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Two-asset minimum-variance weight with long-only bounds

When the variance of the return difference is positive, the quadratic stationary weight is (σB²−Cov(A,B))/(σA²+σB²−2Cov(A,B)).

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State the model objective

Minimize the variance of wRA+(1−w)RB for a fixed compatible covariance record. This is a variance-only calculation, without expected returns, costs or an investor suitability objective. A long-only model restricts w to [0,1].

Differentiate the variance quadratic

w∗=σB2−cσA2+σB2−2c,c=Cov(A,B)w^*=\frac{\sigma_B^2-c}{\sigma_A^2+\sigma_B^2-2c},\qquad c=\mathrm{Cov}(A,B)

The denominator is Var(RA−RB) and must be positive for this expression.

Check the permitted interval

For a coherent covariance model with positive denominator the quadratic is convex. If its stationary weight is outside [0,1], the constrained minimum is the nearer endpoint. For standard deviations 10%/20% and correlation 0.9, c=0.018, denominator=0.014 and the unconstrained weight is 0.022/0.014≈1.571429; the long-only minimum occurs at w=1.

Handle the zero denominator explicitly

If Var(RA−RB)=0, the second-moment dispersion of the two returns is identical and the variance quadratic is flat. Do not divide by zero or invent a unique stationary weight. Under equal positive dispersions and correlation +1, every fully invested weight has the same model variance.

Keep this reference separate from the tool

The public calculator reports entered weights and a sampled curve; it does not optimize or choose an allocation. This formula explains a model boundary, not a personal investment recommendation.

Further references