One relationship that works well in the long run is purchasing power parity (PPP), which says when one currency is exchanged for another, the ability to purchase goods and services forgone in one country should be equal to the purchasing power gained in the other. A well-known example is The Economist's Big Mac index. According to this measure, the British pound is overvalued (i.e. too expensive) relative to the dollar because a Big Mac costs, on average, $3.57 in the US, and £2.29 in Britain, but $3.57 (i.e., enough to by a Big Mac in the US) exchanged into pounds at the exchange rate of $2 per £ would only give £1.79, not enough to buy a Big Mac in Britain.
The Economist has a chart of the latest update of the index (alongside a very distressed-looking - Grimace-ing? - Ronald McDonald). The Norwegian Kroner is the most over-valued currency, with the Euro, Swiss Franc and Argentine Peso (!) also among the dear ones. On the other side, the currencies that are cheaper than their Big Mac parity values include the Yen, Rouble and Yuan.
Of course, "purchasing power" is more than Big Macs (hopefully!), but since national price indexes are made up of many different goods with different weights, etc, making a similar calculation with a complete bundle of goods and services is highly impractical, so we turn to the "relative" form of PPP, which links changes in exchange rates to changes in price levels (i.e. inflation rates). If the exchange rate, e, is the "home" price of "foreign" currency, and ph and pf denote the home and foreign price levels, respectively, relative PPP says %Δe = %Δph - %Δpf.
A bit of my own research helps illustrate that this works well in the long run:
The real exchange rate is q = e x pf/ph, so if relative PPP holds, %Δq = %Δe + %Δpf - %Δph = 0, which means the graph should be a straight line if it holds constantly. Clearly it doesn't: there are substantial and persistent deviations from PPP, but they do tend to die out, and, remarkably, the total change from January, 1794 to December, 2005 is about 3%.
PPP is one of the tools used by Agnes Benassy-Quere, Sophie Bereau and Valerie Mignon to ponder the Euro-Dollar rate at VoxEU. They write:
In the “very long run”, the real exchange rate (the relative price of the same basket of goods across two countries) is expected to come back to a constant level, due to the convergence of the prices of both traded goods (due to international arbitrage) and non-traded goods (due to productivity equalisation all over the world). In the jargon, this is called purchasing power parity, and it acts as a very long run attractor.They believe this implies "in the long and very-long run, the equilibrium value of the euro [currently $1.57] is found much lower – around 1.10. Thus, our work suggests that the euro may have peaked and be due to fall."