Exponential
Lags
(Exponential Averages)
(c)
2005 by Ed Seykota
Summary
In
this section we examine one of the basic tools for detecting trends, the
Exponential Lag. Traders sometimes refer to the exponential lag as
an exponential average  although a lag is not, strictly, an average. Exponential
Lags differ from one another
only in their Time Constants. The Time Constant determines the
speed with which the Lag tracks the price. An
Exponential Lag with a shorter Time Constant tracks the Price more
closely. Traders spend considerable time and effort running simulations
to find Time Constants that optimize the profitability of their trading
systems. For theoretical, very smooth longterm price moves (ones
without short term noise and fluctuation) a very short Time Constant
optimizes profit. For real price moves (ones that do include a lot
of noise) very short Time Constants produce devastating strings of
shortterm whipsaws. In general, Exponential Lag systems do
not work very well for any kind of cyclic markets. They work well
for markets with sustaining trends. 
Your
Car Bumper is a Physical Analog
If
you grasp the front bumper of your car, and try to push it downward, and
lean heavily on it, you might be able to move it a couple
inches. Or, if you try to lift it with all your might, you might
be able to raise it a couple inches. If you are feeling energetic
and also like exercise machines, you might be able to rock the car up
and down by alternating between pushing down and pulling up. You might
be able to move the bumper a few inches every couple seconds.
Now,
if you try to alternate between pushing and pulling faster (with a
higher frequency, say twice per second), you may find you can only move
it slightly. If you try still higher frequencies, say 10 times per
second or more, you may find you can't seem to budge it at all.
In
this example, the car acts as a "lowpass" filter. That
is, it passes or responds to low frequencies and does not pass or
respond to high frequencies. 
LowPass
Filter
Exponential
averages, as we use them in trading, are also lowpass filters.
They respond to the low frequency secular trend of prices and do not
respond to the high frequency noise. Long term trend followers
want to filter out the high frequency noise and follow the long term
trend, so longterm trend traders all use some form or another of
lowpass filtering. Many of them use exponential averages. Other
methods include moving average, weekly rules, support and resistance
lines and trend lines.
Most
methods of lowpass filtering are similar. The type of method matters
less than the value of the time constant. 
How
to Compute it
The
computational form of the exponential lag or exponential tracker is:
EL_{t}
= EL_{t1} + dt * (P  EL_{t1}) / TC
EL_{t}

current
Exponential Average

EL_{t1}

previous
Exponential Average

dt

delta
time since last computation

P

current
Price

TC

averaging
Time Constant

Say
you compute your 10day TC Exponential average once per day. That
makes dt = 1 day and TC = 10 days. Note, the rampequivalent moving
average has an averaging time of 2*TC  1 = 19 days, from TC =
(MAT + 1) / 2.
Say
the price is historically at $10 and it jumps to $20.
Then, in the equation, (P  EL_{t1}) is the difference between the
price ($20) and the Exponential Lag ($10) or $10. We take one
tenth of this amount (1 / TC) or $1, and add it to the Exponential
Average to get the new Lag which now stands at $11. 
A
Flow Chart of the Computation
Flow
Chart for the Lag (Exponential Average)
The
valve [ (P  EA) / TC ] adjusts the flow rate of correction that
changes the Exponential Average. When the difference between Price
and Exponential Average is large, the correction rate is large. As
the Exponential Average approaches the price, the correction rate
decreases. The Time Constant also determines the correction
rate. The longer the time constant, the slower the correction rate. 
Computation
by Spreadsheet
We
can see this process at work, in a spreadsheet. To get a gut feeling for
this process, I recommend you carry out this computation by hand.
Spreadsheet
Computation
for
Exponential Average
If
you carry this out by hand or on a spreadsheet, you get the classic graph
where the Exponential Average decays toward its new equilibrium
value. The Exponential Average rises toward the goal at a rate
proportional to the distance it still has to go. The closer it gets,
the slower it changes. 
Computation
in a Graph
Graph
of Spreadsheet Computation
for
Exponential Average
Price
(dark blue) suddenly rises from 10 to 20. The Exponential Average of
price (light blue) rises exponentially toward the new value. Traders
use exponential averages (and other forms of tracking) to place stoploss
orders in an attempt to lock in profits. Price and Exponential
Average have units of dollars ($). The correction rate ( [P  EA]/TC)
has units of dollars per day ($/day).
As
long as the price remains above the Exponential Average, the Exponential
Average keeps moving higher, although at exponentially small rates.
If we set a protective sell stop at the current Exponential Average, we
can use it to lock in more and more profit.
Engineers
call this kind of chart a Step Response. It shows how the
Exponential Average responds to a step up in price. We can also show
a chart of how the Exponential Average responds to a series of steps, in
this case a few up, and then a few down.
We
can see that the Exponential Average tracks the price and gives a sell
signal when the price crosses beneath it. 
Exponential
Average can Detect Trend Change
Exponential
Average
as
a Trading Tool
The
Exponential Average (light blue) tracks the price (dark blue). When the
Price crosses below the Exponential Average, the crossover signals a
change in trend from up to down. Traders who place a trailing
stoploss order at the Exponential Average Price receive a fill at the
next price. In this case, they enter an order to sell on a stop at $45 and
get a fill at $40.
The
Exponential Average Tracks best when the prices move slowly and the
Average has time to track the price. The Exponential Average tracks
the price, smoothes the price and lags the price. Some engineers call an
Exponential Average a filter, smoother, tracker or lag. 
Tracking
as a Function of Frequency
One
way to see how the Exponential Average responds to a variety of
frequencies is to draw a Frequency Response curve. This curve shows
how far the Exponential Average moves in response to stimulation at
various frequencies. The result is similar to the result you get
with the car bumper. The Exponential Average has enough time to move
at low frequencies and hardly moves for high frequencies.
10Day
Exponential Average
Tracks
a 50day Cycle
The
EA reaches about 13.5 on the first cycle
10Day
Exponential Average
Tracks
a 25day Cycle
The
EA reaches about 12.5 on the first cycle
10Day
Exponential Average
Tracks
a 10day Cycle
The
EA reaches about 11.5 on the first cycle
These
runs show that, for high frequency price cycles, the Exponential Average
does not have time to track the price. It tracks more fully at low
frequencies. The Exponential Average is essentially a lowpass filter. 
Looking
for the Best Average
Another
way to see this frequency effect, in a way that relates directly to
trading, is to keep the price curve constant and change the Time Constant
for each run. This is the method system designers use to find the
system parameters that work best on historical data.
In
the case of prices that move very smoothly and sinusoidally, of course,
the best exponential average is a very short one, since it tracks very
closely and gives a signal quickly after the price changes direction.
In
actual practice, few traders use very short Exponential Averages, or even
just one Exponential Average since the price may get in a mode in which it
crosses back and forth over the Exponential Average repeatedly and
generates a string of whipsaws on alternating days.
Traders
generally use a fast average and a slow average, and wait for the fast
average to cross over the slow average before executing a trade.
This minimizes the dithering effect, at the expense of waiting for the
price to establish its direction.
In
the next study, I show profit versus two moving averages of various
lengths. To normalize the study, I measure the Time Constants in
fractions of the period of the sinusoidal price curve.
Exponential
Trading System
.5
& .225 ==> 70%
Black:
Price,
Green:
Slow EA (.5). TC = .5 X the period of the price.
Red:
Fast EA (.225). TC = 221/2% of the period of the price.
The
short bars indicate the crossovers. The circles indicate the trade
prices. Note that by the time the red line crosses the green line,
the price is already way past the red line, so the trader does not get
prices close to the point of crossover. In this case, the trader
sells near the bottom and buys near the top of each cycle, losing about
70% of the move on each trade.
Exponential
Trading System
.55
& .075 ==> 10%
Black:
Price,
Green:
Slow EA(.55). TC = .55 X the period of the price.
Red:
Fast EA(.075). TC = .075 X the period of the price.
Note
that by the time the red line crosses the green line, the price is already
somewhat past the red line, so the trader does not get prices close to
either average at the point of crossover. In this case, the trader
sells a little below the midpoint and buys a little above the midpoint
of each cycle, losing about 10% of the move on each trade.
Exponential
Trading System
.10
& .025 ==> + 50%
Black:
Price,
Green:
Slow EA(.1).
Red:
Fast EA(.025).
In
this case, the trader sells a little after the top and buys a little after
the low of each cycle, making about 50% of the move on each trade.
Exponential
Trading System
.05
& 0.0 ==> + 80%
Black:
Price,
Green:
Slow EA(.05).
Red:
Fast EA(0.0).
In
this chart, the red line overlays the black line exactly, hiding it.
This 2Average system becomes a 1Average system. The trader sells
right after the top and buys right after the low of each cycle, making
about 80% of the move on each trade. 
Summary
Chart
The
summary chart below shows that increasing the TC for either filter tends
to diminish profit.
Across:
TC for Slow Average, 0 to 140%, increments of 10%.
Down:
TC for Fast Average, 0 to 37.5%, increments of 2.5%.
Top
left corner (0,0), bottom right corner (140, 37.5).
Percentages
are of the period of price oscillation.
Summary
of Double Exponential Average Tests
for
Smooth, Sinusoidal Price Moves
In
general, fast averages work much better than slow averages for smooth
sinusoidal price moves.
For
single moving average systems (fast average TC = 0) all values of TC for
the single average are profitable. See top line.
For
double moving average systems with the fast TC = 2.5%, values of slow TC
below 100% are profitable. See second line.
For
double moving average systems with the fast TC = 5%, values of slow TC
below 50% are profitable. See third line.
Increasing
either the slow or fast TC reduces profitability. 
Prices
are, however, neither Smooth nor Sinusoidal
If
have a prices series that behaves regularly and smoothly, such as the
computer sinusoidal prices in these studies, then we can simply use the
shortest possible Time Constants to optimize our profits
Prices
do not, however, behave so nicely. Actual price histories contain a
superposition of sine waves with a wide range of frequencies. If you
choose a Time Constant short enough to generate profits at one frequency,
you find it is also long enough to generate losses at a higher
frequency.
In
the limit, a very short Time Constant produces devastating strings of
short term whipsaw losses.
In
general, Exponential Averages do not work very well as a trading system,
for sinusoidal markets. Traders know that trendfollowing systems
tend to lose money in sideways, trading range markets, and make money in
longterm trending markets.

Exponential
Average and Complex Waves
In this study,
we test our double average system on a superposition of two sine
waves. We see that there is no averaging system that shows a profit.

