Portfolio Optimization Methods Research Articles

Discrete level-bundle method for mean-CVaR portfolio optimization with cardinality constraint

Discrete level-bundle method for mean-CVaR portfolio optimization with cardinality constraint

Projected gradient descent method for cardinality-constrained portfolio optimization

Cardinality-constrained portfolio optimization aims at determining the investment weights on given assets using the historical data. This problem typically requires three constraints, namely, capital budget, long–only, and sparsity. The sparsity restraint allows investment managers to select a small number of stocks from the given assets. Most existing approaches exploit the penalty technique to handle the sparsity constraint. Therefore, they require tweaking the associated regularization parameter to obtain the desired cardinality level, which is time-consuming. This paper formulates the sparse portfolio design as a cardinality-constrained nonconvex optimization problem, where the sparsity constraint is modeled as a bounded ℓ0-norm. The projected gradient descent (PGD) method is then utilized to deal with the resultant problem. Different from existing algorithms, the suggested approach, called ℓ0-PGD, can explicitly control the cardinality level. In addition, its convergence is established. Specifically, the ℓ0-PGD guarantees that the objective function value converges, and the variable sequences converges to a local minimum. To remedy the weaknesses of gradient descent, the momentum technique is exploited to enhance the performance of the ℓ0-PGD, yielding ℓ0-PMGD. Numerical results on four real-world datasets, viz. NASDAQ 100, S&P 500, Russell 1000, and Russell 2000 exhibit the superiority of the ℓ0-PGD and ℓ0-PMGD over existing algorithms in terms of mean return and Sharpe ratio.

Portfolio Optimization Based on Rolling Window Using 5 Stocks

This paper aims to explore a method of constructing dynamically adjusted investment portfolios through the combination of rolling windows and mean-variance models. The construction of investment portfolios plays a crucial role in financial markets by effectively diversifying risks and achieving stable investment returns. This study analyzed adjusted closing price data from five stocks (AAPL, JPM, JNJ, XOM, and PG), sourced from Yahoo Finance. The rolling window method was employed to predict future stock prices and construct investment portfolios based on these predictions. This study calculated annualized average return and covariance matrix for each window. The mean-variance model and optimization algorithms were then used to determine the optimal weights for daily portfolio composition. The results demonstrate the high accuracy of the rolling window method in predicting short-term stock prices. Notably, Procter & Gamble (PG) and Johnson & Johnson (JNJ) exhibited the lowest prediction errors. Investment portfolios based on rolling window predictions performed well over a certain period. Nevertheless, the S&P 500 index did not show a substantial disadvantage compared to their long-term performance. This research provides dynamic portfolio optimization methods of practical value for investors in financial markets.

On bank stock return spillovers in South Africa: Implications for portfolio hedging

The purpose of this study is to investigate the return spillovers among the five systemically important banks in South Africa. The study employs a time-varying parameter vector autoregression (TVP-VAR) framework utilising daily data between 2000 and 2024. A minimum return connectedness portfolio is constructed from the spillover indexes and compared with traditional portfolio optimisation methods in terms of risk reduction and reward-to-risk ratios. The research findings reveal that Capitec, the smallest bank in terms of assets, has weaker return connectedness with other banks while ABSA, FirstRand, Nedbank and Standard Bank are moderately connected. Moreover, the two largest banks in terms of assets, Standard and FirstRand, are more connected than any other pairwise connection in the network. During significant events such as national elections, the global financial crisis and the COVID-19 pandemic, system-wide connectedness increases.

Robust Portfolio Mean-Variance Optimization for Capital Allocation in Stock Investment Using the Genetic Algorithm: A Systematic Literature Review

Traditional mean-variance (MV) models, considered effective in stable conditions, often prove inadequate in uncertain market scenarios. Therefore, there is a need for more robust and better portfolio optimization methods to handle the fluctuations and uncertainties in asset returns and covariances. This study aims to perform a Systematic Literature Review (SLR) on robust portfolio mean-variance (RPMV) in stock investment utilizing genetic algorithms (GAs). The SLR covered studies from 1995 to 2024, allowing a thorough analysis of the evolution and effectiveness of robust portfolio optimization methods over time. The method used to conduct the SLR followed the Preferred Reporting Items for Systematic Reviews and Meta-Analysis (PRISMA) guidelines. The result of the SLR presented a novel strategy to combine robust optimization methods and a GA in order to enhance RPMV. The uncertainty parameters, cardinality constraints, optimization constraints, risk-aversion parameters, robust covariance estimators, relative and absolute robustness, and parameters adopted were unable to develop portfolios capable of maintaining performance despite market uncertainties. This led to the inclusion of GAs to solve the complex optimization problems associated with RPMV efficiently, as well as fine-tuning parameters to improve solution accuracy. In three papers, the empirical validation of the results was conducted using historical data from different global capital markets such as Hang Seng (Hong Kong), Data Analysis Expressions (DAX) 100 (Germany), the Financial Times Stock Exchange (FTSE) 100 (U.K.), S&P 100 (USA), Nikkei 225 (Japan), and the Indonesia Stock Exchange (IDX), and the results showed that the RPMV model optimized with a GA was more stable and provided higher returns compared with traditional MV models. Furthermore, the proposed method effectively mitigated market uncertainties, making it a valuable tool for investors aiming to optimize portfolios under uncertain conditions. The implications of this study relate to handling uncertainty in asset returns, dynamic portfolio parameters, and the effectiveness of GAs in solving portfolio optimization problems under uncertainty, providing near-optimal solutions with relatively lower computational time.

A novel portfolio optimization method and its application to the hedging problem

Abstract In this article we will propose a novel, self-financing, dynamic and path dependent portfolio trading strategy which is based on the well known principle “sell high – buy low”. Trading strategies are important also for the hedging problem selling/buying an option. The main problem of the writer of an option is how to invest the amount that she has received selling the option therefore the proposed trading strategy play an important role here. We will see that the hedging problem reduces to an optimization one and therefore the portfolio optimization and the hedging problem are closely related. We will also propose a deterministic portfolio selection method (i.e., without making any assumption-guess about the assets) and a notion of a deterministic fair price of an option.

Portfolio Optimization with Multi-Trend Objective and Accelerated Quasi-Newton Method

We propose a portfolio optimization method with a multi-trend objective and an accelerated quasi-Newton method (MTO-AQNM). It leverages a BFGS-based quasi-Newton algorithm and incorporates an ℓ1 regularization term and the self-funding constraint. The MTO is designed to identify multiple trend reversals. Different trend reversals are asymmetric, and we hoped to extract rich and effective information from them. The AQNM adopts the BFGS method with the Wolfe conditions, which reduces computational complexity and improves convergence speed. We wanted to evaluate the performance of our algorithm through financial markets that were asymmetric in all respects. To this end, we conducted comprehensive experimental approaches on six benchmark data sets of real-world financial markets that were asymmetric in time, frequency, and asset type. Our method demonstrated superior performance over other state-of-the-art competitors across several mainstream evaluation metrics.

LQ45 Stock Portfolio Optimization by considering Return Predictions using the Holt Winter Method

A portfolio represents a selection of assets owned by individuals or groups with the goal of generating profit. Stocks are among the various forms of investments available. Investors typically consider two key factors: expected returns and associated risks. Portfolio optimization endeavors to maximize returns while minimizing risks. With the progression of time, portfolio optimization strategies increasingly incorporate return predictions using machine learning techniques. In this study, the Holt-Winter method is employed to forecast stock prices and returns, known for its accuracy in estimating seasonal time series data. Previous research primarily utilized the Mean- Variance method for portfolio optimization, but the outcomes were deemed unsatisfactory. Therefore, this study adopts a novel approach by integrating the Mean-Variance model with forecasting. Performance testing of the optimal portfolio emphasizes sensitivity, aiming for substantial average returns, minimal standard deviation, and a high Sharpe ratio. Comparative analysis with the LQ45 index portfolio reveals that the sensitivity-driven portfolio exhibits superior performance, yielding higher values in terms of average returns, standard deviation, and Sharpe ratio.

Constructing Portfolio Optimization: Analysis in Indonesia Non-Cyclical Industry (Markowitz Approach and Skewness and Kurtosis)

Purpose: The purpose of this research is to examine the application of portfolio optimization in the context of real-world financial issues, particularly in light of the challenges posed by the COVID-19 pandemic. Traditional portfolio optimization strategies, such as those proposed by Markowitz, often rely on the assumption of normally distributed returns, which may not accurately capture the risks associated with extreme events like the COVID-19 crisis. This study aims to shed new light on portfolio optimization methods by exploring various approaches and considering the implications of non-normally distributed returns on portfolio construction. Methods: This research employs a quantitative approach to analyze portfolio optimization techniques in the face of non-normally distributed returns. Using data from financial markets impacted by the COVID-19 pandemic, the study investigates different portfolio construction methods, including risk-free rate for equally weighted portfolios, optimal risk portfolios, minimum variance weights, and maximum expected returns. Various risk metrics such as variance, Sharpe ratio, and standard deviation are considered to evaluate portfolio performance under different constraints. Results and discussion: The empirical findings highlight the limitations of traditional portfolio optimization techniques, particularly in accurately assessing and managing risk in the presence of non-normally distributed returns. Assets exhibit heavily tailed returns, leading to an underestimation of risk when using standard approaches. The study identifies that certain assets offer high returns but also entail significant risks, necessitating a nuanced approach to portfolio construction. By considering stable distribution models and optimizing for both maximum expected return and minimum variance weight, investors can build more profitable and diversified portfolios while managing risk effectively. Implications of the research: The research findings have important implications for investors and financial practitioners, particularly in navigating uncertain market conditions such as those brought about by the COVID-19 pandemic. By recognizing the limitations of traditional portfolio optimization methods and embracing more sophisticated approaches that account for non-normally distributed returns, investors can make more informed decisions and better manage portfolio risk. These insights can inform the development of robust investment strategies tailored to mitigate the impact of extreme events on portfolio performance. Originality/value: This research contributes to the literature by offering a fresh perspective on portfolio optimization under the backdrop of the COVID-19 pandemic. By systematically evaluating different portfolio construction methods and considering the implications of non-normally distributed returns, the study advances understanding in the field of financial risk management. The identification of stable distribution models and the emphasis on balancing maximum expected return with minimum variance weight provide practical guidance for investors seeking to build resilient and profitable portfolios in turbulent market environments. Overall, this research underscores the importance of adapting portfolio optimization strategies to address the realities of contemporary financial markets.

EvoFolio: a portfolio optimization method based on multi-objective evolutionary algorithms

Optimal portfolio selection—composing a set of stocks/assets that provide high yields/returns with a reasonable risk—has attracted investors and researchers for a long time. As a consequence, a variety of methods and techniques have been developed, spanning from purely mathematics ones to computational intelligence ones. In this paper, we introduce a method for optimal portfolio selection based on multi-objective evolutionary algorithms, specifically Nondominated Sorting Genetic Algorithm-II (NSGA-II), which tries to maximize the yield and minimize the risk, simultaneously. The system, named EvoFolio, has been experimented on stock datasets in a three-years time-frame and varying the configurations/specifics of NSGA-II operators. EvoFolio is an interactive genetic algorithm, i.e., users can provide their own insights and suggestions to the algorithm such that it takes into account users’ preferences for some stocks. We have performed tests with optimizations occurring quarterly and monthly. The results show how EvoFolio can significantly reduce the risk of portfolios consisting only of stocks and obtain very high performance (in terms of return). Furthermore, considering the investor’s preferences has proved to be very effective in the portfolio’s composition and made it more attractive for end-users. We argue that EvoFolio can be effectively used by investors as a support tool for portfolio formation.

Improved Hedge Fund Portfolio Optimization Using 2-Step Covariance Matrix and Fund Transaction Costs

This study focuses on fund portfolio investments in the Chinese market. The application of classic portfolio optimization methods encounters several issues when applied to fund portfolios. For example, issues such as the non-normal distribution of returns on funds or fund portfolios, turnover rate limitations in fund investments, and liquidity constraints of fund assets, which can lead to transaction costs and opportunity costs, are prevalent challenges. The existence of these issues can compromise the effectiveness of classic portfolio optimization methods like Mean-Variance Optimization. This may result in a reduction of accuracy in determining the portfolios optimal weights, a deviation of actual trading results from the models optimal expectation, and may even render the optimal weights impractical in real-world scenarios. To address these challenges, this paper integrates the 2-Step covariance matrix method (2-Step method) and the measurement of fund transaction costs into the portfolio optimization process. The paper finds that the 2-Step method, compared to the baseline, can indeed improve the risk-return indicators of the potimal fund portfolio. The inclusion of the transaction cost can effectively control the turnover frequency of the portfolio. Even after accounting for these costs, the 2-Step method continues to exhibit a significant improvement effect compared to the baseline.

Portfolio Optimization Using ANNs and Mean-Semi Variance Markowitz Model: A Comparative Study of South Asian Economies

This study uses many portfolio approaches to optimize mean semi-variance portfolios using artificial neural networks for South Asian investors. These methods include the mean-semi variance strategy, minimal variance approach, limited portfolios with transaction costs and turnover constraints, constrained portfolios with risk and return diversification limits, and the equally weighted approach. These portfolio strategies are analysed using the excess Sharpe ratio and the Information Ratio for financial efficiency and diversity, respectively, and validated using the 130/30 portfolio strategy. The Pakistan Stock Exchange (PSX), Bombay Stock Exchange (BSE), Dhaka Stock Exchange (DSE), Columbo Stock Exchange (CSE), and Nepal Stock Exchange (NEPSE) supply daily data for empirical research. We use a large data collection from 2017 to 2021. The study found that ANN-generated estimators using a mean semi-variance optimization strategy outperformed alternative portfolio optimization methods in South Asian markets. The research reveals that ANN-based returns outperform correlation statistics, descriptive analysis, and mean square prediction error (MPSE). The study suggests utilizing naive diversification as a benchmark for other portfolio optimization methodologies.

Does Islamic investing modify portfolio performance? Time-varying optimization strategies for conventional and Shariah energy-ESG-utilities portfolio

This paper aims to assess the performance of Islamic portfolios vis-à-vis their conventional counterparts across two distinct periods: the pre-COVID-19 era and the COVID-19 era. Departing from prior studies, this study makes a novel contribution by employing an extensive array of 18 portfolio optimization techniques to construct optimal portfolios for conventional stock indices encompassing energy, utilities, and environmental, social, and governance (ESG) dimensions, as well as their Islamic equivalents. Performance comparisons are made utilizing three risk-adjusted performance measures, namely the Sharpe ratio, the Omega ratio, and the Sortino ratio. Our findings reveal that Shariah portfolios outperform conventional portfolios across all performance measures and risk-aversion levels when the most effective optimization methods are employed, both during pre-crisis and crisis periods. Additionally, our analysis highlights certain methods, namely EWMA, GM, DCC, and SHC, which produce portfolios exhibiting superior performance relative to alternative methods, as assessed by risk-adjusted metrics. Furthermore, Islamic portfolios demonstrate higher average returns compared to their conventional counterparts. Notably, incorporating ESG-related stocks into energy and utilities assets significantly enhances average returns, underscoring the potential of ESG investments. Collectively, our findings have noteworthy implications for investors, as they emphasize the role of Islamic stocks as effective diversifiers, yielding favorable resource allocation opportunities during times of crisis as well as stability. However, investors should exercise caution in selecting the optimal portfolio optimization method, as substantial performance disparities exist among different approaches.

A monotone numerical integration method for mean–variance portfolio optimization under jump-diffusion models

We develop a highly efficient, easy-to-implement, and strictly monotone numerical integration method for Mean-Variance (MV) portfolio optimization. This method proves very efficient in realistic contexts, which involve jump-diffusion dynamics of the underlying controlled processes, discrete rebalancing, and the application of investment constraints, namely no-bankruptcy and leverage.A crucial element of the MV portfolio optimization formulation over each rebalancing interval is a convolution integral, which involves a conditional density of the logarithm of the amount invested in the risky asset. Using a known closed-form expression for the Fourier transform of this conditional density, we derive an infinite series representation for the conditional density where each term is strictly positive and explicitly computable. As a result, the convolution integral can be readily approximated through a monotone integration scheme, such as a composite quadrature rule typically available in most programming languages. The computational complexity of our proposed method is an order of magnitude lower than that of existing monotone finite difference methods. To further enhance efficiency, we propose an implementation of this monotone integration scheme via Fast Fourier Transforms, exploiting the Toeplitz matrix structure.The proposed monotone numerical integration scheme is proven to be both ℓ∞-stable and pointwise consistent, and we rigorously establish its pointwise convergence to the unique solution of the MV portfolio optimization problem. We also intuitively demonstrate that, as the rebalancing time interval approaches zero, the proposed scheme converges to a continuously observed impulse control formulation for MV optimization expressed as a Hamilton–Jacobi–Bellman Quasi-Variational Inequality. Numerical results show remarkable agreement with benchmark solutions obtained through finite differences and Monte Carlo simulation, underscoring the effectiveness of our approach.

The performance of bank portfolio optimization

AbstractGiven a liability structure, the bank portfolio optimization determines an asset allocation that maximizes profit, subject to restrictions on Basel III ratios and credit, liquidity, and market risks. Bank allocation models have not been tested using historical data. Using an optimization model based on turnover constraints, we develop such tests, which document the superior performance of optimization strategies compared to heuristic rules, resulting in an average annual out‐of‐sample outperformance of 15.1% in terms of return on equity using our data set. This outperformance is remarkable and contrasts with the reported underperformance of several portfolio optimization methods in the case of investment management.

Proximal linearized method for sparse equity portfolio optimization with minimum transaction cost

In this paper, we propose a sparse equity portfolio optimization model that aims at minimizing transaction cost by avoiding small investments while promoting diversification to help mitigate the volatility in the portfolio. The former is achieved by including the ell _{0}-norm regularization of the asset weights to promote sparsity. Subjected to a minimum expected return, the proposed model turns out to be an objective function consisting of discontinuous and nonconvex terms. The complexity of the model calls for proximal method, which allows us to handle the objective terms separately via the corresponding proximal operators. We develop an efficient algorithm to find the optimal portfolio and prove its global convergence. The efficiency of the algorithm is demonstrated using real stock data and the model is promising in portfolio selection in terms of generating higher expected return while maintaining good level of sparsity, and thus minimizing transaction cost.

Enhancing portfolio resilience during crisis periods: Lessons from BRICS indices and multi asset strategies

This paper uses Markowitz’s mean-variance model to construct an investment portfolio incorporating multiple assets – BRICS equity indices, Gold, crude oil, bonds, and cryptocurrencies. The optimally created risky portfolios outperform alternative portfolio optimization methods – the naive portfolio and the equal risk contribution portfolio; and established indices – the S&P 500 and the MSCI Emerging Equity Index in terms of metrics – adjusted Sharpe ratio, modified Sharpe ratio, and the modified Value at Risk. The findings are validated across different periods, including the COVID-19 period and the Russian invasion of Ukraine, including various in and out of sample periods. The findings highlight the benefits of portfolio diversity, mainly using BRICS indices, Gold, and Brent Crude oil, and challenge the notion of limited diversification benefits in BRICS indices found in previous studies. This paper further suggests the potential of emerging market bonds ETF as a diversification option during turbulent economic periods and highlights the limitations of cryptocurrencies in optimizing multi asset portfolios. By adopting the recommended multi asset portfolios, investors can enhance their risk-return trade-offs and achieve superior performance compared to the S&P500 and MSCI emerging indices. Lastly, the paper recommends future research opportunities in measuring portfolio performance and hedging strategies considering risk-adjusted return measurements, transaction expenses, and dynamic rebalancing techniques.

Enhancing Portfolio Performances through LSTM and Covariance Shrinkage

Portfolio optimization is a crucial aspect of finance, requiring advanced analytical tools and modeling techniques. This paper proposes a new method for portfolio optimization that combines Long Short-Term Memory (LSTM) forecasting with Covariance Shrinkage and Mean-Variance Optimization (MVO) to construct diversified portfolios that maximize risk-adjusted returns. The study utilizes an LSTM-based model to predict stock prices, evaluating its performance using the RMSE metric. The calculated RMSE of 0.0849 indicates accurate and robust predictions. The portfolio constructed shows different weights each day for different assets based on the minimum variance and maximum Sharpe ratio portfolios. As of January 3rd, 2023, the assets with the largest proportion in the Maximum Sharpe Ratio portfolio and in the Minimum Volatility portfolio, are respectively BA, accounting for 27.64% of the portfolio and PG, accounting for 32.66% of the portfolio. This paper compares the performance of the proposed method and benchmark methods by applying 30 daily portfolio weights to real returns. The portfolio constructed by the proposed method has higher cumulative return with a higher Sharpe ratio and lower maximum drawdown, indicating a better ability to diversify risks and create returns. The proposed method offers a new perspective on portfolio optimization, which can potentially benefit investors and asset managers.

Adaptive consensus reaching process with dynamic weights and minimum adjustments for group interactive portfolio optimization

The mean–variance model (MV) can help companies in an economic downturn identify portfolios that increase revenue while reducing risk. However, the MV optimization model derives a set of optimal portfolios. The limited rationality and experience of the company's manager, as well as the conflict of interest with the shareholders, require several experts to help the manager make investment decisions. Moreover, experts' unique preferences require negotiation through interaction to reach a consensus, which is not considered in existing group interactive optimization algorithms. The social network group decision-making (SNGDM) method is an effective approach for reaching consensus. Notably, in addition to the trust relationships, experts' intrinsic risk preferences and rational levels affect investment decisions and change with the decision-making process. Another important problem is reaching consensus with finite interaction costs. Although dynamic opinion is considered by current minimum cost models, the impact of dynamic trust on consensus is neglected. This research aims to develop a group interactive portfolio optimization method based on SNGDM (SNGDM-PO) to address the mentioned limitations. First, we introduce consensus into group interactive optimization algorithms and develop an adaptive learning model that simulates group preferences. Then, we combine dynamic intrinsic influence and extrinsic trust to generate experts' dynamic weights. To reduce interaction consensus costs, we propose an adaptive consensus feedback mechanism that minimally adjusts for dynamic opinion and trust with a time constraint. Finally, empirical study and comparison findings demonstrate that the SNGDM-PO can efficiently obtain consensus with minimum adjustments and low interaction costs and produce high group satisfaction.

Comparative Analysis of Industry and Country Diversification Benefits of the Equity Portfolio

This study focuses on the construction of investment portfolios using different optimization methods (1/N, risk parity, hierarchical risk parity, mean variance) based on two ways of asset diversification—cross-country and cross-industry—and assesses their effectiveness using a number of metrics. The main research question is which approach to asset diversification turns out to be the most profitable for the investor in terms of return and risk. Research results’ degree of susceptibility of fluctuations in time is assessed by an interperiod analysis with the division of the time interval under consideration into crisis and noncrisis years. According to the results, the industry asset allocation strategy outperforms the asset allocation strategy focused on country diversification. In crisis years, these differences are amplified, but the results are highly dependent on the portfolio optimization method.