Lesson 14 of 25 · 6 min

The mechanism: one common factor dominating everything else

Raise the shared driver's volatility and every pairwise correlation rises, with nothing whatever happening at any company.

The previous lesson measured convergence. This one derives it, and the derivation needs only the split Track 7 lesson 2 already set out: a holding's return is a sensitivity times a common driver, plus a residual belonging to that company alone.

For two holdings with a sensitivity of 1 to the same driver and residuals unrelated to each other and to it, the correlation between them takes a form with no free parameters left in it.

r = (sF x sF) / ((sF x sF) + (se x se))

r: The correlation between the two holdings. Also, in this case, the share of each holding's variance the common driver supplies.
sF: The volatility of the common driver over the period. The shared part.
se: The volatility of each holding's own residual. The unshared part, assumed the same for both here.

Hold the residual volatility at 5.6 percent a month and give the driver a volatility of 3.2 percent. The shared variance is 10.24 and the residual variance is 31.36, so each holding's total variance is 41.60 and its volatility is 6.45 percent a month. The correlation is 10.24 divided by 41.60 = 0.2462. Now raise the driver's volatility to 8.0 percent, a factor of 2.5, and change nothing else. Shared variance becomes 64.00, total variance 95.36, single-name volatility 9.77 percent, and the correlation 64.00 divided by 95.36 = 0.6711.

Read what did not change. No residual volatility moved, so nothing happened at any company. No sensitivity moved. The correlation went from 0.25 to 0.67 because the shared part of the variance grew relative to the unshared part — and that ratio is all a correlation ever was. It can rise because the shared part grew, because the unshared part shrank, or both.

Those two parameter sets generated lesson 13's history, which is why its measured 0.29 and 0.76 sit close to the theoretical 0.25 and 0.67 without matching them. Fifty months and ten months estimate those values with error, and the error in the ten-month figure is large. Lesson 16 puts a number on it.

Key terms

Common factor: A driver whose movements show up in many holdings at once. Each holding has its own sensitivity to it, and that shared part is what creates correlation.
Residual: The part of a holding's return the common factor does not explain. Residuals are unrelated across holdings, which is why they cancel and the factor does not.
Variance share: The proportion of a holding's variance supplied by the common factor. At equal unit sensitivities it equals the pairwise correlation exactly.

Worked example

One driver, two volatility levels

Two holdings, each with unit sensitivity to one common driver and a residual volatility of 5.6% a month. Only the driver's volatility changes. Illustrative monthly figures.

Residual variance, both cases: 31.36
Driver at 3.2%: shared variance and total: 10.24 and 41.60
Correlation, and single-name volatility: 0.2462 and 6.45%
Driver at 8.0%: shared variance and total: 64.00 and 95.36
Correlation, and single-name volatility: 0.6711 and 9.77%
Company-specific parameters changed: None

A 2.5-fold rise in one shared driver's volatility took the pairwise correlation from 0.25 to 0.67. Every company-level input was identical in both columns.

Check yourself

Two holdings each have unit sensitivity to a common driver with 6.1% monthly volatility, and residual volatility of 4.8% a month. What is the correlation between them?

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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.