ABSTRACT This paper aims to provide a novel framework for incorporating investor preferences that integrate sustainability criteria as a third objective in portfolio optimization problems. The approach is based on the concept of preference directions, changing the geometric properties of the objective space and guiding the direction of the multi-objective optimization. This reinterpretation of the dominance concept aids in obtaining the Pareto front solutions. Investors' preferences can be integrated before the optimization process without modifying the algorithm, or at the decision-making stage after obtaining the efficient frontier. We empirically test our approach’s performance on actual socially responsible funds and on the S&P 500 index. Our findings suggest that this approach provides a better understanding of ESG preference integration by refocusing the region of interest, facilitating more effective investor decision-making. The benefits of our approach include improved portfolio sustainable construction and a more comprehensive understanding of investor preferences in sustainable investing.

This paper examines the long-term strategic asset allocation of investors in small open economies, where the domestic financial markets are limited, unstable and vulnerable to significant country-specific shocks. Iceland is used as a case study. All wealth, risk and portfolio outcomes are evaluated in Icelandic krona (ISK). The analysis covers the period from 1992 to 2024, including financial liberalisation, the 2008 banking collapse, years of capital controls, and the subsequent tourism-led recovery. The study considers the following eight asset classes, which are available to Icelandic investors and constitute a realistic investable universe: These are Icelandic government bonds; a short-term, risk-free ISK asset; Icelandic equities; global, developed-market equities (MSCI World); emerging-market equities (MSCI Emerging Markets); US Treasury bills; UK government bonds; and gold. As foreign assets are held unhedged, returns reflect both asset-price movements and exchange-rate changes. Using annual data, the paper first documents the long-term risk and return characteristics of each asset in ISK terms. This includes arithmetic and geometric mean returns, volatility, correlations and behaviour during periods of crisis. It then applies a standard long-only mean–variance framework to characterise efficient portfolios for an ISK-based investor with a long time horizon. The results show that, despite high average one-year returns, domestic equities perform poorly in the long run. The collapse of 2008, in particular, drives their real geometric mean return below zero over the full sample due to extreme downside risk. In contrast, global equities deliver the strongest and most stable long-term growth in ISK, achieving the highest risk-adjusted performance. Foreign safe assets and gold offer robust protection during years of severe currency depreciation, but their average excess returns are insufficient for them to play a pivotal role in long-term efficient portfolios. Consequently, the efficient frontier is reduced to a combination of domestic government bonds, the risk-free ISK asset and global equities, with domestic equities and other foreign assets being largely excluded. The findings emphasise the significant expense of home bias in small, open economies with independent currencies. In the long term, achieving portfolio efficiency requires meaningful exposure to global equity markets, anchored by high-quality domestic bonds rather than heavy reliance on domestic equities.

The reactivity of metal nitrides has attracted considerable attention owing to their proposed role as key intermediates in the critical N-N bond breaking and formation steps in N2 fixation and NH3 oxidation processes. Despite intensive efforts over the past decades, no clear-cut examples of direct N-N coupling initiated by metal-nitrides have yet been reported. Herein, we describe the reaction of a transient iron(V)-nitrido complex, [FeV(N)(cyclam-ac)][BArF4] (2, cyclam-ac = 1,4,8,11-tetraazacyclotetradecane-1-acetate), with pentafluoronitrosobenzene (PFPhNO), which quantitatively affords an iron(III)-phenyldiazene-N-oxide [FeIII(NN(O)PFPh)(cyclam-ac)][BArF4] (3). Complex 3 has been structurally and spectroscopically characterized using EPR, Mössbauer, and UV-vis-NIR techniques, coupled to wave function-based ab initio calculations, to contain a low-spin ferric center coordinated by a -N═N(O)PFPh monoanionic ligand featuring a N═N double bond. Further in situ measurements reveal that analogous reactions can occur upon treatment of 2 with PhNO and 4-NO2-PhNO. Detailed electronic structure analyses suggest that exceedingly efficient mutual frontier orbital interactions between 2 and PhNO render the title reaction barrierless and highly exothermic. This can be traced back to the unusual electronic feature of 2, viz., the 2-fold orbital near-degeneracy of the Fe-N π* orbitals and the dominant N-2p character thereof, both of which maximize stabilizing HOMO-LUMO overlaps. Furthermore, under vacuum, 3 was found to release N2, quantitatively furnishing a low-spin ferric phenolate complex, [FeIII(OPFPh)][BArF4] (4). The present work reveals previously unrecognized reactivity of high-valent iron-nitrido species and sheds further light on the N-N coupling reaction mediated by metal-nitrides.

Global environmental issues are becoming increasingly severe, with climate change imposing varying degrees of economic impact on different cities. It is crucial for cities to pursue efficient, low-carbon, and sustainable development pathways to cope with climate change. Carbon emission efficiency (CEE) is an essential indicator for assessing their performance and progress toward low-carbon growth. However, traditional CEE assessments have yet to integrate regional differences in the socioeconomic costs of climate change. To fill this gap, we have built a combined efficient frontier Data Envelopment Analysis (DEA) model based on the weighted carbon emissions of each city’s climate costs to evaluate the CEEs of 252 cities in China from 2006 to 2021. Meanwhile, city classification and spatial Markov chains are used for spatio-temporal heterogeneity analysis, and finally, the efficiency is decomposed to determine the impact of different factors on carbon efficiency. The results indicate that the average CEE of coastal cities (0.57) is lower than that of inland cities (0.63), mainly due to higher climate costs and unbalanced development. In contrast, megacities and super-large cities in coastal areas have the highest CEE levels because of economies of scale and technological advantages. Efficiency decomposition shows that pure technical efficiency (PTE) is the primary driver of CEE differences, contributing 33.37% to inefficiency differences. Our findings emphasize the need for targeted, differentiated policies to address unique urban challenges. Green technology investments should be prioritized in areas with high emission reduction potential, while cross-regional technology diffusion mechanisms should be established in areas with medium reduction potential to foster innovation. Overall, this study could offer valuable insights into the sustainable and low-carbon transition of urban development.

Abstract We propose that investment strategies should be evaluated based on their net-of-trading-cost return for each level of risk, which we term the “implementable efficient frontier.” While numerous studies use machine learning return forecasts to generate portfolios, their agnosticism toward trading costs leads to excessive reliance on fleeting small-scale characteristics, resulting in poor net returns. We develop a framework that produces a superior frontier by integrating trading-cost-aware portfolio optimization with machine learning. The superior net-of-cost performance is achieved by learning directly about portfolio weights using an economic objective. Further, our model gives rise to a new measure of “economic feature importance.” (JEL C5, C61, G00, G11, G12)

The mean-variance rule (M-V) conforms with the expected utility paradigm only in limited and economically unacceptable scenarios. Thus, the most widely employed portfolio-selection rule seemingly loses ground. We show with the commonly employed utility functions in economics, with a preference for a positive skewness, that choosing from the M-V efficient frontier conforms with expected utility maximization even with long investment horizon and skewed distributions of returns. The economic loss induced by choosing from the M-V frontier is negligible. Thus, the M-V rule is universal, or almost universal, provided that the commonly employed utility functions in economics are employed. This is an astonishing result that even Markowitz has not dreamed of.

In the financial market, the investment portfolio is constructed in various ways, with different components, and under some constraint factors. The Markowitz model, invented by Harry Markowitz, as well as the modern portfolio theory, contributes to the effective construction of portfolios. While the model points out the most efficient portfolios, which are greatly balanced between volatility and return, the change in constraints, which strongly affects the results of the solver, can make some of those potential portfolios impossible, or unfeasible. This study focuses on the setting of constraints on each component’s weight in the whole portfolio, as well as the evaluation of the differences of risk and return rate between the two portfolios under different constraints. The Markowitz model is used to calculate the most efficient portfolios and the efficient frontier under each setting of the constraints. The results of the study show that additional constraints on weights, which are based on the total value of the portfolio and average daily trading value of the stock, make the range of potential volatility rate more limited by controlling the weights of some stocks in the portfolio.

The quest for the optimal balance between risk and return remains a pivotal area of research within modern portfolio theory. As two traditional approaches, the Markowitz model and the single-index model embody distinct philosophies: precise calculation versus simplified approximation. The Markowitz model delves deeply into the covariance relationships between assets, laying the foundation for modern portfolio theory; the single-index model enhances practical applicability by streamlining the estimation process of the covariance matrix. This study constructs the efficient frontier, global minimum variance portfolio, and optimal risky portfolio under both the Markowitz model and single index model using data from 21 selected sample stocks. Empirical testing and comparisons are conducted across various scenarios using metrics such as the Sharpe ratio and Sortino ratio. Empirical results indicate that the single-index model demonstrates superior risk-adjusted returns and robustness across most market conditions. Whilst the Markowitz model theoretically achieves greater diversification, cumulative estimation errors may lead to comparatively weaker performance in empirical outcomes. These findings not only assist investors in model selection across dynamic market environments but also inform future research on expanding constraints and incorporating factor models.

The financial market situation is unpredictable. Markowitz's investment theory is the cornerstone of the financial investment field, aiming to diversify risks and improve returns through precise parameter estimation. Entering the 21st century, with the advent of the big data era, Python can leverage its powerful computational capabilities to utilize this model more efficiently and accurately, helping the financial market operate stably. This review synthesizes findings from empirical studies that use Python tools to select real market data, analyze this data to obtain the optimal investment portfolio with the highest Sharpe ratio and the smallest variance, and conduct a comparative analysis of their expected returns, standard deviations, ultimately presenting the efficient frontier of the investment portfolio. The analysis concludes that using Python to analyze the Markowitz investment theory model can fully demonstrate a more intuitive, scientific, and precise quantitative financial research approach, assisting leaders in making rational investment decisions and enhancing risk management capabilities.

Exchange-traded funds (ETFs) provide low-cost, liquid access to broad equity and fixed-income exposures, including rapidly growing Asian and Asia-focused markets. Yet the academic evidence on Asian ETF portfolio construction remains fragmented, often limited to narrow country samples and centered on mean–variance trade-offs and standard performance statistics, with comparatively less emphasis on downside tail risk and on implementable long-only versus long–short designs under leverage constraints. This study examines the performance and risk characteristics of 29 Asian and Asia-focused ETFs over 2014–2025 and evaluates whether optimization using variance-based and tail-sensitive risk measures improves portfolio outcomes relative to a simple, implementable benchmark. We construct Markowitz mean–variance and conditional value-at-risk (CVaR) efficient frontiers and implement six optimized portfolios at the 95% and 99% tail levels under long-only and long–short configurations with leverage up to 30%. Performance is evaluated relative to an equally weighted Asian ETF benchmark using the Sharpe ratio and tail-sensitive measures, including the Rachev ratio and the stable tail adjusted return (STARR), complemented by fat-tail diagnostics based on the Hill tail-index estimator. The empirical results show that optimization improves efficiency relative to equal weighting in risk-adjusted terms and that moderate leverage can increase returns but typically amplifies volatility, dispersion, and drawdowns. Taken together, the evidence indicates that risk-measure choice materially affects portfolio composition and realized outcomes, with tail-based optimization generally producing more robust allocations than mean–variance approaches when downside risk is a primary concern.

In modern financial markets, decision-makers increasingly rely on quantitative methods to navigate complex trade-offs among multiple, often conflicting objectives. This paper addresses constrained multi-objective optimization (MOO) with an application to portfolio optimization for minimizing risk and maximizing return. To this end, and to address existing gaps, we propose a novel reinforcement learning (RL)-guided non-dominated sorting genetic algorithm II (NSGA-II) enhanced with gray relational coefficients (GRC), termed RL-NSGA-II-GRC, which combines an RL agent controller and GRC-based selection to improve the convergence and diversity of the Pareto-optimal fronts. The agent adapts key evolutionary parameters online using population-level metrics of hypervolume, feasibility, and diversity, while the GRC-enhanced tournament operator ranks parents via a unified score simultaneously considering dominance rank, crowding distance, and geometric proximity to ideal reference. We evaluate the framework on the Kursawe and CONSTR benchmark problems and on a NASDAQ portfolio optimization application. On the benchmarks, RL-NSGA-II-GRC achieves convergence metric improvements of about 5.8% and 4.4% over the original NSGA-II, while preserving a well-distributed set of non-dominated solutions. In the portfolio application, the method produces a smooth and densely populated efficient frontier that supports the identification of the maximum Sharpe ratio portfolio (with annualized Sharpe ratio = 1.92), as well as utility-optimal portfolios for different risk-aversion levels. The main contributions of this work are three-fold: (1) we propose an RL-NSGA-II-GRC method that integrates an RL agent into the evolutionary framework to adaptively control key parameters using generational feedback; (2) we design a GRC-enhanced binary tournament selection operator that provides a comprehensive performance indicator to efficiently guide the search toward the Pareto-optimal front; (3) we demonstrate, on benchmark MOO problems and a NASDAQ portfolio case study, that the proposed method delivers improved convergence and well-populated efficient frontiers that support actionable investment insights.

Abstract In the vast landscape of financial markets, identifying potential investment assets such as stocks can be overwhelming and time-consuming. For portfolio managers, focusing on a specific selection of stocks through an effective filtering process can streamline this task. This paper introduces an efficient stock preselection method using multidimensional non-dominated sorting of selected return statistics. Unlike previous research, our approach leverages statistics derived from approximated return series through nonparametric regression and principal component analysis (PCA). We further explore the impact of this preselection on mean-variance and the newly proposed mean-trend risk large-scale portfolio selection strategies. By examining the efficient frontier of portfolios from various return and risk perspectives, our empirical analysis on US stock market data provides both ex-post and ex-ante results for 40 portfolio strategies. The findings suggest that for most risk-averse investors, mean-trend risk strategies with preselection significantly outperform both the same strategies without preselection and traditional mean-variance strategies.

Optimal decumulation of a Defined Contribution (DC) pension plan can be viewed as a problem in optimal stochastic control, which requires specification of an objective function, a combination of reward and risk. An intuitive specification of reward is the sum of withdrawals over the retirement period. This article investigates three tail risk measures for running out of savings in a DC plan decumulation strategy, which includes (i) expected shortfall, (ii) linear shortfall, and (iii) probability of shortfall. From the perspective of all optimal solutions, we establish that, under suitable regularity assumptions, the set of optimal controls corresponding to all expected reward expected shortfall Pareto efficient frontier curves is identical to the set of optimal controls associated with all expected reward and linear shortfall Pareto efficient frontier curves. To better understand the impact of a chosen risk measure, we compare its optimal controls across all three combinations of risk measures and reward performance criteria, while fixing the values of the risk aversion and wealth/probability level parameters at reasonable levels. This comparison reveals a clear preference for the linear shortfall risk measure, which yields more desirable optimal strategies. From a practical point of view, we show that allowing variable withdrawals has a large effect on reducing risk, compared to dynamic asset allocation.

Using Machine Learning to Forecast Market Direction with Efficient Frontier Coefficients

Tangency portfolios using graph neural networks.

Forecasting Tangency Portfolios and Investing in the Minimum Euclidean Distance Portfolio to Maximize Out-of-Sample Sharpe Ratios †

Asset Allocation Using a Markov Process of Clustered Efficient Frontier Coefficients States

Machine Learning Meets Markowitz

Recently, deep reinforcement learning has gained attention for solving routing problems due to its ability to learn complex patterns and optimize sequential decision-making. Despite their success in various routing problems, the applicability of learning-based methods on pickup and delivery problems (PDPs) is limited and only addresses one-to-one PDPs and their variants. However, the many-to-many PDP addressed in this paper is a practical variant of the routing problems and finds several applications in logistics and supply chains. We propose a novel reinforcement learning framework empowered by a multi-head heterogeneous attention mechanism (MHHA) and a decoder capable of exploring a diverse set of solutions, namely HAP, to generate efficient solutions for many-to-many PDP. The proposed framework incorporates an encoder-decoder structure designed explicitly for many-to-many PDP. In particular, the encoder consists of the MHHA mechanism to capture the many-to-many relationships and effectively model the flow constraints. Moreover, it is integrated with the multi-solution generator, polynet and masking scheme to generate high-quality diverse solutions. Additionally, we improve the solution quality of HAP using a warm starting variable neighbourhood search. As extensive experimental results demonstrate, the proposed method outperforms state-of-the-art metaheuristic and learning-based approaches. Additionally, in bi-objective (time and gap) comparison, the HAP lies on the Pareto efficient frontier, proving its effectiveness. Moreover, the HAP effectively generalizes to diverse problem sizes, unseen data distributions, benchmark datasets, and also solves one-to-one PDP more effectively than baseline methods, showing its adaptability and robustness. Finally, we conduct ablation studies to justify the proposed design.

The Green Divide: Divergent Climate Risk Pricing in Global Capital Market Assumptions