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Dotted Representations of Mean-Variance Efficient Frontiers and their Computation

Abstract This paper is about dotted representations of efficient frontiers. Dotted representations, as in portfolio selection, can often be the most practical way of communicating an efficient frontier. The most popular method is to minimize variance subject to fixed levels of expected return. However, even when the fixed levels are evenly dispersed, one can not count on the resulting dots being evenly dispersed. Another method uses fixed values of a risk tolerance parameter, but with this method the resulting dots are even less controllable. In this paper we develop a third approach applicable to what we call Markowitz problems (mean-variance problems with all linear constraints). The approach utilizes the results of algorithms that can compute all hyperbolic segments of a Markowitz efficient frontier. Then the approach can place dots on the hyperbolic segments of the efficient frontier in a variety ways including equally spaced. The advantage of the approach is the speed at which dotted representations can be produced and modified, particularly on large applications.