## Calculating foreign exchange cross-rates

##### Published: May 2002

Executing a foreign exchange transaction between the major international currencies is usually straightforward, as most banks will be trading and making a price between all the major currencies. However, for less common currencies, many banks will not run books and therefore rates are not always quoted or easily available.

In order to establish the appropriate rate of exchange, the cross-rate between the two currencies has to be calculated. This works by translating the first currency into a common currency (often the US Dollar) and then translating the common currency into the second currency. This then determines the cross-rate between the two currencies.

All foreign exchange rates are quoted base currency/variable currency – in other words, an amount of the variable currency in exchange for one unit of the base currency. So for USD/CAD, the US Dollar is the base currency and the rate quoted will be the amount of Canadian Dollars for one US Dollar.

In most cases, the base currency will be US dollars for both currencies and this will enable the cross-rate to be calculated quite easily. However, for some currencies, the FX market convention is that the US Dollar is the variable currency, such as in the trade GBP/USD when the pound becomes the base currency and the amount of dollars varies.

Finally we must remember that for all foreign exchange trades, the dealer can quote two numbers – the first is the bid rate (the rate at which the trader will buy the currency), the second is the offer rate (the rate at which the trader will sell the currency). The dealer always wants more currency if selling units of the base currency than will be given away if currency is being provided for units of the base currency. The difference is the margin and is one of the sources of profit to the dealer.

#### Calculating cross-rates where there is a common base currency quoted for both currencies

Where there are two currencies Y and Z both of which are quoted against X, the two exchange rates are X/Y and X/Z and the cross-rates will be:

• $$Y\:/\:Z = \frac{X \:/ \: Z}{X\: / \: Y}$$
• and $$Z\:/\:Y = \frac{ X\: / \: Y}{X\: / \: Z}$$
##### Example

To calculate the cross-rate between the Canadian Dollar (CAD) and the South African Rand (ZAR), using the US Dollar as the common currency. Let us assume that the Canadian Dollar and Rand are quoted as:

• $$USD\: / \: CAD = \frac{1.58850}{1.58880}$$
• $$USD \: / \: ZAR = \frac{11.0500}{11.1250}$$

The first number is the rate at which the bank sells the currency being quoted against the US dollar and the second is the rate at which the bank buys the currency being quoted against the US dollar. So the cross rates can be calculated as:

• $$CAD\: / \: ZAR = \frac{USD \: / \: ZAR}{USD \: / \: CAD}$$
• $$ZAR \: / \: CAD = \frac{USD\: / \: CAD}{USD \: / \: ZAR}$$
##### For CAD/ZAR – to buy a variable amount of ZAR per 1 CAD:
• $$Bid = \frac{11.05000}{1.58880} = 6.955 \:\:$$ the bank buys CAD and sells ZAR
• $$Offer = \frac{11.12500}{1.58850} = 7.003\:\:$$ the bank sells CAD and buys ZAR
• So CAD / ZAR = 6.955 / 7.003
##### For ZAR/CAD – to buy a variable amount of CAD per 1 ZAR:
• $$Bid = \frac{1.58850}{11.12500} = 0.1428 \:\:$$  the bank buys ZAR and sells CAD
• $$Offer = \frac{1.58880}{11.05000} = 0.1438\:\:$$ the bank sells ZAR and buys CAD
• So ZAR / CAD = 0.1428 /0.143

As you might expect these rates are the reciprocal of the CAD/ZAR rates.

#### Calculating cross-rates where the common currency is the base currency in one pair and the variable currency in the other

Once again there are two currencies Y and Z both of which are quoted against X, but the exchange rates are Y/X and X/Z, not X/Y and X/Z so the cross-rates are:

• Y /  Z = Y / X  x  X / Z
• And $$Z \: / \: Y = \frac{1}{Y\: / \: X \: \times \: X \: / \: Z}$$
##### Example

To calculate the cross-rate between Sterling (GBP) and the Mexican Peso (MXN), using the US Dollar (USD) as the common currency. Let us assume that rates are quoted as follows:

• GBP / USD = 1.43130 / 1.43160
• USD / MXN = 9.02000 / 9.03000
• GBP / MXN = GBP / USD x USD / MXN
• $$MXN\: / \:GBP \:=\:\frac{1}{GBP \: / \:USD \:\times\: USD \: / \: MXN}$$
##### For GBP/MXN – to buy a variable amount of MXN per 1 GBP:
• Bid = 1.43130 x 9.02000 = 12.91 the bank buys GBP and sells MXN
• Offer = 1.43160 x 9.03000 = 12.93 the bank sells GBP and buys MXN
• So GBP / MXN = 12.91 / 12.93
##### For MXN/GBP – to buy a variable amount of GBP per 1 MXN:
• $$Bid = \frac{1}{1.43160 \: \times \: 9.03000} = 0.07735\:\:$$ the bank buys MXN and sells GBP
• $$offer = \frac{1}{1.43130 \: \times \: 9.02000} = 0.07746\:\:$$ the bank sells MXN and sells GBP
• So MXN / GBP= 0.07735 / 0.07746