Lesson 15 of 25 · 7 min

Correlation asymmetry: rising markets against falling ones

Eighteen up months measured about zero and eleven sharply down months measured 0.70, and the sign of the return was not the cause.

Measured on the same history, correlation in rising months and correlation in falling months are different numbers. The gap is real and reproducible. The usual explanation for it is wrong, and the right one changes what you do with the figure.

Return to lesson 13's sixty-month history of six holdings. Restrict the sample to the eighteen months in which the broad market returned more than plus 0.8 percent and the average pairwise correlation is minus 0.0031 — effectively nothing. Restrict it instead to the eleven months in which the market fell more than 6.0 percent and the figure is 0.7006. Two subsets of one history, seventy points apart.

The common explanation is that falling prices cause holdings to move together, as though decline itself were a mechanism. That is testable. Restrict the sample to the fifty ordinary months only and split those by direction: the seventeen up months give 0.0220 and the thirteen down months give minus 0.0125. Within a single regime, the sign of the market does essentially nothing.

So the asymmetry is a composition effect. The down-side sample is where the high-factor-volatility regime lives, because that is what a drawdown is — lesson 14's shared driver becoming large, and large in the negative direction. Measured downside correlation is high not because returns were negative but because the negative returns and the high shared volatility are the same event. The figure is still the right one for a stress calculation; the story attached to it is what is wrong, and it matters because it determines where the effect is expected to appear. It does not appear in an ordinary bad month.

One measurement trap sits underneath this. Conditioning on a narrow band of market outcomes truncates the shared driver's variation inside the sample, which pushes the measured correlation down. That is why the up-month figures come out near zero rather than near the 0.25 the ordinary regime carries. Conditional correlations are biased downward by construction, so the honest comparison is always between two conditional figures, never between a conditional one and a full-sample one.

Key terms

Conditional correlation: A correlation computed only on periods meeting some condition, such as the market falling. Comparable with other conditional figures, not with a full-sample one.
Correlation asymmetry: The observation that downside-conditioned correlations exceed upside-conditioned ones. Produced by which regime occupies the downside sample, not by the direction of returns.
Conditioning bias: The downward pull on a correlation measured inside a narrow band of outcomes, caused by the shared driver having little room to vary within the band.

Worked example

Splitting one history by direction, twice

Lesson 13's illustrative sixty-month history of six holdings, ten months of which are a drawdown. Average of the fifteen pairwise correlations in each subset.

All sixty months: 0.6029
18 months with the market above +0.8%: -0.0031
11 months with the market below -6.0%: 0.7006
Ordinary months only: 17 above +0.8%: 0.0220
Ordinary months only: 13 below -3.0%: -0.0125
Asymmetry surviving once regime is held fixed: About nil

The seventy-point gap in the first split is entirely which regime each subset contains. Inside one regime, up months and down months correlate the same.

Check yourself

On one history, months when the market rose correlate at about 0.02 and months when it fell sharply correlate at 0.70. Restricted to ordinary months only, the up and down figures are 0.02 and -0.01. What produced the 0.70?

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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.