**Leslie Lipschitz and V. Sundararajan**

In a world of generalized floating exchange rates, many countries have sought to peg their currencies to some relatively stable standard. By December 31, 1979, 94 out of the 140 Fund members were classified as having pegged rates; of these, 60 were pegged to a single currency; the rest were pegged to a basket of currencies. In 14 cases, the basket chosen was the special drawing right (SDR) which consists of 16 currencies. In the case of the SDR, the currency composition of the basket is determined by the Fund, is widely known, and is not likely to be changed suddenly. Moreover, the value of the SDR against major currencies is calculated every day, and this information is readily available. The developing countries as a group have generally preferred some form of pegging arrangement, and an increasing number of them have abandoned single currency pegs and have fixed their exchange rates against currency baskets.

There are various reasons for countries to peg the value of their currencies to some standard. Exchange rates are determined in an asset market—a market for different monies—and even in a relatively stable world, asset market prices tend to fluctuate sharply. It is widely believed that real economic costs are associated with such fluctuations; they inhibit trade, harm domestic price stability, increase uncertainty, and serve generally to frustrate economic decision-making. If there is not much international trading of a particular currency, so that occasional large transactions dominate the market, fluctuations of its exchange rate are likely to be even more volatile. Most developing countries fall into this category. In addition, these countries generally do not have well-developed forward exchange markets to allow transactors to protect themselves against future exchange rate changes. In this situation, the risks associated with international transactions are likely to discourage foreign trade. Also, as prices of traded goods are linked to the exchange rate, changes in the relative price of traded goods to home goods are likely to distort the allocation of investment within the local economy, and may lead to investment in less productive sectors.

These factors are sufficient to induce many countries to peg their exchange rates. An additional, though less often discussed, argument for fixing the exchange rate is that a fixed rate system has a builtin cushioning effect that tends to reduce the impact of short-term real shocks. For example, in bad years when there are harvest shortfalls, the authorities will have to sell foreign reserves in the market to support the exchange rate. These reserves will be used to finance imports to make up the shortfall. In good years of bountiful supplies, the authorities will have to buy foreign exchange to prevent an appreciation of the exchange rate, and reserves will be built up against future needs. This cushioning mechanism is particularly important for stability in those developing countries in which domestic consumption and investment are extremely sensitive to output and market conditions for the few primary commodities that are produced and exported.

While for many countries there are good reasons to fix the value of the currency, in most cases the appropriate standard is not apparent. Moreover, there are costs involved in adopting the wrong one. In a world where the major currencies are floating independently, a fixed exchange rate with any particular currency implies a joint float with that currency against all others. If the value of a currency is fixed for a sustained period to some inappropriate standard, periodic adjustments of the exchange rate vis-à-vis the standard would probably be necessary to avoid or deal with the inevitable balance of payments crises. The timing and extent of these adjustments would not necessarily be ideal, but they would be subject to the discretion of the authorities and would, therefore, be influenced by domestic political considerations.

It may be possible, however, to find a standard against which to peg the value of a currency that might itself embody a rule for the automatic adjustment of bilateral exchange rates whenever necessary. With this object in mind, many economists in different countries have advocated fixing the exchange rate in terms of a basket of currencies.

The shares of the various currencies in the basket have usually been determined on the basis of trade elasticities or the currency composition of the country’s international trade. It is argued in this article that while such trade-based weights are suitable for combining data on different trading partners into aggregate exchange rate indices, they are generally not the best weights for constructing a currency basket. The assumed objective of the currency basket in this exercise is to reduce the effects of exchange rate fluctuations on the domestic economy. This can be achieved by minimizing unwarranted variations of the real (price-adjusted) exchange rate—that is, distortions in purchasing power parity—over the future period for which the basket peg is being chosen. The problem becomes one of finding a practical rule for determining the relative shares of the various currencies in the basket that best serves this end.

*This article is based on “The Optimal Basket in a World of Generalized Floating,” by the same authors in* IMF Staff Papers, *March 1980*.

The first step toward arriving at such a rule is to measure the real exchange rate and to make some judgment about its long-run equilibrium value. In principle, of course, it is possible to adjust the nominal exchange rate frequently for changes in relative prices among countries—the so-called crawling peg—and thereby avoid large deviations from the equilibrium value. In practice, while data on exchange rates are almost continuously available to policymakers, price data are available only sporadically and after a considerable delay. In addition, it may be unwise to adjust the nominal exchange rate for every transitory, and possibly reversible, change in relative prices.

This article shows how these difficulties can be overcome in the following fashion. It is possible to construct a basket that takes into account, in the weights allocated to different currencies, all the information available on past relationships between each relative price and the corresponding exchange rate. Provided these relationships are stable, by fixing the nominal exchange rate to this basket there should be automatic adjustment of the exchange rate such that, on average, distortions of purchasing power parity are minimized. While the particular optimal basket will differ for each country, the general discussion of the problem that follows arrives at some practical rules of thumb.

## Objectives

The currencies of the basket are generally selected to minimize the effects of exchange rate fluctuations on the domestic economy. Some changes in flexible exchange rates may simply reflect the adjustment toward purchasing power parity among currencies of economies with different domestic financial policies and hence different inflation rates. Such changes are stabilizing in character—by offsetting different domestic inflation rates they prevent temporary shifts in trading competitiveness. Exchange rate changes that are systematically smaller or larger than those necessary to maintain purchasing power parity, and which, at the same time, do not reflect structural changes in the economies involved, are destabilizing and may induce temporary shifts in competitiveness among countries.

The composition of the basket ought therefore to be chosen to minimize the real—that is, the price-adjusted—exchange rate changes that occur as a result of nominal exchange rate fluctuations among trading partners that are not warranted either by changes in the purchasing power of their respective currencies or by structural changes in their economies. In other words, a reasonable objective for a basket peg is to minimize the fluctuations of the real exchange rate index over some future reference period, rather than to focus on nominal exchange rates.

Because it is impossible—given delays in obtaining price data—and often inadvisable to adjust the nominal exchange rate continuously to compensate for price changes among countries, it is useful to find a rule, based on nominal exchange rate data alone, that serves, on average, to minimize fluctuations of the real exchange rate. This rule should use all the available past information on the relationship between prices and exchange rates to compensate for the unavailability of relevant current price data. A currency basket that is designed to obviate the need for current price data and discretionary exchange rate adjustments provides such a rule and, in this article, is termed the optimal currency basket. It is optimal only in terms of the specific objective of the peg—in this case, to minimize variations in some real exchange rate index.

While this is the principal objective of the peg, another important consideration is that the level of the real exchange rate should not move too far from its long-run equilibrium. There is no reason to alter the nominal exchange rate for every transitory deviation, but sustained deviations from equilibrium are likely to produce effects on the balance of payments (BOP) that will eventually require adjustments. Consequently, an additional consideration in choosing the weights for the optimal basket is that the expected value of the real exchange rate index over the reference period should be within some acceptable range around its equilibrium value. It will be shown that, under identifiable conditions, such a basket is feasible and can be computed.

## Definitions

The real exchange rate index can be defined as a weighted average of relative prices between the home country and each of its trading partners, where trading partner prices have been expressed in domestic currency units by multiplying the foreign price level by the corresponding bilateral nominal exchange rate. The weights assigned to the relative prices and exchange rates comprising the real exchange rate index should reflect the fact that certain bilateral real exchange rate changes are more important than others to a particular country in their effects on the target of the stabilization policy. These weights are elasticities that must be derived from a model of trade; they are real world parameters that must be estimated and cannot be known a priori. In the present exercise, they are taken as given. In practice, however, such elasticity weights are often approximated by weights based on the geographic concentration of trade or on the currency composition of BOP settlements. Because we are concerned with variations of the real exchange rate index around its “equilibrium” value, it is convenient to base the relative price and exchange rate indices on unit values in some equilibrium year—that is, a year during which the external accounts of the country were in reasonable balance.

Pegging to a currency basket can be defined as follows. If the home currency is pegged to a basket of currencies—with weights that are distinct from the elasticity weights—the percentage change in the exchange rate of the home currency against some numeraire currency will be the weighted average of exchange rate changes of each of the trading partner currencies against the same numeraire. It is possible then to derive an expression for the index of the real exchange rate of the home currency that embodies the assumptions that the home currency is fixed to a basket of currencies with one set of weights, and that the real exchange rate index is constructed on the basis of a separate set of relevant elasticity weights. This formulation makes it clear that the set of weights used in the currency basket that minimizes the variance of the real exchange rate may be different from the set of weights used in measuring the real exchange rate index.

### How will the optimal basket work?

The proposition developed in this article is that an optimal set of weights can be derived to provide greater real exchange stability than most commonly used weighting schemes. To illustrate this, some ex ante tests were done for a hypothetical country that sought to peg its currency to a composite of the currencies of its major trading partners—the United Kingdom, the United States, the Federal Republic of Germany, and Japan.

The period for the pegging arrangement was from the third quarter of 1976 until the third quarter of 1978, at which time the authorities planned to review their exchange rate policy. The authorities considered the relative prices that prevailed in the third quarter of 1976 to be consistent with BOP equilibrium. Moreover, they were expecting a steady 2.3 per cent domestic price inflation per quarter, which was the elasticity-weighted average of inflation rates in the trading partner countries during 1974-76. Their objective was to choose a currency basket for the reference period which would minimize the deviations of the country’s real exchange rate from equilibrium. They also wanted to ensure that the real exchange rate remained in equilibrium with respect to the average prices and exchange rates that they expected to prevail during the reference period. The relative importance of movements in each of the bilateral relative prices was known to the authorities and was reflected in the weighting scheme used in the computation of the real exchange rate index. For our illustration, these (arbitrarily chosen) elasticity weights were set at 0.50 for the United States, 0.25 for Japan, 0.20 for the Federal Republic of Germany, and 0.05 for the United Kingdom.

The parameters required to compute the optimal basket were estimated from historical data for the two-year period preceding the third quarter of 1976, when the basket peg was assumed to begin. These parameters are regression coefficients, that were obtained from the linear regression equations relating relative prices (between each country and the United Kingdom) to corresponding exchange rates for the historical period. The derived optimal weights were 0.15 for the United States and 0.85 for the United Kingdom.

Two types of tests are conceivable in this numerical experiment. First, it is possible to apply formal statistical tests to each of the estimated parameters to determine whether they exhibit any instability between the pre-peg period and the basket-peg period. If the parameters remain stable, then the weights based on parameters estimated from data of the pre-peg period should definitely yield optimal weights for the actual period of the peg. Second, it is possible that the first test might yield mixed results—some stable and some unstable parameters—and it is consequently worthwhile to construct an “optimal” basket on the basis of the parameters estimated from historical data and to check whether the basket as a whole leads to a smaller variance of the real exchange rate about the equilibrium than does a basket based solely on elasticity weights.

### The real exchange rate index 1

1 Home country price level relative to trading partner price level, adjusted for exchange rate changes.

The first test, not unexpectedly, produced mixed results. While there was no instability in the relationships between the pound sterling (the numeraire) and either the deutsche mark or the Japanese yen, there was instability between the pound and the U.S. dollar. The second test was more conclusive. As is clear from the chart, the real exchange rate index based on the derived basket remained closer to unity than the index based on the elasticity-weighted basket for most of the reference period. The latter index increased (appreciated) steadily during most of the reference period, while the former index fluctuated around the value in the base period. The basket based on the derived weights led, on average, to substantially less deviation from the equilibrium real exchange rate than the basket based simply on the elasticity weights, mainly because of the stability of the relationships (between the pound sterling and both the deutsche mark and the Japanese yen) used in deriving the weights.

## Criteria for basket weights

The problem of minimizing the variance of the real exchange rate over some reference period is a fairly straightforward mathematical exercise. Although the precise quantitative solution will be different for each specific situation, convenient general rules of thumb can be provided for the selection of the basket weights.

The general rule for choosing optimal basket weights requires that the available elasticity weights be adjusted to take account of the fact that the relative prices and exchange rates of trading partners are, in general, systematically related. The common procedure of using unadjusted elasticity weights, or some trade shares as proxies for these elasticities, in determining the currency basket is not usually ideal. Simple elasticities will help to eliminate only the fluctuations in the elasticity-weighted nominal exchange rate and thus limit the variations of the real exchange rate index to those arising from relative price variations alone. However, in general, a different set of weights—called here the optimal weights—can be chosen so that the nominal exchange rate is allowed to fluctuate together with prices in an offsetting fashion and thereby further reduce the real exchange rate variance. This is possible because relative prices and exchange rates among trading partners often tend to move, at least partly, to offset one another. Insofar as these offsetting movements—the so-called purchasing power parity relationships—can be expected to remain stable over the reference period, they constitute important information that should be used in determining the basket weights. Thus, using such information, it is possible to eliminate some of the variation in the real exchange rate arising from price fluctuations by taking advantage of the extent to which purchasing power parity holds among trading partners.

The larger the weight assigned to a currency in the basket, the smaller will be the fluctuations of the exchange rate of that currency vis-à-vis the domestic currency. At one extreme, with a weight of unity, there would be a single currency peg and no fluctuations in the bilateral exchange rate. However, if the fluctuations in the exchange rate serve to offset the relative price fluctuations, then the weight of the currency in the basket ought to be reduced in order to decrease the variance of the real exchange rate. If there is no tendency at all toward equalizing purchasing power among currencies of trading partners, the best method for constructing a basket is to use simple elasticity weights. If, however, there is some tendency toward purchasing power parity—that is, offsetting movements of exchange rates and relative prices—between the numeraire currency and each of the currencies of the other trading partners, then to obtain optimal basket weights the elasticity weights for each of these other currencies should be adjusted downward according to a simple formula. As the weights have to add up to 1, the weight for the numeraire currency is derived as a residual.

The formula employed involves a parameter representing the relationship between the exchange rate and the relative price at the margin, which can be computed from historical data. If this parameter is zero for a specific trading partner, then purchasing power parity holds exactly, and the weight for the currency of that trading partner should be set at zero. If it is 1, then purchasing power parity does not hold, and the weight for that currency should be just the elasticity weight. If it is greater than zero but less than 1, then the optimal basket weight will simply be the product of the elasticity weight and the parameter. If relative price fluctuations are much larger than the corresponding exchange rate fluctuations (so that the exchange rate variations, while in the right direction, are inadequate to offset these relative price changes), then the weight of the currency should be as small as possible, so as to maximize the offsetting exchange rate movements. Given the constraint that weights be either zero or positive, the weight for the currency should be set at zero in this case.

Thus, pegging to a single currency is best either when purchasing power parity holds among trading partners, or when relative price fluctuations are very large for all trading partners. The former conclusion is consistent with the common perception that it is best to peg to the currency of a country with a rate of inflation similar to one’s own if purchasing power parity holds among trading partners.

There are, of course, other considerations that might prompt a country to peg to a single currency. For example, a country without well-developed financial services might peg to a major currency so that domestic traders could utilize the forward exchange cover and other financial services available for that major currency. However, the benefits that derive from this sort of arrangement should be weighed against the costs of nongermane changes in relative prices—changes that are unrelated to domestic economic considerations and result only from pegging to an inappropriate standard.

The computation of optimal weights in the manner described does not automatically take care of the problem of ensuring that the average value of the real exchange rate remains within an acceptable range around its equilibrium value. If the value of the real exchange rate deviated from the equilibrium value, the weights would have to be recalculated in order to ensure that such deviations did not persist over the period of the peg. This recalculation would require projections of exchange rates, domestic prices, and prices of trading partners and would involve a more complicated formula for devising the weights. The weights based on such projections might, of course, need to be altered with changes in the outlook. It is quite conceivable that given the economic outlook at a particular time, there is no basket against which to peg the home currency which could reasonably be expected to maintain the real exchange rate; in this case a basic change in the value of the currency vis-à-vis the basket would be required before pegging began anew.

These complications do not weaken the general approach to deriving optimal basket weights: as long as there is some known relationship between relative prices and the corresponding exchange rates, this information should be used in formulating a rule for exchange rate management. Of course, it is essential that the information be reliable—that is, that the relationships relied upon be stable.

It is worth sounding a note of caution about the reliance of the foregoing analysis on appropriate elasticity weights and appropriate price indices. Both are needed to define the real exchange rate index that is basic to the exercise. Deriving the correct set of elasticities is likely to pose the most difficult problem. However, this is a crucial step; as already noted, where there are no stable purchasing power parity relationships among trading partners or where the relationships are very weak, the elasticity weights themselves constitute the appropriate basket weights. The models of multilateral trade such as the multilateral exchange rate model (MERM) and its later extensions to developing countries, all developed at the Fund, are helpful in finding elasticities that are appropriate to a given stabilization objective. The choice of price indices for the real exchange rate computation should prove less difficult—many indices are usually available—but equally important. It should depend upon the objective of exchange rate policy as well as on various other specific considerations, such as the degree of monopoly power of the particular country over its exports and imports.

In essence, the foregoing analysis provides a basis for exchange rate policy in the face of uncertainty about future relative price movements among countries. No policymaker would hazard a guess about future rates of inflation in his own country and its principal trading partners and then proceed to base policy solely on that guess. But, despite uncertainty, exchange rate policy must be made, and frequent changes in exchange rates or in the standards against which they are fixed might prove costly. This article argues that for a country that seeks to peg its currency to a currency basket, all the relevant information, including that derived from the recent history of exchange rates and relative prices, should be incorporated into the determination of the basket weights. Clearly, trade elasticities, the share of each major trading partner in total trade, and the currency composition of settlements are all important bits of information in this context. But another piece of relevant information, that is frequently ignored, is the historical relationship between the exchange rate and the corresponding relative price level against each of the major trading partners. If systematic relationships of this sort exist—and economic theory leads one to expect they do—there is then a firmer basis for deriving a set of basket weights able to deal with future price level uncertainties. This means that more attention should be given to the choice of weights; simply resorting to some proxy for elasticity weights—most commonly, trade weights of one sort or another—might well lead to suboptimal results.