Triple Exponential Moving Average

In this round of testing we are looking at the Double Exponential (D-EMA) and Triple Exponential Moving Averages (T-EMA). We have already tested the D-EMA and found that it wasn’t as effective as the EMA but wanted to test it over longer periods and compare it to the T-EMA.

In conducting these tests we measured the performance of each indicator going Long and Short, using Daily and Weekly data, taking End Of Day (EOD) and End Of Week (EOW) signals with smoothing periods varying from from 5 – 400 days or 80 weeks.~ These tests were carried out over a total of 300 years of data across 16 different global indexes (details here).

Note – Due to the huge lead in period required for the T-EMA, 240 weeks of data was ‘left in’ on each market. As a result the average buy and hold annualized return for the test markets was 4.94%. In our previous tests we only ‘left in’ 104 weeks and the subsequent buy and hold annualized return for the test markets was 6.32%. For this reason the results for these tests are not directly comparable to our other tests results. This is also why the returns for the D-EMA displayed below are lower than those previously published.

Triple and Double Exponential MA Annualized Return

Above are the return statistics when going long using daily, end of day signals. As you can see the Triple EMA under performs the Double EMA by a significant margin. Due to the fact that we have already established that the D-EMA is not worthy of use in a trading system the same can be said for the T-EMA and therefore there is no point in displaying any more statistics for these indicators. See also – Simple Vs Exponential Moving Averages

~ 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 for Daily data and EOW signals only for Weekly 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 and vice versa.

The Double and Triple Exponential Moving Average were created by Patrick Mulloy and first published in the February 1994 issue of Technical Analysis of Stocks & Commodities magazine – Smoothing Data With Less Lag. Mulloy stated in his article:

“Moving averages have a detrimental lag time that increases as the moving average length increases. The solution is a modified version of exponential smoothing with less lag time.”

Like an EMA, the D-EMA and T-EMA apply more weight to the most recent data in an attempt to smooth out noise while still remaining highly reactive to changes in the data. This is not achieved by simply double and triple smoothing as one may assume. To do so results in weighting that resembles a backwards log-normal distribution, rather like a Triangular Moving Average but smooth and shifted forward. Below you can see how the weighting is allocated by a single, double and triple smoothed exponential moving average compared to a standard EMA and SMA:

Double and Tripple Smoothed EMA Weighting .

As you can see by double and triple smoothing an EMA the weighting no longer focuses on the latest data. The actual Double and Triple Exponential Moving Average applies the weighing very heavily to the most recent data as illustrated in the chart below:

Double and Tripple EMA Weight

How To Calculate a Double Exponential Moving Average and T-EMA

Double Exponential MA Formula:

D-EMA = 2*EMA – EMA(EMA)

Triple Exponential MA Formula:

T-EMA = (3*EMA – 3*EMA(EMA)) + EMA(EMA(EMA))

Where:

EMA = EMA(1) + α * (Close – EMA(1))

α = 2 / (N + 1)

N = The smoothing period.

Here is an example of a 3 period Double Exponential Moving Average and Triple EMA:

Double and Triple Exponential Moving Average Formula .

Triple Exponential Moving Average and D-EMA Excel File

We have built a spreadsheet to calculate the D-EMA and T-EMA and have made it available for free download. Find the file at the following link near the bottom of the page under Downloads – Technical Indicators: Double (D-EMA) and Triple Exponential Moving Average (T-EMA).

Double EMA, Triple EMA and a Simple Moving Average

Double and Tripple EMA Vs a Simple MA .

Double and Triple Exponential Moving Average Test Results

We ran them through tests through over 300 years of data across 16 different global markets. Here are the results:

Double Exponential Moving average Vs Simple and Exponential Moving average

Double Vs Triple Exponential Moving Average

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.