Correlation and diversification

In short

• Portfolio variance = squared-weight variance terms + covariance terms. Only the covariance terms depend on co-movement.
• Independent (zero-covariance) assets leave only the variance terms; negative covariances cut risk further.
• Expected return does not depend on correlation, so lower correlation means less risk for the same expected return: the diversification benefit.
• At $\rho$ = +1 there is no benefit: $\sigma_p$ is the weighted average of the $\sigma$s. The lower $\rho$, the bigger the benefit.
• Changing weights traces out a curve of risk–return combinations; with low correlation, adding some of a riskier asset can even lower total risk.