The Centre for the Analysis of Risk and Optimisation Modelling Applications (CARISMA), Brunel University
Abstract
In this chapter, we first provide an overview of a number of portfolio planning models
which have been proposed and investigated over the last forty years. We revisit the
mean-variance (M-V) model of Markowitz and the construction of the risk-return
efficient frontier. A piecewise linear approximation of the problem through a
reformulation involving diagonalisation of the quadratic form into a variable
separable function is also considered. A few other models, such as, the Mean
Absolute Deviation (MAD), the Weighted Goal Programming (WGP) and the
Minimax (MM) model which use alternative metrics for risk are also introduced,
compared and contrasted. Recently asymmetric measures of risk have gained in
importance; we consider a generic representation and a number of alternative
symmetric and asymmetric measures of risk which find use in the evaluation of
portfolios. There are a number of modelling and computational considerations which
have been introduced into practical portfolio planning problems. These include: (a)
buy-in thresholds for assets, (b) restriction on the number of assets (cardinality
constraints), (c) transaction roundlot restrictions. Practical portfolio models may also
include (d) dedication of cashflow streams, and, (e) immunization which involves
duration matching and convexity constraints. The modelling issues in respect of these
features are discussed. Many of these features lead to discrete restrictions involving
zero-one and general integer variables which make the resulting model a quadratic
mixed-integer programming model (QMIP). The QMIP is a NP-hard problem; the
algorithms and solution methods for this class of problems are also discussed. The
issues of preparing the analytic data (financial datamarts) for this family of portfolio
planning problems are examined. We finally present computational results which
provide some indication of the state-of-the-art in the solution of portfolio optimisation
problems
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