A target zone with an explicit band for the exchange rate implies bounds on the amount by which the exchange rate can be depreciated or appreciated, since the exchange rate cannot move further than the edges of the band. Given foreign interest rates, these limits on the size of a depreciation or appreciation imply limits on domestic currency rates of return to foreign investment. These limits define a “rate-of-return band” around the foreign interest rates. The rate-of-return bands are narrower for longer terms (that is, longer maturities), since the maximum amount of appreciation and depreciation per unit of the term decreases with the term.

Suppose there is sufficiently free international capital mobility, so that it can be assumed that no international arbitrage possibilities remain. If the domestic interest rate for a certain 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; investors will perceive a risk of a change in the regime for instance, a devaluation before maturity. If, however, the target zone were considered completely credible, arbitrage would be completely safe.

Therefore, a simple and robust test of the credibility of a target zone is whether or not the domestic interest rates are within the rate-of-return bands. If they are within the rate-of-return bands, it does not necessarily follow that the target zone is credible, but if they fall significantly outside the band, it definitely follows that the target zone is not credible.^{1,}^{2}

Under the additional assumption of uncovered interest 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.

In particular, under the assumption of uncovered interest parity, the lack of credibility of the target zone can be quantified, in that the expected rate of depreciation for different maturities, 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.^{3}

Section I makes the assumption of no arbitrage, defines the rate-of-return bands, and discusses the corresponding test of target *zone*credibility. Section II adds the assumption of uncovered interest parity, defines expected future exchange rates, and shows how they can be used to test and quantify target zone credibility. Section III uses these credibility tests on data from the Swedish target zone during the period January 1987-August 1990. Section IV presents conclusions.

## I. No Arbitrage: The Rate-of-Return Band

Let *S _{t}* denote the spot exchange rate in period

*t*(in units of domestic currency per unit of foreign currency);

*i*, the domestic currency interest rate in period

^{τ}_{l}*t*for term-τ loans in domestic currency; and

*if*, the foreign currency interest rate in period

*t*for term-τ loans in foreign currency. (The “foreign currency” may be a particular currency, or a basket of several currencies.) Let the term τbe measured in months, and let the interest rates be expressed as annualized rates of return. In terms of the domestic currency, the annualized rate of return, ex post, on a foreign currency investment in period t of duration t

*, R*, is then given by

^{τ}_{t}This expression can be explained as follows. Investing one unit of domestic currency means investing 1/5, units of foreign currency. This, invested in a τ-month foreign currency bond, results in

Suppose the exchange rate is restricted to a band with lower and upper bounds, __S__ and *:*

The exchange rate band implies bounds on the size of a depreciation or appreciation of the domestic currency, which, in turn, implies that the rate of return

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

and

The bounds decrease with the current exchange rate: a higher exchange rate means a weaker domestic currency, which increases the scope for an appreciation. This lowers the domestic currency rate of return on foreign investments and shifts the rate-of-return band downward. The width of the rate-of-return band decreases 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 decreases with the term, and the tower bound increases.^{4}

Let us now make the assumption of *no arbitrage*. Under a completely credible exchange rate regime, the no-arbitrage assumption implies that the domestic interest rate

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). Thus, the simplest test of whether the exchange rate band lacks credibility is to determine whether, in different periods and for different maturities, domestic interest rates are outside the rate-of-return band.

## II. Uncovered Interest Parity: Expected Exchange Rates

Let us now make the additional assumption of *uncovered interest parity* that is, the expected depreciation of the home currency 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.^{5} Uncovered interest parity can be written as

where, S_{t+T} denotes the expected value in month *t* of the reigning exchange rate in month *t + τ*. Hence, the expected exchange rate τ months in the future can be computed from a particular month’s exchange rate and the domestic and foreign interest rates for bonds with τ months to maturity.

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 test the credibility of a target zone is to compute the expected future exchange rates according to equation (5), and then determine whether the expected future exchange rates are inside or outside the exchange rate band.

Uncovered interest rate parity makes it possible to quantify the degree of credibility, and to measure the lack of credibility, by quantifying the distance between the expected future exchange rates and the edges of the band, as will be seen below.

## III. 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 U.S. 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. Up until June 1985, the width of the band, which was not announced, was ±2.25 percent around the benchmark. In June 1985 the width was reduced to ±1.5 percent (between 130 and 134) and made public^{6}

The data consist 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 maturities above 12 months where Eurorates 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 the month of January 1987 are incomplete and should be viewed with caution. Long-term interest rates for some important basket currencies are not readily available for the period before February 1987.)^{7}

Figure 1 shows the rate-of-return band and the domestic and foreign (basket) interest rates for a 12-month term. The location of the rate-of-return band around the foreign interest rate depends on the position of the exchange rate inside the exchange rate band. From the figure, it can be seen that for most of the period the 12-month domestic interest rate was inside the rate-of-return band until autumn of 1989 when it moved outside the band. Under the assumption of no arbitrage, one can therefore conclude that the Swedish target zone has definitely lacked credibility within a 12-month horizon since the autumn of 1989.^{8}

For longer terms, the domestic interest rate fell outside the rate-of-return band throughout the period. For shorter terms it fell within the band, except for the 6-month interest rate, which reached the edge of its rate-of-return band during autumn 1989.^{9,}^{10}

Figure 1. The Rate-of-Return Band and Domestic and Foreign Interest Rates, 12-Month Term

Sources: Sveriges Riksbank; Finance Department, Stockholm School of Economics.

Note: R__S__^{12} and ^{12} denote the lower and upper edges of the rate-of-return band; i^{12} and ^{12} denote the krona and basket interest rates.

Let us next make the assumption of uncovered interest rate parity and discuss expected future exchange rates. Figure 2 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. Large index values indicate a weak krona.) It can be seen that the expected future exchange rate was lowest in January 1989, and that it reached its peak 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.

Figure 2. Spot and Expected Future Exchange Rates

Sources: Sveriges Riksbank; Finance Department, Stockholm School of Economics.

Note: The krona spot exchange rate and the expected 12-month, 24-month, and 60-month exchange rates are expressed in currency index units. A higher index indicates a weaker krona. The exchange rate band is between 130 and 134 index units.

The situation in January and August 1990 was different. The expected exchange rate 6 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.

Let us examine the situation in August 1990 more closely and demonstrate how the devaluation risk can be quantified. In August 1990 the exchange rate was in the middle of its band, at 132. The total expected change in the currency index in 60 months is 24 (156 - 132). To stay within the band the exchange rate can depreciate at most 1.5 index units. The (minimum) expected accumulated devaluation in 60 months is therefore 22.5 index units (24 — 1.5). This is an expected annual devaluation of 17 percent in 60 months (22.5/132), or an average annual rate of devaluation for the next 60 months of 3.4 percent. This expected annual rate can be interpreted as the product of the expected devaluation size and the probability per year of a devaluation. Suppose the expected devaluation size is 10 percent. Then the probability per year of a devaluation is 34 percent; that is, the expected time to the next devaluation is three years *(≈ 1/0.34). ^{11}*

In this context, the implications of a sizable foreign exchange risk premium can also be demonstrated. Suppose that there is a foreign exchange risk premium, say a large one, equal to plus 1 percent a year (Swedish currency assets are thus considered riskier than foreign currency assets, a far from obvious assumption) (see Svensson (1990a)). A 1 percent risk premium implies that the expected annual rate of devaluation in August 1990 is not 3.4 percent but 2.4 percent. Then, with a 10 percent devaluation, the probability intensity of a devaluation is 24 percent a year, which corresponds to an expected time between devaluations of four years. This is still a sizable devaluation risk, even with the large risk premium.

## IV. Conclusions

A simple but robust test of target zone credibility has been developed in a step-by-step fashion. First, under the assumption of sufficient international capital mobility and, hence, a no-arbitrage assumption, target zone credibility was tested by examining whether domestic interest rates fall outside rate-of-return bands for different terms. Second, under the additional assumption of uncovered interest rate parity, expected future exchange rates can be computed and compared with the exchange rate band. With this assumption, the lack of credibility can also be quantified.

This simple test was applied to the Swedish target zone, revealing that the target zone had no credibility within a 24-month horizon or longer, and that it occasionally lacked credibility within a 12-month horizon. The loss in credibility was particularly evident in the winter of 1989-90, and again in August 1990. For an expected devaluation of 10 percent, the annual probability of a devaluation on these occasions becomes as large as 34 percent.

The no-arbitrage assumption requires sufficiently free international capital mobility. It is a somewhat open question how efficient and liquid the world capital market is for debt instruments in different currencies for longer terms. For simplicity, this paper has assumed that international investment for terms between 1 month and 60 months was available in the important currencies in the Swedish currency basket. This assumption is surely valid for terms up to 12 months, but 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 abolition of exchange controls and increasing international financial integration, the assumption may also hold for longer terms.

It should be noted that the test presented here is one-sided. If the domestic interest rate is inside the rate-of-return band, or the expected future exchange rate is inside the exchange rate band, it does *not*necessarily follow that the target zone is credible. The rate-of-return bands become very wide for short maturities. The probability of short-run 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. A more precise two-sided test would obviously be desirable. With the assumption of uncovered interest parity, more precise, as well as more elaborate, tests can be constructed, as suggested by Bertola and Svensson (1990) and undertaken by Rose and Svensson (1991) for the exchange rate between the franc and the deutsche mark in the exchange rate mechanism (ERM) of the European Monetary System.

This simple test of target zone credibility seems able nevertheless to convey a fair amount of interesting information. It would be worthwhile to collect and compare similar information for longer time periods and for other target zones, both multilateral target zones like those in the ERM and unilateral ones like those in the Nordic countries other than Denmark.^{12}

BertolaGiuseppe and Lars E.O.Svensson“Stochastic Devaluation Risk and the Empirical Fit of Target Zone Models,”NBER WorkingPaper No. 3576 (Cambridge, Massachusetts: National Bureau of Economic ResearchJanuary1991); also published as CEPR DiscussionPaper 513 (London: Centre for Economic Policy ResearchFebruary1991).

FloodRobert P.Andrew K.Rose andDonald J.Mathieson“An Empirical Exploration of Exchange Rate Target Zones,”IMF WorkingPaper 91/15 (Washington: International Monetary FundFebruary1991); forthcoming in Carnegie-Rochester Series on Public Policy (New York; Amsterdam: North-Holland1991).

GronvikGunnvald“The Forward Foreign Exchange Market: Is the Growth of Bank Lending a Result of a Large Demand for Kronor on the Forward Market? Or the Reverse?” (in Norwegian) Sosial0konomenVol. 40 No. 6 (1986) pp. 19–29.

HedmanMalm“Foreign Exchange Management in Some Swedish Corporations? An Empirical Study January-October 1985” (in Swedish) OccasionalPaper No. 3 (Stockholm: Sveriges Riksbank1986).

LindbergHansLars E.O.Svensson andPaulSoderlind“Devaluation Expectations: The Swedish Krona 1982-1991,”Working Paper (Stockholm: Institute for International Economic Studies1991).

PorterMichael G.“A Theoretical and Empirical Framework for Analyzing the Term Structure of Exchange Rate Expectations,”Staff Papers International Monetary FundVol. 18 (November1971) pp. 613–45.

RingstromOlle“The Exchange Rate Index: An Instrument for Monetary and Exchange Rate Policy,”Sveriges Riksbank Quarterly ReviewNo. 4 (1987) pp 16–26.

Rose Andrew K. and Lars E.O.Svensson“Expected and Predicted Realignments: The FF/DM Exchange Rate During the EMS,”NBER WorkingPaper No. 3685 (Cambridge, Massachusetts: National Bureau of Economic ResearchApril1991).

SvenssonLars E.O. (1990a) “The Foreign Exchange Risk Premium in a Target Zone with Devaluation Risk,”NBER WorkingPaper No. 3466 (Cambridge, Massachusetts: National Bureau of Economic ResearchOctober).

SvenssonLars E.O. (1990b) “The Simplest Test of Target Zone Credibility,” IMF Working Paper90/106 (Washington: International Monetary FundNovember).

Lars E.O. Svensson is Professor of International Economics at the Institute for International Economic Studies, Stockholm University, Stockholm. He is a graduate of Stockholm University.

Part of this paper, which is an abbreviated version of Svensson (1990b), was prepared while the author was a Visiting Scholar in the Research Department. He is grateful to Gunnvald Gr0nvik, Michael Dooley, Thomas Franzen, Morris Goldstein, Lars Horngren, Peter Norman, Mats Persson, and participants in a seminar at Sveriges Riksbank for comments and suggestions. The author also thanks John Hassler for expert research assistance, and Sveriges Riksbank and the Finance Department of the Stockholm School of Economics for providing data.

The assumption about no arbitrage may be weakened somewhat by allowing for some small friction in international capital flows, so that arbitrage possibilities may remain for short periods of time but 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), one may not necessarily conclude that the target zone lacks credibility. However, with small capital flows, or large capital flows in the wrong direction, it can indeed be concluded that the target zone is not credible.

Gronvik (1986), discussing rate-of-return bands (or “interest rate corridors”) in a study of the Norwegian forward foreign exchange market, 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 through 1985. Gr0nvik used this fact to explain the large growth of the Norwegian forward exchange market and increasing attempts to circumvent Norwegian capital controls.

Mats Persson, in unpublished lecture notes, examined the credibility of the Swedish target zone by computing expected future exchange rates. In an early paper, Porter (1971) showed 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. In contrast to the present paper, Porter made the assumptions of no uncertainty and perfect foresight.

For longer-term bonds with yearly coupons, the rate-of-return band (equations (4a) and (4b)) still applies if the exchange rate depreciates/appreciates gradually, so as to reach the edge of its band at maturity. Alternatively, one can assume that the exchange rate has moved to the edge of the exchange rate band 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 Svensson (1990b, Appendix) for details.

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

For details, see Ringstrom (1987).

Because of a labor market conflict in the banking industry, interest rates for January 1990 are not available. For convenience, the missing interest rates are set equal to the average of the December and February rates.

In the last few years Swedish capital controls have generally been considered very ineffective and easy to circumvent. Since January 1989 they have been gradually dismantled, and since June 1989, for all practical purposes, they have been abolished. Therefore, the assumption of no arbitrage should be valid, at least for the latter part of the period.

For graphs of rate-of-return bands and interest rates for other maturities, see Svensson (1990b).

In situations with large capital flows, one might argue that the interest rate differential may not be precisely at its equilibrium level according to uncovered interest rate parity. Therefore, one may also want to examine the volume and direction of capital flows during periods when the domestic interest rate is outside the rate-of-return band. For details, see Svensson (1990b).

For further details on these calculations, see Svensson (1990b).

Flood, Rose, and Mathieson (1991) have reported expected future exchange rates for ERM countries; and Lindberg, Svensson, and Soderlind (1991) did an in-depth study of devaluation expectations for the Swedish krona using a variety of methods.