Probabilistic Portfolio Selection

Abstract

Probabilistic portfolio selection handles portfolio selection problem with random returns by means of probability theory. It was researched earliest and began to get rapid development since Markowitz in 1952. Before Markowitz, there were no measurable terms for risk. Mean-variance model, proposed by Markowitz [66], opened the door for mathematical analysis of portfolio selection problem. Mean-semivariance model, also proposed by Markowitz [67], served as an improvement of mean-variance selection model. As an alternative definition of risk, Roy [83] proposed probability of a preset loss level as risk, and the selection idea of minimizing the probability of a specific loss level came to be known. The nowadays popular VaR is in fact another version of Roy’s risk definition. Recently, Huang [36] defined risk curve and proposed a mean-risk selection idea.

This chapter will start with review of some fundamentals of probability theory concerning probabilistic portfolio selection. Since the main concepts and results of probability theory are well-known, the credit references are not provided. Then the chapter will focus on a spectrum of portfolio selection models from different perspectives on risk and return. After that, a hybrid intelligent algorithm is documented as a general solution algorithm for the probabilistic portfolio selection model problems.