Portfolio Analysis of Five Stocks Under Different Constraints Based on Markowitz Model & Index Model

An extension of portfolio theory in selecting projects to construct a preferred portfolio of petroleum assets

Markowitz Model (MM) and Sharpe’s Single Index Model (SIM) are two classical and practical models in portfolio theory. Currently the Chinese stock market is booming, therefore, it is worth testing whether MM and SIM can effectively spread the risk in Chinese security market. This paper establishes the MM and SIM based on the monthly observations of 10 stocks and CSI300. During the analysis process, there is an additional optimization constraint that selling short is not allowed. Finally, some conclusions are obtained: first, the constraint has a significant impact on the optimal portfolio, while the minimum-variance portfolio is very little affected. What’s more, although both MM and SIM can effectively disperse the specific risk, SIM have better performance in the Chinese stock market by providing higher rewards given the same risk level and bearing lower risk given a certain return. These conclusions are of great significance to investors in the Chinese security market, which can help them obtain the maximum utility through asset diversification.

In the financial market, the investment portfolio is always a popular topic that investors are interested in and look forward to exploring as well. Since it does affect the expected return and risk of their investment. In this paper, focusing on two of the main models that are Markowitz model and the index model. The paper also collect data from the stock market to analyze different models under different constraints that have their own practical meaning. The result of 5 constraints is concluded below by conducting an efficient frontier and quick ratio. And it finds out that the index model simplified the calculation of variance matrix but assumption under the Markowitz model is more sustainable in reality. The initial result suggests that both models have their advantages and shortcomings, suggesting it is hard to determine which one is more favorable. However, after further discussion between models. the study found that as long as the original data is accurate, Markowitz model gives better results than index model. If investors want to attain very accurate data, Markowitz model is the best choice. In this paper, the research on the two models of investment portfolios can be a consideration when people are choosing their portfolios. It can be used to create a higher return portfolio as well as a lower risk one, which brings people more thoughts on investing portfolios in promoting the finance market in sense.

It is important to investigate the different impact factors on the establishment of investment portfolio. In order to maximize the profit of a portfolio, this research selects six stocks: Adobe (ADBE), International Business Machines Corp (IBM), Bank of America Corporation (BAC), Citigroup (C), Southwest Airlines Co (LUV) and Alaska Air Group Inc. (ALK) as an empirical case to conduct investment decision. This research compares different results of two models (Markowitz model and Index Model) by analyzing the data of the portfolio such as minimum risk portfolio and maximum Sharpe Ratio portfolio. This research finds that the Markowitz Model is more appropriate to the portfolio establishment, which is presented by its sharper efficient frontier. Under all the conditions the study considered, the Markowitz Model always gets more returns than the Index Model when the risk increases by the same amount. Therefore, this research concludes that the Markowitz Model is more suitable for this portfolio. By analyzing the portfolio in two models, the study has found a better model for this portfolio, thus make more rational investments.

To compare the differences between Markowitz Model and the Single Index Model, we have used historical return data for ten stock which belong in groups to three different equity sectors to practically implement the Markowitz Model and the Single Index Model. We find the stocks in the same equity sector being noticeably positively correlated even after eliminating systemic risks, which violates the formal condition of the IM. Also, as a result of very detailed comparison, we can conclude that the IM serves as an accurate approximation of the MM in practical applications for large enough number of risky assets. And adding a broad index such as S&P 500 to an existing individual equities portfolio is improving its properties.

Portfolio optimization is an essential task for investors to maximize the profit and minimize the risk in trading, which can be reflected in return and standard deviation respectively. This article selects ten well-known firms and S&P 500 index as the portfolio and calculates their basic descriptive data of annualized return, annualized standard deviation, and correlations. Next, this article applies Markowitz model and Single-index model to the portfolio optimizations, as well as setting five constraints for comparison. After using the above methods, this paper gives the results in tabular and graphic ways. It finds that optimizing with no restriction can generate the biggest permissible regions of the minimum-variance frontier; investors only long stocks under constraint 4 can guarantee a finite risk and a finite positive return, which has a robust performance. Moreover, Markowitz model is flexible in covariances, but the number of estimates could complicate the calculating process; Index model is simple to calculate in linear regression form, but it’s not highly sensitive to the covariance matrix. The purpose of this article is to prove the feasibility of optimization in future portfolio management and helps investors to construct the portfolios with the fundamental theory of financial models.

This paper examines two widely used portfolio optimization models: the Markowitz Model (MM) and the Index Model (IM). We collected and processed historical data on ten stocks for 2001-2021 that included four different sectors, one P500 stock index, and a proxy for a risk-free rate (1-month Fed Funds rate). By computing all properly optimized inputs for the full Markowitz and index models, we find the regions allowing the portfolio under five additional realistic constraints. The study shows that the minimum variance combination and the maximum Sharp ratio are better than the Index Model. In addition, this paper further complements the empirical research of two models and provides some valuable investment suggestions for building the portfolio.

The choice of a portfolio optimization of shares has been for so long a subject of major interest in the field of investment and financial mathematics. This article aims to create a portfolio optimization which follows the objective of maximized return and minimized risk under given constraint among S&P 500 and the ten stocks, utilizing data collected from Yahoo! Data. The article ought to elucidate the Markowitz Model and Single Index Model in terms of generating a better result via comparing their distinct mechanisms in calculating their return percentage, standard deviation and Sharpe ratio, alongside with the corresponding efficiency frontier to compare their sharp ratio to determine a better model for each scenario. The result illustrates a better performance of the Markowitz Model in the aspect of producing a better portfolio optimization contrasting the Index Model in all the constraint except constraint three. Eventually backtesting was utilized to verify the feasibility.

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

Tren investasi di pasar saham syariah Indonesia mengalami peningkatan dari tahun 2015 sampai dengan 2017. Peningkatan ini juga harus diimbangi dengan sikap investor yang mampu memilih kombinasi saham terbaik dengan tingkat pengembalian dan risiko yang terukur melalui portofolio optimal. Tujuan penelitian ini adalah untuk menghitung portofolio optimal saham syariah Jakarta Islamic Index menggunakan Model Markowitz dengan preferensi risiko terendah, Model Markowitz dengan sharpe ratio optimal, dan Single Index Model. Analisis kemudian dilakukan untuk membandingkan tingkat pengembalian yang diharapkan dan tingkat risiko portofolio yang terbentuk dari model tersebut, dan memberikan rekomendasi portofolio optimal kepada investor yang dapat dijadikan sebagai bahan pertimbangan pengambilan keputusan dalam berinvestasi pada saham syariah Jakarta Islamic Index. Penelitian ini menggunakan data harga saham harian, data dividen, IHSG, dan data sukuk. Pemilihan 18 objek saham diperoleh dengan menggunakan metode purposive sampling. Berdasarkan hasil perhitungan, portofolio optimal diperoleh menggunakan model Markowitz dengan sharpe ratio optimal yang menghasilkan kombinasi saham terbaik dengan tingkat pengembalian yang diharapkan dan tingkat risiko sebesar 33,74 persen dan 22 persen dalam setahun.

Portfolio management plays a significant role in the world of finance and investing, offering benefits and contributions to both individual investors and institutions. Researchers use many mathematical models to make informed decisions about how to allocate and manage investments within a portfolio to optimize risk versus return trade-off based on the investors preference and constraints. This paper primarily employs the literature review methodology and comparative analysis methodology. Firstly, it collects, summarizes, and analyzes multiple papers on portfolio management models, including Markowiz Mean-Variance Model, Capital Market Line (CML), Arbitrage Pricing Theory (APT) and Capital Asset Pricing Model (CAPM) with the origin of the models, their key assumptions, components, and applications. Additionally, this paper also utilizes comparative analysis by comparing the similarities, differences, and respective downsides and benefits of the four main research models. Through this comparison, the paper investigates the relationships and application distinctions among the models. As a result, these models evolve and refine over time, with some building on others. CAPM extends the Markowitz Model by introducing the risk-free rate and market portfolio as benchmarks, simplifying the risk-return relationship, and introducing systematic risk. The Capital Market Line (CML), derived from CAPM, illustrates efficient portfolios made up of both market and risk-free assets and shows the risk-return trade-off. APT, a later model, can be seen as an extension of CAPM.

This study examines the application of the Markowitz model, an innovative portfolio optimization technique developed in the 1950s, in the context of the Russian stock market. The main objectives of this study are to study the theoretical foundations of the Markowitz model, analyze the current state and prospects of the Russian stock market and, ultimately, use the model to build an optimal portfolio structure. By integrating the Markowitz model with the dynamics of the Russian stock market, this study aims to contribute to the understanding of portfolio management and offer practical recommendations to portfolio managers and investors. The conclusions and recommendations obtained from the study will provide valuable information about the effectiveness and suitability of the Markowitz model in the Russian financial landscape. The study used Excel and Google Colab for comprehensive data analysis and portfolio optimization, providing a robust and data-driven approach.

PurposeThis study aims to investigate the association between Muslim individuals’ portfolio allocation choice and Islamic religiosity (levels and dimensions), controlling for risk tolerance and sociodemographic factors.Design/methodology/approachThe study uses primary data collected via survey questionnaires from a sample of 751 Muslim working individuals in Kuala Lumpur, Malaysia. Owing to the ordinal nature of the dependent variable, which reflects the levels of proportions of risky assets in portfolios, the data were analyzed using an ordered probit regression model.FindingsThe findings reveal that Islamic religiosity levels in general were insignificantly related to portfolio allocation, but that two dimensions of religiosity (virtue and obligation) significantly impact the allocations of risky assets in the portfolio. The higher the level of virtue, the lower the propensity to allocate risky assets into the portfolio. On the contrary, the higher the level of obligation, the higher the propensity to allocate risky assets in the portfolio. Meanwhile, individuals with higher risk tolerance, income and education levels show greater propensity to allocate risky assets in the portfolio.Research limitations/implicationsThe sample is restricted to Muslims in Kuala Lumpur; hence, the findings are not easily generalized to Muslim investors in general. Findings may differ between Muslims across the world, so future research needs to expand from a country specific to an international analysis. In addition, future studies could include other determinants of portfolio allocation, such as financial literacy.Practical implicationsThe findings of this study may assist financial planners and policymakers to better understand the drivers of portfolio allocation among their Muslim clients.Originality/valueWhile other studies have tended to focus on the impact of religiosity on the holdings of specific financial assets, such as Islamic bank accounts or Takaful, the present study explores the effect of Islamic religiosity dimensions on the allocations of risky assets in the portfolio. The study also develops an ordinal measure of portfolio allocation and makes a methodological contribution by using an ordered probit regression analysis.

ESG (Environmental, Social, and Governance) investments are becoming popular because they may provide competitive financial returns while promoting sustainable development. However, the uptake of ESG investments is limited due to barriers such as a lack of understanding and insufficient information. Indonesia has ESG-oriented indexes, such as SRI-KEHATI, ESG Leaders, ESG Sector Leaders IDX KEHATI, and ESG Quality 45 IDX KEHATI serve as a basis for ESG stock selection. This research uses the Markowitz model to optimize an ESG portfolio from four Indonesian green indices. The Markowitz model is employed to determine the optimal weights for establishing a portfolio on the efficient frontier. Using monthly adjusted closing prices from 2019 to 2024, the optimal portfolio suggested by four green indexes offers a balance of risk/return ratio. The optimal portfolio's performance comparison, which is measured by using the Treynor measure, Sharpe ratio, and Jensen's alpha is done to indicate which portfolio provides better risk-adjusted returns. The findings may provide optimal ESG portfolio choices to investor depending on their financial goals and risk tolerance. This research may also inspire other investors who have not yet invested in ESG companies to consider green investing, as ESG stocks provide investors with many benefits.

The application of the Markowitz model is analyzed from five financial assets, a matrix of logarithmic returns was constructed to estimate the parameters. Through computational resolution, the optimal portfolio was obtained that complies with an established minimum return, simultaneously minimizing the level of risk. The results were visualized across the efficient frontier, identifying the optimal combinations between risk and return. It is concluded that the Markowitz model continues to be a valid and valuable tool for the rational management of investments, especially when it is complemented with computational resources that allow it to be adapted to various financial contexts.