A new perspective for optimal portfolio selection with random fuzzy returns

The aim of this paper is to solve the portfolio selection problem when security returns contain both randomness and fuzziness. Utilizing a different perspective, this paper gives a new definition of risk for random fuzzy portfolio selection. A new optimal portfolio selection model is proposed based on this new definition of risk. A new hybrid intelligent algorithm is designed for solving the new optimization problem. In the proposed new algorithm, neural networks are employed to calculate the expected value and the chance value. These greatly reduce the computational work and speed up the process of solution as compared with the random fuzzy simulation used in our previous algorithm. A numerical example is also presented to illustrate the new modelling idea and the proposed new algorithm.     

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 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.     

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。     

Uncertain programming models for portfolio selection with uncertain returns

In an indeterminacy economic environment, experts’ knowledge about the returns of securities consists of much uncertainty instead of randomness. This paper discusses portfolio selection problem in uncertain environment in which security returns cannot be well reflected by historical data, but can be evaluated by the experts. In the paper, returns of securities are assumed to be given by uncertain variables. According to various decision criteria, the portfolio selection problem in uncertain environment is formulated as expected-variance-chance model and chance-expected-variance model by using the uncertainty programming. Within the framework of uncertainty theory, for the convenience of solving the models, some crisp equivalents are discussed under different conditions. In addition, a hybrid intelligent algorithm is designed in the paper to provide a general method for solving the new models in general cases. At last, two numerical examples are provided to show the performance and applications of the models and algorithm.     

Diversified models for portfolio selection based on uncertain semivariance

Since the financial markets are complex, sometimes the future security returns are represented mainly based on experts’ estimations due to lack of historical data. This paper proposes a semivariance method for diversified portfolio selection, in which the security returns are given subjective to experts’ estimations and depicted as uncertain variables. In the paper, three properties of the semivariance of uncertain variables are verified. Based on the concept of semivariance of uncertain variables, two types of mean-semivariance diversified models for uncertain portfolio selection are proposed. Since the models are complex, a hybrid intelligent algorithm which is based on 99-method and genetic algorithm is designed to solve the models. In this hybrid intelligent algorithm, 99-method is applied to compute the expected value and semivariance of uncertain variables, and genetic algorithm is employed to seek the best allocation plan for portfolio selection. At last, several numerical examples are presented to illustrate the modelling idea and the effectiveness of the 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).     

Minimax mean-variance models for fuzzy portfolio selection

This paper discusses fuzzy portfolio selection problem in the situation where each security return belongs to a certain class of fuzzy variables but the exact fuzzy variable cannot be given. Two credibility-based minimax mean-variance models are proposed. The crisp equivalents of the models to linear programming ones are given in three special cases. In addition, a general solution algorithm is also provided. To help understand the modeling idea and to illustrate the effectiveness of the proposed algorithm, one example is presented.     

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.     

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.     

Mean-Entropy Models for Fuzzy Portfolio Selection

This short paper proposes two types of credibility-based fuzzy mean-entropy models. In the short paper, entropy is used as the measure of risk. The smaller the entropy value is, the less uncertainty the portfolio return contains, and thus, the safer the portfolio is. Furthermore, as a measure of risk, entropy is free from reliance on symmetrical distributions of security returns and can be computed from nonmetric data. In addition, the short paper compares the fuzzy mean-variance model with the fuzzy mean-entropy model in two special cases and presents a hybrid intelligent algorithm for solving the proposed models in general cases. To illustrate the effectiveness of the proposed algorithm, the short paper also provides two numerical examples.      

Mean-variance model for fuzzy capital budgeting

In an uncertain economic environment, it is usually difficult to predict accurately the investment outlays and annual net cash flows of a project. In addition, available investment capital sometimes cannot be accurately given either. Fuzzy variables can reflect vagueness of these parameters. In this paper, capital budgeting problem with fuzzy investment outlays, fuzzy annual net cash flows and fuzzy available investment capital is studied based on credibility measure. One new mean-variance model is proposed for optimal capital allocation. A fuzzy simulation-based genetic algorithm is provided for solving the proposed optimization problem. One numerical example and an experiment are also presented to show the optimization idea and the effectiveness of the algorithm.     

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.     

Optimal run lengths in deteriorating production processes in random fuzzy environments

This paper presents a random fuzzy economic manufacturing quantity (EMQ) model in a deteriorating process. It is assumed that the setup cost and the average holding cost are characterized as fuzzy variables and the elapsed time until shift is a random fuzzy variable. As a function of these parameters, the average total cost is also a random fuzzy variable, and the unimodality of its expected value is studied. To obtain the optimal run length and the minimum average cost, simultaneous perturbation stochastic approximation (SPSA) algorithm based on random fuzzy simulation is provided. Random fuzzy EMQ models with fuzzy deterioration, fuzzy linear deterioration and fuzzy exponential deterioration are presented, respectively. These models can be solved by the proposed algorithm. Numerical examples are presented in the end.     

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).     

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。     

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.     

Modeling data envelopment analysis by chance method in hybrid uncertain environments

This article first presents several formulas of chance distributions for trapezoidal fuzzy random variables and their functions, then develops a new class of chance model (C-model for short) about data envelopment analysis (DEA) in fuzzy random environments, in which the inputs and outputs are assumed to be characterized by fuzzy random variables with known possibility and probability distributions. Since the objective and constraint functions contain the chance of fuzzy random events, for general fuzzy random inputs and outputs, we suggest an approximation method to compute the chance. When the inputs and outputs are mutually independent trapezoidal fuzzy random variables, we can turn the chance constraints and the chance objective into their equivalent stochastic ones by applying the established formulas for the chance distributions. In the case when the inputs and the outputs are mutually independent trapezoidal fuzzy random vectors, the proposed C-model can be transformed to its equivalent stochastic programming one, in which the objective and the constraint functions include a number of standard normal distribution functions. To solve such an equivalent stochastic programming, we design a hybrid algorithm by integrating Monte Carlo (MC) simulation and genetic algorithm (GA), in which MC simulation is used to calculate standard normal distribution functions, and GA is used to solve the optimization problems. Finally, one numerical example is presented to demonstrate the proposed modeling idea and the efficiency in the proposed model.     

Fuzzy random programming models for location-allocation problem with applications

Three types of fuzzy random programming models based on the mean chance for the capacitated location-allocation problem with fuzzy random demands are proposed according to different criteria, including the expected cost minimization model, the α-cost minimization model, and the chance maximization model. In order to solve the proposed models, some hybrid intelligent algorithms are designed by integrating the network simplex algorithm, fuzzy random simulation, and genetic algorithm. Finally, some numerical examples about a container freight station problem are given to illustrate the effectiveness of the devised algorithms.