### Why the tracking error for levered ETFs can be large

Exchange traded funds (ETFs) are popular investment instruments. The first generation of ETFs were designed to trade on exchanges and offer returns that track an index. Over time new types of ETFs have been developed. These include levered and reverse ETFs which promise investors returns that are multiples of the index return or are returns that are the negative of the index returns.

However, investors are warned that in the offering documents that these investments may not track the index performance over a period of time. Why is this so? This article will describe some of the situations in which the returns deviate from those that investors might expect and will show some of the mathematics of the return calculations that cause these situations to arise.

The cumulative returns of levered and reverse EFTs.

Levered and reverse ETFs allow investors the opportunity to gain levered returns or hedge their portfolio exposures while satisfying certain portfolio constraints such as the use of leverage or the requirement for long-only investments. Currently, most levered ETFs offer returns that are either 2 times or 3 times the daily returns of the underlying index. Most reverse ETFs offer returns that are either -2 or -3 times the daily returns of the underlying index. So on a day when the index returns 1 percent, the 2 times levered ETF would return 2 percent on the principal. The two-times reverse ETF would return -2 percent.

However, over several days, the cumulative returns from the levered ETF may deviate from the originally specified multiple of the underlying index.  For instance, assume an investor is invested in a index that returned 10% in the first day and then declines 9.0909…% in the second day. Over the two day period, the index has neither a gain nor a loss. If a second investor holds a two-times levered ETF, that investor would receive a 20% gain in the first day followed by an 18.1818…% decline in the second day. This results in a 1.81818…% decline over the two days.

(1 + 0.20)(1 – 0.181818) = 0.981818 = 1 – 0.181818

The levered ETF underperforms the index in this case. In other cases, the levered ETF can outperform the index return times two.

The example above illustrates how a levered ETF is sensitive to the changes in the direction of returns. Each change in direction lowers the return when compared to the return if the index times two.  When the index moves in a consistent direction, the levered ETF will outperform the index times two. Mathematically, we can see this by assuming that the index returns a% in day one and b% in day two. Therefore, the index return in these two days is

Index-two-day-return = (1 + a)( 1 + b) = 1 + (a + b + ab)

The two-times levered ETF returns

Levered-ETF-two-day-return = (1 + 2a)(1 + 2b) = 1 + 2(a + b) + 4ab

= 1 + 2(a + b + ab) + 2ab

If a and b are the same sign, then the levered ETF outperforms twice the index. If a and b have opposite signs, then the levered ETF underperforms twice the index. In other words, reversals hurt the performance of the levered ETF and continuations help performance.

As a result, levered ETFs are exposed to directional changes so they are often recommended as tools to be traded within a day, but not held overnight or for the long term.

The mathematics for the reverse ETF is similar but show even a stronger sensitivity to volatility. In the index example above, the two-times reverse ETF has a 20% decline in day one followed by a 18.18…% gain in day two. The cumulative two day return is a loss of 5.4545…%:

(1 – 0.20)(1 + 0.181818) = 0.94545 = 1 – 0.05454

So, as in the two-times levered ETF, the two-times reverse ETF underperforms the index times two.

Reverse-ETF-two-day-return = (1 – 2a)(1 – 2b) = 1 – 2(a + b) + 4ab

= 1 – 2(a + b + ab) + 6ab

The result is below the anticipated cumulative return when a and b are have different signs and above the anticipated cumulative return when a and b have the same sign.

The figure above shows how repeated reversals can degrade the performance of the levered and reverse ETF. The blue line shows the index performance and the pink and brown lines show the performance of the levered and reverse ETFs.

These calculations show that simply multiplying the daily returns creates an ETF whose returns track the index on a daily basis (times the leverage), but do not consistently track the index (times the leverage) over extended time periods.

As an illustration of how a levered ETF and a reverse ETF can perform in a volatile market, I examine the price history of three real estate ETFs: IYR, URE, and SRS. They are the Dow Jones U.S. Real Estate Index Fund, theProShares Ultra Real Estate (2 time levered), and the ProShares UltraShort Real Estate (2 times levered inverse) respectively. All are tied to the daily performance of the Dow Jones U.S. Real Estate IndexSM.

The period from October 2008 to February 2009 was a period of concerns about the real estate market accompanied by high market volatility. The figure below shows how the three funds fared during that period. It is not surprising that both the index fund, (IYR) and the levered ETF (URE) suffered losses during the period. What is less intuitive is why the reverse ETF (SRS) profited during much of this period, but ultimately ended lower than at the start of the period.

This was due to the large number of market reversals that took place during this period. Another way to examine how market reversals impacted the prices of the levered and reverse ETFs is to plot how their prices moved relative to each other during this time period. The figure below plots the daily prices for the levered ETF (URE) versus the price on the same day for the reverse ETF (SRS). The points at the upper left corner of this graph indicate the prices for both ETFs at the start of the time period. The points at the lower right are the prices at the height of the crisis. The points at the lower left are the prices at the end of the period. Note that for any given short period of time, higher prices in the levered ETF are matched with lower prices for the reverse ETF as indicated by lines running from the upper-left to the lower-right or vice versa. However, each reversal in direction is accompanied by a general decrease in the prices for both ETFs. Over time this creates a sawtooth pattern that slows grinds down the prices for both funds.

This was a time period of high uncertainty with a large number of market reversals which hurt both the levered and the reverse ETFs. When the market is more directional,  it is possible that both the levered and reverse ETFs would be helped by the market momentum. In either case, however, the funds can have large tracking errors with respect to the index they are following, even adjusting for leverage.