Article: Trend Line Slope

In this article, I will develop the mathematics for the slope of a trend line using the price and time relationship presented in the previous article. Let's work with the model illustrated in this figure.

From the previous article, the next time curve will be t bars away for a given price P. At a time t+1 price would meet the curve at price P1. Now, lets solve for the slope of the trend line shown in blue which connects P and P1.

P = t ^ 2
P1 = (t + 1) ^ 2 = t ^ 2 + 2 t + 1 = P + 2 t + 1

Slope = (Change in price) / (Change in time)

Change in price = P1 - P = P + 2 t + 1 - P = 2 t + 1 = 2 t + 2 - 1 = 2 [ t + 1] -1
Change in time = t + 1

Therefore, slope of P to P1 is = (2 [ t + 1] - 1) / (t+1) = 2 - 1 / (t+1) = 2 - 1 / sqrt( P1 )

If we normalize all prices to consider three significant digits, then all prices will fall in the range of [100 ... 1000]. By substituting the price boundaries into the slope formula, we can get a range of slopes as follows.

For a P1 of 100, the slope of the up trend line to 100 = 2 - 1 / 10 = 1.9
For a P1 of 1000, the slope of the up trend line to 1000 = 2 - 1 / 100 = 1.99
The slope of the up trend line at the midpoint of this price range is 2 - 1 / sqrt(500) = 1.96

Let's call this trend line a 45 degree line because we developed the slope using one unit of price change from P to P1 with one unit of time t. For this 45 degree line, the slope is basically 2. I think this is strong justification as to why Gann used 2 cents as the price grid interval of his daily grain charts. Such a scale layout would naturally give Gann 45 degree angles with a slope of 2 cents per daily bar. I have shown that 2 is the slope of the upward 45 degree trend line that develops from the price and time relationship given by the formula: P = t ^ 2.

One can solve for the slope of the downward trend line from P1 to P to obtain this result:

Slope of P1 to P = (-2 t - 1) / (t-1) = (-2 [t - 1] - 3 ) / (t-1) = -2 - 3 / (t-1) = -2 - 3 / (sqrt( P ) - 1)

For a P of 100, the slope of the down trend line to 100 = -2 - 3/9 = -2.33
For a P of 1000, the slope of the down trend line to 1000 = -2 - 3/99 = -2.03

Again, the slope of the down trend line approaches a value of -2. Therefore, -2 is a good approximation for the slope of a downward 45 degree trend line.