Abstract

We review our recent results on applying ideas of distributionally robust optimization to index tracking problems.

Index tracking is a popular form of asset management. The tracking error is expressed as an expectation of a function of the difference between the index returns and the portfolio. We assume that there is a model uncertainty in the distribution of the assets, hence a distributionally robust approach is appropriate. We use phi-divergences in measuring the deviation between the actual and the nominal distribution of the index components and arrive at a semi-analytical form of the solution of the robust index-tracking problem.

Further, in the specific case of the Kullback-Leibler (KL) divergence, we use ideas of the exponential cone representability of the KL divergence to demonstrate that a regularized distributionally robust index tracking problem can be reformulated as a nonlinear conic programming problem. We further simplify the issue when the regularization is convex so that an effective numerical solution can be delivered. More generally, if the regularization can be written as a difference of convex functions, a solution can still be obtained by solving a sequence of conic linear programming problems.

We consider both cases of discrete modeling of the ambiguity on the model's residuals and continuous modeling of the functional neighborhood around the nominal distribution of the data.

The proposed approaches are applied to real financial data and to simulated data sets to demonstrate the superiority of the methods in comparison to some non-robust methods.