Cross rates — how to calculate them and when it matters
A cross rate is simply the exchange rate between two currencies that does not involve the US dollar — pairs such as EUR/JPY, GBP/CHF or AUD/NZD. Despite the somewhat exotic name, cross rates account for the bulk of currency pairs quoted today, far more than the classic dollar majors. Every cross is derived from two USD-based pairs through either multiplication or division, depending on how the quotes line up. The pages below walk through how the calculation works, why a retail trader should understand it, and what it does not give you in practice.
What a cross rate really is
A cross rate is the price of one currency expressed in another when neither is the US dollar. By that test, pairs like EUR/JPY, GBP/CHF, AUD/NZD and EUR/GBP are crosses, while EUR/USD, USD/JPY and GBP/USD remain majors. Historically every exchange rate in the world was quoted against the dollar because the dollar was the reserve currency of the interbank market, and according to the BIS Triennial Survey 2022 roughly 88 per cent of the daily 7.5 trillion dollars of FX turnover still involves the dollar on one side of the trade.
The practical consequence is that even when your broker shows you a direct EUR/JPY price, in the interbank market that price is built up from two trades — EUR/USD and USD/JPY — stitched into a single quote. That is why the spread on a cross like EUR/JPY is typically wider than the sum of the two underlying major spreads: the market maker has to cover both legs plus liquidity risk on the cross itself.
The multiplication case
The simpler scenario is one in which the dollar appears once as base and once as quote — the dollar then cancels in the middle of the product. The textbook example is EUR/JPY. Given EUR/USD = 1.0850 and USD/JPY = 150.00, the cross is EUR/USD times USD/JPY, that is 1.0850 times 150.00, which gives 162.75. That is the theoretical midpoint of EUR/JPY around which the market maker builds the actual quote.
The same formula works for the majority of yen crosses. GBP/JPY at GBP/USD = 1.2700 and USD/JPY = 150 comes out at 190.50. AUD/JPY at AUD/USD = 0.6500 lands at 97.50. EUR/CHF, with USD/CHF = 0.9000, is 1.0850 times 0.9000, which gives 0.9765. The mnemonic: dollar once as base and once as quote — multiply.
The division case
The second pattern applies to pairs in which both currencies sit on the same side of the dollar. EUR/USD and GBP/USD are the canonical example: in both the dollar is the quote currency. To get EUR/GBP, you divide EUR/USD by GBP/USD. With EUR/USD = 1.0850 and GBP/USD = 1.2700, EUR/GBP equals 1.0850 divided by 1.2700, which gives 0.8543.
AUD/NZD works the same way: AUD/USD = 0.6500 divided by NZD/USD = 0.6000 gives 1.0833. The mnemonic mirrors the previous one — dollar twice in the same role, divide. In universal form, for any pair A/B, the cross equals A/USD times USD/B; if you only have B/USD, then USD/B is simply one divided by B/USD, which is why the second case becomes a division. To apply these formulas without errors it helps to be clear on what the base and quote currency are in each notation.
Four practical use cases for a retail trader
The first is correlation. Once you see that EUR/JPY is built from EUR/USD and USD/JPY, every move in EUR/JPY is a combination of two independent stories — euro strength and yen strength. Some sessions show EUR/USD up fifty pips, USD/JPY up fifty pips, and EUR/JPY breaking out aggressively. Other sessions show the euro stronger but the dollar stronger against the yen by even more, and EUR/JPY barely moves. Without that intuition, it is easy to build a "bullish euro" thesis around a trade that actually requires a hawkish yen view as well.
The second is avoiding double exposure. A trader who buys EUR/USD and at the same time buys USD/JPY in the same notional is effectively holding a EUR/JPY position (the dollar nets out). Without an understanding of cross-rate arithmetic, this looks like two independent trades. In reality there is one larger risk on a cross that was never deliberately chosen — a classic beginner mistake from reader correspondence at MyBank.pl.
The third is verifying broker pricing. If your broker shows EUR/JPY at 162.80 while EUR/USD and USD/JPY from the same minute imply 162.75, the broker is taking five pips above midpoint. The effective spread on the cross equals the broker's quote minus the midpoint of the two majors. The broader mechanics of how interbank desks build cross quotes are explained in forex basics on ForexMechanics.
The fourth is choosing the right instrument for a macro thesis. If you expect euro strength against the yen, you have two routes — one EUR/JPY trade, or two trades (buying EUR/USD plus buying USD/JPY). The second pays two spreads but gives flexibility. The first is cheaper but ties the thesis to a single position. The choice depends on how confident you are about both legs at the same time.
Cross volatility — the differences are enormous
A common error is the assumption that all crosses behave similarly. The numbers argue the opposite. EUR/CHF in low-volatility periods has a daily ATR of roughly thirty pips — one of the calmest instruments on the market, partly because the SNB has historically suppressed volatility. GBP/JPY sits at the other extreme with an ATR of one hundred fifty to two hundred pips a day. EUR/JPY lands in the middle, around one hundred to one hundred twenty pips. AUD/NZD is the classic carry-neutral cross at around sixty pips.
The consequence is that the same stop gives a very different chance of survival. A thirty-pip stop on EUR/CHF is an entire day's ATR, so the risk of being knocked out by random noise is high. The same stop on GBP/JPY is a fraction of a normal move and can be reached in ten minutes without any change in market structure. Match the cross's volatility profile to your horizon and stop size.
"Most cross exchange rates are constructed in the interbank market from two underlying transactions denominated in US dollars. On the retail side only one ticker is visible, but the spread paid by the client always reflects the cost of both legs." — John Williams, President of the Federal Reserve Bank of New York, 2023
Triangular arbitrage — why retail traders have no edge
In theory there is triangular arbitrage. If the theoretical EUR/JPY from the two majors equals 162.75 while the broker quotes 162.80, you can in principle sell EUR/JPY at the broker and buy a synthetic EUR/JPY via EUR/USD and USD/JPY — five pips of difference times the notional. On a 100 000 euro position that is roughly 50 dollars of supposedly risk-free profit.
In practice the operation is out of reach for retail. Deviations vanish in milliseconds because HFT bots scan the market continuously. Retail spreads run at two to three pips on the cross and one to one and a half pips on each major, so the combined cost of three trades exceeds the available profit. Brokers also have anti-arbitrage logic (slippage, requote) that further eats the theoretical margin.
For a retail trader the real value lies in reading divergences as macro signals. When EUR/CHF deviates from the theoretical cross by more than ten or fifteen pips, central-bank intervention or a liquidity shock is almost always the reason — as in January 2015, when the SNB removed the EUR/CHF floor and the pair fell from 1.20 to 0.85 within minutes.
The most common mistakes in calculating a cross rate
- Confusing the direction "divide or multiply". Dollar once as base and once as quote — multiply; dollar twice in the same role — divide. EUR/JPY should land near 160, not near 0.01.
- Mixing quotes from different moments. EUR/USD from 12:00 and USD/JPY from 14:00 gives a theoretical EUR/JPY that does not match any real market moment. Use rates from the same minute.
- Confusing the midpoint with the client-facing quote. The theoretical cross is the midpoint between bid and ask. The broker's actual bid or ask is half a spread away from that midpoint.
- Assuming a cross behaves like an average of its legs. Some sessions show EUR/USD higher, USD/JPY higher and EUR/JPY barely moving, because a fund is hedging directly on the cross. The cross has its own liquidity dynamic.
- Computing cross rates for unusual pairs without checking the broker's offering. EUR/CHF is on every platform; EUR/CZK already is not; TRY/ZAR is essentially nowhere outside specialised desks. A missing quote means extreme spreads.
Your next steps if you start trading crosses
- Understand the two stories behind every cross. Before entering EUR/JPY, check the euro thesis (ECB, PMI, HICP) and the yen thesis (BoJ, Japanese CPI). A trade on a cross is the sum of two views — if only one lines up, prefer the corresponding major.
- Compute the theoretical cross and compare it with the broker spread. Multiply EUR/USD by USD/JPY and compare with the EUR/JPY quote. The midpoint vs. broker-ask gap is your real entry cost.
- Size the stop to the volatility of the specific cross. EUR/CHF needs different stop sizes than GBP/JPY. Check the ATR over the last fourteen days and set the stop so it will not be hit by ordinary market noise — at least a third of ATR, no more than half. A broader treatment of stops and invalidation lives at risk management on ForexMechanics.
- Do not attempt triangular arbitrage. HFT bots are faster by six orders of magnitude. Use cross-rate divergence as a macro signal, not an invitation to arbitrage.
- Check what crosses your broker actually offers. The instrument specification should list every cross with a typical (not minimum) spread. If a broker fails to publish the average spread on EUR/JPY or GBP/JPY for the last month, that is a signal something is being concealed.
Sources & bibliography
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Bank for International Settlements Triennial Central Bank Survey of foreign exchange and OTC derivatives markets in 2022 · Globalny dzienny obrót na rynku walutowym 7,5 biliona dolarów, struktura par walutowych, udział dolara w 88 procentach transakcji. www.bis.org ↗
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Swiss National Bank Discontinuation of minimum exchange rate against the euro — press release, 15 January 2015 · Komunikat banku centralnego o zniesieniu floor 1,20 na EUR/CHF, kontekst historyczny dla rozdzielenia teoretycznego cross rate od kwotowania brokera. www.snb.ch ↗
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Investopedia Cross rate — definition, example and uses in foreign exchange · Encyklopedyczna definicja cross rate w kontekście rynku detalicznego, formuły mnożenia i dzielenia z przykładami liczbowymi. www.investopedia.com ↗
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Federal Reserve Bank of New York Foreign exchange market and intraday liquidity — staff working paper · Analiza struktury kwotowań krzyżowych w handlu interbankowym i mechaniki market makingu na crossach. www.newyorkfed.org ↗
Frequently asked
What exactly is a cross rate and which pairs belong to this group?
A cross rate is simply the price of one currency expressed in another when neither is the US dollar. This group includes all popular yen pairs (EUR/JPY, GBP/JPY, AUD/JPY, CAD/JPY), the Swiss franc pairs without the dollar (EUR/CHF, GBP/CHF), and various European or Antipodean combinations (EUR/GBP, AUD/NZD, EUR/SEK). The remaining pairs that have the dollar on one side — EUR/USD, USD/JPY, GBP/USD, USD/CHF, AUD/USD, USD/CAD and NZD/USD — are called majors and do not count as crosses. Historically every exchange rate in the world was quoted through the dollar because the dollar was the reserve currency of the interbank market, and according to the BIS Triennial Survey 2022 the dollar still participates in roughly 88 per cent of the daily 7.5 trillion dollars of FX turnover. That is why even a directly quoted cross is built up in the market from two underlying dollar trades.
When do we multiply and when do we divide when calculating a cross rate?
The decisive factor is how the slashes line up in the two underlying pairs. You multiply when the dollar appears once as base and once as quote currency — in that arrangement the dollar cancels out in the middle of the product. Example: EUR/JPY is calculated as EUR/USD multiplied by USD/JPY. With EUR/USD at 1.0850 and USD/JPY at 150.00 you get 1.0850 times 150.00, which equals 162.75. The same logic applies to GBP/JPY, AUD/JPY, EUR/CHF and most other yen crosses or Swiss franc pairs. You divide when both currencies have the dollar on the same side — both quoted with the dollar as base or both with the dollar as quote. Example: EUR/GBP is calculated as EUR/USD divided by GBP/USD. With EUR/USD at 1.0850 and GBP/USD at 1.2700 you get 0.8543. The short mnemonic runs: dollar once in the base and once in the quote — multiply, dollar twice in the same role — divide. Always sanity-check the answer numerically. EUR/JPY should come out near 160, not near 0.01.
Why should a retail trader bother with cross rate calculations at all?
Four practical reasons. The first is correlation. Every move on EUR/JPY is the sum of two independent stories — the euro thesis and the yen thesis — so seeing the cross immediately tells you which side of the market needs verification in your analysis. The second is avoiding double exposure. A trader who buys EUR/USD and at the same time buys USD/JPY in the same notional size is effectively holding a EUR/JPY position, even though it feels like two separate trades. Without an understanding of cross-rate arithmetic, the trader does not notice that the real risk is twice as large as assumed. The third reason is verifying broker quotes — if the theoretical EUR/JPY is 162.75 while the broker shows 162.80, your effective spread is five pips wider than the marketing material suggests. The fourth is choosing the right instrument for a macro thesis. If you expect simultaneous euro strength and yen weakness, one trade on EUR/JPY costs less in spread than two trades on the underlying majors, but it ties you to a single move — the choice depends on how confident you are about both assumptions at once.
Is triangular arbitrage on crosses a real opportunity for a retail trader?
In theory yes, in practice no. The theoretical opportunity appears when the broker quote on a cross diverges from the midpoint calculated from the two underlying majors. Example: EUR/USD at 1.0850, USD/JPY at 150, theoretical EUR/JPY at 162.75, while the broker shows EUR/JPY at 162.80. A five-pip difference on a 100 000 euro notional is roughly 50 dollars of supposedly risk-free profit. The catch is that such deviations vanish in milliseconds, because institutional HFT bots scan the market continuously and close every pricing gap before a human can submit an order. On top of that, retail clients pay spreads of two to three pips on the cross and one to one and a half pips on each major, so the combined cost of three trades eats the entire theoretical arbitrage profit. And brokers often run anti-arbitrage mechanisms (slippage, requote logic) that further compress the margin. For a retail trader the real value of cross rate arithmetic is not arbitrage but reading divergences as macro signals — when the theoretical cross persistently moves more than ten or fifteen pips from the broker quote, an intervention or a liquidity gap is almost always the reason.
How do crosses differ in volatility and what does it mean for risk management?
The differences are large, and selecting the right cross for a strategy starts with the average true range (ATR). EUR/CHF in low-volatility periods has a daily ATR of roughly thirty pips — one of the calmest instruments on the market, partly because the Swiss National Bank has historically suppressed volatility on this pair. GBP/JPY sits at the opposite extreme with a daily ATR of one hundred fifty to two hundred pips, which is why it is a favourite among intraday traders looking for movement. EUR/JPY lands somewhere in the middle with an ATR of around one hundred to one hundred twenty pips. AUD/NZD is a classic carry-neutral cross with an ATR of around sixty pips. Practical consequences for stops: a thirty-pip stop on EUR/CHF is an entire day of ATR, so the risk of being knocked out by random noise is high. The same stop on GBP/JPY is a fraction of a normal move and can be reached in ten minutes without any change in market structure. Match the cross to your strategy so that its volatility profile fits both your time horizon and the risk you can carry — low-volatility crosses suit range trading, high-volatility ones suit trend strategies or scalping.