Portfolio Selection with Imprecise Future Data

Abstract

In 1952 Markowitz [247] published his pioneering work which laid the foundation of modern portfolio analysis. The mean-variance methodology for the portfolio selection problem has played an important role in the development of modern portfolio selection theory. It combines probability with optimization techniques to model the behavior investment under uncertainty. The basic 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. Most of existing portfolio selection models are based on probability theory. Though probability theory is one of the main techniques used for analyzing uncertainty in finance, the financial market is also affected by several non-probabilistic factors such as vagueness and ambiguity. In many important cases it might also be easier to estimate the possibility distributions of rates of return on securities rather than the corresponding probability distributions. Decision makers are commonly provided with information which is characterized by linguistic descriptions such as high risk, low profit, high interest rate, etc. [298] With the introduction of fuzzy set theory by Zadeh [351], it was realized that imperfect knowledge of the returns on the assets and the uncertainty involved in the behavior of financial markets may be captured by means of fuzzy quantities and/or fuzzy constraints. Since the early seventies fuzzy set theory has been widely used to solve many problems including financial risk management. By using fuzzy approaches, the experts’ knowledge and the investors’ subjective opinions can be better integrated into a portfolio selection model.