When you combine two investments, the risk of the basket is not simply the average of the two risks. It depends on how the assets move relative to each other. Portfolio variance quantifies that combined risk in a single number, and this calculator computes it for a two-asset portfolio in one step. Enter the weights, each asset's variance, and the covariance between them, and you immediately see the portfolio variance, the resulting standard deviation, and how much risk you removed through diversification.
Understanding this number is the foundation of Modern Portfolio Theory: Harry Markowitz showed in 1952 that you can reduce risk without sacrificing expected return simply by pairing assets that do not move in lockstep. This tool puts that formula to work.
How to Use
- Enter the Weight of Asset 1 as a decimal fraction between 0 and 1. A value of 0.6 means 60% of your money sits in asset 1.
- Enter the Weight of Asset 2. For a fully invested two-asset portfolio, the two weights should add up to 1 (for example 0.6 and 0.4).
- Enter the Variance of Asset 1. If you know the standard deviation instead, square it: a 20% volatility means a variance of 0.04.
- Enter the Variance of Asset 2 the same way.
- Enter the Covariance between the two assets. A positive covariance means they tend to rise and fall together; a negative covariance means they offset each other.
- Read the result card: portfolio variance, portfolio standard deviation, the naive weighted-average risk you would have without diversification, and the implied correlation.
The result updates as you type. Compare a few covariance values from strongly positive to negative to feel how sensitive your risk is to the relationship between the two holdings.
The 2-Asset Portfolio Variance Formula
For a portfolio holding two assets, the variance of the overall return is:
σₚ² = w₁²·σ₁² + w₂²·σ₂² + 2·w₁·w₂·σ₁₂
Where:
- w₁ and w₂ are the portfolio weights (fractions of money in each asset).
- σ₁² and σ₂² are the variances of each asset's returns.
- σ₁₂ is the covariance between the two assets.
The first two terms are each asset's own contribution to risk, weighted by the square of its allocation. The third term is the diversification adjustment — it doubles the weighted covariance because it captures the interaction in both directions. The portfolio standard deviation is simply the square root of this variance: σₚ = √σₚ².
A quick worked case: split money 50/50 between two assets with variances of 0.04 and 0.09 and zero covariance. The portfolio variance is 0.25·0.04 + 0.25·0.09 = 0.0325, and the standard deviation is about 0.180, or 18%. Notice this is lower than the naive average of 20% and 30%, which is 25% — the difference is the free lunch of diversification.
How Covariance Drives the Result
Covariance is the engine of diversification. Think of three regimes:
- Positive covariance (assets move together). The cross term is positive, so portfolio variance stays high. If correlation reaches +1.0, the two assets are perfect substitutes and diversification gives you nothing — the portfolio standard deviation equals the weighted average of the two individual deviations.
- Zero covariance (assets are independent). The cross term disappears. You get the full sqrt benefit: combining independent risks shrinks the standard deviation below the weighted average.
- Negative covariance (assets offset). The cross term becomes negative and subtracts from total variance. This is why investors pair stocks with government bonds or gold: when one falls, the other often rises, pulling portfolio risk down further.
You can convert covariance to correlation with ρ = σ₁₂ / (σ₁·σ₂), which always lands between -1 and +1. The calculator shows this implied correlation so you can sanity-check your inputs against typical values — most broad stock indices correlate between 0.3 and 0.7.
Examples
Example 1 — A 60/40 stock-and-bond mix. You put 60% in a diversified equity fund (variance 0.04, i.e. 20% volatility) and 40% in a bond fund (variance 0.09, i.e. 30% volatility — here a higher-risk long-duration bond). The covariance is 0.012, implying a mild 0.20 correlation. The portfolio variance works out to 0.03456, and the portfolio standard deviation is about 18.6%. The naive weighted-average risk would be 24%, so diversification removes roughly 5.4 percentage points of volatility.
Example 2 — Uncorrelated assets. With a 50/50 split, the same two variances, and a covariance of 0, the portfolio variance drops to 0.0325 and the standard deviation to about 18.0%. The diversification benefit grows to nearly 7.0 percentage points because the assets no longer move together.
Example 3 — Perfectly correlated holdings. If the covariance rises to 0.06 (correlation = 1.0), the portfolio variance becomes 0.0625 and the standard deviation exactly equals the weighted average of 25%. The diversification benefit falls to zero — you have effectively doubled up on the same risk.
Example 4 — An ultra-volatile satellite position. A 50/50 split pairing a low-risk asset (variance 0.09) with an extremely volatile one (variance 1,000,000, equivalent to 1000% annualized volatility) still computes cleanly: the portfolio variance is about 250,000.02 and the standard deviation about 500.0, showing that a single wild position dominates the whole basket no matter how you weight it.
FAQ
What is portfolio variance in simple terms?
Portfolio variance is a single number that measures how much the returns of your whole two-asset portfolio are expected to swing. It squares the individual asset risks, weights them by allocation, and then adds an adjustment for how the two assets move together. Its square root is the portfolio standard deviation, which most investors read as annualized volatility.
How is portfolio variance calculated for two assets?
Use the formula σₚ² = w₁²·σ₁² + w₂²·σ₂² + 2·w₁·w₂·σ₁₂. Multiply each weight by itself, multiply by that asset's variance, then add twice the product of the two weights and the covariance. Taking the square root gives the portfolio standard deviation.
What is the difference between variance and standard deviation?
Variance is the average squared deviation from the mean, expressed in squared units — hard to interpret directly. Standard deviation is its square root, expressed in the same units as returns (for example percent per year). Investors usually report standard deviation; variance is the intermediate value the math needs.
How does covariance affect diversification?
Positive covariance raises portfolio variance because the assets amplify each other. Negative covariance lowers variance because the assets offset. When covariance is zero, the cross term vanishes and you still get a sqrt benefit from combining independent risks. Diversification benefit is largest when correlation is negative and shrinks to zero at correlation = +1.
What is a good covariance or correlation for diversification?
For maximum diversification you want assets with low or negative correlation. Most stock-and-bond pairs sit near 0.0 to 0.2, which delivers a meaningful risk reduction. Two broad stock market indices often correlate 0.6–0.9, giving little benefit. Assets that correlate near -1 (such as certain long-short or hedged positions) can nearly eliminate portfolio risk.
Can portfolio variance be negative?
No. Variance is a squared quantity, so it is always non-negative. If an input gives you a negative variance, that is a data error — the calculator rejects it. The covariance between two assets can be negative, but the resulting portfolio variance still cannot drop below zero.
Do the two weights have to add up to 1?
For a fully invested two-asset portfolio, yes — weights of 0.6 and 0.4 sum to 1. If you hold cash, the weights sum to less than 1, and the leftover cash contributes zero variance. Negative weights represent short positions and are allowed in more advanced models, but this calculator assumes long-only allocations between 0 and 1.
Disclaimer
This portfolio variance calculator is provided for informational purposes only and does not constitute investment, financial, or tax advice. Results depend entirely on the variance and covariance inputs you enter, which are historical or assumed values; future returns and risks will differ. Past volatility is not a guarantee of future behavior. Always build and verify your investment plan with a qualified financial professional before making allocation decisions.