**TWO STEPS BACK**

The downward drift has been discussed and theorized by many sources, from retail investors on blogs, to reports and papers from institutional investors. All have the convergent opinion that ETFs do in fact lose value over time. However, the confluence ends there.

Some state that only inverse ETFs have a particular downward bias; others say that both leveraged and inverse ETFs express the trend. One thing is clear — if there were a drift, it would affect bull ETFs in the opposite way that it would affect bear ETFs, with one decreasing in value and the other increasing. This, however, is not observed. The explanation must lie in some process that does not have a directional bias and that decreases the value of both bull and bear ETFs over time.

The most highly regarded explanation for the downward drift is the negative convexity that comes from re-balancing. Specifically, leveraged and inverse ETFs must be, as part of their construction, re-balanced on a daily basis to produce the promised leveraged returns of the underlying index. An illustration of this effect is as follows: suppose the asset in question was a double-leveraged ETF with an initial net asset value (NAV) of $100. The index that this ETF tracks begins at 100, falls 10% the one day, and then rebounds 10% the ensuing day. The result is the index declines by 1% over a span of two-days (down to 90, then up to 99). You then might conclude that the double-leveraged ETF would have fallen 2%, or twice the amount of the index, when in fact it declines by 4%. By doubling the index’s 10% fall, the NAV is driven to $80. A 20% rise the following day, however, puts the NAV at $96. Although an interesting exercise, the example does not consider all other possible paths that the index could have taken, leaving the fund in an entirely different position.

**MEAN & MEDIAN RETURNS**

When you consider all the possible paths, a more complete picture emerges. Consider some examples. The first (see “Mind your median,” page 35) demonstrates both the 10% up and the 10% down movement of an ETF over a period of two days.

The results in the first table show that although the average price at the end of two days remains at 100 = (121 + 99 + 99 + 81) / 4, the most likely price (median price) is 99. Meanwhile, the double-leveraged ETF (in the second table) will have up or down movements of 20%.

The same path that leads the traditional ETF to 99 (or a 1% loss) yields 96 (or a 4% loss) for the double-leveraged ETF. The average price, however, remains the same 100 = (144 + 96 + 96 + 64) / 4. This is a simple demonstration that increased leverage (or increased volatility) makes the average price unchanged, but results in a lower median return.

Though the investor is most likely to observe a downward drift in a leveraged ETF (compared with its underlying ETF), that does not necessarily mean that the expected return to the leveraged ETF investor is any lower. To further demonstrate this point, consider the hypothetical five-times-leveraged ETF displayed in the third table in “Mind your median.”

The median return is 25% lower than the starting price, but the average still retains the price of 100 = (225 + 75 + 75 + 25) / 4, mainly because of the highest return of 225. This leads to the conclusion that leveraged ETFs do not provide returns that are inferior to regular ETFs. They just simply change the probability distribution of future returns. More specifically, the higher the volatility, the lower the median, while the average remains unchanged. The same analysis is applicable to inverse ETFs, but the binomial trees would be plotted upside down. Regardless, the mean and median returns would remain the same, and the median would be smaller than the mean.

The formula for the median can be obtained easily. Starting with price S and holding it for two periods, we can calculate the price will be S * (1 + return) * (1 - return), which in this case is 100 * (1 + 0.1) * (1 - 0.1) = 100 * 1.1 * 0.9 = 99. After four periods, the value would be S * (1 + return) * (1 - return) * (1 + return) * (1 - return) = S * [(1 + return) * (1 - return)]^{2}, which can then be generalized into S * [(1 + return) * (1 - return)] ^{N / 2}, where N is the number of days.

We can further examine the same binomial tree, this time with many small additional steps. Taking this idea further to its (mathematical) limit, the statistical distribution of prices will converge to a lognormal distribution — inarguably the most widely used distribution in finance. Lognormal distribution means that the logarithm of stock prices has normal distribution, and is a result of the same percent-random walk used in the binomial tree (see “Lognormal view,” above).

The formula for this median is:

where S is the current day’s stock price, exp is exponential.

Using the data from the previous examples, we would calculate the median

100 * exp (-0.5 (0.1^{2}) * 2) = 99.00

100 * exp (-0.5 * (0.2^{2}) * 2) = 96.08

100 * exp (-0.5 * (0.5^{2}) * 2) = 77.88

The results from using lognormal distribution are similar to those obtained from the simple binomial tree. The relationship holds true that, if volatility increases it results in -1/2 volatility^{2} decreasing. This then leads to the decreasing of the entire equation of...

...so the median price is lower than the average price. The Cheng and Madhavan paper on leveraged and inverse ETFs has data analysis using regression, and has empirically confirmed this particular dependence of returns on volatility.

The formula suggests that an investor who is long a leveraged or inverse ETF has time decay associated with the realized volatility of the underlying ETF. When realized volatility is high, even if the price of the underlying ETF does not change substantially, the unlucky investor of the leveraged or inverse ETF may underperform the corresponding multiple of return of the underlying ETF. There are, however, circumstances when increased volatility results in the leveraged or inverse ETF producing a substantial profit to the investor.

The usage of the formula can be illustrated on real ETF data. Take IYF as an example. IYF, a regular ETF tracking the Dow Jones U.S. Financial Sector index, has a historical annualized volatility of 51%. Its double-inverse SKF has an annualized volatility of 104%, while the double leveraged UYG has an annualized volatility of 102%. Both come close to doubling IYF’s volatility. To determine the expected decline that is likely to happen because of rebalancing in the next year, we must calculate

exp(-0.5 * (2 * 51%)^{2} * 1 year) = 0.59 for an expected 41% decline.

To expand on the influence of volatility over returns, a model-free analysis of leveraged ETFs using bootstrap was conducted. Bootstrap is a statistical technique used to infer the properties of the distribution by resampling from the original distribution. This allows the construction of multiple alternative histories for the ETF in question. The results are displayed in “Re-booted.”

The top graphs depict the resampling of actual returns of SKF. In the first of these two, the red line is the historical path. As is apparent, the high volatility has pushed the vast majority of the possible alternative paths of SKF downward. The infrequent situations when SKF took an outcome to the upside, however, would have either made the long investor a substantial profit, or would have taken the unlucky short-seller out of the market. Because the distribution has an extreme upside tail, the ETF will have frequent moderate declines and infrequent but astronomical returns (this is, of course, generalized).

The bottom graphs establish that volatility is the dominant force that produces the downward drift in the leveraged ETFs. The graph shows resamplings of SKF returns scaled by one-quarter to produce one-quarter of the volatility (or half of the volatility of the underlying index, the opposite of the double-leverage). As it is shown in the illustration, the lower the volatility, the more symmetric the alternative outcomes will be.

As we have seen from this analysis, the perceived downward drive in bull and bear leveraged ETFs is rather a result of non-symmetric distribution caused by heightened volatility. As with regular ETFs or other stocks, investors face the possibility of a negative return, but there is a higher probability of that occurring with leveraged and inverse ETFs, bull or bear.

However, even with this increased chance of negative returns, it would be incorrect to state that leveraged ETFs have an expected negative return, or negative drift. Such conclusion would ignore the considerable upside potential that leveraged and inverse ETFs have compared to regular ETFs.

**Dennis Dzekounoff is a financial engineer and Roxanne Militaru is an analyst at Kinetic Strategic Group, LLC, a private investment firm. Contact them at info@kinetic-sg.com.**