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MACD – Test Results

The MACD is one of the most widely used technical indicators in the world and is included in every charting program worth owning. Unfortunately however, reliable data on its performance is almost non-existent. Are the standard settings of 26, 12, and 9 the best? To reveal the answer we tested 2000 different combinations through 300 years of data across 16 different global markets~. Stand by for the results below…

Download A FREE Spreadsheet With Data, Charts

And Results For all 2,000 MACDs Tested

MACD Conclusion

Our Testing Strategy Explained

Because there are so many different possible settings for a MACD we started by testing a broad range with the hope this would reveal the areas to focus on more closely. To cast our testing range wide but strategically, we progressed in a liner fashion through the Fast Moving Averages (FC) and set the Slow Moving Averages (SC) as of multiple of the FC:

Fast Moving Averages (FC) = 10, 20, 30, 40, 50
Slow Moving Averages (SC) = 2 * FC, 3 * FC, 4 * FC, 5 * FC, 6 * FC

So each of the five FC settings were tested against five SC settings based on a multiple of the FC. e.g A SC of 50 would be tested against a FC of 100, 150, 200, 250, 300 as these are equal to 50 multiplied by 2, 3, 4, 5 and 6.

Each of these were tested against 10 different Signal Line settings:

Signal Line (SL) = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20

Trading Rules:

An entry signal to go Long for each MACD tested was generated when the MACD Line was above zero AND above the Signal Line. The position was closed when the MACD Line moved below zero OR below the Signal Line (vice versa when going short)^.

If what you have read so far does not make much sense, please read more about the MACD before continuing 🙂

MACD Test Sets – Broad

MACD, EOD Long - Annualized Return During Exposure

Above you can see the annualized return during the time each MACD was exposed Long to the market. Clearly the Signal Line setting is far more influential than the ‘Fast’ and ‘Slow’ Moving Averages (MACD Line). To my surprise having the Signal Line as fast as just 2 days produced the the best results and even more surprising is that the trades produced are not prohibitively short (8 – 27 days on average from the table above, see spreadsheet for full stats).

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MACD Test Sets – Short Term Trading

MACD EOD, Long - Annualized Return During Exposure

After refining the tests down several times we uncovered some interesting findings. Firstly the most efficient returns came from a MACD with a ‘Fast’ Moving Average of 1, which isn’t actually a MACD at all (a Moving Average with a period of 1 is equal to the price itself). So the best results come from measuring the Convergence and Divergence between an MA and the price, with the addition of a Signal Line. What is really exciting however is this also works exceptionally well on the Short side of the market:

MACD EOD, Short - Annualized Return During Exposure

For me, when assessing a trading system, I am more interested in the return during exposure than the return overall. A system may have you exposed to the market for 30%, 70% or 99% of the time but the more time you are exposed to the market the more time your money is at risk. While my money is at risk, if it is not working hard, if I am not getting a high return, then I would rather sit in cash!

The Short side of the market is often not worth trading because decent returns during exposure are difficult to get from a mechanical system. What we see with the MACD however are returns during exposure then exceed even the best that the FRAMA could produce when Long…. Sooooo, what is the catch? Lets have a closer look:

MACD EOD 1, 56 Long and Short, Sig 2

The pink line on the chart above is the performance, taking signals both Long and Short from a MACD with a ‘Fast’ MA of 1 (price), a ‘Slow’ MA of 56 and a Signal Line of 2. I have included the results from the best FRAMA for comparison.

The impressiveness of this MACD can’t be doubted; consider the fact that it achieves these returns while only being exposed to the market 56% of the time and delivers a 42% probability of profit for each trade. But can you see the problems? With an average return of just 0.25% and an average trade duration of 4 days, a MACD with these settings is limited in its practical applications.

Firstly you would need near frictionless trading, such as that offered by some index mutual funds (e.g. Rydex, ProFunds, or Direxion)

Secondly you would need to gain exposure to several diverse equity index funds. Part of the reason for the success of this strategy is the fact that it spreads the risk across 16 different global markets, some of which performed better than others in our tests. In the real world frictionless trading is not accessible to such a variety of indices.

Thirdly between 2003 and 2007 while the Global Average was experiencing a very strong bull market the MACD underperformed quite significantly.

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MACD Test Sets – Practical Trades: Long

MACD EOD, Long - Annualized Return During Exposure

The table above is the result of a search for some more practical MACD settings. As you can see the best returns localise around the the 21/81 mark area:

MACD EOD 21, 81 Long, 2

Above we are looking at the performance of a MACD going Long with a ‘Fast’ Moving Average of 21, a ‘Slow’ Moving Average of 81 and a signal line of 2 compared to the best FRAMA (also notice the poor performance from the standard MACD of 12, 26, 9 – See Full Stats).

Now when comparing the 21, 81, 2 MACD to the FRAMA it must be taken into consideration that the MACD is only exposed to the market 35% of the time while the FRAMA is exposed 57% of the time. So a side by side, total return comparison is not really fair. What is good to see however is the consistency and stability from the MACD during market declines. What I don’t like though is the familiar under-performance during the strong bull market between 2003 and 2007.

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MACD Test Sets – Practical Trades Short

MACD EOD, Short - Annualized Return During Exposure
The Short side of the market behaves differently to the Long so it is not surprising to see that a more reactive MACD performs better, and the top returns were found around 16/97:

MACD EOD 16, 97 Short, 2

Above we are looking at the performance of a MACD going Short with a ‘Fast’ Moving Average of 16, a ‘Slow’ Moving Average of 97 and a signal line of 2 compared to the best FRAMA (also included is the standard MACD of 12, 26, 9 – See Full Stats).

The 16, 97, 2 MACD is quite exceptional, managing to match the returns from the FRAMA with 2/3 the market exposure and a higher probability of profit. These results are very exciting. It would appear as though the MACD’s true strength is in its ability to go Short.

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MACD Conclusion

I have been a big fan of the MACD for a long time and had high expectations for these test results. But reality has been harsh on the MACD and in many ways the Emperor has no clothes.

In an attempt to limit the length of this article we only published results from trades off the Signal Line when the MACD line was above zero (when Long) or below zero (when short). Please note however that trying the trade the MACD when it is on the wrong side of zero will lead to an unhappy bank account, an unhappy wife and an unhappy life.

As a tool for long term trading the MACD fails and can’t compete with its less evolved relative the Moving Average Crossover.

As a tool for short term trading (4 days on average) the MACD is very powerful in theory but with such a small average return the practical applications are limited.

As a tool for medium term trading the MACD should not be your first choice on the Long side of the market BUT on the Short side the MACD is simply outstanding! Using a ‘Fast’ Moving Average of 16, a ‘Slow’ Moving Average of 97 and a signal line of 2 you have a powerful indicator for taming the bear.

More in this series:

We have conducted and continue to conduct extensive tests on a variety of technical indicators. See how they perform and which reveal themselves as the best in the Technical Indicator Fight for Supremacy.

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~The data used for these tests is included in the results spreadsheet and more details about our methodology can be found here.
^ No interest was earned while in cash and no allowance has been made for transaction costs or slippage. Trades were tested using End Of Day (EOD) signals on Daily data. All Moving Averages (MA) in these tests were Exponential (EMA).

Oscillator Classification

Absolute Price Oscillator (APO) deals with moving averages of actual prices such as the MACD.
Percentage Price Oscillator (PPO) computes the difference between two moving averages on a normalized basis by percentage. There are several different methods for this including:

Taking calculations on price percentage changes.
Taking the difference between two moving averages and dividing them by the longer moving average value.
Taking the difference between two moving averages and dividing them by the Average True Range (ATR)

All of the PPO methods produce the same signals however they allow you to compare securities of different prices or the same security during different time periods. Dividing by the ATR is particularly useful when comparing different asset classes or securities of vastly different volatility.

Detrended price oscillator (DPO), ignores long term trends while emphasizing short term patterns.

Golden Cross – Which is the best?

The Golden Cross typically referrers to the crossing of the 50 and 200 Day Simple Moving Averages. When the shorter term average moves above the longer term average this is seen by many as the beginning of a sustained bullish period and vise versa. It is not wise however to risk your money in the market on the assumption that such a theory is true.

One has to ask, which is better, a SMA Golden Cross or an EMA Golden Cross? Are the settings of 50 & 200 really the best? What is the profile of the trades that this strategy generates as far as duration, probability of profit, draw downs etc. In order to answer these questions we applied some brute mathematical force and tested 1750 different combinations through 300 years of data across 16 different global markets~. We have done the hard work and you get the benefits for free… aren’t you lucky.

Michael Stokes over at MarketSci has also written a great series on Trading The Golden Cross.

Download A FREE Spreadsheet With Data, Charts

And Results For all 1750 Moving Average Crossovers Tested

Golden Cross, Moving Average Crossover – Test Results:

Our Testing Strategy Explained
Simple vs Exponential Moving Average Crossover
Daily vs Weekly Moving Average Golden Cross
Golden Cross – Refined Test Sets
The True Golden Cross

Golden Cross Conclusion

Our Testing Strategy Explained

There are endless combinations of moving averages that we could test in search of the best. To cast our testing range wide but intelligently we have used progressions of a ratio; slow/fast MA:

Fast Moving Averages (FC) = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
Slow Moving Averages (SC) = 1.2 * FC, 1.4 * FC, 1.6 * FC, …….. 5.6 * FC, 5.8 * FC, 6 * FC

So each of the ten FC settings were tested against twenty five SC settings based on a multiple of the FC. e.g The traditional Golden Cross with a SC of 50 and a FC of 200 has a multiple of 4 (because 50 * 4 = 200). The tests against a FC of 50 had a multiple as low as 1.2… (50 * 1.2 = 60) and as high as 6… (50 * 6 = 300).

Hopefully by using this tactic we can identify the multiples or ratios that deserve more targeted testing.

Simple vs Exponential Moving Average Crossover

In our original MA test; Moving Averages – Simple vs. Exponential we revealed that the Exponential Moving Average (EMA) was superior to the Simple Moving Average (SMA). If the same proves to be true with the ‘Golden Moving Average Crossover’ then this will further validate the EMA as being of higher-caliber than the SMA.

Simple vs Exponential MA Crossover Returns

The chart above fades between the results from the EMA and the SMA crossover tests. As you can see the EMA outperforms the SMA by well over a percentage point on average. This unequivocally confirms that the EMA is superior to the SMA. Further more it should be noted that every single EMA combination tested (and most SMAs) outperformed the buy and hold annualized return of 6.32%^ during the test period (before allowing for transaction costs and slippage). But “technical analysis doesn’t work” they say.

Below are the all the individual Annualized returns from the EMA chart above:

EMA Crossover, EOD Long, Annualized Return During Exposure

The best returns came from an fast EMA of 10 days with a slow EMA of 50 (ratio of 5 because 10 * 5 = 50). Based on these results we will run more refined tests on fast moving averages in the range of 8 – 17 and slow moving averages 20 – 56.

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Daily vs Weekly Moving Average Golden Cross

What about Weekly data you ask? We didn’t test with Weekly data but we did test using End Of Week (EOW) signals on Daily data (previous tests on the EMA revealed that the two produce almost identical results). When we used EOW signals the returns dropped by 0.5% on average while the trade duration increased by just 10 days and the probability of profit increased by only 2%. In other words; you are better to use daily data and EOD signals on a Moving Average Crossover Strategy.

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Golden Cross – Refined Test Sets

After getting a better idea of the ‘sweet’ spot from our first round of testing, we refined our range and instead of continuing with ratios of fast and slow EMA crossovers, we progressed in a liner fashion. So what is the “True Golden Cross” that proved the best returns during our tests?

EMA Crossover, EOD Long, Annualized Return

There is a zone of dark green on the grid above but the very best from our tests, the True Golden Cross has a slow EMA of 48.5 and a fast EMA of 13. Reality is very different from the 50/200 SMA Golden Cross that someone made up once upon a time and that is why we must test everything.

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The True Golden Cross

The profile of the trades produced by the true Golden Cross have many very desirable features; a significant average trade duration (94 days), a high probability of profit (45%) and solid returns across the board (even on the difficult, bear savaged Nikkei 225). While it does not produce returns any where near as good as the best FRAMA, it certainly out performs the traditional Golden Cross of 50 / 200. Plus with the long trade duration, it may be more desirable than the slower FRAMA for use as a long term indicator as one part of a complete trading system:

252 Day FRAMA, EOW 40, 250 Long

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Golden Cross Conclusion

Moving average crossovers have proven themselves to be a powerful and effective form of technical analysis, however the so called “Golden Cross” of the 50 and 200 day SMA is far from the best. Our testing revealed that the EMA produces better results than the SMA and the best settings are that of a 13 / 48.5 EMA Crossover. The long duration of the trades produced, ability to sidestep bear markets and the high probability of profit make it worth testing as a major component in a complete trading system.

The moving average crossover is a component of the popular Moving Average Convergence Divergence (MACD), see the completed test results here.

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More in this series:

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~ An entry signal to go long for each crossover tested was generated when the faster moving average of each pair closed above the slower moving average (the opposite closed the position or triggered a signal to go short. No interest was earned while in cash and no allowance has been made for transaction costs or slippage. Trades were tested using End Of Day (EOD) and End Of Week (EOW) signals for Daily data. Eg. Daily data with an EOW signal means that only the signals at the end of each week were taken.
^ This was the average annualized return of the 16 markets during the testing period. The data used for these tests is included in the results spreadsheet and more details about our methodology can be found here.

Moving Average Convergence Divergence (MACD)

MACD stands for Moving Average Convergence Divergence and was first developed by Gerald Appel in the late 1970s. It is an Absolute Price Oscillator (APO) and can be used in an attempt to identify changes in market direction, strength and momentum.

It calculates the convergence and divergence between a ‘fast’ and a ‘slow’ Exponential Moving Average (EMA) known as the MACD Line. A signal EMA is then plotted over the MACD Line to show buy/sell opportunities. Appel specified the MA lengths as the following percentages:

Slow EMA = 7.5%   (25.67 period EMA)
Fast EMA   =   15% (12.33 period EMA)
Signal EMA =   20% (9 period EMA)

Usually however these are rounded to EMAs of 26, 12 and 9 respectively. Many charting packages will also plot the difference between the Signal Line and MACD Line as a Histogram.

One of the biggest challenges when dealing with financial data is noise or erratic movements that cause false signals. By smoothing data out you can reduce the number of false signals. But this comes at a cost, and causes an increase in the lag of your signals. The genius of the MACD is that it begins by smoothing data (thus causing lag) and then speeds up the signals from the smoothed data. This combination helps to reduce false signals while minimising the lag.

By comparing EMAs of different lengths the MACD can help to identify subtle changes in the trend and momentum of a security. It is a great visual representation of the acceleration or rate of change in a trend.

How to Calculate a MACD

MACD Formula:

MACD Line = EMA,12 – EMA,26
Signal Line = EMA[MACD,9]
MACD Histogram = MACD – Signal Line
Histogram Trigger = EMA[MACD Histo,5]

Obviously you can change the parameters to any value of your choice.

MACD Excel File

We have put together an Excel Spreadsheet that will automatically adjust to the MACD settings you desire. Find it at the following link near the bottom of the page under Downloads – Technical Indicators: Moving Average Convergence Divergence (MACD)

Test Results

Is the MACD an effective indicator? We are putting it into the ring for the Technical Indicator Fight for Supremacy. It will be tested through 300 years of data across 16 global markets to discover which settings produce the best results and how it performs compared to other indicators:

Moving Average Crossovers – Completed – Golden Cross – Which is the best?
Moving Average Convergence Divergence (MACD) – Completed – Results
ZeroLag MACD (ZL-MACD)
MACD Z Score (MAC-Z)

– Completed – Results

Log-Normal Adaptive Moving Average (LAMA)

The Log-Normal Moving Average (LAMA) is the name I have given to an Adaptive Moving Average that uses the adaptive process developed by John F Ehlers for use in his FRAMA. Stock prices are said to be Log-Normal so Ehlers used EXP to relate his Volatility Index (The Fractal Dimension) to Alpha. The LAMA is designed so that any VI can easily be incorporated as long as it shifts between a range of 1 – 0 where high readings indicate high volatility.

How to Calculate an Log-Normal Moving Average

Seed it with the Close price then after that the LAMA is calculated according to the following formula:

LAMA = LAMA(1) + New α * (Close – LAMA(1))

Where:

New α = 2 / (New N + 1)

New N = ((SC – FC) * ((N – 1) / (SC – 1))) + FC

SC = Your choice of a Slow moving average > FC

FC = Your choice of a Fast moving average < SC

N = (2 – α) / α

α = EXP(W * (1 – VI))

W = LN(2 / (SC + 1))

***If the above formula does not make a lot of sense to you and you would like a more in depth explanation then please read this article on the Fractal Adaptive Moving average.

Log-Normal Adaptive Moving Average Excel File

I have put together an Excel Spreadsheet containing the Log-Normal Adaptive Moving Average and made it available for FREE download. It contains a ‘basic’ version that shows all the working for the formula and a ‘fancy’ one that will automatically adjust to the length as well as the Volatility Index you specify. Find it at the following link near the bottom of the page under Downloads – Technical Indicators: Log-Normal Adaptive Moving Average (LAMA)

How EXP affects Alpha and the Smoothing Period:

Ehlers used EXP to relate the Volatility Index (VI) to Alpha (α) so lets have a look at what affect this has:

LNMA - EXP Affect on Alpha and SmoothingLNMA - EXP Affect on Alpha and Smoothing .

In the top chart you can see Alpha taken directly from the the Fractal Dimension and also taken after it has been modified by applying EXP. In the bottom chart you can see the smoothing period that results from each version of Alpha. Clearly by applying EXP, Alpha is reduced creating an significantly faster Smoothing Period.

The use of EXP results in not just a slower LAMA overall but one that exponentially slows as alpha decreases. This affect is similar to that of raising Alpha to a power as seen in the Adaptive Moving Average (AMA). In fact, it turns out that the LAMA is identical to the AMA if you were to raise it to the power of about 988,869,997.798!!!!!! That is not a typo. The LAMA and therefore the FRAMA is identical to the AMA raised the power of almost 100 million….

In discovering this there is little point in running the tests on this indicator because previous tests on the AMA already reveal it will not be able to out perform. Oh well, that saves some work! That is why we take the time to look closer at these things and try not to make too many uneducated assumptions.

Vertical Horizontal Filter Adaptive Moving Average (VHF-AMA) – Test Results

The Adaptive Moving Average (AMA) modifies the amount of smoothing it applies to data in an attempt to adjust to the changing needs of a dynamic market. It makes these adjustments based on the readings from a Volatility Index (VI). Any measure of volatility or trend strength can be used, however in this article we will focus on how the AMA performs using the Vertical Horizontal Filter (VHF).

The VHF-AMA requires four user selected inputs: A Vertical Horizontal Filter period, a High – Low smoothing period range for the AMA and a power that Alpha is raised to. With four variables there are thousands of possible combinations so we had to make some educated assumptions based on our previous tests to narrow the choices down.

In our tests on the Vertical Horizontal Filter in a VHF-VMA we revealed that VHF periods of 126, 252 and 80 produced the best results. Because Volatility Index settings have proven to produce similar results in both the VMA and the AMA, testing these three settings should be sufficient to capture the best results. Also they corresponded with the approximate number of trading days in standard calendar periods: 80 days = ⅓ year, 126 days = ½ year and there are 252 trading days in an average year, so:

VHF = 80, 126, 252

In previous tests we have seen that a moving average range produces the best results when it can move to as little as 4 periods or less, therefore we will test:

AMA Actual Fast Moving Average (FN) = 1, 4, 10

With the slow moving average we have consistently seen 300 produce the best results while changing this setting hasn’t usually made a big impact. However we still ran tests through several settings:

AMA Actual Slow Moving Average (SN) = 200, 250, 300

For the Alpha Power we also tried several variables:

Alpha Power (P) = 0.5, 0.75, 1, 1.5, 2, 2.5

We tested trades going Long, using Daily data, taking End Of Day (EOD) signals~ analyzing all combinations of the above settings.

Each time the Alpha Power was adjusted the SC and FC had to be modified to account for the change but the actual FN and SN stayed the same.

For instance a SC – FC range of 1 – 24 with alpha ^ 2 has an actual FN – SN range of about 1 – 300 due to the effect of squaring alpha. Here is a table that shows the SC – FC ranges used so that the FN – SN ranges stayed constant regardless of ‘P’:

SC and FC values used to keep FN and SN constant as P was changed.

If that doesn’t make a lot of sense then please read our explanation of the Adaptive Moving Average. A total of 162 different averages were tested and each one was run through 300 years of data across 16 different global indexes (details here).

Download A FREE Spreadsheet With Raw Data For

All 162 VHF-AMA Test Results

Vertical Horizontal Filter Adaptive Moving Average – Test Array

Vertical Horizontal Filter AMA - Ann Return as Alpha Power is Changed

Above we have charted the annualized returns achieved from each VHF with Alpha raised to different powers along the X axis. The chart on the left shows the results when the FN = 1 and SN = 300 while on the right FN = 4 and SN = 300. Clearly keeping the FN at 1 is important to achieve the best returns. A VHF period of 126 stood out as the best performer and this echoes previous tests. Finally, when FN = 1, raising Alpha to the power of 1.5 yielded the best results.

Best Vertical Horizontal Filter Adaptive Moving Average

126 Day VHF-AMA, EOD 1, 56 ^ 1.5 L - Performance

Included on the above chart is the performance of the 126 Day FRAMA, EOD 4, 300 Long because so far this has been the best performing Moving Average. The 126 Day VHF-AMA, EOD 1, 56 Long ^ 1.5 produced extremely similar results and even had the same average trade duration of 14 days. However it did slightly underperform the FRAMA by most measures but lets take a quick look under the hood to see what makes it tick:

126 Day VHF-AMA, EOD 1, 56 ^ 1.5 – Smoothing Period Distribution

126 Day VHF-AMA, EOD 1, 56 ^ 1.5 - Smoothing Distribution

As you can see the VHF-AMA does not have nearly as much of a spread with its smoothing range as the FRAMA. A larger range makes the FRAMA more able to adapt to different market environments.

126 Day VHF-AMA 1, 56 ^ 1.5 – Alpha Comparison

To get an idea of the readings that created these results we charted a section of the alpha for the 126 Day VHF-AMA 1, 56 ^ 1.5 and compared it to the best performing FRAMA and the best VHF-VMA to see if we could learn what makes a good volatility index for use in an AMA:.

126 Day VHF-AMA, EOD 1, 56 ^ 1.5 - Alpha Comparison

The VHF does not look as though it can change as nimbly as the FRAMA while both the AMA and VMA using a VHF look very similar.

Excel Spreadsheet

The VHF is outstanding for use in an AMA and we have build an excel spreadsheet for you to download free so you can have a play. Simply use the flowing link and you will find it under Downloads – Technical Indicators: Adaptive Moving Average (AMA).

For more in this series see – Technical Indicator Fight for Supremacy

~ An entry signal to go long for each average tested was generated with a close above that average and an exit signal was generated on each close below that moving average. No interest was earned while in cash and no allowance has been made for transaction costs or slippage. Trades were tested using End Of Day (EOD) signals on Daily data. Eg. Daily data with EOD signals would require the Daily price to close above a Daily Moving Average to open a long and to close below that Average to close the position.
We used the average annualized return of the 16 markets during the testing period. The data used for these tests is included in the results spreadsheet and more details about our methodology can be found here.

252 Day ER-AMA, 9 – AMA Indicator Equivalent

Relative Volatility Index Adaptive Moving Average (RVI-AMA) – Test Results

The RVI-AMA requires five user selected inputs: A Standard Deviation period, a Wilder’s Smoothing period, a High – Low smoothing period range for the AMA and a power that Alpha is raised to. With five variables there are thousands of possible combinations so we had to make some educated assumptions based on our previous tests to narrow the choices down.

In our tests on the Relative Volatility Index in a RVI-VMA we revealed that a Wilder’s Smoothing (WS) period of 14 worked the best and there is no reason to suggest that this will not also be true for a RVI-AMA so:

WS = 14

We selected the SD lengths that corresponded with the approximate number of trading days in standard calendar periods: 10 days = two weeks, 20 days = 1 month, 40 days = 2 months, 80 days = ⅓ year, 126 days = ½ year and there are 252 trading days in an average year:

SD = 10, 20, 40, 80, 126, 252

In previous tests we have seen that a moving average range produces the best results when it can move to as little as 4 periods or less, therefore we will test:

AMA Actual Fast Moving Average (FN) = 1, 4, 10

AMA Actual Slow Moving Average (SN) = 100, 150, 200, 250, 300

For the Alpha Power we also tried several variables:

Alpha Power (P) = 0.5, 0.75, 1, 1.5, 2, 2.5

We tested trades going Long, using Daily data, taking End Of Day (EOD) signals~ analyzing several combinations of the above settings.

Each time the Alpha Power was adjusted the SC and FC had to be modified to account for the change but the actual FN and SN stayed the same.

SC and FC values used to keep FN and SN constant as P was changed.

If that doesn’t make a lot of sense then please read our explanation of the Adaptive Moving Average. A total of 321 different averages were tested and each one was run through 300 years of data across 16 different global indexes (details here).

Download A FREE Spreadsheet With Raw Data For

All 321 RVI-AMA Test Results

Relative Volatility Index Adaptive Moving Average – Test Array

Relative Volatility Index AMA - Ann Return as Alpha Power is Changed

Above we have charted the annualized returns achieved from each RVI with Alpha raised to different powers along the X axis. The chart on the left shows the results when the FN = 1 and SN = 300 while on the right FN = 4 and SN = 300. Clearly keeping the FN at 1 is important to achieve the best returns. There was no SD period that really stood out so we shall go with 126 because of how it has performed in past tests. Finally, when FN = 1, raising Alpha to the power of 0.5 clearly yielded the best results.

Best Relative Volatility Index Adaptive Moving Average

126 Day RVI-AMA, EOD 1, 45300 ^ 0.5 (WS 14) - Performance

Included on the above chart is the performance of the 126 Day FRAMA, EOD 4, 300 Long because so far this has been the best performing Moving Average. The 126 Day RVI-AMA, EOD 1, 45300 Long ^ 0.5 (WS 14) produced an extremely fast moving average with a typical trade duration of just 4 days. This makes it unpractical for a real world application. Add to this the fact that it underperformed the best the FRAMA and this indicator is hardly worthy of further testing. However lets take a quick look under the hood to see what makes it tick and the causes of its weaknesses:

126 Day RVI-AMA, EOD 1, 45300 ^ 0.5 (WS 14) – Smoothing Period Distribution

126 Day RVI-AMA, EOD 1, 45300 ^ 0.5 (WS 14) - Smoothing Distribution

Instantly you can see a big problem; there isn’t really any smoothing distribution at all from the 126 Day RVI-AMA, EOD 1, 45300 ^ 0.5 (WS 14), instead it is basically a 2 day EMA. The far better performing FRAMA 0n the other hand has a wide spread of smoothing making on it much more adaptive to changing market conditions.

126 Day RVI-AMA 1, 45300 ^ 0.5 (WS 14) – Alpha Comparison

To get an idea of the readings that created these results we charted a section of the alpha for the 126 Day RVI-AMA 1, 45300 ^ 0.5 (WS 14) and compared it to the best performing FRAMA and the best RVI-VMA to see if we could learn what makes a good volatility index for use in an AMA:.

126 Day RVI-AMA, EOD 1, 45300 ^ 0.5 (WS 14) - Alpha Comparison

Higher alpha readings result in a faster average and instantly you can see the RVI-AMA has a very high Alpha compared to the best RVI-VMA and FRAMA. Remember the RVI-AMA and the RVI-VMA both use the same volatility index but the different ways that the two modify Alpha result in a very different outcome.

Excel Spreadsheet

The RVI-AMA is not very useful but should you want to test it or another volatility index then we have build an excel spreadsheet for you to download free. Simply use the flowing link and you will find it under Downloads – Technical Indicators: Adaptive Moving Average (AMA).

For more in this series see – Technical Indicator Fight for Supremacy

~ An entry signal to go long for each average tested was generated with a close above that average and an exit signal was generated on each close below that moving average. No interest was earned while in cash and no allowance has been made for transaction costs or slippage. Trades were tested using End Of Day (EOD) signals on Daily data. Eg. Daily data with EOD signals would require the Daily price to close above a Daily Moving Average to open a long and to close below that Average to close the position.
We used the average annualized return of the 16 markets during the testing period. The data used for these tests is included in the results spreadsheet and more details about our methodology can be found here.

252 Day ER-AMA, 9 – AMA Indicator Equivalent

Standard Deviation Ratio Adaptive Moving Average (SDR-AMA) – Test Results

The SDR-AMA requires five user selected inputs: SD1, SD2, a High – Low smoothing period range for the AMA and a power that Alpha is raised to. With five variables there are thousands of possible combinations so we had to make some educated assumptions based on our previous tests to narrow the choices down.

First of all we have seen that nearly all of the performance characteristics exhibited by a VI have rung true in tests on both a VMA and an AMA. When we tested the SDR in a VMA we found that it was best if SD1 was around half of SD2. We also selected SD lengths that corresponded with the approximate number of trading days in standard calendar periods: 10 days = two weeks, 20 days = 1 month, 40 days = 2 months, 80 days = ⅓ year, 126 days = ½ year and there are 252 trading days in an average year:

SD1/SD2 = 10/20, 40/80, 80/126, 126/252

Second we have seen that a moving average range produces the best results when it can move to as little as 4 periods or less, therefore we will test:

AMA Actual Fast Moving Average (FN) = 1, 4

AMA Actual Slow Moving Average (SN) = 100, 150, 200, 250, 300

For the Alpha Power we also tested several variables:

Alpha Power (P) = 0.5, 0.75, 1, 1.5, 2, 2.5

We tested trades going Long, using Daily data, taking End Of Day (EOD) signals~ analyzing several combinations of the above settings.

Now each time the Alpha Power was adjusted the SC and FC had to be modified to account for the change but the actual FN and SN stayed the same.

SC and FC values used to keep FN and SN constant as P was changed.

If that doesn’t make a lot of sense then please read our explanation of the Adaptive Moving Average. A total of 240 different averages were tested and each one was run through 300 years of data across 16 different global indexes (details here).

Download A FREE Spreadsheet With Raw Data For

All 240 SDR-AMA Test Results

Standard Deviation Ratio Adaptive Moving Average – Test Array

Standard Deviation Ratio AMA - Ann Return as Alpha Power is Changed

Above we have charted the annualized returns achieved from each SDR with Alpha raised to different powers along the X axis. The chart on the left shows the results when the FN = 1 and SN = 300 while on the right FN = 4 and SN = 300. Clearly extending the FC to 4 had a positive effect and the best returns were achieved with a SDR of 126/252 where Alpha was raised to the power of 2.

Best Standard Deviation Ratio Adaptive Moving Average

126/252 Day SDR-AMA, EOD 2, 24 Long ^ 2

Included on the above chart is the performance of the 126 Day FRAMA, EOD 4, 300 Long becuase so far this has been the best performing Moving Average. The 126 Day SDR-AMA, EOD 2, 24 Long ^ 2 performed OK but could not best the FRAMA and has a much shorter average trade duration; just 8 days compared to 14 for the FRAMA. For these reasons the FRAMA remains our preferred moving average and the SDR-AMA does not warrant further testing. But lets take a quick look under the hood:

126 Day SDR-AMA, EOD 2, 24 ^ 2 – Smoothing Period Distribution

126/252 Day SDR-AMA, EOD 2, 24 ^ 2 Smoothing Period Distribution

The smoothing distribution of the 126 Day SDR-AMA, EOD 2, 24 ^ 2 is much more localised around the 4 – 20 range than the FRAMA which explains the shorter trade duration. The FRAMA on the other hand allows the average to move much slower at times, presumably when the trend is weak.

126 Day SDR-AMA 2, 24 ^ 2 – Alpha Comparison

To get an idea of the readings that created these results we charted a section of the alpha for the 126 Day SDR-AMA 2, 24 ^ 2 and compared it to the best performing FRAMA and the best SDR-VMA to see if there were any similarities that would reveal what makes a good volatility index:.

126/252 Day SDR-AMA, EOD 2, 24 ^ 2 - Alpha Comparison

Remember higher alpha readings result in a faster average. The SDR-AMA and the SDR-VMA are clearly both much faster than the FRAMA. However the SDR-AMA does slightly outperform the SDR-VMA and notice that the SDR-AMA’s Alpha moves through a greater range from high to low. This greater ‘adaptability’ is likely to have been a key factor in its better performance.

Excel Spreadsheet

Want to use this indicator? Get a free Excel spreadsheet at the flowing link under Downloads – Technical Indicators: Adaptive Moving Average (AMA). It will automatically adjust to your choice of many different VIs including the Standard Deviation Ratio used in this article.

For more in this series see – Technical Indicator Fight for Supremacy

~ An entry signal to go long for each average tested was generated with a close above that average and an exit signal was generated on each close below that moving average. No interest was earned while in cash and no allowance has been made for transaction costs or slippage. Trades were tested using End Of Day (EOD) signals on Daily data. Eg. Daily data with EOD signals would require the Daily price to close above a Daily Moving Average to open a long and to close below that Average to close the position.
We used the average annualized return of the 16 markets during the testing period. The data used for these tests is included in the results spreadsheet and more details about our methodology can be found here.

252 Day ER-AMA, 9 – AMA Indicator Equivalent

Fractal Dimension Adaptive Moving Average (D-AMA) – Test Results

We did have to make one slight modification to the Fractal Dimension however. The Volatility index in an AMA needs to shift through a 0 – 1 range where higher readings indicate a stronger trend. The Fractal Dimension shifts through a 1 – 2 range where lower readings indicate a stronger trend. Therefore we shall use = ABS(D – 2).

The D-AMA requires four user selected inputs: A Fractal Dimension Period, a High – Low smoothing period range for the AMA and a power that Alpha is raised to. We tested trades going Long, using Daily data, taking End Of Day (EOD) and End of Week (EOW) signals~ analyzing combinations of:

D = 40, 80, 126, 252

Alpha Power (P) = 0.5, 0.75, 1, 1.5, 2, 2.5

AMA Actual Fast Moving Average (FN) = 1, 4, 10, 20, 40, 60

AMA Actual Slow Moving Average (SN) = 100, 150, 200, 250, 300

The D lengths were selected due to the fact that they correspond with the approximate number of trading days in standard calendar periods: 40 days = 2 months, 80 days = ⅓ year, 126 days = ½ year and there are 252 trading days in an average year. In many of out past tests we have also tested VI lengths of 10 and 20 days, however these setting have always failed to yield the best results so we felt that it would be safe to omit them from this set of tests.

The AMA ranges were selected because they should capture the best results based on what we know from previous research into moving averages. Each time the Alpha Power was adjusted the SC and FC had to be modified to account for the change but the actual FN and SN stayed the same.

For instance a SC – FC range of 1 – 24 with alpha ^ 2 has an actual FN – SN range of 1 – 300 due to the effect of squaring alpha. Here is a table that shows the SC – FC ranges used so that the FN – SN ranges stayed constant regardless of ‘P’:

SC and FC values used to keep FN and SN constant as P was changed.

If that doesn’t make a lot of sense then please read our explanation of the Adaptive Moving Average. A total of 960 different averages were tested and each one was run through 300 years of data across 16 different global indexes (details here).

Download A FREE Spreadsheet With Raw Data For

All 960 D-AMA Test Results

Fractal Dimension Adaptive Moving Average – Modifying Alpha by Raising to a Power

Kaufman had a theory that by squaring Alpha and thus causing the AMA to slow rapidly when the data lacked a strong trend he would achieve better results. When we tested this theory on the ER-AMA we found it to be false but with a different VI we may reach a different conclusion. So lets look at the affect of raising Alpha to different powers:

Fractal Dimension – AMA, Alpha to the Power of 1 – Annualized Return

Fractal Dimension - AMA ^ 1 - Annualized Return

With Alpha ^1 there is no modification to Alpha at all. Clearly as the FC is increased the returns decline and as the FC gets higher the change in SC has more impact. Generally it appears as though a SC of 100 is best on a D-AMA with an unmodified Alpha. ER lengths of 80 and 126 yielded the best returns, this finding is similar to that of our previous tests on other ‘intelligent’ moving averages.

Fractal Dimension – AMA to the Power of 2 – Annualized Return

Fractal Dimension - AMA ^ 2 - Annualized Return

By raising Alpha to the power of 2, returns at almost all the data points increased which is just the opposite of what we experienced when testing the ER-AMA. This shows that Kaufman’s theory of rapidly slowing the AMA during times where a trend is lacking had merit but is dependent on the VI being used.

The best results again came from ER lengths of 80 and 126 although an ER length of 40 did produce some notable returns. Changing the SC did not have as much of an effect with Alpha ^2 compared to Alpha ^ 1 however a SC of 24 (SN equivalent of 300) and a short FC tends to produce the best results. So lets rework the charts to focus on what we now know works best:

D-AMA Annualized Return with Alpha to Different Powers

Now we are only looking at ER periods of 40, 80 and 126 with a FN of 1 and 4 and a SN of 300. Each data point plots the change in returns with Alpha raised to different powers. As you can see, the best returns resulted from an ER period of 126 with alpha raised to the power of 2. Therefore when using the Fractal Dimension in an Adaptive Moving average you are best to square alpha as suggested in the original formula..

Best EOD Fractal Dimension Adaptive Moving Average

126 Day D-AMA, EOD 2, 24 Long ^ 2

I have included on the above chart the performance of the 126 Day FRAMA, EOD 4, 300 Long becuase so far this has been the best performing Moving Average. The 126 Day D-AMA, EOD 2, 24 Long ^ 2 put up a good fight against the FRAMA but ultimately underperformed by most measures to a small degree. On the Short side, the the D-AMA also underperformed slightly.

126 Day D-AMA, EOD 2, 24 ^ 2 – Smoothing Period Distribution

126 Day D-AMA 2, 24 ^ 2 Smoothing Period Distribution

Looking at the smoothing distribution you can see the 126 Day D-AMA, EOD 2, 24 ^ 2 is very similar to that of the 126 Day FRAMA, EOD 4, 300 but the FRAMA allows the average to slow down more often.

126 Day D-AMA 2, 24 ^ 2 – Alpha Comparison

To get an idea of the readings that created these results we charted a section of the alpha for the 126 Day D-AMA 2, 24 ^ 2 and compared it to the best performing FRAMA and the best D-VMA to see if there were any similarities that would reveal what makes a good volatility index:.

126 Day D-AMA 2, 24 ^ 2 - Alpha Comparison

You can clearly see that all three use the same VI, the only difference is how they manipulate Alpha. Remember, higher readings result in a faster average so the D-AMA is obviously the fastest of the three while the FRAMA appears to shift through the widest range.

Best EOW Fractal Dimension Adaptive Moving Average

There are times when an average with a longer trade duration better suits ones needs so we also ran the tests looking for the best average using EOW signals, here is the one that came out trumps:

252 Day D-AMA. EOW 111, 372 Long ^ 0.75

We have included on the above chart the performance of the 252 Day FRAMA, EOW 40, 250 Long becuase so far this has been the best performing EOW Moving Average. The 252 Day D-AMA, EOW 111, 372 Long ^0.75 is almost identical to the FRAMA but does outperform it by a fraction. They are so similar in fact that they may as well be the same average. Performance on the short side tells the same story.

252 Day D-AMA, EOW 111, 372 ^ 0.75 – Smoothing Period Distribution

252 Day D-AMA, EOW 111, 372 ^ 0.75 - Smoothing Period Distribution

The smoothing distribution for the 252 Day D-AMA 111, 372 ^ 0.75 has a smaller range than that of the 252 Day FRAMA 40, 250 but the median, lower quartile and minimum are almost identical. You can view an Alpha Comparison Here.

Conclusion

In our tests on the ER-AMA we came to the conclusion that the squaring of alpha as suggested in the standard AMA formula was actually detrimental to performance. However when using the Fractal Dimension as the VI, squaring Alpha was beneficial. Therefore the best Power to use in manipulating alpha varies depending on the VI in use.

Overall the D-AMA produced results that were near identical to that of the FRAMA but the D-AMA is a slightly faster average. The best performing EOD D-AMA was the 126 Day ER-AMA, EOD 2, 26 ^ 2 while the best EOW or ‘slower’ moving average was the 252 Day D-AMA, EOW 111, 372 ^ 0.75.

It is very difficult to pick between the FRAMA and the D-AMA but becuase the FRAMA offers a slightly longer trade duration it the best Moving Average we have tested so far.

For more in this series see – Technical Indicator Fight for Supremacy

~ An entry signal to go long (or exit signal to cover a short) for each average tested was generated with a close above that average and an exit signal (or entry signal to go short) was generated on each close below that moving average. No interest was earned while in cash and no allowance has been made for transaction costs or slippage. Trades were tested using End Of Day (EOD) and End Of Week (EOW) signals on Daily data. Eg. Daily data with an EOW signal would require the Week to finish above a Daily Moving Average to open a long or close a short while Daily data with EOD signals would require the Daily price to close above a Daily Moving Average to open a long or close a short and vice versa.
We used the average annualized return of the 16 markets during the testing period. The data used for these tests is included in the results spreadsheet and more details about our methodology can be found here.

252 Day ER-AMA, 9 – AMA Indicator Equivalent

Efficiency Ratio Adaptive Moving Average (ER-AMA) – Test Results

The ER-AMA requires four user selected inputs: An Efficiency Ratio Period, a High – Low smoothing period range for the AMA and a power that Alpha is raised to. We tested trades going Long, using Daily data, taking End Of Day (EOD) and End of Week (EOW) signals~ analyzing combinations of:

ER = 10, 20, 40, 80, 126, 252

Alpha Power (P) = 0.5, 0.75, 1, 1.5, 2, 2.5

AMA Actual Fast Moving Average (FN) = 1, 4, 10, 20, 40, 60

AMA Actual Slow Moving Average (SN) = 100, 150, 200, 250, 300

The ER lengths were selected due to the fact that they correspond with the approximate number of trading days in standard calendar periods: 10 days = 2 weeks, 20 days = 1 month, 40 days = 2 months, 80 days = ⅓ year, 126 days = ½ year and there are 252 trading days in an average year.

SC and FC values used to keep FN and SN constant as P was changed.

If that doesn’t make a lot of sense then please read our explanation of the Adaptive Moving Average. A total of 1020 different averages were tested and each one was run through 300 years of data across 16 different global indexes (details here).

Download A FREE Spreadsheet With Raw Data For

All 1020 ER-AMA Test Results

ER Adaptive Moving Average – Modifying Alpha by Raising to a Power

Kaufman had a theory that by squaring Alpha and thus causing the AMA to slow rapidly when the data lacked a strong trend he would achieve better results. Here we will put this theory to the test and be looking at the affect of raising Alpha to different powers:

Efficiency Ratio – AMA, Alpha to the Power of 1 – Annualized Return

Efficiency Ratio - AMA ^ 1 - Annualized Return

With Alpha ^1 there is no modification to alpha at all and the results are impressive. It can be said that in most cases as the FC increased the returns declined while changing the SC did not have much of an impact. ER lengths of 80 and 126 yielded the best returns, this finding is similar to that of our previous tests on other ‘intelligent’ moving averages.

Efficiency Ratio – AMA to the Power of 2 – Annualized Return

Efficiency Ratio - AMA ^ 2 - Annualized Return

By raising Alpha to the power of 2 the returns drop almost across the board which immediately brings into question the need to include this function in the AMA formula and what would happen if we used a power below 1? It would appear as though the ER length needs to be at least 40 to be of value in this context with the best results again coming from ER lengths of 80 and 126. Clearly the FC is best when kept short so lets rework the charts to focus on what we now know works best:

ER-AMA Annualized Return with Alpha to Different Powers

Now we are only looking at ER periods of 80 and 126 with a FN of 1 and 4 and a SN of 300. Each data point plots the change in returns with Alpha raised to different powers. As you can see, the best returns resulted from an ER period of 126 with alpha raised to the power of 0.75. As the Power was increased beyond this point, the returns decreased almost across the board. Therefore when using an Efficiency Ratio in an Adaptive Moving average you definitely should not square alpha as suggested in the original formula..

Best EOD Efficiency Ratio Adaptive Moving Average

126 Day ER-AMA EOD 1, 1600 Long ^ 0.75

We have included on the above chart the performance of the 126 Day FRAMA, EOD 4, 300 Long becuase so far this has been the best performing Moving Average. The 126 Day ER-AMA, EOD 1, 1600 Long ^ 0.75 actually outperformed the best FRAMA up until 2008 when the market had a big pull back. As a result, over the full term of the test the FRAMA did perform slightly better. Also the FRAMA has a few added benefits such as turning a profit on the bear ravaged Nikkei 225 and having a 40% longer average trade duration (14 vs 10 Days).

On the Short side, the the ER-AMA also underperformed over the full term but again outperformed until 2008. This makes it very difficult to pick which moving average is the better of the two. But because our personal preference leans towards a longer trade duration we will stick with the FRAMA as being the best moving average we have found so far. (See the results on the short side)

126 Day ER-AMA, EOD 1, 1600 ^ 0.75 – Smoothing Period Distribution

126 Day ER-AMA, EOD 1, 1600 ^ 0.75 - Smoothing Period Distribution

Looking at the smoothing distribution you can see the 126 Day ER-AMA, EOD 1, 1600 ^ 0.75 is quite similar to that of the 126 Day FRAMA, EOD 4, 300 but the FRAMA spends more time as a slow average which explains the longer trade duration.

126 Day ER-AMA, 1, 1600 ^ 0.75 – Alpha Comparison

To get an idea of the readings that created these results we charted a section of the alpha for the 126 Day ER-AMA, 1, 1600 ^ 0.75 and compared it to the best performing FRAMA and the best ER-VMA to see if there were any similarities that would reveal what makes a good volatility index:.

126 Day ER-AMA, 1, 1600, ^ 0.75 - Alpha Comparison

Because the ER-AMA and the ER-VMA both use the same volatility index, obviously their Alpha is identical apart from the slight modifications caused by the separate method of manipulating alpha. Remember, the higher the Alpha the faster the resulting average so you can see why the best ER-AMA was faster moving than the best ER-VMA. The Alpha patterns of the best FRAMA and best ER-AMA do have strong similarities but notice how the FRAMA is far less volatile. It is always preferable to work with indicators that generate clean readings with low levels of noise assuming they still produce good results.

Best EOW Efficiency Ratio Adaptive Moving Average

There are times when an average with a longer trade duration better suits ones needs so we also ran the tests looking for the best average using EOW signals:

252 Day ER-AMA, EOW 10, 100 Long ^ 1

We have included on the above chart the performance of the 252 Day FRAMA, EOW 40, 250 Long becuase so far this has been the best performing EOW Moving Average. The 252 Day ER-AMA, EOW 10, 100 Long ^1 underperforms by a fraction in most measures and while almost identical to the FRAMA it certainly is not superior. Performance on the short side tells the same story.

252 Day ER-AMA, EOW 10, 100 ^ 1 – Smoothing Period Distribution

252 Day ER-AMA, EOW 10, 100 ^ 1 - Smoothing Period Distribution

Looking at the smoothing distribution for the 252 Day ER-AMA 10, 100 ^ 1 you can see that it is far faster than the 252 Day FRAMA 40, 250 and has a more limited range despite the median, upper and lower quartiles being almost identical.

252 Day ER-AMA 10, 100 ^ 1 – Alpha Comparison

252 Day ER-AMA 10, 100 ^ 1 - Alpha Comparison We have included on this chart the best EOW FRAMA and EOW ER-VMA. The ER-VMA and ER-AMA use the same volatility index but the 252 Day ER-AMA 10, 100 ^1 results in a higher Alpha which explains the faster average. The Alpha for the FRAMA and the AMA does stay in a similar zone but the FRAMA is far less volatile which is preferable.

Conclusion

In the standard formula for the AMA, alpha is squared to force the average to slow rapidly during times when there is a lack of trend. When using an Efficiency Ratio as the Volatility Index (which is most commonly used VI in an AMA) we have clearly shown that squaring Alpha has a detrimental effect on returns. We suggest instead not modifying alpha at all or raising it to the power of 0.75

Overall the ER produced some impressive returns during out tests as the VI in an AMA as it did when we used in a VMA. The best performing EOD ER-AMA was a 126 Day ER-AMA, EOD 1, 1600 Long ^ 0.75 which did show periods of out performance over our current best performing MA the 126 Day FRAMA, EOD 4, 300 Long. However due to a longer trade duration we still rate the FRAMA as superior.

When it comes to an EOW or ‘slower’ moving average the 252 Day ER-AMA, EOW 10, 100 ^ 1 is almost identical to our current best ‘slow’ MA the 252 Day FRAMA, EOW 40, 250 Long but certainly does not offer any benefits.

Both the Efficiency Ratio and the Adaptive Moving Average have proven themselves against some formidable competition but based on our findings so far the FRAMA still remains slightly superior.

For more in this series see – Technical Indicator Fight for Supremacy

~ An entry signal to go long (or exit signal to cover a short) for each average tested was generated with a close above that average and an exit signal (or entry signal to go short) was generated on each close below that moving average. No interest was earned while in cash and no allowance has been made for transaction costs or slippage. Trades were tested using End Of Day (EOD) and End Of Week (EOW) signals on Daily data. Eg. Daily data with an EOW signal would require the Week to finish above a Daily Moving Average to open a long or close a short while Daily data with EOD signals would require the Daily price to close above a Daily Moving Average to open a long or close a short and vice versa.
We used the average annualized return of the 16 markets during the testing period. The data used for these tests is included in the results spreadsheet and more details about our methodology can be found here.