How this calculator works
A cross rate is the exchange rate between two currencies worked out from their rates against a third. Most currencies quote against the US dollar, so a cross rate lets you price, for example, Canadian dollars in euros without using a market that trades that pair directly. The arithmetic is a ratio: if one unit of C buys 1.37 units of A and 0.92 units of B, then one unit of B buys 1.37 / 0.92 = 1.489 units of A. The same identity rearranges to recover any of the three rates. In practice the derived rate is only as good as the two inputs, and both must be quoted against the same third currency at the same moment. Banks add a spread, so the rate you can actually trade at is worse in both directions than the mid-market cross rate. Also watch the quoting convention: if one source quotes USD per euro and another quotes euro per USD, invert one before combining them. This calculator takes all rates as units of the first currency per one unit of the second, so A/C means units of A per one unit of C.
Formula
A/B = (A/C) / (B/C). A/C = (A/B) x (B/C). B/C = (A/C) / (A/B). The inverse rates are one divided by each rate: B/A = 1 / (A/B), C/A = 1 / (A/C) and C/B = 1 / (B/C).
Worked example
With C as the US dollar, suppose one US dollar buys 1.37 Canadian dollars and 0.92 euros. Then one euro buys 1.37 / 0.92 = 1.4891 Canadian dollars. If you knew the euro buys 1.4891 Canadian dollars and the dollar buys 0.92 euros, the implied rate of the Canadian dollar against the US dollar is 1.4891 x 0.92 = 1.37. The inverse rates are 0.6715 euros per Canadian dollar and 0.7289 Canadian dollars per euro.
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Frequently asked questions
Why are cross rates useful?
Because not every pair trades directly. A cross rate lets you compare or price two currencies through a common, liquid currency such as the US dollar, and it reveals whether a quoted pair is out of line with the others.
Do cross rates allow arbitrage?
In theory a mismatch between a direct quote and a derived cross rate allows arbitrage. In practice liquid markets close the gap within milliseconds, and transaction costs exceed any leftover difference for ordinary buyers.
Which third currency should I use?
Whichever currency both rates are actually quoted against. The US dollar is the most common, but the euro also serves as a base for many pairs. Do not mix rates quoted against different third currencies.
Is the result the rate a bank will give me?
No. It is a mid-market rate. A bank or exchange service quotes a buy rate and a sell rate around it, and the difference is its margin.