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Exponential Moving Average (EMA) Exponential Smoothing The following is an abbreviated excerpt from our all new, completely revised, 820-page research book, Colby, Robert W., The Encyclopedia of Technical Market Indicators, Second Edition, McGraw-Hill Publishing, 2003 (click here for a description). The Exponential Moving Average (EMA) is the best of the moving average techniques, and it is increasingly preferred by technical analysts over other moving average methods. Behaviorally, in its responsiveness to new data being generated by the markets, the EMA represents an excellent compromise between the overly sensitive weighted moving average and the overly sluggish simple moving averages. Compared to other averaging techniques, the EMA follows the trend of the current data smoothly and seamlessly, minimizing jumps, wiggles, and lags. Computationally, the EMA is the simplest and most streamlined of all moving average techniques. The EMA requires the fewest calculations, the least data handling, and the least data history. The EMA requires numerical values for only two data periods: the most recently available raw data and the immediate past period’s EMA. For example, working with daily data, we need only today’s observed, unprocessed data and yesterday’s EMA in order to calculate today’s EMA. Thus, the EMA eliminates the need to keep and handle long lists of historical data. A significant advantage of this superior computational method is that the EMA is never distorted by old data suddenly dropping out of the calculation. Old data is never suddenly dropped because it is not actually part of the calculation. For practical purposes, the effect of past data fades away gradually due to the ever decreasing weighting of yesterday’s EMA. The EMA’s method of calculation correctly avoids the problem of erratic current movement caused solely by irrelevant and obsolete data dropping out of the calculation. An Exponential Moving Average is calculated as follows: EMA = (C - Ep)K + Ep where EMA = the Exponential Moving Average for the current period. C = the closing price for the current period.

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Ep = the Exponential Moving Average for the previous period. K = the exponential smoothing constant, equal to 2 / (n+1). n = the total number of periods in a simple moving average to be roughly approximated by the EMA. The exponential smoothing constant formula, K = 2/(n+1), allows an approximate comparison of any EMA to the more sluggish Simple Moving Average of length n. As the number of days n increases, the value of K grows ever smaller, and the EMA becomes increasingly less sensitive to the newer data. [Our table for converting from simple n days to exponential smoothing constants (K), and back, can be seen on page 262 of the book.] When first starting a new EMA, it takes approximately n days of calculations for an accurate reading. For a quick startup of a EMA, on the first day of calculation we may use a n-day simple moving average to approximate the previous day’s EMA (Ep) in the formula, EMA = (C - Ep)K + Ep. After that first day, we will never need any data other than yesterday’s EMA and today’s fresh data to maintain our EMA. [Our table illustrating how to compute an EMA of four periods, which is also known as a 40% EMA, named for the exponential smoothing constant, K, can be seen on page 263 of the book.] Indicator Strategy Examples for Exponential Moving Average (EMA) Crossover Based on the daily closing prices for the Dow-Jones Industrial Average from 1900 to 2001, Exponential Moving Average Crossover Strategies of all lengths from 1 day to 300 days would have been profitable and would have beaten the passive buy-and-hold-strategy by at least 69%. The five-, three- and two-day EMA would have produced maximum net profits in excess of six billion dollars, assuming we start with one hundred dollars in 1900. All EMA period lengths of 1 to 20 days would have produced net profits in excess of ten million dollars, and all 20 lengths would have outperformed buy-and-hold by more than 540 to one. The 5-Day Exponential Moving Average (EMA) crossover is the best simple trend-following indicator we tested against daily DJIA daily closing data from 1900 to 2001. Starting with $100 and reinvesting profits, total net profits for this 5-day EMA Crossover Strategy would have been $16 billion, assuming a fully-invested strategy, reinvestment of profits, no transactions costs and no taxes. This would have been 78 million percent better than buy-and-hold. Short selling would have been profitable. Trading frequency would have been hyperactive with one trade every 5.88 calendar days. There would have been 2417 profitable trades and 3889 losing trades, for a winning percentage of only 38.33% profitable. [Our Equity graph and Equis MetaStock® "System Report" (profit and loss summary statistics) table can be seen on page 268 and 269 of the book.]



All EMA period lengths of 1 to 60 days would have produced net profits in excess of one million dollars, and all 60 lengths would have outperformed buy-and-hold by more than 64 to one. Of the "intermediate-term" lengths, the 44-day EMA would have produced the best results, net profit of $3,251,721, which would have been more than 162 times the buy-and-hold-strategy’s $20,105. Performance deteriorated as the moving average period length increased. The popular 200-day EMA Crossover Strategy would have produced much less profit of $109,158, which would have been only 5.4 times the buy-and-hold-strategy’s $20,105 net profit. Of the "long-term" EMA period lengths of 150 days or more, the 171-day EMA Crossover Strategy would have produced the maximum profit on a purely mechanical trend-following signal basis with no subjectivity, no sophisticated technical analysis, and no judgement. Of all the EMA period lengths in excess of 100 days, the 120-day EMA Crossover Strategy would have produced the maximum profit. Starting with $100 and reinvesting profits, total net profits for this 120-day EMA Crossover Strategy would have been $508,772.91, assuming a fully-invested strategy, reinvestment of profits, no transactions costs and no taxes. This would have been 2,430.53 percent better than buy-and-hold. Short selling would have been profitable, but not since the Crash of ‘87. Trading frequency would have been moderate with one trade every 33.57 calendar days. There would have been 240 profitable trades and 862 losing trades, for a winning percentage of only 21.78% profitable. But because this trend-following strategy cuts losses and lets profits run, it makes money despite being wrong on most of its signals. This is typical of the longer-term trend-following strategies. Such a strategy may be used alone, and it also can be useful as a filter to other trading systems. [Our Equity graph and Equis MetaStock® "System Report" (profit and loss summary statistics) table can be seen on page 266 and 267 of the book.] Trading Rules for EMA Crossover Strategy Enter Long (Buy) at the current daily price close of the Dow-Jones Industrial Average when this close is greater than the previous day’s 120-day exponential moving average of the daily closing prices. Close Long (Sell) at the current daily price close of the Dow-Jones Industrial Average when this close is less than the previous day’s 120-day exponential moving average of the daily closing prices. Enter Short (Sell Short) at the current daily price close of the Dow-Jones Industrial Average when this close is less than the previous day’s 120-day exponential moving average of the daily closing prices. Close Short (Cover) at the current daily price close of the Dow-Jones Industrial



Average when this close is greater than the previous day’s 120-day exponential moving average of the daily closing prices. The Equis International MetaStock® System Testing rules are written as follows: Enter long: CLOSE > Ref(Mov(CLOSE,opt1,E),-1) Close long: CLOSE < Ref(Mov(CLOSE,opt1,E),-1) Enter short: CLOSE < Ref(Mov(CLOSE,opt1,E),-1) Close short: CLOSE > Ref(Mov(CLOSE,opt1,E),-1) where the number of days used for opt1 EMA can be allowed to vary from 1 day to 300 days, by a step size of 1 day. . . . The following material is not included in my book. . . . The TradeStation ™ Easy Language Code is written as follows: { Strategy #XA1, Easy Language Code for Evolving Exponential Moving Average Crossover Strategy with One Input Length Copyright © 2000 by Robert W. Colby. All rights reserved. } Inputs: N(1), ShowText(False), Length(200); Variables: Number(1); Number=(100000/Close[1])*(100000+NetProfit)/100000; IF BarNumber > 1 Then Begin If Close Crosses Above XAverage(Close,Length)[1] Then Buy Number of Shares This Bar on Close; If Close Crosses Below XAverage(Close,Length)[1] Then Sell Number of Shares This Bar on Close; IncludeSystem: "LastDay"; End . . . Settings for TradeStation Easy Language Code: start with Initial Capital of $100,000; invest $100,000 Per Transaction; allow $15 for commission per transaction; execute this bar market on close; and load 80 years of daily data from Historybank.com for the Dow-Jones Industrial Average (INDU).



We set Maximum Bars Back for 2, because that is all we need to calculate exponential smoothings, which are also known as exponential moving averages, and the Easy Language name is Xaverage. No matter how large our value for the XAverage, Omega Research TradeStation software needs only 2 bars to calculate any Xaverage. Therefore, we simply click continue to bypass the warning dialog that tells us our input value for our XAverage exceeds Maximum Bars Back. Net Profit for Strategy Optimization using TradeStation: In testing I conducted in year 2000 using TradeStation, which is not shown in my book, I again found that the EMA Crossover Strategy results were consistently profitable, although not as profitable as my test using MetaStock. Variations in results obviously were caused by differences in the date ranges loaded, the initial software settings, transactions costs ($0 for MetaStock versus $15 for TradeStation), and reinvestment rules (100% of profits reinvested for MetaStock versus a limit of $100,000 Per Transaction for TradeStation) used with the two different software programs. I used TradeStation with daily data from the Omega Research HISTORYBANK.COM Financial Database CD for the Dow-Jones Industrial Average (INDU) over the 80 years from 1920 to 2000, sampling daily closing prices only. With MetaStock, I used an entirely different database sampling daily closing prices over 101 years from 1900 to 2001. All these differences obviously add up to substantial differences in the end results. I tested all XAverages from 50 to 3000 days, with an increment of 50 days. A Net Profit graph (not shown here) showed that the profits of the EMA Crossover Strategy (which I named #XA1) were robust, with all tested lengths profitable. The best Net Profits were found at the shorter lengths, 200 days and below. Further fine tuning revealed that a length of just 2 days produced the highest Net Profit of $816 million on an original $100,000 investment. Trading for the 2-day length would have been very active: 7011 trades, one trade every four calendar days on average, given 28,734 calendar days. Of these 7011 trades, 2836 or 40.45% would have been profitable, a fairly typical winning percentage for trend following strategies. With trading this active, the strategy is highly dependent on low transactions costs, that is, low commissions and slippage. Fortunately, low transactions costs are widely available through a large number of highly competitive deep discount online brokers. Long trades would have made 91% of the profit, and shorts would have made 9%. This result reflects the strong long-term price uptrend over the period studied. The annual rate of return would have been 12.12%, the Profit Factor would have been 1.41, and the Return on Account would have been 1365.06%, or 17.28% annually.



Of course, all these numbers are from an optimization run, so we would expect lesser results in actual trading, which is seldom optimal. Nevertheless, these studies do offer encouragement for your further research, which also should include extensive walk-forward simulation. (See my book, pages 10-26.) . . . The EMA Crossover Strategy is one of many technical momentum strategies included in my book. . . . “The majority of the newsletters at the top of the Hulbert Financial Digest's rankings for long-term performance are those that, in one way or another, employ momentum strategies. Momentum strategies work because the markets detect trends before they are widely known among the investment public. We only discover in hindsight what is causing a move.” -- Mark Hulbert . . . CFTC Rule 4.41: Hypothetical or simulated performance results have certain limitations. Unlike an actual performance record, simulated results do not represent actual trading. Also, since the trades have not been executed, the results may have under- or-over compensated for the impact, if any, of certain market factors, such as lack of liquidity. Simulated trading programs in general are also subject to the fact that they are designed with the benefit of hindsight. No representation is being made that any account will or is likely to achieve profit or losses similar to those shown. Copyright © 2000-2008 by www.robertwcolby.com. All rights reserved. Except as permitted under the United States Copyright act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. Trading and investing involve risk of significant loss. Your use of this site means that you have read, understood, and accepted our Disclaimer. return to home page: www.robertwcolby.com









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