An effective branch and bound algorithm for generalized risk parity portfolio optimization

Quantitative asset allocation remains a critical challenge in modern finance, particularly due to the inherent uncertainty of expected returns (μ) and the sensitivity of portfolio outcomes to the stability of portfolio weights. This study conducts a comparative empirical analysis of three portfolio strategies—MVO, Static RP, and Dynamic RP—using a long-only portfolio of eleven highly liquid assets, consisting of U.S. large-cap equities and gold, over the period 2015–2025. Results from historical backtesting indicate maintaining a competitive Sharpe ratio (1.418) and the lowest Maximum Drawdown (−0.2770) relative to Markowitz MVO (−0.3120) and Static RP (−0.2788). Although Markowitz delivers the numerically highest Sharpe ratio (1.655), this advantage is largely driven by in-sample optimization, with limited robustness under realistic implementation settings. In contrast, Dynamic RP demonstrates superior downside risk management, weight stability, and adaptability to changing market conditions, suggesting a more practical and resilient framework for real-world investment applications. Overall, the findings indicate that Dynamic Risk Parity provides an effective and robust alternative to traditional mean-variance optimization, offering investors a strategy that balances return potential, risk mitigation, and portfolio stability, while addressing key limitations of classical MVO approaches.

ABSTRACT This study examines the dynamic interconnections and portfolio implications of clean energy ETFs, artificial intelligence (AI) indices, crude oil, and Bitcoin within sustainable and technology-driven financial markets. Using a Time-Varying Parameter Vector Autoregression (TVP-VAR) framework and daily data from January 2019 to December 2024, we analyze time-varying spillovers and construct optimal portfolios based on dynamic connectedness measures. The results show that clean energy and AI-related assets display relatively stable portfolio weights, whereas Bitcoin exhibits highly volatile and generally limited allocations, particularly under risk-averse strategies. Conventional approaches such as the Minimum Variance and Risk Parity portfolios tend to favor traditional assets, while the Maximum Connectedness Portfolio enhances diversification by allocating more weight to weakly connected assets, including Bitcoin and green ETFs. The findings offer practical insights for resilience-oriented and innovation-driven portfolio construction.

Regime-Aware Risk Parity: Conditioning the Covariance Matrix on Macroeconomic and Stock Market Regimes

Research on asset allocation strategies based on robust risk parity model

This research presents a novel hybrid portfolio optimization framework that combines the Hierarchical Risk Parity (HRP) algorithm with two Multi-Criteria Decision-Making (MCDM) methods, MEREC and WEDBA, specifically to overcome fundamental shortcomings in the standard HRP model. The central goal is to alleviate the chaining problem and resolve HRP’s difficulty in identifying the optimal number of clusters, issues known to negatively affect portfolio diversification and risk allocation. To achieve this structural improvement, the Elbow method is integrated directly into the HRP process, ensuring a robust cluster structure is defined before any weight allocation occurs. The MEREC method is then utilized to calculate objective criterion weights, while the WEDBA approach is employed to assess the financial performance of individual assets within each cluster generated by HRP. This HRP–MCDM algorithm is tested using daily closing price data for stocks on the BIST 100 Index covering the 2018–2022 period. The performance of portfolios generated across seven distinct linkage methods (Ward, single, complete, average, weighted, centroid, and median) is rigorously benchmarked against the outcomes from the traditional HRP approach. Findings demonstrate that the HRP–MCDM framework significantly boosts both return levels and risk-adjusted metrics, especially when using the single and Ward linkage method, thereby surpassing the standard HRP algorithm in the majority of test cases. By strategically blending machine-learning-based risk clustering with objective, multi-criteria evaluation, this study makes a vital methodological contribution to the portfolio optimization domain, equipping investors with a more stable, transparent, and performance-focused asset allocation instrument.

An Orthogonal Hierarchical Risk Parity Allocation Method for Improved Portfolio Out-of-Sample Performance

Text based hierarchical risk parity (TBHRP)

In view of the increasing dynamics and complexity of the financial market, traditional quantitative investment models are difficult to adapt to the high-frequency and changeable trading environment, while deep reinforcement learning (DRL) has gradually become a hot topic in portfolio optimization research with its adaptive decision-making advantages. This study combines the strategy stability of Nearest Neighbor Strategy Optimization (PPO) with the value evaluation ability of Deep Q Network (DQN), aiming to solve the problems of large fluctuations in strategy updates and difficult risk-return balance in dynamic asset allocation. The model combines the clipping mechanism of PPO with the experience replay of DQN to optimize long-term value prediction and limit the scope of strategy updates based on historical experience, thereby improving the robustness of investment decisions. The experiment of constructing a dynamic portfolio based on 15 Chinese A-share stocks (backtest period 2020-2025) shows that the cumulative return of the improved PPO algorithm with the introduction of the invalid action shielding mechanism is 74.8% and the annualized return is 33.7%, which is significantly higher than the original PPO (annualized only 2.3%). In terms of risk control, the maximum drawdown of the model is 5.85%, and the annualized Sharpe ratio is stable at 1.555, which is better than the traditional risk parity model (maximum drawdown of 11.86%). By adjusting the configuration of the neural network hidden layer, the cumulative return of PPO increased to 33.7% after adding a single hidden layer, which verified the effectiveness of structural optimization. Compared with traditional machine learning models (such as random forests), the framework has an annualized return increase of about 12%, and it recovers faster and is more resilient to risks during periods of extreme volatility. The data was normalized by Z-score and corrected by 3σ outliers, divided by 7:1.5:1.5 (rolling window 252 trading days); PPO module with 3-layer fully connected network (128/64/32), γ=0.95, λ=0.9, clipping range [0.8, 1.2]; DQN was used with a dual network (playback pool 106, batch size 256, initial ε=0.9), combined with 4-head attention fusion, alternating training for 500 rounds (200 episodes per round, 60 decisions per step), and using Adam optimization. Research shows that the PPO-DQN synergy framework can continuously optimize investment portfolios by dynamically weighing returns and risks, providing innovative solutions for smart financial decision-making.

The infant mortality rate in Indonesia based on the 2012 Indonesian Demographic and Health Survey (SDKI) was 32 per 1,000 live births and the Neonatal Mortality Rate (AKN) in 2012 was 19 per 1,000 live births.This research is a quantitative analytical study with a case-control design using secondary data from medical record. The population size in this study was 104 mothers who gave birth with low birth weight (LBW).Padangsidimpuan City Hospital in 2025 period from January to September 2025. Researchers can draw the following conclusions 51% of mothers who gave birth to LBW were at risk, 79.8% had low education, 78.8% had high risk parity, 41.3% were working mothers and 65.4% experienced complications during pregnancy. Factors related to the incidence of LBW are maternal age (p value 0.000), education (p value 0.002), parity (p value 0.002) and pregnancy complications (p value 0.000). The dominant variable is maternal age with an OR value of 5.042 (95% CI 2.782-9.132) which means that mothers who are at risk (<20 and >35 years) have a 5.04 times higher chance of giving birth to LBW babies compared to mothers who are not at risk (20-35 years)

ABSTRACT This study explores the effectiveness of factor-based, sector-based, and hybrid investment strategies in the dynamic Chinese stock market, employing six advanced portfolio optimization techniques: equally weighted, risk parity, minimum variance, mean-variance, Bayes-Stein, and Black-Litterman. By analysing five key factors and ten industry sectors, we demonstrate that hybrid strategies, which integrate factor and sector allocations, consistently deliver superior risk-adjusted returns across diverse market conditions. While factor-based approaches excel in stable periods, sector-based strategies provide resilience during financial turbulence. Our findings reveal that hybrid portfolios not only mitigate extreme risks but also exhibit lower turnover, enhancing cost efficiency. These results underscore the strategic value of combining factor and sector insights for robust asset allocation in emerging markets.

This paper evaluates the performance and portfolio role of Artificial Intelligence (AI) and Blockchain exchange-traded funds (ETFs) based on monthly returns from 2010 to 2025. The findings show that both AI and Blockchain ETFs generate positive alpha and high standalone returns but also display considerable drawdown risk. Their weak correlations with each other and with broad indices highlight diversification benefits, particularly when combined with U.S. benchmarks. Portfolio optimization reveals that Global Minimum Variance (GMV) and Tangency portfolios ascribe lower weights to these ETFs, while Risk Parity portfolios have a more balanced exposure, helping to diversify risks. Efficient frontier analysis highlights that the inclusion of AI and Blockchain ETFs improves the attainable risk–return profiles, even if they are not a dominant allocation. The findings stress that AI and Blockchain ETFs are suitable as satellite holdings. When applied judiciously, they offer the potential to improve diversification and risk-adjusted performance; however, concentrated bets subject investors to undue downside risks. Positioning portfolios around broad-based indices and overlaying modest thematic tilts emerges as a prudent approach to capturing innovation-driven upsides without compromising long-term portfolio resilience.

This paper introduces a novel machine learning framework for dynamic risk-based asset allocation that addresses fundamental limitations in traditional portfolio optimization methods. The proposed architecture integrates Long Short-Term Memory networks for volatility forecasting with differentiable risk budgeting layers and regime-switching mechanisms, enabling end-to-end training of portfolio weights under adaptive risk constraints. Unlike conventional approaches that rely on static risk budgets and historical covariance estimates, our methodology dynamically adjusts risk targets based on real-time market indicators, including volatility expectations, credit spreads, and yield curve dynamics. The framework achieves three primary research objectives: first, it demonstrates superior risk-adjusted performance with a Sharpe ratio of 1.38 during the out-of-sample period (2017-2022), representing a 55% improvement over traditional risk parity strategies and a 23% enhancement over contemporary deep learning approaches. Second, the architecture maintains computational efficiency through sparse attention mechanisms, scaling linearly with asset count while processing 50-asset portfolios in under 25 milliseconds. Third, the model preserves interpretability via SHAP-based risk attribution, providing transparent insights into allocation decisions across different market regimes. Empirical results reveal particularly strong performance during volatile market conditions, with maximum drawdowns reduced by 41% during stress periods compared to conventional methods. The framework's proactive risk management capabilities were evidenced during the COVID-19 crisis, where it began reducing equity exposure two weeks before the market trough, demonstrating genuine predictive ability rather than reactive adjustment. Robustness checks confirm performance persistence under varying transaction costs, rebalancing frequencies, and alternative risk measures. These findings establish a new paradigm for portfolio optimization that successfully bridges theoretical finance with practical implementation. The framework's ability to navigate complex market environments while maintaining computational efficiency and interpretability suggests readiness for widespread institutional adoption. This research contributes to the evolving literature on differentiable finance while providing portfolio managers with a robust tool for constructing adaptive, risk-aware investment strategies.

This study examines the dynamic volatility spillovers and portfolio implications of clean and dirty cryptocurrencies (DCs) over recent crisis periods, employing a two-stage approach. First, we model the evolving connectedness structure using a time-varying parameter vector autoregression (TVP-VAR) framework. Second, we assess portfolio performance through back testing four strategies, namely minimum variance portfolio (MVP), minimum correlation portfolio (MCP), minimum connectedness and risk parity, focusing on hedging effectiveness (HE) and dynamic Sharpe ratios (SRs). The analysis covers the period from 16 September 2022 to 27 February 2025, a time marked by heightened market volatility and uncertainty. Our findings reveal that clean cryptocurrencies (CCs) such as Binance Coin (BNB), Ethereum (ETH) and Cardano consistently enhance portfolio diversification, risk mitigation and stability compared to traditional dirty assets like Bitcoin (BTC), Monero and Dash. Robustness checks across alternative asset combinations confirm the consistency of these results. These insights highlight the strategic value of integrating clean digital assets to achieve both financial resilience and sustainable investment objectives in the evolving cryptocurrency market.

Most reinforcement learning (RL) methods for portfolio optimization remain limited to single markets and a single algorithmic paradigm, which restricts their adaptability to regime shifts and heterogeneous conditions. This paper introduces a generalized version of the Modular Portfolio Learning System (MPLS), extending beyond its initial PPO backbone to integrate four RL algorithms: Proximal Policy Optimization (PPO), Deep Q-Network (DQN), Deep Deterministic Policy Gradient (DDPG), and Soft Actor-Critic (SAC). Building on its modular design, MPLS leverages specialized components for sentiment analysis, volatility forecasting, and structural dependency modeling, whose signals are fused within an attention-based decision framework. Unlike prior approaches, MPLS is evaluated independently on three major equity indices (S&P 500, DAX 30, and FTSE 100) across diverse regimes including stable, crisis, recovery, and sideways phases. Experimental results show that MPLS consistently achieved higher Sharpe ratios—typically +40–70% over Minimum Variance Portfolio (MVP) and Risk Parity (RP)—while limiting drawdowns and Conditional Value-at-Risk (CVaR) during stress periods such as the COVID-19 crash. Turnover levels remained moderate, confirming cost-awareness. Ablation and variance analyses highlight the distinct contribution of each module and the robustness of the framework. Overall, MPLS represents a modular, resilient, and practically relevant framework for risk-aware portfolio optimization.

CVaR-based risk parity model with machine learning

This study introduces a deep learning-based framework for portfolio optimization tailored to different investor risk preferences. We combine two prediction models, Long Short-Term Memory (LSTM) and One-Dimensional Convolutional Neural Network (1D-CNN), with three portfolio frameworks: Mean-Variance with Forecasting (MVF), Risk Parity Portfolio (RPP), and Maximum Drawdown Portfolio (MDP). Each framework represents a distinct risk profile: return-seeking, moderate-risk and conservative. The dataset is constructed from daily returns of VN-100 stocks in Vietnam, covering the period from 2017 to 2024. Forecasts from the deep learning models are integrated into each optimization approach. Results from the 2023–2024 test period showed that LSTM outperforms 1D-CNN in both accuracy and stability. Portfolios using LSTM achieved better performance. LSTM+MVF delivers the best risk-adjusted returns, while LSTM+MDP achieves the highest total return. The study highlights the value of aligning predictive models with appropriate optimization strategies for improved investment outcomes. Future work may include other asset classes, transaction cost modeling, and dynamic rebalancing. Combining deep learning with macroeconomic or alternative data could also improve forecasting and portfolio outcomes.

Risk Parity with Trend-Following

This paper investigates the dynamics of volatility spillovers in the South African foreign exchange market across calm and crisis periods, with particular attention paid to the pre- and post-COVID-19 eras. Employing daily exchange rate returns from 2015 to 2025, we apply a Quantile Vector Autoregression (QVAR) model to uncover asymmetries in spillover transmission across the distribution of returns. We evaluate the implications of these spillovers for portfolio performance under three canonical strategies: risk parity, tangency, and naïve equal-weighting. Our findings indicate that the COVID-19 shock intensified volatility spillovers and exacerbated their asymmetry, especially in the lower tail, while the pre-COVID period portrayed higher volatility compared to the post-COVID period under calm market conditions. While risk-based strategies dominate in tranquil markets, equal-weighted portfolios exhibit superior downside resilience under stress, although they ignore risk exposure. These results underscore the importance of accounting for tail-risk-driven interconnectedness in portfolio construction and risk management. This study contributes to the growing literature on volatility spillovers and offers practical insights for managing currency exposure in emerging markets under nonlinear dependence structures.

Abstract In this paper, we propose a general bi-objective model for portfolio selection, aiming to maximize both a diversification measure and the portfolio expected return. Within this general framework, we focus on maximizing a diversification measure recently proposed by Choueifaty and Coignard for the case of volatility as a risk measure. We first show that the maximum diversification approach is actually equivalent to the Risk Parity approach using volatility under the assumption of equicorrelated assets. Then, we extend the maximum diversification approach formulated for general risk measures. Finally, we provide explicit formulations of our bi-objective model for different risk measures, such as volatility, Mean Absolute Deviation, Conditional Value-at-Risk, and Expectiles, and we present extensive out-of-sample performance results for the portfolios obtained with our model. The empirical analysis, based on five real-world data sets, shows that the return-diversification approach provides portfolios that tend to outperform the strategies based only on a diversification method or on the classical risk-return approach.