Quantitative Methods & RiskProbability & Distributions for Traders
Correlation, Covariance & Diversification
How positions move together
A single position's risk is its volatility. A portfolio's risk is not just the sum of its parts — it depends on how the parts move together. Covariance measures the joint variability of two return series; correlation is covariance normalized to the range −1 to +1, which makes it comparable across pairs. Diversification — the only genuine free lunch in finance — is the mathematical consequence of combining assets that are not perfectly correlated.
Covariance and correlation
Cov(X,Y) = E[(X−μ_X)(Y−μ_Y)] ρ(X,Y) = Cov(X,Y) / (σ_X · σ_Y)
Two-asset portfolio variance
σ²_p = w₁²σ₁² + w₂²σ₂² + 2·w₁·w₂·ρ·σ₁·σ₂
Correlation is not stable — it spikes in a crash
The single most dangerous assumption in portfolio risk is that the correlations you measured in calm markets hold in a crisis. They don't. In a panic, almost everything sells off together — correlations rush toward +1 and the diversification you thought you had evaporates exactly when you need it. Stress-test your portfolio with correlations forced to +1, not just the historical estimate.
Diversification
Reducing portfolio variance by combining assets that are not perfectly correlated. It lowers risk without necessarily lowering expected return — the only 'free lunch' in finance. Its benefit is exactly the cross-correlation term in the portfolio-variance formula, and it shrinks toward zero as correlations rise toward 1.
Correlation ≠ causation, and ≠ tail dependence
Two assets can be linearly uncorrelated (ρ ≈ 0) yet crash together — linear correlation misses tail dependence. Copulas exist to model joint tail behaviour separately from the marginals. For a working trader the takeaway is simpler: never assume a hedge holds in the tail just because the correlation looked favourable in the body of the distribution.
Knowledge check
You hold equal weights in two assets, each with 20% volatility. If their correlation is 0, the portfolio volatility is about 14%. What happens to portfolio volatility if the correlation rises to +1?