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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.

http://www.robertwcolby.com/EMAreport.html 10/27/2008


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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