Archive for June, 2009

Leveraged ETFs and Compounding

June 25th, 2009 Kevin Comments off

There has been plenty of coverage in the past few months regarding leveraged ETFs and the concept of decay when these ETFs are held longer than a day. Besides increased risk, decay is certainly one of the downsides of leveraged ETFs. However, there is an equal upside to leveraged ETFs that has not had much coverage which is the topic of compounding during trends. Before compounding is explained, it can be helpful to revisit some fundamentals in order to have a better understanding of how leveraged ETFs may behave in various scenarios.

Volatility vs Trend

When a stock or ETF is moving in the same direction day after day, it is said to be in a trend.


Conversely, when a stock or ETF is moving in opposite directions, it is said to be volatile.


Since in the real world stocks and ETFs have elements of both, they can look like the following.


And since some have a low volatility to trend ratio, their chart may look smoother.


While others have a high volatility to trend ratio (like the financials during the 2008-2009 credit crisis).


Magnification of Characteristics

Leveraged ETFs track an underlying instrument (typically a stock index). When the underlying index is trending, the leveraged ETF will amplify the trend. When the underlying index is volatile the leveraged ETF will amplify this volatility. The problem of decay arises from this amplification of volatility. Decay occurs due to very simple math and has nothing to do with how leveraged ETFs are run or managed. On the other hand when an index is trending, a leveraged ETF amplifies the trend by a multiple (such as 3x) and also by compounding.

  • Leveraged ETF + Volatility = Decay
  • Leveraged ETF + Trend = Increased Compounding

Compounding Explained

Compounding affects both non leveraged and leveraged stocks and ETFs. Consider this simple example.


When the ETF is at $10 a 10% gain results in an increase of $1. Each time the ETF goes up, a 10% gain results in a larger and larger dollar increase. By the time the ETF reaches $20, a 10% increase ends up being $2 which is double the $1 from the start of the trend.

The following is a chart that shows a trend of daily 3% gains.


A chart of the dollar amount gained each day shows days at the end of the trend to have higher dollar increases than days at the beginning of the trend. This is essentially the concept compounding.


Leveraged Compounding

Since leveraged ETFs multiply daily % increases by a factor such as 3, the effects of compounding are significantly increased. Rather than an ETF increasing by 3% each day, the 3x leveraged ETF increases by 9%. This alone causes a significant increase, but during a trend, each percentage gain becomes increasingly more significant as the trend continues.

Here is a chart showing an index increasing 3% each day and a 3x ETF increasing 9% each day.


Not only does a leveraged ETF increase faster than the index, but a chart of the daily dollar increases shows the leveraged ETF’s rate of daily gains increasing, which is from the amplified compounding.


If the previous descriptions and charts are not sufficient for understanding leverage ETFs and compounding, another way to visualize the increased compounding is to view a comparison between the underlying index, 3:1 leverage using a margin account, and a 3x ETF.


The 3:1 margin account starts and ends at 3 times the value of the underlying index. Here is a chart showing the profit.


The 3:1 margin will grow at exactly 3 times the rate of the underlying index since its compounding increases at the same rate as the index. However, the 3x leveraged ETF will grow at a much faster rate due to the amplified compounding. This is a huge advantage of leveraged ETFs. If you can guess the trend correctly, there is potential for significant profits.

Bulls, Bears, and Trends

Bull and Bear leveraged ETFs have the same compounding characteristics, but since not all trends are the same, these two opposite ETFs have some interesting behavioral differences during trends. Two types of trends will be considered for demonstration purposes.

  • +50% trend (1% increments)
  • -50% trend (-1% increments)

Guessing the trend correctly means choosing a bull in the positive trend and choosing a bear in the negative trend.

  • Correct Guess
  • o Bull 3x ETF with +50% trend
  • o Bear 3x ETF with -50% trend
  • Incorrect Guess
  • o Bull 3x ETF with -50% trend
  • o Bear 3x ETF with +50% trend


It is important to realize is that it takes fewer days to move 50% in the positive direction at 1% per day than it does to go 50% down in a negative trend. This is because compounding occurs in the positive direction making each +1% gain bigger and bigger, and the opposite occurs on the way down. There is ‘reduced exposure’ in a negative trend therefore each -1% change has less effect. All in all, it takes 69 days to go -50% versus only 41 days to go +50%, when +/- 1% daily increments are used.


This has very important implications because it seems bears will have a huge advantage during downward trends than bulls during a positive trend. Charts and data will indicate whether this holds true.

Bulls in a Bull Trend


In the 50% positive trend, the index gains 50% as expected, the 3:1 margin account gains 150%, and the 3x ETF gains almost 250% due to compounding.

Bears in a Bear Trend


Shorting an index that drops 50% results in a 100% profit (the index was sold at twice the price it was bought). Using 3:1 margin to short the index results in a 300% profit, and the 3x bear ETF has a massive 668% profit.

Even if the trend is over a shorter time period (such as -3% daily changes) the 3x bear profits over 620%. It is not the number of negative days that makes the bear so profitable; it is the ‘opposite of compounding’ effect.


Bulls in a Bear Trend

If the trend is guessed incorrectly, obviously significant losses will occur.


The 3:1 margin account gets crushed by losing 150%, which is more than the value of the index. Even though leveraged ETFs can move significantly in either direction, one advantage they have over margin accounts is that the most they can lose is 100% (the index goes to 0). With a 3:1 margin account, an index or stock that drops to 0 results in a 300% loss. In the chart above, the bull lost 88%.

Bears in a Bull Trend


The bear 3x ETF loses 71% which is less than the bull’s loss of 88%.

Comparing Gains and Losses

The final numbers are as follows.

  • Correct Guess
  • o Bull 3x ETF with +50% trend [+236%]
  • o Bear 3x ETF with -50% trend [+668%]
  • Incorrect Guess
  • o Bull 3x ETF with -50% trend [-88%]
  • o Bear 3x ETF with +50% trend [-71%]


When it comes to profitability during trends the bears win hands down. Not only do the bears gain more during correctly guess trends, they lose less during incorrectly guessed trends. From the numbers presented it seems like there is little reason to use the bull 3x ETFs. In reality this is not the case. Even though leveraged bear ETFs have the upper hand when it comes to trends, they are severely affected by decay, much worse than the bulls. Future articles will cover the advantages and disadvantages of the bull and bear leveraged ETFs.

Categories: Compounding Tags:

Leveraged ETFs – Links and Resources

June 7th, 2009 Kevin 2 comments

Leveraged ETFs are a relatively new instrument in the financial world. Although 2x ETFs have been around for several years now, the 3x ETFs by Direxion have really sparked people’s interest in using these leveraged ETFs as trading vehicles. Unfortunately, many mistake them as long term investment vehicles. While it is possible to buy and hold these leveraged ETFs for the long run, it is important for such investors to know the fundamental risks involved with such a strategy.

This page (Leveraged ETFs – Link and Resources) is an attempt to become a central resource for links to information regarding leveraged ETFs. As new information and articles are made available, those links will be added to the page.

List of Leveraged ETFs

Leveraged ETF List – A single list of most leveraged ETFs
Direxion ETFs – Popular 3x ETFs
ProShares ETFs – Popular 1x and 2x ETFs
Rydex ETFs – More 2x ETFs
PowerShares ETNs – 1x and 2x ETNs

Essential Links

Leveraged Decay – A simple introduction to the risks of leveraged ETFs that uses charts and images to describe the long term effects of leveraged decay. If you intend to hold on to leveraged ETFs for longer than a day, this is a must read article (I am biased, I wrote it).
Trend, Volatility, and Returns – Another 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,, and
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.

Research Documents on Leveraged ETFs

The Dynamics of Leveraged and Inverse Exchange-Traded Funds
Path-dependence of Leveraged ETF returns
Long Term Performance of Leveraged ETFs

Popular Papers on Leveraged ETFs

Leveraged ETFs vs Futures
Warning: Leveraged and Inverse ETFs Kill Portfolios
Understanding Leveraged ETFs
Under the Hood of Leveraged ETFs

Other Articles

How Long Can You Hold Leveraged ETFs?
UltraShort ETFs – They work if you follow the rules
The Truth About 3x ETFs and Long Term investing
What Happens When You Hold Leveraged ETFs for More than One Day?
Leveraged and Short ETFs – 3 Flaws You Should Know
Explaining Inverse and Leveraged ETFs
FAZ 3x Financial Bear Fund Crushed

Categories: Uncategorized Tags:

Historical Daily Volatility

June 2nd, 2009 Kevin No comments

It can be interesting to see the historical day to day change percent of ETFs. It is no suprise that financials have been the most volatile in the past year. Here is a chart that shows a 50 day SMA of day to day percent moves for several ETFs:

(click for a large image)

The volatility calculation is just a comparison between each day. So if XLF was at 10 and the next day it moves to 11, that is a gain of 10%. If the next day it is still 11 then the change is 0%. Using a 50 day SMA makes the plot more smooth.

One of the interesting similarities between each ETF is that the drop in the volatility seems to ‘confirm’ the 2003 bull market. This may be obvious in hindsight, but perhaps the still heightened daily % changes we are still experiencing is an indication we are not yet out of the woods. The market became very calm at the end of 2003, even at its low levels.

The XLE chart is a great example:

Charts for each individual ETF can be viewed here:


Categories: Uncategorized 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: