Diversification

Core Concepts

Portfolio Variance (2 Assets)

For a portfolio of two assets with weights w_1 and w_2, volatilities sigma_1 and sigma_2, and correlation rho_12:

sigma^2_p = w_1^2 * sigma_1^2 + w_2^2 * sigma_2^2 + 2 * w_1 * w_2 * sigma_1 * sigma_2 * rho_12

Diversification benefit arises whenever rho_12 < 1, because the portfolio volatility will be less than the weighted average of individual volatilities.

Portfolio Variance (n Assets)

In matrix notation for n assets with weight vector w and covariance matrix Sigma:

sigma^2_p = w' * Sigma * w

This generalizes to any number of assets and captures all pairwise correlations.