How to solve triangular arbitrage problems

Once students learn that arbitrage in the wrong direction end in a riskless loss, they often ask how do we know which direction to take?

Assume the following information:

                             Quoted Prices from different banks

              Bank 1) Value of 1 CAD in USD= USD 0.90

              Bank 2) Value of 1 NZD in USD= USD 0.30

              Bank 3) Value of 1 CAD in NZD= NZD 3.02

              Given this information, is triangular arbitrage possible?  If so, explain the steps that would reflect triangular arbitrage, and compute the profit from this strategy if you had $1,000,000 to use.

The first step is to determine if there is an arbitrage opportunity. Start calculating the “implied” cross rate from Banks 1 and 2.

(USD/CAD) 0.9 / (USD/NZD) 0.3 = NZD/CAD 3

Given that Bank 3 offers more NZD per CAD, there is an imbalance, we can arbitrage the difference.

We would like to buy CADs for 3 NZDs sell CADs for 3.02 NZDs. Since we cannot directly buy CAD for @ NZD at the implied rate, we need to sell CADs to Bank 3)… In order to sell CADs we need to buy them from somewhere.  The problem tells us that we have USD and therefore we have to get the CADs from bank 1. Once we get them we can sell them to Bank 3 in exchange for NZD, which we then sell in Bank 2 for USD.

[$1,000,000/$.90 = C$1,111,111 × 3.02 = NZ$3,355,556 × $.30 = $1,006,667]