Correlation and Covariance Between Assets

Diversification only works when the things you hold actually move differently. Correlation is how you check.

Covariance: raw co-movement

Covariance measures whether two assets move together:

Cov(X,Y) = Σ (xi − μx)(yi − μy) ÷ (n − 1)

• Positive → they tend to rise and fall together
• Negative → one rises when the other falls
• Near zero → no linear relationship

The problem: covariance has units and scale, so you can't compare it across pairs. A covariance of "12" means nothing on its own.

Correlation: the normalized version

Pearson correlation fixes the scale problem:

ρ(X,Y) = Cov(X,Y) ÷ (σx · σy)

The result is bounded between −1 and +1:

ρ	Relationship
+1	Perfect positive
+0.7 to +0.9	Strong positive
0	No linear relationship
−0.7 to −0.9	Strong negative
−1	Perfect negative

A trader holding two assets with ρ = 0.95 is, in risk terms, holding the same thing twice.

Why it matters for portfolios

Portfolio variance depends on correlation:

σp² = w1²σ1² + w2²σ2² + 2·w1·w2·ρ·σ1·σ2

With ρ = 0, risk falls sharply by combining assets. With ρ = 1, there is no benefit at all. Lower correlation → better diversification → smoother equity curve.

The dangerous catch: correlation is not constant

Correlations measured in calm markets (say, ρ = 0.2 between stocks and bonds) often spike to +0.8 or higher in crashes. The exact moment you need diversification most is when it disappears. This is why "diversified" portfolios still suffer deep drawdowns in crises.

Spearman: when the relationship isn't linear

Pearson only captures linear relationships. If asset B tends to follow A with a curve or lag, Pearson may read near zero even when they're tightly linked. Spearman correlation ranks the data first, capturing monotonic (rank-based) relationships — often a better choice for noisy financial series.

Practical use

1. Compute rolling correlations, not a single full-history number
2. Watch for correlation regime changes — a breakdown in a stable ρ is a signal itself
3. Stress-test diversification under historical crisis windows where ρ jumped
4. Never assume two assets are uncorrelated just because they were last year

Summary

Covariance gives you raw co-movement; correlation makes it readable. Together they let you build portfolios whose risk actually diversifies — and warn you when that diversification is an illusion that will vanish in the next crash.