不断有人对双倍ETF的误差对价格的影响产生了疑问，为此特转贴有关文章解答：

Tristan Yates and Lye Kok (IndexRoll) submit: The Leveraged ETF offensive is under way. A year ago, there were no leveraged ETFs in existence. Today, there are at least fifty leveraged ETF products in the marketplace and another fifty in the SEC/AMEX pipeline. By this time next year, perhaps every traded ETF will have a 2x leveraged counterpart. Are these leveraged ETFs suitable for retail investors? No, they are not.

In this article, we lay out the case against these products, based upon popular misconceptions of what exactly these ETFs provide, a hidden trap related to leverage, and the poor performance of related funds and of the ETFs themselves.

Note that this article updates a SeekingAlpha article posted about six weeks ago, and we’d like to thank the many readers who were kind enough to provide us with additional research and commentary.

The Daily Double

Leveraged ETFs are exchange-traded funds that are based upon well-known indexes, but that provide investors with additional leverage by using borrowed money. Their goal is to increase the return of the underlying index and provide a better return for the fund’s investors. Typically they provide $1 of debt for every $1 of investor equity, and are marketed as 2X funds.

Leveraged ETFs are implemented using financial derivatives, such as options, swaps, and index futures. All of these tools are available to individual investors, but are much more complex than traditional share buying and selling and require larger amounts of capital. Thus, the advantage of the leveraged ETFs for many investors is a reduction of complexity and lower capital requirements.

Two companies, Rydex and ProShares, dominate the leveraged ETF marketplace. They have offered leveraged investment funds for many years, and have recently repackaged these products into ETFs.

A listing of some of the more popular ProShares leveraged funds:

A widely held misconception about these funds is that they will offer twice the return of the underlying index, which means that if the S&P 500 returns about 10% a year, then the SSO should return 20%. But that’s not true, because these funds only double the daily return, and there’s a big difference between doubling the daily return and doubling the annual return.

What’s the difference? Let’s say that one day the market goes up 10%, and the next day it falls 10%. The two-day loss for the index is 1%, but the loss for the leveraged fund is 4%. Here’s why:

Index: (1 + 10% ) x (1 – 10%) = 1.1 x 0.9 = 0.99, 1% loss
X2 Fund: (1 + 20%) x (1 – 20%) = 1.2 x 0.8 = 0.96, 4% loss

Thus over a two day period, this fund’s losses are 4x the amount of the index, not 2x. This example comes from the ProShares prospectus, and is a clear indication that investors in 2X funds should not expect their investment to provide double the return of the S&P 500 for any period longer than one day.

Historical Returns

How much do daily doubled returns diverge from doubled annual returns? To find out, we created an imaginary fund called X2. X2 simply provides double the daily return of the S&P 500 index. We invested $1 in the fund and compared it to a $1 investment in the SPY over the fourteen-year period.

Note that this fund is imaginary by definition – there are no management fees, transaction costs, or capital costs. It is simply used to provide an illustration of the potential of a 2x leveraged fund.

The annual returns are close to double the SPY for the most part, but some years are off. In 1994, the X2 fund only provided 1.35x the return of the SPY, rather than 2X as expected. In 2001, the X2 fund provided 4.17x the loss of the SPY.

As a long-term investment, the X2 fund provides very good returns. Most years have positive double-digit returns. $1 invested in the X2 fund in 1993 would return $11.08 by the end of 2006, as compared to only $4.07 for the SPY.

However, if the investor had the misfortune to invest at the peak of the market, at the end of 1999, the investor would have lost 61% of the investment in the next three years, and would still have a loss by the end of 2006. Our X2 fund is a solid performer in the good years, but suffers disproportionately during the downturn and can’t recover its losses in the next rally.

Constant Leverage Trap

Let’s take a moment to see how a leveraged ETF works. In order to deliver the 2x results that the fund’s investors expect, fund management has to hold equal proportions of debt and equity at all times.

In other words, if there’s $100m invested in the fund, it has to borrow an additional $100m and make a $200m investment in the underlying index. That’s the only way that the fund can provide 2x the underlying daily return of the index.

Of course the fund doesn’t go to the local bank and borrow money every day and then invest it. It uses financial derivatives, such as swaps, options, and futures. But the overall effect is the same.

However, every day the market moves and the assets in the fund either increase or decrease in value, throwing off the leverage ratio because total assets are no longer equal to total debt. By the end of the market day, the fund’s leverage is either too high, or too low, and some kind of corrective action is required to bring it back to 2x.

In order to maintain the target leverage ratio, our fund has to buy or sell millions of dollars worth of shares every day. Not only does this increase expenses, transaction costs, and short-term capital gains taxes, but it’s also just a bad investment strategy.

Whenever the market makes a big move downward, the fund sells shares and reduces its debt level in order to maintain its target leverage ratio. This locks in losses and reduces the afund’s sset base, making it much harder to recover gains in the next market upturn.

Note that this situation is called the Constant Leverage Trap and is a well-known problem in financial modeling. Investment portfolios that try to maintain constant levels of leverage over time perform very poorly in bad market conditions because they sell off large percentages of their assets. It’s similar to a margin maintenance call.

X2 Implementations

Our X2 fund is imaginary for one important reason – nobody will loan us $100m to invest without charging interest.

In the real world, trading costs money, administration costs money, and money costs money. Funds like the ProShares ETFs have transaction costs, expenses, and costs of capital. Put all these costs together, and a lag is created between an investor’s expected and actual results.

We’ll look at two ETFs: SSO and QLD by ProShares. Neither has been around that long, just about nine months. We compared the funds' nine-month performance to a doubled daily return of the underlying index. Note that the prices used are adjusted for dividends.

Both ETFs have a big lag over the expected doubled daily performance. It’s not clear why the lag is so large, but it’s likely due to high transaction costs and management fees. Maintaining the required portfolio of financial derivatives may be more costly than expected.

Because the QLD lost about 0.7% every month due to lag, the fund’s investors had a hidden expense in their investment. This expense lowers short-term gains, long-term gains, and most importantly, future gains from investment compounding.

Note that some lag will always be unavoidable because the fund will be charged short-term interest on its leveraged investments. The current interest rate used in options calculation is approximately 5.25%, and it is reasonable to expect that a fund would pay that rate. Because this rate would only apply to the half of the portfolio that is debt, we would expect a 2.63% annual lag, minimum, on our fund, based upon just the interest cost alone.

Leveraged Funds

In order to learn more about leveraged ETFs, we have to look at original set of funds that they’re derived from. ProShares, Rydex, and Direxion all have offered leveraged funds based upon the indexes for several years. We reviewed the long-term performance records of some of these funds from their websites.

Note that there are at least fifty different leveraged funds for an investor to choose from, but these funds listed below seem representative of the larger population of leveraged funds.

The leveraged fund returns are shocking. The two Rydex leveraged funds are still in negative territory after almost seven years. Two other funds have returns that look like Japanese bonds. Only the ProFunds Ultra Midcap fund has an acceptable long-term return.

What happened? All of these funds were started just before the stock market downturn in 2000. Due to the use of constant leverage, they sold shares in the down market, and didn’t have enough of an asset base to recover. That’s why even with their leverage, they’re still poor performers.

What if we were able to avoid the downturn? Would we have sidestepped the losses and racked up huge gains in the next bull market?

For almost five years, we have been in a bull market, and the leveraged funds have performed well, but not exceptionally. They certainly haven’t provided double the performance. Three unleveraged low-cost Vanguard funds are provided below for comparison.

The fact is that the track record of leveraged index funds is poor. During the bull markets, they perform a little better than low-cost unleveraged funds, but during the downturns, they really suffer. We should expect the current crop of ETFs to perform the same way.

Conclusions

1. Many investors are misled by these leveraged ETFs and believe that they’ll get twice the daily return of the underlying index over the long term. In other words, if the index returns 10% next year, they’ll get 20%. But doubling a string of daily returns is not the same thing as doubling the annual returns, so investors should not expect that level of performance, unless they somehow rebalance their portfolios every single day.

2. Even just looking at X2 funds as a theoretical concept, the idea has some problems. The only way for a fund to maintain a constant leverage ratio is to buy shares whenever prices go up and then sell them when prices go down. This buying and selling activity increases the underlying volatility, and can lead to huge sell-offs in down markets that are impossible to recover from in the next bull market. This effect can be seen in every leveraged fund that went through the 2000 to 2002 downturn.

3. The current ETFs do not even deliver twice the daily performance of the underlying index. In just the nine months that the products have available, a lag has emerged between theoretical and actual performance. This lag is 3.3% for the SSO and 6.4% for the QLD. Given this lag, we could extrapolate that over several years, these funds would greatly underperform their theoretical X2 counterparts and at some points, even the underlying indexes. Note that a lag is unavoidable because of interest costs.

4. Investors should compare these new ETFs to the leveraged funds that were offered by these same companies seven years ago, because the long-term performance will most likely be similar. These funds were hit hard during the downturn and never really recovered during the next bull market.

I want to make it clear that I am not saying that leveraged indexing is a bad idea. I’ve written many articles on the subject and find that it has tremendous potential. The underlying concept, borrowing cheaply to make long-term investments in a total stock market index, is based upon both solid historical data and Nobel-prize winning academics.

But until new products are available, based upon the short-term results of SSO and QLD, the long-term results of the leveraged funds, and the mathematical pitfalls of constant leverage, I would suggest that investors avoid holding these leveraged ETFs as investments. If a leveraged indexed investment is desired, the best solution is still call options, index futures, or conventional index ETFs held in a margin account.