Correlation

The sample calculation begins with the sample covariance: for each observation, take the deviation of the first variable from its mean, multiply it by the corresponding deviation of the second variable, sum these products across all paired observations, and divide by the number of observations minus one. That figure is then divided by the product of the two sample standard deviations.

Dividing by the standard deviations rescales the statistic so that the units cancel, forcing the result into the interval from negative one to positive one. The sign comes entirely from the covariance term, since standard deviations are never negative. The magnitude answers a different question from the sign: it measures how tightly the paired observations cluster around a straight line, not how steep that line is. A slope of 0.1 and a slope of 10 can both accompany a correlation of 0.95.

A sample correlation is an estimate, and small samples produce large estimates by chance. The standard test of the null hypothesis that the population correlation equals zero uses a t-statistic formed by multiplying the sample correlation by the square root of the sample size minus two, then dividing by the square root of one minus the squared sample correlation. The statistic is compared against a t-distribution with degrees of freedom equal to the sample size minus two.

The structure of that expression carries a practical lesson: significance rises with both the magnitude of the correlation and the number of observations. A correlation of 0.30 may be insignificant across twenty monthly returns and decisively significant across four hundred daily returns. Candidates should also note that outliers exert heavy leverage on the estimate, so a single extreme observation can manufacture or destroy an apparent relationship.

Correlation enters portfolio variance directly. For a two-asset portfolio, variance equals the squared weight of the first asset times its variance, plus the squared weight of the second times its variance, plus twice the product of the two weights, the two standard deviations and the correlation. Because that final term scales linearly with correlation, the diversification benefit is entirely a function of how far the correlation sits below positive one. At a correlation of positive one, portfolio standard deviation is simply the weighted average of the components; below that, it is strictly lower.

The same statistic underpins hedge ratios, tracking error against a benchmark, factor models where residuals are assumed uncorrelated, and multiple regression, where high correlation among independent variables produces multicollinearity and unstable coefficient estimates.

Expect to convert between covariance and correlation in either direction: given a correlation and two standard deviations, the covariance is their triple product. Expect to read scatter plots and match them to candidate correlation values, and to recognize patterns where a strong deterministic relationship exists but the linear measure is near zero because the association is curved or symmetric.

Two further limitations deserve attention. Spurious correlation arises when two series move together through pure chance, through a shared third variable, or through a common trend such as inflation or the passage of time; the statistic itself cannot distinguish this from a genuine economic link. And correlation never establishes direction of causation. Sound practice pairs the number with an economic rationale for why the relationship should persist out of sample.