On November 5, 2008--the height of the financial crisis--Direxion introduced the first 3x leveraged ETFs, revolutionizing investing and garnering a fair share of criticism in the process. While ETFs date all the way back to SPY in 1993 followed by 2x and inverse ETFs in 2006, it was the arrival of the triple-leveraged product that truly brought the ETF market to the forefront. These products were the ultimate hot potato. One day, a 3x ETF could rally 20% and then fall by 30% the next day, a feat that few, if any, stocks can replicate. Obviously, such volatility was a day traders' paradise. Additionally, over time, 3x ETFs could boast staggering returns, with the Direxion Daily Junior Gold Minor 3x ETF (JNUG) climbing from $9 to $120--a 1200% return--in just 5 months between January and June 2016.

However, investors who trade these products regularly have likely noticed that, more often than not, gains fall short or that losses are magnified compared to those predicted by the ETF's 1x counterpart, particularly over a period of time. This concept of "leverage-induced decay" is a well-established phenomenon. To take a recent example, the popular US Natural Gas Fund (UNG) rallied 15% between April 2016 and April 2017, but its 3x counterpart, the VelocityShares ETF UGAZ, despite a predicted 45% return, actually fell 12% during the same time, a massive underperformance that can be chalked up to leverage-induced decay. This underperformance is shown below in Figure 1.

**Figure 1: Performance of UGAZ versus UNG over the 1-year period between April 2016 and April 2017 showing a significant underperformance, highlighting leverage-induced decay.** [Source: YahooFinance Historical Data]

Unfortunately, many investors in pursuit of outsized gains are ignorant of the risks of a buy-and-hold strategy for 3x leveraged ETFs and their portfolio suffers. For every JNUG circa 2016, there are probably 10 UGAZ 2016-2017s. This article analyzes the phenomenon of leverage-induced decay and attempts to quantify the underperformance that an investor holding a given 3x ETF can expect by deriving a simple set of equations.

In the broadest sense, leveraged ETFs underperform because the market resets at the end of each trading day and 3x ETF performance is based on the daily move of its 1x counterpart and not that ETF's absolute share value. The math behind why these products underperform instead of outperform is complicated and difficult to explain in a single paragraph. Instead, consider two examples.

First, consider a hypothetical 1x ETF that opens at $100/share and gains $0.10/share daily, each and every day. This 1x ETF has a 3x leveraged counterpart. Were the ETF to be a true 3x product, its share value would always be 3x its 1x underlying ETF. Instead, it actually gains slightly more than that such that, when the ETF's performance is determined by 3x the daily move of the 1x ETF rather than its absolute value. Provided the underlying 1x ETF goes up every single day, the leveraged product will accelerate its gains as time goes by and outperform its 3x predicted performance. Figure 2 below plots a hypothetical ETF that does up this $0.10/day allotment and compares it to the predicted performance of a 3x ETF that tracked the 1x ETF perfectly 3:1.

**Figure 2: Hypothetical performance of a 3x ETF that tracks a 1x ETF that trades linearly showing rare outperformance.** [Source: YahooFinance Historical Data]

As the graph shows, after 300 trading days, the 1x ETF is up 30%, as expected. The 3x ETF would be expected to have gained 90% but it is actually up 119%, outperforming by nearly 30%.

Now consider another 1x ETF that also opens at $100/share and falls to $99/share on one day and then rises to $100/share on the next day. This cycle repeats with the ETF alternating between $100/share and $99/share. The corresponding 3x ETF would therefore then be expected to alternative between $100/share and $97/share were it to maintain a strict 3:1 performance. Figure 3 below similarly plots this 1x hypothetical ETF and compares it to the predicted performance of the leveraged ETF and the observed performance of the leveraged product.

**Figure 3: Hypothetical performance of a 3x ETF that tracks a 1x ETF that trades laterally alternating between $99 and $100 daily that underperforms.** [Source: YahooFinance Historical Data]

At the end of 300 trading days, the 1x ETF, having alternated between $99/share and $100/share, is unchanged as is the "predicted" 3x ETF were it to maintain perfect 3:1 tracking. However, in reality, the 3x ETF finishes down 9%, a substantial underperformance, due to leverage-induced decay.

It is important to note that the magnitude of this underperformance is dependent on the size of the daily move of the ETF. Consider the same 1x ETF discussed above but, instead of alternating between $100/share and $99/share, it instead alternates between $100/share and $98/share. Figure 4 below compares it to the predicted and observed performance of its corresponding 3x ETF.

**Figure 4: Hypothetical performance of a 3x ETF that tracks a 1x ETF that trades laterally alternating but with an increased daily volatility between $98 and $100 daily that underperforms much more than the example in Figure 2.** [Source: YahooFinance Historical Data]

By increasing the daily move from $1/day to $2/day, the underperformance of the 3x ETF versus its predicted performance increases exponentially from 9% to a whopping 31%.

Based on this analysis, we see that 3x ETFs outperform when the underlying 1x ETF moves linearly but underperforms when it trades horizontally. Now, which type of trading pattern is more common? While stocks and ETFs may experience quick spikes, over an extended period of time, the vast majority trade much more wave-like with upturns and downturns rather than continuous movement in a single direction. Thus, most ETFs trade more closely to the up-and-down hypothetical ETF rather than the linear hypothetical ETF. It is for this reason that, given enough time, 3x leveraged ETFs almost always underperform.

In summary, there are 3 important takeaway points from these examples. First, leveraged ETFs underperform versus their underlying 1x ETF when they move laterally in choppy, directionless trading. Second, leveraged ETF underperformance increases over time. Lastly, and most importantly, the degree of underperformance is directly related to the magnitude of the daily move in the ETF. That is, 3x leveraged ETFs based on more volatile underlying indexes, commodities, or sectors underperform to a greater extent.

Now let's consider some real-world examples that we will use to build a model predicting 3x ETF underperformance. The 5 3x ETFs chosen cross a spectrum from least volatile to most volatile.

First up is the oldest ETF, the SPDR S&P 500 (SPY) and its 3x leveraged counterpart, the S&P 500 Direxion 3x ETF (SPXL). This pair is the least volatile of the five, with SPXL moving an average of +/-1.7% per day since 2012. This daily volatility is calculated by taking the absolute value of the daily percent gain or loss and averaging it over the given timeframe. Analyzing SPXL performance versus SPY over the past 5 years, SPXL averages an underperformance of -1.9% versus the predicted performance based on the movement of SPY. This undervaluation is determined thusly: If, for a given 120-day period, SPY rose by 4%, SPXL would be expected to have risen by 12%. If, instead, SPXL only rose 9%, the underperformance would be said to be 3%. The average of all the 120-day periods since January 12 (~2100 periods) are averaged together to arrive at the -1.9% undervaluation.

Next up is the gold sector and the 1x ETF, the SPDR Gold Shares (GLD) and its 3x counterpart, the VelocityShares Gold ETN (UGLD), the second least volatile pair. Since 2012, UGLD has moved an average of +/-2.1% per day and, at 120 days, underperformed its GLD-predicted performance by an average of -2.9%.

Third on the list is the oil sector and the 1x ETF, the United States Oil Fund (USO) and the 3x VelocityShares Oil Bull ETN (UWT). Since 2012, UWT has moved an average of +/-4.8% per day and underperformed USO by -3.2% at 120 days. Of note, because UWT replaced the old UWTI this past winter, values for UWT going back to 2012 were derived via simulation and but should not differ significantly from actual values.

The second most volatile sector on the list belongs to natural gas. It includes the 1x United Natural Gas Fund (UNG) and the 3x VelocityShares Natural Gas Bull ETN (UGAZ). Since 2012, UGAZ moved an average of +/-5.4% per day and underperformed UNG by a sizable 10.2% at 120 days.

The single most volatile ETFs that I examined were the junior gold miners. The 1x VanEck Vector Junior Gold Miners ETF (GDXJ) was compared to the 3x Direxion Daily Junior Gold Miners ETF (JNUG). Since 2012, JNUG moved an average of +/-7.3% per day and underperformed GDXJ by a huge 16.0% at 120 days.

Figure 5 below plots the 30, 60, 90, and 120 day underperformances of each of these 5 3x ETFs versus their 1x underlying ETF.

**Figure 5: Percent underperformance of 5 ETFs at 30, 60, 90, and 120 day intervals** [Source: YahooFinance Historical Data]

Unsurprisingly, JNUG, the most volatile ETF, underperformed the most while SPXL, the least volatile of the five, underperformed the least.

Figure 6 below plots the 120 day underperformance of each of the 5 3x ETFs versus its average daily movement on a scatterplot.

**Figure 6: Comparison of underperformance at 120 days of 5 ETFs versus each ETF's average daily volatility showing a linear relationship.** [Source: YahooFinance Historical Data]

The plot shows a substantial linearity supporting the idea that daily price movement is directly related to underperformance. Only UWT fell slightly off-line.

We can now use the data from this set of 5 ETFs that encompasses the range of leveraged ETF volatility to construct a model predicting the expected underperformance of all the other 3x ETFs. While each individual 1x ETF and 3x ETF pair could be analyzed individually, this would take an enormous amount of time and it certainly would be easier if a simple-to-calculate variable could be plugged into an equation that outputs expected underperformance.

This is a three-step process. First, we can use the data from Figure 6 to calculate the predicted 120-day underperformance of a given 3x ETF based on its average daily volatility. We then take this predicted 120-day underperformance and use the data from Figure 5 to generate a curve of projected underperformance versus time. Finally, we plug in a target level of underperformance--which I arbitrarily set at 5%--to determine the length of time required to reach this underperformance. I used the 5% underperformance threshold as a reasonable loss that an investor would be willing to take before the trade stops becoming worthwhile.

The raw equation derived from these steps is:

**Raw Equation 1:**

**# Days Until 5% Underperformance = [-0.05 x {(-31589.4 x [(Average Daily Volatility x -2.30)+0.0291])-5129.37}]+6.02**

Simplifying this equation leads to the much more manageable.

**Final Equation 1:**

**# Days Until 5% Underperformance = 308 - (Average Daily Volatility x 3,633)**

We can check the accuracy of this equation by using it to calculate the 5% underperformance time of our known 5 3x ETFs. This results in an average error of 5.3 days, which is reasonable.

With this equation we can calculate the projected time to a 5% undervaluation of any 3x ETF. Figure 7 below lists the time to a 5% underperformance for the top 15 3x ETFs by market cap (excluding inverse ETFs or repeat ETF products launched by different companies). While I use the average daily volatility since 2012 to be consistent, you could really use any timeframe, as little as a week or even a day.

**Figure 6: Number of days that it takes 15 of the top 3X ETFs to underperform by 5% based on Equation 1** [Source: YahooFinance Historical Data]

A trader can hold the majority of these ETFs including TQQQ, FAS, TNA, SPXL, ERX, SOXL, TECL, USLV, EDC, and YINN for 150-250 days before suffering a 5% underperformance although a few, like NUGT, JNUG, UGAZ, UWT, and LABU are more volatile and suffer a 5% underperformance in less than 130 days and, in the case of JNUG, in as few as 43 days.

As mentioned above, the 5% underperformance was rather arbitrary and represented this author's opinion of what constituted an unacceptable loss. Others might have a tighter or looser threshold. Equation 2 below includes a second variable where you can enter your own underperformance threshold.

**Equation 2:**

**Days Until Specified Underperformance = {[6049 - (72,655 x Average Daily Volatility)] x Underperformance Threshold}+6**

For example, if you wanted to see how long it would take on average for JNUG to lose 8%, you would plug in -0.08 (a percent loss target converted to decimal form) to the "Underperformance Threshold" variable and 0.073, the ETF's average daily volatility, to that variable. The result would be a mere 176 days.

Finally, this equation can be reworked to predict underperformance per a given period of days. This equation is:

**Equation 3:**

**% Underperformance = (# Days ETF Held - 6)****/[6049 - (72655 x Average Daily Volatility)]**

For example, if you were planning to hold onto UGAZ for 2 months--around 45 days--and were wondering how badly this ETF would be expected to underperform during that time, you would plug in 45 to "# Days ETF Held" and 0.054 as the ETF's calculated Average Daily Volatility. The answer would come out to a modest -1.9% underperformance.

Future analysis may also take into account the long term movement of the 3x ETF. For example, a 5% underperformance might be a very big deal for an ETF like SPXL which might be expected to only move 10%-15% in a year. However, such an underperformance might be less of a concern for JNUG which can easily move over 100% over the course of a year and this loss might be worth it for the added volatility.

In conclusion, leveraged ETFs that underperform the most are those that tend to trade laterally with frequent spikes and pullbacks and those with a consistently large day-to-day volatility. There is a linear relationship between a leveraged ETF's day-to-day volatility and its underperformance versus that predicted by its 1x counterpart. Using 5 3x ETFs that include both low-volatility and high-volatility products, a simple model was created to calculate the expected underperformance of a given ETF based on its day-to-day volatility. I hope that this model proves to be useful for ETF traders and serves as a guide to how long an ETF can safely be held without comprising profits.

It is, of course, an oversimplified model. 3x ETF leverage-induced decay arises due to more than just average daily variation. For example, the propensity of a given index, commodity or sector to trade linearly versus laterally and the period of time over which spikes and pullbacks occur also influences the magnitude of its underperformance. However, accounting for these factors would increase the complexity of this discussion enormously and this article is intended to be a quick guide to an investor considering buying a 3x ETF, not to force them to relive their darkest days of undergraduate linear algebra.

**Disclosure:** I am/we are short UGAZ. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.