How do I get portfolio volatility?

Three-Asset Portfolio Variance

Variance of a portfolio of three assets from their weights, volatilities and the three pairwise correlations. Each pair contributes 2 w_i w_j σ_i σ_j ρ_ij, so low or negative correlations pull the total below the weighted sum of variances.

When to use: Use to see how much diversification a third holding adds, and how the pairwise correlations, not just the volatilities, drive total risk. Take the square root for portfolio volatility.

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Formula

\sigma_p^2 = \sum_{i=1}^{3} w_i^2 \sigma_i^2 + 2 \sum_{i<j} w_i w_j \rho_{ij} \sigma_i \sigma_j

Variables

Symbol	Name	Description	Unit
PortVar	Portfolio Variance	Variance of a 2-asset portfolio return as a decimal²	variance
w1	Weight 1	Portfolio weight in asset 1 as a decimal	%
w2	Weight 2	Portfolio weight in asset 2 as a decimal (typically 1 − w1)	%
w3	Weight of Asset 3	Fraction of the portfolio in asset 3	%
Sigma1	Volatility 1	Standard deviation of asset 1 returns as a decimal	%
Sigma2	Volatility 2	Standard deviation of asset 2 returns as a decimal	%
Sigma3	Volatility of Asset 3	Standard deviation of asset 3 returns	%
Rho12	Correlation 1–2	Correlation between assets 1 and 2	integer
Rho13	Correlation 1–3	Correlation between assets 1 and 3	integer
Rho23	Correlation 2–3	Correlation between assets 2 and 3	integer

Real-Life Examples

Example 1: Stocks, Bonds and a Diversifier

Weights 50/30/20 in assets with volatilities 18%, 10% and 25%; correlations 0.3 (1–2), 0.5 (1–3), 0.1 (2–3).

Given

w1 = 50.0000%w2 = 30.0000%w3 = 20.0000%Sigma1 = 18.0000%Sigma2 = 10.0000%Sigma3 = 25.0000%Rho12 = 0.30Rho13 = 0.50Rho23 = 0.10

Step-by-Step

1.Own-variance terms: 0.25×0.0324 + 0.09×0.01 + 0.04×0.0625 = 0.01150

2.Cross terms: 2×(0.5×0.3×0.3×0.18×0.10 + 0.5×0.2×0.5×0.18×0.25 + 0.3×0.2×0.1×0.10×0.25) = 0.00642

3.σ_p² = 0.01792000

Result:0.01792000 (return²)

Portfolio volatility is √0.01792 = 13.4%, well below the 16.7% weighted average of the three volatilities. The imperfect correlations did that.

Example 2: A Hedge Pair Plus Cash

Two 20%-volatility assets at −0.5 correlation, 40% each, plus 20% in a 5% asset uncorrelated with both.

Given

w1 = 40.0000%w2 = 40.0000%w3 = 20.0000%Sigma1 = 20.0000%Sigma2 = 20.0000%Sigma3 = 5.0000%Rho12 = -0.50Rho13 = 0.00Rho23 = 0.00

Step-by-Step

1.Own-variance terms: 0.16×0.04 + 0.16×0.04 + 0.04×0.0025 = 0.01290

2.Cross term: 2×0.4×0.4×(−0.5)×0.2×0.2 = −0.00640

3.σ_p² = 0.00650000

Result:0.00650000 (return²)

Volatility 8.1% from assets that are 20% volatile each: the negative correlation cancels half the risk. A correlation of +1 would have given 16.2%.

Frequently Asked Questions

Take the square root of the variance. Variance is reported because it is what adds up; volatility is what people quote.

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