Mean-variance Model Research Articles

Carbon Emissions Pricing: Linkages Between EU ETS Spot and Future Prices and Completeness of EU ETS Market

The study examines the relationship and linkages between spot and future prices of European Union Emission Trading Systems (EU-ETS) during 2019–2021. Dynamic conditional correlation specification of multivariate generalized autoregressive conditional heteroskedasticity (MGARCH) model was employed to examine the time-varying correlation between spot and future prices. Also, vector autoregression mean model and MGARCH-Baba-Engle-Kroner-Kraft volatility model was jointly estimated to model the spillover between EU ETS spot and future prices in the first and second moments. Lastly, we utilize the variance-covariance matrix of joint mean-variance model estimation to derive the optimal conditional hedge ratio as well as the hedge effectiveness of EU ETS future contracts. Our findings reveal a high conditional correlation and significant spillover between carbon spot and future markets in EU. Further, our study uncovers a high degree of hedge effectiveness for EU ETS future contracts. This is possibly the first study that examines the linkages between EU ETS spot and future prices pertaining to the recent transition stage of phase III and the initial stage of ongoing phase IV of the ETS market. Our findings pinpoint to ETS markets becoming more complete and in turn offering optimal hedging avenues. JEL Codes: G15, C58, Q38

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.

APPLICATION OF QUADRATIC PROGRAMMING ON PORTFOLIO OPTIMIZATION USING WOLFE’S METHOD AND PARTICLE SWARM OPTIMIZATION ALGORITHM

Stock portfolios can be modeled into quadratic programming problems using the Markowitz mean-variance model. Quadratic programming problems can be solved using two methods, namely classical and heuristic methods. In this research, the classical method uses Wolfe’s method, while the heuristic method uses the particle swarm optimization (PSO) algorithm. This research aims to determine optimal results in portfolio problems using two methods, namely Wolfe’s method and the PSO algorithm. The data used in this research is data from 10 stock companies that distribute the highest dividends in the IDX High Dividend 20 category for the 2022 period. The research results discuss the portfolios of PT Astra International Tbk and PT. Indo Tambangraya Megah Tbk. Based on the result, using Wolfe’s method, the ASII and ITMG stock portfolios are obtained, namely the optimal proportion of ASII shares = 0.76401 or 76.401% and ITMG shares = 0.23598 or 23.598%, while the PSO algorithm obtains a portfolio of ASII and ITMG shares, namely ASII shares = 75.02% and ITMG shares = 24.98%. Compared to Wolfe’s method, the PSO algorithm has a smaller Z value 5.7.

An uncertain bi-objective mean-entropy model for portfolio selection with realistic factors

In the imprecise investment environment, there are many indeterminate factors impacting security returns. This paper introduces a portfolio optimization problem where cross-entropy is utilized to control portfolio risk within the framework of uncertainty theory and presents an uncertain bi-objective mean-entropy portfolio selection model. To be more realistic, some realistic factors such as minimum transaction lots, dividend factors and tax factors are also considered. By introducing a risk preference coefficient, the bi-objective model is converted into a single-objective model and some equivalents are discussed. Additionally, a hybrid intelligent algorithm integrating a genetic algorithm with uncertain estimation is designed to solve the proposed model. Finally, a case study is executed to confirm the practicability of the model and the performance of the algorithm, and an empirical analysis based on the proposed model and the uncertain mean–variance model is developed to illustrate the advantage of the uncertain mean-entropy model in practical investment.

Optimization of Investment Portfolio Mean-Variance Model Using Genetic Algorithm

Optimization of Investment Portfolio Mean-Variance Model Using Genetic Algorithm

A Bibliometric Analysis of Possibilistic Portfolio Selection Models

Possibility theory is one of the most used uncertainty theories in decision-making. This study aims to examine possibilistic portfolio selection models. In this context, we perform their bibliometric analysis with the Web of Science (WOS) data, using the Bibliometrix, without limiting the timespan. We get many results by analyzing the data of 303 documents, of which timespan is from 1995 to 2023. We see that W. G. Zhang is the most influential author in this field. The paper introducing the possibilistic mean-variance (MV) model is the most influential document in this field. The paper introducing Markowitz’s MV model is the most influential reference. China is the most productive country in this field, whereas The South China University of Technology is the most productive institution in this field. Fuzzy Sets and Systems is the most influential journal in this field. Variance originated from Markowitz’s MV model is the most critical keyword plus in this field. It has also maintained its trend topic position for a long time. To the best of our knowledge, this is the first paper making a bibliometric analysis of possibilistic portfolio selection models.

Optimal Portfolio Analysis of Listed Companies in IDX 30

Investors face the dual considerations of return and risk when making investment decisions. Therefore, proper analysis is crucial, especially during the COVID-19 crisis, to achieve maximum returns while minimizing risk. This research used three portfolio optimization models, the Mean-Variance Model, the Mean-Absolute Deviation Model and the Value-at-Risk Model, to construct a stock portfolio. The findings indicated that the Mean-Variance Model can yield an expected return of 16.55% and a portfolio risk of 258.66%. The result from the Mean-Absolute Deviation Model was that the target return is 16%, along with a portfolio risk of 282.43%. Keywords: Portfolio Optimization, Mean-Variance, Mean Absolute Deviation, Value at Risk, R Language, IDX30

The Tail Mean–Variance optimal capital allocation under the extended skew-elliptical distribution

In this paper we introduce a novel Tail Mean-Variance (TMV) model intended to optimize capital allocation in financial decision-making. The TMV model has many intriguing properties, such as considering the variability and tail risk of the loss function simultaneously. We specifically consider the multivariate extended skew-elliptical (ESE) distributions, which are widely applicable in financial and insurance data modeling. Additionally, we explore the probabilistic properties of the multivariate extended skew-elliptical (ESE) distributions and present explicit formulas for the TMV model within this distribution. Finally, a numerical example is provided to illustrate the results.

The Comparison of Investment Portfolio Optimization Result of Mean-Variance Model Using Lagrange Multiplier and Genetic Algorithm

Investment portfolio optimization is carried out to find the optimal combination of each stock with the aim of maximizing returns while minimizing risk by diversification. However, the problem is how much proportion of funds should be invested in order to obtain the minimum risk. One approach that has proven effective in building an optimal investment portfolio is the Mean-Variance model. The purpose of this study is to compare the results of the Mean-Variance model investment portfolio optimization using Lagrange Multiplier method and Genetic Algorithm. The data used are stocks that are members of the LQ45 index for the period February 2020-July 2021. Based on the research results, there are five stocks that form the optimal portfolio, namely ADRO, AKRA, BBCA, CPIN, and EXCL stocks. The optimal portfolio generated by the Lagrange Multiplier method has a risk of 0.000606 and a return of 0.000726. Meanwhile, using the Genetic Algorithm resulted in a risk of 0.000455 and a return of 0.000471. Thus, the Genetic Algorithm method is more suitable for investors who prioritize lower risk. Meanwhile, the Lagrange Multiplier method produces a relatively higher risk, making it less suitable for investors who expect a small risk.

Research on the Optimal Strategy of Investment Portfolio Based on Markowitz Model

Nowadays, investment is becoming increasingly common. Under the globalization of the economy, investors are given more investment opportunities and choices. Investors need to select excellent assets and allocate the selected asset portfolio on weight during the investment process. The mean-variance model proposed by Markowitz plays an important guiding role in investment and risk management. This model can effectively evaluate investors portfolio risk and return decisions and significantly impact their decision-making choices. This study uses the data of the annual average return and variance from 11 risky assets and 1 risk-free asset in about 20 years to construct an investment portfolio in a fixed group based on the Markowitz Model. The study calculates the changes in the allocation of the maximum Sharpe ratio and the minimum variance portfolios under three different constraints and one condition with the addition of risk-free assets, respectively, and then analyzes changes in the outcome of data. The capital allocation lines are introduced to analyze the outcome of an investment portfolio with risk-free assets. Compared with the changes from three constraints, reasons are explained why these changes happen under three constraints. Then, the above limitations are taken as a premise, and the study proposes recommendations for the optimal investment portfolio selection for different investors according to the investment cycle, the preference of investors based on the situation of different investors, and market constraints. The study guides investors future investment decisions.

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.

Robust portfolio selection with smart return prediction

Return prediction and portfolio decision are closely connected steps in forming optimal portfolios. However, existing studies often separate return prediction from portfolio decisions or overlook the high sensitivity of optimal portfolios to estimated returns, leading to poor out-of-sample performance. To address these two issues simultaneously, this paper proposes a robust portfolio framework that optimizes parameters in return prediction by maximizing robust mean–variance utility. The proposed framework effectively combines return prediction and portfolio decisions while mitigating the uncertainty in return prediction. Empirical analysis on S&P500 constituents demonstrates that, compared to the benchmarks, the proposed framework achieves lower risk and higher risk-adjusted returns, originating from more stable weight estimation and diversified portfolios. Further, we uncover that the market friction characteristics considerably contribute to the proposed strategies’ performance. Our results suggest that integrating return prediction with the robust mean–variance models can significantly improve portfolio performance.

Optimisasi Portofolio Saham Syariah dengan Pendekatan Fuzzy Goal Programming

This study discusses about solving the problem of stock portfolio optimization. The portfolio optimization problem can be modeled as a multiobjective model with two objective functions, they are maximizing return and minimizing risk. This study uses the mean-variance Markowitz model to model the portfolio. The fuzzy goal programming approach is used to solve stock portfolio problems by converting a multi-objective model into a fuzzy mathematical programming model which is solved using a nonlinear programming method. The results of implementing the model on stocks listed on the Jakarta Islamic Index show that the best investment proportion forms an optimal portfolio according to the level of aspirations. Several different optimal portfolios can be formed using the fuzzy goal programming approach. The difference in the results of these portfolios depends on the level of aspiration of investors.

Analyzing the Performance of Diversified Commodity Derivatives Portfolios in Brazil

This paper seeks to investigate how we invest in commodity derivatives by comparing the performance of portfolios constructed using the mean-variance model, hierarchical risk parity, and the naive strategy with the stock market and the risk-free rate. For this purpose, daily closing future prices of ethanol, corn, cattle, gold, coffee, and soybeans traded in Brazil from 2015 to 2023 are considered, in addition to the rate of interbank deposit certificates (CDI or DI) and the performance of the theoretical portfolio. of shares in the Ibovespa index. The constructed portfolios are analyzed using the Sharpe Ratio, CVaR, Beta, and Maximum Drawdown metrics and the comparison of their returns. The results indicate that the portfolio constructed using the mean-variance model presented the lowest risk ratios and generated the highest return, followed by the hierarchical risk parity, the naive strategy, and the Ibovespa index. These findings are useful for scientific literature that investigates finance by bringing empirical evidence of Brazilian commodities to portfolio managers, investors, and other agents in the supply chain.

Small Portfolio Construction with Cryptocurrencies

In this paper, we describe and apply different models of portfolio construction in the selection between a small number of big-cap cryptocurrencies. Our purpose is to select the minimum riskiness between cryptocurrencies, comparing different risk measures and maximum diversification. We build our models without the constraints of the expected returns. Without relying on expected returns, we have the same condition on the comparison between them. Cryptocurrencies are not common stock or other assets indexed in the market but it is interesting to study how diversification can significantly improve investment performance. We first give the methodology to use high-frequency observation data, in the numeral approximation especially in the novel application of the Risk parity models, used with different risk measures we can achieve a very good result, from the position of gaining and variation. Since Risk parity models divide the weights of the asset in equal risk contribution proportion, it is suggested to use a small number of cryptocurrencies, otherwise their performance will be close to the uniform portfolio. To the traditional Mean Variance model, and the alternative, Expected shortfall/Conditional Value at Risk, we use three versions of Risk Parity with two different risk measures and a naive risk parity. The uniform portfolio is used as a benchmark for selection comparison with the other portfolio models. We give the conditions for the Risk Parity with the Expected shortfall/Conditional Value at Risk (CVaR) to guarantee convergence with the numerical approximation. In the end, we study the tradeoff between each model and which is more suitable for a small cryptocurrency portfolio.

Analysis of a portfolio selection model at three-moments

This work performs a numerical simulation for an investment portfolio selection model that considers the three first moments of asset returns distribution – mean return, variance, and skewness. The application of the model, based on data collected on the platform of a Brazilian stockbroker, allowed obtaining portfolios of maximum skewness for fixed values of expected return and weighted variance of the portfolio. The results are analyzed and presented graphically, giving rise to an optimal surface for triples of moments associated with portfolios of maximum skewness. Furthermore, this experiment allowed us to confirm the relevance of considering higher-order moments in the selection of investment portfolios and verifying the efficiency of the Three-Moments model, having as reference the Markowitz solution in his Mean-Variance model.

Portfolio Optimization: Application and Comparison of Markowitz Model and Single Index Model on LQ45 Stocks in Indonesia Stock Exchange

This paper examines the optimization of an Indonesian stock portfolio using two models: the Markowitz Model (Mean-Variance Model) and the Single Index Model. The data comprises historical returns of LQ 45 stocks from January 2016 to December 2021. The focus is on selecting stocks for the portfolio and determining their weights based on the two models. The study compares the performance of both optimized portfolios with the LQ45 Index benchmark, IHSG market, and each other using Sharpe and Treynor measurements. The paper tests whether the stock composition of the optimized portfolio from both models successfully and consistently generates a better performance in the future (1 January 2022 – 31 December 2022) and (1 January 2023 – 31 December 2023) compared to both LQ45 and IHSG. The results reveal a notable contrast in portfolio performance between 2022 and 2023. In 2022, both the Markowitz and Single Index portfolios exhibited remarkable returns, surpassing LQ 45 and IHSG. However, in 2023, both portfolios experienced substantial underperformance, with negative returns and unfavorable risk-adjusted metrics. These findings underscore the dynamic nature of financial markets and the need for continuous portfolio monitoring and adaptation. Investors are encouraged to reevaluate their portfolio strategies in response to changing market conditions. The study contributes valuable insights into the temporal variability of optimized portfolios and their sensitivity to evolving market dynamics.

Cardinality Constrained Portfolio Optimization via Alternating Direction Method of Multipliers.

Inspired by sparse learning, the Markowitz mean-variance model with a sparse regularization term is popularly used in sparse portfolio optimization. However, in penalty-based portfolio optimization algorithms, the cardinality level of the resultant portfolio relies on the choice of the regularization parameter. This brief formulates the mean-variance model as a cardinality ( l0 -norm) constrained nonconvex optimization problem, in which we can explicitly specify the number of assets in the portfolio. We then use the alternating direction method of multipliers (ADMMs) concept to develop an algorithm to solve the constrained nonconvex problem. Unlike some existing algorithms, the proposed algorithm can explicitly control the portfolio cardinality. In addition, the dynamic behavior of the proposed algorithm is derived. Numerical results on four real-world datasets demonstrate the superiority of our approach over several state-of-the-art algorithms.

Research on Optimization of Investment Portfolio in the New Energy Vehicle Industry Chain

The urgent environmental problems caused by global climate change have propelled the rise of the new energy vehicle industry. Nations and business are also focusing on the research and promotion of this industry. Due to the fast development, investors face both promised investment opportunities and complicated challenges. Against this backdrop, a vital issue, which is how to choose the optimal investment portfolio to achieve the maximum possible return while experience the lowest possible risk, arises. This paper collects the closing price of stocks from 5 companies in different sectors of the new energy vehicle industry, with the period from 19 August 2022 to 18 August 2023 and then computes the variance of closing price and average return of the stocks, finally uses the COVAR function in excel to obtain the covariance matrix. After processing the data, the paper applies the Markowitz mean-variance model to obtain the optimal portfolio formation. Results show that this industry may not be a good choice for investors currently since all three portfolios show negative returns, while the negativity do not completely negate the long-term potential of this industry as the development of the industry can be influenced by various factors. It is therefore to conduct deeper research to better understand the industry dynamics and future trend.

Graphical Analysis of the Markowitz Model under Different Constraints

The Markowitz mean-variance model is an important tool in portfolio theory. This study examines the complexity of the model by analyzing its application and performance under different constraints. The study uses Bloomberg data on 10 stocks from 2001 to 2021 to rigorously analyze the impact of limitations on portfolio optimization. The study explores limitations ranging from limiting total allocations and individual weightings to intentional exclusions and long-term strategies to discern the impact on the risk-reward profile. The research uses graphical representations and key metrics such as Sharpe ratios, stock allocation lines, and various portfolio metrics to provide insight into optimal allocation strategies. The results indicate that these constraints have a significant impact on the risk-return profile of a portfolio and that it is essential to take them into account when optimizing a portfolio. The study provides a comprehensive analysis of the Markowitz model and its application under various constraints, which can help investors make informed decisions while optimizing their investment portfolios.