Portfolio Selection Model with Fuzzy Liquidity Constraints

In 1952, Markowitz published his pioneering work which laid the foundation of modern portfolio analysis. Markowitz’s model has served as a basis for development of the modern financial theory over the past five decades. However, contrary to its theoretical reputation, it is not used extensively to construct large-scale portfolios. One of the most important reasons is the computational difficulty associated with solving a large-scale quadratic programming problem with a dense covariance matrix. Konno and Yamazaki (1991) used the absolute deviation risk function, to replace the risk function in Markowitz’s model, and thus formulated a mean absolute deviation portfolio optimization model. It turns out that the mean absolute deviation model retains the useful properties of Markowitz’s model and removes most of the principal computational difficulties in solving Markowitz’s model. Simaan (1997) provided a thorough comparison of the mean variance model and the mean absolute deviation model. Furthermore, Speranza (1993) used semi-absolute deviation to measure the risk and formulated a portfolio selection model.

In this chapter, we will propose portfolio selection models with fuzzy liquidity constraints. At first, we construct a risk function - Minimax semi-absolute deviation risk function. Then we will formulate two optimization models for a portfolio selection problem with fuzzy liquidity constraints, based on the new risk function and semi-absolute deviation risk function, respectively.