Portfolio Variance Calculator
Created by: Olivia Harper
Last updated:
Calculate portfolio variance, standard deviation, expected return, and diversification benefit for a 2 or 3-asset portfolio. Enter each asset's weight, return, standard deviation, and pairwise correlations to quantify how diversification reduces overall portfolio risk.
Portfolio Variance Calculator
FinanceCompute portfolio risk, expected return, and diversification benefit for a 2 or 3-asset portfolio.
Asset Weights, Returns & Standard Deviations (Σ weights: 100% — must equal 100%)
Asset 1
Asset 2
Pairwise Correlations (−1 to +1)
What Is a Portfolio Variance Calculator?
A portfolio variance calculator computes the total risk of a multi-asset portfolio by combining each asset's individual volatility and the correlations between assets.
It shows how much risk is reduced through diversification — one of the foundational concepts of Modern Portfolio Theory (MPT) developed by Harry Markowitz.
By quantifying the diversification benefit, the calculator helps investors understand why holding a mix of imperfectly correlated assets produces a portfolio with lower risk than any single asset held individually — and provides the key inputs needed to optimize portfolio construction.
How Portfolio Variance Is Calculated
For a two-asset portfolio, variance = w1²σ1² + w2²σ2² + 2·w1·w2·σ1·σ2·ρ12, where w = weight, σ = standard deviation, and ρ = correlation.
For three assets, an additional six cross-product terms are added (one for each pair).
Portfolio standard deviation = √(Variance), expressed as a percentage.
The diversification benefit = Weighted Average Standard Deviation − Portfolio Standard Deviation.
The larger this number, the more diversification is reducing portfolio risk.
A correlation of −0.5 between two equally weighted, equal-volatility assets reduces portfolio standard deviation by roughly 30% vs. holding either asset alone.
Portfolio Variance Formula (2-Asset)
σ²p = w1²σ1² + w2²σ2² + 2·w1·w2·σ1·σ2·ρ12
σp = √(σ²p)
E(Rp) = w1·R1 + w2·R2
Diversification benefit = Σ(wi·σi) − σp
Sharpe = (E(Rp) − Rf) / σp
Example Scenarios
Stocks and Bonds — Classic 60/40
Asset 1: US Equities, 60% weight, 10% return, 17% std dev. Asset 2: US Bonds, 40% weight, 4% return, 7% std dev. Correlation: −0.1. Weighted avg std dev: 13%. Portfolio std dev: 11.3%. Diversification benefit: 1.7%. Expected portfolio return: 7.6%. Sharpe (4.5% Rf): 0.27.
Two Stocks with Positive Correlation
Asset 1: Tech ETF, 50% weight, 14% return, 22% std dev. Asset 2: Healthcare ETF, 50% weight, 10% return, 16% std dev. Correlation: 0.6. Weighted avg std dev: 19%. Portfolio std dev: 17.4%. Diversification benefit: 1.6%. At high positive correlation, diversification provides limited risk reduction.
Three-Asset Portfolio with Low Correlation
Stocks (50%, 10%, 17%), Bonds (30%, 4%, 7%), Gold (20%, 6%, 15%). Correlations: stocks/bonds −0.1, stocks/gold 0.1, bonds/gold 0.05. Portfolio std dev drops to ~10.5% vs. 13.1% weighted average — a 2.6% diversification benefit. Adding a low-correlation asset (gold) meaningfully reduces total portfolio volatility.
How People Use This Calculator
- Quantifying the risk reduction from combining assets with different correlations
- Optimizing portfolio weights to maximize the Sharpe ratio
- Understanding why diversification reduces portfolio volatility
- Evaluating whether adding a new asset class improves portfolio risk/return
- Teaching Modern Portfolio Theory concepts with live numerical examples
Portfolio Construction Tips
Diversification's power comes from correlation, not just from holding multiple assets.
Adding 20 stocks in the same industry provides much less diversification than combining stocks with bonds or real estate — the correlations within an industry are high (0.5–0.8), leaving substantial undiversifiable risk.
Correlations are not stable over time.
During market crises (2008, 2020 COVID crash), cross-asset correlations tend to spike toward +1 as investors sell everything — exactly when you need diversification most.
Building portfolios around low average correlations is prudent, but be prepared for correlation convergence in tail events.
The efficient frontier is the set of portfolios that maximize expected return for a given level of risk (or minimize risk for a given return).
To find your optimal portfolio, you can run this calculator multiple times with different weight combinations and plot the expected return vs. standard deviation — the upper-left boundary of those points is the efficient frontier.
Frequently Asked Questions
What is portfolio variance and why does it matter?
Portfolio variance measures the total risk (volatility) of a multi-asset portfolio, accounting for how the assets move relative to each other (correlation). It matters because a portfolio of risky assets can have lower total risk than any individual asset if the assets are imperfectly correlated — this is the mathematical basis of diversification. The portfolio variance formula shows that as correlation decreases toward −1, portfolio risk decreases dramatically, even if individual asset volatilities remain high.
How does correlation affect portfolio risk?
Correlation ranges from −1 to +1. At +1, assets move in perfect lockstep and there is no diversification benefit — portfolio volatility equals the weighted average of individual volatilities. At 0, assets move independently — portfolio volatility is meaningfully lower than the weighted average. At −1, assets move in perfectly opposite directions — a perfectly hedged portfolio can theoretically achieve zero volatility. In practice, correlations between stocks and bonds have historically ranged from −0.2 to 0.3, providing moderate diversification.
What is the diversification benefit?
The diversification benefit is the reduction in standard deviation achieved by combining assets in a portfolio, compared to simply holding each asset in proportion. It is calculated as: Weighted Average Standard Deviation − Portfolio Standard Deviation. A larger number means more risk reduction from combining the assets. The diversification benefit is maximized when assets have low or negative correlations and similar volatility levels.
What correlations should I use for stocks and bonds?
Historical stock-bond correlations have varied significantly over time. From roughly 2000–2021, U.S. stocks and bonds had a mildly negative correlation of around −0.2 to −0.1, providing strong diversification. In 2022, when both fell together in the rate-hiking cycle, the correlation turned positive (+0.4 to +0.6), reducing diversification benefits. For planning purposes, a correlation of 0 to 0.2 between a U.S. equity index and long-term Treasuries is a reasonable current assumption, though this can shift materially in inflationary environments.
How is the portfolio Sharpe ratio calculated here?
This calculator computes the portfolio's Sharpe ratio as: (Portfolio Expected Return − Risk-Free Rate) / Portfolio Standard Deviation, where the risk-free rate defaults to 4.5%. The Sharpe ratio measures the risk-adjusted return of the combined portfolio — a key output of modern portfolio theory. The efficient frontier optimization problem is to find the portfolio weights that maximize the Sharpe ratio.
Sources and References
- Markowitz, H. (1952). "Portfolio selection." Journal of Finance.
- Sharpe, W. F. (1964). "Capital asset prices: A theory of market equilibrium under conditions of risk." Journal of Finance.
- CFA Institute: Portfolio Management — Modern Portfolio Theory
- Fama, E. F. (1991). "Efficient capital markets: II." Journal of Finance.