# In this 3-way currency exchange, who loses, and by how much?

Let's say, hypothetically, we have the following imbalanced exchange rates:

1USD = 1.2EUR

1.2EUR = 1000 JPY

1000 JPY = 1.5 USD

Let's say person A starts off with 1 USD, buys 1.2 EUR from B, then buys 1000 JPY from C with the 1.2 EUR, then finally buys 1.5 USD from D with the 1000 JPY.

Person A now has .5 more USD than they started with. Where did the .5 USD come from? Who lost that equivalent amount? Does it depend on the exchange rates before and after this imbalance happens?

• There is no need for the extra currency, you could just have 1USD = 1EUR and 1EUR = 1.1USD. Where does the extra 0.1 USD come from? This would make the example and the explanation simpler. Commented Jun 14, 2017 at 20:27
• @denesp But that's not the question I'm interested in. I'm fairly certain those exchange rates never happen in practice. I could be wrong, but at any rate I'm interested in what happens in the 3-way exchange
– C_Z_
Commented Jun 22, 2017 at 20:42
• I don't see why my imbalanced exchange rates are any less plausible than yours. I maintain that there is no difference between the two scenarios, 3-way just adds unnecessary complication. Commented Jun 23, 2017 at 8:27

If the starting balances were

A = 1USD
B = 1.2EUR
C = 1000JPY
D = 1.5USD


then the ending balances are

A = 1.5USD
B = 1USD
C = 1.2EUR
D = 1000JPY


so the simple answer to where the 0.5USD came from is D.

A more complicated answer would be some combination of B, C and D depending on how the exchange rates rebalanced to iron out this arbitrage opportunity.

For example if 1000JPY dropped back to 1USD, then D could do some exchanges with B and C leaving them where they started and D with 1USD, i.e. D lost the 0.5USD

But if the EUR/JPY changed so that 1.8EUR = 1000JPY, then C ends up with -0.5USD, if you do exchanges to restore the JPY and EUR to where they started.

If USD/EUR changed so that 1.5USD = 1.2EUR, B ends up with -0.5USD.

Let us assume that there are only these four individuals, and that the original currency they had is all the asset available. Then, B can sell its USD to buy EUR to C, who can then sell its USD to D to buy JPY. This means that B and C end up as before, except for A and D who swapped their currencies. Because your example only allows A to benefit from the arbitrage, and D also had USD, necessarily A has benefited at expenses of D. Notice that in this process, no money was created. Total assets in the group after transactions are 2.5 USD, 1.2 EUR, and 1000 JPY, as initially.

It is evident that if you allow everyone to arbitrage, or allow for further rounds of arbitrage, the exchange rates are unsustainable; the demand for USD would be infinity.

I believe D loses in the exchange because USD in this are more valuable when directly exchanged for USD as opposed to EUR. And, the values between all others hold consistent, so it depends which direction USD and JPY are being exchanged in. As an example, If the person was to exchange 1.5 USD for 1000 JPY, then change 1000 JPY for 1.2 EUR and then 1.2 EUR for 1 USD then he would have a net loss of .5 USD. So, that one inbalance can cause a net loss or profit dependent upon which way it is traded.

I think you have answered your own question in the first line

we have the following imbalanced exchange rates

Keyword here, imbalanced.

What you have written is mathematically impossible, one of your 3 exchange rates could never happen.

• I don't think it is "mathematically impossible" to find a person who is willing to sell two dollars for one dollar. Commented Jun 21, 2017 at 21:16
• This is definitely not true. The market can exist in an imbalanced state, if only for a fraction of a microsecond, but it can happen.
– C_Z_
Commented Jun 22, 2017 at 20:40