Review of Foreign Exchange Rates

A foreign exchange (FX) rate is, simply, the price of one currency in terms of another. An FX rate between US dollars and British pounds can be expressed as either (a) US dollars per British pound or (b) British pounds per US dollar. We use the notation 2 $/£ to mean 2 US dollars ($2) per British pound, or that $2 will buy 1 British pound. Equivalently, we can use the reciprocal, 0.50 £/$, which means 0.50 British pounds (£0.50) per US dollar, or that £0.50 will buy 1 US dollar.

An FX rate expresses the price of the “denominator currency” in terms of the “numerator currency.” The numerator currency is called the pricing currency, or the terms currency. The denominator currency is sometimes called the base currency. Always remember that when we use the expression “FX price of such-and-such currency,” we are thinking in terms of that currency as the “denominator currency,” and we are expressing its price in terms of the “numerator currency.” Thus, 2 $/£ expresses the FX price of the British pound versus the US dollar (the pricing currency), and 0.50 £/$ expresses the FX price of the US dollar versus the British pound (the pricing currency).

In financial markets, FX rate quotes often involve the US dollar as one of the two currencies. The usual convention is to quote the FX rate with the US dollar as the base currency. The common FX market convention to quote the FX price of the US dollar is called European terms, although the pricing currency involved is not necessarily a European currency. The convention to quote most FX rates in European terms emerged after World War II, when the US dollar replaced the British pound as the principal international currency. The FX rates expressed the price of 1 US dollar in terms of the currency of each country, many of which were European. For example, an FX quote of 1.20 in the case of the Swiss franc (the “Swissie”) implies 1.20 Swiss francs per US dollar, or 1.20 Sf/$, and an FX quote of 108 for the Japanese yen means 108 yen per US dollar, or 108 ¥/$.

Although most FX rates are conventionally quoted in European terms, a few important currencies are typically quoted with the US dollar as the pricing currency. This style is referred to as American terms. An FX quote of 1.50 in the case of the British pound means 1.50 US dollars per British pound, or 1.50 $/£, which is the FX price of a British pound (in US dollars). Other significant currencies usually quoted in American terms include the euro (€), the Australian dollar (A$), and the New Zealand dollar (Z$).

From a country’s perspective, an FX rate is said to be in direct terms if the home currency is the pricing currency and in indirect terms if the foreign currency is the pricing currency. Thus, the FX rate of 2 $/£ is in direct terms from the U.S. point of view and indirect terms from the British point of view, because the US dollar is the pricing currency, that is, in the numerator. The FX rate of 0.50 £/$ is in indirect terms from the U.S. point of view, but is in direct terms from the British point of view.

An FX rate for immediate delivery is called a spot FX rate. The notation for a spot FX rate in this text is the capital letter X. To keep things straight, generally we’ll follow X with a two-currency superscript. Thus, X^Sf/$ represents a spot FX rate expressed in Swiss francs per US dollar. X^$/£ would represent a spot FX rate expressed in US dollars per British pound. We’ll often use a subscript to denote time. Thus denotes a current spot FX rate, is the spot FX rate at time 1, is the spot FX rate at time N, and so forth.

To compute the percentage change in the spot FX price of a currency, you use FX rates expressed with that currency in the denominator. The percentage change in the spot FX price of the euro versus the US dollar, over the period from time 0 to time N, is denoted as , and is equal to . (The text often uses lower case letters to denote percentage-change variables.) A shortcut for this calculation that we’ll frequently use is shown in equation (1):

Percentage FX Change: Euro versus US Dollar

For example, if the euro depreciates versus the US dollar from 1.25 $/€ to 1 $/€, the percentage change in the euro versus the US dollar is (1 $/€)/(1.25 $/€) − 1 = −0.20, or −20%.

To compute a currency’s percentage change when the currency is the “numerator” of the FX rate quote, you need think in terms of reciprocal FX rates. For example, if the FX rate for the yen goes from 100 ¥/$ to 80 ¥/$, the percentage change in the yen is not −20%; instead, it is the US dollar that changes by −20% versus the yen. The FX price of the yen changes from 0.01 $/¥ to 0.0125 $/¥, a percentage change of (0.0125 $/¥)/(0.01 $/¥) − 1 = 0.25, or a 25% appreciation of the yen versus the US dollar. The relationship between the percentage FX changes from the two currency directions in equation (2) may come in handy:

Percentage FX Change and Currency Direction

Thus, per equation (2), a 20% depreciation of the US dollar versus the yen implies a percentage change in the yen versus the US dollar of 1/(1 − 0.20) − 1 = 0.25, or 25%.