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Comparison of

Exponential Lags and Moving Averages


(c) 2005 by Ed Seykota






Moving Averages are similar to Exponential Averages in some ways.  In other ways they are very different.  In both cases their names are unfortunate.


To take an average, you take the sum (total) of a number of things and then divide the total by the number.  For example, I have prices at 10, 11 and 15.  I sum them to get 36 and then divide the total by 3 to get the average, 12.


Say the closing price on Monday is 10; on Tuesday it's 11 and on Wednesday it's 15.  Then we say the three-day average is 12.  Now, actually, that is not quite accurate.  More precisely, we are taking a three-price average, not a three day average.  You can only average things you can see in a snapshot, things that are still, things you can measure.  You can not average one day with another.


Furthermore, if you measure the actual time interval between the close on Monday and Wednesday, it's only two days, not three.  So when someone says the three-day average, they likely mean the three-point average (the two-day average).  For the sake of clarity in this book, I do not refer to three-day averages.  I refer to three-point averages.


Now at the close on Thursday, the price is 19.  We can create a new Three Point Average from {T=11, W=15, H=19} and that gives us 15.  So Thursday's three point average is 15.


Moving Averages


We can compute a new three point average every day.  We can plot all these averages on a graph.  Traders call this graph a 3-day moving average.  Actually, the average does not move.  It is a graph of many individual averages. And each average averages three points, not three days. 


To compute the N-Point Average, simply sum up the previous N data points and divide by N.


AN = (Pt-1 + Pt-2  + ... + Pt-N) /N



Exponential Averages


An Exponential Average is also a misnomer. It is actually not an average at all.  (See the section on Exponential Averages, above.)  It is more precisely a lag or a filter.  To compute the Exponential Lag, use the formula:


Lagt = Lagt-1 + dt * (Pricet - Lagt-1) / TC




Lagt = Lagt-1 + dt * Ratet

Ratet = (Pricet - Lagt-1) / TC




The Time Constant (TC), determines the rate at which the Lag travels toward the Price.  The Time Constant does not indicate a number of points.  It has dimensions of time. 


The equation determines the rate at which the Lag seeks the price.  It also uses this rate to compute the change in the Lag over the next time interval, dt.




A 3-day moving average is really a series of  three-point (two day) averages while a 3-day exponential average is really an Exponential Lag with a Time Constant of 3-days.


There is no particular reason for the 3-day moving average and the 3-day exponential average to have anything in common.


They happen to look somewhat similar and we can use them both as indicators.


To get a feel for the similarities and differences between moving averages and exponential averages, we can observe their responses to various types of price patterns, namely: step, ramp and sine.



In this Step Response Test, Price suddenly rises from 0 to 10 in a step-up pattern.


MA-5 rises in a straight line, from 0, and reaches 10 in 5 days.  EA-5 (TC = 5) begins rising with the same slope as MA-5; MA-5 and EA-5 track each other closely for the first two days.  Thereafter, EA-5 lags behind MA-5. At the point where MA-5 reaches its goal at 10, EA-5 is at about 6.28. 


EA-3 initially rises faster than both MA-5 and EA-5.  On day 4, EA-3 and MA-5 are both at about 8.  Thereafter EA-3 lags behind MA-5. When MA-5 reaches 10, EA-3 is at about 8-1/2.


Summary: At first, EA-5 tracks MA-5 better.  Later, EA-3 tracks MA-5 better.




In this Ramp Response Test, Price increases at a rate of one unit per day, in a ramp-up pattern.


On day 5, I have enough data to start computing MA-5.  The initial value is 3.  I also initialize a 3-day Exponential Average on the same date to equal the price at 5.  Over time, the EA-3 converges with MA-5.


Likewise, EA-5 converges with MA-9.


Some people like to use a rule of thumb to find TC for an exponential average to normalize it with an n-day moving average.


TC = (N + 1) / 2


For example, a nine-day moving average has N = 9.  Plugging 9 into the formula above gives TC = 5.  An exponential average with a Time Constant of 5 is equivalent to a 9-Point Moving Average, in the matter of tracking steady ramping prices.


Note again that this adjustment factor is only true in very limiting circumstances, namely for tracking prices that ramp at a steady rate.  Of course, if you can find prices that ramp at a steady rate, you don't need any averages, you just need to get in on it.



In this Sine Response Test, the price oscillates between 0 and 20 in a sinusoidal pattern.  The MA-5 tracks closer than both the EA-5 and the EA-3.




Moving Averages and Exponential Averages track stepping prices and sinusoidal prices differently;  they track ramping prices similarly enough that we can find a Time Constant for an Exponential Average that makes it fit perfectly over a Moving Average.