Optimization in Finance: Portfolio selection models

Portfolio selection problem deals with how to form a satisfying portfolio, taking into account the uncertainty involved in the behavior of the financial markets. Markowitz (1952) established the relationship between the mean and variance of the investment in the framework of risk-return trade-off. Since then a variety of enlarged and improved models have been developed in several directions. Some models of portfolio management combines probability theory and optimization theory to represent the behavior of the economic agents. These representations of return and risk have permitted to apply different optimization tools to the portfolio management. IBERIAN CONFERENCE IN OPTIMIZATION, Coimbra 2006– p. 2/59 Abstract In this talk we provide some new models for portfolio selection in which the returns on the securities are considered as fuzzy numbers rather than random variables. In order to find the portfolio that minimizes the risk in achieving a given level of return we introduce different approaches. In some of them the expected total return is considered otherwise is the fuzzy total return. The return on each asset and their membership functions are described using historical data and the risk of the investment is approximated by using interval-valued means which evaluate the downside risk for a given portfolio. In order to illustrate the performance of our methods we have used weekly returns corresponding to a selection of assets from the Spanish Stock Market.In this talk we provide some new models for portfolio selection in which the returns on the securities are considered as fuzzy numbers rather than random variables. In order to find the portfolio that minimizes the risk in achieving a given level of return we introduce different approaches. In some of them the expected total return is considered otherwise is the fuzzy total return. The return on each asset and their membership functions are described using historical data and the risk of the investment is approximated by using interval-valued means which evaluate the downside risk for a given portfolio. In order to illustrate the performance of our methods we have used weekly returns corresponding to a selection of assets from the Spanish Stock Market. IBERIAN CONFERENCE IN OPTIMIZATION, Coimbra 2006– p. 3/59