What it means
The dollar sits at the centre of the foreign exchange market, so the deepest and cheapest quotes are dollar pairs. A cross rate is what you get when two of those dollar quotes are combined to price a pair that does not involve the dollar at all.
The arithmetic depends on how each rate is quoted. When one pair is quoted with the dollar as the second currency and the other with the dollar as the first, the two rates are multiplied, and when both are quoted the same way round, one is divided by the other.
Cross rates matter commercially because they determine what a business actually receives. A European exporter selling to Japan is paid at a euro to yen cross rate, and the spread built into that rate is a direct cost of doing business.
Liquidity explains why the spread is usually wider on a cross than on a dollar pair. The bank quoting a euro to yen price often manages the risk through two dollar trades, so it charges for both legs plus its own margin.
Cross rates also underpin triangular arbitrage, where a trader checks whether the quoted cross matches the rate implied by the two dollar pairs. In liquid markets any gap closes within moments, but the constant checking is what keeps quotes consistent.
The practical nuance for a treasurer is to always compare a direct cross quote with the rate implied by two dollar trades. Sometimes the direct quote is better, and sometimes routing through the dollar costs less even after two sets of fees.
In practice
Real-world examples.
Example
A Swedish furniture retailer paying Polish suppliers asks its bank for a direct krona to zloty quote and separately prices the same conversion through two dollar trades. The direct cross comes out 0.3% cheaper on a EUR 4,000,000 equivalent, so the treasurer books the direct trade.
Example
A travel money desk publishes a board of thirty currency rates every morning, almost all of them cross rates derived from dollar quotes plus a retail margin. When the dollar moves sharply at lunchtime, every non-dollar rate on the board has to be refreshed even though no customer traded a dollar.
Example
A commodity trading firm buys copper priced in dollars and sells to a Turkish buyer paying in lira. Its risk system converts the position into a euro reporting currency using a euro to lira cross rate rebuilt from dollar quotes every few seconds.
Formula
Calculation
When the dollar is the quote currency in one pair and the base currency in the other:
Cross rate (A per unit of B) = (A per dollar) x (dollars per B)
Suppose the market quotes EUR/USD at 1.10, meaning one euro buys 1.10 US dollars, and USD/JPY at 150.00, meaning one dollar buys 150 yen.
EUR/JPY cross rate = 1.10 x 150.00 = 165.00 yen per euro
A fictional European machinery exporter needs to convert EUR 2,000,000 of receipts into yen to pay a Japanese subcontractor. At the mid-market cross rate the conversion is 2,000,000 x 165.00 = JPY 330,000,000.
In practice the bank quotes a two-way price of 164.80 bid and 165.20 offer. Selling euros at the bid gives 2,000,000 x 164.80 = JPY 329,600,000, so the spread costs the exporter 330,000,000 - 329,600,000 = JPY 400,000 on the trade.
The same method works for other pairs. With GBP/USD at 1.25 and USD/CHF at 0.90, the GBP/CHF cross rate is 1.25 x 0.90 = 1.125 Swiss francs per pound.Case study
Seen in the real world.
Lindwall Machinery is an invented European exporter used here as an illustrative example. It received EUR 2,000,000 a quarter from Japanese customers and paid a Japanese subcontractor in yen, converting through whichever bank happened to answer the phone first.
A review of a year of trades showed the company had been accepting quotes an average of 0.35% away from the mid-market cross rate, which on roughly JPY 1,320,000,000 of annual conversions amounted to about JPY 4,620,000 of avoidable cost. The treasurer had never compared the quoted cross against the rate implied by EUR/USD at 1.10 and USD/JPY at 150.00, which pointed to 165.00.
In this fictional outcome, Lindwall introduced a simple rule: calculate the implied cross before every call, request quotes from three banks, and record the difference between the executed rate and the implied cross. The average gap fell to 0.12% within two quarters, without any change to the underlying business.
Watch out
Common mistakes.
- Multiplying when the quotes call for division. If both pairs are quoted with the dollar in the same position, one rate must be divided by the other, and multiplying produces a nonsensical answer.
- Treating the mid-market cross rate as the price a business will actually receive. The dealing spread on a cross is usually wider than on a dollar pair, and the difference is a genuine cost.
- Assuming the direct cross quote is always cheapest. For less liquid pairs, routing through two dollar trades sometimes costs less even after paying two spreads, so both routes are worth pricing.
Questions
People also ask.
Why is the dollar used as the intermediate currency?
Because dollar pairs carry the deepest liquidity and tightest spreads, so banks manage risk in dollars and derive most other prices from those quotes.
What is triangular arbitrage?
It is the practice of comparing a quoted cross rate with the rate implied by two dollar pairs and trading the difference when they disagree, which in liquid markets keeps quotes aligned within moments.
Is a cross rate the same as a forward rate?
No. A cross rate is a price for immediate exchange between two currencies, while a forward rate applies to an exchange on an agreed future date and reflects the interest rate difference between the two currencies.
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