Building a cross-rate from two quotes

Quotes as fractions

The professional FX market does not use 'direct' and 'indirect' quotes, because those depend on where you live. It quotes fixed currency pairs instead, each written in price/base form. USD/EUR = 1.0850 says that one euro (the base) costs 1.0850 US dollars (the price currency).

The easiest way to avoid mistakes is to treat the quote as an algebraic fraction with units. USD/EUR is 'USD per EUR'. Its inverse, EUR/USD, is 'EUR per USD' and equals 1 ÷ 1.0850 = 0.9217.

A cross-rate is any rate between two currencies that you derive from their quotes against a third, common currency. Because most currencies trade actively against the dollar, the dollar is usually the bridge.

Write the currency codes next to the numbers every time. If the units on the left do not cancel down to the pair you want, the arithmetic is wrong, whatever the calculator says.

Case 1: multiply, and the middle currency cancels

Suppose you have SEK/USD = 10.4200 and USD/EUR = 1.0850 and want SEK/EUR. Written as fractions, $\frac{SEK}{USD} \times \frac{USD}{EUR} = \frac{SEK}{EUR}$. The USD on the bottom of the first quote cancels the USD on top of the second.

So SEK/EUR = 10.4200 × 1.0850 = 11.3057. One euro costs about 11.31 Swedish kronor.

The rule: when the common currency is the base in one quote and the price currency in the other, simply multiply the two quotes.

Case 2: invert one quote first

Now suppose you want JPY/NOK from NOK/USD = 10.5000 and JPY/USD = 147.00. The dollar is the base in both quotes, so multiplying would give $\frac{NOK \cdot JPY}{USD^2}$, which is meaningless.

Invert NOK/USD to get USD/NOK = 1 ÷ 10.5000 = 0.095238. Now $\frac{JPY}{USD} \times \frac{USD}{NOK} = \frac{JPY}{NOK}$ = 147.00 × 0.095238 = 14.00 yen per krone.

Inverting and multiplying is the same as dividing: JPY/NOK = (JPY/USD) ÷ (NOK/USD) = 147.00 ÷ 10.5000. Likewise, if the dollar is the price currency in both quotes, divide: AUD/GBP = (USD/GBP) ÷ (USD/AUD).

Note the decimals: the result involves yen, so it is shown to two decimal places, as yen rates conventionally are.

Who actually calculates cross-rates?

A client who wants to trade NOK against JPY simply asks a dealer or electronic platform for a JPY/NOK quote; she is not handed two dollar quotes to combine herself.

Dealers, however, often compute the cross-rate behind the scenes, because the two dollar pairs may be run by different traders or desks. Electronic dealing systems in both the interbank and client markets do this calculation automatically.

Either way, the quoted cross-rate has to be consistent with the two dollar rates, which is the subject of the next lesson.