Cross Exchange Rate Calculator

What a cross rate is

Almost every currency quote you see is against the US dollar. The dollar is the common leg in the market, so a bank in Tokyo and a bank in Frankfurt both quote against it rather than against each other. A cross rate is the price of one non-dollar currency in another non-dollar currency, built from those two dollar quotes.

The convention that makes the arithmetic work is units of currency per one US dollar. A quote of 150 yen means one dollar buys 150 yen. A quote of 0.92 euros means one dollar buys 0.92 euros. Both numbers share the same denominator, which is the dollar, and that is what lets you divide one by the other.

For euros per yen, divide the euro leg by the yen leg: 0.92 / 150 = 0.006133. Read that as a little over six tenths of a cent for one yen. Turn it around and one euro buys 150 / 0.92 = 163.04 yen. Neither number is a rate anyone quotes directly, and both describe the same relationship.

The Federal Reserve H.10 release publishes dollar rates for major currencies every Monday, which is where a lot of published cross rates start. If you build a cross from those numbers, you are using noon buying rates in New York rather than the rate a bank will hand you at a counter.

The arithmetic: divide one leg by the other

Write the two quotes down with their units and the formula falls out on its own:

• Rate A is units of A per 1 USD.
• Rate B is units of B per 1 USD.
• B per 1 A = rate B / rate A.

The dollar cancels because it sits in the denominator of both quotes. That is the whole trick. One dollar buys rate A units of A, and that same dollar buys rate B units of B, so one unit of A is worth rate B / rate A units of B.

Going the other way, A per 1 B = rate A / rate B. With the same two legs, 150 / 0.92 = 163.04 yen per euro, which is the inverse of 0.006133.

Two checks keep the work honest. First, multiply the cross rate by the amount and see whether the answer is the right size: 100,000 yen at 0.006133 euros per yen is about 613 euros, not 6,133 and not 61. Second, multiply the cross rate by its inverse and confirm you get 1: 0.006133 × 163.04 = 1.000. If either check fails, a leg was flipped.

Flipping a leg is the most common error in this calculation, because a quote can be written either way. A yen quote of 150 is the same as a dollar quote of 0.006667, and mixing the two conventions in one calculation produces an answer that is wrong by a factor of 22,500. Write the units next to every number before you divide.

Worked example: 100,000 yen into euros

Take a yen leg of 150.00 and a euro leg of 0.92, and convert 100,000 yen. Both legs are example values chosen to make the arithmetic easy to follow, not today's market.

• Yen to dollars: 100,000 / 150 = 666.67 USD
• Dollars to euros: 666.67 × 0.92 = 613.33 EUR
• One step check: 100,000 × (0.92 / 150) = 100,000 × 0.006133 = 613.33 EUR

The two routes agree to the cent, which is the point of a cross rate. You do not need to own dollars to convert yen into euros. The dollar is a unit of account in the middle of the calculation, and the two deals happen at once.

Now add a one percent spread on each side, which is what a retail desk might charge. Selling yen, you get fewer dollars than the mid rate: 150 × 1.01 = 151.50 yen per dollar, so 100,000 / 151.50 = 660.07 USD. Buying euros, you get fewer euros per dollar: 0.92 × 0.99 = 0.9108, so 660.07 × 0.9108 = 601.19 EUR.

The exact figures are 613.333 and 601.188, so the gap against the mid rate is 12.15 EUR, which is 1.98 percent of the amount. One percent on each leg did not cost one percent. It cost almost exactly two, and the next section shows why.

Why the spread is charged twice

A dealer does not trade at the mid rate. There is a bid, which is what the dealer pays, and an ask, which is what the dealer charges, and the gap between them is the spread. When you convert A into B through the dollar, you hit the bid on the first leg and the ask on the second, so you pay the spread on both.

With a spread of s on each side, the rate you end up with is:

effective B per A = rate B × (1 - s) / (rate A × (1 + s))

The loss against the mid rate works out to 1 - (1 - s) / (1 + s), which is close to 2s for small spreads but always a little worse. At one percent that is 1.98 percent, not 2.00. At three percent it is 5.83 percent, not 6.00, because the two effects overlap slightly.

This is the number to look for when a bank advertises a small fee and a good rate. The fee is visible and the spread is not, and the spread is charged on every unit you convert rather than once per transaction. A flat 5 euro fee on a 10,000 euro conversion is 0.05 percent, while a one percent spread on each leg is 198 euros. The cheap looking option is the expensive one.

Direct quotes usually beat the two step

A bank that trades both currencies can quote the pair directly, and that quote usually beats anything you can assemble from two dollar legs. The reason is mechanical: one deal means one spread instead of two, and one spread at 1.5 percent costs less than two spreads at 1 percent each.

Compare 1,000 euros into yen at a mid cross of 163.04 yen per euro. The mid rate pays 163,043 yen. Going through the dollar with a one percent spread on each leg pays 163,043 × (0.99 / 1.01) = 159,815 yen. A direct quote of 161.50 yen per euro, which is about one percent below the mid rate, pays 161,500 yen, or 1,685 yen more.

The two step route is not always the wrong choice. If you already hold dollars, converting dollars into a third currency is a single leg with a single spread, and the cross rate is just a way to check whether the quote you were given is fair. The two step only costs double when it describes two actual trades.

Airport desks and hotel front desks sit at the far end of this. Their spread is often wide enough that the two step through a bank card beats their direct quote, which is why the same 1,000 euros can buy thousands more yen depending on where you change it.

Reading a quote board

A board shows pairs, not crosses, and the order of the pair tells you which currency is the base. EUR/USD at 1.0870 means one euro costs 1.0870 dollars, so the euro is the base and the dollar is the quote currency. USD/JPY at 150.00 means one dollar costs 150 yen.

To build a cross from a board, put both quotes in the same direction first. For euros per yen you need the euro quote per dollar and the yen quote per dollar, then divide: 0.92 / 150 = 0.006133. If the board gives you dollars per euro instead, invert it before dividing: 1 / 1.0870 = 0.9200 euros per dollar.

Two rules make a board readable. The base currency is worth more than 1 unit of the quote currency when the number is above 1, so EUR per GBP at 1.1646 says the pound is the stronger of the two. And a cross rate should always sit between the two dollar legs it came from when both are on the same side of 1, which is a fast way to catch a flipped division.

Boards also show two numbers per pair. The first is the bid and the second is the ask, and the gap is the spread. When you sell a currency you get the bid, and when you buy it you pay the ask, so the same pair costs you the wider number in both directions.

Cards, transfers and the rate that lands on your statement

Most travelers meet the cross rate through a card, not a quote board. The card network converts at a wholesale rate and the issuer adds a foreign transaction fee, which appears as a separate line on the statement. That structure is worth knowing because it separates the rate from the fee, and the two are often confused.

Under US rules for remittance transfers, a provider has to show you the exchange rate and the fees before you pay, and the total the recipient will get. The Consumer Financial Protection Bureau guide to sending money walks through the receipt, which is the document to check when a transfer arrives short. It also notes a 30 minute window to cancel a transfer for a full refund, which is short but real.

Dynamic currency conversion is the other trap. A merchant or ATM offers to charge you in your home currency instead of the local one, and the rate behind that offer is set by the merchant rather than the card network. Accepting it removes the uncertainty and adds a markup. Declining it keeps the conversion with your bank, where the spread is disclosed on the statement.

For a large transfer, compare three numbers before you commit: the mid rate from a source like the H.10 release, the rate the provider quotes, and the fee. Convert the fee into a percentage of the amount and add it to the gap between the two rates. That total is the real cost, and it is the only number that lets you compare a bank, a transfer service and a card on equal terms.

Rates used on this page are example values for the arithmetic. Live rates move every business day, and a provider's rate includes a spread that no published table will match.