Portfolio Correlation Matrix

Correlation is a simple but powerful way to understand diversification. When assets are highly correlated, they tend to move together, and diversification benefits are limited. When correlations are low or negative, combining assets can significantly reduce overall portfolio volatility.

1. Correlation definition

Correlation between assets i and j:
ρ(i,j) = Cov(R_i, R_j) / (σ_i σ_j)

Where:
R_i, R_j = returns of assets i and j
Cov = covariance
σ_i, σ_j = standard deviations of returns
        

Correlation is bounded between -1 and 1. It is dimensionless and focuses on co-movement, not magnitude of returns.

2. Correlation matrix structure

For N assets, the correlation matrix is an N × N table:

Diagonal:
ρ(i,i) = 1 for all i

Off-diagonal:
ρ(i,j) = ρ(j,i) for all i ≠ j
        

The diagonal is always 1 because each asset is perfectly correlated with itself. The matrix is symmetric because correlation is the same regardless of order.

3. Diversification intuition

Lower correlations → lower covariance terms in portfolio variance:

Var_p = Σ w_i² σ_i² + Σ Σ w_i w_j σ_i σ_j ρ(i,j)

Holding volatilities and weights fixed, reducing ρ(i,j) reduces Var_p.
        

This is why mixing assets with low or negative correlations can reduce overall risk even if individual assets are volatile.