Portfolio Variance

The two-asset variance formula shows exactly how correlation between assets determines whether diversification meaningfully reduces risk.

1. The idea

For a portfolio of two assets with weights $w_{1}, w_{2}$ (summing to 1), the portfolio variance is $σ_{p}^{2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} ρ σ_{1} σ_{2}$ , where $ρ$ is the correlation between the assets' returns.

When $ρ = 1$ , the assets move together and there is no diversification benefit: $σ_{p}$ is simply the weighted average of the two volatilities. As $ρ$ falls below 1, the cross term shrinks and $σ_{p}$ becomes strictly less than the weighted-average volatility — the mathematical source of diversification.

When $ρ = - 1$ , it's possible to choose weights that make $σ_{p}^{2} = 0$ exactly, constructing a riskless combination from two risky assets, which also connects directly to the minimum-variance hedge ratio.

2. Why it matters

This formula is the two-asset building block for the general $w^{T} Σ w$ portfolio variance, and interviewers use it to test intuition about correlation and diversification quickly with clean numbers.

3. Core mathematics

Two-asset portfolio variance

σ_{p}^{2} = w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 w_{1} w_{2} ρ σ_{1} σ_{2}

Perfect correlation case

ρ = 1 \Rightarrow σ_{p} = w_{1} σ_{1} + w_{2} σ_{2}

Minimum variance weight (two assets)

w_{1}^{*} = \frac{σ _{2}^{2} - ρ σ _{1} σ _{2}}{σ _{1}^{2} + σ _{2}^{2} - 2 ρ σ _{1} σ _{2}}

4. How to recognize it

Think of this when…

Any two-asset variance or standard deviation calculation
'How does correlation affect portfolio risk' conceptual questions
Finding the minimum-variance combination of two assets

5. Worked example

Worked example

Two assets have $σ_{1} = 0.1$ , $σ_{2} = 0.2$ , $ρ = 0$ , with weights $w_{1} = w_{2} = 0.5$ . Find portfolio volatility.

1
Apply the formula with the cross term zero (uncorrelated).
$σ_{p}^{2} = 0.25 (0.01) + 0.25 (0.04) + 0 = 0.0025 + 0.01 = 0.0125$
2
Take the square root.
$σ_{p} = 0.0125 \approx 0.1118$
Answer: Portfolio volatility ≈ 11.2%, below the simple average of 15%

6. Common mistakes

Averaging volatilities directly instead of variances plus the covariance cross term
Forgetting the factor of 2 on the covariance term
Assuming diversification always helps regardless of correlation sign and magnitude

7. Quick check

If correlation between two assets is 1, portfolio volatility equals:

Diversification benefit increases as correlation:

8. Practice

8 QuantSolve questions use this concept.

Practice this concept

9. Related methods

This lesson is conceptual; see the topic hub for methods.

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