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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:

 

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having strength or power greater than average or expected.

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potent: having or wielding force or authority.

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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 up-trend.  If lower, we have a down-trend.  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 minute-to-minute, day-to-day and year-to-year.  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 long-term monotonic up-trend investment. 

 

Say we compound one penny at a three percent per year interest rate from year Zero-AD 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 now-trend (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 up-jump.

 

 

 

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 5-day 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 5-day perspective. Note that all these projections depend on our historical perspective. If we use a 20-day perspective rather than a 5-day 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 time-constantth 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 5-Day 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 = 5-Day Lag

Green = 20-Day Lag

Blue = 50-Day Lag

 

 

 

Chart of Daily Rates of Change

 

Black = DROC, Basis 5-Day Lag

Red = DROC, Basis 20-Day Lag

Green = DROC, Basis 50-Day Lag

 

 

The 5-day trend is initially the highest. On day 12, the 20-day trend becomes the highest.  On day 28, the 50-day trend becomes the highest.

 

 

For More Information

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