Correlation
A measure from −1 to +1 of how closely two investments' returns move together.
σp = √(w1²σ1² + w2²σ2² + 2·w1·w2·ρ·σ1·σ2)
Explanation
The correlation coefficient (ρ, “rho”) describes the relationship between two return series:
- +1: they move perfectly together. No diversification benefit.
- 0: no consistent relationship.
- −1: they move perfectly opposite. Risk can, in theory, be eliminated.
Most real asset pairs fall somewhere in between. US and international stocks are typically highly positively correlated. High-quality bonds and stocks have had low or negative correlation for long stretches, but that relationship changes over time; for example, both fell together in 2022 when inflation and interest rates rose.
Two practical cautions:
- Correlations tend to rise in a crisis, when many risky assets fall at once. Diversification still helps, but often less than calm-period statistics suggest.
- Correlation describes direction, not size. Two assets can be uncorrelated and still both lose money.
In scenarios, Monte Carlo runs generate returns for each asset class with the correlations you set (or sensible defaults), so the simulated portfolio’s volatility reflects how its parts interact rather than just their weighted average.
Example
A 60/40 portfolio of stocks (16% volatility) and bonds (6% volatility). The weighted-average volatility is 0.6 × 16% + 0.4 × 6% = 12%. The actual portfolio volatility depends on correlation:
| Correlation | Calculation | Portfolio volatility |
|---|---|---|
| +1.0 | √(0.0092 + 0.0006 + 0.0046) | 12.0% |
| +0.1 | √(0.0092 + 0.0006 + 0.0005) | 10.1% |
| 0.0 | √(0.0092 + 0.0006) | 9.9% |
| −0.5 | √(0.0092 + 0.0006 − 0.0023) | 8.7% |
Here 0.0092 = 0.36 × 0.0256 (stocks), 0.0006 = 0.16 × 0.0036 (bonds), and 0.0046 = 2 × 0.6 × 0.4 × 0.16 × 0.06 (the cross term at full correlation). Lower correlation shrinks the cross term and the overall risk.