Mean-Variance Portfolio Allocation With A Budget Constraint

Why Portfolio Weights Must Satisfy More Than a Budget

A portfolio allocation is described by the fraction of the total investment assigned to each asset. A budget constraint requires those fractions to sum to one. But investors may also require a particular expected return, and they may want to minimize the portfolio’s variance among all allocations that meet both requirements.

Variance is not generally the sum of each asset’s variance multiplied by its weight. The assets can move together or in opposite directions, so their covariances also affect the result. A precise optimization model must account for these relationships.

Weights, Returns, and Variance

Suppose there are \(n\) assets. Let \(w_i\) be the fraction of the portfolio invested in asset \(i\), and collect the weights in the vector \(\mathbf{w}=(w_1,\ldots,w_n)^\top\). Let \(\mu_i\) be the expected return of asset \(i\), and let \(\boldsymbol{\mu}\) be the vector of expected returns. The expected portfolio return is the weighted average \(\boldsymbol{\mu}^\top\mathbf{w}\).

The covariance matrix \(\mathbf{\Sigma}\) has entry \(\Sigma_{ij}\) equal to the covariance between the returns of assets \(i\) and \(j\). Its diagonal entries are the individual asset variances. The portfolio variance is a quadratic function of the weights.

The objective is the variance, not the standard deviation. Since standard deviation is the nonnegative square root of variance, minimizing variance and minimizing standard deviation select the same feasible portfolio.

A covariance matrix is symmetric and positive semidefinite: for every vector \(\mathbf{v}\), \(\mathbf{v}^\top\mathbf{\Sigma}\mathbf{v}\geq 0\). This condition expresses the fact that the variance of any linear combination of asset returns cannot be negative.

Check Feasibility Before Optimizing

With nonnegative weights summing to one, the expected return is a weighted average of the asset returns. Consequently, a target \(R\) is attainable exactly when it lies between the smallest and largest asset returns, including the endpoints. If \(R\) is strictly between those two values, assets with returns on opposite sides of \(R\) can be mixed to attain it. At either endpoint, an allocation entirely to an asset with that endpoint return attains the target.

If short selling is allowed, negative weights can take the weighted average outside the range of individual asset returns. Feasibility must then be checked using the two equality constraints together, rather than by applying the nonnegative-weight range test.

Finding an Interior Candidate

First consider a candidate for which all weights are unrestricted locally: in particular, none of the nonnegative-weight constraints is binding. The objective is \(Q(\mathbf{w})=\mathbf{w}^\top\mathbf{\Sigma}\mathbf{w}\). Because \(\mathbf{\Sigma}\) is symmetric, its gradient is \(2\mathbf{\Sigma}\mathbf{w}\). Applying the Lagrange multiplier condition to the two equality constraints gives the following system.

The multipliers \(\lambda\) and \(\gamma\) are unknowns along with the weights. Solving these equations produces an equality-constrained stationary candidate. A candidate is an admissible portfolio only if it also satisfies any required nonnegative-weight constraints.

Why a Stationary Candidate Is a Global Minimum

Here is a direct justification of the theorem. Let \(\mathbf{w}\) be any other vector satisfying the same two equality constraints, and put \(\mathbf{d}=\mathbf{w}-\mathbf{w}_*\). Subtracting the constraints for the two vectors shows that \(\mathbf{1}^\top\mathbf{d}=0\) and \(\boldsymbol{\mu}^\top\mathbf{d}=0\). Expand the change in variance:

Stationarity says \(2\mathbf{\Sigma}\mathbf{w}_*=\lambda\mathbf{1}+\gamma\boldsymbol{\mu}\). Therefore the first term in the expansion is

The remaining term is nonnegative because \(\mathbf{\Sigma}\) is positive semidefinite. Hence \(Q(\mathbf{w})-Q(\mathbf{w}_*)=\mathbf{d}^\top\mathbf{\Sigma}\mathbf{d}\geq0\). This proves global minimality. If that quadratic term is strictly positive for every nonzero feasible direction, every other feasible portfolio has strictly larger variance, which proves uniqueness.

Worked Example: An Interior Minimum for Three Assets

Consider three assets with expected returns \(4\%,8\%,12\%\), target return \(8\%\), and covariance matrix \(\operatorname{diag}(1,2,4)\), measured in squared percentage points. With weights \(w_1,w_2,w_3\), the constraints are

Subtract four times the budget equation from the return equation. This gives \(4w_2+8w_3=4\), or \(w_2+2w_3=1\). Set \(w_3=t\). Then \(w_2=1-2t\), and the budget equation gives \(w_1=t\). The variance along the feasible line is

Differentiate: \(Q'(t)=26t-8\). The stationary value is \(t=\frac{8}{26}=\frac{4}{13}\), so \(w_1=\frac{4}{13}\), \(w_3=\frac{4}{13}\), and \(w_2=1-2(\frac{4}{13})=\frac{5}{13}\). Every weight is nonnegative. The weights sum to \(\frac{4+5+4}{13}=1\), and the expected return is \(\frac{16+40+48}{13}\%=8\%\). Direct substitution into the objective gives

Finally, \(Q''(t)=26>0\), so the stationary point is the minimum along the feasible line. Equivalently, completing the square gives \(Q(t)=13(t-\frac{4}{13})^2+\frac{10}{13}\), which verifies the minimum value and its location.

Nonnegative Weights Require Boundary Checks

An equality-constrained stationary point may have a negative weight. If short selling is forbidden, it is then infeasible and cannot solve the stated problem. The minimum may instead occur where one or more weights are zero. A reliable approach is to examine feasible faces: set a suspected boundary weight to zero, impose the remaining constraints, optimize on that face, and compare feasible candidates. For a convex quadratic objective, the minimum over the nonnegative feasible set is global when the appropriate minimum conditions, including boundary conditions, are satisfied.

For a problem with one feasible parameter \(t\geq0\) at a boundary, the one-sided derivative is informative. If the objective increases as \(t\) moves from zero into the feasible region, then that boundary point is the minimum along the feasible set. This is a boundary check, not an assumption that the constraints can be treated as equalities in place of inequalities.

A Common Mistake: Ignoring a Negative Weight

In the preceding boundary example, an incorrect procedure would be to find the unrestricted stationary point and accept it without checking feasibility. Differentiating the displayed quadratic gives \(Q'(t)=8+30t\). Setting this to zero yields \(t=-\frac{4}{15}\). Substituting into the weights gives \(w_3=-\frac{4}{15}<0\), so this candidate violates the no-short-selling requirement. It cannot be the answer to the nonnegative-weight problem.

The correction is to respect the feasible interval \(0\leq t\leq\frac14\). On that interval \(Q'(t)=8+30t>0\), so the variance increases from the left endpoint. The minimum is therefore \(t=0\), as the direct comparison above confirms. An unconstrained stationary point is useful only when it is feasible; otherwise the boundary must be analyzed.

Practical Interpretation

Portfolio variance depends on both individual asset risks and co-movement. An asset with relatively high variance can still affect total portfolio variance through its covariances with other assets. The matrix formula captures these interactions, while the return and budget equations define which allocations are allowed.

The optimization has two distinct checks. First, the equalities must be feasible: the weights must sum to one and deliver the target return. Second, the solution must respect the permitted weight signs. With short selling allowed, a valid stationary solution of the equality-constrained problem gives a global minimum under the positive-semidefinite covariance condition. Without short selling, boundary candidates must also be considered. In either case, substituting the final weights back into the constraints and objective is an essential verification.