Setting optimal bounds on risk in asset allocation – a convex program

The ‘Portfolio Selection Problem’ is traditionally viewed as selecting a mix of investment opportunities that maximizes the expected return subject to a bound on risk. However, in reality, portfolios are made up of a few ‘asset classes’ that consist of similar opportunities. The asset classes are managed by individual ‘sub-managers’, under guidelines set by an overall portfolio manager. Once a benchmark (the ‘strategic’ allocation) has been set, an overall manager may choose to allow the sub-managers some latitude in which opportunities make up the classes. He may choose some overall bound on risk (as measured by the variance) and wish to set bounds that constrain the submanagers. Mathematically we show that the problem is equivalent to finding a hyper-rectangle of maximal volume within an ellipsoid. It is a convex program, albeit with potentially a large number of constraints. We suggest a cutting plane algorithm to solve the problem and include computational results on a set of randomly generated problems as well as a real-world problem taken from the literature.