Archive for the ‘Decay’ Category

Decay Can Affect Cyclists Too

July 29th, 2009 Kevin Comments off

There are plenty of articles on the topic of how decay affects leveraged and inverse ETFs. Just recently there has been news of brokers putting them under review or even not allowing their clients to use them. Critics of leveraged ETFs often claim that they are flawed and they like to point out the decay that has occurred in FAS and FAZ since their inception. They claim leveraged ETFs are a one way ticket to zero (or reverse splits) and they love to mention how the funds have to buy high and sell low to achieve their goals.

I agree that leveraged ETFs may be highly misunderstood and extremely dangerous, but they actually achieve their stated goals (daily leveraged percent change tracking) probably better than most people think. It does not matter that the ETF internals need to buy high and sell low. What matters is that they should work as advertised, and that traders or investors should understand how they work over their respective timeframe before they decide to use them.

Meet Marcus and Larry

In order to illustrate how leveraged ETFs decay for a reason and why it is not a ‘flaw,’ I will demonstrate an alternate scenario outside of the world of finance that also experiences decay. To keep things as simple as possible it will be explained using a 5th grader’ish storyline.

Imagine three bicyclists that are training for a long race. Their names are Igor Index, Marcus Margin, and Larry Leverage. Igor is a veteran and has lots of experience. His training regimen is very strict. He alternates days where he rides a long distance and then a shorter distance.  More specifically, on Monday he rides 100 kilometers, Tuesday he rides 80 km, Wednesday back to 100 km, and so on. Though his short rides are only 20 km less, Igor has found this training schedule has helped him become one of the best cyclists in the world.

Two up and coming young cyclists named Marcus Margin and Larry Leverage idolize Igor and they have been looking for a training schedule that will put them in better shape for competitions. Marcus has studied Igor’s training technique and decided he will do twice the change value of Igor. Rather than do 20 km less on Tuesday and Thursday, Marcus decides to double it and do 40 km less. Larry Leverage has decided to do things a bit differently. Larry wants to double the percent change that Igor does each day. Using this training plan, we can see what each cyclist will do on Tuesday.

It is clear that both Marcus and Larry will be riding less on Tuesday, both by the same amount. However when Wednesday comes around, Igor increases his riding by 20km (25% of the 80 km), Marcus doubles Igor’s 20 km increase for a 40 km increase, and Larry decides to do double Igor’s percent increase for a 50% increase. Since Larry rode 60 km on Tuesday, an extra 50% puts him at 90 km. What he does not realize is that just tracking Igor’s percent change will cause his training to have some weird effects if he continues over multiple days. A chart shows what happens to the Wednesday bike ride.

Over the course of two days Larry’s long ride has become 90 km instead of 100 km all because he decided to track Igor’s changes by percent instead of value. If Larry keeps this up, after one month he will have a long ride of 35 km instead of Igor’s 100 km long ride. This is proof that decay in leverage ETFs does not happen due to buying high and selling low, it is from them working as they should: daily leveraged percent change tracking.

Decay is not a ‘flaw’

Even if the leveraged ETF fund managers had nightly parties where they take turn shoveling money into a furnace, as long as the funds meet their stated goals, it seems irrelevant as to how they achieve them.

To those that think leveraged ETFs are flawed:

  • Decay is by design.
  • Trends can offset the decay.
  • Decay of leveraged ETFs tracking the S&P 500 is roughly 20 times less during normal volatility.
  • Decay of leveraged ETFs tracking the financials is roughly 40 times less during normal volatility.
  • Any instrument that has goals similar to leveraged ETFs is also affected by decay.

The only flaw is when someone expects these instruments to perform 2x or 3x over long time periods. Just like it would be flawed to think a car with a V12 engine is going to use the same amount of gas as a V4 engine. In all fairness, a V12 versus V4 engine gas usage comparison is probably more obvious than understanding why leveraged ETFs are affected by decay over long timeframes. Hence, I can understand people’s frustrations and confusion. That is why this blog attempts to spread information about how leveraged ETFs work; so that traders or investors can be better informed for the decisions they make.

We avoided a disaster

I am not opposed to people that think leveraged ETFs should not be used. That is their opinion and they are entitled to it. It is actually good that leveraged ETFs have become more popular during the past year instead of before the 2008 crash. The abnormally high volatility has amplified the effects of decay and significantly increased awareness of how these ETFs work. Had leveraged ETFs become popular 5 years ago during the bull market, investors probably would have piled into them long term, not prepared for the financial tsunami that was about to hit. Long term investors probably would have been blind-sided by the compounding effects of the market going down coupled with the negative effects of decay during the once in a lifetime volatility. Now that market volatility has subsided, the leveraged ETFs are becoming much less affected by decay. Future articles will go more in depth on just how well leveraged ETFs track, and the kind of decay we should expect to see in the future at normal levels of volatility.

Categories: Decay Tags:

Measuring Leveraged ETF Decay

July 12th, 2009 Kevin Comments off

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.

-Day to day volatility can cause losses from decay.
-Leveraged compounding during downtrends can result in significant losses.
-All other trading/investing risks apply.

-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.



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.

Categories: Decay Tags:

Leveraged Decay

June 1st, 2009 Kevin 29 comments


Welcome to the first of several articles exploring the ups and downs, ins and outs, of leveraged ETFs. These articles will attempt to visually explain the dynamics of leverage as well as go deep into exploring the benefits and pitfalls of extreme volatility that accompany 2x and 3x ETFs. As a software engineer by day, I lack the investment and trading experience of many who do it for a living or on a daily basis. However, the articles and comments that I read online make it clear there are only a rare few that really understand how leveraged ETFs work for periods longer than a day. All that is required is an understanding of some very simple math. To explore long term leveraging effects, custom software was written measure leverage in a variety of scenarios. Questions such as the following will be addressed on this blog:

  • What is leveraged decay?
  • How much do leveraged ETFs lose due to decay?
  • Are leveraged ETFs going to 0?
  • What are the compounding effects of leverage?
  • If I hold on to my leveraged bull ETF, will it eventually ride out the volatility in a bull market?
  • What are the differences between using margin and a leveraged vehicle?

Many investors put money into leveraged ETFs without understanding the risks involved. Aside from the risk of potentially losing money at 2 or 3 times an underlying index, there is also a risk of losing money due to volatility if these ETFs are held for longer than a day. This loss is often called “Leveraged Decay” or “Volatility Decay.”

A Simple Explanation

Consider both a 1x and 3x ETF that have a starting price of $100 per share.

If the 1x ETF were to drop 10% on a particular day, the 3x would drop 30%. That puts the 1x ETF at $90 and the 3x at $70. Then on the next day the 1x ETF increases $10 back to its original value of $100, which is an 11.11% gain (10/90 is 11.11%). The 3x ETF will gain 33.33% of its $70 which is an increase of roughly $23. This puts the 3x ETF at a value of $93, for a loss of $7! Repeat this process again and again and over time the 3x ETF will continue to lose value while the underlying index always returns to $100. Here is a chart that shows this process happening 15 times (30 days).

As you can see, the 1x ETF is still at $100 while the 3x ETF has decayed 65% to end up at $35. A 10% move on a daily basis is not typical market activity, but it does help illustrate how significant decay can be. To demonstrate something more realistic, we can use the average SPY daily percent change from 1993 to 2008 which is 0.8%.

At such low volatility of 0.8%, decay has much less effect than when daily volatility is 10%. The 3x ETF above lost only 0.59% over 30 trading days. Here is a chart showing a full year of 0.8% volatility.

For 250 trading days with 0.8% volatility, the 3x ETF would lose around 5%. This doesn’t seem like the doom and gloom many blogs and articles have predicted for leveraged ETFs. The reason is because 0.8% daily volatility is far from the current average in 2009. A 30-day SMA of daily volatility for various 1x indexes will give us an idea of the volatility we have seen in the past year:

Back in November the indexes were at extremely high volatility levels, averaging between 4% and 6% daily moves. There were several days that reached higher than 10% resulting in +/-30% for the 3x ETFs. Such large daily moves can cause significant decay over time. An important point is that the amount of decay that occurs is not linearly proportional to the daily percent changes. In other words, 2% daily percent changes of an index does not result in twice the decay of 1% daily moves. Instead, it is significantly more. The following chart demonstrates the amount of leveraged decay for a hypothetical 3x ETF over 30 days for various underlying index daily move percentages (the ‘down’ day is not shown in order to make a smooth plot).

1x ETF Daily Change % 3x ETF Loss % due to Decay
1% 0.91%
2% 3.61%
3% 8.03%
4% 13.99%
5% 21.24%
6% 29.44%

It is evident from the table that the loss from decay grows significantly as the daily percent change increases. You will not see 6 times the decay comparing 6% daily moves to 1% daily moves; you will see over 29 times the decay. The bottom line: expect more much more decay from higher daily percent moves. This is a large part of why FAS and FAZ have dropped so much since their inception.

A Closer Look at Decay

Some key points to understand are:

  1. Leveraged ETFs move at a multiple of their underlying index on a daily timeframe.
  2. It is volatility on a daily timeframe (not intraday volatility) that causes decay.
  3. There is no absolute measure for loss due to volatility; a trend significantly affects the loss or gain.
  4. Decay is the result of simple percentage math and is not specific to leveraged ETFs.
  5. Between any two equal points, there is more positive percent gain than negative percent loss.
  6. As a drop % increases, its difference between the gain % needed to reach the original price increases.

Point number 5 says that no matter how many days it takes for an ETF to go down by a percentage, the total percentage to get back up to the original value will always be greater than the percentage down. As proof, from any two equal data points in a graph we can compute the sum of all percent gains and losses, and the gains will always be greater than the losses. Here is a chart of the SPY from the start of 2000 until May 2009 that includes such a calculation.

The light blue line (which is not important and just used as reference) is the % change since the start of 2000. The green data points represent the aforementioned calculation and show that between two equal price points in the SPY, the sum of the percent gains are more than the sum of the percent losses.

Point #6 can be confusing, and basically means that as the magnitude of a percent drop increases, in order to reach the original price, the difference between the drop percent and the gain percent increases. Better explained by an example, consider a 10% drop from $100, which results in $90. To get back to $100, an increase of 11.11% is needed. The difference between these percentages is 1.11%. Now consider a more significant drop of 30% from $100 which results in $70. To get back to $100 a gain of 42% is needed. The difference between 30% and 42% is a much larger 12%.

Drop Gain to Recover Difference
1% 1.0101% 0.0101%
3% 3.0928% 0.0928%
10% 11.111% 1.111%
30% 42.86% 12.86%
(3x ETF)
(3x the 1x ETF)
(9.5% missing!)

For a non-leveraged ETF, this does not mean much. But for a leveraged ETF, the fundamental problem of leveraged decay arises from the fact that as daily percent changes are increased due to leverage, the gain percentages are not enough to make up from the increased loss percentages. The example above proves this point since a 3x ETF drops 30% for an index that drops 10%. When the underlying index gains 11.111% to reach its original value the 3x ETF will increase 33.333% which is not enough for the 3x ETF to reach its original price point.

Doom and Gloom: Bearish ETFs

While there are some positive aspects of bearish ETFs which will be covered in future posts, the math is unfortunately set up against them.  As we have seen previously, a repeated pattern of a 1x ETF of -10% and +11.11% results in the 3x bull ETF moving -30% and +33.33%. The bear does the opposite which is +30% and -33.33%.

Bull 3x: -30%, +33.33% (+3.33 extra percentage points)
Bear 3x: +30%, -33.33% (-3.33 extra percentage points)

Leveraged bear ETFs not only suffer from decay, but they also suffer from point #5 which means in a horizontal market the bear ETFs are always moving down by greater percentages than the percentages they move up. If the market turns into a bull market, or even moves sideways, bear ETFs are a one way ticket to zero. Only a market that continually moves downwards can sustain leveraged bear ETFs. This has significant implications for long term holders. Anyone that has a long term hedge position in a leveraged bear ETF has both decay and loss percentages working against them. A chart of 6% daily moves for a 1x index over 30 days (15 oscillations of -6% then +6.4% for the 1x ETF, resulting in -18% then +19.2% for the 3x bull ETF and +18% then -19.2% for the 3x bear ETF) shows the bear ETF getting destroyed compared to the bull:

Hypothetical Long Term 3x ETF performance

To get an idea of how 3x bull and bear ETF might perform (excluding fees and other costs), this is a chart showing IWM (Russell 2000) during the bear market of 2000 to 2003:

ETF % Gain
IWM -36%
IWM 3x Bull -84%
IWM 3x Bear +44%

For the bull market of 2003 to 2007:

ETF % Gain
IWM 149%
IWM 3x Bull 941%
IWM 3x Bear -97%

This is a good example that demonstrates that 3x bull ETFs are not going to 0, provided there is enough upward trend in the underlying index to offset loss due to decay.

Essential Links

Trend, Volatility, and Returns – A must read study that goes more in depth on leveraged decay, but also covers other aspects such as compounding. Written by Connors Research, (the research division of The Connors Group) which is part of a network of sites such as
Direxion Literature – A fantastic set of 5 articles describing the ins and outs and risks of leveraged ETFs.
Understanding ProShares’ Long Term Performance – A 2-page document that has a simple explanation of the relationship between compounding and volatility in leveraged ETFs.
On-Demand Webinar: Getting Leverage, Going Short – A must watch webinar that covers the history of Leveraged ETFs, an overview of how they work, and information on their tracking properties. Written by Matt Hougan, editor of The link includes a PDF of the slides if you do not want to watch the video.


This post covered the reasons for leveraged decay and the factors that influence the amount of loss due to decay. However, there are many other aspects to decay and leveraged ETFs that will be covered in future posts. Here is a peek at some topics that will be covered:

Categories: Decay Tags: