One formula. Three terms. The middle one is where diversification gets paid.

Portfolio variance is not a weighted average. It has a cross term, and that cross term carries the correlation. Low correlation between two assets shrinks the cross term, which shrinks the whole variance, even when neither asset got any safer on its own.

Two asset portfolio variance . Markowitz 1952

sigma p squared = w1 squared . sigma1 squared + w2 squared . sigma2 squared + 2 . w1 . w2 . sigma1 . sigma2 . rho

First two terms are the individual variances. The third term is the lever. If rho is 1 it adds full second asset variance to the portfolio. If rho is 0 the term vanishes. If rho is negative the term subtracts. Equity vs gold in calm regimes prints rho close to 0.15.

Equity . Gold mix

Return

Vol

Sharpe

Free vol absorbed

100 / 0 Pure equity book

13.0pct

19.0pct

0.32

80 / 20 Gave up 1.0 pp return

12.0pct

15.9pct

0.31

3.1 pp lower vol

70 / 30 Gave up 1.5 pp return

11.5pct

14.7pct

0.31

4.3 pp lower vol

60 / 40 Gave up 2.0 pp return

11.0pct

13.7pct

0.29

5.3 pp lower vol

36 / 64 Minimum variance point

9.8pct

12.6pct

0.22

6.4 pp lower vol

0 / 100 Pure gold book . dominated

8.0pct

15.0pct

0.07

The minimum variance portfolio sits at 12.6 pct vol. That is below pure equity vol of 19.0 pct and below pure gold vol of 15.0 pct. The mix is calmer than either ingredient. That is not a bug in arithmetic. That is the cross term doing its job.

Two asset toy. Equity return 13 pct vol 19 pct, gold return 8 pct vol 15 pct, correlation 0.15. RFR 7 pct. Min variance solved analytically. Numbers illustrative, not a forecast. Markowitz 1952 Journal of Finance 7:1. Past performance . backtest only . not a guarantee.