It is a common refrain that, in a crisis, asset price correlations move towards 1. What was once independent is no longer so, as a large common driver has emerged creating large and often forced flows from leverage unwinds and VaR models that then feed on themselves. This is shorthand – what really happens is correlations move to extremes.
Over the past 40 years or so, generally speaking, stocks and yields have been positively correlated. There were a couple of short-lived hiccups around the taper tantrum and early 2007 which soon reversed. In previous periods of stock weakness, you can see the correlations move decidedly higher.

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