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Portfolio 3: Minimum Variance Portfolios with No Short Sales

Consider now the fully invested, minimum variance portfolio given by the solution to

but with the added (inequality) constraint that the weights must all be non-negative, that is,

This is an example of an inequality constraint; in this case, short sales are prohibited. The portfolio that solves this constrained minimization problem cannot be solved directly as we did for portfolios 1 and 2. Rather, we must iterate using numerical procedures (variants of Newton's method, for example). Excel has a data tool called Solver, which we can use to solve problems such as this. (Recall, we used Solver to find a solution to the linear programming problem of hedging future pension liabilities by selecting among several bonds of varying duration.)

To illustrate, let's take a look at a hypothetical three-asset portfolio with covariances and expected returns given by:

The diagonal of matrix A contains the asset return variances. The two off-diagonal elements indicate that asset 1 and asset 3 returns are positively correlated (, which you ...