Two-Asset Minimum Variance Depends on the Feasible Weights

State the problem before using its formula

A two-asset minimum-variance calculation chooses a weight w in asset A and 1-w in asset B to minimize modeled return variance. It does not maximize expected return, establish suitability or minimize every kind of loss. With variances vA and vB and covariance c, the objective is w squared times vA plus (1-w) squared times vB plus 2w(1-w)c. The full-investment condition is already built into this expression. Whether w can be negative, exceed one or occupy only a narrower interval is a separate question. Different answers to that question define different optimization problems, even with identical risk estimates.

The interior solution and its denominator

Differentiating the quadratic gives the unconstrained stationary weight wA = (vB-c)/(vA+vB-2c). The denominator equals the variance of the return difference A minus B. When it is positive, the quadratic is strictly convex and this stationary point is the unique unconstrained minimum. Correlation enters through c = rho times sigmaA times sigmaB; it is not interchangeable with covariance. A large positive covariance can make the numerator exceed the denominator or become negative. Such a weight is not automatically a numerical error. It may be the mathematically correct answer to a problem that permitted leveraged and short positions.

Bounds turn an interior minimum into a boundary

For a strictly convex quadratic over a nonempty interval [L,U], the constrained solution is the unconstrained weight clipped to that interval. This is exact for this one-dimensional problem, not a general recipe for repairing multivariate optimized portfolios. If both assets have lower and upper bounds, the feasible interval for A is the intersection of A's bounds with [1-upperB, 1-lowerB]. An empty intersection means infeasibility. For example, requiring each asset to hold at least 60% is incompatible with full investment. Returning normalized weights would silently change those constraints. The correct mathematical interpretation is that no permitted portfolio exists.

One covariance matrix, three different optima

In the synthetic comparison, A has 10% annual volatility, B has 20%, and correlation is 0.8. Covariance is therefore 0.016. The unconstrained formula gives A a weight of 4/3 and B a weight of -1/3. Its gross exposure is 5/3 even though net weights sum to one. Long-only weights eliminate that short hedge and put all capital in A. Adding an 80% cap on A forces the remaining 20% into B. The corresponding variances are 0.008, 0.01 and 0.01312. Lower variance in the unrestricted problem does not establish that its leverage and borrowing requirements are acceptable.

Degeneracy is not the same as infeasibility

If vA+vB-2c is exactly zero for a valid covariance matrix, A minus B has zero variance. The quadratic is then flat in the allocation weight, and every feasible weight has the same modeled variance. A division by zero cannot select a unique optimum. This differs from an empty feasible set, where no admissible weight exists at all. A very small positive denominator creates a third situation: a well-defined but sensitive solution. Tiny covariance changes can produce large weight changes. Numerical tolerances should distinguish these situations rather than classify every difficult input as either a valid ordinary solution or a generic failure.

Validate the matrix, not just the answer

For two assets, nonnegative diagonal variances and c squared no greater than vA times vB characterize a positive semidefinite covariance matrix. If this condition fails materially, a computed negative portfolio variance reflects invalid inputs rather than a novel hedge. Correlation must lie between -1 and 1, and both volatilities must use the same return units and frequency. Small rounding errors near a boundary require an explicit tolerance policy; silently changing a correlation or diagonal entry creates a different model. Zero-volatility inputs deserve particular care because correlation with a zero-variance series is not defined, even though a covariance-based problem may remain meaningful.

Interpret the optimum conditionally

The two-asset-minimum-variance reference is most useful when its output is read alongside its feasible interval and objective value. A boundary solution can be fully correct, while a precise interior solution can rely on unstable covariance estimates. Compare nearby admissible weights using the same matrix to understand how much modeled variance is sacrificed by a bound. Financing costs, transaction costs and expected-return requirements would add terms or constraints that this simple objective does not contain. The sources explain the broader optimization framework; the table deliberately remains a transparent quadratic exercise. Reproducing its minimum confirms arithmetic and constraint handling, not an investment strategy.

Primary references

These sources support methods and conventions. The example values are original constructions, not observations extracted from these documents.

• MOSEK Portfolio Optimization Cookbook: Markowitz portfolio optimization
• CVX Group: Portfolio optimization, constraints and leverage limits

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