Correlation and covariance

Diversification works only to the degree that assets move differently. The two numbers that measure that relationship are covariance and correlation.

Covariance: do they move together?

Covariance measures whether two assets tend to be above or below their own averages at the same time.

covariance(A,B) = average of (A return minus A mean) times (B return minus B mean)

• Positive covariance: they usually rise and fall together.
• Negative covariance: when one is up, the other tends to be down.
• Near zero: their movements are largely unrelated.

The trouble with covariance is that its units are unintuitive (return squared), so its size is hard to interpret on its own.

Correlation: the standardized version

Correlation rescales covariance to a clean range from minus 1 to plus 1:

correlation(A,B) = covariance(A,B) / (stdev(A) times stdev(B))

• Plus 1: the assets move in perfect lockstep — no diversification benefit at all.
• 0: unrelated — solid diversification.
• Minus 1: perfect mirror images — in theory you could combine them into a riskless position.

Portfolio variance for two assets

The whole point is what happens to a blend. For weights w in A and (1 minus w) in B:

portfolio variance = w squared times varA
                   + (1-w) squared times varB
                   + 2 times w times (1-w) times correlation times stdevA times stdevB

The final term is where diversification lives. Lower correlation shrinks it, dragging total variance down. With correlation of plus 1 the formula collapses to a simple weighted average and the benefit disappears.

The real-world catch

Correlations are not constants — they are estimates from a past window, and they drift. Worse, they tend to spike toward plus 1 precisely in crises, exactly when you were counting on diversification to protect you. The 2008 crash humbled many portfolios that looked well diversified on paper. Treat every correlation as a fragile estimate, not a law.