Trends
The trend is your friend except at the end when it
bends.
Tutorial Exercise
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Most traders agree they like to be
long strong instruments. Traders agree less on the definition of strong.
Some definitions of strong from the web, include:

having strength or power greater
than average or expected. 

potent: having or wielding force
or authority. 

impregnable: able to withstand
attack. 
None of these definitions seem to fit
stocks, bonds or futures exactly. Trading instruments do not possess muscles or
authority or even a physical structure. So traders who use the term strength,
really mean something else, likely, trending up.
The preference for the word, strength
over trending, may owe to the richness and variety of gut associations people
have with strong and the rather sparse and obscure associations they have
with with trend.
A trend is a general drift or
tendency in a set of data. All measurements of trend involve taking a
current reading and a historical reading and comparing them. If the
current reading is higher than the historical reading, we have an uptrend.
If lower, we have a downtrend. In the improbable event of an exact match,
we have a sideways trend.
The direction of the trend depends
upon the method we use to perform the comparison. Real
instruments fluctuate minutetominute, daytoday and yeartoyear. We
have, therefore an enormous supply of historical points to use to determine
trend. As such, we can determine as many instances of trend as we please,
in any direction that we please.
There is no such thing as the
trend; there are countless trends, depending on the method we use to determine a
trend. People typically pick a method for determining trend that fits with
their current positions and/or view of the market. 
There's No Such Thing
as an Uptrend Forever
Note: In
the case of a monotonically increasing series, such as the balance
of a US dollar bank account with daily compounding interest, every
price is above all its predecessors. You can make a claim, then,
that no matter how you measure it, the trend is always up.
However,
in a larger scope, in a world in which the US dollar fluctuates in
value relative to other currencies, and in which banks sometimes
fail, a bank account may not actually continue to be a longterm
monotonic uptrend investment.
Say we
compound one penny at a three percent per year interest rate from
year ZeroAD to the present. We get about $5.48 * 10^23, or around a
half trillion trillion dollars. Clearly someone in those early
days has a penny earning interest, per stories of money lenders in
temples. That no such investment survives today indicates
severe financial setbacks, from time to time, in which people and
whole societies experience collapse and have to start over.
Compounding interest seems to work pretty well for a few hundred
years at a time. 
All methods of defining trends
compare various combinations of historical price points. All trends are
historical, none are in the present. There is no way to determine the
current trend, or even define what current trend might mean; we can only determine historical trends.
The only way to measure a nowtrend
(one entirely in
the moment of now) would be to take two points, both in the now and compute
their difference. Motion, velocity and trend do not exist in the now. They do
not appear in snapshots. Trend does not exist in the now and the phrase,
"the trend" has no inherent meaning. When we speak of
trends, we are speaking, necessarily, from some or another view of history.
There is no such
thing as a current trend. When we speak of trends we are necessarily
projecting our own definitions.
With that in mind, we can proceed
to examine ways to define, compute and use
trends.
Examples of
Computing Trends
Say we have a stock trading for a
long time at a price of $10 and that on day 5 it suddenly jumps to a price of
$20 and then stays there for a long time. Let's examine what a trend might look
like at and after the upjump. 
1. Pick a current price.
For the
current price, use today's close of $20.
2. Pick a historical price.
For the
historical price, use the 5day exponential lag of historical
closes. The lag is still at $10.
3. Compare (1) and (2) to find the
difference.
Difference =
current_price  historical_price
= $20  $10 = $10.
4. Estimate the rate of change, by
normalizing for the lag.
In this case,
we might use 5 days to normalize the result.
The rate of Change = difference /
5 days.
= $10/5 days
= $2/day.
Percent ROC
= $2/day/$10
= .2 or 20% per day
= 73.05 or 7305% per year of 365.25 days
Thus, we might say the stock is
trending up at a rate of about $2/day, 20% per day or 7305% per
year, basis a 5day perspective. Note that all these projections
depend on our historical perspective. If we use a 20day perspective
rather than a 5day perspective, we get projections about a quarter
as large.
5. Update the Lag:
To update
the lag, use:
Lag = Lag + (Currrent_Close
 Lag) / Time_Constant
= Lag = $10 + ($20  $10) / 5 days
= $12
Thus, every day at the
close, we increment (or decrement) the lag by one timeconstant^{th}
of the difference between the current close and the lag. In this
way, the lag tracks the price with a time constant of 5 days.
The Next Day, repeat the process:
1. Price
= $20, again.
2. Historical Price
= $12, the lag from yesterday
3. Difference
= $8
4. Rate of Change
= $8 / 5 days
= $1.60 / day
= $1.60 / day / $12
= .133 or 13.3% per day
5. Update the Lag: Lag = $12 +
$1.60
= $13.60. 
Day 
Price 
Lag 
0 
10 
10.00 
1 
10 
10.00 
2 
10 
10.00 
3 
10 
10.00 
4 
10 
10.00 
5 
20 
12.00 
6 
20 
13.60 
7 
20 
14.88 
8 
20 
15.90 
9 
20 
16.72 
10 
20 
17.38 
11 
20 
17.90 
12 
20 
18.32 
13 
20 
18.66 
14 
20 
18.93 
15 
20 
19.14 
16 
20 
19.31 
17 
20 
19.45 
18 
20 
19.56 
19 
20 
19.65 
20 
20 
19.72 
Spreadsheet of 5Day Lag Computation
After 5 days (the time constant) the
Lag is about 16.72 or about 2/3 of the way to the new price.
If we were to compound continuously, rather than once per day, the
price would be, after 5 days, one "e"
of the way from $10 to $20, where e, Euler's constant is about
0.5772.


Leonhard Euler (1707  1783)
Wearing Trendy Math Hat
Although to penetrate into the intimate mysteries of
nature and thence to learn the true causes of phenomena is not
allowed to us, nevertheless it can happen that a certain fictive
hypothesis may suffice for explaining many phenomena.  Euler 
Chart of Lags of
Various Lengths
Black = Price: Rises
from $10 to $20 on day 5
Red = 5Day Lag
Green = 20Day Lag
Blue = 50Day Lag
Chart of Daily Rates of
Change
Black = DROC, Basis 5Day
Lag
Red = DROC, Basis 20Day Lag
Green = DROC, Basis 50Day
Lag
The 5day trend is
initially the highest. On day 12, the 20day
trend becomes the highest. On day 28, the 50day trend becomes the highest.

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