Correlation on returns and on price levels

A correlation on daily log returns measures whether two stocks step in the same direction on the same day. A correlation on price levels measures linear co-movement of the two level paths, not whether the prices have similar dollar heights. Shared trends can create high level correlation even when daily moves differ.

The case here is the pair CleanSpark and MARA Holdings, which share 365 sessions inside a window of 365 sessions. On those shared sessions, the Pearson r of the two daily log-return series is 0.838. The Pearson r of the two price-level series over the same sessions is 0.456.

The returns figure is larger in this window. The level figure compares the two price paths. The returns figure removes those paths and compares only the day-to-day steps. The two correlations answer different questions, so either one can be larger.

We can see the gap and the shared variance in a few lines.

Pearson r, daily log returns: 0.838

Pearson r, price levels: 0.456

Gap, levels minus returns: 0.456 - 0.838 = -0.382

R squared on the returns pair: 0.838 * 0.838 = 0.7022

The Spearman rho on the returns pair is 0.835. The p-value is under 0.0001.

So what just happened. We computed a correlation on the price levels and got a number for the association between the two price paths. We computed a correlation on daily log returns and got a number for day-to-day co-movement. Their gap shows that those two questions give different answers in this window. The returns figure is the one that describes whether the two stocks moved together on a given day.