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 you’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 fund’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. Here’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 × $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) × (1 + z)) – 1

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