Lesson 11 of 25 · 7 min

Estimation windows, and the correlation you happen to measure

Two six-month halves each measured about 0.22, and the twelve months containing both measured 0.91.

A correlation is not measured on a pair of holdings. It is measured on a pair and a window, and changing only the window can change the answer past recognition. Here is a case where the pooled figure sits far above both of its own halves.

Twelve monthly returns for holding A: 2.4, 3.3, 3.5, 0.2, 1.5, 3.3, then minus 3.6, minus 3.9, minus 4.1, minus 4.3, minus 3.6, minus 2.6 percent. For holding B over the same months: 1.8, 2.1, 1.1, 0.1, 2.5, 0.4, then minus 2.7, minus 2.7, minus 1.7, minus 4.1, minus 2.9, minus 2.6 percent. The first half is a rising stretch and the second is a falling one.

Compute lesson 2's ratio three times. On months one to six the correlation is 0.2010. On months seven to twelve it is 0.2337. On all twelve months together it is 0.9135. The pooled figure is not between its halves; it is four times either of them.

The mechanism is the mean. Each half's correlation measures deviations from that half's own mean, so within the rising stretch the months scatter above and below plus 2.37 percent for A, and within the falling stretch they scatter around minus 3.68 percent. The pooled calculation measures every month against the full-period means of minus 0.66 and minus 0.73 percent instead. Against those, every first-half month is a positive deviation in both series and every second-half month is a negative deviation in both. What the 0.9135 is measuring is the level shift between the halves, not the month-to-month co-movement inside either.

A correlation computed across a regime change is partly a measurement of the change. It will be high whether or not the two holdings move together within either regime.

Key terms

Pooling effect: The way combining two sub-periods with different average returns raises the measured correlation, because the difference in averages becomes shared deviation.
Sub-period correlation: A correlation computed within one stretch of a longer sample. It answers a different question from the pooled figure and can be far lower.
Level shift: A change in the average return between two stretches of a sample. A correlation computed across one will register it as co-movement.

Worked example

The same twelve months, measured three ways

A: 2.4, 3.3, 3.5, 0.2, 1.5, 3.3, -3.6, -3.9, -4.1, -4.3, -3.6, -2.6. B: 1.8, 2.1, 1.1, 0.1, 2.5, 0.4, -2.7, -2.7, -1.7, -4.1, -2.9, -2.6. Illustrative monthly percentages.

Months 1-6 correlation: 0.2010
Months 7-12 correlation: 0.2337
Half-period means for A: +2.37% then -3.68%
Full-period means: -0.66% for A and -0.73% for B
Months 1-12, Sab and the divisor: 76.5125 and 83.7540
Months 1-12 correlation: 0.9135

Three correlations, one pair of holdings, twenty-four numbers. The only thing that differed between the three was which months were included.

Check yourself

Two holdings correlate 0.20 within a six-month rising stretch and 0.23 within the six-month falling stretch that follows, yet 0.91 across all twelve months. How can the combined figure exceed both parts?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.