Sample Covariance, Correlation, and Bivariate Dispersion

A quant researcher backtesting a factor never observes the population correlation between a forecast score and next-period return — only a finite sample of paired observations, from which a sample correlation must be estimated. That estimate is the information coefficient (IC), the single number that decides whether a signal is worth trading. This note is the bivariate sibling of the previous lesson: exactly as $S^2$ estimated the population variance $\sigma^2$ with an $(n-1)$-corrected divisor, sample covariance and sample correlation estimate the population quantities $\text{Cov}(X,Y)$ and $\rho(X,Y)$ defined in Covariance and Correlation. It proves the sample covariance is unbiased by the identical algebraic move used for $S^2$, then draws a distinction that trips up practitioners routinely: cross-sectional dispersion (how assets differ from one another at a single point in time) versus time-series volatility (how one asset varies across time) — two applications of the same formula, resting on very different independence assumptions.

Learning objectives. By the end of this note you should be able to: