Caught this article on Seeking Alpha on Wednesday on the problems with using inverse ETFs. It reminded me of a topic that I’ve debated quite a bit with Elliot Shmukler over the past two years, and have been meaning to write about here on the blog.

Since I haven’t commented much on personal finance topics lately, this seemed like an opportune time. 🙂

Quick set of definitions, if you aren’t familiar with the topic:

**ETF**: Exchange Traded Fund. These are funds that are similar to mutual funds, except that they are traded like stocks. For the most part, they are index funds with extremely low expenses. In the last five years, there has been an explosion of index-based ETF funds, ranging from the plain vanilla US Stock Market to indexes only for water companies.**Inverse ETF**: This is a fund that makes money based on the opposite direction of an index. Example: An S&P 500 bear ETF, or inverse index, would attempt to give you a positive 20% return in the case that the S&P 500 dropped 20%.**Double/Triple ETFs**: A double (or triple) ETF attempts to give you double (or triple) the returns of a given index. For example, a double oil fund would attempt to give you a 20% return in the case that oil rose in price 10%.

The original discussion Elliot & I had was back in 2007, when we were comparing two hypothetical portfolios:

**Portfolio 1**: 100% invested in the S&P 500**Portfolio 2**: 50% invested in cash, 50% invested in double S&P 500 ETF

At first blush, it might seem like Portfolio 2 is the winner. After all, with Portfolio 2, it would seem that if the stock market were to drop more than 50%, Portfolio 2 would allow you to keep 50% in cash safe and sound. If the stock market returns more than negative 50%, then the portfolio seems to offer exactly the same return as Portfolio 1.

The trick, however, is the nature of the double ETF itself. Double ETFs are not perfect instruments – because they simulate returns using derivative contracts, they try to match their promised returns on a daily basis. (I’m assuming here, by the way, that the simulation of returns is perfect. In real life, these funds have an additional problem of tracking error due to the nature of the way they are implemented.)

So if on Day 1 the S&P 500 goes up 1%, the double ETF tries to return 2%. I say “tries” because it’s rarely perfect. If the next day the S&P 500 drops 2%, the double ETF should drop 4%.

Unfortunately, it turns out that the math for doubling daily returns does not necessarily lead to a long term result that matches the long term double of the index.

An easy example of where this breaks is to take a huge increase followed by a huge drop.

Let’s say you put $1000 into the S&P 500, and $1000 into the double S&P 500 ETF.

On Day 1, the index goes up 40%. Great news! Your $1000 in the S&P 500 is worth $1400. Your $1000 in the double S&P 500 ETF is now $1800, expected, for a 80% gain. Everything is as expected on Day 1.

On Day 2, the index goes down 40%. Bummer. Your $1400 in the S&P 500 is now $840. Your $2000 in the double S&P 500 ETF is now $360. Wow. That’s bad news.

You can see the problem. Technically, over two days, the S&P 500 dropped 16%. But the double S&P 500 ETF dropped 64%. Most people assume it would have dropped 32%, which is double the two-day return. This is the issue with percentages – the product of a series of percentage changes will not equal the product of a series of 2x those same percentage changes. (Covered in Calculus D at most schools, for those in a nostalgic mood).

If you are skeptical, you might be thinking: “Well, Adam just picked the extreme example. Of course a one day 40% move wipes you out. Most stock market moves are small, so the error is small.”

Unfortunately, this isn’t the case either. A long series of small moves can lead to large errors as well.

Let’s assume a stock market where every day is either up 2% or down 2%. In this case, we’ll go up twice for every down.

Let’s say you put $1000 into the S&P 500, and $1000 into the double S&P 500 ETF.

After thirty days of +2%, +2%, -2% (10 times), here is what you are left with:

You’ll have $1214 in the S&P 500, for a 21.4% gain. But you’ll have $1457 in the double S&P 500 ETF for a 45.7% gain. In just thirty days, you’re off by 6.6% (in gains). In this case, it’s a good thing, but obviously in a market like the one we’ve been having over the past six months, it can be a very bad thing.

In fact, over the long term, the errors can be fairly extreme, and whether they are to your benefit or not is based purely on the size and ordering of the volatile movements. The only way to get ride of the errors is to have an absolutely equally distributed, linear progression of the market in one direction. And let me be the first to say that we will never actually see that happen.

So, what’s the takeaway here? Simple.

The errors in inverse and double/triple ETFs grow rapidly based on volatility. In low volatility markets, they can be used for a short period with expected results. Like options and other derivatives, however, their tracking errors make them poor choices for long term allocation or investment.

They do make interesting options for speculative bets in the short term, especially in situations where:

- You want to keep liquidity (ie, cash available)
- You want to limit your downside to 50%

But watch out for those error rates…