Work out cross-rates between currencies and explain what they mean
The idea
A cross-rate is the exchange rate between two currencies derived through a third, usually the US dollar. Quotes follow the price currency/base currency convention, so USD/EUR = 1.08 means 1.08 dollars per euro. Treat each quote as a fraction and arrange them so the common currency cancels. If it is the price currency in one quote and the base in the other, multiply: JPY/USD x USD/EUR = JPY/EUR. If it is the base in both, divide: MXN/USD / CHF/USD = MXN/CHF. Checking the units is the surest guard against inverting the answer. When a dealer's quoted cross-rate differs from the implied one, triangular arbitrage is possible: buy the currency where it is cheap, sell it where it is dear and route through the third currency.
A practice question
Quotes are given as price currency/base currency. USD/EUR is 1.08 and JPY/USD is 150. The JPY/EUR cross-rate is closest to:
- 138.89Why not: Dividing 150 by 1.08 leaves the dollars uncancelled; the quotes must be multiplied so USD cancels.
- 162.00 — correct
- 0.0072Why not: This divides 1.08 by 150, which inverts the answer and still fails to cancel the dollar.
JPY/EUR = JPY/USD x USD/EUR = 150 x 1.08 = 162.00 yen per euro. The dollar appears as price currency in one quote and base in the other, so multiplying cancels it.
A harder one
Quotes are price currency/base currency. CAD/USD is 1.3600 and USD/GBP is 1.2700. The CAD/GBP cross-rate is closest to:
- 1.7272 — correct
- 1.0709Why not: Dividing 1.36 by 1.27 does not cancel the dollar, since it is base in one quote and price in the other.
- 0.9338Why not: Dividing 1.27 by 1.36 fails to cancel the dollar and gives a rate in no meaningful units.
CAD/GBP = CAD/USD x USD/GBP = 1.3600 x 1.2700 = 1.7272 Canadian dollars per pound. Multiplication cancels USD because it is the base in one quote and the price in the other.
The hardest tier
Quotes are price currency/base currency. USD/EUR is 1.10 and JPY/USD is 148, while a dealer quotes JPY/EUR at 160. Starting with JPY 1,000,000 and ignoring spreads, triangular arbitrage yields closest to:
- a loss of JPY 17,199Why not: This routes the trade the wrong way, selling yen for dollars first, which buys euros where they are dear.
- a gain of JPY 17,500 — correct
- no gain, as rates agreeWhy not: The implied cross-rate is 1.10 x 148 = 162.80, which differs from the dealer's 160, so the rates are inconsistent.
The dealer sells euros cheaply (160 vs implied 162.80). Buy EUR 6,250 for JPY 1,000,000, sell them for USD 6,875 at 1.10, and sell those for JPY 1,017,500 at 148, a gain of JPY 17,500.
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