I tried a simple correlation experiment in Excel to test this out. First I created perfectly anti-correlated series A = {-1, 1, -1... } and B = {1, -1, 1...}, such that Correl(A,B) = -1. I then added slope to the original series where slope={1,2,3...} to form two new series doing pairwise addition. That is A[0] + slope[0], A[1] + slope[1], etc. These new series express the kind of perfect market pairing result you often see in text books. The resulting correlation is now 0.85. Interesting.

In other words, adding equal slope to a negative correlated series resulted in a highly correlated pair.

Wait, but I actually want a portfolio behavior that's smooth and growing, right? So, maybe the appropriate thing is to detrend the series first and then do correlation.

Cheers,

Kevin

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Adding Slope to Negative Correlated Series

CorrWithSlopeAdded.jpg (28.46 KiB) Viewed 7662 times

Perfect Negative Correlation

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The way people often describe it, is "measure the correlation of the equity curve returns."

If you're testing without positionsizing (such as one-lot tests), returns = ( equity - equity[(i-1)] )

If you're testing with positionsizing (such as fixed fractional), returns = ( (equity / equity[(i-1)]) - 1 )

The idea is to produce a scatter diagram with returns(instrument_1) on the horizontal axis, and returns(instrument_2) on the vertical axis. The correlation function measures the grouping-ness [or non-grouping-ness] of the returns in the scatter diagram.

If the points are perfectly smeared out on the scatter diagram (math talk: uniformly distributed over the domain), then the correlation is zero. But if the points are grouped together in something approximating a straight line, then the correlation is large. If the straight line slopes upward, the correlation is a large positive number (such as +0.8733), and if the straight line slopes downward, the correlation is a large negative number (such as -0.93698).

BTW: this suggests partitioning the equity curve return data into four pieces, corresponding to the four quadrants of the scatterplot.
1. (return_A > 0), (return_B > 0)
2. (return_A < 0), (return_B > 0)
3. (return_A < 0), (return_B < 0)
4. (return_A > 0), (return_B < 0)

The ideal holy grail would be to find A and B such that there are no points at all in the 3rd quadrant. The third quadrant is the Double Whammy zone, the two-stage rocket to bankruptcy.

Thanks Stan. Focus on holding period returns (HPR's) is the approach suggested in literature, and follows what you describe for a period of one unit of time. But I would think that the same comment applies ... you want low correlation, not negative correlation of HPRs. Perfect negative correlation of HPRs would leave the account churning.

After shooting my self in the foot, Kevin eloquently described what I would a call desirable objective function for correlation:

In other words, adding equal slope to a negative correlated series resulted in a highly correlated pair.

Wait, but I actually want a portfolio behavior that's smooth and growing, right? So, maybe the appropriate thing is to detrend the series first and then do correlation.

I am puzzling over how William Eckhardt derived (mathematically, or seat of the pants) the correlation categories (Closely, Loosely). I notice that they are not numeric. So I assume that correlation need not be precise, however the concept of diversification is the dominate objective.

I had to read your post twice, since the words and graphics didn't mix. Then I checked your spreadsheet. You must have meant portfolio (A+B) was most preferable, since it had 50% of the MaxDD of (A+C). Also, (A+B) is most negatively correlated (-0.67). Per the same net result (A+B) gives the smoothest growth curve.

Hiramhon wrote:If A and B each have positive expectation, it's more desirable for them to be negatively correlated than zero correlated.

That may be true only if you are a unidirectional trader. An example of this are the people who are IBD (Investor's Business Daily) traders who are usually long-only position traders.

Many futures/commodity traders are long/short: if you are concerned about correlation factors, then you might want zero magnitude correlations, NOT + or - 1.00 correlations. If you trade a simple trend-following breakout system, when one instrument goes long, the -1.0 instrument will go short .. and your daily equity changes will be twice as large.

I do not see why Hiramhon's example would not work for short or longs. If one looks at his spread sheet, it considers returns from an equity regardless of position. From this perspective longs and shorts trend in the same direction (north). Correlation does make sense.

I am not convinced correlation is something to worry about. I get signals every day. The signals are taken at different days , weeks, time intervals (all are not taken on the same day) thus by just letting trend following system run naturally, it takes care of correlation issues by itself -- auto-majik. (The assumption being that one is trading from a large set of instruments)