Minimum-variance portfolio

Minimum-variance portfolio is the combination of risky assets with the lowest possible portfolio variance, marking the leftmost point on the Markowitz efficient frontier.

ByHoang TruongUpdated July 11, 2026

FrameworkModern portfolio theory

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With Asset A at 20% volatility and Asset B at 10%, uncorrelated, the minimum-variance weight on Asset A is 20%. Of €10,000 invested, €2,000 goes to Asset A and €8,000 to Asset B. That mix gives a portfolio standard deviation of about 8.94%, lower than either asset held alone, including Asset B's own 10%.

Where it fits

TopicRisk, Return & the CAPMAdvancedSubjectCorporate FinanceAdvanced

The formula

LaTeX

w_A^{*} = \frac{\sigma_B^2 - \text{Cov}_{AB}}{\sigma_A^2 + \sigma_B^2 - 2\,\text{Cov}_{AB}}

Variables

$w_{A}^{*}$: Minimum-variance weight on Asset A (decimal)
$σ_{A}^{2}$: Variance of Asset A's return (decimal squared)
$σ_{B}^{2}$: Variance of Asset B's return (decimal squared)
$Cov_{A B}$: Covariance between Asset A's and Asset B's returns (decimal squared)

The closed-form weight on Asset A that minimises the variance of a two-asset portfolio; the weight on Asset B is 1 − wA*.

LaTeX

\sigma_p^2 = w_A^2 \sigma_A^2 + w_B^2 \sigma_B^2 + 2 w_A w_B \, \text{Cov}_{AB}

Variables

$σ_{p}^{2}$: Portfolio variance (decimal squared)
$w_{A}$: Weight on Asset A (decimal)
$w_{B}$: Weight on Asset B (decimal)
$σ_{A}^{2}$: Variance of Asset A's return (decimal squared)
$σ_{B}^{2}$: Variance of Asset B's return (decimal squared)
$Cov_{A B}$: Covariance between Asset A's and Asset B's returns (decimal squared)

The variance of a two-asset portfolio given each asset's weight, variance, and covariance with the other.