Abstract
The purpose of this paper is to solve the portfolio problem when security returns are uncertain variables. Two types of portfolio selection programming models based on uncertain measure are provided according to uncertain theory. Since the proposed optimization problems are generally difficult to solve by conventional methods, the models are converted to their crisp equivalents when the return rates are adopted some special uncertain variables such as linear uncertain variable, trapezoidal uncertain variable and normal uncertain variable. Thus the transformed models can be completed by the conventional methods. In the end of the paper, one numerical experiment is provided to illustrate the effectiveness of the method.
Highlights
The theory of portfolio selection was initially provided by Markowitz (1952, p.77) and has been greatly developed since
When the return rates are considered as uncertain variables, the chance measure in the conventional chance-constrained programming model becomes the uncertain measure in the sense of uncertainty theory
M{xT ξ ≤ r} ≤ β x1 + x2 +L+ xn =1 xi ≥ 0, i = 1,2,L, n where max f is the α return which means the maximal investment return the investor can obtain at confidence level α, here it is the α − optimistic value to the total return rate xT ξ, and r is the minimum return that the investor can accept satisfying M{xT ξ ≤ r} ≤ β in which xT ξ ≤ r means the investment risk
Summary
The theory of portfolio selection was initially provided by Markowitz (1952, p.77) and has been greatly developed since . It is concerned with selecting a combination of securities among portfolios containing large number of securities to reach the goal of obtaining satisfactory investment return. Returns of individual securities are assumed to be stochastic variables, and many researchers were focused on extending Markowitz’s mean-variance models and on developing new mathematical approaches to solve the problems of computation. When the return rates are considered as uncertain variables, the chance measure in the conventional chance-constrained programming model becomes the uncertain measure in the sense of uncertainty theory.
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