## Measuring Leveraged ETF Decay

The past year has been an extremely volatile period for the market. During this time leveraged ETFs have grown in popularity and there have been hundreds of articles covering how volatility causes these leveraged ETFs to decay. Some are scared of the concept of decay and thus walk away. Some *think* they understand decay and believe it will cause all leveraged ETFs to go to zero (which is incorrect). And then there are those that understand decay, but they have no simple way to measure how much their positions are losing from decay or gaining from compounding.

**Leveraged ETF Swing Trading**

Any trader or investor can buy and hold a long term position in a leveraged ETF just like someone can buy a sports car and drive it 150 mph into a wall. I am not advocating either one of these, but the point is that should someone choose to swing trade leveraged ETFs, it is important to understand the risks and rewards of the tool (leveraged ETFs) for the job (swing trading). Here are a few quick points.

** Risks
**-Day to day volatility can cause losses from decay.

-Leveraged compounding during downtrends can result in significant losses.

-All other trading/investing risks apply.

** Rewards
**-Leveraged compounding during uptrends can result in gains that exceed the ETF’s daily objectives.

**Inventing a Conceptual Framework for Decay Estimations**

There have been articles that include decay equations (see here or here), but this math probably serves little practical use to most swing traders. Rather than present complex math equations for calculating decay, I propose an alternate conceptual framework for *estimating* decay. Coupled with software to automatically chart calculations, this framework will make it easier for swing traders to understand how volatility is affecting their trades. I suggest reading my two previous articles on decay and compounding for some background before trying to understand this decay estimation.

Of all the articles that cover the concept of decay, I do not see any that describe a reference point for which to calculate decay. To measure decay we need to calculate the difference between a reference point and the decayed point. Identifying the decayed point is easy: it is just a point from the data of the actual leveraged ETF. But what will serve as the reference point so we can calculate the difference?

**The Optimum Path**

I propose using a leveraged ETF’s optimum path between two separate days as the reference point for decay calculations. The optimum path can be thought of as essentially how a leveraged ETF would have performed if there was no volatility. It makes sense to calculate decay by measuring the difference between data with volatility and data without volatility. Consider the following data as an example. Over the course of 10 days a tracking index moves down 6% one day then up 10% the next.

Index: +18.2%

3x ETF: +31.3%

This hypothetical 3x ETF that tracks the index is properly tracking on a daily basis, but due to volatility it does not track it at 3x over the course of multiple days. Instead, for this data over 10 days the 3x ETF performed at roughly 1.7x. This is nothing new and many articles have covered how leveraged ETFs only track on a daily basis and not over longer periods. However, in order for us to estimate the decay amount we need to see how the optimum path for the leveraged ETF would have performed. Remember, the optimum path is when there is no volatility.

Index: +18.2%

3x ETF: +63.8%

Without the volatility in the underlying index, the hypothetical 3x ETF would have had a return of 63.8% which is 3.5x the underlying index. This is the optimum path and is the best case scenario. The extra 0.5x is due to the extra compounding.

We can now calculate the decay by comparing the optimum path with the path that includes volatility.

**Calculating the Optimum Path**

Removing the volatility between two points is quite simple. If the start value (*PV*), end value (*FV*), and the number of periods (*n*) between the two points is known, then the daily change percent (*i*) needed to go from start to end without volatility can be calculated using the following equation:

(image and equation from Wikipedia)

Once the daily change percent is known (*i*), it is just a matter of applying that daily change percent for the desired number of periods and the optimum path will be found. No matter how much volatility exists between two points over a given number of periods, the optimum path will always been the same. The amount of decay, however, will depend on the volatility between those two points.

**Calculating the Decay**

As stated previously, the decay estimation is just the difference between the optimum path (no volatility) and the actual path with volatility. However, it is important to stress that each day requires a new optimum path calculation in order to calculate the decay up to that day. The Decay Estimation chart above only shows the estimated decay for the period from day 1 to day 11. The decay for any of the previous timeframes in that chart is not known unless the optimum path is recalculated specifically for that day. Decay is something that happens over multiple periods (days) and hence a decay value is a representation of a period’s decay. There is no decay for just a single day.

**Bad News Bears**

In my previous article on compounding I showed the advantages bears have over bulls during trends. Unfortunately, the bears are at a disadvantage compared to bulls when it comes to volatility. Bears have the same amount of decay due to volatility, but they are also affected by a different kind of decay. I am not aware of any official name for this decay so I call it Inversion Decay.

**Inversion Decay**

A simple mathematic principle is that for any percentage loss %L, a higher percent gain %G is required to get back to the original value. For example, a 20% drop from 10 results in 8. To get back to 10 a 25% gain is required. No matter how you slice it or dice it, over any period of data, between any two equal points there is more positive gain percentages than loss percentages. I am truly fascinated by this fact, but unfortunately it means bad news for the bear ETFs. Since bear ETFs move at opposite the bull, this means between any two equal points in the market, bears are going to have more loss percentages than gain percentages. This is covered in more detail in my previous article on decay. In summary, leveraged bear ETFs are not only affected by volatility decay, but also inversion decay. Therefore in a volatile horizontal market, a leveraged bear will have worse long term performance than its bull counterpart.

**Silver Lining**

The good news is that just because volatility is causing leveraged ETFs to decay, if there is enough positive trend, it can offset and outgain the losses from decay. You can think of trends and decay as two forces. An uptrend is a positive force that fights against the negative force of decay.

**Real-world Examples with QLeverageSim**

*Warning: These examples are simulations and estimations and do not include losses due to fees, transaction costs, taxes, other factors.*

Using QLeverageSim we can calculate decay estimations for the purpose of understanding how much volatility is affecting an ETF. Again, the decay is not the actual ETFs performance, but rather a loss calculation from a value had the ETF had no volatility.

Using TNA (300% daily tracking of the Russell 2000) as an example, we can show a chart of TNA versus each day’s optimum path (had there been no volatility).

From Jan 1, 2009 to July 1, 2009 TNA went from 34.09 to 29.17 which is a 14.43% loss. The decay estimation is 23.49%, because had there been no volatility in the underlying index during this period, TNA would have been 38.12.

Even with decay it is possible to have incredible returns with leveraged ETFs, as a chart of FAS (tracks Russell 1000 financial services index) from March 9, 2009 to May 8, 2009 demonstrates.

Even though significant volatility during this period resulted in an estimated 29.52% decay, the uptrend was so strong that FAS gained 375% versus the underlying index’s 90% gain. That’s a factor of 4.16x. If there was no volatility the gain would have been much more.

**When Knowing the Decay is Handy**

When an underlying index of a leveraged ETF has a period where the end point is the same as the start point, the decay amount ends up matching the actual loss of the leveraged ETF. Between February 9, 2009 and May 28, 2009, the Russell 1000 energy index was approximately flat. But a 3x leveraged ETF that tracks this index (ERX) was down by almost 15%. The QLeverageSim decay calculation shows the decay estimation matching the loss from the leveraged ETF (-15%).

This is useful because the decay estimation can be interpreted as knowledge that if an underlying index were to revert back to its original value, the decay estimation will be roughly how much the leverage ETF will have lost (only applies to bull ETFs).

**Simulating Leveraged ETFs**

QLeverageSim is a free utility that allows users to choose a non-leveraged ETF (such as SPY) and simulate a bullish or bearish 2x or 3x hypothetical ETF for a given period. It also provides a decay percentage estimation for identifying how volatility affects leveraged ETFs.

**Summary**

The daily tracking of leveraged ETFs causes interesting behaviors for periods longer than a day. If a trader decides to swing trade leveraged ETFs it is important for them to understand how trends, volatility, and holding periods affect returns. This article has shown that volatility has a significant negative effect on leveraged ETFs, but that trends can still cause leveraged ETFs to outperform their daily objectives. The technique provided in this article for measuring decay is just a calculation between the hypothetical value of an ETF with no day to day volatility and the actual ETF that includes volatility. It is not a demonstration that leveraged ETFs are flawed. In fact, they meet their objectives very well: daily leveraged tracking of an index.