Covariance & Correlation

Covariance & Correlation

Covariance measures how two variables move together. A positive covariance means they tend to move in the same direction; negative means opposite.

Sample covariance between $X$ and $Y$: $\text{Cov}(X, Y) = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})$

Pearson Correlation normalizes covariance to the range $[-1, 1]$: $\rho = \frac{\text{Cov}(X, Y)}{s_X \cdot s_Y}$

where $s_X$ and $s_Y$ are the sample standard deviations.

A correlation of 1 means perfect positive linear relationship, -1 means perfect inverse, and 0 means no linear relationship.

In portfolio construction, correlation determines diversification benefit: assets with low or negative correlation reduce portfolio volatility.

Your Task

Implement:

• covariance(xs, ys) — sample covariance
• correlation(xs, ys) — Pearson correlation coefficient