An algorithmic approach to deriving the minimum-variance zero-beta portfolio

This paper examines the problem of deriving Black's (1972) minimum-variance zero-beta portfolio. Long's (1971) methods, used by Morgan (1975), are briefly mentioned. Then the complementary pivot algorithm of Lemke (1965), which has been shown to be capable of deriving the optimal solution to certain quadratic programming problems that are subject to a non-negativity constraint, is described. Finally, Lemke's algorithm is shown to be capable of deriving the minimum-variance zero-beta portfolio efficiently from samples of risky assets where both long and short positions are allowed by reformulating the problem so as to avoid the difficulties encountered by having a non-negatively constraint.