Abstract
This paper studies the portfolio selection problem in hybrid uncertain decision systems. Firstly the return rates are characterized by random fuzzy variables. The objective is to maximize the total expected return rate. For a random fuzzy variable, this paper defines a new equilibrium risk value (ERV) with credibility level beta and probability level alpha. As a result, our portfolio problem is built as a new random fuzzy expected value (EV) model subject to ERV constraint, which is referred to as EV-ERV model. Under mild assumptions, the proposed EV-ERV model is a convex programming problem. Furthermore, when the possibility distributions are triangular, trapezoidal, and normal, the EV-ERV model can be transformed into its equivalent deterministic convex programming models, which can be solved by general purpose optimization software. To demonstrate the effectiveness of the proposed equilibrium optimization method, some numerical experiments are conducted. The computational results and comparison study demonstrate that the developed equilibrium optimization method is effective to model portfolio selection optimization problem with twofold uncertain return rates.
Highlights
Based on mean-variance criterion, Markowitz [1] first established portfolio theory
In the case that the randomness of uncertain return rates follows normal distributions with deterministic covariance matrix, and the fuzziness is characterized by trapezoidal fuzzy variables, triangular fuzzy variables, or normal fuzzy variables, the proposed expected value (EV)-equilibrium risk value (ERV) model is transformed into its deterministic convex programming models, which can be solved by general purpose optimization software
Under the equilibrium risk criterion, a new EVERV portfolio optimization model was built for portfolio selection problems, where the return rates are characterized by both probability distributions and possibility distributions
Summary
Based on mean-variance criterion, Markowitz [1] first established portfolio theory. In Markowitz’s mean-variance model, the returns of individual securities are taken as random variables and the expected value and variance of the random return are taken as the investment return and risk, respectively. Based on the above works, this paper models the portfolio selection problem where randomness and fuzziness are considered simultaneously. This paper employs the expected value (EV) [33] to represent the investment return and introduces a new index called ERV for random fuzzy return rate to measure the investment risk. The adopted risk measure quantifies the uncertainties of randomness and fuzziness simultaneously This method shows the qualitative and quantitative analysis about the uncertainty of return rate. In the case that the randomness of uncertain return rates follows normal distributions with deterministic covariance matrix, and the fuzziness is characterized by trapezoidal fuzzy variables, triangular fuzzy variables, or normal fuzzy variables, the proposed EV-ERV model is transformed into its deterministic convex programming models, which can be solved by general purpose optimization software.
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