The inclusion of transaction costs is an essential element of any realistic portfolio optimization. In this paper, we consider an extension of the standard portfolio problem in which convex transaction costs are incurred to rebalance an investment portfolio. In particular, we consider linear, piecewise linear, and quadratic transaction costs. The Markowitz framework of mean-variance efficiency is used. If there is no risk-free security, it may be possible to reduce the measure of risk by discarding assets, which is not an attractive practical strategy. In order to properly represent the variance of the resulting portfolio, we suggest rescaling by the funds available after paying the transaction costs. This results in a fractional programming problem, which can be reformulated as an equivalent convex program of size comparable to the model without transaction costs. An optimal solution to the convex program can always be found that does not discard assets. The results of the paper extend the classical Markowitz model to the case of convex transaction costs in a natural manner with limited computational cost. Computational results for two empirical datasets are discussed.
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180 USA. http://www.rpi.edu/~mitchj/papers/transcostsconvex.html, December 2004.