This paper shows that there can be equilibria in which exchange rates display randomness unrelated to fundamentals. This is demonstrated in the context of a two-currency, one-good model, with three agent types and cash-in-advance constraints. A crucial feature is that the type i agents, for i = 1, 2, must satisfy a cash-in-advance constraint by holding currency i, while type 3 agents can satisfy it by holding either currency. It is shown that real allocations vary across the multiple equilibria if markets for hedging exchange risk do not exist and that the randomness is innocuous if complete markets exist.