Portfolio selection models based on interval numbers

A portfolio selection model with return as triangular intuitionistic fuzzy number is developed in this study to assess and select portfolios on China Stock Exchange. Although the portfolio selection has been widely investigated, and most studies have regarded return and risk as the main decisive criteria, there are many uncertainties in the financial market, such as social, political and human psychological factors, which makes it difficult for us to describe the risks in line with empirical evidence. To fill this gap, first, a literature review was conducted to clarify the current situation and shortcomings of portfolio selection. Second, the triangular intuitionistic fuzzy number was used to fuzzify the coefficients of the objective function and the constraints in the portfolio model. Third, the triangular intuitionistic fuzzy number model was transformed into a linear programming model by using the selected exact ranking function, and the model was solved by MATLAB. Finally, the historical monthly returns of 10 stocks in China’s Stock Exchange from December 2017 to November 2020, lasting 36 months, were selected to demonstrate the model. The results indicate that the portfolio selection model with triangular intuitionistic fuzzy number can better meet the uncertainty of the real securities market, and more reasonable investment decision guidance can be provided for investors. Besides, the limitations of this study are pointed out, and implications and directions for future research are discussed.

Purpose This paper aims to propose two portfolio selection models with hesitant value-at-risk (HVaR) – HVaR fuzzy portfolio selection model (HVaR-FPSM) and HVaR-score fuzzy portfolio selection model (HVaR-S-FPSM) – to help investors solve the problem that how bad a portfolio can be under probabilistic hesitant fuzzy environment. Design/methodology/approach It is strictly proved that the higher the probability threshold, the higher the HVaR in HVaR-S-FPSM. Numerical examples and a case study are used to illustrate the steps of building the proposed models and the importance of the HVaR and score constraint. In case study, the authors conduct a sensitivity analysis and compare the proposed models with decision-making models and hesitant fuzzy portfolio models. Findings The score constraint can make sure that the portfolio selected is profitable, but will not cause the HVaR to decrease dramatically. The investment proportions of stocks are mainly affected by their HVaRs, which is consistent with the fact that the stock having good performance is usually desirable in portfolio selection. The HVaR-S-FPSM can find portfolios with higher HVaR than each single stock and has little sacrifice of extreme returns. Originality/value This paper fulfills a need to construct portfolio selection models with HVaR under probabilistic hesitant fuzzy environment. As a downside risk, the HVaR is more consistent with investors’ intuitions about risks. Moreover, the score constraint makes sure that undesirable portfolios will not be selected.

A model for portfolio selection with order of expected returns

A new expected utility (EU) portfolio selection model is proposed under the assumption that the trading volume has the box constraints. In the model, the expected utility function is quadratic. The model is solved by a pivoting algorithm which needn’t be added any slack, surplus and artificial variable, and is easy to operate and works efficiently. A numerical example of a portfolio selection problem is given to compare the new model and the EU portfolio selection model not considering upper bounds. The comparison shows that when the risk preference coefficient is greater than a critical value, the risk and expected return don’t increase as the coefficient increases; The relationship between the risk preference coefficient and the expected return (or risk) is nonlinear while that is linear in the case when short sales are allowed; the efficient portfolio selection considering the box constraints is subset of that not considering the constraints.KeywordsPortfolio selectionUtilityPivoting algorithmThe box constraints

In this paper, the optimal portfolio selection problem with transaction costs is studied. In the previous study, the transaction cost is generally assumed as a V-shaped function of difference between the existing and the new portfolio. But, in this study, a portfolio selection model with quadratic subsection concave transaction costs is presented. Due to proposed model is a complex quadratic programming problem which can't be solved by exact algorithms efficiently, an improved particle swarm optimization (IPSO) is designed to solve it. Finally, a numerical example is given to illustrate our proposed effective approach and the performances of IPSO and standard genetic algorithm (SGA) are compared. Experiment results show that IPSO is clearly superior compared to a SGA.

Criteria, Models and Strategies in Portfolio Selection.- A Model for Portfolio Selection with Order of Expected Returns.- A Compromise Solution to Mutual Funds Portfolio Selection with Transaction Costs.- Optimal Portfolio Selection of Assets with Transaction Costs and No Short Sales.- Portfolio Frontier with Different Interest Rates for Borrowing and Lending.- Multi-period Investment.- Mean-Variance-Skewness Model for Portfolio Selection with Transaction Costs.- Capital Asset Pricing: Theory and Methodologies.- Empirical Tests of CAPM for China's Stock Markets.

Mean-risk models have been widely used in solving portfolio selection problems in the last years. Since the mean-variance theory of Markowitz, an enormous amount of papers have been published extending or modifying the basic model in several directions like the simplification of the type and amount of input data, the introduction of alternative measures of risk, the incorporation of additional criteria or constraints. Recently, risk measures concerned with the left tails of distributions that evaluate the extremely unfavourable outcomes are used. The most used risk measure for such purposes is Value-at-Risk (VaR). In this paper we concentrate on the second direction of incorporating of a new risk measure in portfolio modeling. We define a new risk measure, which take into consideration the values exceeding a certain threshold in the extreme tail of the loss distribution, called Limited Value-at-Risk (LVaR). We study the properties of this risk measure. We build a new model for portfolio selection, named mean-LVaR model, in which risk is evaluated using LVaR risk measure. We study the properties of the new mean-risk model and compare it with the classical mean-VaR model. We derive the analytical form of LVaR risk measure in the case of normal distribution. We provide computational results and analyze the implications of using the mean-LVaR risk model in portfolio optimization problem.

In 1952 Markowitz [247] published his pioneering work which laid the foundation of modern portfolio analysis. The mean-variance methodology for the portfolio selection problem has played an important role in the development of modern portfolio selection theory. It combines probability with optimization techniques to model the behavior investment under uncertainty. The basic principle of the mean-variance model is to use the expected return of a portfolio as the investment return and to use the variance of the expected returns of the portfolio as the investment risk. Most of existing portfolio selection models are based on probability theory. Though probability theory is one of the main techniques used for analyzing uncertainty in finance, the financial market is also affected by several non-probabilistic factors such as vagueness and ambiguity. In many important cases it might also be easier to estimate the possibility distributions of rates of return on securities rather than the corresponding probability distributions. Decision makers are commonly provided with information which is characterized by linguistic descriptions such as high risk, low profit, high interest rate, etc. [298] With the introduction of fuzzy set theory by Zadeh [351], it was realized that imperfect knowledge of the returns on the assets and the uncertainty involved in the behavior of financial markets may be captured by means of fuzzy quantities and/or fuzzy constraints. Since the early seventies fuzzy set theory has been widely used to solve many problems including financial risk management. By using fuzzy approaches, the experts’ knowledge and the investors’ subjective opinions can be better integrated into a portfolio selection model.

This paper discusses a portfolio selection problem under the mean-variance-skewness framework wherein the security returns are obtained through evaluation of the experts instead of historical data. By treating security returns as the uncertain variables, an uncertain mean-variance-skewness model is proposed for portfolio selection under consideration of the transaction costs, bounds on holdings, cardinality of the portfolio, and minimum transaction lots constraints. To solve the resultant portfolio selection problem, which is an NP-Complete nonlinear integer programming problem, a hybrid solution method termed the FA-GA is developed by combining features of the firefly algorithm (FA) and genetic algorithm (GA). In the proposed method, the crossover and mutation operators of the GA are integrated into the FA to strike an optimal balance between the exploration and exploitation. A numerical example of portfolio selection is given to demonstrate effectiveness of the proposed model and solution algorithm. Furthermore, a detailed performance analysis and comparison are done to establish superiority of the proposed model and solution method.

Modern Portfolio Theory (MPT) is based upon the classical Markowitz model which uses variance as a risk measure. A generalization of this approach leads to mean‐risk models, in which a return distribution is characterized by the expected value of return (desired to be large) and a “risk” value (desired to be kept small). Portfolio choice is made by solving an optimization problem, in which the portfolio risk is minimized and a desired level of expected return is specified as a constraint. The need to penalize different undesirable aspects of the return distribution led to the proposal of alternative risk measures, notably those penalizing only the downside part (adverse) and not the upside (potential). The downside risk considerations constitute the basis of the Post Modern Portfolio Theory (PMPT). Examples of such risk measures are lower partial moments, Value‐at‐Risk (VaR) and Conditional Value‐at‐Risk (CVaR). We revisit these risk measures and the resulting mean‐risk models. We discuss alternative models for portfolio selection, their choice criteria and the evolution of MPT to PMPT which incorporates: utility maximization and stochastic dominance. Copyright © 2009 Wilmott Magazine Ltd

Optimization problem of cardinality constrained mean-variance (CCMV) model for sparse portfolio selection is considered. To overcome the difficulties caused by cardinality constraint, an exact penalty approach is employed, then CCMV problem is transferred into a difference-of-convex-functions (DC) problem. By exploiting the DC structure of the gained problem and the superlinear convergence of semismooth Newton (ssN) method, an inexact proximal DC algorithm with sieving strategy based on a majorized ssN method (siPDCA-mssN) is proposed. For solving the inner problems of siPDCA-mssN from dual, the second-order information is wisely incorporated and an efficient mssN method is employed. The global convergence of the sequence generated by siPDCA-mssN is proved. To solve large-scale CCMV problem, a decomposed siPDCA-mssN (DsiPDCA-mssN) is introduced. To demonstrate the efficiency of proposed algorithms, siPDCA-mssN and DsiPDCA-mssN are compared with the penalty proximal alternating linearized minimization method and the CPLEX (12.9) solver by performing numerical experiments on realword market data and large-scale simulated data. The numerical results demonstrate that siPDCA-mssN and DsiPDCA-mssN outperform the other methods from computation time and optimal value. The out-of-sample experiments results display that the solutions of CCMV model are better than those of other portfolio selection models in terms of Sharp ratio and sparsity.

We first deduce the variance formulas of normal, triangular and trapezoidal fuzzy variables in credibility theory. Then two classes of fuzzy portfolio selection models are built based on credibility measure, the expected value and variance of a fuzzy variable. To solve the proposed models, a genetic algorithm is employed. Finally, two numerical examples are provided for the proposed portfolio selection models to test the designed algorithm.KeywordsGenetic AlgorithmReturn RatePortfolio SelectionFuzzy VariablePossibility DistributionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the complex financial market, there are situations where the security returns have to be evaluated by experienced experts due to the lack of historical data. In this paper, within the framework of uncertainty theory, we propose a multi-period bi-objective regret minimization model for portfolio selection, in which bankruptcy risk and liquidity risk are both considered. Furthermore, in order to solve the proposed multi-objective optimization problem, a novel hybrid algorithm named MFA-SCA is proposed by combining the advantages of the firefly algorithm (FA) and sine cosine algorithm (SCA). Finally, a numerical example is given to illustrate the effectiveness of the proposed approaches.

With the nor-normal distribution and nor-binomial utility function of investors, higher-order moments should be taken into consideration with portfolio selection. So far, studies on portfolio selection with higher-order moments are mainly conducted under the framework of mean-variance-skewness while kurtosis is seldom taken into account. But many literatures have pointed out kurtosis play more important role in portfolio selection than variance and skewness. So, in order to fill up the study gap, portfolio selection with higher-order moments will be studied, under the framework of mean-variance-skewness-kurtosis, in this paper. Firstly, the optimization model of portfolio selection with kurtosis minimization is set up. Then, semidefinite programming relaxation is suggested based on research results of Lasserre and Waki to resolve the difficulty of model solution from the higher order and non-convexity of objective function and the efficient frontier of portfolio selection model minimizing kurtosis is deduced theoretically. Finally, the theoretical efficient frontier and the effectiveness of semidefinite programming relaxation are verified through empirical analysis.

In daily life events, there are many complexities arising from lack of information and uncertainty. Fuzzy linear programming model has been developed to reduce or eliminate this complexity. Fuzzy linear programming is the process of choosing the optimum solution from among the decision alternatives to achieve a specific purpose in cases where the information is not certain. One of the fields where the lack of information or uncertainty makes it difficult to decide is financial markets. Investors who have a certain amount of accumulations are aiming to increase in various ways as well as protecting the value of their income. While doing this, encounter the problem of deciding to which investment vehicle they need to invest in what extent. Therefore, investors apply to fuzzy linear programming model to eliminate this uncertainty and to create the optimal portfolio. In the portfolio selection process suggestions in the literature, the determination of criteria weights is based on triangular fuzzy numbers. In this study, as an alternative to the Enea and Piazza's portfolio selection model, which uses the triangular fuzzy numbers for criteria weighting, a new model that uses the trapezoidal fuzzy numbers for the same aim was proposed. With the solution of the linear programming model which is based on the determined weights, an alternative solution has been produced to the problem of which investment instrument will be invested at what proportion. The results obtained from the existing methods and the results obtained from the proposed model were compared.