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The objective of this paper is to demonstrate how to use the Elton, Gruber, and Padberg [1978] model to construct optimal portfolios and to facilitate the use of this paradigm by providing an example of how the technique is used. The EGP model uses the risk‐adjusted, excess return for an asset to determine the optimal portfolio for a given risk‐free rate of return. This paper shows exactly how to calculate the optimal portfolio and provides a True Basic@ program to do so. The data used are constructed from Capital International Indexes taken from various issues of Barrons from March 1978 to December 1986.

Purpose: The present study aims to test the Quadratic Programming model for Optimal Portfolio selection empirically.

Need for the Study: All the investors who buy financial products are motivated to obtain higher profits or, in other words, to maximise their returns. However, the high returns are often accompanied by higher risks, and avoiding such risks has become the primary concern for all investors. There is a great need for such a model to maximise profits and minimise risk, which can help design an investment portfolio with minimum risk and maximum return. The Quadratic Programming model is one such model which can be applied for selected shares to build an optimised portfolio.

Methodology: This study optimises the stock samples using a two-level screening of correlation coefficient and coefficient of variation. The monthly closing prices of the NSE-listed Indian pharmaceutical stocks from December 2019 to January 2022 have been used as sample data. The Lagrange Multiplier method is used to apply the model to achieve the optimal portfolio solution. Based on the market reality, the transaction costs have also been considered. The Quadratic programming model is further optimised to achieve the optimal portfolio for the select stocks.

Findings: The traditional portfolio theory and the modified quadratic model gives similar and consistent results. In other words, the modified quadratic model asserts the accuracy of the conventional portfolio model. The portfolio constructed in the present study gives a return much higher than the return of the benchmark portfolio of Nifty Fifty, indicating the usefulness of applying the Quadratic Programming model.

Practical Implications: The construction of an optimal portfolio using the traditional or modified Quadratic model can help investors make rational investment decisions for better returns with lower risks.

The purpose of this paper is to evaluate two different asset selection methodologies and further examine these by forming optimal portfolios.

This paper deals with the problem of portfolio formation, broadly in two steps: asset selection and asset allocation by using the two different approaches for the first step and then well‐known mean variance portfolio optimization. In addition, the resulting portfolios are compared using Sharpe ratio.

The empirical observations prove the applicability of the methodology adopted in the research design, ordered weighted averaging (OWA)‐heuristic algorithm gives us a better portfolio from the sample observations. Also the asset selection procedures adopted in the research proves to be of help when an investor has to narrow down the number of assets to invest in.

The analysis provides two different methodologies for portfolio formation – though the asset allocation is based on the mean variance portfolio optimization, the asset selection methods adopted provide a systematic approach to select the efficient securities.

This paper shows that OWA can be used to decide the order of inputs for the heuristic algorithm. Also an attempt is made to use data envelopment analysis to find a solution to the problem of portfolio formation.

The purpose of this study is to provide a novel portfolio asset prediction by means of the modified deep learning and hybrid meta-heuristic concept. In the past few years, portfolio optimization has appeared as a demanding and fascinating multi-objective problem, in the area of computational finance. Yet, it is accepting the growing attention of fund management companies, researchers and individual investors. The primary issues in portfolio selection are the choice of a subset of assets and its related optimal weights of every chosen asset. The composition of every asset is chosen in a manner such that the total profit or return of the portfolio is improved thereby reducing the risk at the same time.

This paper provides a novel portfolio asset prediction using the modified deep learning concept. For implementing this framework, a set of data involving the portfolio details of different companies for certain duration is selected. The proposed model involves two main phases. One is to predict the future state or profit of every company, and the other is to select the company which is giving maximum profit in the future. In the first phase, a deep learning model called recurrent neural network (RNN) is used for predicting the future condition of the entire companies taken in the data set and thus creates the data library. Once the forecasting of the data is done, the selection of companies for the portfolio is done using a hybrid optimization algorithm by integrating Jaya algorithm (JA) and spotted hyena optimization (SHO) termed as Jaya-based spotted hyena optimization (J-SHO). This optimization model tries to get the optimal solution including which company has to be selected, and optimized RNN helps to predict the future return while using those companies. The main objective model of the J-SHO-based RNN is to maximize the prediction accuracy and J-SHO-based portfolio asset selection is to maximize the profit. Extensive experiments on the benchmark datasets from real-world stock markets with diverse assets in various time periods shows that the developed model outperforms other state-of-the-art strategies proving its efficiency in portfolio optimization.

From the analysis, the profit analysis of proposed J-SHO for predicting after 7 days in next month was 46.15% better than particle swarm optimization (PSO), 18.75% better than grey wolf optimization (GWO), 35.71% better than whale optimization algorithm (WOA), 5.56% superior to JA and 35.71% superior to SHO. Therefore, it can be certified that the proposed J-SHO was effective in providing intelligent portfolio asset selection and prediction when compared with the conventional methods.

This paper presents a technique for providing a novel portfolio asset prediction using J-SHO algorithm. This is the first work uses J-SHO-based optimization for providing a novel portfolio asset prediction using the modified deep learning concept.

This paper aims to explore the impact of Sharīʿah-compliant stocks on other investor risk preferences beyond the risk aversion, namely, prudence and temperance.

This paper uses the non-parametric model data envelopment analysis with the shortage function as a measure of performance. The model uses three specifications considering skewness and kurtosis that describe according to expected utility theory, prudence and temperance.

Results show that first, efficient portfolios consist mainly of conventional stocks in the three-model specification. Second, the skewness improvement is achieved only when considering conventional stocks while Sharīʿah-compliant assets do not exhibit any impact on the third moment. Finally, diversification through both conventional and Sharīʿah-compliant stocks does not lead to kurtosis reduction. Sharīʿah-compliant stocks in efficient portfolios are sensitive to return and risk solely, and hence, prudence and temperance as related to skewness and kurtosis measures can be ignored in optimal portfolio selection during normal market conditions.

Findings suggest the same conclusions for four Islamic screening methods; however, readers should be prudent due to the limited sample. Results show that Sharīʿah-compliant assets do not have an impact on higher-order moments optimal portfolio returns, and hence, question the commonly admitted assumption of non-normality return distributions at least for Sharīʿah-compliant stocks.

The research findings suggest that Islamic investor preferences are described only by return and risk along with Sharīʿah criteria for stock selection and portfolio allocation. Portfolio managers should not care about higher-order moments to manage Sharīʿah-compliant funds. The traditional mean-variance Markowitz framework will be sufficient for investment or allocation decision-making. Description of Sharīʿah-compliant portfolio returns with only the first two order moments gives such asset more resiliency to extreme events like a crisis.

This research is the first in literature exploring whether prudence and temperance defined by higher-order moments can be drivers, besides Sharīʿah criteria, in portfolio allocation decision-making. This study is unique in terms of methodology and application. It uses individual stock data on the Casablanca Stock Exchange.

This chapter presents the survey of selected linear and mixed integer programming multi-objective portfolio optimization. The definitions of selected percentile risk measures are presented. Some contrasts and similarities of the different types of portfolio formulations are drawn out. The survey of multi-criteria methods devoted to portfolio optimization such as weighting approach, lexicographic approach, and reference point method is also presented. This survey presents the nature of the multi-objective portfolio problems focuses on a compromise between the construction of objectives, constraints, and decision variables in a portfolio and the problem complexity of the implemented mathematical models. There is always a trade-off between computational time and the size of an input data, as well as the type of mathematical programming formulation with linear and/or mixed integer variables.

This study aims to develop a model for project portfolio selection considering organizational goals such as budgets, sustainability cash flow and reinvestment strategy under an uncertain environment.

A multi-objective mathematical programming model is proposed for project selection, which takes the social, environmental and financial aspects into account as the objectives of the project portfolio selection problem. The project evaluation and selection process in one of the large capitals in the Middle East with numerous urban construction projects was considered as a real case study, in which the subjects of environmental and social sustainability are of great importance. Then, the most significant criteria for project evaluation and selection based on sustainability were identified and ranked using the fuzzy best-worst method (BWM).

The criterion of “defining clear and real objectives” was ranked first, “project investment return period” was ranked second, “minimum changes in the predicted range” was ranked third, and the other ten sustainability indicators were ranked as well. Next, the presented mathematical programming model was solved using the augmented e-constraint method. The sensitivity analysis indicated that increasing the amount of investments in projects would increase their net present value. Also, increased investment had no effect on sustainability, while decreased investment caused sustainability to not being optimal.

This study focuses on the impact of the amount of investments on projects, and the associated costs of sustainable projects. Further to the authors' knowledge, there has been no relevant study taking uncertainty into account. Also, very few studies proposed a mathematical programming model for the project portfolio selection problem. Moreover, this research uses the brainstorming and Delphi method to identify the sustainability indicators influencing the organization and screens the evaluation indicators. Furthermore, the weights of the evaluation indicators are determined using the fuzzy BWM based on the consistency of opinions.

Under the classical mean-variance framework, the purpose of this paper is to investigate the properties of the instability of minimal variance portfolio and then propose a novel penalized expected risk criterion (PERC) for optimal portfolio selection.

The proposed method considers not only a portfolio’s expected risk, but also its instability that is quantified by the variance of the estimated portfolio weights. This study tests the out-of-sample performance of various portfolio selection methods on both China and US stock markets.

It is very useful to control portfolio stability in real application of portfolio selection. The empirical results on both US and China stock markets show that PERC portfolio effectively controls turnover and consequently the transaction cost, and that is why it is so competing compared with other alternative methods.

The findings suggest that the rebalancing turnover and the associated transaction cost that is usually ignored in theoretical analysis play a very important role in real investment.

For investors, especially institutional investors, the rebalancing turnover and corresponding transaction cost must be carefully addressed. The variance of the estimated portfolio weights is a good candidate to quantify portfolio instability.

This study addresses the important role of portfolio instability and proposes a novel expected risk criterion for portfolio selection after the quantitative definition of portfolio instability.

This paper aims to examine from commodity portfolio managers’ perspective the performance of liquidity adjusted risk modeling in assessing the market risk parameters of a large commodity portfolio and in obtaining efficient and coherent portfolios under different market circumstances.

The implemented market risk modeling algorithm and investment portfolio analytics using reinforcement machine learning techniques can simultaneously handle risk-return characteristics of commodity investments under regular and crisis market settings besides considering the particular effects of the time-varying liquidity constraints of the multiple-asset commodity portfolios.

In particular, the paper implements a robust machine learning method to commodity optimal portfolio selection and within a liquidity-adjusted value-at-risk (LVaR) framework. In addition, the paper explains how the adapted LVaR modeling algorithms can be used by a commodity trading unit in a dynamic asset allocation framework for estimating risk exposure, assessing risk reduction alternates and creating efficient and coherent market portfolios.

The optimization parameters subject to meaningful operational and financial constraints, investment portfolio analytics and empirical results can have important practical uses and applications for commodity portfolio managers particularly in the wake of the 2007–2009 global financial crisis. In addition, the recommended reinforcement machine learning optimization algorithms can aid in solving some real-world dilemmas under stressed and adverse market conditions (e.g. illiquidity, switching in correlations factors signs, nonlinear and non-normal distribution of assets’ returns) and can have key applications in machine learning, expert systems, smart financial functions, internet of things (IoT) and financial technology (FinTech) in big data ecosystems.

Should we include cryptocurrency in risky portfolio investing? Bitcoin, given its status as the leader of cryptocurrencies and a speculative asset due to its non-dividend-paying trait and high volatility as well as high returns, poses an interesting question whether it can also be beneficial in a portfolio of risky assets. In order to find an answer, we revisit the conventional dual objective of minimizing risk and maximizing expected return for risky assets. Various models are tested to analyze the risk-return trade-off of risky portfolios including Bitcoin. Given an initial budget for a finite portfolio, the cumulative filtration yields the expected return and the covariance matrix. With the addition of Bitcoin, we compare the performance of the portfolio generated from the optimization models and technical analysis. The main implications are follows: (1) risk tolerance and diversification constraints are the key factors in portfolio optimization; (2) including cryptocurrency enhances portfolio returns; and (3) the Markowitz model (Kataoka’s and conditional value-at-risk models) recommends to fully weigh (unload) Bitcoin in (from) the portfolio.