Calculate and Interpret Currency Cross-rates

Spot Exchange Rates

It is possible to back out the cross rates given two exchange rates involving three currencies. Considering a foreign exchange market with the exchange rate between the South African rand and the Chinese yuan, this market can also quote the exchange rate between the South African rand and the Russian ruble (RUB). It is, therefore, possible to back out the cross-rates between the Chinese yuan and the Russian ruble, which is quoted as (RUB/CNY) according to market conventions and can be represented as:

$$\frac{RUB}{ ZAR}\times \frac{ZAR}{CNY} = \frac{RUB}{CNY}$$

For example, the RUB/ZAR exchange rate is 1.4876, and the ZAR/CNY exchange rate is 1.6459. We can calculate the RUB/CNY exchange rate using sample spot exchange rates as follows:

$$\frac{RUB}{ZAR}\times \frac{ZAR}{CNY} = 1.4876\times 1.6459 = 2.4484  \quad \text{Russian Ruble per Chinese Yuan}$$

Inversion

Sometimes, it is important to invert one of the quotes to get the intermediary currency to cancel out in the equation and get the cross rate. For example, to get the Russian ruble–Japanese yen (JPY/RUB) quote, we first invert the South African rand–Russia ruble (RUB/ZAR) quote before multiplying it by the South African rand–Japanese yen (JPY/ZAR).

Example: Currency Cross-rates

Let’s assume that we have spot exchange rates of 1.4876 for South African rand–Russian ruble (RUB/ZAR) and 70.74 for South African rand–Japanese yen. The South African rand–Russia ruble exchange rate of 1.4876 inverts to:

$$\frac{1}{1.4876} = 0.6722$$

Multiplying this figure by the South African rand–Japanese yen quote of 70.74 gives us the Russian ruble–Japanese yen quote of:

$$0.6722\times 70.74 = 47.5531\quad \text{Japanese yen per Russia ruble}$$

Question

A forex trader noticed that the USD/EUR spot rate is 1.3960. Similarly, the CHF/USD spot rate is 0.9587. Calculate the spot CHF/EUR cross-rate.

A. 0.7422.

B. 1.3383.

C. 1.4561.

Solution

The correct answer is B.

The spot rate is:

\(\frac{CHF}{EUR}=\frac{CHF}{USD} \times \frac{USD}{EUR}=1.3960×0.9587=1.3383\)