On This Page
- What is a minimum variance portfolio?
- How is the minimum variance portfolio calculated?
- Minimum variance portfolio example: an 82/
18 mix - How correlation changes the minimum variance portfolio
- Minimum variance vs equal risk contribution and maximum Sharpe
- Limitations of the minimum variance portfolio
- Bottom line
- FAQ
Key Takeaways
- The minimum variance portfolio is the lowest-variance combination available from a set of assets, and it marks the leftmost point of the efficient frontier.
- For an invertible positive-definite covariance matrix and fully invested unrestricted weights, the closed-form solution uses inverse covariance and no expected-return forecast.
- When correlations are low enough, the portfolio volatility falls below the standalone volatility of every asset it holds.
- Set the asset universe, covariance inputs and weight limits before calculating the minimum.
- Compare sample and shrunken covariance estimates, then apply the portfolio’s position and short-sale limits.
- Minimum variance and maximum Sharpe are different objectives. Compare both using the same asset universe, constraints and risk-free return.
What is a minimum variance portfolio?
A minimum variance portfolio, or global minimum variance (GMV) portfolio, is the mix of assets with the lowest possible return variance within a selected asset universe and set of weight constraints. It needs only the covariance matrix and the constraints, not expected-return forecasts, and it marks the leftmost point of the efficient frontier. Its volatility is the square root of that minimum variance.
On a risk-return chart, the GMV portfolio sits at the leftmost feasible risk level. Expected returns locate it vertically, while the covariance matrix and constraints determine its allocation.
How is the minimum variance portfolio calculated?
With a positive-definite covariance matrix Sigma and weights that sum to one, the unrestricted solution is:
w = Sigma⁻¹1 / (1ᵀSigma⁻¹1).
Its variance is 1 / (1ᵀSigma⁻¹1). Short positions are permitted in this closed-form problem. Add long-only or holding limits and solve the corresponding constrained problem.
For two assets, the first asset's weight is:
wA = (varianceB − covarianceAB) / (varianceA + varianceB − 2 × covarianceAB).
Formula
w = (Sigma^−1 × 1) / (1^T × Sigma^−1 × 1), with minimum variance = 1 / (1^T × Sigma^−1 × 1). Two assets: w1 = (σ2² − rho × σ1 × σ2) / (σ1² + σ2² − 2 × rho × σ1 × σ2)- Sigma is the covariance matrix. This inverse formula uses an invertible positive-definite matrix and fully invested weights without additional position constraints.
- 1 is a column vector of ones with one entry per asset; the denominator is the sum of all entries of Sigma^−1, which normalizes the weights to sum to one.
- w is the vector of portfolio weights; entries can be negative when short selling is allowed.
- σ1 and σ2 are the standard deviations of the two asset returns, and rho is the correlation between them.
- The global minimum variance calculation uses covariance and constraints. Expected returns locate its vertical position on a risk-return chart.
Minimum variance portfolio example: an 82/
An allocator blends two assets. Asset A has 12% annual volatility, Asset B has 20%, and the estimated correlation between them is 0.30. Expected returns are not used, because the minimum variance solution does not need them.
- Covariance term: rho × σ1 × σ2 = 0.30 × 12 × 20 = 72
- Weight in A: (400 − 72) / (144 + 400 − 2 × 72) = 328 / 400 = 0.82, leaving 0.18 in B
- Portfolio variance: 0.82² × 144 + 0.18² × 400 + 2 × 0.82 × 0.18 × 72 = 96.83 + 12.96 + 21.25 = 131.04
- Portfolio volatility: square root of 131.04 = 11.45%, below the 12% volatility of Asset A on its own
- Now assume the true correlation was 0.60, not 0.30. The same 82/18 mix carries variance 96.8256 + 12.96 + 42.5088 = 152.2944, or 12.34% volatility
At correlation 0.30, the minimum uses 82% A and 18% B, with 11.45% volatility. Revalue that same allocation at correlation 0.60 and volatility becomes 12.34%. Reoptimizing with 0.60 selects 100% A and restores 12.00% volatility. These are three distinct portfolio comparisons.
For the chart, add expected returns of 5% for A and 9% for B. The 82/18 mix has 5.72% expected return. It improves on A alone on both displayed dimensions: lower volatility and higher expected return.
How correlation changes the minimum variance portfolio
Holding the same 82/18 weights at correlation 0.60 raises variance to 152.2944 and volatility to 12.34%. Reoptimizing at 0.60 selects 100% A and gives 12.00% volatility.
The correlation threshold here is 12 / 20 = 0.60. Above it, the unrestricted minimum uses a short position in B; a long-only minimum remains entirely in A.
At perfect positive correlation, unequal volatilities allow unrestricted offsetting positions to cancel the common fluctuation. Equal volatilities instead leave every fully invested mix with the same fluctuation. Treat singular cases separately from the matrix-inverse formula.
Minimum variance vs equal risk contribution and maximum Sharpe
Minimum variance minimizes total volatility. Equal risk contribution balances its sources. Maximum Sharpe combines expected excess returns with volatility. Run each comparison on the same universe and position rules.
Use holding limits, sector limits and turnover rules to define the investable result. Compare the resulting weights and portfolio volatility across the selected covariance estimates. Review the portfolio’s concentration and scenario results alongside its variance.
Limitations of the minimum variance portfolio
- Stress the correlation input before you trade the weights. In the example, a correlation of 0.60 instead of 0.30 lifts the 82/18 mix from 11.45% to 12.34% volatility, and reoptimizing moves it to 100% A.
- Above the 0.60 correlation threshold, the unrestricted solution shorts B. Long-only and holding limits change the answer, so solve under the limits you will actually trade.
- The lowest-volatility mix ignores expected return, so run maximum Sharpe on the same universe when risk-adjusted return is the goal.
- Variance scores gains and losses alike, so check tail losses and drawdowns separately before you size the allocation.
- The inverse formula needs a positive-definite covariance matrix, so repair a singular or near-singular estimate, with shrinkage for example, before you invert it.
Bottom line
The minimum variance portfolio is the least volatile mix a set of assets and rules allows, found from covariance alone. Use it as a low-volatility anchor or benchmark, estimate covariance carefully, for example with shrinkage, and apply realistic holding limits before treating its weights as tradable.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- CFA Institute: Portfolio Risk and Return, Part I
- Jagannathan and Ma, Risk Reduction in Large Portfolios: Why Imposing the Wrong Constraints Helps (NBER Working Paper 8922)
- Ledoit and Wolf: Honey, I Shrunk the Sample Covariance Matrix
How entries are sourced, calculated and updated: editorial methodology.