What Is the Cross Rate Formula and How Is It Used in Finance?

Currency exchange rates are essential for international trade, investing, and travel. While major currency pairs like EUR/USD or USD/JPY are widely quoted, direct exchange rates between some currencies are not always available. In these cases, cross rates determine the exchange value using a common third currency, usually the U.S. dollar.

Understanding how to calculate and apply cross rates is important for businesses managing foreign transactions, investors trading in forex markets, and travelers converting money efficiently.

The Cross Rate Formula

When a direct exchange rate between two currencies is unavailable, a cross rate can be determined using exchange rates that share a common intermediary currency, often the U.S. dollar. The formula is:

Cross Rate = (Currency A to USD Rate) / (Currency B to USD Rate)

For example, if the exchange rate for the British pound (GBP) to U.S. dollar (USD) is 1.30 and the exchange rate for the Japanese yen (JPY) to USD is 0.0075, the GBP/JPY cross rate is:

1.30 / 0.0075 = 173.33

This means one British pound is equivalent to 173.33 Japanese yen.

Cross rates are widely used in forex trading and international business transactions. Traders use them to identify arbitrage opportunities, where discrepancies in exchange rates can be exploited for profit. Businesses with multinational operations rely on cross rates to price goods and services in different currencies without needing direct exchange rate quotes.

Direct vs. Indirect Currency Quotes

Exchange rates can be presented as direct or indirect quotes. A direct quote expresses the value of a foreign currency in terms of the domestic currency, while an indirect quote shows how much foreign currency is needed to buy one unit of the domestic currency.

For example, if a U.S. resident sees an exchange rate of 1.25 USD/CAD, this is a direct quote because it states how many Canadian dollars are needed to buy one U.S. dollar. Conversely, if the same rate were expressed as 0.80 CAD/USD, it would be an indirect quote from the perspective of a Canadian resident. The two are mathematical reciprocals, meaning one can be derived from the other by taking the inverse.

The method used to quote exchange rates varies by country. The U.S. typically expresses most foreign exchange rates as direct quotes, whereas the U.K. often uses indirect quotes for certain currencies. This can cause confusion for businesses and investors operating across multiple regions, requiring careful attention to how rates are presented.

Financial markets standardize exchange rate formats to avoid misinterpretation. The interbank forex market, where large financial institutions trade currencies, generally follows a convention where major currencies like the euro, British pound, and Australian dollar are quoted in direct terms against the U.S. dollar. Less commonly traded currencies may follow different conventions depending on historical and market-driven factors.

Triangular Conversion Method

Not all currency pairs are actively traded. When a direct exchange rate between two currencies is unavailable, an indirect path through a third currency can be used. This approach, known as triangular conversion, allows for an accurate exchange rate calculation using two known currency pairs.

For example, converting Swiss francs (CHF) to South Korean won (KRW) without a direct exchange rate requires using the U.S. dollar as an intermediary. If the CHF/USD rate is 0.92 and the USD/KRW rate is 1,300, the CHF/KRW rate is:

0.92 × 1,300 = 1,196

This means one Swiss franc is equivalent to 1,196 South Korean won.

Banks and forex traders use triangular conversion to identify pricing inefficiencies. If discrepancies exist between quoted exchange rates, traders can execute a series of transactions to profit from mispriced currency values. If the implied cross rate from the triangular calculation differs from the market rate, arbitrageurs can buy the undervalued currency and sell the overvalued one, bringing exchange rates back into equilibrium.

Spot vs. Forward Cross Rates

Exchange rates fluctuate constantly due to market dynamics, but not all transactions settle immediately. Spot cross rates refer to the exchange rate at which a currency pair can be traded for immediate delivery, typically within two business days. These rates are influenced by real-time supply and demand, central bank interventions, and macroeconomic data releases. Businesses rely on spot rates for immediate conversions, such as settling international invoices or executing same-day forex trades.

Forward cross rates apply to currency contracts with a future settlement date, often ranging from one month to a year or more. These rates are derived from the spot rate but adjusted for interest rate differentials between the two currencies. The difference between a currency’s spot and forward rate is known as the forward premium or discount, reflecting expected changes in value based on interest rate parity. If the euro has a higher interest rate than the U.S. dollar, the EUR/USD forward rate may trade at a discount compared to the spot rate.