Fuzzy Evolutionary Programming for Portfolio Selection in Investment

The problem of portfolio selection in investment concerns with minimizing the risk for a prespecified level of return. In this paper, the constraint on the level of return is fuzzified and the technique of fuzzy evolutionary programming is employed to select an optimal portfolio of securities with low risk and with highly acceptable level of total return. Experimental results show the method is highly effective. The problem of selecting a portfolio with low risk and with high probability of expected return is resolved in the same manner. INTRODUCTION Portfolio selection is a crucial task in investment. An investor is faced with a choice from among an enormous number of securities. The decision on which securities to invest in and the proportions of investment in those securities is highly complicated. The two principal factors that concern a portfolio selection are its risk and its expected total return. As broadly acknowledged, most investors hold strong aversion to risk and rather accept a modest return for low risk. Therefore, Markowitz [4] has formulated the problem of portfolio selection into minimizing the risk provided that its expected total return is above some prespecified threshold. Thus, portfolio selection is a constrained optimization problem. Theoretically, this problem can be solved mathematically by solving systems of linear equations. In practice, however, due to the large number of securities, the mathematical approach requires a great deal of programming computation (see e.g. [2], [5]). Recently, Watanabe et al. [6] employed a Boltzmann machine to solve the portfolio selection problem by maximizing the energy function Expected Return Risk. In reality, for most investors, the constraint on expected total return is tolerable, particularly in the case where investment risk is expectedly too high and needs to be lowered. Thus, one may relax the constraint and allow it to be satisfied to certain degrees in order to find a compromised optimal solution. In [3], I have presented an evolutionary approach to solving this type of constrained optimization problems, which is based on a combination of fuzzy logic and evolutionary programming. In this paper, the method of fuzzy evolutionary programming is employed to solve the portfolio selection problem, according to which the degree of satisfying the total return expectation is used as a weight factor for potential solutions. PORTFOLIO SELECTION IN INVESTMENT Consider N securities having expected returns ri with variances σi and covariances σij, 0 ≤ i, j ≤ N. The portfolio selection problem is expressed as Minimize x x x i i i j ij j i j N