Heuristics for cardinality constrained portfolio optimisation

Abstract

In this paper we consider the problem of finding the efficient frontier associated with the standard mean–variance portfolio optimisation model. We extend the standard model to include cardinality constraints that limit a portfolio to have a specified number of assets, and to impose limits on the proportion of the portfolio held in a given asset (if any of the asset is held). We illustrate the differences that arise in the shape of this efficient frontier when such constraints are present. We present three heuristic algorithms based upon genetic algorithms, tabu search and simulated annealing for finding the cardinality constrained efficient frontier. Computational results are presented for five data sets involving up to 225 assets. Scope and purpose The standard Markowitz mean–variance approach to portfolio selection involves tracing out an efficient frontier, a continuous curve illustrating the tradeoff between return and risk (variance). This frontier can be easily found via quadratic programming. This approach is well-known and widely applied. However, for practical purposes, it may be desirable to limit the number of assets in a portfolio, as well as imposing limits on the proportion of the portfolio devoted to any particular asset. If such constraints exist, the problem of finding the efficient frontier becomes much harder. This paper illustrates how, in the presence of such constraints, the efficient frontier becomes discontinuous. Three heuristic techniques are applied to the problem of finding this efficient frontier and computational results presented for a number of data sets which are made publicly available.