Optimal Programming Models for Portfolio Selection with Uncertain Chance Constraint

Limei Yan

doi:10.5539/mas.v3n9p84

Abstract

The paper is concerned with the portfolio selection problem about how to assign one’s money in security market in order to obtain the maximal profit. One type expected maximization programming model with chance constraint in which the security returns are uncertain variables are proposed in accordance with uncertainty theory. Since the provided models can not be solved by the traditional methods, the crisp equivalents of the corresponding models are discussed when the uncertain returns are chosen as some special cases such as linear uncertain variables, trapezoidal uncertain variables and normal uncertain variables. Two numerical examples with different types of uncertain variables are given in order to demonstrate the effectiveness and feasibility of the proposed programming models. Finally, the paper gives the conclusion.

Highlights

Portfolio selection is concerned with an investor who is trying to allocate one’s wealth among alternative securities so that the investment goal can be achieved
There are three models to deal with the portfolio selection problems with uncertain return rates
To use the theory of chance-constrained programming, the author tries to do something in portfolio selection problems when the return rates are assumed to be uncertain variable which is proposed by Liu (2009)

Summary

Uncertain programming models for portfolio selection

There does exist situations that security returns may be uncertain variable parameters. Let xi denote the investment proportion in the ith security, ξi represents uncertain return of the ith security, i = 1, 2,L, n , respectively, and a the minimum return level that the investor can tolerate. If we want to maximize the expected value or minimize risk of the total return subject to some chance constraints, to express it in mathematical formula, the models are as follows: max E[x1ξ1 + x2ξ 2 + L + xnξ n ]. If the investor wants to minimize the investment risk with some chance constraints, we have the following model, min V[x1ξ1 + x2ξ 2 + L + xnξ n ]. Where V denotes the variance of the total return which represents the risk of the investment

Crisp equivalents

Models for linear uncertain variable

Models for trapezoidal uncertain variable

Models for normal uncertain variable

Numerical Examples

Conclusions