the Federal Reserve System and the IMF’s International Financial Statistics database.

The cost of holding reserves is the difference between the opportunity cost of borrowing long term and the return on reserves invested in safe, short-term liquid assets. In the benchmark calibration, this premium consists only of the term premium, set at 1.5 percent—that is, the mean difference between the yield on 10-year U.S. treasury bonds and the federal funds rate during 1987–2005. Variants of the model consider the case where the opportunity cost of holding reserves also includes the sovereign spread. The coefficient of risk aversion is allowed to vary in the 1–10 range, as is customary in existing studies on growth and business cycles. It is assumed that the authorities are benevolent and maximize the welfare of the representative domestic consumer. The risk-free rate is set at 5 percent. The potential output growth rate is set at 3.3 percent, the mean real GDP growth rate in middle-income countries during 1980–2003 excluding sudden-stop years.

To analyze the role of the parameters in determining the optimal level of reserves, the model is repeatedly simulated by letting each key parameter vary (one at a time) over a plausible range while keeping the other parameters fixed at their baseline values as in Table A3.1. For the sake of comparison, in each plot (Figure A3.1) the optimal level of reserves is presented alongside the level of reserves predicted by the Greenspan-Guidotti rule.

When the size or probability of a sudden stop is sufficiently low, the optimal ratio of reserves is zero. (The costs of reserves exceed their benefits.) Optimal reserves rise near-linearly as the size of the sudden stop increases (beyond 2.5 percent of GDP), and non-linearly as the probability of a sudden stop increases. For some starting values of the sudden-stop probability, a relatively modest rise can imply a substantial adjustment in the optimal ratio of reserves to GDP: for example, an increase in the annual probability from 5 to 10 percent leads optimal reserves to rise from 4 to 10 percentage points of GDP. Changes in the opportunity cost of holding reserves (regardless of whether they stem from a change in the term premium or, when included, the sovereign spread) have a substantial impact on optimal reserve levels. For example, an increase in the opportunity cost by 1.5 percentage points above its baseline value reduces the optimal reserve ratio by more than 6 percentage points of GDP. Finally, the impact of changes in the degree of risk aversion on the optimal level of reserves is major when starting from relatively low values of the coefficient, but limited when starting from higher values.

### Determinants of Vulnerability to Sudden Stops

The probability of a sudden stop can be estimated as a function of a country’s economic fundamentals by running a probit estimation of the probability of sudden stops in a sample of 33 middle-income countries over 1980–2003. The explanatory variables are selected using a general-to-specific approach—that is, by initially considering a large set of potential explanatory variables and iteratively eliminating those that are statistically less significant to arrive at the preferred specification reported in Table A3.2. The complete set of variables considered is listed in Jeanne and Rancière (2006). All explanatory variables are averages of the first and second lags, and are thus predetermined with respect to the sudden stop. The results are robust to the inclusion of time effects and fixed effects.

Table A3.2.Probit Estimation of Probability of Sudden Stop
(1)(2)(3)(4)(5)
Real effective exchange rate deviation from HP trend1.521.521.851.78
(3.1)***(2.9)***(3.4)***(3.1)***
GDP growth–1.25–1.56–1.42–1.87
(1.6)*(1.7)*(1.7)*(1.8)*
Public debt/GDP0.810.780.721.020.85
(3.3)***(3.5)***(2.6)***(2.4)**(1.6)
Ratio of foreign liabilities to money in banking sector0.230.200.220.220.20
(3.0)***(2.7)***(2.8)***(2.2)**(1.9)*
Financial openness as (|gross inflows|)/GDP9.1110.049.8211.15
(5.7)***(5.5)***(4.9)***(4.9)***
Dummy for fixed exchange rate regime0.29
(1.8)*
Constant–2.31–1.80–2.41
(12.6)***(12.9)***(5.3)***
Observations707690707537537
Pseudo R20.140.060.17
Time effectsNoNoYesNoYes
Fixed effectsNoNoNoYesYes
Sources: IMF, International Financial Statistics database; World Bank, Global Development Finance database; and IMF staff calculations.Note: One asterisk (*) denotes significance at 10 percent; two asterisks (**) denote significance at 5 percent; and three asterisks (***) denote significance at 1 percent. Absolute values of z statistics are in parentheses. All explanatory variables are taken as averages of first and second lags. The fixed exchange rate regime is a “fix” or “peg” in the Reinhart and Rogoff (2004) classification.
Sources: IMF, International Financial Statistics database; World Bank, Global Development Finance database; and IMF staff calculations.Note: One asterisk (*) denotes significance at 10 percent; two asterisks (**) denote significance at 5 percent; and three asterisks (***) denote significance at 1 percent. Absolute values of z statistics are in parentheses. All explanatory variables are taken as averages of first and second lags. The fixed exchange rate regime is a “fix” or “peg” in the Reinhart and Rogoff (2004) classification.

The probability of a sudden stop decreases with the pre-crisis growth performance and increases with the currency’s real appreciation, the ratio of public debt to GDP, openness to financial flows, and the ratio of foreign liabilities to money in the banking sector.32 The last two determinants suggest that the vulnerability to sudden stops rises with the degree of international financial integration.33 A fixed exchange rate regime is associated with a higher probability of a sudden stop, though only if financial openness and exchange rate overvaluation are omitted from the regression.

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