  Mean-reverting portfolios with volatility and sparsity constraints are of
prime interest to practitioners in finance since they are both profitable and
well-diversified, while also managing risk and minimizing transaction costs.
Three main measures that serve as statistical proxies to capture the
mean-reversion property are predictability, portmanteau criterion, and crossing
statistics. If in addition, reasonable volatility and sparsity for the
portfolio are desired, a convex quadratic or quartic objective function,
subject to nonconvex quadratic and cardinality constraints needs to be
minimized. In this paper, we introduce and investigate a comprehensive modeling
framework that incorporates all the previous proxies proposed in the literature
and develop an effective unifying algorithm that is enabled to obtain a
Karush-Kuhn-Tucker (KKT) point under mild regularity conditions. Specifically,
we present a tailored penalty decomposition method that approximately solves a
sequence of penalized subproblems by a block coordinate descent algorithm. To
the best of our knowledge, our proposed algorithm is the first for finding
volatile, sparse, and mean-reverting portfolios based on the portmanteau
criterion and crossing statistics proxies. Further, we establish that the
convergence analysis can be extended to a nonconvex objective function case if
the starting penalty parameter is larger than a finite bound and the objective
function has a bounded level set. Numerical experiments on the S&P 500 data set
demonstrate the efficiency of the proposed algorithm in comparison to a
semidefinite relaxation-based approach and suggest that the crossing statistics
proxy yields more desirable portfolios.
