Cross-Currency Consistency, Three-Part SDF Factorizations, and an Impossibility Theorem for the Stationarity of Exchange Rates in International Economies

We consider an incomplete markets international economy in discrete-time. The first result is an impossibility theorem showing that if cross-currency no-arbitrage is to hold, the exchange rate cannot be a stationary process in levels. The second result is a system of stochastic discount factor (SDF) factorizations that enforce no-arbitrage in international economies. For example, the domestic SDF admits a new three-part multiplicative factorization: the inverse of (gross) exchange rate growth, a martingale, and a non-martingale component which is the reciprocal of the gross return of the foreign long-term bond. Examples and empirical work showcase our theoretical characterizations and economic insights