The cross exchange rate calculator above derives the rate between two currencies that are not quoted directly against each other, using their quotes against a shared third currency. It carries the bid and ask through the division in the correct order, so the resulting two-sided price reflects what you would actually transact at rather than a mid-market number that nobody deals on.
Every rate on this page is an input. Arb Digital publishes no exchange rates, because a rate is a live market price that changes second by second and would be stale before you read it. The numbers in the fields are round placeholders chosen so the worked example below is easy to follow. Take real two-sided prices from your bank, broker or payment provider, and treat the output as arithmetic on those numbers rather than as a quote you can deal on.
What a Cross Rate Is
Most currency pairs in the world are not quoted directly. The interbank market concentrates liquidity in a handful of pairs against a small number of vehicle currencies, and everything else is derived. If you want to price one currency against another and no direct market exists, you take each one's price against a common third currency and divide.
The vehicle currency is the one that appears in both quotes and cancels out in the division. The US dollar plays that role for most of the market: the Bank for International Settlements Triennial Central Bank Survey of foreign exchange markets is the standard reference for how turnover is distributed across currencies and pairs, and it documents the dollar's position on one side of the great majority of trades.
A cross rate derived this way is often called a synthetic rate, because it is constructed rather than observed. Where a direct market does exist, the direct price is the one that matters, and any difference between it and the synthetic rate is what arbitrage exists to remove. In practice the two agree closely for liquid pairs, and the residual difference is smaller than most retail spreads.
How to Use It
- Name the three currencies. The vehicle is whichever currency appears in both of your quotes and will cancel out.
- Enter each leg's two-sided price. Bid is where the market buys from you, ask is where it sells to you, and the ask is always the higher of the two.
- Set each leg's convention. A quote can be given either way round, and choosing the wrong one inverts the result.
- Add any markup. A retail conversion typically carries a percentage margin on top of the quoted spread, and that margin usually dwarfs the spread itself.
- Read the bid and the ask, not just the mid. The mid is a reference point; the two outer numbers are what a transaction happens at.
The Arithmetic, Worked Through
With both legs quoted as vehicle per unit, the cross rate expressing the second currency per unit of the first is leg one divided by leg two. Applying the two-sided prices correctly requires thinking about the sequence of trades. To go from the first currency into the second you sell the first for the vehicle at leg one's bid, then buy the second with the vehicle at leg two's ask. So the cross bid is leg one bid divided by leg two ask, and the cross ask is leg one ask divided by leg two bid.
Work the placeholders. Leg one is 1.2500 bid and 1.2505 ask; leg two is 1.6000 bid and 1.6008 ask. The cross bid is 1.2500 ÷ 1.6008 = 0.780860, and the cross ask is 1.2505 ÷ 1.6000 = 0.781562. The mid of those two is 0.781211, and the spread of 0.000702 is about 0.090 per cent of the mid. Selling 10,000 units of the first currency at the cross bid yields 7,808.60 units of the second.
Notice what happened to the spread. Each leg was quoted with a spread of about 0.04 and 0.05 per cent respectively, and the cross carries roughly the sum of the two. That is the general result: spreads add when you cross through a vehicle, because you pay one on each transaction. It is the single most important practical fact on this page.
Why the Spread Decides the Real Cost
People compare providers on the rate and are then surprised by the amount received. The reason is almost always that the headline rate is a mid-market reference while the transaction happens at the bid or the ask, and the distance between them is the provider's revenue. A conversion advertised as commission-free is not free; the charge has been moved into the spread, where it is harder to see.
The honest comparison is to compute the effective rate: take the amount you received, divide by the amount you sold, and compare that with an independent mid-market reference for the same moment. The gap, expressed as a percentage, is the total cost of the conversion however it was labelled. That single number makes an explicit fee and a widened spread directly comparable.
Reference mid rates are published by central banks. The European Central Bank's euro foreign exchange reference rates are published each working day, and the Federal Reserve's H.10 foreign exchange rates release serves the same purpose for the dollar. Both are explicitly reference data for statistical and accounting use, not dealing rates, which is exactly the point: even central banks distinguish between a published reference and a price you can transact at.
Getting the Quote Convention Right
Currency quotes name two currencies, and the convention determines which is priced in terms of the other. Some pairs are conventionally quoted one way in the market and the opposite way in retail, and some retail applications display whichever direction gives a number greater than one. The result is that a quote read from a screen can mean either of two reciprocal things.
The test is arithmetic rather than memory. Ask which currency you would receive more units of, and check that the number is consistent with it. If a quote of 1.25 is meant to be dollars per euro then a euro is worth more than a dollar; if it is euros per dollar then the reverse. Inverting a quote is simply taking its reciprocal, but with two-sided prices you must also swap bid and ask, because the reciprocal of the higher number is the lower one.
This tool handles that swap for you when you change a leg's convention, and the note under the results always states the direction of the answer in words. If the number looks wrong by roughly a factor of the rate squared, a convention is inverted somewhere.
What Moves a Cross Rate
Because a cross is built from two prices, it moves when either of them moves. A currency can weaken sharply against the dollar while barely moving against a neighbour that weakened equally, and the cross would be unchanged despite two large moves in the underlying legs. That is why cross rates are often less volatile than either leg, and occasionally far more.
Interest rate differentials are the structural driver for forward rates rather than spot ones. A forward exchange rate is not a forecast; it is the spot rate adjusted by the interest rate difference between the two currencies over the period, because otherwise a riskless profit would exist. That relationship, covered interest parity, is what links the money markets to the currency markets, and it is why forward points look large for pairs whose interest rates differ widely.
None of that is modelled here. This page performs arithmetic on the spot quotes you supply. For value dates beyond spot, for forward points, or for anything involving settlement risk, the numbers come from a dealer rather than from a division. If you are converting business revenue, the profit margin calculator and the markup calculator show how quickly a conversion cost eats a margin.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Dividing mid by mid — that produces a rate nobody deals at and hides the cost of both legs.
- Using bid with bid — the sequence is sell one currency and buy the other, so a bid pairs with an ask.
- Reading a quote in the wrong direction — inverting a two-sided quote also swaps bid and ask, not just the number.
- Comparing providers on the rate alone — compute the effective rate from the amount actually received, which captures spread and fees together.
- Treating a derived cross as a dealable price — it is arithmetic on two quotes, and a dealer's own price for the pair may differ.
Related Free Tools From Arb Digital
For digital assets rather than fiat currencies, the crypto converter does the coin-to-coin equivalent with your own prices and swap fees. Display the result correctly for a locale with the currency format converter. For the business impact of conversion costs see the profit margin calculator and the markup calculator, and for changes over time the percentage change calculator and the inflation calculator. The full free online tools hub lists everything.
Frequently Asked Questions
A rate between two currencies derived from each one's quote against a common third currency rather than observed directly. The third currency, called the vehicle, cancels out in the division. Such a rate is also described as synthetic, because it is constructed rather than quoted.
By following the sequence of trades. Going from the first currency to the second means selling the first at its bid and buying the second at its ask, so the cross bid is leg one bid divided by leg two ask. The cross ask is leg one ask divided by leg two bid.
Because you pay a spread on each transaction. Crossing through a vehicle currency means two conversions, so the costs add rather than replacing each other. A directly quoted pair usually costs less than a synthetic one built from two legs.
No. It publishes no rates at all. Every number is an input you supply, and the values in the fields are round placeholders chosen to make the worked example readable. Take live two-sided prices from your own bank, broker or payment provider.
Compute the effective rate: divide the amount you received by the amount you sold, then compare that with an independent mid-market reference for the same moment. The percentage gap is the total cost however it was labelled, which makes an explicit fee and a widened spread directly comparable.
Change that leg's convention and the tool takes the reciprocal, swapping bid and ask as it does so. That swap matters: the reciprocal of the higher number is the lower one, so an inverted quote's ask becomes the bid.
No. It works on spot quotes only. A forward rate is the spot rate adjusted by the interest rate differential between the two currencies over the period, and those points come from a dealer rather than from dividing two spot prices.
This tool performs arithmetic on rates you supply and is provided for educational use. It is not a quote, an offer or financial advice, it does not access live market data, and the rate you actually transact at is set by your bank, broker or payment provider along with their spread and fees.
