Covariance

Two computational routes exist, and the exam expects both. With a joint probability distribution, multiply each deviation of the first variable from its mean by the corresponding deviation of the second variable, weight each product by the probability of that joint outcome, and sum across all outcomes. Equivalently, take the expected value of the product of the two variables and subtract the product of their individual expected values.

With historical observations, pair the two series period by period, multiply the paired deviations from their respective sample means, sum the products, and divide by the number of observations minus one. The degrees-of-freedom adjustment mirrors sample variance and produces an unbiased estimate. Note that the covariance of a variable with itself is simply its variance, which is why variances sit on the diagonal of a covariance matrix.

Covariance carries the units of both variables multiplied together, so a stock-return pair measured in percentage points and a pair measured in decimals produce numbers that differ by a factor of ten thousand while describing an identical relationship. Rescaling one variable rescales the covariance proportionally, which means the raw figure says nothing about strength on its own.

Correlation solves this by dividing covariance by the product of the two standard deviations, yielding a dimensionless value bounded between negative one and positive one. The relationship runs both ways and is heavily examined: given a correlation and two standard deviations, covariance equals the correlation multiplied by both standard deviations. Only the sign of a covariance is directly interpretable; magnitude requires the standardised form.

Portfolio variance for two assets equals the sum of each weight squared times its variance, plus twice the product of the two weights and the covariance between the assets. That final cross term is the entire mechanism of diversification: holding standalone volatilities fixed, lower or negative covariance reduces portfolio risk, and the benefit disappears only when assets are perfectly positively correlated. Extending to many assets, the covariance matrix holds variances on the diagonal and pairwise covariances off it, and the number of distinct covariance estimates grows roughly with the square of the number of assets, which is the practical constraint on mean-variance optimisation.

Covariance also drives regression and beta. A simple regression slope equals the covariance between the independent and dependent variables divided by the variance of the independent variable; an asset's beta is its covariance with the market divided by market variance.

A candidate should be able to build a covariance from a joint probability table, convert freely between covariance and correlation, insert the covariance term into a two-asset portfolio variance calculation, and derive a beta or regression slope from covariance and variance inputs. Recognising that the covariance of any variable with itself is its own variance often shortcuts a matrix question.

Two caveats matter conceptually. First, covariance captures only linear co-movement; a strong curved dependence can produce a covariance near zero, so zero co-movement and statistical independence are not equivalent claims. Second, covariance is estimated from historical samples and is unstable across regimes, meaning optimisers built on backward-looking matrices can understate risk precisely when correlations converge in a crisis.