Absolute return portfolios C.A. Valle a , N. Meade b , J.E. Beasley a,c,n a Mathematical Sciences, Brunel University, Uxbridge UB8 3PH, UK b Business School, Imperial College, London SW7 2AZ, UK c JB Consultants, Morden SM4 4HS, UK article info Article history: Received 11 February 2012 Accepted 18 December 2013 Processed by B. Lev Available online 3 January 2014 Keywords: Absolute return portfolio Linear regression Mixed-integer programming abstract In this paper we consider the problem of selecting an absolute return portfolio. This is a portfolio of assets that is designed to deliver a good return irrespective of how the underlying market (typically as represented by a market index) performs. We present a three-stage mixed-integer zero-one program for the problem that explicitly considers transaction costs associated with trading. The first two stages relate to a regression of portfolio return against time, whilst the third stage relates to minimising transaction cost. We extend our approach to the problem of designing portfolios with differing characteristics. In particular we present models for enhanced indexation (relative return) portfolios and for portfolios that are a mix of absolute and relative return. Computational results are given for portfolios derived from universes defined by S&P international equity indices. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction Absolute return portfolios (henceforth ARPs) are financial portfolios that aim to produce a good return regardless of how the underlying market performs. This (clearly) is a relatively easy task when the market is performing well, a much less easy task when the market is performing poorly. Essentially investors are interested in ARPs either because  they believe that the market will perform poorly, and so wish to focus on portfolios that will not perform as poorly or  they are unsure of how the market will perform and wish to hold an ARP as insurance against market deterioration. ARPs are a relatively popular strategy amongst managers of some hedge funds, which, as their name suggests, often seek to hedge some of the risks inherent in their investments using a variety of methods. Their objective is to achieve absolute returns by balancing investment opportunities and risk of financial loss. Al-Sharkas [1], Connor and Lasarte [14], Jawadi and Khanniche [24] and Till and Eagleeye [39], discuss the various strategies that hedge funds can adopt. ARPs are sometimes called market neutral portfolios as they are designed to have a low correlation with overall market return. Whilst, due to this strategy, ARPs may be able to achieve positive returns in falling markets, on the other hand they may not perform as well as market indices or other types of investments in rising markets. However, the fear of significant financial events (we have seen the 2008 subprime financial crisis; in the near future will we see a Eurozone default?) makes ARPs popular amongst investors, who see them as a reasonable strategy to adopt given market uncertainty and volatility. In this paper, we present a three-stage mixed-integer zero-one program for the problem of designing an ARP. Our formulation includes transaction costs associated with trading, a constraint limiting the number of assets that can be held and a limit on the total transaction cost that can be incurred. The first two stages relate to a regression of portfolio return against time, whilst the third stage relates to minimising transaction cost. One feature of note in our ARP approach is that we do not specify the return that the ARP should achieve; rather that emerges as a result of an optimisation. The original contribution of our model/formulation relates not to the constraints adopted (which are in fact standard and have been seen before in the literature, e.g. in Canakgoz and Beasley [10]). Rather the original contribution of our model relates to a clear definition of an ARP via the three-stage objective function. Because our approach is flexible we are able to extend it to the problem of designing portfolios with differing characteristics. In particular we present models for enhanced indexation (relative return) portfolios and for portfolios that are a mix of absolute and relative return. In general terms this paper addresses a financial problem via mathematical modelling and optimisation. This is a common Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/omega Omega 0305-0483/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.omega.2013.12.003 n Corresponding author. E-mail addresses: cristiano.arbexvalle@brunel.ac.uk (C.A. Valle), n.meade@imperial.ac.uk (N. Meade), john.beasley@brunel.ac.uk, john.beasley@jbconsultants.biz (J.E. Beasley). Omega 45 (2014) 20–41