A Dual Concept and Associated Algorithm in Mean-Variance Portfolio Analysis

We consider in this paper the problem of allocation by an investor of his funds between a set of risky assets and a single safe asset, under conditions in which the separation theorem holds. Focussing in particular on the computationally more difficult case where no short sales are permitted, we exhibit a dual problem which arises naturally out of the saddle-point property implied in the Kuhn-Tucker optimising conditions. By showing how to solve this dual we derive a computationally straightforward method of solution for the optimum portfolio proportions and obtain a condition for every security to be held. We show briefly how the treatment can be extended to general homogeneous restrictions on portfolio proportions.