Subject 3. Cross-Rate Calculations

A cross rate is the exchange rate between two countries computed from each country's exchange rate against a third country. For example, since most currencies are quoted against the U.S. dollar, sometimes we need to work out the cross rates for currencies other than the U.S. dollar.

If we interpret a:b as a "divide" sign, then a:b is actually b/a. Assume we have currencies a, b and c. If a:b and b:c are both known, then a:c = a:b x b:c. For example, if the Mexico peso (MXN) is selling for $0.0923 (MXN:USD = 0.0923) and the buying rate for the EUR is $0.7928 (USD:EUR = 0.7928), then the MXN/EUR cross rate is MXN:EUR = 0.0923 x 0.7928 = 0.0732.

Cross-Rate Calculations with Bid-Ask Spreads

Example

The rate between Japanese ¥ and the U.S. $ is $:¥ = 119.05 - 121.95 and the rate between the euro and the U.S. $ is $:€ = 0.7920 - 0.7932. The direct quote between the yen and the euro in Japan will be: (¥119.05/$)/(€0.7932/$) = ¥150.0883/€, and (¥121.95/$)/(€0.7920/$) = ¥153.9773/€.

The lower rate is the bid, and the higher rate is the ask. Therefore, the rate between yen and euro is €:¥ = 150.0883 - 153.9773.

In fact, each cross-currency transaction is the combination of two trades:

  • The bid price: a bank will buy U.S. dollars with yen low ($:¥ = 119.05), and sell U.S. dollars for euro high ($:€ = 0.7932). Thus, the bid price is €:¥ = 150.0883.
  • The ask price: a bank will sell U.S. dollars for yen high ($:¥ = 121.95), and buy U.S. dollars with euro low ($:€ = 0.7920). Thus, the ask price is €:¥ = 153.9773.

Note that in calculating the cross rates you should always assume that you have to sell a currency at the lower (or bid) rate and buy it at the higher (or ask) rate, giving you the worst possible rate. This method of quotation is how dealers make money in foreign exchange.

Similarly, the direct quote in France or Germany is ¥:€ = 0.006494 - 0.006663

User Contributed Comments 5

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Lambo83: Good explanation but the CFA material states the rate 119.05 as a direct quote is JPY:USD making the Yen the price currency. The ratio is JPY:USD not USD:JPY. USD:JPY would be 0.0084 not 119.05
Dabuya: The textbook says: "There is no overall consistency in this mixture of direct and indirect quoting conventions in the professional FX market".

It also says: "the actual number quoted is the number of yen per sterling (JPY/GBP) ... the codes may appear in a variety of formats that all mean the same thing. For example, GBPJPY might instead appear as GBP:JPY or GBP-JPY."

I won't say the notes is wrong of saying "USD:JPY".
gc1210: My understanding was that a:b = a/b isn't it?
MattG: @gc1210 a:b = b/a always.
tyjz: @gc1210 @MattG a:b = a/b. Take MXN : USD = 0.0923 for example. If MXN : USD = USD/MXN = 0.0923, does it mean 1 USD = 0.0923? So gc1210 is right.