Cross Rate

A cross rate is an exchange rate between two currencies that is worked out from the rate of each against a common third currency. Because most of the world's currency trading passes through the US dollar, the third currency is almost always the dollar. If the price of euros in dollars and the price of dollars in yen are known, the price of euros in yen follows by arithmetic.

In market usage the term also describes the currency pair itself: any pair that does not include the dollar, such as euro against yen or sterling against Swiss franc, is called a cross. Cross rates matter to anyone converting between two non-dollar currencies, including importers, travellers, remittance senders and payment companies, since they determine both the rate obtained and the cost of obtaining it.

The foreign exchange market is organized around a small number of heavily traded pairs. With more than 150 currencies in use, quoting every possible combination directly would require thousands of markets, most with little trading. Instead, dealers quote each currency against the dollar, and to a lesser extent the euro, and derive other rates from those. The dollar acts as a vehicle currency: a bank asked to convert Thai baht to Mexican pesos will usually sell baht for dollars and then dollars for pesos, because each of those markets has far more liquidity than a direct baht-peso market would.

Crosses between major currencies are an exception. Pairs such as euro-yen, euro-sterling and euro-Swiss franc are traded directly in large volumes and have their own quoted prices, which arbitrage keeps in line with the rates implied by the dollar pairs.

The method depends on how the two dollar rates are quoted. A quotation states how many units of the quote currency buy one unit of the base currency. Where the dollar is the quote currency in one pair and the base currency in the other, the rates are multiplied:

If one euro costs 1.10 dollars and one dollar costs 150 yen, then one euro costs 1.10 times 150, or 165 yen.

Where the dollar is the quote currency in both pairs, one rate is divided by the other:

If one euro costs 1.10 dollars and one pound sterling costs 1.25 dollars, then one euro costs 1.10 divided by 1.25, or 0.88 pounds. The same division applies when the dollar is the base currency in both pairs, with the order of the terms reversed.

Real quotations have two prices, a bid at which the dealer buys the base currency and an offer at which the dealer sells it. A derived cross rate must use the side of each dollar quote that the customer would actually face. To find the price at which a customer can buy euros with yen, the calculation uses the offer for euros against dollars and the offer for dollars against yen, since the customer is in effect buying dollars with yen and then euros with dollars.

The result is that the bid-ask spread on a derived cross is roughly the sum of the spreads on the two underlying pairs. This is why converting between two minor currencies is noticeably more expensive than converting either one to or from dollars, and why crosses involving an exotic currency carry the widest spreads.

Where a cross is quoted directly, its price cannot stray far from the rate implied by the dollar pairs. If it did, a trader could convert through the three currencies in a loop and end with more than the starting amount, a form of currency arbitrage called triangular arbitrage. Automated trading systems watch for such gaps continuously, and in the major currencies they close within fractions of a second, leaving differences smaller than transaction costs.

For a business or consumer sending money between two non-dollar currencies, the practical question is how many conversions the provider performs and at what margin. A bank routing a payment through dollar correspondent accounts may convert twice, each time applying its own FX spread, an outcome sometimes described as double currency conversion. Providers that hold balances in many currencies, or that can match customers' opposite flows internally, can quote a single cross rate closer to the mid-market rate.

Card networks apply a similar process when a purchase is made in a currency different from the card's billing currency: the amount is converted using the network's daily rates, which for less common pairs are derived through the dollar. Published reference rates from central banks for minor currencies are frequently cross rates calculated from dollar or euro fixings.

In some countries the phrase is used from a domestic viewpoint to mean any rate between two foreign currencies, neither of which is the home currency. Financial newspapers traditionally print a cross rate table, a grid giving the rate of each G10 currency or other major currency against every other.