Portfolio analysis based on Markowitz model

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.

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.

Despite the fact any investor prefers lower risk and higher return, investors may have different preferences about what would be an acceptable risk or a minimal return. For instance, some investors prefer to have a lower bound risk rather than gaining a higher return. In portfolio theory, it is commonly assumed the existence of one risk free asset that offers a positive return. This theoretical risk free asset combined with a risky portfolio creates a new portfolio that presents a linear relation between risk and return as the risk free asset weight (w f ) changes. Hence, any level of risk or of return is easy to achieve separately, just by changing w f . However, in a world without any risk free assets, the combination between assets creates nonlinear portfolios. Achieving a specific level of risk or return is not a trivial task. In this paper, we assume a risky world rather than the existence of a risk free asset, in order to model an automated asset management system. Furthermore, some automated asset managers give very different results when evolving in different contexts: hence, a very profitable manager can have very bad results in other market situations. This paper presents a multiagent architecture, aiming to tackle these problems. The architecture, named COAST (COmpetitive Agent SocieTy), is based on competitive agents that act autonomously on behalf of an investor in financial asset management. It allows the simultaneous and competitive use of several asset analysis techniques currently applied in the finance field. Some dedicated agents, called advisors, apply a particular technique to a single asset. The results provided by these advisors are then submitted to and analyzed by a special agent called coach, who evaluates its advisors' performance and defines an expectation about the future price of one specific asset. Within COAST, several coaches negotiate to define the best money allocation among different assets, by using a negotiation protocol defined in this paper. We also propose an investor description model that is able to represent different investors' preferences with defined acceptable limits of risk and/or return. The COAST architecture was designed to operate adequately with any possible investor's preference. It was implemented using a financial market simulator called AgEx and tested using real data from the Nasdaq stock exchange. The test results show that the architecture performed well when compared to an adjusted market index.

This study evaluates the use of futures contracts for precious metals to hedge against stock market risks and their hedging effectiveness on the Indonesian Stock Exchange (IDX) and the Kuala Lumpur Stock Exchange (KLSE). This study found that gold was the most effective hedging instrument, since it produced the highest hedging effectiveness both on the IDX and the KLSE among the other precious metals. None of the hedged portfolios had a higher Sharpe’s ratio than the unhedged one on the IDX; however, all the hedged portfolios on the KLSE had a higher Sharpe’s ratio than the unhedged ones. Almost all the hedged portfolios could produce a higher Treynor’s ratio than the unhedged portfolios, both on the IDX and the KLSE. In general, this study concluded that studying some precious metals could reduce the investment risk, which was shown through the variance produced by the smaller portfolios, while gold can improve the risk-adjusted performance.

Modeling Tail Risk in Portfolio Selection: Selected Iranian Food Industry Companies

This research intends to understand the dynamic integration of the Indonesian capital market with the ASEAN Capital Market before and after the implementation of the MEA (ASEAN Economic Community) or AEC. The sample used in this observation is a series of data collected in the period January 2011 to December 2015 before the MEA was implemented and January 2016 to December 2022 after the MEA was implemented, while the capital markets studied were the Indonesian Stock Exchange, Kuala Lumpur Stock Exchange, Stock Exchange Singapore, Philippine Stock Exchange and Thai Stock Exchange. This research analyzes the dynamic integration of the Indonesian capital market towards ASEAN after the implementation of the MEA using the Orthogonal Generalized Autoregressive Conditional Heteroscedasticity (OGARCH) method which can manifest the level of integration in a measurable way. The results of this research are that there is co-movement among the ASEAN capital markets studied, however, not all ASEAN capital markets are fully integrated. The results of this observation also found that the Indonesian Stock Exchange, Kuala Lumpur Stock Exchange, Thailand Stock Exchange and Singapore Stock Exchange were integrated but the Philippine Stock Exchange was not. The Philippine Stock Exchange tends to be segmented rather than integrated.

The article focuses on the consideration of econometric models of stock quotes of large domestic companies based on modeling the securities portfolio and predicting its behavior using mathematical modeling using elements of probability theory and mathematical statistics. It is also shown how the problem of choosing the optimal portfolio can be reduced to the problem of convex quadratic programming. In this article, based on the Markowitz model, a model of an optimal investment portfolio with bilateral restrictions on variables associated with the requirements of the law is developed. An example is considered in which 10 types of stocks and bonds of large Kazakhstani companies are selected, which generates an optimal set of assets and calculates the risk of an optimal portfolio for a given level of expected return. Based on the results obtained, companies and entrepreneurs can build a strategy for investing and buying shares, knowing the probable income from a portfolio of certain types of securities.

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.

Asset allocation with distorted beliefs and transaction costs

The current financial market is affected by interest rate adjustments, policy changes, and other factors, resulting in frequent phase transitions. Chinese funded enterprise investment portfolios are facing problems such as income fluctuations and difficulty in risk management. Traditional optimization models are difficult to adapt to dynamic market environments. This article focuses on the phase transition characteristics of financial markets and the investment allocation needs of Chinese enterprises. It proposes an optimization scheme that couples TOPSIS and Markowitz models. Firstly, TOPSIS multi index screening is used to adapt assets, and risk parameters are adjusted based on phase transition prediction. Then, relying on the Markowitz model, weight optimization is completed to form a closed-loop system of "screening adjustment configuration". By sorting out the laws of market transformation, analyzing the limitations of a single model, clarifying the coupling logic and verifying the adaptability in combination with industry historical data, it provides an operational path for Chinese enterprises to achieve controllable portfolio risk and stable returns in volatile markets. Research has shown that coupled models can compensate for the shortcomings of traditional methods in considering multiple objectives and ignoring market fluctuations. They are in line with the investment preferences of Chinese enterprises for low risk and stable returns, and have strong practical application value.

In this chapter, a risk-free asset is added to the set of investable securities and the optimal portfolios are now derived in this augmented economy. One of the results obtained is that, in the risk/return analysis, the parameters (σ, m) of any investment portfolio lay either on or inside a certain cone \(\mathcal {C}(\sigma , m)\). The upper side of the cone represents the efficient investment portfolios and is called the Capital Market Line. We show that the portfolios on the Capital Market Line can be built as an allocation between the risk-free asset and a particular investment portfolio, made of risky assets only, called the Tangent Portfolio. The Tangent Portfolio appears to define the tangent point between the cone \(\mathcal {C}(\sigma , m)\) and the hyperbola \(\mathcal {F}(\sigma , m)\) delimiting the (σ, m) of all investment portfolios made of risky assets only. Based on some economic reasoning, the Tangent Portfolio is sometimes assimilated to the Market Portfolio, for which the investment in each asset is proportional to its relative market capitalisation. We also show in this chapter that the problem of optimal allocation can be segmented into two steps. First, the investor decides on the risk exposure he is ready to take, secondly, he calculates the allocation to the Tangent Portfolio which gives him this level of risk (the rest of the money being invested in the risk-free asset). This paradigm for investing is known as the Separation Theorem of James Tobin (Nobel Prize in Economics in 1981, see Tobin) and shows that whatever the risk appetite is, there is only one way to take risk exposure efficiently.

A note on portfolio selection with restrictions on leverage

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.

Since the dawn of the 21st century, while there has been an abundance of talented fund managers, truly exceptional asset portfolios have been rare. Addressing this phenomenon, the study delves into the current state of asset management, employing two renowned financial models: the Markowitz model and the Index model. These models are utilized to analyze the returns and risks of five high-performing assets. Additionally, acknowledging the diversity of investor preferences, this paper incorporates five different constraints to cater to a wider spectrum of client needs. The crux of the research is the optimization of the asset portfolio, grounded in these models and constraints. The paper findings indicate that the integration of the Markowitz and Index models with the capital market line and the efficient frontier proves highly effective in the management of risk asset portfolios. In this paper, the Markowitz model consistently excels over the index model in both minvariance and maxsharpe scenarios. This study has important reference value for investors in related fields.