Robust and sparse portfolio model for index tracking

In the context of index tracking, the tracking error measures the difference between the return an investor receives and that of the benchmark he was attempting to imitate. In this paper, we use the weighted $\ell_{2}$ and $\ell_{p}$ $(0 < p < 1)$ norm penalties as well as the shortsale constraints ($\ell_2-\ell_p$ model for short) to the tracking portfolio model in order to get a robust and sparse portfolio for index tracking. The $\ell_{2}$ norm penalty imposes smoothness to alleviate the effect of the existence of highly correlated variables and hence has better out-of-sample performance and the $\ell_{p}$ norm penalty achieves sparsity to account for transaction costs. We enroll in the model explicitly the non-negativity constraints, that is, the shortsale constraints appeared in practice. The $\ell_p$ norm penalty is non-Lipschitz, nonconvex which leads to computational difficulty. We adopt the smoothing projected gradient (SPG) method to solve the robust and sparse portfolio model. We show that any accumulation point of the SPG method is a special limiting stationary point. We find our proposed $\ell_2-\ell_p$ model outperforms the $\ell_2 +\ell_0$ model proposed by Takeda et al. [26] for real stock data set S&P500 in terms of in-sample and out-of-sample errors.