Correlation and Covariance: Compare Aligned Return Series

Align the observations first

Choose matching timestamps, return intervals and currency bases. A daily source-account return paired with a follower's next-day return answers a lagged question rather than a same-period one. Price levels, account balances and returns are also different variables; label the chosen series before computing any statistic.

R's statistics documentation describes covariance, correlation and missing-value choices. For sample covariance, sum products of paired deviations from each series mean and divide by n − 1. Pearson correlation divides covariance by the product of the corresponding sample standard deviations.

Calculate a simple linear example

Take three hypothetical paired returns in percentage-point units: A = −1, 0, +1 and B = −2, 0, +2. Both means are zero. The products of paired deviations are 2, 0 and 2, summing to 4. Sample covariance is 4 ÷ 2 = 2 squared percentage points.

A has sample standard deviation 1 and B has sample standard deviation 2. Correlation is 2 ÷ (1 × 2) = 1. B's movements are twice A's in this illustration, showing that perfect correlation does not mean equal monetary or percentage exposure. With decimal returns, covariance becomes 0.0002 while correlation remains 1.

See why zero correlation is narrower than independence

Keep A = −1, 0, +1 and define C = 1, 0, 1 by squaring each A observation. C is determined entirely by A, so the variables are not independent. Yet their sample covariance is zero: the negative and positive paired deviation products cancel. Their Pearson correlation is therefore zero despite that nonlinear relationship.

This small constructed example is a mathematical demonstration, not a market model. It explains why a single correlation coefficient cannot rule out shared nonlinear or tail behavior.

Inspect data choices and instability

Dropping unmatched dates pair by pair can give different samples for different entries of a correlation matrix. Filling absent valuations with zeros can also change the result. Preserve the missing-data rule and the number of paired observations for each calculation. A constant series has zero standard deviation, making its Pearson correlation undefined.

Historical correlation can vary by time window and conditions. A high positive relationship between two assets does not mean a long and short position in them will cancel every relevant risk; quantity, volatility scale, currency, basis and contract differences still matter. A low sample correlation can also miss joint extreme losses.

For copied-account review, compare actual return construction before interpreting correlation as replication quality. Common deposits, stale marks or duplicated source data can create misleading similarity. Report the paired series, date range, count, scatterplot and meaningful exceptions so the coefficient remains an auditable description rather than a standalone claim of diversification or execution fidelity.