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Diversification and Correlation Matrix

Diversification works because portfolio variance depends on correlations while expected return does not. It is the closest thing in finance to a free lunch, and it has three limits worth understanding precisely.

How far it goes

For nn equally weighted assets with equal variance σ2\sigma^2 and equal pairwise correlation ρ\rho:

σp2=σ2n+n−1nρσ2  →n→∞  ρσ2\sigma_p^2 = \frac{\sigma^2}{n} + \frac{n-1}{n}\rho\sigma^2 \;\xrightarrow{n \to \infty}\; \rho\sigma^2

The first term is idiosyncratic risk, which vanishes. The second is systematic risk, which does not.

With ρ=0.3\rho = 0.3 and σ=20%\sigma = 20\%, the floor is 0.3×20%≈11%\sqrt{0.3}\times 20\% \approx 11\%. No number of stocks gets below it.

Start at a correlation of 0.2 and push it toward 1, which is what happens to a portfolio in a crisis. The bow flattens and the lowest available risk climbs back toward the safer asset held alone, so the diversification you were counting on disappears exactly when you need it.

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