Cross rates: calculating them and understanding triangular relationships
Calculate cross rates and their bid/ask prices from two dollar pairs, and use triangular relationships to see what really drives a cross.
What you will learn
- How to calculate a cross rate by multiplying or dividing two dollar pairs.
- How to build the bid and ask of a cross correctly, and why cross spreads are wider.
- What triangular arbitrage is and why retail traders cannot use it.
- How to split a cross's move into its two "legs" to understand which currency is driving it.
Explanation
A cross rate is the exchange rate between two currencies that is derived from their rates against a third currency, usually the US dollar. Even if your broker quotes EUR/JPY directly, its price is held in line with EUR/USD and USD/JPY by banks and algorithms.
Rule 1: the dollar is in different positions → multiply
If USD is the quote currency in one pair and the base currency in the other, multiply: EUR/JPY = EUR/USD × USD/JPY. The USD "cancels out" like in a fraction: (USD per EUR) × (JPY per USD) = JPY per EUR.
Rule 2: the dollar is in the same position → divide
If USD is the quote currency in both pairs, divide: EUR/GBP = EUR/USD ÷ GBP/USD. If USD is the base in both, also divide: CAD/JPY = USD/JPY ÷ USD/CAD.
Bid and ask of a cross
The bid is the price at which you can sell the base currency; the ask is the price at which you can buy it. A cross is built from two real trades, so you must use the side that you would actually get:
- Multiply case: cross bid = bid × bid; cross ask = ask × ask.
- Divide case (EUR/GBP from EUR/USD and GBP/USD): cross bid = EUR/USD bid ÷ GBP/USD ask; cross ask = EUR/USD ask ÷ GBP/USD bid.
Because both legs' spreads are combined, a synthetic cross has a wider spread than either leg. Directly quoted crosses like EUR/GBP are liquid enough to have their own tight market, but less traded crosses usually stay more expensive.
Triangular arbitrage
Triangular arbitrage means exploiting a quoted cross that differs from the implied cross by trading all three pairs in a loop. In modern markets, such gaps close within milliseconds through bank algorithms. For a retail trader, spreads, delays and broker rules make it impossible; brokers may also cancel trades made on obviously wrong prices. The concept is still useful because it explains why crosses cannot drift away from their legs.
Decomposing a cross
For small moves, the percentage change of a multiplied cross is roughly the sum of the percentage changes of its legs: %EUR/JPY ≈ %EUR/USD + %USD/JPY. This tells you whether a rise in EUR/JPY comes from euro strength, yen weakness, or both. A setup driven only by yen weakness is effectively a yen trade, and may be correlated with your other yen positions.
Worked example
Hypothetical example: EUR/USD 1.0999/1.1001, USD/JPY 149.98/150.02, GBP/USD 1.2498/1.2502. Fees, slippage and taxes are excluded.
- EUR/JPY mid: 1.1000 × 150.00 = 165.00.
- EUR/JPY bid: 1.0999 × 149.98 ≈ 164.963; ask: 1.1001 × 150.02 ≈ 165.037. The spread is about 0.074, or 7.4 pips, wider than either leg's spread (2 pips on EUR/USD, 4 pips on USD/JPY), because both spreads are combined.
- EUR/GBP mid: 1.1000 ÷ 1.2500 = 0.8800. Bid: 1.0999 ÷ 1.2502 ≈ 0.87978; ask: 1.1001 ÷ 1.2498 ≈ 0.88022.
- Arbitrage thought experiment: suppose a dealer wrongly bids EUR/JPY at 165.50. Start with $100,000. Buy EUR at 1.1001: 100,000 ÷ 1.1001 ≈ €90,900.83. Sell EUR for JPY at 165.50: ≈ ¥15,044,087. Buy USD with JPY at 150.02: ≈ $100,280.54. Gross gain ≈ $280.54, which in real markets would disappear before a retail order arrives.
- Decomposition: if EUR/USD rises 0.2% and USD/JPY rises 0.5% on a day, EUR/JPY rises about 0.7%, and most of the move comes from yen weakness.
Common mistake
Practice activity
On paper, calculate: (1) AUD/NZD mid from AUD/USD 0.6500 and NZD/USD 0.6000. (2) GBP/JPY mid from GBP/USD 1.2500 and USD/JPY 150.00. (3) If GBP/USD falls 0.3% and USD/JPY rises 0.4%, what is the approximate change of GBP/JPY, and which currency drove it?
Show solution
(1) USD is the quote currency in both, so divide: 0.6500 ÷ 0.6000 ≈ 1.0833.
(2) USD in different positions, so multiply: 1.2500 × 150.00 = 187.50.
(3) −0.3% + 0.4% ≈ +0.1%. GBP/JPY barely moved: the pound weakened against the dollar, but the yen weakened even more.
Limitations: the addition rule is an approximation that works for small moves; for large moves multiply the factors (1 + change).
Key takeaways
- Multiply when the dollar is in different positions; divide when it is in the same position.
- Build cross bid/ask prices from the sides you would really trade; the spreads add up.
- Triangular arbitrage keeps crosses in line, but it is not a retail strategy.
- Split a cross's move into its legs to see which currency really drives it.
Next step
In the next lesson you will see how interest-rate differentials create overnight swaps and the carry trade.
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