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An investor is expected to analyze the market risk while investing in equity stocks. This is because the investor has to choose a portfolio which maximizes the return with a minimum risk. The mean-variance approach by Markowitz (1952) is a dominant method of portfolio optimization, which uses variance as a risk measure. The purpose of this paper is to replace this risk measure with modified expected shortfall, defined by Jadhav et al. (2013).

Modified expected shortfall introduced by Jadhav et al. (2013) is found to be a coherent risk measure under univariate and multivariate elliptical distributions. This paper presents an approach of portfolio optimization based on mean-modified expected shortfall for the elliptical family of distributions.

It is proved that the modified expected shortfall of a portfolio can be represented in the form of expected return and standard deviation of the portfolio return and modified expected shortfall of standard elliptical distribution. The authors also establish that the optimum portfolio through mean-modified expected shortfall approach exists and is located within the efficient frontier of the mean-variance portfolio. The results have been empirically illustrated using returns from stocks listed in National Stock Exchange of India, Shanghai Stock Exchange of China, London Stock Exchange of the UK and New York Stock Exchange of the USA for the period February 2005-June 2018. The results are found to be consistent across all the four stock markets.

The mean-modified expected shortfall portfolio approach presented in this paper is new and is a natural extension of the Markowitz’s mean-variance and mean-expected shortfall portfolio optimization discussed by Deng et al. (2009).

This paper aims to propose an improved version of portfolio optimization model through the prediction of the future behavior of stock returns using a combined wavelet-based long short-term memory (LSTM).

First, data are gathered and divided into two parts, namely, “past data” and “real data.” In the second stage, the wavelet transform is proposed to decompose the stock closing price time series into a set of coefficients. The derived coefficients are taken as an input to the LSTM model to predict the stock closing price time series and the “future data” is created. In the third stage, the mean-variance portfolio optimization problem (MVPOP) has iteratively been run using the “past,” “future” and “real” data sets. The epsilon-constraint method is adapted to generate the Pareto front for all three runes of MVPOP.

The real daily stock closing price time series of six stocks from the FTSE 100 between January 1, 2000, and December 30, 2020, is used to check the applicability and efficacy of the proposed approach. The comparisons of “future,” “past” and “real” Pareto fronts showed that the “future” Pareto front is closer to the “real” Pareto front. This demonstrates the efficacy and applicability of proposed approach.

Most of the classic Markowitz-based portfolio optimization models used past information to estimate the associated parameters of the stocks. This study revealed that the prediction of the future behavior of stock returns using a combined wavelet-based LSTM improved the performance of the portfolio.

Optimum utilization of investments has always been considered one of the most crucial aspects of capital markets. Investment into various securities is the subject of portfolio optimization intent to maximize return at minimum risk. In this series, a population-based evolutionary approach, stochastic fractal search (SFS), is derived from the natural growth phenomenon. This study aims to develop portfolio selection model using SFS approach to construct an efficient portfolio by optimizing the Sharpe ratio with risk budgeting constraints.

This paper proposes a constrained portfolio optimization model using the SFS approach with risk-budgeting constraints. SFS is an evolutionary method inspired by the natural growth process which has been modeled using the fractal theory. Experimental analysis has been conducted to determine the effectiveness of the proposed model by making comparisons with state-of-the-art from domain such as genetic algorithm, particle swarm optimization, simulated annealing and differential evolution. The real datasets of the Indian stock exchanges and datasets of global stock exchanges such as Nikkei 225, DAX 100, FTSE 100, Hang Seng31 and S&P 100 have been taken in the study.

The study confirms the better performance of the SFS model among its peers. Also, statistical analysis has been done using SPSS 20 to confirm the hypothesis developed in the experimental analysis.

In the recent past, researchers have already proposed a significant number of models to solve portfolio selection problems using the meta-heuristic approach. However, this is the first attempt to apply the SFS optimization approach to the problem.

This paper aims to analyze the impact of transaction costs in portfolio optimization in Peru. The study aims to compare the transaction costs structure applied in Peru with respect to the ones applied in the USA, and over a few dimensions.

The paper opted for an empirical study analyzing the cost of rebalancing portfolios over a set period and dimensions. Stocks have been carefully selected using Bloomberg terminals, and portfolio designed then rebalanced using VBA programming. Over a few dimensions as type and number of stocks, holding period and trading strategy, the behavior of these different transaction costs has been compared. The analysis has been done for four different portfolios.

The paper provides empirical insights about how a retail investor actively trading in Peru can pay up to 14 times more in transaction costs than trading the same portfolio in the USA. These comparatively high transaction costs prevent retail investors to trade in the Peruvian stock market while fueling illiquidity to this market.

The paper deals with a limited amount of Peruvian stocks. Researchers are encouraged to test the proposition further, including other dimensions.

The paper includes implications for any retail investor that wants to invest in Peruvian stocks, giving an insight about how expensive it is to actively rebalance a portfolio in Peru.

This paper fulfils an identified need to study how much it costs to actively invest on the stock market in Peru.

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.

The paper uses 101 years of Chilean and international financial assets returns to investigate mean-variance optimal portfolio allocations. The key conclusion is that the share of international unhedged investments is substantial even in minimum risk portfolios (20%), unless the period 1980–2002 is assumed to be drawn from a different distribution and previous history is disregarded. In addition to that, the paper finds that mean-variance optimal investors would have generated substantial demand for an asset replicating the return profile of an efficient pay-as-you-go pension scheme. Labour income and departures from log-normality of returns might, however, affect the latter conclusion.

The purpose of this paper is to describe some optimization exercises which have proved to be very useful for introducing students to Markowitz‐style mean‐varience optimization.

This paper describes two exercises that walk students through the process of gathering security price and dividend data, estimating the parameters of the joint distribution of asset returns, and then using a portfolio optimizer to construct mean‐variance efficient portfolios. It describes the basic methodology, and the more complex formulations of the portfolio optimization problem that are used in practice.

Portfolio selection is typically taught in finance courses as an abstract solution to a system of equations, and does little to connect the portfolio construction process to Exchange Traded Funds, stocks, bonds and other assets that are traded in markets. This study offers a practical approach to teaching portfolio optimization, that starts with gathering market data and shows how a quadratic optimization system is used to construct mean‐variance optimal portfolios.

The exercises in this case study prepare students to construct mean‐variance efficient portfolios for asset allocation with Exchange Traded Funds, and for building stock and bond portfolios, using market data and a portfolio optimizer.

In this study, we analyze the power of the individual return-to-volatility security performance heuristic (ri/stdi) to simplify the identification of securities to buy and, consequently, to form the optimal no short sales mean–variance portfolios. The heuristic ri/stdi is powerful enough to identify the long and shorts sets. This is due to the positive definiteness of the variance–covariance matrix – the key is to use the heuristic sequentially. At the investor level, the heuristic helps investors to decide what securities to consider first. At the portfolio level, the heuristic may help us find out whether it is a good idea to invest in equity to begin with. Our research may also help to integrate individual security analysis into portfolio optimization through improved security rankings.

This study aims to utilize the mean–variance optimization framework of Markowitz (1952) and the generalized reduced gradient (GRG) nonlinear algorithm to find the optimal portfolio that maximizes return while keeping risk at minimum.

This study applies the portfolio optimization concept of Markowitz (1952) and the GRG nonlinear algorithm to a portfolio consisting of the 30 leading stocks from the three different sectors in Amman Stock Exchange over the period from 2009 to 2013.

The selected portfolios achieve a monthly return of 5 per cent whilst keeping risk at minimum. However, if the short-selling constraint is relaxed, the monthly return will be 9 per cent. Moreover, the GRG nonlinear algorithm enables to construct a portfolio with a Sharpe ratio of 7.4.

The results of this study are vital to both academics and practitioners, specifically the Arab and Jordanian investors.

To the best of the author’s knowledge, this is the first study in Jordan and in the Arab world that constructs optimum portfolios based on the mean–variance optimization framework of Markowitz (1952) and the GRG nonlinear algorithm.

One of the agency conflicts between investors and managers in fund management is reflected by risk‐taking behaviors led by their different goals. The investors may stop their investments in risky assets before the end of the investment horizon to minimize risk, while the managers may do so to entrench their reputation so as to pursue better opportunities in the labor market. This study aims to consider a one principal‐one agent model to investigate this agency conflict.

The paper derives optimal asset allocation strategies for both parties by extending the traditional dynamic mean‐variance model and considering possibilities of optimal early stopping. Doing so illustrates the principal‐agent conflict regarding risk‐taking behaviors and managerial investment myopia in fund management.

This paper not only paves the way for further studies along this line, but also presents results useful for practitioners in the money management industry.

According to the theoretical analysis and numerical simulations, the paper shows that potential early stop can make the agency conflict worsen, and it proposes a way to mitigate this agency problem.

As one of the exploratory studies in investigating agency conflict regarding risk‐taking behaviors in the literature, this study makes multiple contributions to the literature on fund management, asset allocation, portfolio optimization, and risk management.