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My permanent residence is in the USA and almost all of the money to name is in US dollars (USD) in US accounts. This month I'm in India and may end up having withdrawn more rupees (INR) than I need as a tourist here. From India I'll stopover in Germany for one week, before finally returning to the USA. Naturally, I will need to get some euros (EUR) for this short trip, and I already know that amount is more than the equivalent value of the extra INR I'm holding.

I could either convert my extra INR to EUR, or to USD. In either case the (maybe additional) euros I need would be converted from my USD account, being careful not to withdraw more EUR than I'll need for Germany. I'm wondering if there are any considerations that might favor one or the other conversions, in terms of my overall remaining US funds?

I know there are a lot of factors here, such as different places' exchange fees and that and we cannot predict future exchange rates. I'm not trying to "game" the market's future activity. I think I'm not expecting a straight answer from SE to "please decide for me which currency to covert my rupees" but rather what I really wonder is more theoretical about how foreign exchange markets work:

If 1 INR = p EUR and 1 EUR = q USD, then does 1 INR = pq USD? (assuming fair rates, no fees, and p, q, & pq being measured at the same time.) Would this also be true for any trio (or more, by extension) of world currencies? If not, why not?

(If there is a mathematical name for this property I'm asking about, please enlighten us!)

Jeff G
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    I'm voting to close this question as off-topic because it's not a travel question. – JonathanReez Feb 28 '17 at 12:43
  • It's likely you are looking for the term "cross-currency arbitrage" and the game is to find where pq falls in your favour. It's difficult unless you are playing in the wholesale markets. But the question is off-topic here, sorry. – Gayot Fow Feb 28 '17 at 13:17
  • Fair enough. I figured there was a chance because I really am going to make this conversion decision this week as part of my travel/tourism, but I won't push the point. – Jeff G Feb 28 '17 at 13:40
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    Currency traders can an do make money from this situation. However, it requires trading a huge amount of money to exploit very small differences in exchange rates between different sequences of currencies. For the amounts of money that you'll be changing, 1INR is so close to pqUSD that it doesn't matter what you do. – David Richerby Feb 28 '17 at 14:48

1 Answers1

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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.

Calchas
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  • "Not permitted in real markets": is this a legal prohibition? – phoog Feb 28 '17 at 18:09
  • @phoog No, it is an axiom of the standard theory underpinning quantitative finance. Essentially it means "there's no such thing as a free lunch". In reality, arbitrage opportunities may exist briefly, but are eliminated very quickly by the market. The presence of an apparent arbitrage opportunity is usually disguising hidden risk. – Calchas Feb 28 '17 at 18:32
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    @phoog, no, it's a practical prohibition. If you see a situation where you could perform arbitrage, rest assured that a professional trader (or their high-frequency trading computer) saw it before you did, and has already moved enough money to eliminate it. – Mark Mar 01 '17 at 02:53