In this paper, the main goal is to tackle a sustainable portfolio optimization problem in which we aim to minimize a tail-dependence risk measure. More specifically, the considered risk measure is represented by the Delta Conditional Value-at-Risk, a tail-dependence measure meant to quantify the potential losses of a portfolio due to the riskiness associated with an individual asset or a group of assets. In addition, in the portfolio construction, we take into account some real-world trading constraints. On the one hand, we impose stock market restrictions through buy-in thresholds and budget constraints. Moreover, a minimum level of guaranteed expected return is required. On the other hand, a turnover threshold restricts the total amount of trades allowed in the rebalancing phases. Finally, in order to meet the growing appetite for sustainable investment, we introduce a green threshold into the portfolio’s design. To deal with these asset allocation models, in this paper we develop an improved hybrid Particle Swarm Optimizer (PSO) that is dynamically adjusted by a neural network architecture embedded with a suitable constraint-handling procedure. The neural network paradigm is fundamental for enhancing the basic PSO’s performance and improving the quality of estimating the Delta Conditional Value-at-Risk. Finally, we conduct empirical tests on two different American datasets to illustrate the effectiveness of the proposed strategies and evaluate the performance of our investments as the sustainable preferences vary.

In this paper, we tackle a bi-objective optimization problem in which we aim to maximize the portfolio diversification and, at the same time, minimize the portfolio volatility, where the ESG (Environmental, Social, and Governance) information is incorporated. More specifically, we extend the standard portfolio volatility framework based on the financial aspects to a new paradigm where the sustainable credits are taken into account. In the portfolio’s construction, we consider the classical constraints concerning budget and box requirements. To deal with these new asset allocation models, in this paper, we develop an improved Multi-Objective Particle Swarm Optimizer (MOPSO) embedded with ad hoc repair and projection operators to satisfy the constraints. Moreover, we implement a deep learning architecture to improve the quality of estimating the portfolio diversification objective. Finally, we conduct empirical tests on datasets from three different countries’ markets to illustrate the effectiveness of the proposed strategies, accounting for various levels of ESG volatility.