Dependence, Covariance & Correlation

Two observations are dependent when learning one changes what should be believed about the other. Correlation is a useful linear summary, but “uncorrelated” is not the same as “independent” except under special models.

Covariance and correlation #

For variables $X$ and $Y$,

$$Cov(X,Y)=E[(X-E[X])(Y-E[Y])],$$

$$\rho_{XY}=\frac{Cov(X,Y)}{\sigma_X\sigma_Y}, \qquad -1\le\rho_{XY}\le1.$$

Covariance has product units; correlation is dimensionless. The variance of a sum exposes why dependence matters:

$$Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y).$$

Positive covariance makes combined outcomes swing together; negative covariance can offset them.

Life example #

Travel time by car and bus can both rise on a snow day. Averaging two transport estimates does not create two independent pieces of evidence when weather drives both.

Trading example #

Suppose two positions each have variance 0.2275. If independent, their sum has variance 0.455. If covariance is 0.15, variance becomes $0.455+0.30=0.755$. The expected profit can add normally while the risk grows much faster than an independence assumption suggests.

Application to no-hedge #

No-hedge positions commonly share the same reference feed and execution logic. A stale or biased reference can move every estimated $q$ in the same direction. Aggregate exposure by common settlement event, signal, venue, and time window before sizing; never apply a single-trade limit separately to correlated copies.