Operational Aspects of the Siegel Paradox

Abstract

In a recent article, Siegel has called attention to a paradox involving forward foreign exchange rates.' This paradox has been partially resolved in two comments by Roper and Boyer, who assume that national economies are highly interdependent. In the present paper by taking into account the payment mechanism for collection of speculative profits, we are able to resolve the paradox even when national economies are nearly independent. We then investigate the sample size necessary for the effect in question to influence empirical conclusions and find, using historical variances, that we would need hundreds of years of data. Siegel's paradox is based on the fact that for purely mathematical reasons, we cannot simultaneously have the forward dollar price of pounds dtf equal to the expected value E[dta] of the corresponding anticipated future spot rate and the forward pound price of dollars ctf= 1/dtf equal to the expected value E [Cta], even though Cta= 1/dta. As a consequence, there are always expected profit opportunities to speculation on future exchange rates. These profits arise from Jensen's inequality from probability theory and have no apparent economic cause. Roper2 points out that there are two relevant numeraires, dollars and pounds. Jensen's inequality does not guarantee that there will be a speculative strategy with positive expected profits in terms of both numeraires. In fact, if the forward rate lies within the bounds, (1) 1/E [Cta] < 1/Ctf <E [ l/Cta], the strategy that makes expected dollar profits is exactly the strategy that makes expected pound losses and vice versa. Boyer 3 has developed this line further by demonstrating that under simple assumptions (1) must hold. These assumptions are that the dollar and pound price levels are independently distributed, that the foreign exchange rate is determined by purchasing power