# Linear Trend Lines, Part I, from Phil McDonnell

April 3, 2009 |

The Chair has issued a challenge for anyone who can prove of disprove the existence of linear trend lines as suggested by Jim Sogi.

The first issue in doing a market study is to develop an adequate definition of what a trend is. Given that the idea is widely and perhaps first used in Technical Analysis it is good to start there. Tom DeMark, a widely known and respected TA guru, defines a trend line as the line connecting two bottoms in a price series. This, of course assumes one has a good definition of a bottom. He defines a bottom as a daily low which has the property that the low of the previous day and the low of the next day are higher than the bottom low. Thus it takes three days to define a bottom day. The most recent bottom cannot be known until one day after it has happened.

The idea of a trend line is to find the most recent bottom and then go back to the next most recent bottom. The whole pattern takes at least 6 days to work out and can be much longer because there can be an arbitrary number of days between the two most recent bottoms. Our own John Bollinger has confirmed that this is essentially what his understanding is of how trend lines are used by practitioners.

The theory of a trend line based on lows is that it acts as support. In other words the market will tend to stay above the line more often than would be expected by chance. It should be noted that nothing in the above definition presupposes an uptrend or a down trend. In an uptrend the second low point is higher than the first low point. And the difference per day defines the slope of the line. In a down trend the second low is lower than the first.

There is another type of trend line which is based on the highs of the day. A high point is defined as the high day between two adjacent days, whose highs are both lower. Again two high point days are required to draw a well defined trend line.

Mathematically it is always possible to draw a line between two points, so one should not be surprised to find trend line patterns in random data as well as real market data. The real challenge is to test whether they occur more frequently or less frequently. More importantly do trend lines have any predictive value either as measured by higher probability of a successful trade or a higher average return?

The study looked at 1800 trading days of SPY, the S&P ETF. This period started 100 days ago and went back 1800 days. It should be noted that the SPY was down 1.5 points during this period resulting in an average daily return of 0.00% to 5 decimal places. The study was further classified into four categories based on whether it was a high point or low point trend and whether it was an up or down trend:

Low  Up
Low  Down
High Up
High Down

The measure of profitability was the simple next day close to close return for the trade.

For trends based on Low points we have:

Trend   n     # Up   % Up      Avg Profit     Total Profit
Up     204    109    53.4        -0.212        -43.23
Down    202    115    56.9        +0.031          6.33

For trends based on High points we have:

Trend    n     # Up   % Up      Avg Profit     Total Profit
Up     182     86    47.3        -0.143        -25.98
Down    225    106    47.1        -0.050        -11.22

Most of the results showed about 200 pairs of trend points. This means that the typical trend point pair took about 9 days to form and thus had about 3 days in between points. TA practitioners would expect more days to be up after a trend point pair is put in place. There appears to be weak support for this idea because for both cases based on Low points the % Up seems slightly favorable. However this is belied by the fact that the avg profit for Up trends is actually strongly negative. Thus the preferred strategy is probably to fade the appearance of and up trend pattern.

For High Points the expectation is that the market has reached an extreme. Thus we expect it to fall back. In fact all the above does qualitatively agree with that premise because both the % Up and the Avg Profit are negative.

The key to all of this is to test the results for significance. Often one has noted that when a trend line is broken there is sometimes a dramatic drop. The import of this is not whether it is true or not. Rather if true then it implies a negatively skewed distribution. Thus the standard normal / log-normal assumptions may be too far off. For something like this then that argues that it would be better to choose a bootstrap test for significance so that we do not have to worry about the normality of our data.

That is the subject of Part 2.

Dr. McDonnell is the author of Optimal Portfolio Modeling, Wiley, 2008

## Jim Sogi writes:

Thanks Phil. The problem with defining a 'trend line' is that a rather random number of bars may or may not form the trend 'line'. More than two points would be needed to define a trendline as any two points forms a line, so it become totally random as to which points when only two define it. Then when not all lines touch the support line, then it turns random again. By the way, I was wrong. Random walks DO regularly form 'trend lines' to the naked eye.

One idea is to follow the a variant to the solution to the math problem Buffon's Needle  which determines the probability of a needle of a given length touching two parallel lines when its thrown down. The problem must be restated to determine the probability of a needle of fixed or variable length touching three or more points on a grid of timeseries points. Then the time series could be randomly simulated from actual data, and probability determined or randomly generated and compared to actual data. Here is a nice trailhead with the R code to visualize the problem.

If this code could be altered to solve the above variation, this might help solve this problem.

The solution may require limiting the time period to some defined time period such as a day, week or month so that the 'straw' has a defined length and require that the touches, touch within 1 point of an interval low to give a little slack as you do when eyeballing. However, there appear to be solutions to Buffon's Needle allowing for various or random length needle's. Perhaps Buffon's 'points' could also be lengthened to be short parallel lines that make up the time series, and use the formula to determine the probability that the needle (trendline) crosses three or more time series points to create the trendline.

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