A penalty decomposition algorithm with greedy improvement for mean‐reverting portfolios with sparsity and volatility constraints

Abstract

Mean-reverting portfolios with few assets, but sufficient variance, are of great interest for investors in financial markets. Such portfolios are straightforwardly profitable because they include a small number of assets whose prices not only oscillate predictably around a long-term mean but also possess enough volatility. Roughly speaking, sparsity minimizes trading costs, volatility provides arbitrage opportunities, and mean-reversion property equips investors with ideal investment strategies. Finding such appealing portfolios can be formulated as a nonconvex quadratic optimization problem with an additional sparsity constraint. To the best of our knowledge, there is no method in the literature that directly solves this problem with its solution achieving favorable theoretical properties. In this paper, we develop an efficient two-stage algorithm for solving this problem which, in the worst case, provably obtains a stationary point and further, demonstrates numerical efficiency. In the first stage of our algorithm, we apply a tailored penalty decomposition method for finding a stationary point of this nonconvex problem. For a fixed penalty parameter, the block coordinate descent method is utilized to find a stationary point of the associated penalty subproblem. In the second stage, we improve the result from the first stage via a greedy scheme that solves restricted nonconvex quadratically constrained quadratic programs (QCQPs). We show that the optimal value of such a QCQP can be obtained by solving their semidefinite relaxations. Numerical experiments on S&P 500 are conducted to demonstrate the efficiency of the proposed algorithm.

Model and efficient algorithm for the portfolio selection problem with real‐world constraints under value‐at‐risk measure

“…Furthermore, as pointed out by Yoshimoto (1996) and Najafi and Mushakhian (2015), transaction costs (including fixed and variable components) are one of the main concerns of today's portfolio managers, and ignoring them may result in unsatisfactory outcomes. Another main extension of the Markowitz model is the risk controlling by other measures except for the variance such as worst‐case (Chen and Korn, 2019; Korn and Leoff, 2019), Omega‐ratio (Sehgal and Mehra, 2021; Yu et al., 2023), Sharp‐ratio (Jing et al., 2022), value‐at‐risk (VaR) (Wozabal, 2012), conditional VaR (CVaR) (Sehgal and Mehra, 2019; Filippi et al., 2020), multivariate CVaR (Noyan and Rudolf, 2013), mean‐reversion (Mousavi and Shen, 2022), as well as the idea of index‐tracking (Wu et al., 2017; Hooshmand and MirHassani, 2022; Soares Silva et al., 2022).…”

The performance of bank portfolio optimization

“…To the best of our knowledge, there have been no historical tests of bank portfolio optimization methodologies, unlike, for instance, on investment portfolio optimization (Juszczuk et al., 2022; Mousavi and Shen, 2023; Salas‐Molina et al., 2023; Hooshmand et al., 2023), where numerous papers have addressed the historical performance of different optimization methodologies. As many articles have shown in the case of investment portfolios (see, for instance, DeMiguel et al., 2009; Chaves et al., 2011), optimization methods guarantee best returns ex ante (or in‐sample) but often do not outperform heuristic strategies ex post (out‐of‐sample).…”

ESG portfolio for TDFs with time‐varying higher moments and cardinality constraint

“…Some articles add cardinality constraints (or sparse constraints) to the portfolio model controlling the number of underlying assets and improving portfolio performance (Rui et al., 2019; Ahmad and Jinglai, 2023; Przemyslaw et al., 2023). Due to the double charge of the TDFs' constituent funds, we also introduce a cardinality constraint to control the number of constituent funds and minimize over‐diversification and high costs.…”