Decision Support International portfolio management with affine policies Raquel J. Fonseca ⇑ , Berç Rustem Department of Computing, Imperial College of Science, Technology and Medicine, 180 Queen’s Gate, London SW7 2AZ, UK article info Article history: Received 23 August 2011 Accepted 1 June 2012 Available online 15 June 2012 Keywords: Linear decision rules Robust optimization Multistage portfolio optimization Semidefinite programming Worst case value-at-risk abstract While dynamic decision making has traditionally been represented as scenario trees, these may become severely intractable and difficult to compute with an increasing number of time periods. We present an alternative tractable approach to multiperiod international portfolio optimization based on an affine dependence between the decision variables and the past returns. Because local asset and currency returns are modeled separately, the original model is non-linear and non-convex. With the aid of robust optimization techniques, however, we develop a tractable semidefinite programming formulation of our model, where the uncertain returns are contained in an ellipsoidal uncertainty set. We add to our formu- lation the minimization of the worst case value-at-risk and show the close relationship with robust opti- mization. Numerical results demonstrate the potential gains from considering a dynamic multiperiod setting relative to a single stage approach. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction Since Markowitz’s seminal work in portfolio theory [22], investors have selected their portfolios by considering a trade off between the investment return and its associated risk. Tradition- ally, the variance is taken as a risk measure and minimized for some expected target return over a single period of time. One of the disadvantages of this approach is the consideration of a single period of time, as opposed to a longer term investment planning strategy. To avoid this ‘‘myopic’’ view, one possible formulation would be to minimize the investment risk over the entire period, while imposing a final target return and further constraints on the portfolio composition at the intermediate stages. The standard framework to solve such multistage problems is stochastic programming [4], in which the decision space is discretized and a scenario tree representative of the many paths herein is constructed. There are many examples in the literature of stochastic programming applications to portfolio optimization models. Topaloglou et al. have used scenario trees to model an international portfolio optimization problem with the conditional value-at-risk as risk measure [26,27]. They further complement their study with forwards and options. Gulpinar and Rustem [17] extend the mean–variance optimization framework to a multiperi- od model, where rival scenarios of risk and return are considered, and assume a minmax approach in order to guarantee the perfor- mance of the portfolio. In [19], Kuhn et al. also consider a robust approach to multiperiod portfolio optimization by minimizing the worst case variance over a set of possible return distributions. The consideration of uncertainty directly in the model results from the awareness of a lack of robustness in the Markowitz framework, as pointed out in Broadie [5]. Many approaches have been sug- gested to reduce the effect of such parameter estimation errors, see, for example, Rustem and Howe [24], Goldfarb and Iyengar [15] or Ceria and Stubbs [8]. A survey on the application of robust optimization techniques to portfolio management may be found in Fabozzi et al. [11]. Gregory et al. [16] study the impact of robust- ness on optimal portfolios. In this paper, we propose an alternative tractable formulation of computing multiperiod optimization models which does not rely on scenario trees, but is instead based on linear decision rules. One of the drawbacks of stochastic programs is their numerical intractability. While one would like a scenario tree that truly represents the uncertainty of the parameters, an increasing num- ber of stages and branches results in an exponential growth of the problem size. On the other hand, if one were to consider a smaller tree to facilitate computational implementation, the ob- tained results might not be useful, as the reality is not adequately represented. Furthermore, when dealing with financial problems, the number of branches must be greater than the number of assets considered to avoid arbitrage in the market. Linear decision rules, however, are quite effective at overcoming computational complexity, as the resulting problems are either linear, second order cone or semidefinite programs. All of these are convex and can be efficiently solved with standard optimization techniques. In this new paradigm, decisions made at each stage are an affine combination of the past uncertain parameters. Affine polices have first been studied by Gartska and Wets in 1974 [14] and later on used in the context of robust optimization 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ejor.2012.06.001 ⇑ Corresponding author. Present address: Department of Statistics and Opera- tional Research, Faculty of Sciences, University of Lisbon, Portugal. E-mail address: rjfonseca@fc.ul.pt (R.J. Fonseca). European Journal of Operational Research 223 (2012) 177–187 Contents lists available at SciVerse ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor