Cross Rates

It is possible to back out the cross rates given two exchange rates involving three currencies. Consider 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 follows:

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

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:

$$\frac{RUB}{ZAR}\times\frac{ZAR}{CNY}=1.4876\times1.6459=2.4484\ \text{Russian Rubble per Chinese Yuan}$$

Inversion

Sometimes, it is important to invert one of the quotes to get the intermediary currency to cancel out 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 we have spot exchange rates of RUB/ZAR =1.4876 and JPY/ZAR  = 70.74. The South African rand–Russia ruble (RUB/ZAR)  ruble exchange rate of 1.4876 inverts to:

$$\left(\frac{RUB}{ZAR}\right)^{-1}=\left(\frac{ZAR}{RUB}\right)=\ \frac{1}{1.4876}=0.6722$$

Multiplying this figure with the JPY/ZAR quote of 70.74 gives us the JPY/RUB.

$$\left(\frac{ZAR}{RUB}\right)\times\left(\frac{JPY}{ZAR}\right)=0.6722\ \times\ 70.74\ =\ 47.5531\ \text{JPY per RUB}$$

Triangular Arbitrage in Cross Rate Calculations

Market participants can access both cross-rate quotes (e.g., JPY/CAD for Japan yen–Canada) and the underlying component exchange rate quotes (e.g., CAD/USD for dollar–Canada and JPY/USD for dollar–yen). These cross rates must align with their respective calculations; if not, traders will exploit the discrepancy through arbitrage. This type of profit-seeking, termed triangular arbitrage (given its involvement with three currencies), would persist until the price imbalance is corrected.

Example: Illustrating Triangular Arbitrage

To illustrate, consider a JPY/CAD rate derived at 85.98 based on the underlying CAD/USD and JPY/USD rates of 1.3020 and 111.94, respectively:

$$\frac{JPY}{CAD}=\left(\frac{CAD}{USD}\right)^{-1}\times\left(\frac{JPY}{USD}\right)={(1.3020)}^{-1}\times111.94=85.98$$

If a misinformed dealer simultaneously offers a JPY/CAD rate of 86.20, it presents a different price for the same service, which, in this case, is converting yen to Canadian dollars. A savvy trader could purchase CAD1 for JPY85.98 and immediately sell it for JPY86.20, making a risk-free profit of JPY0.22 per CAD1.

In practice, such discrepancies in cross-rates are infrequent. Both human traders and automated trading algorithms vigilantly monitor the markets for any pricing inefficiencies, ensuring swift corrections.

Question

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

1. 7422.
2. 3383.
3. 4561.

Solution

The correct answer is B.

The spot rate is:

$$\frac{CHF}{EUR}=\frac{CHF}{USD}\times\frac{USD}{EUR}=1.3960\times0.9587=1.3383$$