In a world of generalized floating exchange rates, many countries have sought to peg their currencies to some relatively stable standard; some of these have chosen to peg to a weighted currency composite. 1 This paper provides an operational definition for the optimal weighted currency composite and sets up criteria for selecting such a composite. It argues that the real exchange rate rather than the nominal exchange rate is the important variable for policymakers to monitor and that, although exchange rate data are available daily, price data are available only after some delay. Consequently, continuous discretionary management of the real exchange rate is impossible, and a rule is needed for fixing the nominal exchange rate so that the real exchange rate is stabilized. The optimal weighted currency basket (or composite) provides such a rule. A method is developed for assigning weights to the currencies in the composite, in order to minimize variations of the real exchange rate around its equilibrium level over some reference period, and to ensure that the mean value of the real exchange rate remains close to its equilibrium level over the same period. 2 The optimal basket weights differ, in general, from the predetermined weights used in the construction of the real exchange rate index, and the differences are related primarily to the amplitude of deviations from purchasing power parity (PPP) among the trading partner countries. Based on the formulas for optimal weights, the conditions are described under which a single currency peg is optimal.
Section I discusses the background to the problem. In Section II the formulas for the optimal selection of the currency composite are discussed and numerically illustrated. In addition, the results of stability tests on some of the important constituent parameters of the formulas are presented and discussed in this section. Section III presents some concluding comments. The full algebraic derivation of the formulas is given in the Appendix.
In a world of generalized floating, there are various reasons why a country might wish to peg the value of its currency in terms of 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. 3 It is widely believed that real economic costs are associated with such fluctuations; they inhibit trade, increase uncertainty, and serve generally to frustrate economic decision making. If the market for a particular currency is thin, the exchange rate fluctuations are likely to be even more volatile, and, for a country without a well-developed financial market, the hedging costs for transactors may be prohibitively high. These factors are sufficient to induce many countries to peg their exchange rates. An additional, although less often discussed, argument for fixing the exchange rate is that a fixed rate system has a built-in reserve-cushioning effect that tends to reduce the impact of short-term real (as opposed to financial) shocks. 4 This is particularly important for some developing countries in which the whole economy is extremely sensitive to output and market conditions for a few primary commodities.
While it is apparent that for many countries there is a good case for fixing the value of the currency, for most of them the appropriate standard against which to fix is not immediately apparent. In a world of generalized floating, a fixed exchange rate with any particular currency implies a joint float with that currency against all others. For this reason, many countries have chosen to fix their exchange rates in terms of baskets of currencies, with the composition of the basket determined for each country by the relative importance of the various component currencies in the external transactions of the country concerned.
The composition of the basket is generally related to the stabilization objective of minimizing the effects of exchange rate fluctuations on the economy. It ought therefore to be chosen to minimize the real exchange rate changes that occur as a result of nominal exchange rate fluctuations among trading partners. The appropriate real exchange rate index will reflect the fact that a country is more sensitive to certain bilateral real exchange rates than to others. The elasticities necessary for a proper weighting scheme may be derived from a model of trade; they are real world parameters that must be estimated and cannot be known a priori. 5
In the following section, an optimal basket is constructed and compared with the alternative of using the known elasticities as weights in the currency basket. It is optimal only in the sense of the specific objective of the peg—in this case to minimize variations in the real exchange rate index that are due to transitory deviations from PPP among trading partners. The construction of the real exchange rate index is based on a set of predetermined bilateral elasticities that are assumed given and are assumed to be appropriate to the objectives of the authorities.
II. The Optimal Basket
The objective of the authorities is to minimize the variance of the real exchange rate index over some reference period or time horizon. It is assumed that the authorities are given an appropriate set of elasticities (ηi), that is, they have full knowledge of the contributions of changes in each bilateral real exchange rate to the overall index.
Assume that the small country with which we are concerned trades with n partner countries, i = 1, …, n. For convenience, let the pound sterling (i = 1) be treated as the numeraire currency and the domestic currency be called the rupee.
|=||pounds per unit of the ith currency|
|=||pounds per rupee|
|=||ith currency units per rupee|
|P||=||the price level, for the home country, if no subscript is given, or for the subscripted partner country|
The subscript t refers to the time period, and the subscript 0 refers to the base date for indices, that is, t = 0. Two indices are defined, a relative price index,
and an exchange rate index,
The authorities have decided to fix the value of the rupee to some (as yet undetermined) weighted combination of partner countries’ currencies. In the limit, of course, a single currency peg might be chosen, in which case one weight will be set at unity and all others at zero. This, however, is a special case, and in the general formulation the only initial constraint should be that the weights are all nonnegative and that they sum to unity.
If the rupee is pegged to a log-linear basket of n currencies, with weights wi, i = 1, … n, one can write
The elasticity-weighted real exchange rate index can be expressed as
where lowercase letters indicate logarithms of the corresponding uppercase letters, and
The task of the authorities is to choose a set of weights (wi) that minimizes the variance of the expression in equation (3) over the reference period. In defining this variance, covariances among the relative prices of the partner countries, as well as those among their exchange rates, may be ignored, because the concern is only with the set of bilateral relations between the rupee and the currency of each partner country. However, the covariance of each bilateral exchange rate and the corresponding bilateral relative price is an important component of the PPP relationship between the rupee and each of the other currencies and cannot be ignored. With this in mind, the relevant variance may be written as
The variance with which we are concerned is not the variance about any mean value, but the variance about the equilibrium. For this reason, the indices are constructed about unit values in some “normal” year, that is, a year during which the purchasing power of the rupee was in a sustainable relationship with that of the aggregate of external currencies.
While the objective of the peg is to minimize the variance of the real exchange rate about this equilibrium, in the absence of real changes in the structure of the economies involved, the authorities are also concerned that the level of the real exchange rate should not move too far from the equilibrium level. There are good reasons not to alter the nominal exchange rate for every transitory deviation from PPP; indeed, if the deviation occurs because of some real shock, maintaining the real exchange rate could exacerbate the impact of the shock. 6 Sustained deviations, however, are likely to produce balance of payments effects that will require adjustment. Consequently, a constraint on the minimization of the variance in equation (4) is that the expected value of the real exchange rate index should be within some acceptable range around unity (that is, its logarithm should be in the range – α to + α) over the reference period. This constraint may be characterized as
where the middle term of the inequality is the real exchange rate defined in equation (3), and (–) denotes the expected value over the reference period. The preceding inequality may alternatively be expressed as
It will always be technically possible to find a set of nonnegative weights that minimizes the variance in equation (4) subject to the foregoing constraint and the condition that the weights sum to unity,
provided that a sufficiently broad range of acceptability is specified for the constraint in equation (5). However, if the authorities seek to maintain the average real exchange rate over the reference period fairly close to its equilibrium value (that is, if a is small), for various sets of expected future exchange rates and relative price movements, it might not be feasible to fix the exchange rate to any basket. In such a case, it will be necessary to change the exchange rate vis–à–vis any basket in order to offset a sustained deviation of the real exchange rate from equilibrium.
The problem of minimizing the expression in equation (4) subject to the constraints (5) and (6) may be solved by standard quadratic programming techniques. Details of the general solution are provided in the Appendix; some particular solutions that are amenable to verbal description and are of likely practical importance are discussed in the next section.
Consider the case in which the optimal solution falls within the range specified in constraint (5)—that is, the expected real exchange rate over the reference period is acceptably close to equilibrium—and all the weights are positive. In this case, the optimal weights are
If, however, the expected real exchange rate over the reference period falls at either limit of the acceptable range,
It is clear from equations (7) and (7’) that the use of the elasticity-weighted basket (i.e., wi = ηi) is not generally optimal. Equation (4) helps to clarify this point. In the case where both covariance terms are zero, elasticity weights will be optimal. While it is not unlikely that the second covariance term (zi)—that is, the covariance between the relative price of the home country vis–à–vis the numeraire currency country and the exchange rates of each of the other partner country currencies vis–à–vis the numeraire currency—will be close to zero, it is unlikely that the first covariance term (xi)—that is, the covariance between partner countries’ prices and exchange rates vis–à–vis the numeraire currency—will be that small. If, however, these covariances are zero, the best result possible is to limit the variance of the real exchange rate to that of relative prices; as may be seen from equation (4), this is achieved by adopting simple elasticity weights. In general, as is clear from either equation (4) or equation (7), if the variance of other countries’ currencies against the numeraire currency is large relative to the covariance terms, a simple elasticity-weighted basket is optimal. Larger covariances lead to weights different from the elasticity weights, even if they indicate that exchange rates have tended to exacerbate rather than to offset relative price movements.
The question of whether the optimal basket, in terms of the specified criterion, is a single currency peg or a broader weighted basket can be examined in terms of equations (5) and (7). A necessary condition for a single currency peg emerges directly from the constraint (5), which may be written
Clearly, if all the weights from 2 to n are zero, B must fall within the range -α to + α or, in the strictest case, B must equal zero. This condition is quite intuitive. B is nothing more than the domestic price relative to the weighted average of partner country prices expressed in sterling. Thus, a necessary condition for a single currency peg is that the weighted average of partner country prices in terms of that currency does not deviate, on average, beyond narrowly prescribed limits from the domestic price index. This is not, however, a sufficient condition.
Various sufficient conditions may be described with reference to equation (7). For a single currency peg to be optimal, all the other weights derived must be zero. In our programming framework, it is easy to compute each of the weights for i = 2, … n from equation (7). Where inclusion of the currency (i) in the basket reduces the variance of the real exchange rate, its weight will be positive; where it does not, the relevant weight will be negative or zero. Where the computed weight is negative, it should simply be set equal to zero and the currency excluded from the basket. 7 On this basis, several situations can be described in which a single currency peg is optimal.
First, consider the (not unlikely) case in which the covariance between the relative price of the home country and the United Kingdom and the exchange rate of the United Kingdom vis–à–vis all other partner countries is zero—that is, zi = Cov(rpi, qi) = 0. If, in addition, PPP holds continuously among all the trading partners, so that xiyi = 1, all the weights, from 2 to n, in equation (7) are zero and a single currency peg is optimal. In this case, all partner country currencies may be aggregated and considered as a single currency—in the Hicksian composite commodity sense. As long as price inflation in the domestic currency does not deviate on average by more than an acceptable limit from that in the numeraire currency, it is optimal to peg to the numeraire currency. While sterling has thus far been used as the numeraire currency, where PPP holds among partner countries it would be optimal to peg to whichever currency of the partner countries that inflates at about the same rate as the home country.
Second, consider the case in which, as before, zi = 0 but xiyi > 1. Thus, the computed weight from equation (7) is negative, and hence the optimal weight for the ith currency is set at zero. If this is so for all the currencies, from 2 to n, then a single currency peg is optimal. In this case, the relative prices of other partner countries and the numeraire currency country are much more variable than are the corresponding exchange rates. 8 While the exchange rates do move in the right direction to offset the relative price movements, the amplitude of their movement is insufficient. Given the necessary condition that domestic prices are expected to move in line with the weighted average of partner country prices expressed in terms of the numeraire currency, the optimal basket would be one for which the average exchange rate movement between the domestic currency and the other partner country currencies was no smaller than the movement between the numeraire currency vis–à–vis the average of other partner country currencies. Clearly, to the extent that any of these currencies were included in the basket, the amplitude of the former would be lower than that of the latter. Consequently, the optimal basket is a single currency peg.
This result is made most accessible by considering equation (4), from which it can be seen that if
A third example of an optimal single currency peg arises when ziyi > 0, and, as a result, wi from equation (7) is negative. In the easiest case, PPP holds between the trading partner currency concerned and the numeraire currency, so that xiyi = 1. In such a situation, the currency (i) will have to be excluded from the basket—that is, assigned a zero weight. If this is true for each of the currencies i = 2, … n, a single currency peg is optimal.
This case is best described by reference to an example. Suppose that the currency under consideration for inclusion in the basket is the U. S. dollar and that the U. S. inflation rate is 20 per cent. Suppose further that the inflation rate in the United Kingdom is 5 per cent and that, since PPP holds, the dollar is depreciating by 15 per cent per annum against sterling. If domestic prices relative to U. K. prices have a positive covariance with the price of dollars in terms of sterling, dollars are best excluded from the basket. In our example, a domestic inflation rate of less than 5 per cent would make the covariance positive. In such circumstances, the dollar should be excluded from the basket. If the domestic inflation rate is close to 5 per cent, then a sterling peg would result in a low variance of the real exchange rate. However, if the domestic inflation rate is only 2 per cent, then the best available option is to peg to the currency of the partner with the slowest rate of inflation—in this example, sterling. Of course, eventually the PPP constraint will be violated (unless α is very large), and an exchange rate adjustment will be required.
It is possible to describe a few simplified cases at a general level, but to fully understand the operational significance of the formulas, it is best to proceed to an example of an actual computation.
Consider a hypothetical country that chose to peg its currency to a composite of currencies of its major trading partners—the United Kingdom, the United States, the Federal Republic of Germany, and Japan—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 balance of payments equilibrium. Moreover, they were expecting a steady domestic price inflation of 2.3 per cent per quarter. 9 Their objective was to choose a currency basket for the reference period that 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. Table 1 shows these elasticity weights—arbitrarily chosen for illustrative purposes—and the relevant statistics on exchange rates and relative prices.
|Trading Partners||Elasticity Weights ηi (1)||Ratio of Covariance to Variance ||Optimal Weights wi (3)|
|United States||0.50||0.70 (5.0)||0.15|
|Germany, Fed. Rep.||0.20||1.05 (6.8)||0|
This table illustrates the computation of the optimal basket weights. The parameters—variances and covariances—required to compute the optimal basket are estimated from historical data for the two-year period preceding the third quarter of 1976 when the basket peg is assumed to begin.
Column (2) shows the ratios of covariance to variance (xiyi), relating relative prices
Evaluation of the last term in equation (7’)
Since the optimal basket was derived on the basis of historical correlations, and since those parameters for which there was no reason to expect a stable value over the reference period were simply omitted, it is of interest to compare the performance of the estimated “optimal” basket with the elasticity-weighted basket over the reference period. 14Chart 1 shows the time paths of the real exchange rate under the two different weighting schemes, and Table 2 presents some comparative measures of stability of the two baskets.
Chart 1.The Real Exchange Rate Index, Third Quarter 1976—Third Quarter 19781
1 Home country price level relative to trading partner price level, adjusted for exchange rate changes.
|Some Statistics Relating to the Real Exchange Rate 1|
|Mean||Mean absolute deviation from unity||Mean squared deviation from unity||Variance around the mean|
It is clear from Chart 1 that during most of the reference period, the real exchange rate index based on the optimal basket remained closer to unity than did the index based on the elasticity-weighted basket. The real exchange rate index based on the elasticity-weighted basket increased (appreciated) steadily during most of the reference period, while the index based on the optimal basket fluctuated around the base period value. As shown in Table 2, use of the optimal basket resulted in a substantially lower mean squared deviation from unity than did use of the elasticity-weighted basket; this measure of stability was lower by nearly 80 per cent.
The stability of the estimated parameters is critical to the usefulness of such estimates based on past data for computing optimal weights for a future period. Table 3 provides some tests of stability.
|Regression Equations 1|
|2d quarter 1974-2d quarter 1976||3d quarter 197-3d quarter 1978||F-Statistics (with Degrees of Freedom 2d quarter 1976 (2, 14)) 2|
|Germany, Fed. Rep.||(–10.6)||(6.8)||(–2.8)||(3.4)|
The PPP relationship between the United Kingdom and the Federal Republic of Germany and between the United Kingdom and Japan remained stable, while the relationship between the United Kingdom and the United States was extremely unstable. The optimal basket, based on past data, outperformed the elasticity-weighted basket because of the stability of the coefficients for Japan and the Federal Republic of Germany. Thus, insofar as it is reasonable to expect stable relationships of this sort, the derivation of useful optimal basket weights is feasible.
III. Summary and Conclusion
During the past few years there has been a great deal of discussion at the policy-making level of the best, or optimal, basket peg for a country seeking relative stability in a world of generalized floating. This paper has sought to define the term “optimal peg” by specifying precise criteria for optimality. It has been argued that the real rather than the nominal exchange rate is the variable deserving of policymakers’ attention. Although exchange rate data are available daily, price data are usually available only after a lag of some months. Continuous, discretionary fine tuning of the real exchange rate is, therefore, impossible. Consequently, what is sought is a rule for fixing the nominal exchange rate so that the real exchange rate is stabilized. Fundamental to the proposed methodology is the idea that this rule, or optimal basket, should employ all the available information, including any systematic relationships between bilateral exchange rates and the corresponding relative prices. The optimal basket, as defined in this paper, is one that minimizes the variance of the real exchange rate about its equilibrium, while maintaining the average value of the real exchange rate close to its equilibrium level over the reference period. Besides its basis in economic theory, an advantage of this definition is that it is amenable to quantification, so that the optimal basket weights for trading partners’ currencies can be established by standard quadratic programming techniques.
A solution for the optimal basket is derived. It is found that, in general, the optimal weight of the currency in the basket will differ from the preassigned elasticity weight that denotes the importance of the currency in the real exchange rate index. The reason is that the variances and covariances of exchange rates and relative prices have an important effect on the real exchange rate index, and weights in the optimal basket are chosen to maximize the contribution of this effect to stability. A number of interesting specific cases of the general solution are discussed, and particular emphasis is given to the conditions under which a single currency peg is optimal.
A numerical illustration of the solution is provided, and a comparison, in terms of relative stability, is made between an optimal basket peg and a basket peg in which simple elasticity weights are used. The question remains as to whether the variances and covariances required to derive the optimal weights are stable over time and therefore useful in setting an exchange rate rule for the future. It turns out that these parameters are of the nature of regression coefficients, so that their stability can be tested. The results of such tests are presented in the paper. It is argued that insofar as the parameters required to calculate the optimal basket weights may be estimated from historical data, and may reasonably be expected to be stable, the derivation of useful optimal basket weights is feasible.
The problem is to choose a set of weights w1, … wn that minimizes the variance of the real exchange rate.
Subject to the constraints,
Σ wi = 1
wi ≥ 0, i = 1, …, n
This is a quadratic programming problem that can be solved using a variety of computational algorithms. 15 However, to highlight the nature of the optimal solution, the Kuhn-Tucker conditions characterizing the solution are examined.
The Lagrangian expression for this problem can be written as
where λi, i = 1, 2, 3 are the Lagrangian multipliers. The multiplier λ1 relates to the lower bound of the range for the real exchange rate specified in equation (9); λ2 relates to the upper bound; λ3 relates to the condition that the weights sum to unity.
The Kuhn-Tucker conditions are as follows:
The conditions (10) to (13) state that the first derivative either vanishes at an interior solution where the optimal weights are positive or remains positive at a corner solution where the optimal weight is zero. The inequalities in equation (14) represent the lower and upper limits specified for the mean real exchange rate over the reference period. Equation (15) states that the weights sum to unity. Equation (16) states that the multipliers are nonnegative. The equations in (17) state that if either of the multipliers, λl or λ2, is positive, then the mean real exchange rate over the reference period attains its lower or upper bound, and if the mean real exchange rate is within the bounds, then the multipliers are zero.
Using these conditions, solutions to some useful special cases can be derived and rules of thumb can be evolved for deciding when a currency should be excluded from the optimal basket. In all that follows, it is assumed that the optimal weight of the numeraire currency is positive, so that X3 = 0 (from equations (10) and (13)).
First, consider the situation when the optimal weights lead to an average real exchange rate falling within the bounds specified in equation (9). In this case, the conditions in inequalities (9) (and (14)) hold strictly; therefore, the multipliers λ1 and λ2 become zero and the inequality (11) becomes
Moreover, from equation (13) we note that if
then wi = 0. Thus, the inequality (18) provides a sufficient condition for the optimal weight of the ith currency to be zero. Using wi = 0, one can rewrite (18)
ηi(xiyi − 1) + ziyi > 0
The interpretation of this condition is contained in the text.
(wi − ηi) + ηixiyi +ziyi = 0
wi = ηi(1 −xiyi) − ziyi i=2,…, n
These are the optimal weights shown in equation (7) and constitute an interior solution that can also be derived by the classical optimization methods.
Consider the case when the optimal weights lead to an average real exchange rate that falls on the boundary of the range specified in equation (9). For definiteness, let us first assume that the upper limit is reached, so that λ2 is non-negative, and λ1 = 0. In this case, the inequality (11) becomes
If the preceding expression holds as a strict inequality, then from equation (13), wi = 0. Thus, a sufficient condition for the exclusion of currency i from the optimal basket is
This condition is either more or less restrictive than the corresponding condition in case I, depending on whether
An expression for λ2 can be derived by substituting from equation (19) in the constraint,
These are the solutions presented in equation (7’). If α = 0, then the foregoing solution applies on substituting α = 0. However, if the lower limit specified in (9) is reached, then we have –λ1 in place of λ2 in equation (19), with
Artus, Jacques R., and Rudolf R.Rhomberg, “A Multilateral Exchange Rate Model,” Staff Papers, Vol. 20 (November1973), pp. 591–611.
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Lipschitz, Leslie (1979), “Exchange Rate Policy for a Small Developing Country, and the Selection of an Appropriate Standard,” Staff Papers, Vol. 26 (September1979), pp. 423–49.
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Simmons, Donald M.,Nonlinear Programming for Operations Research (Englewood Cliffs, N. J., 1975).
Mr. Lipschitz, an economist in the Asian Department, is a graduate of the London School of Economics and Political Science.
Mr. Sundararajan, also an economist in the Asian Department, is a graduate of the Indian Statistical Institute and Harvard University.
On June 30, 1979, of 138 Fund members, 92 were classified as having pegged rates: 18 were pegged to a currency composite, 13 to the special drawing right, and 61 to a single currency.
Lipschitz (1979) discusses the allocation, distribution, and balance of payments implications of various types of currency composite, and suggests the criterion of minimizing real exchange rate fluctuations.
For developed countries, the multilateral exchange rate model developed in Artus and Rhomberg (1973) provides some elasticities. For developing countries, Bélanger (1976) and Feltenstein and others (1979) use a commodity-by-commodity approach to developing elasticities for primary producing countries.
The analysis in this paper begs the question of the appropriate price index to use in computing the real exchange rate index. Sundararajan (1976) discusses a method of combining import and export prices, and comparing the resultant index with a nontraded goods price index, when the objective is maintaining equilibrium in the trade balance.
In principle, in deriving a rule for setting the exchange rate so that the real exchange rate is stabilized, there is no reason for not including negative weights. However, negative weights cannot be readily interpreted in the context of the currency composition of an optimal basket.
It can be easily shown that
so that for |xiyi| > 1 the variance of the relative price must be larger then that of the corresponding exchange rate.
This was the elasticity-weighted average of inflation rates in the trading partner countries during the period 1974-76.
All relative prices used in the regressions were ratios of wholesale price indices. The exchange rates and relative prices were expressed in sterling per ith currency unit and converted into indices with the third quarter of 1976 as the base period. The logarithms of these indices were used to estimate the variances, covariances, and the regression coefficients. The regression equations were estimated with an intercept term. Strictly, to measure deviations from base period values, the intercept should have been constrained to zero. However, the inclusion of the intercept is justified in that the condition that the real exchange rate should be close to the base period equilibrium is imposed separately. Notably, none of the conclusions would require alteration if regressions without intercept terms were used.
The t-statistics on the significance of deviations from unity of the estimated xiyi are as follows:
In addition, separate estimates of variances of the exchange rates will be required to compute X.
The average deviation was only 0.023 in the first year. Had this not been the case, it would have been necessary to re-estimate optimal weights to include the last term in equation (7’). As it turned out, the average deviation was close to zero over the entire reference period.
The word “optimal” is used loosely here. Clearly, it is possible that the elasticity-weighted basket may outperform the “optimal” basket in terms of the specified criterion. In this case, the “optimal” basket would obviously be suboptimal.
For a description of the available algorithms, see Simmons (1975).