A Cross Rate is a foreign exchange rate between two currencies that are both then valued against a third currency. This third currency is usually the U.S. Dollar and, hence any currency pair that doesn't involve the dollar is considered a cross-currency pair. EURGBP, EURCHF, AUDJPY, GBPCAD are all examples of cross-currency pairs.
If you are new to this topic, please read our blog on the currency exchange rate that does not assume any prior knowledge of foreign exchange rates.
Before we move ahead, let's refresh what we need to calculate cross rates.
What is a base and quote currency?
In a currency pair, the base currency is on the left side and, the quote currency is on the right side. In EURGBP, EUR (Euro) is the base currency and, GBP (British Pound) is the quote currency.
How to inverse a currency rate?
To convert GBPUSD to USDGBP, divide the rate by 1. If the GBPUSD rate is 1.3708, the USDGBP rate is 0.7295 (1 divided by 1.3708).
How to calculate cross-currency rates?
Given we have a basic understanding of the forex terminology, let's calculate the EURGBP rate (a cross rate) from EURUSD and USDGBP. Before we move to the formula, let's understand what we are doing in essence. We are converting Euros to British Pounds. Let's assume we have no price for EURGBP but, we have prices for EURUSD and USDGBP, we have to convert 100 Euros to British Pounds. EURUSD is at 1.1425 and, GBPUSD is at 0.7295. To convert 100 Euros to British Pounds, we have to convert 100 Euros to Dollars to British pounds. 100 Euros should get us 114.25 dollars (100 x 1.1425), 114.25 U.S. Dollars should get us 83.34 pounds (114.25 x 0.7295). We get 83.34 British Pounds when we convert 100 Euros. In other words, the EURGBP rate is 0.8334 (1.142 x 0.7295).
Notice we multiplied the above two rates. Why is that? If you read our above-recommended blog, you will know why. But if you didn't, here is a quick recap:
If you are converting currency from left to right (base to quote), multiply. Alternatively, if you are converting currency from right to left (base to quote), divide.
Because we are converting the dollars to pounds in the final conversion using USDGBP quote, we multiple as we converting U.S. Dollar (base currency) to British Pound (Quote Currency). Now assume we have no rate for USDGBP (0.7295) but only have a rate for GBPUSD (1.3708). Then what you would do? You guessed right! The EURGBP rate will still be 0.8334 (1.1425 / 1.3708 ), but instead of multiplying 1.1425 with 0.7295, you would have divided 1.1425 by 1.3708.
What is the formula for calculating Cross-Currency Rates?
The formula can be generalised in two ways:
Do we divide the two rates or, do we multiply.
That depends! If the currency rates are both quoted in dollar terms or the currency rates are both (quoted) in non-dollar terms, we divide. On the other hand, if one currency pair is (priced) in dollar term and the other pair is (priced) in a non-dollar, we multiply.
Let see that with the following examples:
For example, EURUSD and GBPUSD are both quoted in dollar terms as USD is the currency on the right in both those pairs. Hence, we divide.
EURGBP = EURUSD / GBPUSD - 0.8334 = 1.1425 / 1.3708
GBPEUR = GBPUSD / EURUSD - 1.1998 = 1.3708 / 1.1425
In above examples we are first converting EUR to USD (left to right) so we multiply EURUSD then we convert USD to GBP (right to left) so we divide GBPUSD.
EURUSD is a dollar quoted currency while USDGBP is a British Pound quoted currency pair. Hence, we multiply.
EURGBP = EURUSD x USDGBP - 0.8334 = 1.1425 x 0.7295
GBPEUR = GBPUSD x USDEUR - 1.1998 = 1.3708 x 0.8753
In above examples we are first converting EUR to USD (left to right) so we multiply EURUSD then we convert USD to GBP (left to right) so we multiply USDGBP.
Now notice we used USDEUR when calculating GBPEUR. What if we don't have a USDEUR rate? Then use the divide method or inverse EURUSD to get the USDEUR rate. You can also inverse the calculated EURGBP rate to get the GBPEUR rate. Remember multiplying the two (dollar) rates to get a third (Cross-Currency) rate can be confusing, so always fall back to the rule: multiply when converting left to right.
Now, some of you may ask, what if we have USDGBP rate and USDEUR rate and want to calculate the GBPEUR rate. Given both rates have the dollar as the base currency, we divide.
GBPEUR = USDEUR / USDGBP - 1.1998 = 0.8753 / 0.7295
EURGBP = USDGBP / USDEUR - 0.8334 = 0.7295 / 1.1998
We know it's a lot of material to digest for beginners, not familiar with the forex market. However, we assure you some practice will help you master the topic. If you have any questions or suggestions, do let us know, we are always keen to hear from our users.