FX Market: Cross-Rate Calculations

Cross-rates in FX markets are found by using information in the other given exchange rates. Assume that there are three currencies you are interested in: USD, EUR, and GBP. Now assume you have the quote for USD/EUR and GBP/USD. This means that you have information about the exchange rate for two pairs of currencies, the USD and EUR and the GBP and USD. What other pair of currencies do you now have a quote for?

Incorrect. Although the example did not quote the pair as EUR/USD, you were given USD/EUR which is just the inverse of EUR/USD. In other words, you already have all the information you would need about the exchange rate between the euro and the dollar.

You want to know the JPY/AUD exchange rate, but only know that USD/AUD = 0.90 and USD/JPY = 0.20. What is the cross-rate in this example?

Incorrect. This answer is found by taking $$ 0.90 \times 0.20 = 0.18 $$. If you do this, here is the currency ordering you have: $$\displaystyle \frac{USD}{AUD} \times \frac{USD}{JPY}$$ This does not result in USD canceling or with JPY in the numerator.

Incorrect. This answer is found by taking 0.20/0.90. If you do this, here is the currency ordering you have: $$\displaystyle \frac{AUD}{USD} \times \frac{JPY}{USD}$$ This does not result in USD canceling or with AUD in the denominator.

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Incorrect. You already have the quote for one of these currency pairs.

That's it! You were not given information about the euro and pound in the quotations given. However, you can back out what you need to know about the EUR and GBP exchange rate from the quotations given. Doing this will give you what is called the cross-rate.

Calculating the cross-rates depends on correctly organizing the quotes so that the right units cancel and the right units remain. It is usually helpful to start with what you want as an end result. Thinking back to the example above, assume you want to find the EUR/GBP exchange rate quoted as it is here. That means that in the end you want EUR in the numerator, GBP in the denominator and USD to cancel out of the equation.

Assume now that the quotations are as follows: USD/EUR = 1.20, GBP/USD = 0.75 and you want to find EUR/GBP. To do this, just combine the given quotes in such a way that you have the right units in the right place like this: $$\displaystyle \frac{USD}{EUR} \times \frac{GBP}{USD}$$ Here the USDs cancel and you are left with GBP/EUR: $$\displaystyle \frac{GBP}{EUR} = 1.20 \times 0.75 = 0.90 $$. However, you want the EUR/GBP rate, which is $$\displaystyle \frac{EUR}{GBP} = \frac{1}{\frac{GBP}{EUR}} = \frac{1}{0.90} \approx 1.11$$.

Good work. Since you want to find JPY/AUD you need to make sure that USD cancels, JPY is in the numerator and AUD is in the denominator. You will need to make use of the fact that exchange rate quotations are the inverse of themselves. This answer is found by taking 0.90/0.20 = 4.50. If you do this, here is the currency ordering you have: $$\displaystyle \frac{USD}{AUD} \times \frac{JPY}{USD}$$ This results in USD canceling, with JPY in the numerator and AUD in the denominator.

EUR/USD

EUR/GBP

EUR/USD and EUR/GBP

JPY/AUD = 0.18

JPY/AUD = 0.22

JPY/AUD = 4.50

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