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.

Interesting.. but, since I have to purchase EWZ in US$, my CAD$ are converted to US an then the returns in CAD$ reflect the strengthening of the US$ in the period, correct? if that’s the case, the US/CAD fluctuation will take a bite in the returns by a 21% at least, excluding the purchase commission and the sell commission (if the investment is sold at the end of the period being studied). As long as the fund is US$ based, we cannot ignore the US/CAD fluctuations, since the valuation in our portfolio is in CAD.

@tony: Your assumptions are very common, but they’re not correct. This is exactly the misunderstanding I am trying to clear up. Canadian Financial DIY has a detailed post on this, including a spreadsheet with some examples that may clarify the idea:

http://canadianfinancialdiy.blogspot.com/2007/05/clarification-of-foreign-exchange-risk.html

I used to add the local return to currency return also (with signs). It’s more intuitive.

One learns something new everyday!

The MSCI EAFE returned 2.4% in 2010 ($CAD).

The Canadian Dollar appreciated 5.7% against the $USD.

VEA and EFA returned around 8.0% ($USD).

That’s good enough for me.

@Eric: Yes, if you held VEA or EFA in a taxable account in USD, then you would have earned an 8% return in US dollar terms. But your US dollars are now worth 5.7% less in your home currency. So the return on your EAFE investments, in CAD terms, is right around 2.4%. In other words, it’s the same result as if you had bought the EAFE stocks with Canadian dollars.

If you have $1,000 USD in cash and the USD goes down 10% against the loonie, have you lost money? Depends on your perspective, I guess. You seem to be saying no, but I would argue that you have.

Very well written Dan.

I have been enjoying your blog. Keep up the good work.

Lance

Thanks, Lance, much appreciated.

@Dan

I agree, the returns are the same (minus the MER).

TD intern (e) returned 1.7%

Good post !

Good post. What about the idea (or myth) that long term currency fluctuations do not really matter and that it all washes out in the end? Would make an interesting study…

@Michel: This is good question. For most individuals, even over 20 or 30 years, it’s unlikely that currency fluctuations will even out. The point is that currencies have no “expected return.” You can reasonably expect that, over the long term, stocks will outperform bonds, but you can’t make a good argument that the Canadian dollar will “outperform” the US dollar or any other currency.

If hedging were free and precise, I would have little problem with it. The reason I choose not to hedge is that frictional costs and the high tracking error erode returns over the long term.

Great post– very helpful.

I’ve used the hedged ETF for short periods. In 2008, when the market crashed, the Canadian dollar rapidly plummetted against the USD. Since the USD was so expensive at that time, I used the iShares Canada version of the S&P 500 and Russell 2000 to get my US equity exposure. I held it until the dollar hit parity and have since sold the hedged versions and moved into the USD based Vanguard version.

Here’s my own math-nerd take: if you use log points, it all works out nicely. The stocks rose 51% = 41 log points, and the currency appreciated 14% = 13 log points, so you got a total of 54 log points = 72%.