Rebalancing an Investment Portfolio in the Presence of Transaction Costs

Note: This paper has been superseded by our paper on "Rebalancing an Investment Portfolio in the Presence of Convex Transaction Costs".

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Authors:

John E. Mitchell
Department of Mathematical Sciences
Rensselaer Polytechnic Institute
Troy, NY 12180 USA
mitchj@rpi.edu

Stephen Braun
Warren & Selbert, Inc.
Santa Barbara, CA 93101
brauns2@alum.rpi.edu

November 28, 2002. Revised December 16, 2003.

Abstract:

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 linear transaction costs are incurred to rebalance an investment portfolio. The Markowitz framework of mean-variance efficiency is used with costs modelled as a percentage of the value transacted. Each security in the portfolio is represented by a pair of continuous decision variables corresponding to the amounts bought and sold. 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 quadratic programming problem. We show that this fractional quadratic programming problem has a structure that allows it to be reformulated as an equivalent quadratic programming problem of size comparable to the model without transaction costs. Theoretically, one way to reduce the measure of risk is to buy and sell the same security, which is not an attractive practical strategy. We show that an optimal solution to the quadratic programming reformulation can always be found that does not simultaneously buy and sell a single security. The results of the paper extend the classical Markowitz model to the case of proportional transaction costs in a natural manner with limited computational cost. Computational results for two empirical datasets are discussed.

Keywords: Portfolio optimization, transaction costs, rebalancing, quadratic programming.

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Note: This paper has been superseded by our paper on "Rebalancing an Investment Portfolio in the Presence of Convex Transaction Costs".

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