Markowitz Mean-variance Research Articles

Urban-scale power decarbonization using a modified power purchase agreements framework based on Markowitz mean-variance theory

Urban power decarbonization is essential in the fight against climate change, yet current research often neglects the financial risks faced by investors and the shifting demands of consumers in liberalized electricity markets. This study addresses these gaps by proposing a modified Markowitz Mean-Variance Portfolio (MVP) theory, integrated with the Low Emissions Analysis Platform (LEAP), and a deep learning model. On this basis, an urban energy transition framework centered on Power Purchase Agreements (PPAs) is proposed and developed. The framework is validated considering a case study in Kitakyushu, Japan, highlighting its potential in accelerating power sector decarbonization and achieving net-zero emissions by 2038. Additionally, the internal rate of return (IRR) remains stable between 14.5 % and 19.6 % across seven other cities. While the framework reduces long-term cash flow volatility, its effectiveness hinges on industrial electrification efficiency and regional energy self-sufficiency. The findings indicate that relying solely on renewable energy for low-carbon transitions is unrealistic. Furthermore, green hydrogen could emerge as a viable alternative to fossil fuels, potentially replacing batteries for long-term energy storage. Future research should explore cross-regional energy trade and establish legal frameworks for long-term energy transactions to bolster urban energy transition resilience across diverse geographic and economic contexts.

International Portfolio Diversification Benefits: An Empirical Investigation of the 28 European Stock Markets During the Period 2014–2024

Abstract This study investigates the benefits of international diversification in the stock markets of the 28 European countries (the EU and the UK) over two five-year periods: a stable period from 2014 to 2019 and a turbulent period from 2019 to 2024. The analysis draws on the Markowitz mean-variance, Sharpe reward-to-variability, and naive diversification models, based on which different investment strategies were developed and implemented. We find that actively managed portfolios perform significantly better than naively diversified portfolios. The analyzed markets exhibit positive short-term associations, with an average correlation coefficient of 0.29 in the first period and 0.46 in the second period. However, these markets do not show long-term cointegration. Recent crises have reduced diversification benefits, yet significant opportunities for diversification remain. Diversification benefits are almost halved in the second period: average single-market standard deviation can be reduced by 60.5% with investments in 20-indices portfolios in the stable period, and only by 33.7% with the same portfolio size in the turbulent period.

Optimizing Portfolio Construction Using Nifty India Manufacturing Index: A Risk-Return Analysis with Python

This paper presents a comprehensive analysis of portfolio construction strategies aimed at maximizing risk-adjusted returns for investors. Utilizing historical data on stock returns and risks, a meticulous selection process was employed to identify 15 stocks with superior risk-return profiles. These stocks were chosen based on their outperformance relative to the dataset's average measures, prioritizing returns higher than the dataset average and risks lower than average risk levels. The selected stocks, representing a diverse range of manufacturing-related businesses, formed the cornerstone of an optimal portfolio designed to minimize sector-specific risks while maximizing growth potential. Through rigorous portfolio optimization techniques, including Markowitz's mean-variance optimization, the study identified an optimal portfolio allocation with a return of 50.22% and volatility of 13.33%. Insights from the study offer valuable perspectives for investors seeking realistic and sustainable wealth accumulation strategies. The research also identifies avenues for future exploration, including the integration of alternative asset classes, behavioral finance insights, and advancements in risk management technologies. This study demonstrates the effectiveness of Python in financial analysis and portfolio optimization while furthering the theory of portfolios and providing valuable guidance for making investment decisions. Keywords: Returns, Risk, Volatility, Optimal Portfolio Construction, Manufacturing Sector, Markowitz Portfolio Theory,

Optimizing Portfolio Allocation in the Indian Defense Sector: A Python-Based Approach

This paper presents a detailed analysis of portfolio construction strategies aimed at maximizing risk-adjusted returns for investors, utilizing Python-based methodologies. By leveraging historical data on stock returns and risks, a meticulous selection process identified 5 stocks with superior risk-return profiles, outperforming the dataset's average measures with higher returns and lower risks. The selected stocks from the defense industry formed the foundation of an optimal portfolio designed to minimize sector-specific risks while maximizing growth potential. Through rigorous portfolio optimization techniques, including Markowitz's mean-variance optimization, an optimal portfolio allocation was identified with a return of 62.40% and volatility of 27.74%. Insights from the study provide valuable perspectives for investors seeking realistic and sustainable wealth accumulation strategies. The research also highlights future avenues for exploration, such as integrating alternative asset classes, incorporating behavioral finance insights, and leveraging advancements in risk management technologies. This study contributes to advancing portfolio theory and offers practical guidance for investment decision-making.This study demonstrates the effectiveness of Python in financial analysis and portfolio optimization while furthering the theory of portfolios and providing valuable guidance for making investment decisions. Keywords: Returns, Risk, Volatility, Optimal Portfolio Construction, Defense Sector, Markowitz Portfolio Theory,

A mean field game model for optimal trading in the intraday electricity market

AbstractIn this study, we provide a simple one period mean-field-games setting for the joint optimal trading problem for electricity producers in the electricity markets. Based on the Markowitz mean-variance approach from stock trading, we consider a decision problem of an electricity provider when determining the optimal fractions of production that should be traded in the day-ahead and in the intraday markets. Moreover, all such providers are related by a ranking criterion and each one wants to perform as good as possible in this ranking. We first start with a simple model where only the price risk in the intraday market is present and subsequently extend the problem to the cases involving either production and/or demand uncertainty. The key technique is to reduce the optimality conditions to a first order non-linear ordinary differential equation. We will illustrate our findings by various numerical examples. Our findings will in particular be important for electricity producers using renewable resources.

Enhancing portfolio management using artificial intelligence: literature review.

Building an investment portfolio is a problem that numerous researchers have addressed for many years. The key goal has always been to balance risk and reward by optimally allocating assets such as stocks, bonds, and cash. In general, the portfolio management process is based on three steps: planning, execution, and feedback, each of which has its objectives and methods to be employed. Starting from Markowitz's mean-variance portfolio theory, different frameworks have been widely accepted, which considerably renewed how asset allocation is being solved. Recent advances in artificial intelligence provide methodological and technological capabilities to solve highly complex problems, and investment portfolio is no exception. For this reason, the paper reviews the current state-of-the-art approaches by answering the core question of how artificial intelligence is transforming portfolio management steps. Moreover, as the use of artificial intelligence in finance is challenged by transparency, fairness and explainability requirements, the case study of post-hoc explanations for asset allocation is demonstrated. Finally, we discuss recent regulatory developments in the European investment business and highlight specific aspects of this business where explainable artificial intelligence could advance transparency of the investment process.

Deep Reinforcement Learning Model for Stock Portfolio Management Based on Data Fusion

Deep reinforcement learning (DRL) can be used to extract deep features that can be incorporated into reinforcement learning systems to enable improved decision-making; DRL can therefore also be used for managing stock portfolios. Traditional methods cannot fully exploit the advantages of DRL because they are generally based on real-time stock quotes, which do not have sufficient features for making comprehensive decisions. In this study, in addition to stock quotes, we introduced stock financial indices as additional stock features. Moreover, we used Markowitz mean-variance theory for determining stock correlation. A three-agent deep reinforcement learning model called Collaborative Multi-agent reinforcement learning-based stock Portfolio management System (CMPS) was designed and trained based on fused data. In CMPS, each agent was implemented with a deep Q-network to obtain the features of time-series stock data, and a self-attention network was used to combine the output of each agent. We added a risk-free asset strategy to CMPS to prevent risks and referred to this model as CMPS-Risk Free (CMPS-RF). We conducted experiments under different market conditions using the stock data of China Shanghai Stock Exchange 50 and compared our model with the state-of-the-art models. The results showed that CMPS could obtain better profits than the compared benchmark models, and CMPS-RF was able to accurately recognize the market risk and achieved the best Sharpe and Calmar ratios. The study findings are expected to aid in the development of an efficient investment-trading strategy.

Naïve Markowitz policies

AbstractWe study a continuous‐time Markowitz mean–variance portfolio selection model in which a naïve agent, unaware of the underlying time‐inconsistency, continuously reoptimizes over time. We define the resulting naïve policies through the limit of discretely naïve policies that are committed only in very small time intervals, and derive them analytically and explicitly. We compare naïve policies with pre‐committed optimal policies and with consistent planners' equilibrium policies in a Black–Scholes market, and find that the former achieve higher expected terminal returns than originally planned yet are mean–variance inefficient when the risk aversion level is sufficiently small, and always take strictly riskier exposure than equilibrium policies. We finally define an efficiency ratio for comparing return–risk tradeoff with the same original level of risk aversion, and show that naïve policies are always strictly less efficient than pre‐committed and equilibrium policies.

Optimization of national grain imports to balance risk and return: a portfolio theory approach

Abstract Global grain trade plays a key role in food security. Many nations rely on imported grain to meet their dietary requirements. Grain imports may be at risk due to weather shocks, economic crises, or international conflicts. Countries aim to balance import risk with the expected return of their grain supplies. This research brings these dual objectives together in an innovative modern portfolio theory framework. Modern portfolio theory provides a set of concepts to formulate the trade-off between risk and expected return in national grain imports. Using Markowitz’s mean-variance optimization model, we identify opportunities to reduce risk in existing national grain import accounts, without increasing costs under realistic supply mass constraints of trade partners. Several major grain importers may be able to reduce risk in their grain imports without increasing cost, such as wheat imports in Egypt, maize imports in Vietnam, and rice imports in Saudi Arabia. However, some countries would indeed have to pay more to achieve more stable grain supplies, such as wheat imports in Turkey. This study provides a framework to quantify the different costs, benefits, and levels of risk in grain trade that can inform future research and decision-making.

Financial market shocks and portfolio rebalancing

PurposeThe intention of the empirics is to contribute to the general understanding of investor responses to market price shocks. The authors review assumptions about investor behavior in response to price shocks and investigate alternative rebalancing heuristics.Design/methodology/approachThe authors use market data over 40 years to define market shocks. Portfolio rebalancing implements constrained Markowitz mean-variance (MV) heuristics.FindingsMomentum rebalancing in portfolio management outperforms contrarian rebalancing in the study interval. Sensitivity analysis by decade, sector constraints and proportion of security holdings bought or sold continue to support momentum rebalancing.Research limitations/implicationsThe results are consistent with under-responding to price shocks at consensus levels in financial markets. The theoretical background provides a basis for experimental lab studies of shocks of different magnitudes under conditions in which participants have information on the levels of other participants and a condition in which they can only observe their previous estimates.Practical implicationsManaging portfolios in the face of price disturbances of different magnitudes is informed by empirical studies and their implications for investor behavior.Originality/valueThis is the first study the authors can locate that uses market data with alternative rebalancing heuristics to estimate price returns from the respective heuristics over a time interval of 40 years. The authors support the results with sensitivity estimates and consider implications for the underlying agent heuristics in light of background studies.

Uncertain mean-CVaR model for portfolio selection with transaction cost and investors’ preferences

Portfolio selection is a common practice among investors, scholars often rely on probability statistics to construct portfolios based on the Markowitz mean–variance model. However, due to the complexity of the stock market and the limitations of historical data, it may not be possible to fully predict the future market. In such situations, it is necessary to incorporate expert knowledge and judgments to obtain estimates of security return rates. This paper proposes a class of portfolio models suitable for uncertain environments. We employ the uncertainty theory and Arbitrage Pricing Theory (APT) to measure tail risk and construct a new uncertain mean-Conditional Value-at-Risk (CVaR) model. Additionally, to better match investors’ preferences, this paper takes into account transaction costs and liquidity needs. To solve this problem, we design a hybrid algorithm that incorporates these factors. Finally, we present a numerical experiment to illustrate the performance of the proposed model. In summary, our study offers a novel approach to portfolio selection that considers uncertainty theory, tail risk, and investors’ preferences. The results demonstrate that the proposed uncertain mean-CVaR model with hybrid algorithm optimization can improve portfolio performance in uncertain market conditions.

Mixture Design of Experiments as Strategy for Portfolio Optimization

Portfolio analysis is widely used by financial investors to find portfolios producing efficient results under various economic conditions. Markowitz started the portfolio optimization approach through mean-variance, whose objective is to minimize risk and maximize the return. This study is called Markowitz Mean-Variance Theory (MVP). An optimal portfolio has a good return and low risk, in addition to being well diversified. In this paper, we proposed a methodology for obtaining an optimal portfolio with the highest expected return and the lowest risk. This methodology uses Mixture Design of Experiments (MDE) as a strategy for building non-linear models of risk and return in portfolio optimization; computational replicas in MDE to capture dynamical evolution of series; Shannon entropy index to handle better portfolio diversification; and desirability function to optimize multiple variables, leading to the maximum expected return and lowest risk. To illustrate this proposal, some time series were simulated by ARMA-GARCH models. The result is compared to the efficient frontier generated by the traditional theory of Markowitz Mean-Variance (MVP). The results show that this methodology facilitates decision making, since the portfolio is obtained in the non-dominated region, in a unique combination. The advantage of using the proposed method is that the replicas improve the model precision.

A LINEAR-PROGRAMMING PORTFOLIO OPTIMIZER TO MEAN–VARIANCE OPTIMIZATION

In the Markowitz mean–variance portfolio optimization problem, the estimation of the inverse covariance matrix is not trivial and can even be intractable, especially when the dimension is very high. In this paper, we propose a linear-programming portfolio optimizer (LPO) to solve the Markowitz optimization problem in both low-dimensional and high-dimensional settings. Instead of directly estimating the inverse covariance matrix [Formula: see text], the LPO method estimates the portfolio weights [Formula: see text] through solving an [Formula: see text]-constrained optimization problem. Moreover, we further prove that the LPO estimator asymptotically yields the maximum expected return while preserving the risk constraint. To offer a practical insight into the LPO approach, we provide a comprehensive implementation procedure of estimating portfolio weights via the Dantzig selector with sequential optimization (DASSO) algorithm and selecting the sparsity parameter through cross-validation. Simulations on both synthetic data and empirical data from Fama–French and the Center for Research in Security Prices (CRSP) databases validate the performance of the proposed method in comparison with other existing proposals.

Regularization methods for sparse ESG-valued multi-period portfolio optimization with return prediction using machine learning

In the context of global sustainable development, environmental, social, and governance (ESG) investment has become a frontier topic in the field of asset management. This paper focuses on ESG-valued multi-period portfolio selection problem which incorporates ESG factor into traditional portfolio optimization. Using ESG-valued returns instead of traditional returns, we propose an ESG-valued fused LASSO model to promote an optimal sparse multi-period portfolio strategy. The objective of the model is to minimize a combination of returns and risks with respect to ESG ratings based on classical Markowitz mean–variance framework. The sparsity of the portfolio at each period and the turnover across periods is achieved by the ℓ1 regularization approach. To improve the out-of-sample performance of the ESG-valued model, we introduce two machine learning methods, namely random forest and support vector regression, to predict the ESG-valued returns. We develop a symmetric alternating direction method of multipliers to solve the regularized optimization model. Additionally, some other sparsity-driven penalty functions are discussed which results in a general framework of multi-period portfolio optimization. Finally, numerical experiments on several real datasets from both in-sample and out-of-sample demonstrate the positive effect of ESG in multi-period portfolio optimization.

Portfolio Analysis and Investment Value Based on US Stock Market

An investment portfolio can contain a variety of assets, including cash, stocks, bonds, financial derivatives, precious metals, and cash. You shouldn't invest all of your money in the same thing, similar to the adage "don't put all your eggs in one basket." Put your eggs in a variety of baskets so that even if one breaks, they will remain safe. In this study, five equities—Eli Lilly, Amazon, FedEx, J.P. Morgan Chase Bank, and Pfizer—are constructed using the Markowitz Mean-Variance Model and Effective Frontier. The portfolio is built using data from these five stocks during the previous three years. Given that stocks can be shorted, the portfolio has a Sharpe Ratio of 9.07%, a yield of 2.99%, and a ceiling of negative 1 for each stock. The risk of this best answer, which was determined by solving combinations, is 9.63%. The usual 10-year Treasury bond's average return is 2.12%, which is less than the ideal yield.

Effectiveness and Limitation of Markowitz Mean-variance Model: Evidence from Hang Seng Index

As China's economy has expanded rapidly, its financial system has gradually improved, new financial businesses have evolved, and new theories and technologies have also been developed. The Markowitz mean-variance model, the cornerstone of asset pricing theory, is utilized as an example in this paper to demonstrate how the Markowitz portfolio theory may be applied to assess a portfolio of five stocks in the Hang Seng data, a significant indicator of the Hong Kong stock market. Empirically, the optimal portfolios with the highest Sharpe ratios and lowest variances are identified. Their expected returns, standard deviations, and Sharpe ratios are then evaluated by comparing them to equal-weight portfolios and giving the effective frontier. The author further demonstrates the guiding significance of Markowitz's portfolio theory in the Chinese investment market through the assessment of the findings. Finally, in the context of the Chinese securities market, identify and analyze the model's constraints and offer suggestions.

Portfolio Construction Based on Optimization Model and Combination of Various Assets

Since the fifties of the last century, the concept of portfolio management is proposed to enrich the portfolio theory by Markowitz et al., as well as the improvement of portfolio theory by later generations, forming a modern portfolio management theory. This paper uses Markowitz's mean variance portfolio theory to perform a portfolio return maximization analysis on selected assets with assumptions that investors are risk-averse according to mean variance theory. Based on mean variance portfolio theory, as well as Python computational simulation modeling, Monte Carlo simulation is used to find the effective set and effective frontier of the portfolio, and the optimal solution of the portfolio is filtered according to the two constraints of maximum Sharpe ratio and minimum volatility. According to the analysis, the portfolio return of the maximum Sharpe ratio is relatively high, and the maximum Sharpe ratio combination is more worthwhile than the minimum volatility combination. Overall, these results provide enlightenment for further exploration and implementation of portfolio theory.

Markowitz-based Shariah compliant portfolio model with stochastic purification and probabilistic compliance screening constraints

PurposeOne important study in the portfolio investment is the study of the optimal asset allocations. Markowitz is the pioneer of modern portfolio theory that analyses the performance of portfolio based on the mean (reward) and variance (risk). Motivated by the Markowitz's mean variance model, the purpose of this paper is to propose a new portfolio optimization model that takes into consideration both processes of purification and screening, which are key to constructing a Shariah-compliant portfolio. In practice, this paper introduces a stochastic purification variable and a probabilistic screening constraint into a portfolio model.Design/methodology/approachFirst, the authors study the stochastic nature of purification variable and apply it to both investment and dividend purification. Second, recognizing that the importance of on-going screening could adversely affect the portfolio strategy, the authors impose probabilistic constraints to control the risk of compliance change. They evaluate the proposed model by formulating the screening constraints at both asset and portfolio levels, together with three different financial screening divisors that are broadly used by the international Shariah boards. The authors also conduct an extensive empirical study using a sample of Shariah-compliant public companies listed on the Indonesia Stock Exchange.FindingsBased on the empirical example presented in this paper, the authors found that the purification variable in the proposed model is closer to the practice in the Sharia capital market in terms of the nature of the non-constant data, and this variable reduces the total income of portfolio which has not been captured in the previous literature. The authors also have successfully derived the portfolio screening constraint to mitigate the risk of the asset change to be non-compliant in the future.Originality/valueBased on the authors’ knowledge, this is the first paper that proposed the stochastic purification and the dynamic of screening processes into the Shariah portfolio model. This paper also examines the impact of non-short-selling, purification and screening policies to the performance of Shariah portfolio.

Does the substitution effect lead to feedback effect linkage between ethanol, crude oil, and soft agricultural commodities?

Despite increased demand for cleaner fuel alternatives such as ethanol in recent decades, portfolio weight allocation has become challenging due to the complex interlinkage amongst crude, ethanol and soft agricultural commodities that form part of the value chain. As a result, portfolio returns face three trade-offs in terms of risk: dispersion across mean, risk arising due to market interconnectedness, and risk arising due to global shocks for assets sharing common macroeconomic fundamentals. This study proposes an optimal weight allocation portfolio strategy, encapsulating the three risk measures and returns, estimated using state-of-the-art multi-objective elitist Non-Dominated Sorting Algorithm II (NSGA-II). Our proposed strategy performs well for newly constituted objectives against the Markowitz Mean-Variance approach and Global Minimum Variance. A balanced diversification escapes the feedback spillover loop trap at the same time. Our results indicate that soybean oil, sugar, and rice offer a better reward to risk, aiding portfolio immunisation to extreme market movements. Furthermore, using GJR-GARCH volatility to capture the volatility asymmetry effect, the Generalized Forecast Error Variance Decomposition (GFEVD) shows the existence of a strong triplet pair Crude-Ethanol-Soybean as a breeding ground for the feedback effect to occur. Moreover, replacing crude weight with ethanol depicts a fall in spillover risk up to a threshold of 30% Ethanol weight, after which the feedback effect kicks in.

Robustifying Markowitz

Markowitz mean–variance portfolios with sample mean and covariance as input parameters feature numerous issues in practice. They perform poorly out of sample due to estimation error, they experience extreme weights together with high sensitivity to change in input parameters. The heavy-tail characteristics of financial time series are in fact the cause for these erratic fluctuations of weights that consequently create substantial transaction costs. In robustifying the weights we present a toolbox for stabilizing costs and weights for global minimum Markowitz portfolios. Utilizing a projected gradient descent (PGD) technique, we avoid the estimation and inversion of the covariance operator as a whole and concentrate on robust estimation of the gradient descent increment. Using modern tools of robust statistics we construct a computationally efficient estimator with almost Gaussian properties based on median-of-means uniformly over weights. This robustified Markowitz approach is confirmed by empirical studies on equity markets. We demonstrate that robustified portfolios reach the lowest turnover compared to shrinkage-based and constrained portfolios while preserving or slightly improving out-of-sample performance.