In my last post, I explained that US-listed ETFs that hold overseas stocks do not expose Canadians to fluctuations in the US dollar. This is an important idea to understand if youâ€™re comparing ETFs that hold international equities.

Vikash Jain, portfolio manager at the Toronto investment firm archerETF, was kind enough to dig into the data and provide a real-world example of this principle. In 2009 and 2010, the MSCI Brazil Index returned a total of 51% in its local currency, the Brazilian real (BRL). During that same two-year period, the following currency movements took place:

• The BRL strengthened against the US dollar (USD) by 38%
• The BRL strengthened against the Canadian dollar (CAD) by 14%
• The CAD strengthened against the USD by 21%

Jain explained how all of this would have affected the returns of investors who held the iShares MSCI Brazil Index Fund (EWZ), which is traded in US dollars:

• An American holding EWZ would have received the 51% index return, plus a big boost because the BRL shot up 38% against the greenback. In USD terms, he would have earned 109%.
• A Canadian holding EWZ would also have received the 51% index return, plus a boost because the BRL appreciated by 14% against the loonie. In CAD terms, she would have earned 72%.
• Although Canadians must buy and sell EWZ in USD, the fact that the CAD strengthened by 21% against the USD during this period is irrelevant.

### A mathematical footnote

I used to assume that you could calculate an international fundâ€™s total return by simply adding the gain/loss on the stocks to the movement of the currency. In the Brazil example, I expected that if the stocks rose 51% and the BRL appreciated by 14%, then a Canadian would earn 65% from EWZ. But the correct return, as you can see above, is actually 72%.

Jain corrected my bad math and explained that you have to multiply the figures, not add them. Hereâ€™s the logic behind the calculation:

• Assume that 1 BRL = CAD \$0.50.
• Your CAD \$100 buys BRL 200, which are then invested in EWZ.
• EWZ goes up 51% in local currency, so your 200 BRL have become 302 BRL.
• Over the same period, the BRL appreciates 14% versus the CAD. Now one BRL is worth \$0.57.
• At this new exchange rate, your 302 BRL are now worth \$172.14 (302 Ă— \$0.57).
• Your original investment of \$100 is now worth \$172.14, so your return in CAD is 72.14%.

For the math geeks in the audience, the formula for this calculation is as follows, where y is the local return and z is the gain (or loss) on the currency:

((1 + y) Ă— (1 + z)) â€“ 1

Many thanks to Vikash Jain for patiently explaining this complicated but important topic.