A constrained portfolio selection model at considering risk-adjusted measure by using hybrid meta-heuristic algorithms

Portfolio selection is a key issue in the business world and financial fields. This article presents a new decision making method of portfolio optimization (PO) issues in different risk measures by using new evolutionary computing method and cardinality constrains which is mentioned as hybrid meta-heuristic algorithms. Based on mean–variance (MV) Method by Markowitz we collected three risk levels; mean absolute deviation (MAD), semi variance (SV) and variance with skewness (VWS). The developed algorithms are Electromagnetism-like algorithm (EM), particle swarm optimization (PSO), genetic algorithm (GA), genetic network programming (GNP) and simulated annealing (SA). Also a diversification mechanism strategy is implemented and hybridized with the developed algorithms to increase the diversity and overcome local optimality. The sustainability of this proposed model is verified by 50 factories on the Iranian stock exchange. Finally, experimental results of proposed algorithms with cardinality constraint are compared with each other by four effective metrics in which the algorithms performance for achieving the optimal solution discussed. In addition, we have done the analysis of variance technique to confirm the validity and accurately analyze of the results which the success of this method was proved.     

Combined forecasts in portfolio optimization: A generalized approach

In this paper a general mathematical model for portfolio selection problem is proposed. By considering a forecasting performance according to the distributional properties of residuals, we formulate an extended mean-variance-skewness model with 11 objective functions. Returns and return errors for each asset obtained using different forecasting techniques, are combined in optimal proportions so as to minimize the mean absolute forecast error. These proportions are then used in constructing six criteria related to the mean, variance and skewness of return forecasts of assets in the future and forecasting errors of returns of assets in the past. The obtained multi-objective model is scalarized by using the conic scalarization method which guarantees to find all non-dominated solutions by considering investor preferences in non-convex multi-objective problems. The obtained scalar problem is solved by utilizing F-MSG algorithm. The performance of the proposed approach is tested on a real case problem generated on the data derived from Istanbul Stock Exchange. The comparison is conducted with respect to different levels of investor preferences over return, variance, and skewness and obtained results are summarized.     

Fuzzy mean–variance–skewness portfolio selection models by interval analysis

In portfolio selection problem, the expected return, risk, liquidity etc. cannot be predicted precisely. The investor generally makes his portfolio decision according to his experience and his economic wisdom. So, deterministic portfolio selection is not a good choice for the investor. In most of the recent works on this problem, fuzzy set theory is widely used to model the problem in uncertain environments. This paper utilizes the concept of interval numbers in fuzzy set theory to extend the classical mean–variance (MV) portfolio selection model into mean–variance–skewness (MVS) model with consideration of transaction cost. In addition, some other criteria like short and long term returns, liquidity, dividends, number of assets in the portfolio and the maximum and minimum allowable capital invested in stocks of any selected company are considered. Three different models have been proposed by defining the future financial market optimistically, pessimistically and in the combined form to model the fuzzy MVS portfolio selection problem. In order to solve the models, fuzzy simulation (FS) and elitist genetic algorithm (EGA) are integrated to produce a more powerful and effective hybrid intelligence algorithm (HIA). Finally, our approaches are tested on a set of stock data from Bombay Stock Exchange (BSE).     

A memetic algorithm for cardinality-constrained portfolio optimization with transaction costs

A memetic approach that combines a genetic algorithm (GA) and quadratic programming is used to address the problem of optimal portfolio selection with cardinality constraints and piecewise linear transaction costs. The framework used is an extension of the standard Markowitz mean–variance model that incorporates realistic constraints, such as upper and lower bounds for investment in individual assets and/or groups of assets, and minimum trading restrictions. The inclusion of constraints that limit the number of assets in the final portfolio and piecewise linear transaction costs transforms the selection of optimal portfolios into a mixed-integer quadratic problem, which cannot be solved by standard optimization techniques. We propose to use a genetic algorithm in which the candidate portfolios are encoded using a set representation to handle the combinatorial aspect of the optimization problem. Besides specifying which assets are included in the portfolio, this representation includes attributes that encode the trading operation (sell/hold/buy) performed when the portfolio is rebalanced. The results of this hybrid method are benchmarked against a range of investment strategies (passive management, the equally weighted portfolio, the minimum variance portfolio, optimal portfolios without cardinality constraints, ignoring transaction costs or obtained with L₁ regularization) using publicly available data. The transaction costs and the cardinality constraints provide regularization mechanisms that generally improve the out-of-sample performance of the selected portfolios.     

Random fuzzy mean-absolute deviation models for portfolio optimization problem with hybrid uncertainty

Absolute deviation is a commonly used risk measure, which has attracted more attentions in portfolio optimization. The existing mean-absolute deviation models are devoted to either stochastic portfolio optimization or fuzzy one. However, practical investment decision problems often involve the mixture of randomness and fuzziness such as stochastic returns with fuzzy information. Thus it is necessary to model portfolio selection problem in such a hybrid uncertain environment. In this paper, we employ random fuzzy variable to describe the stochastic return on individual security with ambiguous information. We first define the absolute deviation of random fuzzy variable and then employ it as risk measure to formulate mean-absolute deviation portfolio optimization models. To find the optimal portfolio, we design random fuzzy simulation and simulation-based genetic algorithm to solve the proposed models. Finally, a numerical example for synthetic data is presented to illustrate the validity of the method.     

The mean–variance cardinality constrained portfolio optimization problem: An experimental evaluation of five multiobjective evolutionary algorithms

This paper compares the effectiveness of five state-of-the-art multiobjective evolutionary algorithms (MOEAs) together with a steady state evolutionary algorithm on the mean–variance cardinality constrained portfolio optimization problem (MVCCPO). The main computational challenges of the model are due to the presence of a nonlinear objective function and the discrete constraints. The MOEAs considered are the Niched Pareto genetic algorithm 2 (NPGA2), non-dominated sorting genetic algorithm II (NSGA-II), Pareto envelope-based selection algorithm (PESA), strength Pareto evolutionary algorithm 2 (SPEA2), and e-multiobjective evolutionary algorithm (e-MOEA). The computational comparison was performed using formal metrics proposed by the evolutionary multiobjective optimization community on publicly available data sets which contain up to 2196 assets.     

考虑投资者心理特征的模糊投资组合优化模型

投资者在实际金融市场中的决策行为往往会受到主观心理认知的影响.考虑参照依赖、敏感性递减和损失厌恶等影响投资决策的心理特征,研究模糊环境下的投资组合选择问题.首先,假设资产的收益为梯形模糊数,依据前景理论中的价值函数,将组合收益转化为体现投资者心理特征的感知价值;然后,以感知价值的可能性均值最大化和可能性下半方差最小化为目标,建立考虑心理特征的模糊投资组合优化模型;接着,为了有效地求解模型,设计一个多种群遗传算法;最后,通过实例分析表明模型和算法的有效性.结果表明,与传统的遗传算法相比,所设计的多种群遗传算法可更有效地求解模型,考虑心理特征的模糊投资组合优化模型能够提升投资者的满意程度,可为实际的投资活动提供决策支持.     

Constrained Portfolio Selection using Particle Swarm Optimization

This paper presents a novel heuristic method for solving an extended Markowitz mean–variance portfolio selection model. The extended model includes four sets of constraints: bounds on holdings, cardinality, minimum transaction lots and sector (or market/class) capitalization constraints. The first set of constraints guarantee that the amount invested (if any) in each asset is between its predetermined upper and lower bounds. The cardinality constraint ensures that the total number of assets selected in the portfolio is equal to a predefined number. The sector capitalization constraints reflect the investors’ tendency to invest in sectors with higher market capitalization value to reduce their risk of investment.The extended model is classified as a quadratic mixed-integer programming model necessitating the use of efficient heuristics to find the solution. In this paper, we propose a heuristic based on Particle Swarm Optimization (PSO) method. The proposed approach is compared with the Genetic Algorithm (GA). The computational results show that the proposed PSO effectively outperforms GA especially in large-scale problems.     

A class of multi-period semi-variance portfolio for petroleum exploration and development

Variance is substituted by semi-variance in Markowitz's portfolio selection model. For dynamic valuation on exploration and development projects, one period portfolio selection is extended to multi-period. In this article, a class of multi-period semi-variance exploration and development portfolio model is formulated originally. Besides, a hybrid genetic algorithm, which makes use of the position displacement strategy of the particle swarm optimiser as a mutation operation, is applied to solve the multi-period semi-variance model. For this class of portfolio model, numerical results show that the mode is effective and feasible.     

An optimization model of the portfolio adjusting problem with fuzzy return and a SMO algorithm

Based on possibilistic mean and variance theory, this paper deals with the portfolio adjusting problem for an existing portfolio under the assumption that the returns of risky assets are fuzzy numbers and there exist transaction costs in portfolio adjusting precess. We propose a portfolio optimization model with V-shaped transaction cost which is associated with a shift from the current portfolio to an adjusted one. A sequential minimal optimization (SMO) algorithm is developed for calculating the optimal portfolio adjusting strategy. The algorithm is based on deriving the shortened optimality conditions for the formulation and solving 2-asset sub-problems. Numerical experiments are given to illustrate the application of the proposed model and the efficiency of algorithm. The results also show clearly the influence of the transaction costs in portfolio selection.     

Evaluating methods of investment project and optimizing models of portfolio selection in fuzzy uncertainty

This paper deals with the problems of both project valuation and portfolio selection under the assumption that the investment capitals and the net cash flows of the projects are fuzzy variables. Using the credibilistic expected value and the credibilistic lower semivariance of fuzzy variables, this paper proposes both the credibilistic return index and the credibilistic risk index, which are measures of investment return and investment risk with annuity form for evaluating single project. Moreover, a composite risk-return index for selecting the optimal investment strategy is also presented. Then, we set up a general project portfolio optimization model with fuzzy returns and two specific models: triangle and interval fuzzy returns. Furthermore, we provide two algorithms: the improved heuristic rules based on genetic algorithm and the traversal algorithm. Finally, two numerical examples are presented to illustrate the efficiency and the effectiveness of these proposed optimization methods.     

Neural network-based mean–variance–skewness model for portfolio selection

In this study, a novel neural network-based mean–variance–skewness model for optimal portfolio selection is proposed integrating different forecasts and trading strategies, as well as investors’ risk preference. Based on the Lagrange multiplier theory in optimization and the radial basis function (RBF) neural network, the model seeks to provide solutions satisfying the trade-off conditions of mean–variance–skewness. The feasibility of the RBF network-based mean–variance–skewness model is verified with a simulation experiment. The experimental results show that, for all examined investor risk preferences and investment assets, the proposed model is a fast and efficient way of solving the trade-off in the mean–variance–skewness portfolio problem. In addition, we also find that the proposed approach can also be used as an alternative tool for evaluating various forecasting models.     

A new Chance-Variance optimization criterion for portfolio selection in uncertain decision systems

The Markowitz’s mean-variance (M-V) model has received widespread acceptance as a practical tool for portfolio optimization, and his seminal work has been widely extended in the literature. The aim of this article is to extend the M-V method in hybrid decision systems. We suggest a new Chance-Variance (C-V) criterion to model the returns characterized by fuzzy random variables. For this purpose, we develop two types of C-V models for portfolio selection problems in hybrid uncertain decision systems. Type I C-V model is to minimize the variance of total expected return rate subject to chance constraint; while type II C-V model is to maximize the chance of achieving a prescribed return level subject to variance constraint. Hence the two types of C-V models reflect investors’ different attitudes toward risk. The issues about the computation of variance and chance distribution are considered. For general fuzzy random returns, we suggest an approximation method of computing variance and chance distribution so that C-V models can be turned into their approximating models. When the returns are characterized by trapezoidal fuzzy random variables, we employ the variance and chance distribution formulas to turn C-V models into their equivalent stochastic programming problems. Since the equivalent stochastic programming problems include a number of probability distribution functions in their objective and constraint functions, conventional solution methods cannot be used to solve them directly. In this paper, we design a heuristic algorithm to solve them. The developed algorithm combines Monte Carlo (MC) method and particle swarm optimization (PSO) algorithm, in which MC method is used to compute probability distribution functions, and PSO algorithm is used to solve stochastic programming problems. Finally, we present one portfolio selection problem to demonstrate the developed modeling ideas and the effectiveness of the designed algorithm. We also compare the proposed C-V method with M-V one for our portfolio selection problem via numerical experiments.     

A robust mean absolute deviation model for portfolio optimization

In this paper we develop a robust model for portfolio optimization. The purpose is to consider parameter uncertainty by controlling the impact of estimation errors on the portfolio strategy performance. We construct a simple robust mean absolute deviation (RMAD) model which leads to a linear program and reduces computational complexity of existing robust portfolio optimization methods. This paper tests the robust strategies on real market data and discusses performance of the robust optimization model empirically based on financial elasticity, standard deviation, and market condition such as growth, steady state, and decline in trend. Our study shows that the proposed robust optimization generally outperforms a nominal mean absolute deviation model. We also suggest precautions against use of robust optimization under certain circumstances.     

Bacterial Foraging Optimization Approach to Portfolio Optimization

In this paper we propose a heuristic approach based on bacterial foraging optimization (BFO) in order to find the efficient frontier associated with the portfolio optimization (PO) problem. The PO model with cardinality and bounding constraints is a mixed quadratic and integer programming problem for which no exact algorithms can solve in an efficient way. Consequently, various heuristic algorithms, such as genetic algorithms and particle swarm optimization, have been proposed in the past. This paper aims to examine the potential of a BFO algorithm in solving the PO problem. BFO is a new swarm intelligence technique that has been successfully applied to several real world problems. Through three operations, chemotaxis, reproduction, and elimination-dispersal, the proposed BFO algorithm can effectively solve a PO problem. The performance of the proposed approach was evaluated in computational tests on five benchmark data sets, and the results were compared to those obtained from existing heuristic algorithms. The proposed BFO algorithm is found to be superior to previous heuristic algorithms in terms of solution quality and time.     

Portfolio selection using neural networks

In this paper we apply a heuristic method based on artificial neural networks (NN) in order to trace out the efficient frontier associated to the portfolio selection problem. We consider a generalization of the standard Markowitz mean-variance model which includes cardinality and bounding constraints. These constraints ensure the investment in a given number of different assets and limit the amount of capital to be invested in each asset. We present some experimental results obtained with the NN heuristic and we compare them to those obtained with three previous heuristic methods. The portfolio selection problem is an instance from the family of quadratic programming problems when the standard Markowitz mean-variance model is considered. But if this model is generalized to include cardinality and bounding constraints, then the portfolio selection problem becomes a mixed quadratic and integer programming problem. When considering the latter model, there is not any exact algorithm able to solve the portfolio selection problem in an efficient way. The use of heuristic algorithms in this case is imperative. In the past some heuristic methods based mainly on evolutionary algorithms, tabu search and simulated annealing have been developed. The purpose of this paper is to consider a particular neural network (NN) model, the Hopfield network, which has been used to solve some other optimisation problems and apply it here to the portfolio selection problem, comparing the new results to those obtained with previous heuristic algorithms.     

A portfolio selection model with borrowing constraint based on possibility theory

Compared with the conventional probabilistic mean-variance methodology, fuzzy number can better describe an uncertain environment with vagueness and ambiguity. In this paper, the portfolio selection model with borrowing constraint is proposed by means of possibilistic mean, possibilistic variance, and possibilistic covariance under the assumption that the returns of assets are fuzzy numbers. And a quadratic programming model with inequality constraints is presented when the returns of assets are trapezoid fuzzy numbers. Furthermore, Lemke algorithm is utilized to solve the model. Finally, a numerical example of the portfolio selection problem is given to illustrate our proposed effective means and variances. The results of the numerical example also show that the investor can make different decisions according to different requirements for the values of expected returns. And the efficient portfolio frontier of the model with borrowing constraints can be easily obtained.     

Expected model for portfolio selection with random fuzzy returns

This paper researches portfolio selection problem in combined uncertain environment of randomness and fuzziness. Due to the complexity of the security market, expected values of the security returns may not be predicted accurately. In the paper, expected returns of securities are assumed to be given by fuzzy variables. Security returns are regarded as random fuzzy variables, i.e. random returns with fuzzy expected values. Following Markowitz's idea of quantifying investment return by the expected value of the portfolio and risk by the variance, a new type of mean–variance model is proposed. In addition, a hybrid intelligent algorithm is provided to solve the new model problem. A numeral example is also presented to illustrate the optimization idea and the effectiveness of the proposed algorithm.     

基于粒子群位移转移的混合遗传算法及其应用

基于粒子群位移转移的思想,改变遗传算法的变异规则,提出了一种新的混合遗传算法。利用3个benchmark函数测试了新的混合算法的性能,并将测试结果与标准遗传算法进行了比较。提出了一种多阶段半方差投资选择模型,并将混合算法应用在多阶段半方差投资选择问题的求解上。     

多期证券投资组合问题的区间多目标规划求解方法

投资组合问题主要研究如何将有限的资金合理地分配到不同的金融资产中,以实现收益最大化与风险最小化之间的均衡.然而,证券市场往往具有很强的不确定性,投资者对于证券的期望收益率和风险损失率难以用精确数值描述,区间规划则是处理这类不确定性问题的有力工具.鉴于此,首先基于区间多目标规划建立一个以预期收益率、风险损失率和流动性为目标函数的多期投资组合选择模型;然后通过设计一个定向变异算子,改进基于偏好多面体的交互式遗传算法,并将上述算法的运算机制与所建模型的多期特性相结合以求解模型;最后在不确定交互进化优化系统上进行实证分析.实验结果表明,所提出算法能够根据投资者的不同需要得到相应最满意的多期资产组合.