the Simplest Test of Target Zone Credibility

International Monetary Fund

doi:10.5089/9781451947007.001.A001

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the Simplest Test of Target Zone Credibility

Author: International Monetary Fund

Publication Date:: 01 Jan 1990

eISBN:: 9781451947007

ISBN:: 9781451947007

Language:: English

Keywords:: WP; rate-of-return band; annualized interest rate differential; krona rates of return; exchange rate regime; spot exchange rate; krona Treasury Bill interest rate

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Under the assumption of no arbitrage exchange rate target zone credibility is tested by whether domestic interest rates fall within “rate-of-return bands” between the maximum and minimum home-currency rate of return on a foreign investment absent a devaluation. Under the assumption of uncovered interest rate parity credibility is tested by whether expected future exchange rates fall within the exchange rate band. These tests are applied on data about the Swedish target zone during January 1987-August 1990.

Abstract

I. Introduction

An exchange rate target zone with an explicit band for the exchange rate implies bounds on the amount of depreciation and appreciation of the exchange rate, since the exchange rate cannot move further than to the edges of the band. Given foreign interest rates, these bounds on the amount of depreciation and appreciation imply bounds on domestic currency rates of return to foreign investment. These rate-of-return bounds define a rate-of-return band around the foreign interest rates. The rate-of-return bands are narrower for longer terms (times to maturity), since the maximum amount of appreciation and depreciation per unit term is decreasing in the term.

Suppose there is sufficiently free international capital mobility, so we can assume that there remain no international arbitrage possibilities. If the domestic interest rate for some term is outside the rate-of-return band for that term, the exchange rate regime cannot be completely credible within the horizon given by the term. That is, investors must perceive a risk of a change in the regime, for instance a devaluation, before maturity. For if the target zone was considered completely credible there would be completely safe arbitrage.

We may weaken the assumption about no arbitrage somewhat by allowing for some small friction in international capital flows so that arbitrage possibilities may remain for short periods of time, but that they are eventually removed by large capital flows. Then, if the domestic interest rate for some term is outside the rate-of-return band and capital flows are large (and in the right direction) we may not necessarily conclude that the target zone lacks credibility. However, with small capital flows, or large capital flows in the wrong direction, we can indeed conclude that the target zone is not credible.

Therefore, whether or not the domestic interest rates are within the rate-of-return bands can be used as a very simple and most straightforward test of the credibility of a target zone. If the interest rates are within the rate-of-return bands it does not necessarily follow that the target zone is credible, but if the interest rates fall significantly outside the band it definitely follows that the target zone is not credible (if there is sufficient international capital mobility and capital flows are either not large or in the wrong direction).

Grønvik (1986) discussed rate-of-return bands (called “interest rate corridors”) in a study of the Norwegian forward foreign exchange market. He argued that they should be interpreted as constraints on domestic monetary policy and domestic interest rates. He computed a three-month rate-of-return band for Norway during the period 1983-85 and showed that the Norwegian three-month interest rate was outside and above the rate-of-return band during the end of 1984 and 1985. This fact was used to explain the large growth of the Norwegian forward exchange market and increasing attempts to circumvent the Norwegian capital controls.

Here we shall assume that there is sufficient capital mobility between Sweden and the rest of the world so that the no-arbitrage assumption can be used, and we shall consequently apply the rate-of-return bands as a simple test of the credibility of the Swedish target zone. Separately, we shall also examine the volume and direction of capital flows. ^2/

Under the additional assumption of uncovered interest rate parity, expected future exchange rates can be computed from current spot exchange rates and domestic and foreign interest rates for different maturities. Whether or not the domestic interest rate for the corresponding maturity is inside its rate-of-return band is then equivalent to whether or not the expected future exchange rate at maturity is inside the exchange rate band. Then target zone credibility can be tested by examining whether the expected future exchange rate is inside or outside the exchange rate band. ^3/

Also, under the assumption of uncovered interest rate parity the lack of credibility of the target zone can be quantified, in that the expected rate of depreciation for different terms, adjusted for the rate of depreciation consistent with a credible exchange rate band, can be used as a measure of the expected rate of devaluation. Under additional assumptions about the stochastic process of devaluations and the size of devaluations, the perceived probability of devaluations per unit of time can be estimated.

Section 2 and 3 define rate-of-return bands, expected rates of depreciation and expected rates of devaluation. Section 4 examines data from the Swedish target zone during the period January 1987-August 1990 (the period is determined by the availability of interest rate data). Section 5 concludes. An appendix presents some technical details.

II. No Arbitrage: The Rate-of-Return Band

Let S_t, $i_{t}^{τ}$ , and $i_{t}^{* τ}$ denote, respectively, the spot exchange rate in period t (in units of domestic currency per unit of foreign currency), the domestic-currency interest rate in period t for term-τ loans in domestic currency, and the foreign-currency interest rate in period t for term-τ loans in foreign currency. (The “foreign currency” may be a particular foreign currency, or a basket of several foreign currencies.) Let us measure the term τ in months, and let the interest rates be annualized effective interest rates. The annualized effective domestic-currency ex post rate of return on a foreign currency investment in period t of duration τ, $R_{t}^{τ}$ , is then given by

\begin{matrix} (1) & R_{t}^{τ} = (1 + i_{t}^{* τ}) {(S_{t + τ} / S_{t})}^{12 / τ} - 1 \end{matrix}

This expression can be understood as follows. Investing one unit of domestic currency means investing 1/S_t units of foreign currency. This invested in a τ-month foreign currency bond results in ${(1 + i_{t}^{* τ})}^{τ / 12} / S_{t}$ units of foreign currency after τ months (recall that $i_{t}^{* τ}$ is the annualized effective interest rate). This is ${(1 + i_{t}^{*})}^{τ / 12} S_{t + τ} / S_{t}$ units of domestic currency, which equals ${(1 + R_{t}^{τ})}^{τ / 12}$ , where $R_{t}^{τ}$ is the annualized effective domestic-currency rate of return. The result can be written as in equation (1).

Suppose the exchange rate is restricted to a band with lower and upper bounds S and $\bar{S}$ ,

\begin{matrix} (2) & \underline{S} \end{matrix} \leq S_{t} \leq \bar{S} .

The exchange rate band implies bounds on the amount of depreciation and appreciation of the domestic currency. This implies that the rate of return $R_{t}^{τ}$ will also be restricted to a band,

\begin{matrix} (3) & {\underline{R}}_{t}^{τ} \leq R_{t}^{τ} \leq \end{matrix} {\bar{R}}_{t}^{τ},

which we will call the rate-of-return band. The lower and upper bounds on the rates of return are given by

\begin{matrix} (4 a) & {\underline{R}}_{t}^{τ} \end{matrix} = (1 + i {\underset{t}{*}}^{τ}) (\underline{S} / S_{t})^{12 / τ} - 1

and

\begin{matrix} (4 b) & {\bar{R}}_{t}^{τ} \end{matrix} = (1 + i {\underset{t}{*}}^{τ}) (\bar{S} / S_{t})^{12 / τ} - 1 .

The bounds are decreasing in the current exchange rate: A higher exchange rate means a weaker domestic currency, which increases the scope for domestic currency appreciation. This lowers the domestic currency rate of return on foreign investments and shifts down the rate-of-return band. The width of the rate-of-return band is decreasing in the term: A given relative change in the exchange rate during a longer time period implies a smaller relative change per unit of time. Therefore the upper bound of the rate of return is decreasing in the term, and the lower band is increasing.

Under a completely credible exchange rate regime, the no-arbitrage assumption implies that the domestic interest rate $i_{t}^{τ}$ must lie inside the rate-of-return band (3). For if the domestic interest rate is outside the rate-of-return band there is completely safe arbitrage: If the domestic interest rate is above (below) the rate-of-return band, an agent can borrow (lend) abroad and lend (borrow) at home and make a safe profit. Such safe arbitrage would not be compatible with an equilibrium in the world capital market.

Therefore, if indeed the domestic interest rate in some period and for some maturity is outside the rate-of-return band (3), the no-arbitrage assumption implies that the exchange rate regime cannot be completely credible. That is, investors must perceive a risk of a change in the exchange rate regime, for instance a devaluation (a shift in the band). Therefore, the simplest test of whether the exchange rate band is completely credible is to check whether domestic interest rates are inside the rate-of-return band, in different periods and for different maturities. This we shall do below, for the Swedish target zone.

III. Uncovered Interest Rate Parity: Expected Exchange Rates

Let us now make the additional assumption of uncovered interest rate parity: That the expected home currency depreciation compensates for the interest rate differential between home and foreign interest rates such that the expected rate of return on a home-currency investment equals the expected rate of return on a foreign-currency investment. ^4/ We can write uncovered interest rate parity on the form

\begin{matrix} (5) & \begin{matrix} {{}_{t}S}_{t + τ} = S_{t} {[\frac{1 + i_{t}^{τ}}{1 + i_{t}^{* τ}}]}^{τ / 12}, \end{matrix} \end{matrix}

where _tS_t+τ denotes the expected value in month t of the exchange rate to rule in month t+τ. Hence, from a particular month’s exchange rate and domestic and foreign interest rates for bonds with τ months to maturity, we can compute the month’s expectation of the exchange rate τ months later.

Whether the domestic interest rate in month t is inside or outside the rate-of-return band for a particular term of τ months is then equivalent to whether the month’s expectation of the exchange rate in month t+τ is inside or outside the exchange rate band. Therefore, an alternative way to illustrate the credibility of a target zone is to compute the expected future exchange rates according to equation (5), and then examine whether the expected future exchange rates are inside or outside the exchange rate band. This we shall also do below, for the Swedish target zone.

Given expected future exchange rates computed by equation (5), we can compute expected annualized rates of depreciation $d_{t}^{τ}$ from month t to month t+τ according to

\begin{matrix} (6) & d_{t}^{τ} = \end{matrix} {({{}_{t}S}_{t + τ} / S_{t})}^{12 / τ} - 1.

Replacing the expected exchange rate in equation (6) by the lower and upper bounds for the exchange rate, S and $\bar{S}$ , we get the minimum and maximum rates of depreciation compatible with a credible exchange rate band, ${\underline{d}}_{t}^{τ}$ and ${\bar{d}}_{t}^{τ}$ . By equation (5) the annualized expected rates of depreciation in equation (6) are of course just annualized interest rate differentials and fulfill

\begin{matrix} (7) & d_{t}^{τ} = \end{matrix} \frac{1 + i_{t}^{τ}}{1 + i_{t}^{* τ}} - 1.

We shall also compute and discuss these expected rates of depreciation. We shall see that they need to be interpreted with some care:

More specifically, we make the additional assumption that devaluations are expected to occur regularly over time with a probability intensity λ per unit of time. This implies that devaluations are assumed to follow a Poisson stochastic process. ^5/ Furthermore, we assume that devaluations, if and when they occur, are expected to be of a given size g (measured in percent). Then the expected rate of devaluation per unit time, d (measured in percent per unit of time), is simply the product of the probability per unit time of a devaluation and the size of a devaluation,

\begin{matrix} (8) & d \end{matrix} = λ g .

Excess of the expected rate of depreciation over the maximum rate of depreciation compatible with the target zone indicates a positive expected rate of devaluation. From this expected rate of devaluation, assuming a given size of a devaluation, the corresponding probability per unit of time of a devaluation can be computed.

IV. Swedish Data

Sweden has a unilateral exchange rate target zone. An exchange rate index is defined as the exchange rate between the krona and a currency basket consisting of the trade-weighted currencies of Sweden’s 15 largest trade partners (with double weight for the dollar). The basket exchange rate is restricted to a band around a benchmark rate. The benchmark rate has been fixed at 132 since the latest devaluation in October 1982. The bandwidth was first kept secret at ±2.25 percent around the benchmark. In June 1985 the bandwidth was reduced to ±1.5 percent (between 130 and 134) and made public. ^6/ The data consists of monthly observations (last trading day of the month) for the period January 1987 to August 1990 of the basket exchange rate, krona interest rates for Swedish Treasury Bills and government bonds of terms from 1 to 60 months, corresponding Euro interest rates for most of the currencies in the basket and, for a few currencies and terms above 12 months where Euro rates are not available, national bond rates. The available interest rates for terms up to 12 months make up about 95 percent of the currency basket. The available interest rates for longer terms up to 60 months make up about 75 percent of the currency basket. (The interest rates for January 1987 are incomplete and should not be taken seriously. Long term interest rates for some important basket currencies are not easily available for the period before February 1987.)

1. Rate-of-return bands

Using the weights of the currency basket, foreign basket interest rates have been computed. Then the lower and upper bounds of the krona rates of return on foreign basket investment have been computed according to equation (4), for each month and for the different maturities.

The resulting rate-of-return bands’ dependence on the exchange rate is illustrated in Figure 1. The bottom solid curve shows the basket exchange rate’s percentage deviation, S, from the benchmark rate during the period January 1987-August 1990. ^7/ That is, in Figure 1 the benchmark rate is at 0 percent, and the exchange rate band is between plus and minus 1.5 percent, shown as dashed horizontal lines. (Positive values indicate a weak krona.) The two dashed curves in the upper part of the figure show the upper and lower bounds on the rate of return on a 12-month foreign-currency basket investment. The top solid curve shows the 12-month basket interest rate. We see that when the exchange rate is near its benchmark rate, for instance in December 1987, the rate-of-return is symmetric around the basket interest rate, since then the exchange rate can either appreciate or depreciate 1.5 percent. We also see that when the exchange rate is at the lower edge of its band (when the krona is strong), for instance in April 1988, the rate-of-return band shifts up and is between 0 and +3 percent relative to the basket interest rate, since then the home currency can only depreciate. For a 24-month term, the width of the rate-of-return band will decrease to about a half, and for a six-month term it will increase to about double the width for a 12-month term (“about” rather than “exactly” because effective annualized rates of return are used).

Figure 2 shows the six-month rate-of-return band (the top and bottom dashed curves) and the six-month basket interest rate (the bottom solid curve). Also shown is the six-month krona Treasury Bill interest rate (the top solid curve). We see that the krona interest rate is above the basket interest rate, but that the krona interest rate is well into the rate-of-return band, except during January and February 1990 and in August 1990. From Figure 2 we cannot here find any evidence of a devaluation risk within a six-month horizon except possibly at these two occasions. The three- and one-month interest rates (not shown here) are completely inside their rate of return bands, which is not surprising given that the rate-of-return bands are (respectively) about two and six times the width of the six-month rate-of-return band.

Figure 3 shows the same variables for a 12-month term. Here we see that the krona interest rate was slightly above the band during October 1987-January 1988, and that it rose much above the band from November 1989. Hence we find clear evidence of a perceived risk of a devaluation within a 12-month horizon during the latter period, possibly also an indication of perceived devaluation risk during the earlier period. ^8/

Figure 4 shows the same variables for a 24-month term. Here we find evidence of a perceived devaluation risk throughout the period, except during January 1989 (which was when the government announced that foreign exchange controls where going to be abolished). We note the dramatic rise above the band since September 1989.

Figure 5, for a 60-month term, gives the same impression as Figure 4, except that now the rate-of-return band is narrower and the 60-month domestic rate is further above the band, and above the band also in January 1989. ^9/

2. Capital flows

The reasoning above is under the assumption of free international capital mobility, which motivates the no-arbitrage assumption. If the domestic interest rate happens to move outside the rate-of-return band for some term, large capital flows would quickly push the domestic interest rate inside the rate-of-return band, if the target zone is credible. If no big capital flows arise, in spite of the domestic interest rate being outside the rate-of-return band, this is evidence of a lack of credibility of the target zone.

Let us therefore check whether any large capital flows have been observed during periods when krona interest rates have been outside their rate-of-return bands. Let us more specifically examine the amount of foreign exchange intervention done by Sveriges Riksbank. Figure 6 shows monthly foreign exchange interventions (in billion kronor) during the period January 1987-August 1990. Positive values denote foreign exchange inflows (purchases of foreign exchange). The Riksbank reports weekly and monthly net foreign currency flows, consisting of spot foreign exchange interventions less government net foreign borrowing. These spot foreign exchange interventions are plotted for each month in Figure 6 as the dashed curve, denoted “Spot”. These interventions do not, however, include changes in the Riksbank’s position in the forward foreign exchange market, the forward foreign exchange interventions. (These forward positions are not included in the official definition of foreign exchange reserves, although they should of course be included in an economically meaningful definition of foreign exchange reserves.) Changes in the forward foreign exchange positions are reported separately, in Sveriges Riksbank Quarterly Review, with a three-month lag. When these forward interventions are added to the-spot interventions, we get the total foreign exchange interventions, plotted as a solid curve and denoted “Total” in Figure 6.

Let us compare the total interventions in Figure 6 to the 12-month interest rate in Figure 3. The 12-month interest rate is slightly outside its rate-of-return band from the end of October 1987 to the end of January 1988. During these months we do not see any large foreign exchange inflows in Figure 6; on the contrary we see some outflows. This lends some support to the interpretation that these months might have been a period with a lack of credibility of the target zone on a 12-month horizon. ^10/ We also see no large inflows in August 1988 and October 1988, when the interest rate is at the edge of its rate-of-return band. The 12-month interest rate again moves outside its rate-of-return band from the end of November 1989 in which month we see a large foreign exchange outflow, indicating a definite lack of credibility of the target zone. Total interventions are small during December 1989-February 1990, although previously reported spot foreign exchange flows are positive. This does not contradict a continuing lack of credibility of the target zone. During March-May 1990 there are large inflows and the 12-month krona interest rate approaches its band, pointing to a possible improvement in the target zones credibility. At the time of writing this, in October 1990, changes in the Riksbank’s forward foreign exchange positions have not been released for months after May; August shows a large outflow with regard to the spot interventions, and the 12-month interest rates moves further above the rate-of-return band, consistent with credibility deteriorating again.

3. Expected exchange rates

The expected exchange rate index is computed according to equation (5) and plotted as a function of the calendar month t and the horizon month τ in Figure 7. (The exchange rate index is here measured in absolute units, rather than in percentage deviation from the benchmark value as in Figure 1.) The spot exchange rate is plotted for a zero horizon and can be read off the front edge of the box (the bottom of the box corresponds to an exchange rate index of 130, the lower edge of the exchange rate band). For each calendar month, running from month 1 (January 1987) to month 44 (August 1990), the expected future exchange rate is plotted parallel to the left edge of the box, towards the back edge of the box which corresponds to a 60-month horizon ahead of the calendar month.

Figure 8 shows the spot exchange rate and the expected future rate in 12, 24, and 60 months plotted against the calendar month. (The horizontal dashed line for the currency index at 134 shows the upper edge of the exchange rate band.) We see that the expected forward exchange rates were lowest in January 1989, and that they reach their peaks in January 1990 and again in August 1990.

In January 1989, the expected exchange rate 12 months ahead was well inside the band. The expected exchange rate 60 months ahead was outside the band, a little above 139, which corresponds to an expected depreciation of 6.5 percent in five years, that is only 1.3 percent per year.

The situation in January and August 1990 was different. The expected exchange rate only six months ahead was at the edge of the band (not shown in the figure), the expected exchange rate 12 months ahead was well outside the band, and the expected exchange rate 60 months ahead was at 156, which corresponds to an expected depreciation of about 18 percent in five years, a good 3.6 percent per year.

4. Expected rates of depreciation

In Figure 9 we have plotted the expected annualized rate of depreciation $d_{t}^{60}$ , using each month’s expectation of the exchange rate 60 months later. We have also plotted the minimum and maximum rates of depreciation consistent with a credible exchange rate band, ${\underline{d}}_{t}^{60}$ and ${\bar{d}}_{t}^{60}$ . (These variables are of course by equation (7) nothing but the annualized interest rate differential between the 60-month krona interest rate and the 60-month basket interest rate, and the maximum and minimum 60-month rates of return less the 60-month basket interest rate, cf. Figure 5.)

We see that the expected rate of depreciation reached a minimum, 1.3 percent per year, in January 1989. From September 1989 it rose to the highest level during the whole period, 3.6 percent per year, in March 1990. In August 1990 the expected rate of depreciation again rose to almost the same high level. (The peak for the expected future exchange rate in 60 months in Figure 8 occurs for January 1990, whereas the peak for the expected depreciation in 60 months in Figure 9 occurs for March 1990; the difference is because the the spot exchange rate was lower in March 1990 than in January 1990, as is apparent in Figure 8.)

The excess of the total expected rate of depreciation over the expected rate of depreciation within the band can be interpreted as the expected rate of devaluation. What is the expected rate of depreciation within the band? We know that the expected rate of depreciation within the band is bounded by the minimum and maximum rates of depreciation. If the unconditional probability distribution of future exchange rates is symmetric, the expected depreciation within the band is for sufficiently long horizons given by depreciation to the middle of the band. This would correspond to the rate of depreciation given by the middle of the band between d⁶⁰ and ${\bar{d}}^{60}$ in Figure 9. Since the band is narrow anyhow, let us simplify by setting the expected depreciation within the band equal to zero, and hence identify the expected rate of depreciation in Figure 9 with the expected rate of devaluation. ^11/

If we then assume that devaluations are expected to occur regularly over time according to a Poisson process, we know from equation (8) that the expected rate of devaluation is equal to the product of the probability intensity per unit of time of a devaluation and the size of a devaluation. Suppose for simplicity that a devaluation, if it occurs, is expected to equal 10 percent. Then we can interpret an expected rate of depreciation of 1.3 percent per year in Figure 9 in January 1989 as indicating expectations with a probability of devaluations of 13 percent per year ( $λ = d_{t}^{60} / g = 1.3 / 10$ ), that is, an expected length of time to the next devaluation of almost eight years. ^12/ Similarly, when the expected rate of depreciation rose to 3.6 percent per year in March 1990, we can interpret this as indicating expectations with a probability of devaluations of 36 percent per year, that is, an expected length of time to the next devaluation of less than three years.

If we instead assume that a devaluation, if it occurs, is expected to be 20 percent rather than 10 percent, the probabilities above are halved and the expected length of time to the next devaluation is doubled.

In this context, we can also demonstrate what the implications of a sizable risk premium are. Suppose that counter to what we assume there is foreign exchange risk premium, say equal to plus 1 percent per year (Swedish currency assets are thus considered riskier than foreign currency assets, far from an obvious assumption). This is probably a very large risk premium (see Svensson (1990a) for details). In any case, a 1 percent risk premium here implies that the expected rate of depreciation in March 1990 is not 3.6 percent per year but 2.6 percent per year. Then, with a 10 percent devaluation, the probability intensity of a devaluation is 26 percent per year, which corresponds to an expected time between devaluations of four years. This is still a sizable devaluation risk, even with the risk premium.

V. Conclusions

In conclusion, we have applied very simple tests of credibility to the Swedish target zone during January 1987-April 1990, in a step-by-step fashion. First, under the assumption of sufficient international capital mobility and hence a no-arbitrage assumption, we have tested target zone credibility by examining whether krona interest rates fall outside rate-of-return bands for different terms. As an additional check, we have also examined the volume and direction of foreign exchange interventions at times when krona interest rates have been outside their rate-of-return bands.

This simple test reveals that the Swedish target zone never had credibility within a two-year horizon or longer, and that it occasionally has lacked credibility within a 12-month horizon. The loss in credibility in the winter of 1989-90, and again in August 1990, is particularly evident. On these occasions there is an indication of a perceived devaluation risk even within close to six months, since the six-month krona interest rate then reaches the edge of the six-month rate-of-return band.

That the interest rates for shorter maturities than six months fall inside the bands does not, of course, imply that the target zone is necessarily credible for short horizons. The rate-of-return bands become very wide for short maturities. The probability of exchange rate movements to the edges of the band may be rather small, so the expected depreciation or appreciation in a short period within a credible band is much smaller. Therefore, short term interest rates may be well inside the rate-of-return bands as calculated here and still indicate devaluation risks. This is the case for the target zone models originated by Krugman (1988), where structural assumptions allow an explicit computation of the expected depreciation within a credible exchange rate band. In Svensson (1990b) such computations are used to specify interest rate bands for different maturities that are much narrower than the simple rate-of-return bands calculated here. The empirical results on the Swedish target zone in that paper for the period February 1986-October 1989 (hence, excluding the winter 1989-90!) indicate statistically significant average expected rates of devaluation within the range between 1.2 and 2 percent per year for horizons down to one month. The first simple test used in the present paper has, however, the attractive feature of relying on a minimum of structural assumptions. ^13/

Second, under the additional assumption of uncovered interest rate parity, expected future exchange rates have been computed. Then target zone credibility has been illustrated in greater detail by plotting these expected future exchange rates in relation to the exchange rate band. The assumption also allows the lack of credibility of the target zone to be quantified: Expected krona depreciation for different terms in excess of the maximum krona depreciation compatible with the exchange rate band indicates expected positive devaluation for those terms. The expected devaluation within a 60-month horizon varies between 6.5 and 18 percent (the peak is observed in the winter 1990).

Third, by assuming that devaluations follow a Poisson process, the expected rate of devaluation of the krona can be interpreted as the product of the probability per unit of time of a devaluation and the size of the devaluation. The expected rate of depreciation within a 60 month horizon has usually fluctuated between 1.3 and 3 percent per year. The peak is 3.6 percent in March 1990. For a 10 percent devaluation, that corresponds to a probability of devaluation of 36 percent per year, or an expected length of time to the next devaluation of less than three years.

These very simple tests of target zone credibility seem able to convey a fair amount of interesting information. It should be worthwile to collect and compare similar information for longer time periods and for other target zones, both multilateral target zones like those in the Exchange Rate Mechanism within the European Monetary System and unilateral ones like those in the Nordic countries other than Denmark.

Of special interest is the issue of how efficient and liquid the world capital market is for debt instruments in different currencies for longer terms. For simplicity, we have in this paper assumed that international investment for terms between 1 and 60 months is available in the important currencies in the Swedish currency basket. This assumption is surely all right for terms up to 12 months. It may be doubtful for longer maturities and earlier periods, since exchange controls were probably more binding for longer term assets and since some longer term bonds are less liquid on international capital markets. With the abolishing of exchange controls and increasing international financial integration, the assumption may be all right also for longer terms. ^14/

Appendix: The Rate-of-Return Band with Uneven Depreciation

For the 60-month interest rates, $i_{t}^{* 60}$ , let us assume that interest is paid yearly, in months t+12, t+24, t+36, t+48 and t+60. We then define the ex post rate of return, $R_{t}^{60}$ , as fulfilling the equation

\begin{matrix} (A 1) & 1 = i_{t}^{* 60} Σ_{n = 1}^{5} \frac{s_{t + 12 n} / S_{t}}{{(1 + R_{t}^{60})}^{n}} + \end{matrix} \frac{s_{t + 60} / S_{t}}{{(1 + R_{t}^{60})}^{5}} .

Let us assume that the home currency appreciates (depreciates) gradually at an even rate until the exchange rate reaches the edge in month t+60. That is, S_t+12n = S_t(S/S_t)^n/5, n = 1, 2, .., 5 ( $S_{t + 12 n} = S_{t} {(\bar{S} / S_{t})}^{n / 5}$ , n = 1, 2, .., 5). This still gives rise to lower and upper bounds according to equations ^(4a) and ^(4b).

Suppose we define the lower (upper) bound on the rate of return under the alternative assumption that the exchange rate appreciates (depreciates) to the lower (upper) edges of the exchange rate band already in one year and then stays there. That is, S_t+12n = S, n = 1, 2, .., 5) ( $S_{t + 12 n} = \bar{S}$ , n = 1, 2, .., 5). Under this assumption the lower and upper bounds for the rate of return fulfill

\begin{matrix} (A 2 a) & 1 = \end{matrix} [\frac{i_{t}^{* 60}}{{\underline{R}}_{t}^{60}} [1 - \frac{1}{{(1 + {\underline{R}}_{t}^{60})}^{5}}] + \frac{1}{{(1 + {\underline{R}}_{t}^{60})}^{5}}] \bar{S} / S

and

\begin{matrix} (A 2 b) & 1 = [\frac{i_{t}^{60}}{{\bar{R}}_{t}^{60}} [1 - \frac{1}{{(1 + {\bar{R}}_{t}^{60})}^{5}}] + \frac{1}{{(1 + {\bar{R}}_{t}^{60})}^{5}}] \bar{S} / S . \end{matrix}

These expressions can be rewritten according to

\begin{matrix} (A 3 a) & {\underline{R}}_{t}^{60} \frac{{(1 + {\underline{R}}_{t}^{60})}^{5} S / \underline{S} - 1}{{(1 + {\underline{R}}_{t}^{60})}^{5} - 1} \end{matrix} = i_{t}^{* 60}

and

\begin{matrix} (A 3 b) & {\bar{R}}_{t}^{60} \frac{{(1 + {\bar{R}}_{t}^{60})}^{5} S / \bar{S} - 1}{{(1 + {\bar{R}}_{t}^{60})}^{5} - 1} \end{matrix} = i_{t}^{* 60} .

These equations are easy to solve numerically.

The rate-of-return band arising from equations (A3a) and (A3b) is wider than the one resulting from equations ^(4a) and ^(4b), although the difference is small and hardly visible in ^{Figure 5}.

References

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Krugman, Paul, “Target Zones and Exchange Rate Dynamics,” Quarterly Journal of Economics (Cambridge), Vol. 105 (1990, forthcoming).
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Porter, Michael G., “A Theoretical and Empirical Framework of Analyzing the Term Structure of Exchange Rate Expectations,” Staff Papers, International Monetary Fund (Washington), Vol. 18 (1971), pp. 613–642.
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Ringström, Olle, “The Exchange Rate Index: An Instrument for Monetary and Exchange Rate Policy,” Sveriges Riksbank Quarterly Review: 4 (Stockholm: Sveriges Riksbank, 1987), pp.16–26.
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Svensson, Lars E. O. (1990a), “The Foreign Exchange Risk Premium in a Target Zone with Devaluation Risk,” Working Paper (Stockholm: Institute for International Economic Studies, 1990).
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Svensson, Lars E. O., (1990b), “The Term Structure of Interest Rate Differentials in a Target Zone: Theory and Swedish Data,” Working Paper (Stockholm: Institute for International Economic Studies, 1990).
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Part of the work for this paper was done while I was a Visiting Scholar at the Research Department at the International Monetary Fund. I thank the Research Department for its hospitality. This paper was initiated after a discussion with Gunnvald Grønvik, Norges Bank, who gave me the idea to compute rate-of-return bands. I am also grateful for comments from Michael Dooley, Bernard Dumas, Thomas Franzén, Morris Goldstein, John Hassler, Lars Hörngren, Peter Norman, Peter Sellin, Ingrid Werner, participants in a seminar at Sveriges Riksbank, and, in particular, Mats Persson who gave me the idea to also plot expected future exchange rates. I thank John Hassler for expert research assistance, Molly Akerlund and Rosalind Oliver for editorial and secretarial assistance, and Sveriges Riksbank and the Finance Department of the Stockholm School of Economics for providing data. Remaining errors and obscurities are my own.

Swedish capital controls the last few years have generally been considered very ineffective, with many ways of circumventing them. The capital controls have been gradually dismantled, in particular since January 1989, and since June 1989 they are for all practical purposes abolished.

Mats Persson has in lecture notes independently examined the credibility of the Swedish target zone by computing expected future exchange rates.

Porter (1971), in a remarkable paper pointed out to me by Michael Dooley and Morris Goldstein after the first version of this paper was written, shows how the implicit expected time path of exchange rates between two currencies can be inferred from the term structure of interest rate differentials, under the assumption of uncovered interest rate parity. Porter also presents an empirical analysis of the term structure of Canadian-U.S. interest rate differentials and the implied expected time path of future CAD-USD exchange rates.

Porter makes the assumption of no uncertainty and perfect foresight, and observes that with uncertainty a foreign exchange risk premium will enter and violate uncovered interest rate parity. In contrast, the present paper does not assume certainty and perfect foresight. Instead the paper relies on Svensson (1990a) which shows that the foreign exchange risk premium is likely to be small for relatively narrow target zones, also when there is devaluation risk. Therefore uncovered interest rate parity should indeed be a good approximation for narrow target zones.

Svensson (1990a) demonstrates that any foreign exchange risk premium in a relatively narrow target zone should be small, also when there is devaluation risk. Therefore uncovered interest rate parity should be a good approximation for narrow target zones.

See Svensson (1990a,b) and Bertola and Svensson (1990) for further details on a model of a target zone with devaluations being a Poisson process.

For details, see Ringström (1987).

Because of a labor market conflict in the banking industry Swedish interest rates are missing for January 1990. Therefore the January Swedish interest rates are set equal to the average of the December and February rates.

For October 1987-January 1988 it has been suggested that what we can call “institutional friction” (for instance a bias in corporations’ and banks’ accounting practices), rather than devaluation risks, may have reduced the amount of interest arbitrage. See Hedman (1986) for a report on some special aspects of foreign exchange management by Swedish corporations. For instance, corporations may prefer to clear their books before the end of the year in order to get a good solidity rating, since accounting rules do not allow the netting of foreign currency liabilities against domestic currency assets.

For longer term bonds with yearly coupons the rate-of-return band (4a) and (4b) still applies if the exchange rate depreciates/appreciates gradually so as to reach the edge if its band at maturity. Alternatively, we can assume that the exchange rate moves to the edge of the exchange rate band already in one year. That assumption causes a slightly wider rate-of-return band than the one given by (4a) and (4b), but the change is small. See Appendix for details.

10/

See, however, footnote 1 on p. 7.

11/

See Svensson (1990b) for a rigorous derivation of the expected depreciation within the band for arbitrary horizons.

12/

For a Poisson process with a probability intensity λ per unit of time, the expected length of time to the next event is 1/λ units of time.

13/

Bertola and Svensson (1990) formulate a model with explicit stochastic devaluation risk and indicate a more precise way of extracting the expected rate of devaluation from data on exchange rates and interest rate differentials.

14/

The role of institutional friction should also be considered, cf. footnote 1 on p. 7.

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