Fuzzy Returns Research Articles

Credibilistic Multi-Period Mean-Entropy Rolling Portfolio Optimization Problem Based on Multi-Stage Scenario Tree

This study considers a time-consistent multi-period rolling portfolio optimization issue in the context of a fuzzy situation. Rolling optimization with a risk aversion component attempts to separate the time periods and psychological effects of one’s investment in a mathematical model. Furthermore, a resilient portfolio selection may be attained by taking into account fuzzy scenarios. Credibilistic entropy of fuzzy returns is used to measure portfolio risk because entropy, as a measure of risk, is not dependent on any certain sort of symmetric membership function of stock returns and may be estimated using nonmetric data. Mathematical modeling is performed to compare the Rolling Model (RM) and the Unified Model (UM). Two empirical studies from the Tehran stock market (10 stocks from April 2017 to April 2019) and the global stock market (20 stocks from April 2021 to April 2023) are utilized to illustrate the applicability of the suggested strategy. The findings reveal that RM can limit the risk of the portfolio at each time, but the portfolio’s return is smaller than that of UM. Furthermore, the suggested models outperform the standard deterministic model.

Mean-AVaR in credibilistic portfolio management via an artificial neural network scheme

ABSTRACT This paper focuses on the computation issue of portfolio optimisation with scenario-based mean-Average Value at Risk (AVaR) in credibilistic environment. The portfolio optimisation problem is designed in two cases: risk taker model and risk-averse model. The main idea is to replace the portfolio selection models with linear programming (LP) problems. Since the computing time required for solving LP greatly depends on the dimension and the structure of the problem, the conventional numerical methods are usually less effective in real-time applications. One promising approach to handle online applications is to employ recurrent neural networks based on circuit implementation. Hence, according to the convex optimisation theory and some concepts of ordinary differential equations, a neural network model for solving the LP problems related to portfolio selection problems is presented. The equilibrium point of the proposed model is proved to be equivalent to the optimal solution of the original problem. It is also shown that the proposed neural network model is stable in the sense of Lyapunov and it is globally convergent to an exact optimal solution of the portfolio selection problem with fuzzy returns. Some illustrative examples are provided to show the feasibility and the efficiency of the proposed method in this paper.

Fuzzy time series model based on red–black trees for stock index forecasting

Forecasting data is still an extensively investigated area of research specially in stock markets. The subjectivity of the elements that influence the market oscillation is the main challenge that any forecasting model faces. In this context, existing fuzzy models have attempted to increase forecasting accuracy in financial markets over the years. Fuzzy returns of the phenomenon under investigation helps to mitigate the subjective part of the financial market, specially regarding the human feeling influence over it. Although there are several data structures that can help to define the proper clusters from the universe of discourse of a fuzzy model, this paper proposes a novel fuzzy model from which the universe of discourse is based on a red–black tree (RBT) data structure so as to increase the possibilities of obtaining better predictions. The RBT data structure is a binary search three data structure that promotes a better balance, which allows a better accuracy in the forecasting results. The proposed model is compared to well known fuzzy models in the literature showing better forecasting results.

Adjustable Security Proportions in the Fuzzy Portfolio Selection under Guaranteed Return Rates

Based on the concept of high returns as the preference to low returns, this study discusses the adjustable security proportion for excess investment and shortage investment based on the selected guaranteed return rates in a fuzzy environment, in which the return rates for selected securities are characterized by fuzzy variables. We suppose some securities are for excess investment because their return rates are higher than the guaranteed return rates, and the other securities whose return rates are lower than the guaranteed return rates are considered for shortage investment. Then, we solve the proposed expected fuzzy returns by the concept of possibility theory, where fuzzy returns are quantified by possibilistic mean and risks are measured by possibilistic variance, and then we use linear programming model to maximize the expected value of a portfolio’s return under investment risk constraints. Finally, we illustrate two numerical examples to show that the expected return rate under a lower guaranteed return rate is better than a higher guaranteed return rates in different levels of investment risks. In shortage investments, the investment proportion for the selected securities are almost zero under higher investment risks, whereas the portfolio is constructed from those securities in excess investments.

A novel fuzzy dominant goal programming for portfolio selection with systematic risk and non-systematic risk.

In this paper, we consider a fuzzy portfolio selection problem with systematic risk and non-systematic risk simultaneously. These two kinds of risks are measured by beta coefficient and random error variance obtained by Sharp Single Index Model. The total risk as the objective of portfolio decision is obtained by weighting the two kinds of risk. Among them, the weight of systematic risk lambda is regarded as the degree of investors' attention to system risk in economic sense. In addition, the fuzzy return and the degree of diversification are measured by triangular fuzzy number and entropy, respectively. And they are also considered the goal of investment decisions. Hence, a tri-objective portfolio is proposed in this paper. For the fuzzy objectives in the model, a goal programming method based on fuzzy dominance is proposed, which can help investors better capture the ideal point of fuzzy returns according to their risk preference. Finally, combined with the systematic impact of COVID-19 on the financial market, we make an empirical analysis based on our proposed model. The results show that the total risk will be on the high side when lambda value is too large or too small. That means paying too much or little attention to the systematic risk will lead investors to bear more risk. In addition, when investors ignore the systematic risk; that is, the lambda value is low, and investors will concentrate their funds in the same industry.

Multiobjective Fuzzy Portfolio Performance Evaluation Using Data Envelopment Analysis Under Credibilistic Framework

In this article, two different multiobjective fuzzy portfolio selection models are presented. The significant criteria considered for portfolio selection are risk (variance or conditional value at risk), return, liquidity, and entropy. Here, the return of the portfolio is considered to be satisfied by a minimum return threshold constraint. Also, to introduce some degree of diversification in the model, a lower and upper bound constraint on investment in an asset is used along with the capital budget and no short selling constraints. Trapezoidal fuzzy returns are considered to incorporate the inherent uncertainty of the stock market, which is handled by using the credibility theory. The weighted sum approach is used to aggregate the objectives and characterize different investor attitudes. Random sample portfolios with progressively increasing sample sizes are generated that obey the constraints of the portfolio models. These random sample portfolios with multiple inputs (risk and entropy) and multiple outputs (return and liquidity) are evaluated in terms of their performance by using data envelopment analysis. Furthermore, a frontier improvement technique existing in the literature is used to rebalance the inefficient random sample portfolios to make them efficient, so that an investor may have more avenues to select efficient portfolios. A detailed numerical illustration with a simulation study using different sample sizes is presented to substantiate the proposed study.

A Simple View on the Interval and Fuzzy Portfolio Selection Problems.

In this paper, first we show that the variance used in the Markowitz’s mean-variance model for the portfolio selection with its numerous modifications often does not properly present the risk of portfolio. Therefore, we propose another treating of portfolio risk as the measure of possibility to earn unacceptable low profits of portfolio and a simple mathematical formalization of this measure. In a similar way, we treat the criterion of portfolio’s return maximization as the measure of possibility to get a maximal profit. As the result, we formulate the portfolio selection problem as a bicriteria optimization task. Then, we study the properties of the developed approach using critical examples of portfolios with interval and fuzzy valued returns. The -cuts representation of fuzzy returns was used. To validate the proposed method, we compare the results we got using it with those obtained with the use of fuzzy versions of seven widely reputed methods for portfolio selection. As in our approach we deal with the bicriteria task, the three most popular methods for local criteria aggregation are compared using the known example of fuzzy portfolio consist of five assets. It is shown that the results we got using our approach to the interval and fuzzy portfolio selection reflect better the essence of this task than those obtained by widely reputed traditional methods for portfolio selection in the fuzzy setting.

Performance evaluation of portfolios with fuzzy returns

The existing literature on DEA (Data Envelopment Analysis) for evaluating fuzzy portfolios usually takes risk as an input and return as an output. This assumption is actually not congruent with the real investment process, where the input is the initial wealth and the output is the corresponding terminal wealth. As for the risk and return, which are essentially two indicators derived from the terminal wealth, both should be regarded as outputs. In addition, few studies have employed the diversification model (nonlinear DEA) to estimate the fuzzy portfolio efficiency (PE), despite the fact that there are many studies available within the framework of classical probability theory. Further, the relationship between DEA and diversification models needs to be defined. In this paper, we take the initial wealth as an input, while the return and risk of terminal wealth are taken as desirable and undesirable outputs, respectively. We construct different evaluation models under the fuzzy portfolio framework. The relationships among the evaluation model based on a real frontier, the diversification model and the DEA model are investigated. We show the convergence of the diversification and DEA models under the fuzzy theory framework. Some simulations as well as empirical analysis are presented to further verify the effectiveness of the proposed models. Finally, we check the robustness of the evaluation results by using the bootstrap re-sampling approach.

A Hybrid Multiobjective Bat Algorithm for Fuzzy Portfolio Optimization with Real-World Constraints

This paper investigates a fuzzy portfolio selection problem in the framework of multiobjective optimization. A multiobjective mean–semivariance–entropy model with fuzzy returns is proposed for portfolio selection. Specifically, it simultaneously optimizes the return, risk and portfolio diversification, taking into account transaction costs, liquidity, buy-in thresholds, and cardinality constraints. Since this kind of mixed-integer nonlinear programming problems cannot be efficiently solved by the conventional optimization approaches, a new metaheuristic method termed as the hybrid BA–DE is developed by combining features of the bat algorithm (BA) and differential evolution (DE). In order to demonstrate the effectiveness of the proposed approaches, we also provide a numerical example.

Possibilistic Moment Models for Multi-period Portfolio Selection with Fuzzy Returns

The aim of this paper is to investigate the effects of higher moments on multi-period portfolio selection with fuzzy returns. This paper gives the definitions of possibilistic mean and variance about the product of multiple fuzzy numbers. Based on these definitions, three multi-period fuzzy portfolio optimization models are proposed. The proposed models aim to maximize terminal wealth and minimize terminal risk by taking into account some realistic constraints including higher moments, budget constraint, round-lot constraint, cardinality constraint and bound constraint. To ensure the selection of the best solutions, a novel fuzzy programming approach-based self-adaptive differential evolution algorithm is designed to solve the proposed models. A numerical example is given to demonstrate the application of the proposed models. Computational results show that the designed algorithm is effective for solving complex portfolio selection model with realistic constraints.

MULTIPERIOD CREDIBILITIC MEAN SEMI-ABSOLUTE DEVIATION PORTFOLIO SELECTION

In this paper, we discuss a multiperiod portfolio selection problem with fuzzy returns. We present a new credibilitic multiperiod mean semi- absolute deviation portfolio selection with some real factors including transaction costs, borrowing constraints, entropy constraints, threshold constraints and risk control. In the proposed model, we quantify the investment return and risk associated with the return rate on a risky asset by its credibilitic expected value and semi- absolute deviation. Since the proposed model is a nonlinear dynamic optimization problem with path dependence, we design a novel forward dynamic programming method to solve it. Finally, we provide a numerical example to demonstrate the performance of the designed algorithm and the application of the proposed model.

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.

Multi-period mean–variance fuzzy portfolio optimization model with transaction costs

This paper examines the multi-period portfolio optimization problem with transaction costs and fuzzy variables to count for the uncertainty of future returns and liquidities on assets. The portfolio risk is quantified by using the variance of fuzzy returns. Two conflicting optimization objectives, namely, maximizing the terminal wealth and minimizing the cumulative risk of portfolios over the entire investment horizon, are taken into consideration. For solving the proposed model we introduce a new multi-objective evolutionary algorithm. Finally the performance of the proposed algorithm is compared with the NSGAII and MOEA/D with the assistance of real data from FTSE-100 in London.

Credibilistic mean-entropy models for multi-period portfolio selection with multi-choice aspiration levels

This paper deals with fuzzy multi-objective multi-period portfolio selection problems. The major criteria used for portfolio selection and rebalancing are wealth, risk, transaction cost, liquidity, and number of assets held in the portfolio. Considering that entropy as a measure of risk does not rely on any particular type of symmetric membership functions of the asset returns and can be computed from the nonmetric data as well, the portfolio risk is quantified using credibilistic entropy of the fuzzy returns. Furthermore, we use multi-choice aspiration levels for the financial goals to better suit the human perceptions such as “the more the better” or “the less the better” regarding return and risk. Two credibility-based fuzzy optimization models are developed using both the discrete choices and interval ranges for the multi-choice aspiration levels of each financial goal. To obtain efficient investment strategies for the entire investment horizon, we use a fuzzy credibilistic programming approach with multi-choice goal programming embedded in it. A real-world empirical application with data-set from an Indian stock market is presented to demonstrate usefulness of the proposed models and the solution approach for multi-period portfolio selection problem in fuzzy environment.

An integrated multi-objective Markowitz–DEA cross-efficiency model with fuzzy returns for portfolio selection problem

In this paper, a novel multi objective model is proposed for portfolio selection. The proposed model incorporates the DEA cross-efficiency into Markowitz mean–variance model and considers return, risk and efficiency of the portfolio. Also, in order to take uncertainty in proposed model, the asset returns are considered as trapezoidal fuzzy numbers. Due to the computational complication of the proposed model, the second version of non-dominated sorting genetic algorithm (NSGA-II) is applied. To illustrate the performance of our model, the model is implemented for 52 firms listed in stock exchange market of Iran and the results are analyzed. The results show that the proposed model is suitable in compared with Markowitz and DEA models due to considering return, risk and efficiency, simultaneously.

Fuzzy Portfolio Allocation Models Through a New Risk Measure and Fuzzy Sharpe Ratio

A new portfolio risk measure that is the uncertainty of portfolio fuzzy return is introduced in this paper. Beyond the well-known Sharpe ratio (i.e., the reward-to-variability ratio) in modern portfolio theory, we initiate the so-called fuzzy Sharpe ratio in the fuzzy modeling context. In addition to the introduction of the new risk measure, we also put forward the reward-to-uncertainty ratio to assess the portfolio performance in fuzzy modeling. Corresponding to two approaches based on ${\rm T}_{\rm M}$ and ${\rm T}_{\rm W}$ fuzzy arithmetic, two portfolio optimization models are formulated in which the uncertainty of portfolio fuzzy returns is minimized, while the fuzzy Sharpe ratio is maximized. These models are solved by the fuzzy approach or by the genetic algorithm (GA). Solutions of the two proposed models are shown to be dominant in terms of portfolio return uncertainty compared with those of the conventional mean-variance optimization (MVO) model used prevalently in the financial literature. In terms of portfolio performance evaluated by the fuzzy Sharpe ratio and the reward-to-uncertainty ratio, the model using ${\rm T}_{\rm W}$ fuzzy arithmetic results in higher performance portfolios than those obtained by both the MVO and the fuzzy model, which employs ${\rm T}_{\rm M}$ fuzzy arithmetic. We also find that using the fuzzy approach for solving multiobjective problems appears to achieve more optimal solutions than using GA, although GA can offer a series of well-diversified portfolio solutions diagrammed in a Pareto frontier.

A Mean-Variance Hybrid-Entropy Model for Portfolio Selection with Fuzzy Returns

In this paper, we define the portfolio return as fuzzy average yield and risk as hybrid-entropy and variance to deal with the portfolio selection problem with both random uncertainty and fuzzy uncertainty, and propose a mean-variance hybrid-entropy model (MVHEM). A multi-objective genetic algorithm named Non-dominated Sorting Genetic Algorithm II (NSGA-II) is introduced to solve the model. We make empirical comparisons by using the data from the Shanghai and Shenzhen stock exchanges in China. The results show that the MVHEM generally performs better than the traditional portfolio selection models.

New methods for portfolio selection problem with fuzzy random variable returns

In conventional portfolio optimisation models, the market condition is predicted by historical data and the asset returns are random variables. In this paper, a special class of portfolio selection problems is introduced where the asset returns are fuzzy random variables. Then, the proposed problem is formulated and solved by using new methods. In the presented methods, we use the scalar expected value of fuzzy random variables and fuzzy stochastic chance–constrained programming based on possibility and necessity measures. Furthermore, a numerical example is also given to show the efficiency of the methods discussed in this paper.

CAPM with fuzzy returns and hypothesis testing

Over the last four decades, several estimation issues of the beta have been discussed extensively in many articles. An emerging consensus is that the betas are time-dependent and their estimates are impacted by the return interval and the length of the estimation period. These findings lead to the prominence of the practical implementation of the Capital Asset Pricing Model. Our goal in this paper is two-fold. After studying the impact of the return interval on the beta estimates, we analyze the sample size effects on the preceding estimation. Working in the framework of fuzzy set theory, we first associate the returns based on closing prices with the intraperiod volatility for the representation by the means of a fuzzy random variable in order to incorporate the effect of the interval period over which the returns are measured in the analysis. Next, we use these fuzzy returns to estimate the beta via fuzzy least square method in order to deal efficiently with outliers in returns, often caused by structural breaks and regime switches in the asset prices. A bootstrap test is carried out to investigate whether there is a linear relationship between the market portfolio fuzzy return and the given asset fuzzy return. Finally, the empirical results on French stocks suggest that our beta estimates seem to be more stable than the ordinary least square (OLS) estimates when the return intervals and the sample size change.

Entropy Model for Portfolio Selection with Fuzzy Returns

Entropy Model for Portfolio Selection with Fuzzy Returns