RDP 8610: Equilibrium Exchange Rates and a Popular Model of International Asset Demands: An Inconsistency 1. Introduction

Adler and Dumas (1983) and Branson and Henderson (1985) have surveyed the applications to the international setting of Merton's (1969 and 1971) seminal work applying the tools of stochastic calculus to the analysis of the micro-foundations of asset demands. The distinguishing characteristic of these international models is that the real return on a given asset is not perceived to be identical by agents in different countries – i.e., real exchange rates differ (stochastically) across countries. Most of these studies have dealt with the simplest case of geometric Brownian motion for asset prices and exchange rates.^[1] The analyses have typically proceeded in one of two broad directions.

Firstly, much attention has been paid to the form of individual asset demand equations under different assumptions about the menu of assets available, the currency of denomination, the nature of exchange rate and general price level risk, and the links between these risks and those coming directly from asset prices. The main results to emerge from this work have concerned portfolio separations that arise when foreign assets and their associated risks are introduced into the model, and the larger number of mutual funds that are required to span the investment opportunity space when it is perceived heterogeneously by different agents – i.e., a full set of mutual funds is typically required in each country.^[2]

The other branch of analysis has involved using the partial equilibrium constraints imposed by market clearing on the parameters of the asset price processes to develop an International Capital Asset Pricing Model.^[3] For example, in their survey Adler and Dumas (1983) solve for the equilibrium relationships between the instantaneous expected returns on the assets under the assumption that the supplies of bonds and other securities are fixed. The results here have been more limited. At best, these models are capable of pricing the assets only when the exchange rate parameters are given. These models appear to be a translation to many countries of the earlier Capital Asset Pricing Model results in that one “security” (in effect the exchange rate) remains unpriced in each country. At worst, the restrictions obtained are essentially non-testible due to unobservability problems that arise form the aggregation of demands by agents who perceive real returns heterogeneously.

Attempts at addressing these issues often bypass the aggregation problems by ignoring the heterogeneity of the purchasing power of different monies, either by assuming purchasing power parity or a global numeraire. However, this is the very property that seems to distinguish international finance from its closed economy counterpart.

All these models assume market clearing, yet focus solely on clearing in asset markets and asset price determination in terms of a single currency; they ignore the fundamental issue of exchange rate determination. Endogenising the exchange rate would allow assets to be priced in terms of every currency, by pricing the currencies themselves. The central contribution of this paper is the explicit inclusion of equilibrium conditions for foreign exchange markets (balance of payments equilibrium).

Applying this analysis to the model which is commonly encountered in the international literature reveals a fundamental flaw. The assumption of geometric Brownian motion for exchange rates is inconsistent with the constraints imposed by balance of payments equilibrium.^[4] Moreover, real exchange rates do not posses idiosyncratic risk – instantaneous exchange rate risk is no more than a combination of the risks on individual asset prices.

The remainder of the paper is set out in six sections. Section 2 presents the model of individual agents' asset demands and the aggregate market clearing conditions in fairly general terms and outlines the proof methodology that is used in the following sections. The two sections that follow illustrate the inconsistency of the typically assumed exchange rate process with the restrictions derived from the equilibrium conditions. These examples both involve a single asset in each of two countries. The more general case is presented in Sections 5 and 6. Concluding comments may be found in Section 7. The continuous time budget constraints and equilibrium conditions are derived from discrete time in the Appendix.

Footnotes

This paper does not deal directly with the work of Hodrick (1981), Stulz (1981) or others who follow Merton (1973) or Breeden (1979) in postulating more complicated processes for asset prices. It does conclude that this extension is a necessary (but not sufficient) requirement for a consistent model in the international setting. [1]

These results are discussed in both Adler and Dumas (1983) and Branson and Henderson (1985). [2]

The analysis in this paper, as in almost all the associated literature, is partial equilibrium. In particular, it takes asset supply behaviour as given. The general equilibrium work of Cox, Ingersoll and Ross (1985a) and (1985b) remains to be extended to an international setting. Such an extension will require the explicit treatment of multi-good economies. [3]

Rosenburg and Ohlson (1976) point out a problem with the assumption of geometric Brownian motion for asset prices in the domestic setting. [4]