About the Portfolio Variance Calculator
A portfolio is usually less risky than the average of its parts, because assets that do not move in lockstep partly offset each other. This portfolio variance calculator applies the Markowitz formula to two or three assets: enter each asset’s weight, standard deviation (volatility) and expected return, plus the correlation between each pair, and it returns the portfolio variance, standard deviation and expected return.
It is built for investors comparing allocations such as 60/40 stocks and bonds, for finance and CFA students checking homework, and for advisers explaining why adding a lower-correlated asset can cut risk. The calculator also shows the diversification benefit — the gap between the weighted-average volatility and the actual portfolio volatility — and how much of the total risk each asset contributes.
Inputs should be on the same time basis (normally annualised). Variance is a backward-looking or estimated measure; correlations change over time and tend to rise in market sell-offs, so treat the result as an estimate of typical risk rather than a worst case.
With the default inputs, the portfolio standard deviation is 11.61%. Change any value above to recalculate instantly.
How to use the portfolio variance calculator
- 1Choose two or three assets.
- 2Enter each asset’s portfolio weight, annual standard deviation and expected return.
- 3Enter the correlation for each pair of assets (between −1 and +1).
- 4Read the portfolio standard deviation, variance and expected return.
- 5Compare capital weights with risk shares in the chart to see what really drives risk.
Formula and method
Portfolio variance adds up every pair of assets’ covariance, weighted by how much of the portfolio each represents. The covariance of two assets is their standard deviations multiplied by their correlation. For three assets the formula has three variance terms plus three cross terms (each counted twice): w₁²σ₁² + w₂²σ₂² + w₃²σ₃² + 2w₁w₂σ₁σ₂ρ₁₂ + 2w₁w₃σ₁σ₃ρ₁₃ + 2w₂w₃σ₂σ₃ρ₂₃. The portfolio standard deviation is the square root of the variance.
Unless all correlations are +1, the portfolio standard deviation is below the weighted average of the individual standard deviations; the difference is the diversification benefit. Each asset’s share of risk is wᵢ × Cov(i, portfolio) ÷ σp², and these shares sum to 100%. Expected return is simply the weighted average of the asset returns. Weights that do not total 100% are scaled proportionally.
- wᵢ
- Weight of asset i (fraction of the portfolio)
- σᵢ
- Standard deviation (volatility) of asset i
- ρᵢⱼ
- Correlation between assets i and j (−1 to +1)
- σp
- Portfolio standard deviation
Worked examples
Stocks, bonds and gold (60/30/10)
Weighted-average volatility is 14.1%, but low correlations bring the portfolio standard deviation down to about 11.61% (variance 0.013477). Stocks are 60% of the money yet about 90.6% of the risk, and the expected return is 6.5%.
Classic 60/40 with negative stock–bond correlation
0.6²×0.2² + 0.4²×0.08² + 2×0.6×0.4×0.2×0.08×(−0.2) = 0.0144 + 0.001024 − 0.001536 = 0.013888, so σp = 11.78%. That is 3.4 points below the 15.2% weighted average, for a 7% expected return.
Perfectly correlated assets: no diversification
When correlation is +1 the assets move together, so the portfolio standard deviation equals the 25% weighted average and the diversification benefit is zero.
Frequently asked questions
How do you calculate portfolio variance?+
Multiply each pair of weights by the pair’s covariance (σᵢ × σⱼ × ρᵢⱼ) and add up all the terms, including each asset with itself. For two assets that is w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁σ₂ρ₁₂.
What is the difference between portfolio variance and standard deviation?+
Standard deviation is the square root of variance. Variance is in squared units, which makes it convenient for maths, while standard deviation is in the same percentage units as returns and is what people usually quote as volatility.
Why does correlation matter so much?+
Correlation controls the cross terms. With a correlation of +1 there is no risk reduction; at 0 the cross terms vanish; and below 0 they subtract from variance. Lower correlation means more of each asset’s ups and downs cancel out.
Where do I find standard deviations and correlations?+
Estimate them from historical returns (for example five years of monthly returns, annualised) using a spreadsheet or our standard deviation and correlation tools, or use capital-market assumptions published by fund managers.
Can I use this for more than three assets?+
The same double-sum formula works for any number of assets, but the number of correlations grows quickly (n × (n − 1) ÷ 2). For larger portfolios, group holdings into asset classes or use a spreadsheet covariance matrix.
Results are estimates for educational purposes and are not financial advice. Rates, fees and terms vary — confirm with your lender or a licensed advisor before making decisions.