Two new models for portfolio selection with stochastic returns taking fuzzy information

This paper proposes two new models for portfolio selection in which the security returns are stochastic variables with fuzzy information. A hybrid intelligent algorithm is designed to solve the optimization problem which is otherwise hard to solve with the existing algorithms due to the complexity of the return variables. To illustrate the modelling idea and to show the effectiveness of the proposed approach, two numerical examples are provided.     

Mean-variance-skewness model for portfolio selection with fuzzy returns

Numerous empirical studies show that portfolio returns are generally asymmetric, and investors would prefer a portfolio return with larger degree of asymmetry when the mean value and variance are same. In order to measure the asymmetry of fuzzy portfolio return, a concept of skewness is defined as the third central moment in this paper, and its mathematical properties are studied. As an extension of the fuzzy mean-variance model, a mean-variance-skewness model is presented and the corresponding variations are also considered. In order to solve the proposed models, a genetic algorithm integrating fuzzy simulation is designed. Finally, several numerical examples are given to illustrate the modelling idea and the effectiveness of the proposed algorithm.     

随机市场系数的M-V最优投资组合选择:一个鞅方法

通过引进凹函数U(x)以及等价鞅测度p^-，应用鞅的性质得到了随机市场系数情形下的M—V模型的最优投资策略以及有效前沿．     

随机市场系数条件下不连续股价的M-V投资组合选择

在假定市场系数为随机过程并且股票价格服从跳跃扩散过程的市场条件下应用鞅方法讨论一个M-V模型的最优投资组合选择问题.通过引进凹函数U(x)以及等价鞅测度,应用鞅方法以及贝叶斯定理得到了最优投资策略以及有效边界表达式.     

A portfolio selection model using fuzzy returns

We study a static portfolio selection problem, in which future returns of securities are given as fuzzy sets. In contrast to traditional analysis, we assume that investment decisions are not based on statistical expectation values, but rather on maximal and minimal potential returns resulting from the so-called α-cuts of these fuzzy sets. By aggregating over all α-cuts and assigning weights for both best and worst possible cases we get a new objective function to derive an optimal portfolio. Allowing for short sales and modelling α-cuts in ellipsoidal shape, we obtain the optimal portfolio as the unique solution of a simple optimization problem. Since our model does not include any stochastic assumptions, we present a procedure, which turns the data of observable returns as well as experts’ expectations into fuzzy sets in order to quantify the potential future returns and the investment risk.     

Mean-semivariance models for fuzzy portfolio selection

This paper discusses portfolio selection problem in fuzzy environment. In the paper, semivariance is originally presented for fuzzy variable, and three properties of the semivariance are proven. Based on the concept of semivariance of fuzzy variable, two fuzzy mean-semivariance models are proposed. To solve the new models in general cases, a fuzzy simulation based genetic algorithm is presented in the paper. In addition, two numerical examples are also presented to illustrate the modelling idea and the effectiveness of the designed algorithm.     

Characterization of order dominances on fuzzy variables for portfolio selection with fuzzy returns

Peng et al (Int J Uncertain Fuzziness Knowl Based Syst 15:29–41, 2007) introduced, by means of the credibility measure, two dominance relations on fuzzy variables, namely the first- and the second-order dominances. In this paper, we characterize each of these dominance relations, and we justify that they satisfy six well-known properties of comparison methods. We propose a Game Theory approach for the determination of optimal portfolios when returns are fuzzy by introducing the set of best portfolios with respect to the first- and the second-order dominances. Based on the characterization of the first-order dominance, we numerically display some of the best portfolios of the classical set of portfolios of seven independent assets described by triangular fuzzy numbers.     

A hybrid intelligent algorithm for portfolio selection problem with fuzzy returns

Portfolio selection theory with fuzzy returns has been well developed and widely applied. Within the framework of credibility theory, several fuzzy portfolio selection models have been proposed such as mean–variance model, entropy optimization model, chance constrained programming model and so on. In order to solve these nonlinear optimization models, a hybrid intelligent algorithm is designed by integrating simulated annealing algorithm, neural network and fuzzy simulation techniques, where the neural network is used to approximate the expected value and variance for fuzzy returns and the fuzzy simulation is used to generate the training data for neural network. Since these models are used to be solved by genetic algorithm, some comparisons between the hybrid intelligent algorithm and genetic algorithm are given in terms of numerical examples, which imply that the hybrid intelligent algorithm is robust and more effective. In particular, it reduces the running time significantly for large size problems.     

Bayesian portfolio selection with multi-variate random variance models

We consider multi-period portfolio selection problems for a decision maker with a specified utility function when the variance of security returns is described by a discrete time stochastic model. The solution of these problems involves a dynamic programming formulation and backward induction. We present a simulation-based method to solve these problems adopting an approach which replaces the preposterior analysis by a surface fitting based optimization approach. We provide examples to illustrate the implementation of our approach.     

Mean-variance portfolio selection for a non-life insurance company

We consider a collective insurance risk model with a compound Cox claim process, in which the evolution of a claim intensity is described by a stochastic differential equation driven by a Brownian motion. The insurer operates in a financial market consisting of a risk-free asset with a constant force of interest and a risky asset which price is driven by a Lévy noise. We investigate two optimization problems. The first one is the classical mean-variance portfolio selection. In this case the efficient frontier is derived. The second optimization problem, except the mean-variance terminal objective, includes also a running cost penalizing deviations of the insurer’s wealth from a specified profit-solvency target which is a random process. In order to find optimal strategies we apply techniques from the stochastic control theory.     

The mean-absolute deviation portfolio selection problem with interval-valued returns

In real-world investments, one may care more about the future earnings than the current earnings of the assets. This paper discusses the uncertain portfolio selection problem where the asset returns are represented by interval data. Since the parameters are interval valued, the gain of returns is interval valued as well. According to the concept of the mean-absolute deviation function, we construct a pair of two-level mathematical programming models to calculate the lower and upper bounds of the investment return of the portfolio selection problem. Using the duality theorem and applying the variable transformation technique, the pair of two-level mathematical programs is transformed into a conventional one-level mathematical program. Solving the pair of mathematical programs produces the interval of the portfolio return of the problem. The calculated results conform to an essential idea in finance and economics that the greater the amount of risk that an investor is willing to take on the greater the potential return.     

Forecasting portfolio returns using weighted fuzzy time series methods

We propose using weighted fuzzy time series (FTS) methods to forecast the future performance of returns on portfolios. We model the uncertain parameters of the fuzzy portfolio selection models using a possibilistic interval-valued mean approach, and approximate the uncertain future return on a given portfolio by means of a trapezoidal fuzzy number. Introducing some modifications into the classical models of fuzzy time series, based on weighted operators, enables us to generate trapezoidal numbers as forecasts of the future performance of the portfolio returns. This fuzzy forecast makes it possible to approximate both the expected return and the risk of the investment through the value and ambiguity of a fuzzy number.We incorporate our proposals into classical fuzzy time series methods and analyze their effectiveness compared with classical weighted fuzzy time series models, using historical returns on assets from the Spanish stock market. When our weighted FTS proposals are used to point-wise forecast portfolio returns the one-step ahead accuracy is improved, also with respect to non-fuzzy forecasting methods.     

Robust-based interactive portfolio selection problems with an uncertainty set of returns

This paper considers a robust portfolio selection problem with an uncertainty set of future returns and satisfaction levels in terms of the total return and robustness parameter. Since the proposed model is formulated as an ill-defined problem due to uncertainty and is bi-objective, that is, to maximize both the abovementioned satisfaction levels, it is difficult to solve the model directly without introducing some criterion of optimality for the bi-objective functions. Therefore, by introducing fuzzy goals and an interactive fuzzy satisficing method, the proposed model is transformed into a deterministic equivalent problem. Furthermore, to obtain the exact optimal portfolio analytically, a solution method is developed by introducing the auxiliary problem and performing equivalent transformations. In order to compare the proposed model with previous useful models, numerical examples are provided, and the results show that it is important to maximize the robustness parameter and total return using the interactive process for adjusting investor’s satisfaction levels.     

Mean-variance portfolio rebalancing with transaction costs and funding changes

Investment portfolios should be rebalanced to take account of changing market conditions and changes in funding. Standard mean-variance (MV) portfolio selection methods are not appropriate for portfolio rebalancing, as the initial portfolio, change in funding and transaction costs are not considered. A quadratic mixed integer programming portfolio rebalancing model, which takes account of these factors is developed in this paper. The transaction costs in this portfolio rebalancing model are composed of fixed charges and variable costs, including the market impact costs associated with large market trades of individual securities, where these variable transaction costs are assumed to be non-linear functions of traded value. The use of this model is demonstrated and it is shown that when initial portfolio, funding changes and transaction costs are taken into account in portfolio construction and rebalancing, MV efficient portfolios that include risk-free lending do not have the structure expected from portfolio theory.     

Probability maximization models for portfolio selection under ambiguity

This paper considers several probability maximization models for multi-scenario portfolio selection problems in the case that future returns in possible scenarios are multi-dimensional random variables. In order to consider occurrence probabilities and decision makers’ predictions with respect to all scenarios, a portfolio selection problem setting a weight with flexibility to each scenario is proposed. Furthermore, by introducing aspiration levels to occurrence probabilities or future target profit and maximizing the minimum aspiration level, a robust portfolio selection problem is considered. Since these problems are formulated as stochastic programming problems due to the inclusion of random variables, they are transformed into deterministic equivalent problems introducing chance constraints based on the stochastic programming approach. Then, using a relation between the variance and absolute deviation of random variables, our proposed models are transformed into linear programming problems and efficient solution methods are developed to obtain the global optimal solution. Furthermore, a numerical example of a portfolio selection problem is provided to compare our proposed models with the basic model.     

Multiobjective expected value model for portfolio selection in fuzzy environment

In this paper, we propose a credibilistic framework for portfolio selection problem using an expected value multiobjective model with fuzzy parameters. We consider short term return, long term return, risk and liquidity as key financial criteria. A solution procedure comprising fuzzy goal programming and fuzzy simulation based real-coded genetic algorithm is developed to solve the model. The proposed solution approach is considered advantageous particularly for the cases where the fuzzy parameters of the problem may assume any general functional form. An empirical study is included to illustrate the usefulness of the proposed model and solution approach in real-world applications of portfolio selection.     

A multi-objective genetic algorithm for cardinality constrained fuzzy portfolio selection

This paper presents a new procedure that extends genetic algorithms from their traditional domain of optimization to fuzzy ranking strategy for selecting efficient portfolios of restricted cardinality. The uncertainty of the returns on a given portfolio is modeled using fuzzy quantities and a downside risk function is used to describe the investor's aversion to risk. The fitness functions are based both on the value and the ambiguity of the trapezoidal fuzzy number which represents the uncertainty on the return. The soft-computing approach allows us to consider uncertainty and vagueness in databases and also to incorporate subjective characteristics into the portfolio selection problem. We use a data set from the Spanish stock market to illustrate the performance of our approach to the portfolio selection problem.     

A fuzzy goal programming approach to portfolio selection

Portfolio selection is a usual multiobjective problem. This paper will try to deal with the optimum portfolio for a private investor, taking into account three criteria: return, risk and liquidity. These objectives, in general, are not crisp from the point of view of the investor, so we will deal with them in fuzzy terms. The problem formulation is a goal programming (G.P.) one, where the goals and the constraints are fuzzy. We will apply a fuzzy G.P. approach to the above problem to obtain a solution. Then, we will offer the investor help in handling the results.     

Exact and heuristic procedures for solving the fuzzy portfolio selection problem

We propose a fuzzy model for the portfolio selection problem which takes into account the vagueness of the investor’s preferences regarding the assumed risk. We also describe an exact method for solving it as well as a hybrid meta-heuristic procedure which is more adequate for medium and large-sized problems or in cases in which a quick solution is needed. As an application, we solve several problems based on data from the IBEX35 index and the Spanish Stock Exchange Interconnection System.