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.     

具有不确定收益的新的投资组合优化模型的研究

解决了具有不确定收益的投资组合问题。从一个新的视角给出了不确定投资组合的风险定义,在此基础上,提出了新的投资组合优化模型,并设计出新的混合智能算法来解决这一新的优化问题。在新的算法中,99方法被用来计算期望值和机会值,与之前的算法相比,大大减少了计算的工作量,加快了求解过程。最后,提出一个数值例子来验证新的优化模型和所提算法的可行性和正确性。     

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.     

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.     

A multiple objective stochastic portfolio selection problem with random Beta

When selecting a portfolio, we need to consider, in general, the portfolio return and portfolio risk. Many risk measures have been used in portfolio selection problems as the Beta risk measure, introduced by the capital asset pricing model. Most of the existing research papers suppose that security's Beta has a deterministic value. Recently, many researchers argued that in selecting the optimal portfolio, securities’ Beta should be considered as an uncertain parameter. In this paper, we set up fundamentals to model the portfolio's Beta as a random variable and propose a multiple objective stochastic portfolio selection model with random Beta. To solve the proposed model, we apply a stochastic goal programming approach. A numerical example from the US stock exchange market is reported.     

风险效用投资组合模型的遗传算法求解

给出一个折衷考虑风险最小化和收益最大化的单目标决策方法,以单位风险收益最大化为决策目标建立了投资组合的非线性分式规划模型,考虑到分式规划问题的求解难度,利用遗传算法求解模型,并给出算法步骤。最后,给出了数值算例,结果表明该算法是简单有效的。     

基于战略对应的IT项目组合选择模型

把信息技术项目当作组合来管理可以通过平衡风险和收益来促进企业目标和IT应用的结合,但由于决策信息的不确定性和IT项目目标与企业战略的难以对应,企业面临IT项目组合选择的挑战。构建基于战略对应的IT项目组合选择模型,其中模糊集和模糊层次分析法用来刻画不确定信息和评估IT项目风险、成本及收益,关键成功因素法用来提高IT项目与企业战略的对应,并建立模糊0－1整数规划。利用定性可能性理论把模糊组合选择模型转化为一般可求解的整数规划形式,最后用一个案例说明模型的用法。     

Portfolio selection with the effect of systematic risk diversification: formulation and accelerated gradient algorithm

In this paper, we propose an optimal trade-off model for portfolio selection with the effect of systematic risk diversification, measured by the maximum marginal systematic risk of all the risk contributors. First, the classical portfolio selection model with constraints on allocation of systematic risk is shown to be equivalent to our trade-off model under certain conditions. Then, we transform the trade-off model into a special non-convex and non-smooth composite problem equivalently. Thus a modified accelerated gradient (AG) algorithm can be introduced to solve the composite problem. The efficiency of the algorithm for solving the composite problem is demonstrated by theoretical results on both the convergence rate and the iteration complexity bound. Finally, empirical analysis demonstrates that the proposed model is a preferred tool for active portfolio risk management when compared with the existing models. We also carry out a series of numerical experiments to compare the performance of the modified AG algorithm with the other three first-order algorithms.     

Fuzzy R&D portfolio selection of interdependent projects

Global competition of markets has forced firms to invest in targeted R&D projects so that resources can be focused on successful outcomes. A number of options are encountered to select the most appropriate projects in an R&D project portfolio selection problem. The selection is complicated by many factors, such as uncertainty, interdependences between projects, risk and long lead time, that are difficult to measure. Our main concern is how to deal with the uncertainty and interdependences in project portfolio selection when evaluating or estimating future cash flows. This paper presents a fuzzy multi-objective programming approach to facilitate decision making in the selection of R&D projects. Here, we present a fuzzy tri-objective R&D portfolio selection problem which maximizes the outcome and minimizes the cost and risk involved in the problem under the constraints on resources, budget, interdependences, outcome, projects occurring only once, and discuss how our methodology can be used to make decision support tools for optimal R&D project selection in a corporate environment. A case study is provided to illustrate the proposed method where the solution is done by genetic algorithm (GA) as well as by multiple objective genetic algorithm (MOGA).     

Credibilistic multi-period portfolio optimization model with bankruptcy control and affine recourse

Avoiding the possibility of bankruptcy during the investment horizon is very important to multi-period portfolio management. This paper considers a multi-period fuzzy portfolio selection problem with bankruptcy control. A multi-period portfolio optimization model imposed by a bankruptcy control constraint in fuzzy environment is proposed on the basis of credibility theory. In the proposed model, a linearly recourse policy is used to reflect the influence of historical predication basis on current portfolio decision. Three optimization objectives, viz., maximizing the terminal wealth and minimizing the cumulative risk and the cumulative uncertainty of the returns of portfolios over the whole investment horizon, are taken into consideration. For solving the proposed model, a fuzzy programming approach is applied to transform it into a single objective programming model. Then, a hybrid particle swarm optimization algorithm is designed for solution. Finally, an empirical example is presented to illustrate the application of the proposed model and solution comparisons are also given to demonstrate the effectiveness of the designed algorithm.     

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).     

Fuzzy portfolio model with fuzzy-input return rates and fuzzy-output proportions

In the finance market, a short-term investment strategy is usually applied in portfolio selection in order to reduce investment risk; however, the economy is uncertain and the investment period is short. Further, an investor has incomplete information for selecting a portfolio with crisp proportions for each chosen security. In this paper we present a new method of constructing fuzzy portfolio model for the parameters of fuzzy-input return rates and fuzzy-output proportions, based on possibilistic mean–standard deviation models. Furthermore, we consider both excess or shortage of investment in different economic periods by using fuzzy constraint for the sum of the fuzzy proportions, and we also refer to risks of securities investment and vagueness of incomplete information during the period of depression economics for the portfolio selection. Finally, we present a numerical example of a portfolio selection problem to illustrate the proposed model and a sensitivity analysis is realised based on the results.     

A random fuzzy minimum spanning tree problem through a possibility-based value at risk model

This paper considers a minimum spanning tree problem under the situation where costs for constructing edges in a network include both fuzziness and randomness. In particular, this article focuses on the case that the edge costs are expressed by random fuzzy variables. A new decision making model based on a possibility measure and a value at risk measure is proposed in order to find a solution which fully reflects random and fuzzy information. It is shown that an optimal solution of the proposed model is obtained by a polynomial-time algorithm.     

Interactive multiobjective fuzzy random programming through the level set-based probability model

This paper considers a multiobjective linear programming problem involving fuzzy random variable coefficients. A new fuzzy random programming model is proposed by extending the ideas of level set-based optimality and a stochastic programming model. The original problem involving fuzzy random variables is transformed into a deterministic equivalent problem through the proposed model. An interactive algorithm is provided to obtain a satisficing solution for a decision maker from among a set of newly defined Pareto optimal solutions. It is shown that an optimal solution of the problem to be solved iteratively in the interactive algorithm is analytically obtained by a combination of the bisection method and the simplex method.     

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.     

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.     

A Modified Genetic Algorithm for solving uncertain Constrained Solid Travelling Salesman Problems

In this paper, a Modified Genetic Algorithm (MGA) is developed to solve Constrained Solid Travelling Salesman Problems (CSTSPs) in crisp, fuzzy, random, random-fuzzy, fuzzy-random and bi-random environments. In the proposed MGA, for the first time, a new ‘probabilistic selection’ technique and a ‘comparison crossover’ are used along with conventional random mutation. A Solid Travelling Salesman Problem (STSP) is a Travelling Salesman Problem (TSP) in which, at each station, there are a number of conveyances available to travel to another station. Thus STSP is a generalization of classical TSP and CSTSP is a STSP with constraints. In CSTSP, along each route, there may be some risk/discomfort in reaching the destination and the salesman desires to have the total risk/discomfort for the entire tour less than a desired value. Here we model the CSTSP with traveling costs and route risk/discomfort factors as crisp, fuzzy, random, random-fuzzy, fuzzy-random and bi-random in nature. A number of benchmark problems from standard data set, TSPLIB are tested against the existing Genetic Algorithm (with Roulette Wheel Selection (RWS), cyclic crossover and random mutation) and the proposed algorithm and hence the efficiency of the new algorithm is established. In this paper, CSTSPs are illustrated numerically by some empirical data using this algorithm. In each environment, some sensitivity studies due to different risk/discomfort factors and other system parameters are presented.     

A fuzzy portfolio selection method based on possibilistic mean and variance

This paper deals with the portfolio selection problem when the returns of assets obey LR-type possibility distributions and there exist the limits on holdings. A new possibilistic mean–variance model to portfolio selection is proposed based on the definitions of the possibilistic return and possibilistic risk, which can better integrate an uncertain decision environment with vagueness and ambiguity. This possibilistic mean–variance model can be regarded as extensions of conventional probabilistic mean–variance methodology and previous possibilistic approaches since it contains less parameter and has a more extensive application. A numerical example of a possibilistic fuzzy portfolio selection problem is given to illustrate our proposed effective means and approaches. This project was supported by NCET (No.06-0749) and The National Natural Science Foundation of China (No.70571024).     

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.     

Genetic algorithm-based fuzzy goal programming for class of chance-constrained programming problems

This paper presents a procedure for solving a multiobjective chance-constrained programming problem. Random variables appearing on both sides of the chance constraint are considered as discrete random variables with a known probability distribution. The literature does not contain any deterministic equivalent for solving this type of problem. Therefore, classical multiobjective programming techniques are not directly applicable. In this paper, we use a stochastic simulation technique to handle randomness in chance constraints. A fuzzy goal programming formulation is developed by using a stochastic simulation-based genetic algorithm. The most satisfactory solution is obtained from the highest membership value of each of the membership goals. Two numerical examples demonstrate the feasibility of the proposed approach.