**Averages**

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.

**A**_{N}
= (P_{t-1} + P_{t-2 }+ ... + P_{t-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:

**Lag**_{t}
= Lag_{t-1} + dt * (Price_{t} - Lag_{t-1}) / TC

**or**

**Lag**_{t}
= Lag_{t-1} + dt * Rate_{t}

**Rate**_{t}
= (Price_{t} - Lag_{t-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.

**Summary**

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.