Foreign Exchange Markets and Currency Forwards › Cross Rates

A cross rate is an exchange rate between two currencies that doesn't involve USD, calculated from each currency's rate against the dollar. Cross rates are essential when you need to exchange between two non-USD currencies. Because the US dollar is the world's primary reserve and trading currency, most FX transactions involve the dollar on one side. When you need to exchange euros for Japanese yen, however, you need to determine the appropriate cross rate.

Given two exchange rates involving USD, you can derive the cross rate between the other two currencies. This calculation ensures that consecutive conversions are equivalent to a direct conversion. In other words, converting euros to dollars and then dollars to yen must yield the same result as a direct euro-to-yen exchange at the proper cross rate.

If you know EUR/USD and USD/JPY, you can find EUR/JPY:

$$\text{EUR/JPY} = \text{EUR/USD} \times \text{USD/JPY}$$

where:

• $\text{EUR/JPY}$: cross rate (Yen per Euro)
• $\text{EUR/USD}$: exchange rate of USD per EUR
• $\text{USD/JPY}$: exchange rate of JPY per USD

To understand why this multiplication works, consider the dimensional analysis. We want to find yen per euro, and we can trace through the currency units step by step:

$$\frac{\text{JPY}}{\text{EUR}} = \frac{\text{USD}}{\text{EUR}} \times \frac{\text{JPY}}{\text{USD}}$$

The USD units cancel in the multiplication, leaving yen per euro. Canceling intermediate currency units is the basis for cross rate calculations.

When both rates have USD as the quote currency (e.g., EUR/USD and GBP/USD), the calculation requires division rather than multiplication. To calculate GBP/EUR:

$$\text{GBP/EUR} = \frac{\text{GBP/USD}}{\text{EUR/USD}}$$

• $\text{GBP/EUR}$: implied cross rate (Euros per Pound)
• $\text{GBP/USD}$: exchange rate of USD per GBP
• $\text{EUR/USD}$: exchange rate of USD per EUR

We can verify the units cancel correctly by treating the exchange rates as fractions of units. Dimensional analysis confirms the formula:

$$\begin{aligned} \frac{\text{GBP/USD}}{\text{EUR/USD}} &= \frac{\text{USD per GBP}}{\text{USD per EUR}} \\ &= \frac{\text{USD}}{\text{GBP}} \times \frac{\text{EUR}}{\text{USD}} \\ &= \frac{\text{EUR}}{\text{GBP}} \\ &= \text{EUR per GBP} \end{aligned}$$

Since the pair GBP/EUR is defined as Euros per Pound, the derived units match perfectly.

The calculated cross rate of 0.8579 EUR/GBP indicates that one Euro purchases roughly 0.86 Pounds, consistent with the direct EUR/USD and GBP/USD rates. This internal consistency is maintained by arbitrageurs who continuously monitor cross rates and exploit any deviations.

In efficient markets, cross rates derived from USD pairs should equal directly quoted cross rates. If they don't, an arbitrage opportunity exists. This arbitrage keeps FX markets consistent and prevents pricing contradictions.

Consider this triangular arbitrage example: if derived EUR/JPY differs from the quoted EUR/JPY, you can profit by executing a series of trades that takes advantage of the mispricing. The strategy works as follows:

1. Starting with USD
2. Converting to EUR at the spot rate
3. Converting EUR to JPY at the (mispriced) EUR/JPY rate
4. Converting JPY back to USD
5. If you end with more USD than you started, you've captured the arbitrage

This arbitrage requires no capital commitment and no risk, only the ability to execute three trades simultaneously. The profit is locked in at initiation.