# Cross Rate

The cross rate is the currency exchange rate between currency A and currency C derived from exchange rate between currency A and currency B and between currency B and currency C.

Currency vendor provides quotes for only the most liquid currencies such as the US dollar, Euro, Pound Sterling, Swiss Franc, etc. Exchange rates between other currencies is normally calculated as the cross rates using the quotes for major currencies.

## Formula

Using mid-point exchange rate quotes, the mid-point cross exchange rate can be derived using the following formula:

$$ \frac{\text{A}}{\text{C}}=\frac{\text{A}}{\text{B}}\times\frac{\text{B}}{\text{C}} $$

Where A/C is the exchange rate between A and C expressed as units of currency A per unit of currency C and A/B and B/C are exchange rates between currnecy A and B and currency B and C respectively.

If instead of B/C we had an exchange rate in terms of C/B (i.e. direct quote vs indirect quote), we would need to take a raciprocal of the exchange rate to get the exchange rate in B/C form.

If we have exchange rates in the form of bid-ask quote, we can derive the bid-ask cross rate by multiplying the bid rate of one currency with the ask of the other such that the common currencies cancel out and vice versa. This point is illustrated in Example 2 below.

## Example

### Cross rate with mid-point exchange rate

As at 27 December 2012, the exchange rate between Euro and US dollar is €0.75 per US$. Exchange rate between US$ and Swiss Franc is 1.09 US$ per Swiss Franc. Find the exchange rate between Euro and Swiss Franc in € per Swiss Franc.

Euro per Swiss Franc = €0.75 per US$ × US$ 1.09 per Swiss Franc = €0.8175 per Swiss Franc.

Finding the same exchange rate in Swiss Franc per Euro would involve taking a reciprocal of the exchange rate calculated above. Swiss Franc per € exchange rate would be 1.223 Swiss Francs per € (=1 ÷ (€0.8175 per Swiss Franc)).

### Cross rate with bid-ask quote

You are in UK and $/£ exchange rate is 1.540- 1.560 and ¥/£ is 149.06 – 149.50. Calculate the $/¥ exchange rate.

We need the cross exchange rate in the form of $/¥, i.e. $ in numerator and ¥ in denominator. The exchange rates we already have are in the form of $/£ and ¥/£. We need to take a reciprocal of any one of the exchange rates so that the British pound cancels out when we multiply. The reciprocal exchange rate for ¥/£ is £/¥ which equals 0.006689 – 0.006709 calculated by taking reciprocal of the ¥/£ exchange rate's bid and ask legs and then switching their positions i.e. (1/149.50 – 1/149.06). We have switched their position between the bid must be always lower than the ask.

The bid leg of the $/¥ cross rate is 0.0103 (=1.540 × 0.006689) and the ask leg is 0.0104 (=1.560 × 0.006709). The cross-rate quote is 0.0103-0.0104.

by Obaidullah Jan, ACA, CFA and last modified on

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