Optimization Methods for Financial Index Tracking: From Theory to Practice

Abstract

Index tracking is a very popular passive investment strategy.Since an index cannot be traded directly, index trackingrefers to the process of creating a portfolio that approximatesits performance. A straightforward way to do that isto purchase all the assets that compose an index in appropriatequantities. However, to simplify the execution, avoidsmall and illiquid positions, and large transaction costs, it isdesired that the tracking portfolio consists of a small numberof assets, i.e., we wish to create a sparse portfolio.Although index tracking is driven from the financial industry,it is in fact a pure signal processing problem: a regression ofthe financial historical data subject to some portfolio constraintswith some caveats and particularities. Furthermore, the sparse index tracking problem is similar to many sparsityformulations in the signal processing area in the sense thatit is a regression problem with some sparsity requirements.In its original form, sparse index tracking can be formulatedas a combinatorial optimization problem. A commonly usedapproach is to use mixed-integer programming MIP tosolve small sized problems. Nevertheless, MIP solvers are notapplicable for high-dimensional problems since the runningtime can be prohibiting for practical use.The goal of this monograph is to provide an in-depth overviewof the index tracking problem and analyze all the caveats andpractical issues an investor might have, such as the frequentrebalancing of weights, the changes in the index composition,the transaction costs, etc. Furthermore, a unified frameworkfor a large variety of sparse index tracking formulations isprovided. The derived algorithms are very attractive forpractical use since they provide efficient tracking portfoliosorders of magnitude faster than MIP solvers.