Can three or more currency exchange rates simply be multiplied?

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Accepted answer

The mathematical property is called transitivity.

At first it seems intuitively obvious that the rates must be transitive. If the rates are not transitive, it establishes a situation where it is possible to generate unlimited money by trading money in a circle in the appropriate direction (or lose unlimited money by going the other way). This is called arbitrage. Under the No Arbitrage Principle of quantitative finance, this situation is not permitted in real markets and so currency rates are transitive.

However, the situation is slightly complicated by the fact that there exist separate buy and sell rates (sometimes called bid and ask or bid and offer) for each currency pair. The buy and sell rates have a non-zero spread between them. This spread is how the money changers make money and cover the costs of the transaction. So even if you take 1 USD and convert it into INR at the best possible rate, and then immediately convert it back to USD, you will have slightly less than 1 USD.

Another way of saying this is that the trades are not frictionless; there is a cost to each exchange.

In general it is best to do as few transactions as possible. Therefore a straight conversion is usually going to be most economic. The exception might be between two highly illiquid currencies where it is necessary to use an intermediary.

In practice for a traveller, it is hard to budget precisely how much of each currency you will require; and at the low volume of currency a traveller typically trades, it is harder still to get a rate very close to the spot rate (the rate that the consensus of the market believes is the true exchange rate). Both of these inefficiencies will probably outweigh the inefficiency of trading your currency twice instead of once, unless your single trade is done at a really bad rate. But in general, one trade is better than two.

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