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With increasing profit in securities investment, portfolio analysis has become a major topic for investors. We propose a fuzzy portfolio model as it is an efficient portfolio selection method associated with uncertain or vague returns. Although many researchers focus on studying the fuzzy portfolio model, they do not consider excess investment based on the selected guaranteed rates of return for some securities. To manage such an investment, a new fuzzy return function—where some securities are considered for excess investment based on the selected guaranteed rate of return—is introduced to improve the possibilistic mean and variance values, leading to a revised fuzzy portfolio model. Accordingly, to set certain securities for excess investment in the fuzzy return function, efficient portfolios for each selected guaranteed rate of return can be obtained under different levels of investment risk. Finally, we present a numerical example of a portfolio selection problem to illustrate the proposed model. This example shows that the expected rate of return of a lower guaranteed rate of return is larger than that of a higher guaranteed rate of return under different levels of investment risks. The portfolio analysis with some guaranteed rate of returns can provide more invested risk selection.

Uncertain information in the securities market exhibits fuzziness. In this article, expected returns and liquidity are considered as trapezoidal fuzzy numbers. The possibility mean and mean absolute deviation of expected returns represent the returns and risks of securities assets, while the possibility mean of expected turnover represents the liquidity of securities assets. Taking into account practical constraints such as cardinality and transaction costs, this article establishes a fuzzy portfolio model with cardinality constraints and solves it using the differential evolution algorithm. Finally, using fuzzy c-means clustering algorithm, 12 stocks are selected as empirical samples to provide numerical calculation examples. At the same time, fuzzy c-means clustering algorithm is used to cluster the stock yield data and analyse the stock data comprehensively and accurately, which provides a reference for establishing an effective portfolio.

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.

Portfolio selection is a major topic for investors to allocate their assets and maximize their profit under constrained risk. For uncertain investment behavior in a vagueness environment, some researchers have devoted themselves to this field of fuzzy portfolio models for portfolio selection. Especially, Tsaur, Chiu and Huang in 2021 defined guaranteed return rates to excess investment for securities whose return rates are bigger than the guaranteed return rates in the fuzzy portfolio selection. However, an independent investor has original ideas in investment, and thus we need to consider more types of risk attitudes for an investor’s portfolio selection when the guaranteed return rates are used to excess investment. To manage the excess investment by the risk preference, a new concept of s dimensions of excess investment is introduced to perceive the risk attitude of an investor for portfolio selection. Finally, we present a numerical example of a portfolio selection problem to illustrate the proposed model. This example shows that the higher dimensions of excess investment derive lower expected return rates with lower constrained risk than that of dimension s = 1; and we suggest lower risk preference should select a higher dimension of excess investment. Then, the dimension of excess investment s = 2 can be applied for portfolio selection when the risk preference is lower.

While the international lockdown for the COVID-19 pandemic has greatly influenced the global economy, we are still confronted with the dilemma about the economy recession when the stock market hits record highs repeatedly. As we know, since portfolio selection is often affected by many non-probabilistic factors, it is of not easiness to obtain the precise probability distributions of the return rates. Therefore, fuzzy portfolio model is proposed for solving the efficient portfolio when the economy environment remains in vagueness for the future. To manage such an investment, we revise the Chen and Tsaur’s fuzzy portfolio model by using fuzzy goal programming model. Then, two numerical examples are illustrated by the proposed model which shows that the portfolio selection can be solved by the linguistic imprecise goal of the expected return. With different linguistic descriptions for the imprecise goal of expected return for the future stock market, the optimal portfolio selection that can be solved under different investment risks which is considered better than Chen and Tsaur’s model in real world situations. The sensitivity analysis with some parameter groups can be provided for more invested risk selection in the portfolio analysis.

Fuzzy portfolio selection has resulted in many researchers to focus on this field. Based on the risk attitudes, this study discusses the risk attitudes in a decision group for portfolio selection. Therefore, we adopt the risk attitudes to describe the experts’ risk preferences and subjective judgments, and then we suppose that the risk seeker considers a higher return for an excess investment based on the selected guaranteed rate of return; the risk averter considers a shortage in investment for the securities whose return rates are smaller than the selected guaranteed rate of return; and finally, the risk neutral pursues the regular return rate. In order to solve the multi-objective return rate functions under the corresponding investment risks, the SMART-ROC weighting method is used to hybridize the multi-objective programming model to a linear programming model for solving the portfolio selection. Finally, we illustrate a numerical example and two risk scenarios to show the optimal portfolio selection under different investment risks. The results show that the proposed model can obtain a more robust portfolio than the compared models under different risk priorities in a decision group.

Investment portfolios are typically selected to reduce investment risk. In an economic recession or depression, investment strategies tend to be short term, subtle, and uncertain. When the economy is recovering or booming, investors should approach portfolio selection differently in response to the varying investment return and risk. Therefore, this study posits that different portfolios should be selected in different stages of the business cycle. An improved function for weighting possibilistic mean and variance is proposed, and a weighted fuzzy portfolio model for various investment conditions is then derived. Finally, a numerical example is presented to illustrate that the proposed models can obtain the optimal proportion of an investment throughout the business cycle to meet investors’ expectations.

This paper proposes a new fuzzy model for portfolio selection problem, which takes into account the vagueness of the investor’s preferences. The model proposed in this paper also regards the elastic increment of decision-making risk, background risk, and other financial risks. In order to solve this model, we present a modified evolutionary algorithm called modified chaos fruit fly optimization algorithm, which is more adequate when a quick and efficient solution is needed. Numerical examples are given to illustrate the effect of the background risk on portfolio selection.

Purpose This paper aims to propose two portfolio selection models with hesitant value-at-risk (HVaR) – HVaR fuzzy portfolio selection model (HVaR-FPSM) and HVaR-score fuzzy portfolio selection model (HVaR-S-FPSM) – to help investors solve the problem that how bad a portfolio can be under probabilistic hesitant fuzzy environment. Design/methodology/approach It is strictly proved that the higher the probability threshold, the higher the HVaR in HVaR-S-FPSM. Numerical examples and a case study are used to illustrate the steps of building the proposed models and the importance of the HVaR and score constraint. In case study, the authors conduct a sensitivity analysis and compare the proposed models with decision-making models and hesitant fuzzy portfolio models. Findings The score constraint can make sure that the portfolio selected is profitable, but will not cause the HVaR to decrease dramatically. The investment proportions of stocks are mainly affected by their HVaRs, which is consistent with the fact that the stock having good performance is usually desirable in portfolio selection. The HVaR-S-FPSM can find portfolios with higher HVaR than each single stock and has little sacrifice of extreme returns. Originality/value This paper fulfills a need to construct portfolio selection models with HVaR under probabilistic hesitant fuzzy environment. As a downside risk, the HVaR is more consistent with investors’ intuitions about risks. Moreover, the score constraint makes sure that undesirable portfolios will not be selected.

In this work a fuzzy multi-criteria model for portfolio selection is proposed which includes together with the classical financial risk-return bi-objective problem a new non-financial criterion. The proposed model will allow the analyst to offer the investor not only the financially good solutions but also some alternative solutions. In fact, the investor will be allowed to introduce in the model information about how far he or she is willing to go from the financially efficient portfolios knowing about the financial cost of these alternative solutions. A numerical example is presented in order to illustrate the proposed model. The social responsibility of the portfolio is considered as an additional secondary non-financial goal in the mean-variance portfolio selection model. Social responsibility is by its nature a vague and imprecise concept and will be handled by means of fuzzy set tools.