Abstract
This paper studies the Mexican and Israeli experience with a target zone. The first part of the paper develops a model of exchange rate determination under a target zone regime with stochastic realignments, and examines the conditions under which the adoption of the target zone, instead of a fixed exchange rate, reduces the volatility of the interest rate differential. We conclude that if the variance of the expected realignment is sufficiently large, then the target zone will be useful. The second part of the paper is an empirical study that shows that the target zone regime helped reduce interest rate variability in Israel and Mexico by absorbing part of the shocks to the expected realignment with movements of the exchange rate inside the band.
I. Introduction
After having stabilized their inflation rates from the three digit level to a moderate range of between 10 and 30 percent, Mexico and Israel adopted target zones as their exchange rate regime and continued using the exchange rate as the nominal anchor for the economy. This decision came after experiencing fixed or crawling exchange rates for a period of time.
Given the lack of credibility of the exchange rate announcements, the wide swings in expectations of a realignment and the high degree of capital mobility these countries experienced, a fixed exchange rate made domestic interest rates extremely volatile. This experience with fixed exchange rates led these countries to switch to a target zone on the grounds that this regime would enable the country to keep the benefits of the exchange rate as a nominal anchor and at the same time provide a degree of flexibility to cope with highly variable capital movements. 1/ An exchange rate band allows for some degree of adjustment of the nominal exchange rate in response to shocks without breaking long-run policy commitments. In contrast to a fixed exchange rate, the variability of the domestic interest rate will be reduced because exchange rate fluctuations inside the band will help to absorb shocks that are speculative in nature.
After adopting the target zone, Israel changed the width of its band a few times and Mexico continuously widened its target zone. This brings us to the question of what are the conditions under which a target zone reduces the variance of the interest rate differential compared to what would be observed under a fixed exchange rate.
The first part of the paper develops a simple model of a target zone with stochastic realignments, and in this framework examines the conditions under which the interest rate variability will be reduced. We show that, when the variance of the expected realignment is larger than a linear function of the variance of the fundamentals, the target zone will be useful in reducing the volatility of the interest rate differential.
The second part of the paper looks at the Mexican and Israeli experience with a target zone. I present evidence supporting the claim that the currency band is a good instrument to reduce the volatility of the interest rate differential. To do this, I construct a measure of the expected realignment by subtracting the expected depreciation inside the band from the interest rate differential (as in Svensson (1992)); then I compare the variance of the expected realignment and the variance of the interest rate differential. The results from this exercise show that the target zone is effective in reducing interest rate variability.
The paper is structured as follows. Section II develops a simple model of the exchange rate in a target zone and looks at the variances of the exchange rate and the interest rate differential. Section III reviews the Mexican and Israeli experience and Section IV concludes.
II. The Model
1. Exchange rate determination
The exchange rate will be determined, following Bertola and Svensson (1993), in a simple log linear monetary model of the exchange rate. The exchange rate at any point in time will be equal to:
where f denotes a measure of fundamentals 1/ and x is the exchange rate, all variables are measured in logarithms. We assume that when a change in the central parity (realignment) takes place the fundamentals also jump by the size of the realignment and the exchange rate inside the band stays in the same place. 2/ Given this assumption, the expected depreciation will be the sum of the expected depreciation inside the band plus the expected realignment.
where a bar over a variable denotes deviations from the central parity and g is the expected rate of realignment (that is the product of the probability of a realignment and the size of it). Using equation (2) we can rewrite equation (1) as:
Subtracting the mean of the fundamentals from both sides of equation (3) we finally get:
Where a bar over a variable denotes deviations from the mean or central parities, as may be appropriate. The exchange rate deviation from central parity will be a function of the combined fundamental h(t):
We assume that the monetary authority controls the combined fundamental, by controlling f, to stay within the boundaries, H and −H. The devaluation expectation will be modeled as a Weiner process; the fundamentals (f) will also be a Weiner process in the absence of intervention. 1/ When the combined fundamental reaches H or −H, the government will control f, to maintain h between H and −H. Formally:
With these assumptions and applying Ito’s Lemma to equation (4) we get a second order differential equation for the exchange rate inside the band.
where
The solution for the exchange rate will be given by the following equation (See Krugman (1991)):
where
and
We can write the expression for the exchange rate inside the band as:
Next we derive the interest differential. Under the assumption of perfect capital mobility and risk neutrality it will be equal to the expected change in the exchange rate:
Where i(t) and i*(t) are the domestic and foreign interest rates respectively. The interest rate differential is thus the sum of the expected realignment plus the expected change in the exchange rate inside the band (the second term on the right hand side). These two terms will be negatively correlated because when the expectations of a realignment increases, this leads to a depreciation of the exchange rate inside the band which itself creates an expected appreciation inside the band, this is due to the usual mean reversion of exchange rates inside a target zone. This second effect is increasing with the expected realignment because we are closer to the upper part of the band. This effect will also be stronger the higher the instantaneous variance of the expected realignment, because in this case the probability of stabilizing intervention is higher.
We see that the response of the interest rate differential to a change in the expected realignment will be smaller than one (the value it has when there is a fixed exchange rate). The value of this response will be:
Given that the interest rate differential is less sensitive to the expectations of a realignment when a band is in place, this effect will help reduce the variance of the interest rate differential when a target zone is adopted. On the other hand, the interest rate differential will now be affected by changes in the fundamentals, f (i.e. velocity shocks), that by affecting the current position of the exchange rate inside the band generate an expected change in the position of the exchange rate inside the band in the future. This source of fluctuations in interest rate differentials is not present under a fixed exchange rate. Given that we have these two different forces that affect the variance of the interest rate differential in opposite directions, we would like to know which effect is stronger. We now turn our attention to this problem.
Given that the interest differential is also a Brownian motion, its instantaneous variance will be equal to:
Under a fixed exchange rate regime the variance of the interest rate differential is (σg2). The next proposition identifies a sufficient condition under which the instantaneous variance of the interest rate differential is reduced when we move from a fixed exchange rate to a target zone.
Proposition 1:
is a sufficient condition for achieving a reduction in the variance of the interest rate differential when a target zone is adopted instead of a fixed exchange rate.
Proof:
Equation (15) will reach a maximum either at h=0 or h=H; this can be seen by calculating the derivative of equation (15) with respect to h:
This equation will be increasing in h and it will positive at h=H. For some parameter values equation (17) can be negative when evaluated at h=0. Due to this, equation (16) can achieve its maximum either at h=0 or h=H. If the maximum is reached at h=0, then for the variance of the interest rate differential to decrease we need equation (16) evaluated at h=0 to be smaller than the variance of the expected realignment. This requirement gives us:
when the maximum is reached at h=H, the condition will give us:
Given that when equation (19) holds equation (18) will always hold then (19) is a sufficient condition to achieve a reduction on the interest rate differential variance when a target zone is adopted instead of a fixed exchange rate.
When the condition in equation (19) holds, the introduction of a target zone will increase the volatility of the nominal exchange rate but will reduce the instantaneous volatility of the interest rate differential. To the extent that interest rate hedging is not widespread in the economy and a large fraction of financial transactions are done at the nominal interest rate, the existence of a volatile expected realignment will generate volatile ex-post real interest rates. This will create an undesirable redistribution of wealth and has the potential to generate a financial crisis. This is why in our view the target zone is a useful instrument to partially shift the volatility away from the interest rate differential towards the nominal exchange rate. The arguments made in this section apply to the conditional volatility, but under some assumptions that make the expected realignment stationary, similar results can be obtained for the case of the asymptotic volatility of the interest rate differential.
III. The Mexican and Israeli Experiences with a Target Zone
1. An overview of the Mexican and Israeli experiences
By the end of 1987 inflation in Mexico had reached 150 percent per year. At this time the authorities implemented a comprehensive stabilization plan. The important fiscal adjustment was supported by price controls and a fixed exchange rate. After a year with a fixed exchange rate the authorities decided to implement a crawling peg regime. This was done mainly to reduce the rate of appreciation of the real exchange rate. After changing the rate of crawl several times, the Government finally adopted a target zone in November of 1991. The floor of the band was fixed and the ceiling was devalued by 2 cents per day (equivalent to 2.4 percent per year). In October 1992 the pace of crawl of the ceiling of the band was raised to 4 cents per day (equivalent to 4.5 percent per year). By the end of 1993 the width of the band was 9.4 percent.
The experience in Israel was similar. Following the stabilization plan of 1985 the New Israeli Shekel was fixed with respect to the U.S. dollar. This regime persisted with periodic devaluations and a change from pegging the currency with respect to the dollar to pegging to a basket of currencies. In January 1989 the government adopted a target zone with a fixed central parity and a 3 percent band around it. The width was increased to 5 percent in March 1990. After five realignments the authorities decided to start a daily devaluation of the central parity at a rate of 9 percent per year. Subsequently, there were two minor realignments and a reduction in the rate of crawl of the central parity.
Figures 1 and 2 show the evolution of the exchange rate and the target zone for the Mexican and Israeli case respectively.
Mexico: Exchange Rate
Mew Pesos per Dollar
Citation: IMF Working Papers 1995, 114; 10.5089/9781451853766.001.A001
Israel: Exchange Rate
New Shekel per Basket of Currencies
Citation: IMF Working Papers 1995, 114; 10.5089/9781451853766.001.A001
In what follows, I will study the behavior of the interest rate differential and show that the target zone was a helpful device to reduce the variance of the interest rate differential.
2. Target zones and expected realignment
A way to detect if the introduction of the target zone reduced the volatility of the interest rate differential is to recover the expected realignment from data on the interest rate differential. If the exchange rate were fixed, the interest rate differential would be equal to the expected realignment. By comparing the variance of the interest rate differential and the expected realignment we will be able to see if the band helped decrease the volatility of the interest rate differential.
In a target zone the interest rate differential adjusted by the preannounced devaluation path of the central parity is equal to the sum of the expected realignment plus the expected change of the exchange rate inside the target zone. 1/
Where δtz is the interest rate differential, g is the expected realignment, and Eds (tilde)/dt is the expected depreciation of the exchange rate inside the band. We can solve for the expected realignment:
Next, we need to estimate the expected change of the exchange rate inside the target zone. Following the literature and using weekly data we regress the observed monthly change in the logarithmic deviation of the exchange rate from the central parity (er(t+4)−ert) on a constant and on the logarithmic deviation of the exchange rate from the central parity (ert). 1/ For the Israeli case, I also included dummies for the different periods between realignments to account for changes in the credibility across regimes. We used weekly data from November 1991 to June 1993 for Mexico and from January 1989 to December 1993 for Israel. The data come from the Banco de Mexico and the Central Bank of Israel. The results for these regressions are shown in Table 1.
Expected Change in the Exchange Rate Inside the Band
Expected Change in the Exchange Rate Inside the Band
| er(t+4)-ert=c+β0ert | |||
| c | β0 | ||
| Mexico | −0.00095 | −0.4718 | |
| 1991.46-1993.26 | (−1.483) | (−4.635) | |
| Israel | −0.01074 | −0.6965 | |
| 1989.01-1993.52 | (−3.207) | (−12.334) | |
| t-statistics in parenthesis | |||
Expected Change in the Exchange Rate Inside the Band
| er(t+4)-ert=c+β0ert | |||
| c | β0 | ||
| Mexico | −0.00095 | −0.4718 | |
| 1991.46-1993.26 | (−1.483) | (−4.635) | |
| Israel | −0.01074 | −0.6965 | |
| 1989.01-1993.52 | (−3.207) | (−12.334) | |
| t-statistics in parenthesis | |||
The coefficients for the dummy variables for each different band for the case of Israel are not reported (They are significant for almost all the regimes). 1/ We see that in both countries the degree of mean reversion inside the band is very significant.
With these estimates we can derive the expected change of the exchange rate inside the band; and subtracting this from the monthly interest differential adjusted by the announced devaluation of the central parity gives us a measure of the expected realignment (See equation (21)). 2/
In Figures 3 and 4 we plot the estimated expected realignment and the interest rate differential adjusted by the depreciation of the central parity (For Mexico and Israel respectively). We see that throughout the period, the expected realignment was fairly high and extremely volatile in both countries, more so in Mexico. We see that the expected realignment was more volatile than the interest rate, and that the movements of the exchange rate inside the target zone were really helpful in isolating the domestic short term interest rates from shocks to the estimated expected realignment. The exchange rate inside the band was not plotted but goes up every time the expected realignment increases. This is a little surprising given the small size of the target zone; but when we look at the expected rate of change of the exchange rate inside the target zone for the following month we realize that it is of the same order of magnitude as the interest rate differential. From Table 1 we see that if the exchange rate is 3 percent higher than the central parity then the monthly expected appreciation inside the band is 1.5 percent and 1.1 percent for Mexico and Israel respectively. Given that this expected change of the exchange rate has a negative correlation with the expected realignment the smoothing effect on interest rates is considerable.
Interest Rates and Expected Realignment
Mexico
Citation: IMF Working Papers 1995, 114; 10.5089/9781451853766.001.A001
Interest Rates and Expected Realignment
Israel
Citation: IMF Working Papers 1995, 114; 10.5089/9781451853766.001.A001
To confirm that the asymptotic and conditional variance of the expected realignment is higher than the interest rate differential variance, we estimated both variances for both countries. 3/ We used the sample variance to estimate the asymptotic variance of the interest rate differential and the expected realignment (See Hamilton (1994)). To estimate the conditional variance we estimated the process driving the interest rate differential and the expected realignment as an AR(5) for both countries. 1/ Table 2 presents the estimates for the asymptotic and conditional variance and the F statistic for each of these variances. We see that in both countries the estimated variance for the interest rate differential is much smaller than the expected realignment estimated variance. This is strong evidence in favor of a target zone over a fixed exchange rate regime, because if a fixed exchange rate regime were in place the interest rate differential would be equal to the expected realignment. For larger maturities this effect will be reduced because the expected rate of change of the exchange rate in the band is limited by the size of the band. Then as the maturity of the interest rate differential increases the expected change of the exchange rate inside the band per unit time decreases, due to this the negative correlation between the expected realignment and the expected change of the exchange rate inside the band also decreases.
Asymptotic and Conditional Variance
Statistically significant at 95 percent.
We conclude that there is supporting evidence to the claim that the introduction of the target zone was a helpful instrument in reducing the volatility of the interest rate differential.
IV. Conclusion
We first studied a model of exchange rate determination under a target zone with stochastic realignments, to find the conditions under which the introduction of the target zone is helpful in reducing the variance of the interest rate differential. The main conclusion from this exercise is that, if the volatility of the expected realignment is sufficiently large, then the target zone will reduce the variance of the interest rate differential.
The second part of the paper looks at the Mexican and Israeli experience with a target zone. The principal result from the empirical study is that the target zone regime helped to reduce the interest rate variability by absorbing part of the shocks to the expected realignment through movements of the exchange rate inside the band. We conclude that the target zone is a useful exchange rate regime to reduce the variance of the interest rate differential.
References
Bertola, Guiseppe, and Ricardo Caballero, (1992), “Target Zones and Realignments,” American Economic Review, Vol. 82, No. 3, pp. 520–35.
Bertola, Guiseppe, and Lars Svensson, (1991), “Stochastic Devaluation Risk and the Empirical Fit of Target Zones Model,” NBER Working Paper No. 3576.
Caramazza, Francesco, (1993), “French-German Interest Rates Differentials and Time Varying Realignment Risk,” IMF Staff Papers, Vol. 40, No. 3, pp. 567–83.
Flood, Robert, Andrew Rose and Donald Mathieson, (1991), “An Empirical Exploration of Exchange Rate Target Zones,” IMF Working Paper No. 15.
Hamilton, James, (1994), “Time Series Analysis,” Princeton University Press.
Helpman, Elhanan, and Leonardo Leiderman, (1991), “Israel’s Exchange Rate Band,” Mimeo.
Krugman, Paul, (1991), “Target Zones and Exchange Rate Dynamics,” Quarterly Journal of Economics, Vol. 56, No. 3, pp. 669–82.
Svensson, Lars, (1991), “The Term Structure of Interest Rate Differentials in a Target Zone: Theory and Swedish Data,” Journal of Monetary Economics Vol. 28, pp. 87–116.
Svensson, Lars, (1991), “Target Zones and Interest Rate Variability,” Journal of International Economics, Vol. 31, pp. 27–53.
Svensson, Lars, (1992), “Assessing Target Zone Credibility: Mean Reversion and Devaluation Expectations in the ERM 1979–92,” European Economic Review (forthcoming).
Werner, Alejandro, (1995), “Exchange Rate Target Zones, Realignments and the Interest Rate Differential: Theory and Evidence,” Journal of International Economics, (forthcoming).
I would like to acknowledge the comments and suggestions of Ricardo Caballero, Gustavo Canonero, Martina Copelman, Rudiger Dornbusch, Stanley Fischer, Luis Herrera and Peter Wickham. I also want to thank Gil Bufman and Leonardo Leiderman for providing the data. This research was partly supported by a grant from the World Economy Laboratory at MIT.
Similar arguments have been made by Helpman and Leiderman (1991) and Svensson (1993).
If we think in terms of a monetary model f will be the sum of the nominal money stock and a velocity shock. Starting with a money demand as follows m−p=v+y−απ, and assuming purchasing power parity, we can arrive at equation (1).
The model will not change if I alternatively assume that a realignment implies a jump of fixed size in the exchange rate independently of the position in the band, and that this level of the exchange rate will be the new central parity. Thus, the central parity is adjusted by different amounts depending on the position of the exchange rate inside the band. At the time of a realignment the money supply will be adjusted to achieve this result.
I will assume that these two stochastic processes are uncorrelated.
Although the model of the previous section did not include a deterministic trend for the devaluation of the central parity, this will not change any of the results. All the results in the paper will only shift by a constant.
The estimation of the expected change of the exchange rate inside the band is done under the assumption that no realignment takes place. Because of this reason in the case of Israel we drop the observations from the months before and after each realignment.
Several specifications were tried and the results presented here were not changed.
The estimation procedure does not incorporate the possibility that the expected future deviations of the exchange rate from the central parity cannot be greater than the width of the band. A logistic transformation that took this into account was tried and the results were similar to those reported here. Another drawback is the lack of offshore interest rates denominated in mexican pesos and new sheckels.
We rejected the null hypothesis of a unit root at 20 percent for the expected realignment and the interest rate differential for both countries.
Where a lower order process was appropriate we estimated that instead. These results are available upon request.
