TABLE OF CONTENTS This chapter began with a simple question: What is the best way to form an investment portfolio given a list of assets, an ordering of relative preferences on those assets, and a covariance matrix? A large part of the motivation came from the observation that existing methods were very ad hoc and had no way of incorporating volatility and correlation information. These methods were therefore incompatible with the main stream of portfolio methods going back to Markowitz, who emphasized the importance of incorporating risk into the construction of the optimal portfolio.
In the course of developing this solution, we were led to develop a very robust and powerful framework for thinking about portfolio optimization problems. This framework includes classic portfolio theory as a special case and provides a natural generalization to a broad class of ordering information. It also includes more modern constructions, such as robust optimization, as we now discuss.
To summarize, our formulation has three ingredients:
Ordering information which gives rise to a cone of consistent returns. This is a set in which the true expected return vector is believed to lie. In the examples considered in this chapter, this cone is always constructed as the intersection of half-spaces corresponding to a finite list of homogeneous inequality beliefs. But more generally, we may specify any convex cone, with curved edges or other more complicated geometrical structure; our construction can in principle be carried out for any such set.
A probability density
