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 expectation of a function of the difference between the returns of the index and of the portfolio. We also assume that there is an 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 distribution of the components of the index. In this scenario, we derive the optimal robust index tracking strategy 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 strategy with the optimal non-robust strategy. We show that, especially during market downturns, the robust strategy can be very advantageous.Comment: To be published in Quantitative Financ

Locally robust methods and near-parametric asymptotics

It has already been shown theoretically and numerically that infusing a little localization in the likelihood-based methods for regression and for density estimation can actually improve the resulting estimators with respect to suitably defined global risk measures. Thus various local likelihood methods have been suggested. In this paper, we demonstrate that a similar effect can also be observed with respect to robust estimation procedures. Localized versions of robust density estimation procedures perform better with respect to global risk measures based on minimization of Bregman divergence measures