Myopic robust index tracking with Bregman divergence
Index tracking is a popular form of asset management. Typically, a quadratic function is used to define the tracking error of a portfolio and the look back approach is applied to solve the index tracking problem. We argue that a forward looking approach is more suitable, whereby the tracking error is expressed as an expectation of a function of the difference between the returns of the index and of the portfolio. We also assume that there is model uncertainty in the distribution of the assets, hence a robust version of the optimization problem needs to be adopted. We use Bregman divergence in describing the deviation between the nominal and actual (true) distribution of the components of the index. In this scenario, we derive the optimal robust index tracking portfolio in a semi-analytical form as a solution of a system of nonlinear equations. Several numerical results are presented that allow us to compare the performance of this robust portfolio with the optimal non-robust portfolio. We show that, especially during market downturns, the robust portfolio can be very advantageous.