A distributionally robust index tracking model with the CVaR penalty: tractable reformulation

TL;DR

This paper introduces a new distributionally robust index tracking model with CVaR penalty, reformulated into a tractable nonsmooth optimization problem, and demonstrates its effectiveness on real financial data.

Contribution

It presents a novel reformulation of a distributionally robust index tracking model with CVaR penalty into a tractable nonsmooth optimization problem and proposes an efficient solution method.

Findings

The model effectively reduces tracking errors on NASDAQ data.

The SPG method converges to a global minimizer of the discretized problem.

Numerical results outperform several state-of-the-art models.

Abstract

We propose a distributionally robust index tracking model with the conditional value-at-risk (CVaR) penalty. The model combines the idea of distributionally robust optimization for data uncertainty and the CVaR penalty to avoid large tracking errors. The probability ambiguity is described through a confidence region based on the first-order and second-order moments of the random vector involved. We reformulate the model in the form of a min-max-min optimization into an equivalent nonsmooth minimization problem. We further give an approximate discretization scheme of the possible continuous random vector of the nonsmooth minimization problem, whose objective function involves the maximum of numerous but finite nonsmooth functions. The convergence of the discretization scheme to the equivalent nonsmooth reformulation is shown under mild conditions. A smoothing projected gradient (SPG) method is employed to solve the discretization scheme. Any accumulation point is shown to be a global minimizer of the discretization scheme. Numerical results on the NASDAQ index dataset from January 2008 to July 2023 demonstrate the effectiveness of our proposed model and the efficiency of the SPG method, compared with several state-of-the-art models and corresponding methods for solving them.