Monetary and Economic Studies Vol.1, No.2 / October 1983

The Theory of Exchange Rate Determination in a Multi-Currency World

Mitsuhiro Fukao

In addition to the dollar, international transactions have increasingly been denominated in other currencies such as the German mark, the Japanese yen, and the Swiss franc. The weight of these different currencies in the reserve assets of each country is also increasing (See Table 1). As a result, a one dimensional perspective of looking at the Japanese yen only in terms of the U.S. dollar has been losing its validity. To analyze the yen-U.S. dollar exchange rate, it has become increasingly necessary to adopt a multi-dimensional approach of examining the exchange rates of the Japanese yen with other major currencies such as the German mark, the Swiss franc, and the U.K. pound in addition to the U.S. dollar. This point has been briefly explained in Fukao (2). Here, the theory of exchange rate determination between multiple currencies will be explained in detail utilizing a model of real exchange risk (Fukao (3), pp. 33-40)
In analyzing the supply and demand of foreign exchange in a multiple currency world, the concepts of substitutability and complementarity among currencies are important. In PartII, it will be shown that the substitutability and complementarity among different currencies are determined by the correlation among exchange rates of these currencies.
In Part III, the micro foundations of the analysis in Part II will be presented rigorously by using the capital asset pricing model (CAPM). Assuming investors are risk averse, the demand function for different currency denominated assets will be derived. Next, it will be shown that the amount of foreign currency denominated assets held by each investor depends upon such factors as the interest rates in each country, current exchange rate levels, and the previously mentioned expected future correlation between exchange rates (mathematically speaking, variance and covariances).
In Part IV, the supply and demand of foreign exchange in a multicurrency world is analyzed from the macro point of view by making use of the international balance of indebtedness table (Fukao (1)) enlarged to n currencies and n countries. An equilibrium equation for supply and demand in the foreign exchange market in which transactions take place among n currencies is introduced. Moreover, an exchange rate determination equation is derived for the case where no intervention occurs in the foreign exchange market by combining this equilibrium equation with the demand function for foreign currency denominated assets presented in Part III. By using this exchange rate determination equation, it is shown that deviations from the long run equilibrium exchange rate as determined by purchasing power parity between the ith country and the arbitrarily chosen base currency country consist of real interest rate differentials between the two countries and the risk premium arising from real exchange rate fluctuations (This risk premium is an additional return which compensates for the risk of holding foreign currency denominated assets). Furthermore, it is shown that this risk premium is proportional to the weighted total value of the world's cumulative current account balance excluding that of the base country.
In Part V, sterilized exchange rate intervention by monetary authorities is introduced in the model. The main conclusions of this analysis are as follows.
1) Intervention changes the amount of different currency denominated assets held by the private sector. Through changes in the risk premium, intervention affects the exchange rate.
2) In order to conduct support buying of the yen, selling dollars and selling marks affects the yen dollar exchange rate differently, with the difference depending on the degree of substitutability and complementarity currencies.
3) When Japan intervenes in the foreign exchange market by selling dollars and buying yen, this affects the dollar-mark exchange rate. Similarly, when Germany intervenes by selling dollars and buying marks, the yen-dollar exchange rate is affected.
4) Cooperative intervention supporting the yen and the mark through simultaneous selling of dollars by both Japan and West Germany has a greater impact on the yen-dollar exchange rate than does independent intervention because of the above stated influence on the third country currency. In other words, comparing the case in which the Bank of Japan sells dollars alone in order to support a weak yen with the case where this occurs in conjunction with the Bundesbank selling dollars and buying marks, the effect of strengthening the yen is greater in the latter case.
Finally, since rather elaborate mathematical exposition is needed to treat the problem of determining n-1 independent exchange rates in an n currency world, all mathematical proofs are relegated to the Appendices.

Views expressed in the paper are those of the authors and do not necessarily reflect those of the Bank of Japan or Institute for Monetary and Economic Studies.

Copyright © 1983 Bank of Japan All Rights Reserved.

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