Rebalancing and Index Tracking

In this chapter, our interest is focused on problems related to portfolio selection and that can still be formulated as LP or MILP models. The first part of the chapter is devoted to the portfolio rebalancing problem, where the investor already owns a portfolio of assets and, due to changed market conditions and possibly to the availability of additional capital, is interested in modifying it by selling/purchasing shares or amounts of some assets. We analyze all the aspects that, at least in practice, cannot be ignored when rebalancing, as the role played by transaction costs and the rebalancing frequency. The second part deals with the index tracking problem, that is the problem of selecting a set of assets that replicates as closely as possible the performance of a market index, while limiting the number of assets held in the portfolio and hence the associated transaction costs. When the goal is to exceed the performance of an index we talk about the enhanced index tracking problem. In this case, the investor aims at exceeding the index performance, possibly by a specified excess return. In this chapter, we provide mathematical formulations and discuss important modeling issues for both index and enhanced index tracking problems. At the end, we briefly analyze the case of long/short positions in portfolio holdings. Considering a portfolio optimization problem that includes long/short positions means to assume that the sign of the investment in an asset is not constrained.