A possibilistic approach to selecting portfolios with highest utility score

The mean-variance methodology for the portfolio selection problem, originally proposed by Markowitz, has been one of the most important research fields in modern finance. In this paper we will assume that (i) each investor can assign a welfare, or utility, score to competing investment portfolios based on the expected return and risk of the portfolios; and (ii) the rates of return on securities are modelled by possibility distributions rather than probablity distributions. We will present an algorithm of complexity o(n) for finding an exact optimal solution (in the sense of utility scores) to the n-asset portfolio selection problem under possibility distributions. 1 A utility function for ranking portfolios The mean-variance methodology for the portfolio selection problem, originally proposed by Markowitz [4], has been one of the most important research fields in modern finance theory [7]. The key principle of the mean-variance model is to use the expected return of a portfolio as the investment return and to use the variance of the expected returns of the portfolio as the investment risk. Following [1] we shall asssume that each investor can assign a welfare, or utility, score to competing investment portfolios based on the expected return and risk of those portfolios. The utility score may be viewed as a means of ranking portfolios. Higher utility values are assigned to portfolios with more attractive riskreturn profiles. One reasonable function that is commonly employed by financial theorists assigns a risky portfolio P with a risky rate of return rP , an expected rate of return E(rP ) and a variance of the rate of return σ(rP ) the following utility score [1]: U(P ) = E(rP )− 0.005×A× σ(rP ), (1) ∗The final version of this paper appeared in: Fuzzy Sets and Systems, 131(2002) 13-21.